Research ArticleInvestigation on a No Trial Weight Spray OnlineDynamic Balancer

Xialun Yun 123 Xuesong Mei 123 Gedong Jiang123 Zhenbang Hu123

and Zunhao Zhang123

1State Key Laboratory for Manufacturing Systems Engineering Xirsquoan Jiaotong University 710048 Xirsquoan China2Shaanxi Key Laboratory of Intelligent Robots Xirsquoan Jiaotong University 710049 Xirsquoan China3School of Mechanical Engineering Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Xuesong Mei xsmeixjtueducn

Received 24 June 2018 Accepted 28 August 2018 Published 1 November 2018

Academic Editor Mario Terzo

Copyright copy 2018 Xialun Yun et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to suppress the spindle vibration with high efficiency and high precision a no without trial weight spray online balancemethod is proposed in this paper By analyzing the relationship between the unbalanced excitation and the unbalanced response ofthe spindle the relationship between the dynamic influence coefficient and the system model is studied A high-speed spindlefinite element analysis model was established and the dynamic influence coefficient matrix was identified A no trial weight sprayonline dynamic balancing system was developed which has the advantages of without trial weight and high-precision loading Anew type of integrated balancing terminal that was formed using 3D printing technology was first proposed by our research groupand its advantages in various aspects are significantly higher than traditional assembly balanced terminals e experimentalverification of the without trial weight spray online dynamic balancing system was performed on a high-speed spindle test standExperiments show that the no trial weight spray online balancing method proposed in this paper can achieve high-efficiency andhigh-precision vibration suppression greatly reducing balance time and cost of the spindle At the same time the online balancetest also verified the reliability of the integrated balanced terminal

1 Introduction

High-speed spindle is the core functional component of high-end CNC machine tools and its dynamic characteristics willdirectly affect the machining level of machine tools Howeverdue to uneven material and manufacturing and assemblyerrors the center of mass of the spindle and the center ofrotation may be inconsistent e unbalance of the spindle isinevitable Unbalance can cause a disastrous accident in manyengineering applications [1 2] erefore strict offline bal-ancing will be performed when the spindle leaves the factory[3 4] However the offline balance cannot compensate theunbalance mass caused by high-speed expansion tool orgrinding wheel wear and the offline balance whose balanceefficiency is low and requires the disassembly of the main shaftor grinding wheelerefore it is very meaningful to study thehigh-speed and precision online dynamic balance [5 6]

Spindle online balancing includes two main processes(1) unbalanced vector identification and (2) vibration onlinesuppression A large number of researchers have studied theunbalanced vector identification method [7] e un-balanced vector identification methods mainly include in-fluence coefficient method (ICM) [8] modal balancemethod (MBM) [9] holospectral balance method (HBM)[10] and no trial weight balance method (NTWBM) [11]Because NTWBM has the characteristics of without trialweight and high balancing precision it has been widelyapplied in the intelligent spindle dynamic balancing [12 13]e research team headed by Mr Qu Liang Sheng studiedthe no trial weight balancing method based on holospectraltechnique [14ndash16] In 2009 Niebsch and Ramlau [17 18]used wind turbines as the research object to study ill-conditioned phenomena in the inverse problem of the notrial weight balance method Khulief et al [19] and Li et al

HindawiShock and VibrationVolume 2018 Article ID 7021215 15 pageshttpsdoiorg10115520187021215

[20] simulated the functional expressions of the principalmodes of the various stages of the rotor based on the finiteelement theory and a good balancing effect is achievedResearch on spindle online vibration suppression technol-ogy began in the middle of the last century Canadianscholars have proposed an active balancing device that isbalanced by controlling the weight position e mass blockis driven by the motor [21] Since then many scholars havestudied the way of changing the rotor mass distribution bybalancing terminals In 2006 Hredzak and Guo [22] de-veloped an electromagnetic automatic balancing device Amovable electromagnetic adsorption device on the outside ofthe disc is used to adjust the position of the steel ball edevice has achieved good results in the experiment Moonet al [23] proposed an electromagnetic balance device basedon the similar principle He achieved the spindle balanceexperiment by means of the influence coefficient method at14400 rpm In 2008 Japan Nakamoto et al [24] designeda new type of balance device that uses magnetic fluid asa balancing mass By changing the magnetic field distri-bution in the periphery of the balance terminal thepurpose of changing the magnetic fluid distribution inthe terminal was verified e device was validated at6000 rpm A new electromagnetic ring balancer for activeimbalance compensation of rotating machinery was pro-posed by Fan et al [25] in 2014 He verified the feasibilityand effectiveness of balancing devices Although they havestudied different implementations most of them use theICM to verify the device e ICM is a kind of test weightmethod Test weight is needed to calculate the unbalancevector of the rotor system and then readjust the coun-terweight to achieve the purpose of correcting the vectorHowever this method does not apply to weighted onlinebalancing due to the test weight such as spray onlinebalancing e no trial weight balance method can effec-tively solve the above problems In addition the spraydynamic balancing balance has high precision and canperform multiple online dynamic balancing which is verysuitable for high-precision loading

e no trial weight spray online dynamic balancingtechnology is studied in this paper is method can quicklyidentify unbalanced vectors and achieve high-precisionloading erefore it can achieve high-efficiency andhigh-precision online vibration suppression An integratedbalanced terminal (IBT) using 3D rapid prototyping tech-nology was first proposed by our research group comparedwith the traditional spray online balancing technology eIBTperfectly solved the problems of sealing and high-speedspin-off of traditional assembled balanced terminals istechnology is important for improving the accuracy ofprecision rotating machinery

2 No Trial Weight Balance Method of Spindle

e influence coefficient method used in the industrial fieldis a kind of method that causes the response of the target

rotor system by adding trial weights e aggravated in-fluence coefficient matrix of the rotor system can be ob-tained However the influence coefficient matrix is nota virtual matrix It is a characteristic parameter that canreflect the response of the rotor system when excited byenergy is shows that this matrix is an inherent charac-teristic of the system In other words the unbalance vector[α] can be calculated by identifying the characteristic matrixof the rotor system We combine the high-speed electricspindle designed by our laboratory to study the WTWBMe structure is shown in Figure 1

is high-speed spindle system was developed specifi-cally for studying online balancingemaximum rotationalspeed of the spindle is 40000 rpm A pair of bearings SNFAVEX35 are mounted on the spindle and arranged in a back-to-back layout and preloaded by two sleeves and a locknute bearings are lubricated with oil mist and the spindle iscooled by circulating water cooling It should be noted that1 7 taper is designed at the front and rear ends of the spindleto mount the counterweight disc

It is necessary to establish the dynamic model of thespindle in order to identify the unbalance vector e rotorsystem of spindle can be discretized into discs elastic shaftsections and bearing units e rotor is divided into 50nodes to form 49 units e coordinate system is establishedwith the rotor axis as the axise finite elementmodel of therotor system is shown in Figure 2

e beam element has a simple structure and is easy toprogram so it is still the first choice for establishing a finiteelement modele Timoshenko beam element consisting oftwo nodes is shown in Figure 3 e motion of each nodeconsists of five degrees of freedom namely three trans-lational degrees of freedom δx δy δz and two rotationaldegrees of freedom cy cz where the X-axis torsional free-dom is not considered [26 27]

Regardless of the internal damping of the beam theequation of motion of the beam can be expressed as follows[28]

Mb

1113960 1113961 euroq1113864 1113865minusΩ Gb

1113960 1113961 _q1113864 1113865 + Kb

1113960 1113961 + Kb

1113960 1113961PminusΩ2 M

b1113960 1113961

C1113872 1113873 q1113864 1113865 F

b1113966 1113967

(1)

where [Mb] is mass matrix [Mb]C is additional mass matrixconsidering effect of centrifugal force [Gb] is symmetricalgyro matrix [Kb] is stiffness matrix [Kb]P is additionalstiffness matrix due to axial loading q is system dis-placement vector and q1113864 1113865 δx δy δz cy czT Fb is ex-ternal force vector e superscript b stands for the beamunit and Ω is the speed

Similarly the equation for the motion of the disk unit isgiven by the following eqaution

Md

1113960 1113961 euroq1113864 1113865minusΩ Gd

1113960 1113961 _q1113864 1113865 Fd

1113966 1113967 (2)

where [Md] is mass matrix [Gd] is gyro matrix the su-perscript b stands for the beam unit Assuming that theunbalanced mass is mD and the eccentricity is e the un-balance force is as follows

2 Shock and Vibration

Fd

1113966 1113967

0

mDeΩ2 cosΩt

mDeΩ2 sinΩt

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

q1113864 1113865

u

v

w

θy

θz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(3)

In addition the generalized forces acting on the bearingscan be written as follows [29]

Cc

1113858 1113859 q1113864 1113865 + Kc

1113858 1113859 q1113864 1113865 Qc

1113864 1113865 (4)

where [Cc] is the bearing damping matrix [Kc] is thebearing stiffness matrix and Qc is the vector of generalizedforces of the bearings

In combination with the above derivation the model ofthe spindle rotor disc and bears is integrated e differ-ential equation of the spindle system is as follows [30]

[M] eurox +[C] _x +[K] x F (5)

where system mass matrix is [M] [Mb] + [Md] and [C] issystem damping matrix which can be obtained by modalanalysis e excitation F is due only to harmonic unbalanceexcitations Fd1113864 1113865 under steady-state condition e un-balance excitation force Fd1113864 1113865 can be expressed as follows

Fd

u middot ejωmiddott

(6)

Vibration response x is

x r middot e[j(ωmiddott+φ)]

(7)

where r is unbalanced response u is unbalanced mass and φis lag angle Substituting Eqs (6) and (7) into Eq (5) resultsin

[K]minusω2[M] + i [C] +[G] ω1113966 1113967 middot r middot exp(jφ) u (8)

e above formula can be written as [α] middot r u where[α] is a aggravated influence coefficient matrix of the rotorsystem which can be derived from the above equation

[α] [K]minusω2[M] + i [C] +[G] ω1113966 1113967

minus1middot exp(minusjφ) (9)

Locknut

Locknut

Counter balanceCounter balance

Front bearing Rear bearing EncoderMotor stator Motor rotorCooling jacket

Sha

Taper 1 7Taper 1 7

Figure 1 High-speed spindle model

SensorMotor

Counter balancedisc

Counter balancedisc

Sensor Sensor

Bearing Bearing Bearing Bearing1 5 10 15 18 21 30 32 35 42 46 50

Figure 2 Finite element model (node number 1sim50)

Y

γy2

δy2

Xδx2

Zδz2

γz2

Y

γy1

δy1

Xδx1

Z δz1

γz1

ΩNode 2Node 1

Figure 3 Timoshenko beam element

Shock and Vibration 3

From the above equation it can be seen that the ag-gravated influence coefficient of the rotor system is de-termined by the mass stiffness lag angle and rotation speedat is to say the aggravated influence coefficient of therotor system is determined by the rotor structure parame-ters It is an inherent parameter to reflect the energy transfercharacteristics of the rotor system

[Q] minus[α]minus1

middot A (10)

where [Q] is balance vector A is unbalanced vibration re-sponse obtained through testing and the negative sign in-dicates that the phase difference between the balance vectorand the imbalance is 180deg After calculating the balance vectorwe apply it to the online dynamic balance by modifying thematching process and the online weighted correction schemecan be obtained by rapid orientation iteration matching edetailed online balance process is described later

3 Online Balancing Experiment System

31 System Structure e spray dynamic balance systemdeveloped by our research group was used to perform ex-perimental research on the no trial weight online balancemethod e structure of the spray online balancing system is

shown in Figure 4 e system is mainly composed of bal-anced terminals hydraulic systems and measurement andcontrol systemse balance terminal is the key component ofthe dynamic balancing system and its function is to store thecorrection mass e main function of the hydraulic system isthe quantitative delivery of liquid e function of the mea-surement and control system is unbalanced vector calculationand hydraulic system control

32 New Integrated Balance Terminal Balancing terminalsare the core components of spray dynamic balancing emain function is to store the correction mass rough in-vestigation it has been found that the balance terminal in theworld is the assembled balanced terminal shown in Figure 5

As can be seen from Figure 5 the traditional assembledbalance terminal is assembled from the inner ring and fromthe outer ring through interference fit However the de-formation of the outer ring of the balanced terminal is largerthan that of the inner ring under the action of centrifugalforce When the balanced terminal rotates with the spindle ata high speed the magnitude of interference between the innerand outer rings of the balanced terminal will decrease iswill lead to a drop in the sealing performance of the balancedterminal or even complete failure According to the theory of

Balanced terminalElectromagnetic value Hydraulic

system

Pump

Tank

Computer

Interactivetouchscreen

Vibrationsuppressioncontroller

Electricalisolation

Electrical isolation

Eddy currentsensor

Accelerometer

Test and control system

Nozzle

Figure 4 Spray online balance system structure

4 Shock and Vibration

elasticity [31] regardless of the tangential displacementcomponent of the inner and outer rings in the rotationprocess the radial centrifugal force of the inner and outerrings is applied as a unit volume force which can beequivalent to the problem of axisymmetric plane stress At anyradius r the deformation displacement ur is [32]

ur 3 + ]8E

ρω2r (1minus ]) B

2+ A

21113872 1113873 +(1 + ])

A2B2

r2minus1minus ]2

3 + ]r2

1113890 1113891

(11)

where ω is angular velocity ρ is material density E is elasticmodulus ] is Poissonrsquos ratio and A and B are the diametersof the inner and outer rings respectively en the mini-mum allowable amount of interference δ is shown as follows

δ uo RF( 1113857minus ui RF( 1113857 3 + v

4Eρw

2RF R

2O minusR

2I1113872 1113873 (12)

where RF is the critical radius of the inner and outer rings(RF 33 mm) RO is the outer radius of the outer ring of thedynamic balance terminal (RO 47 mm) and RI is innerradius of the inner ring of the balancing terminal(RI 28 mm) Balanced terminal material is titanium alloy() When the magnitude of interference is 20 μm the re-lationship between the radial expansion of the inner andouter rings and the rotation speed is shown in Figure 6 Itcan be seen that when the rotational speed reaches the loosecritical speed the interference connection will fail So theassembled balance terminal not only loses the capacity of thechamber seal but also has potential safety hazards Of coursethe magnitude of interference cannot be too large If themagnitude of interference is too large assembly will bedifficult Pressing in force will even cause fatigue damage tothe inner and outer rings

Because of the above problems in the traditional as-sembly balanced terminals an integrated balanced terminalthat combines the needs of spray online dynamic balancingand the technical features of rapid prototyping was proposed

by our research group as shown in Figure 7 e integratedbalancing terminal eliminates the traditional design of theinner and outer rings It is mainly composed of the terminalbody flow path offline weight holes tapered holes andchamberse integrated balanced terminal is formed in onetime using 3D printing technology e tapered holes areused to connect with the spindle e function of offlineweight holes is to eliminate the initial eccentricity of thebalanced terminal in the manufacturing process is willnot cause new additional imbalance to the spindle efunction of the flow path is to direct the fluid to the cor-responding chamber

e capacity of the integrated balanced terminal de-termines its balance capacity In order to calculate thebalance capacity of the balanced terminal the volume ofchamber must be calculated In order to simplify the cal-culation the corner is ignored and the cross section shownin Figure 8(a) is obtained the integral microelement isshown in Figure 8(c)

120

100

80

60

40

20

00 2 4 6

Speed (rpm) (times104)

Radi

al d

ispla

cem

ent (microm

)

Initialinterference

Critical point

Inner ringOuter ring

Figure 6 Relationship between radial displacement and speed

Assembly

Inner ring

Outer ring

Cavity line

Figure 5 Assembled balanced terminal schematic

Flow-path

Offline weighthole

Tapered holeCavity line

Figure 7 Integrated balance terminal diagram

Shock and Vibration 5

en the volume of a single liquid chamber is given by

V1 2πα360

1113946l1

0R1 + l( 1113857 h1 + 2l( 1113857dl

V2 2πα360

1113946l2

0R1 + l1 + l( 1113857 h2 minus 2l( 1113857dl

V V1 + V2

(13)

where h1 145 mm h2 26 mm l1 l2 575 mm R1

33 mm so V asymp 13230 mm3

Assume that the density of the weighted liquid isρ the maximum balance capacity of a single liquidchamber

Q1 2πρα360

1113946l1

0R1 + l( 1113857 h1 + 2l( 1113857(33 + l)dl

Q2 2πρα360

1113946l2

0R1 + l1 + l( 1113857 h2 minus 2l( 1113857 33 + l1 + l( 1113857dl

Q Q1 + Q2

(14)

where ρ 08 gmm3 so Q asymp 413 gmiddotmm It can be knownfrom the balanced terminal cavity distribution structure thatwhen the adjacent two chambers are filled with the balanceliquid and the other chambers are all empty the balancingcapacity of the balanced terminal will reach the maximumvalue at this time e maximum is

2

radicQ 584 gmiddotmm

e balancing terminal rotates with the spindle at a highspeed So balancing the liquid in the terminal chamber willstress the chamber due to centrifugal force Assuming therotational speed is 30000 rpm the cavity of the balanceterminal is filled with the weight fluid with density08 gmm3 and the centrifugal force produced by the weightfluid in a single cavity can be calculated by the followingequation

F mω2r Qω2

(15)

where m is the liquid mass in the cavity r is the radialdistance between the center of gravity of the counterweight

fluid and the axis of rotation ω is the angular velocity andQ m times r hence centrifugal force F 4096 N

When the balanced terminal rotates at a high speed mostof the water pressure generated by the centrifugal force willact on the outer surface of the cavity and the outer surface ofthe cavity is not a regular plane To simplify the calculationthe pressure area on the outer surface of the chamber isequivalent to the projection surface as shown in Figure 9When the balanced terminal is filled with the counterweightfluid the pressure on the outer surface of the chamber iscalculated by the following equation

P F

S

F

(2α360) πR2h2( 1113857 (16)

where centrifugal force F 4076 N α 84deg radius R2

445 mm and thickness h2 26 mm thus the pressureP 24 MPa

In order to select the thickness of the cavity wall of thebalanced terminal the analysis was carried out under theextreme conditions (full-chamber liquid and 30000 rpm)When the pressure is 24MPa different thicknesses of thecavity arc surface force analysis model are built in ANSYS asshown in Figure 10

e relationship between the material thickness and thestress and deformation of the titanium alloy balance ter-minal surface is shown in Figure 11 It can be seen that whenthe material thickness is selected as 2mm under the extremeconditions the stress value and deformation tend to bestable erefore in order to meet the requirement of thebalanced terminal with the spindle acceleration and de-celeration the wall thickness of the force surface should begreater than 2mm We have designed the thickness of thebalanced terminal force surface to be 25mm

e assembled balance terminal of different materialswas trial-produced by us as shown in Figure 12 By analyzingthe requirements of 3D printing technology and strengthtitanium alloy is selected as the manufacturing material forthe integrated balance terminal

So the integrated balanced terminal manufacturedaccording to the design parameters is shown in Figure 13(a)For the nonblocking of the liquid flow and balancing the

h 1l1 l2h 2

(a)α

R 1

R2

(b)

h

Δ1

(c)

Figure 8 Calculation model (a) Section (b) Top view (c) Microelement

6 Shock and Vibration

(a) (b)

Figure 10 Surface force analysis model

P P

Figure 9 Fluid pressure on the cavity surface

0 1 2 3 4 550

100

150

200

250

300

350

400

ickness (mm)

Def

orm

atio

n (μ

m)

(a)

0 1 2 3 4 50

100

200

300

400

500

600

ickness (mm)

Stre

ss (M

pa)

(b)

Figure 11 e relationship between thickness and stress and deformation

(a) (b) (c)

Figure 12 Assembled balance terminal (a) Steel-45 (b) Aluminum alloy-7A04 (c) Titanium alloy-TC4

Shock and Vibration 7

signal test on the terminal surface the flow channel and thesurface were modified as shown in Figure 13(b) e bal-ancing terminal is balanced to eliminate the initial masseccentricity on the balancing machine is will not causenew additional imbalance to the spindle

e 14 model of assembled balanced terminals and in-tegrated balanced terminals is shown in Figure 14 e pa-rameters of the two balanced terminals are shown in Table 1It can be seen that the integrated balanced terminal designedin this paper has obvious advantages in comparison withthe assembled balanced terminal e maximum balancecapacity of balanced terminal increased from 489 gmiddotmm to584 gmiddotmm On the basis of a 194 increase in balance ca-pacity thickness dropped from 38mm to 28mm a drop of263 e volume of balanced terminal dropped from160549 mm3 to 83780 mm3 with 478 drop ratio e massof balanced terminal has dropped from 562 g to 387 g with

a drop ratio of 311 so that the additional mass added by thebalanced terminal to the spindle will be effectively reduced

33 Balanced Procedure and Software e no trial weightbalance method is applied to the spray online balancesystem which can achieve efficient online balance thusmaximizing the protection automation processing link eflow diagram of the no trial weight spray online dynamic isshown in Figure 15 in which three main steps are included(1) acquisition process of rotor structural parameters (2)acquisition process of counterweight scheme and (3) onlinebalance process e balancing procedure is as follows

(1) Establish the physical model and finite elementmodel of the spindle rotor system by separating therotor model

(a) (b)

Figure 13 Titanium alloy integrated balancing terminal (a) Before modification (b) After modification

l2prime = 4mml3prime = 4mm

l1prime = 4mm

(a)

l2 = 25mm

l3 = 2mm

l1 = 2mm

(b)

Figure 14 Comparison of 14 model (a) Assembled terminal (b) Integrated terminal

Table 1 Comparison of two balanced terminal parameters

SpeciesParameters

Balance ability (gmiddotmm) ickness (mm) Volume (mm3) Mass (g) Price (RMB)Assembled terminal 489 38 160549 562 1024000Integrated terminal 584 28 83780 387 580400Variation () 194 263 478 311 433

8 Shock and Vibration

(2) Obtain the structural parameters of the spindle rotorsystem such as mass matrix M stiffness matrix Kand damping matrix C by the finite element simu-lation and simulate the inherent modal character-istics of the spindle

(3) Substitute the correction matrices of structural pa-rameters in Equation (9) to identify the aggravatedinfluence coefficient matrix

(4) Extract the original vibration signal of the spindleand the phase information corresponding to thevibration signal substitute aggravated influencecoefficient matrix into Equation (10) to calculate theunbalance vector

(5) e weighting scheme is searched according to thecavity angle and the unbalance vector

(6) e spray online dynamic balancing system is in-jected into the corresponding cavity to correct theunbalance vector

According to the balance flow chart a measurement andcontrol system based on virtual instruments was developedby our research group e operation interface is shown inFigure 16 e no trial weight online balance measurementand control software consists of vibration signal monitoringtrend of vibration displacement change vibration statusdisplay and monitoring liquid state display and monitoringand online balance information records

Start

Build the spindle physical model

Establish the FEDM of spindlerotor system

Frequency-responsecharacteristic simulation

Obtain the spindle structure parametersM K C G ω

Mode test

Is the inspection errorreasonable

N

Y

Corrected initial value(ndashω2

2M2 + K2 + jω2C2)H2 (ω2) = I

Identify the ETCSRS(α) = (K)ndashω2 (M) + i(C)+(G)ωndash1middotexp (ndashjφ)

Vector synthesis principle for weight matching

Calculate the balance vector (Q)=ndash(α)ndash1middotA

Extract the originalvibration signal A

Get balance scheme

End

Online spray balance

Vibration gt targetvalue

Vibration gt targetvalue

Y

EndN

Y

N

Figure 15 e balance procedure

Shock and Vibration 9

4 Methods and System Validation

41 Experimental Setup In order to verify the application ofno trial weight spray online dynamic on high-speed spindlethe high-speed spindle comprehensive performance test bedwas designed by our research group as shown in Figure 17e test bed comprises a spindle 150 SD40Q7 (Figure 1)whose maximum rotational speed is 40000 rpm and twopairs of SNFA VEX35 bearings lubricated by oil mist aremounted on the spindle with a clearance tolerance to avoidthe preload variation e water cooling system controls thetemperature variation e balancing discs are installed onthe ends of the spindle to suppress vibration A rotatingencoder mounted on the shaft feedbacks the rotationalspeed e spindle rotor is designed with a ratio of 1 7tapered to install the integrated balancing terminal and the

balancing terminal is fixed with a locknut In the front of therotor a HSK C32 tool holder interface is designed emeasuring equipment is composed of a synchronous ac-quisition system developed by our research group A Na-tional Instrument PXI-4472 was used to extract vibrationand phase information German Micro-Epsilon U-05 eddy-current sensor (range 05mm resolution 0025 μm) is usedto measure vibration displacement e NI-PXI6123 is usedto control the hydraulic system A photoelectric sensor wasapplied to measure the phasee layout of the experimentalsetup is shown in Figure 18

42 Measurement Principles e system carries out real-time synchronous acquisition of vibration using eddy-current displacement sensors mounted along the X-axis

Vibrationsignal

Phase signal Vibration state

Data record

Straystate

Figure 16 Measurement and control panel

Diaplacementsensors

Integratedterminal

Sensor supportFixture

Spindle system Phase sensor

Control program

Control card anddata acquisition card

NI-PXI-1003

Nozzle Tool interface

Hydraulic system

24V DC

Control board

Conditioning circuit

Figure 17 Experimental setup

10 Shock and Vibration

X

S1 S2

S3

Z

Y

(a)

90deg

0deg360deg

270deg

180deg

ωY

Xϕθφ

Cavity-2

Cavity-1

Cavity-4

Cavity-3

S2

S1 S3A1

A2

A3

(b)

Figure 18 Measurement principles (a) Sensor layout (b) Angle relation

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

Before balanceAer balance

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

3

25

2

15

1

05

0

286 μm

021 μm

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(c)

287 μm

051 μmAm

plitu

de (μ

m)

3

25

2

15

1

05

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(d)

Figure 19 Continued

Shock and Vibration 11

0 5 10 15 20 25 30Time (seconds)

Am

plitu

de (μ

m)

35

3

25

2

15

Start balance(14286)

Balance end(19021)1

05

0

(e)

Figure 19 Comparison of balance information in the X-Y direction at 12000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

ndash4

0

2

4

ndash2Am

plitu

de (μ

m)

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

(a)

ndash2

ndash3

ndash1

0

1

2

3

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

337 μm

082 μm

(c)

0

05

15

1

2

25

3

Am

plitu

de (μ

m)

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

279 μm

1233 μm

(d)

Figure 20 Continued

12 Shock and Vibration

Am

plitu

de (μ

m)

4

3

2

1

0

Start balance(23337)

Balance end(26082)

0 5 10 15 20 25 3530Time (seconds)

(e)

Figure 20 Comparison of balance information in the X-Y direction at 15000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAfter balance

(b)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

466 μm

052 μm

(c)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

498 μm

131 μm

(d)

Figure 21 Continued

Shock and Vibration 13

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

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Submit your manuscripts atwwwhindawicom

Fd

1113966 1113967

0

mDeΩ2 cosΩt

mDeΩ2 sinΩt

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

q1113864 1113865

u

v

w

θy

θz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(3)

In addition the generalized forces acting on the bearingscan be written as follows [29]

Cc

1113858 1113859 q1113864 1113865 + Kc

1113858 1113859 q1113864 1113865 Qc

1113864 1113865 (4)

where [Cc] is the bearing damping matrix [Kc] is thebearing stiffness matrix and Qc is the vector of generalizedforces of the bearings

In combination with the above derivation the model ofthe spindle rotor disc and bears is integrated e differ-ential equation of the spindle system is as follows [30]

[M] eurox +[C] _x +[K] x F (5)

where system mass matrix is [M] [Mb] + [Md] and [C] issystem damping matrix which can be obtained by modalanalysis e excitation F is due only to harmonic unbalanceexcitations Fd1113864 1113865 under steady-state condition e un-balance excitation force Fd1113864 1113865 can be expressed as follows

Fd

u middot ejωmiddott

(6)

Vibration response x is

x r middot e[j(ωmiddott+φ)]

(7)

where r is unbalanced response u is unbalanced mass and φis lag angle Substituting Eqs (6) and (7) into Eq (5) resultsin

[K]minusω2[M] + i [C] +[G] ω1113966 1113967 middot r middot exp(jφ) u (8)

e above formula can be written as [α] middot r u where[α] is a aggravated influence coefficient matrix of the rotorsystem which can be derived from the above equation

[α] [K]minusω2[M] + i [C] +[G] ω1113966 1113967

minus1middot exp(minusjφ) (9)

Locknut

Locknut

Counter balanceCounter balance

Front bearing Rear bearing EncoderMotor stator Motor rotorCooling jacket

Sha

Taper 1 7Taper 1 7

Figure 1 High-speed spindle model

SensorMotor

Counter balancedisc

Counter balancedisc

Sensor Sensor

Bearing Bearing Bearing Bearing1 5 10 15 18 21 30 32 35 42 46 50

Figure 2 Finite element model (node number 1sim50)

Y

γy2

δy2

Xδx2

Zδz2

γz2

Y

γy1

δy1

Xδx1

Z δz1

γz1

ΩNode 2Node 1

Figure 3 Timoshenko beam element

Shock and Vibration 3

From the above equation it can be seen that the ag-gravated influence coefficient of the rotor system is de-termined by the mass stiffness lag angle and rotation speedat is to say the aggravated influence coefficient of therotor system is determined by the rotor structure parame-ters It is an inherent parameter to reflect the energy transfercharacteristics of the rotor system

[Q] minus[α]minus1

middot A (10)

where [Q] is balance vector A is unbalanced vibration re-sponse obtained through testing and the negative sign in-dicates that the phase difference between the balance vectorand the imbalance is 180deg After calculating the balance vectorwe apply it to the online dynamic balance by modifying thematching process and the online weighted correction schemecan be obtained by rapid orientation iteration matching edetailed online balance process is described later

3 Online Balancing Experiment System

31 System Structure e spray dynamic balance systemdeveloped by our research group was used to perform ex-perimental research on the no trial weight online balancemethod e structure of the spray online balancing system is

shown in Figure 4 e system is mainly composed of bal-anced terminals hydraulic systems and measurement andcontrol systemse balance terminal is the key component ofthe dynamic balancing system and its function is to store thecorrection mass e main function of the hydraulic system isthe quantitative delivery of liquid e function of the mea-surement and control system is unbalanced vector calculationand hydraulic system control

32 New Integrated Balance Terminal Balancing terminalsare the core components of spray dynamic balancing emain function is to store the correction mass rough in-vestigation it has been found that the balance terminal in theworld is the assembled balanced terminal shown in Figure 5

As can be seen from Figure 5 the traditional assembledbalance terminal is assembled from the inner ring and fromthe outer ring through interference fit However the de-formation of the outer ring of the balanced terminal is largerthan that of the inner ring under the action of centrifugalforce When the balanced terminal rotates with the spindle ata high speed the magnitude of interference between the innerand outer rings of the balanced terminal will decrease iswill lead to a drop in the sealing performance of the balancedterminal or even complete failure According to the theory of

Balanced terminalElectromagnetic value Hydraulic

system

Pump

Tank

Computer

Interactivetouchscreen

Vibrationsuppressioncontroller

Electricalisolation

Electrical isolation

Eddy currentsensor

Accelerometer

Test and control system

Nozzle

Figure 4 Spray online balance system structure

4 Shock and Vibration

elasticity [31] regardless of the tangential displacementcomponent of the inner and outer rings in the rotationprocess the radial centrifugal force of the inner and outerrings is applied as a unit volume force which can beequivalent to the problem of axisymmetric plane stress At anyradius r the deformation displacement ur is [32]

ur 3 + ]8E

ρω2r (1minus ]) B

2+ A

21113872 1113873 +(1 + ])

A2B2

r2minus1minus ]2

3 + ]r2

1113890 1113891

(11)

where ω is angular velocity ρ is material density E is elasticmodulus ] is Poissonrsquos ratio and A and B are the diametersof the inner and outer rings respectively en the mini-mum allowable amount of interference δ is shown as follows

δ uo RF( 1113857minus ui RF( 1113857 3 + v

4Eρw

2RF R

2O minusR

2I1113872 1113873 (12)

where RF is the critical radius of the inner and outer rings(RF 33 mm) RO is the outer radius of the outer ring of thedynamic balance terminal (RO 47 mm) and RI is innerradius of the inner ring of the balancing terminal(RI 28 mm) Balanced terminal material is titanium alloy() When the magnitude of interference is 20 μm the re-lationship between the radial expansion of the inner andouter rings and the rotation speed is shown in Figure 6 Itcan be seen that when the rotational speed reaches the loosecritical speed the interference connection will fail So theassembled balance terminal not only loses the capacity of thechamber seal but also has potential safety hazards Of coursethe magnitude of interference cannot be too large If themagnitude of interference is too large assembly will bedifficult Pressing in force will even cause fatigue damage tothe inner and outer rings

Because of the above problems in the traditional as-sembly balanced terminals an integrated balanced terminalthat combines the needs of spray online dynamic balancingand the technical features of rapid prototyping was proposed

by our research group as shown in Figure 7 e integratedbalancing terminal eliminates the traditional design of theinner and outer rings It is mainly composed of the terminalbody flow path offline weight holes tapered holes andchamberse integrated balanced terminal is formed in onetime using 3D printing technology e tapered holes areused to connect with the spindle e function of offlineweight holes is to eliminate the initial eccentricity of thebalanced terminal in the manufacturing process is willnot cause new additional imbalance to the spindle efunction of the flow path is to direct the fluid to the cor-responding chamber

e capacity of the integrated balanced terminal de-termines its balance capacity In order to calculate thebalance capacity of the balanced terminal the volume ofchamber must be calculated In order to simplify the cal-culation the corner is ignored and the cross section shownin Figure 8(a) is obtained the integral microelement isshown in Figure 8(c)

120

100

80

60

40

20

00 2 4 6

Speed (rpm) (times104)

Radi

al d

ispla

cem

ent (microm

)

Initialinterference

Critical point

Inner ringOuter ring

Figure 6 Relationship between radial displacement and speed

Assembly

Inner ring

Outer ring

Cavity line

Figure 5 Assembled balanced terminal schematic

Flow-path

Offline weighthole

Tapered holeCavity line

Figure 7 Integrated balance terminal diagram

Shock and Vibration 5

en the volume of a single liquid chamber is given by

V1 2πα360

1113946l1

0R1 + l( 1113857 h1 + 2l( 1113857dl

V2 2πα360

1113946l2

0R1 + l1 + l( 1113857 h2 minus 2l( 1113857dl

V V1 + V2

(13)

where h1 145 mm h2 26 mm l1 l2 575 mm R1

33 mm so V asymp 13230 mm3

Assume that the density of the weighted liquid isρ the maximum balance capacity of a single liquidchamber

Q1 2πρα360

1113946l1

0R1 + l( 1113857 h1 + 2l( 1113857(33 + l)dl

Q2 2πρα360

1113946l2

0R1 + l1 + l( 1113857 h2 minus 2l( 1113857 33 + l1 + l( 1113857dl

Q Q1 + Q2

(14)

where ρ 08 gmm3 so Q asymp 413 gmiddotmm It can be knownfrom the balanced terminal cavity distribution structure thatwhen the adjacent two chambers are filled with the balanceliquid and the other chambers are all empty the balancingcapacity of the balanced terminal will reach the maximumvalue at this time e maximum is

2

radicQ 584 gmiddotmm

e balancing terminal rotates with the spindle at a highspeed So balancing the liquid in the terminal chamber willstress the chamber due to centrifugal force Assuming therotational speed is 30000 rpm the cavity of the balanceterminal is filled with the weight fluid with density08 gmm3 and the centrifugal force produced by the weightfluid in a single cavity can be calculated by the followingequation

F mω2r Qω2

(15)

where m is the liquid mass in the cavity r is the radialdistance between the center of gravity of the counterweight

fluid and the axis of rotation ω is the angular velocity andQ m times r hence centrifugal force F 4096 N

When the balanced terminal rotates at a high speed mostof the water pressure generated by the centrifugal force willact on the outer surface of the cavity and the outer surface ofthe cavity is not a regular plane To simplify the calculationthe pressure area on the outer surface of the chamber isequivalent to the projection surface as shown in Figure 9When the balanced terminal is filled with the counterweightfluid the pressure on the outer surface of the chamber iscalculated by the following equation

P F

S

F

(2α360) πR2h2( 1113857 (16)

where centrifugal force F 4076 N α 84deg radius R2

445 mm and thickness h2 26 mm thus the pressureP 24 MPa

In order to select the thickness of the cavity wall of thebalanced terminal the analysis was carried out under theextreme conditions (full-chamber liquid and 30000 rpm)When the pressure is 24MPa different thicknesses of thecavity arc surface force analysis model are built in ANSYS asshown in Figure 10

e relationship between the material thickness and thestress and deformation of the titanium alloy balance ter-minal surface is shown in Figure 11 It can be seen that whenthe material thickness is selected as 2mm under the extremeconditions the stress value and deformation tend to bestable erefore in order to meet the requirement of thebalanced terminal with the spindle acceleration and de-celeration the wall thickness of the force surface should begreater than 2mm We have designed the thickness of thebalanced terminal force surface to be 25mm

e assembled balance terminal of different materialswas trial-produced by us as shown in Figure 12 By analyzingthe requirements of 3D printing technology and strengthtitanium alloy is selected as the manufacturing material forthe integrated balance terminal

So the integrated balanced terminal manufacturedaccording to the design parameters is shown in Figure 13(a)For the nonblocking of the liquid flow and balancing the

h 1l1 l2h 2

(a)α

R 1

R2

(b)

h

Δ1

(c)

Figure 8 Calculation model (a) Section (b) Top view (c) Microelement

6 Shock and Vibration

(a) (b)

Figure 10 Surface force analysis model

P P

Figure 9 Fluid pressure on the cavity surface

0 1 2 3 4 550

100

150

200

250

300

350

400

ickness (mm)

Def

orm

atio

n (μ

m)

(a)

0 1 2 3 4 50

100

200

300

400

500

600

ickness (mm)

Stre

ss (M

pa)

(b)

Figure 11 e relationship between thickness and stress and deformation

(a) (b) (c)

Figure 12 Assembled balance terminal (a) Steel-45 (b) Aluminum alloy-7A04 (c) Titanium alloy-TC4

Shock and Vibration 7

signal test on the terminal surface the flow channel and thesurface were modified as shown in Figure 13(b) e bal-ancing terminal is balanced to eliminate the initial masseccentricity on the balancing machine is will not causenew additional imbalance to the spindle

e 14 model of assembled balanced terminals and in-tegrated balanced terminals is shown in Figure 14 e pa-rameters of the two balanced terminals are shown in Table 1It can be seen that the integrated balanced terminal designedin this paper has obvious advantages in comparison withthe assembled balanced terminal e maximum balancecapacity of balanced terminal increased from 489 gmiddotmm to584 gmiddotmm On the basis of a 194 increase in balance ca-pacity thickness dropped from 38mm to 28mm a drop of263 e volume of balanced terminal dropped from160549 mm3 to 83780 mm3 with 478 drop ratio e massof balanced terminal has dropped from 562 g to 387 g with

a drop ratio of 311 so that the additional mass added by thebalanced terminal to the spindle will be effectively reduced

33 Balanced Procedure and Software e no trial weightbalance method is applied to the spray online balancesystem which can achieve efficient online balance thusmaximizing the protection automation processing link eflow diagram of the no trial weight spray online dynamic isshown in Figure 15 in which three main steps are included(1) acquisition process of rotor structural parameters (2)acquisition process of counterweight scheme and (3) onlinebalance process e balancing procedure is as follows

(1) Establish the physical model and finite elementmodel of the spindle rotor system by separating therotor model

(a) (b)

Figure 13 Titanium alloy integrated balancing terminal (a) Before modification (b) After modification

l2prime = 4mml3prime = 4mm

l1prime = 4mm

(a)

l2 = 25mm

l3 = 2mm

l1 = 2mm

(b)

Figure 14 Comparison of 14 model (a) Assembled terminal (b) Integrated terminal

Table 1 Comparison of two balanced terminal parameters

SpeciesParameters

Balance ability (gmiddotmm) ickness (mm) Volume (mm3) Mass (g) Price (RMB)Assembled terminal 489 38 160549 562 1024000Integrated terminal 584 28 83780 387 580400Variation () 194 263 478 311 433

8 Shock and Vibration

(2) Obtain the structural parameters of the spindle rotorsystem such as mass matrix M stiffness matrix Kand damping matrix C by the finite element simu-lation and simulate the inherent modal character-istics of the spindle

(3) Substitute the correction matrices of structural pa-rameters in Equation (9) to identify the aggravatedinfluence coefficient matrix

(4) Extract the original vibration signal of the spindleand the phase information corresponding to thevibration signal substitute aggravated influencecoefficient matrix into Equation (10) to calculate theunbalance vector

(5) e weighting scheme is searched according to thecavity angle and the unbalance vector

(6) e spray online dynamic balancing system is in-jected into the corresponding cavity to correct theunbalance vector

According to the balance flow chart a measurement andcontrol system based on virtual instruments was developedby our research group e operation interface is shown inFigure 16 e no trial weight online balance measurementand control software consists of vibration signal monitoringtrend of vibration displacement change vibration statusdisplay and monitoring liquid state display and monitoringand online balance information records

Start

Build the spindle physical model

Establish the FEDM of spindlerotor system

Frequency-responsecharacteristic simulation

Obtain the spindle structure parametersM K C G ω

Mode test

Is the inspection errorreasonable

N

Y

Corrected initial value(ndashω2

2M2 + K2 + jω2C2)H2 (ω2) = I

Identify the ETCSRS(α) = (K)ndashω2 (M) + i(C)+(G)ωndash1middotexp (ndashjφ)

Vector synthesis principle for weight matching

Calculate the balance vector (Q)=ndash(α)ndash1middotA

Extract the originalvibration signal A

Get balance scheme

End

Online spray balance

Vibration gt targetvalue

Vibration gt targetvalue

Y

EndN

Y

N

Figure 15 e balance procedure

Shock and Vibration 9

4 Methods and System Validation

41 Experimental Setup In order to verify the application ofno trial weight spray online dynamic on high-speed spindlethe high-speed spindle comprehensive performance test bedwas designed by our research group as shown in Figure 17e test bed comprises a spindle 150 SD40Q7 (Figure 1)whose maximum rotational speed is 40000 rpm and twopairs of SNFA VEX35 bearings lubricated by oil mist aremounted on the spindle with a clearance tolerance to avoidthe preload variation e water cooling system controls thetemperature variation e balancing discs are installed onthe ends of the spindle to suppress vibration A rotatingencoder mounted on the shaft feedbacks the rotationalspeed e spindle rotor is designed with a ratio of 1 7tapered to install the integrated balancing terminal and the

balancing terminal is fixed with a locknut In the front of therotor a HSK C32 tool holder interface is designed emeasuring equipment is composed of a synchronous ac-quisition system developed by our research group A Na-tional Instrument PXI-4472 was used to extract vibrationand phase information German Micro-Epsilon U-05 eddy-current sensor (range 05mm resolution 0025 μm) is usedto measure vibration displacement e NI-PXI6123 is usedto control the hydraulic system A photoelectric sensor wasapplied to measure the phasee layout of the experimentalsetup is shown in Figure 18

42 Measurement Principles e system carries out real-time synchronous acquisition of vibration using eddy-current displacement sensors mounted along the X-axis

Vibrationsignal

Phase signal Vibration state

Data record

Straystate

Figure 16 Measurement and control panel

Diaplacementsensors

Integratedterminal

Sensor supportFixture

Spindle system Phase sensor

Control program

Control card anddata acquisition card

NI-PXI-1003

Nozzle Tool interface

Hydraulic system

24V DC

Control board

Conditioning circuit

Figure 17 Experimental setup

10 Shock and Vibration

X

S1 S2

S3

Z

Y

(a)

90deg

0deg360deg

270deg

180deg

ωY

Xϕθφ

Cavity-2

Cavity-1

Cavity-4

Cavity-3

S2

S1 S3A1

A2

A3

(b)

Figure 18 Measurement principles (a) Sensor layout (b) Angle relation

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

Before balanceAer balance

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

3

25

2

15

1

05

0

286 μm

021 μm

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(c)

287 μm

051 μmAm

plitu

de (μ

m)

3

25

2

15

1

05

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(d)

Figure 19 Continued

Shock and Vibration 11

0 5 10 15 20 25 30Time (seconds)

Am

plitu

de (μ

m)

35

3

25

2

15

Start balance(14286)

Balance end(19021)1

05

0

(e)

Figure 19 Comparison of balance information in the X-Y direction at 12000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

ndash4

0

2

4

ndash2Am

plitu

de (μ

m)

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

(a)

ndash2

ndash3

ndash1

0

1

2

3

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

337 μm

082 μm

(c)

0

05

15

1

2

25

3

Am

plitu

de (μ

m)

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

279 μm

1233 μm

(d)

Figure 20 Continued

12 Shock and Vibration

Am

plitu

de (μ

m)

4

3

2

1

0

Start balance(23337)

Balance end(26082)

0 5 10 15 20 25 3530Time (seconds)

(e)

Figure 20 Comparison of balance information in the X-Y direction at 15000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAfter balance

(b)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

466 μm

052 μm

(c)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

498 μm

131 μm

(d)

Figure 21 Continued

Shock and Vibration 13

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

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From the above equation it can be seen that the ag-gravated influence coefficient of the rotor system is de-termined by the mass stiffness lag angle and rotation speedat is to say the aggravated influence coefficient of therotor system is determined by the rotor structure parame-ters It is an inherent parameter to reflect the energy transfercharacteristics of the rotor system

[Q] minus[α]minus1

middot A (10)

where [Q] is balance vector A is unbalanced vibration re-sponse obtained through testing and the negative sign in-dicates that the phase difference between the balance vectorand the imbalance is 180deg After calculating the balance vectorwe apply it to the online dynamic balance by modifying thematching process and the online weighted correction schemecan be obtained by rapid orientation iteration matching edetailed online balance process is described later

3 Online Balancing Experiment System

31 System Structure e spray dynamic balance systemdeveloped by our research group was used to perform ex-perimental research on the no trial weight online balancemethod e structure of the spray online balancing system is

shown in Figure 4 e system is mainly composed of bal-anced terminals hydraulic systems and measurement andcontrol systemse balance terminal is the key component ofthe dynamic balancing system and its function is to store thecorrection mass e main function of the hydraulic system isthe quantitative delivery of liquid e function of the mea-surement and control system is unbalanced vector calculationand hydraulic system control

32 New Integrated Balance Terminal Balancing terminalsare the core components of spray dynamic balancing emain function is to store the correction mass rough in-vestigation it has been found that the balance terminal in theworld is the assembled balanced terminal shown in Figure 5

As can be seen from Figure 5 the traditional assembledbalance terminal is assembled from the inner ring and fromthe outer ring through interference fit However the de-formation of the outer ring of the balanced terminal is largerthan that of the inner ring under the action of centrifugalforce When the balanced terminal rotates with the spindle ata high speed the magnitude of interference between the innerand outer rings of the balanced terminal will decrease iswill lead to a drop in the sealing performance of the balancedterminal or even complete failure According to the theory of

Balanced terminalElectromagnetic value Hydraulic

system

Pump

Tank

Computer

Interactivetouchscreen

Vibrationsuppressioncontroller

Electricalisolation

Electrical isolation

Eddy currentsensor

Accelerometer

Test and control system

Nozzle

Figure 4 Spray online balance system structure

4 Shock and Vibration

elasticity [31] regardless of the tangential displacementcomponent of the inner and outer rings in the rotationprocess the radial centrifugal force of the inner and outerrings is applied as a unit volume force which can beequivalent to the problem of axisymmetric plane stress At anyradius r the deformation displacement ur is [32]

ur 3 + ]8E

ρω2r (1minus ]) B

2+ A

21113872 1113873 +(1 + ])

A2B2

r2minus1minus ]2

3 + ]r2

1113890 1113891

(11)

where ω is angular velocity ρ is material density E is elasticmodulus ] is Poissonrsquos ratio and A and B are the diametersof the inner and outer rings respectively en the mini-mum allowable amount of interference δ is shown as follows

δ uo RF( 1113857minus ui RF( 1113857 3 + v

4Eρw

2RF R

2O minusR

2I1113872 1113873 (12)

where RF is the critical radius of the inner and outer rings(RF 33 mm) RO is the outer radius of the outer ring of thedynamic balance terminal (RO 47 mm) and RI is innerradius of the inner ring of the balancing terminal(RI 28 mm) Balanced terminal material is titanium alloy() When the magnitude of interference is 20 μm the re-lationship between the radial expansion of the inner andouter rings and the rotation speed is shown in Figure 6 Itcan be seen that when the rotational speed reaches the loosecritical speed the interference connection will fail So theassembled balance terminal not only loses the capacity of thechamber seal but also has potential safety hazards Of coursethe magnitude of interference cannot be too large If themagnitude of interference is too large assembly will bedifficult Pressing in force will even cause fatigue damage tothe inner and outer rings

Because of the above problems in the traditional as-sembly balanced terminals an integrated balanced terminalthat combines the needs of spray online dynamic balancingand the technical features of rapid prototyping was proposed

by our research group as shown in Figure 7 e integratedbalancing terminal eliminates the traditional design of theinner and outer rings It is mainly composed of the terminalbody flow path offline weight holes tapered holes andchamberse integrated balanced terminal is formed in onetime using 3D printing technology e tapered holes areused to connect with the spindle e function of offlineweight holes is to eliminate the initial eccentricity of thebalanced terminal in the manufacturing process is willnot cause new additional imbalance to the spindle efunction of the flow path is to direct the fluid to the cor-responding chamber

e capacity of the integrated balanced terminal de-termines its balance capacity In order to calculate thebalance capacity of the balanced terminal the volume ofchamber must be calculated In order to simplify the cal-culation the corner is ignored and the cross section shownin Figure 8(a) is obtained the integral microelement isshown in Figure 8(c)

120

100

80

60

40

20

00 2 4 6

Speed (rpm) (times104)

Radi

al d

ispla

cem

ent (microm

)

Initialinterference

Critical point

Inner ringOuter ring

Figure 6 Relationship between radial displacement and speed

Assembly

Inner ring

Outer ring

Cavity line

Figure 5 Assembled balanced terminal schematic

Flow-path

Offline weighthole

Tapered holeCavity line

Figure 7 Integrated balance terminal diagram

Shock and Vibration 5

en the volume of a single liquid chamber is given by

V1 2πα360

1113946l1

0R1 + l( 1113857 h1 + 2l( 1113857dl

V2 2πα360

1113946l2

0R1 + l1 + l( 1113857 h2 minus 2l( 1113857dl

V V1 + V2

(13)

where h1 145 mm h2 26 mm l1 l2 575 mm R1

33 mm so V asymp 13230 mm3

Assume that the density of the weighted liquid isρ the maximum balance capacity of a single liquidchamber

Q1 2πρα360

1113946l1

0R1 + l( 1113857 h1 + 2l( 1113857(33 + l)dl

Q2 2πρα360

1113946l2

0R1 + l1 + l( 1113857 h2 minus 2l( 1113857 33 + l1 + l( 1113857dl

Q Q1 + Q2

(14)

where ρ 08 gmm3 so Q asymp 413 gmiddotmm It can be knownfrom the balanced terminal cavity distribution structure thatwhen the adjacent two chambers are filled with the balanceliquid and the other chambers are all empty the balancingcapacity of the balanced terminal will reach the maximumvalue at this time e maximum is

2

radicQ 584 gmiddotmm

e balancing terminal rotates with the spindle at a highspeed So balancing the liquid in the terminal chamber willstress the chamber due to centrifugal force Assuming therotational speed is 30000 rpm the cavity of the balanceterminal is filled with the weight fluid with density08 gmm3 and the centrifugal force produced by the weightfluid in a single cavity can be calculated by the followingequation

F mω2r Qω2

(15)

where m is the liquid mass in the cavity r is the radialdistance between the center of gravity of the counterweight

fluid and the axis of rotation ω is the angular velocity andQ m times r hence centrifugal force F 4096 N

When the balanced terminal rotates at a high speed mostof the water pressure generated by the centrifugal force willact on the outer surface of the cavity and the outer surface ofthe cavity is not a regular plane To simplify the calculationthe pressure area on the outer surface of the chamber isequivalent to the projection surface as shown in Figure 9When the balanced terminal is filled with the counterweightfluid the pressure on the outer surface of the chamber iscalculated by the following equation

P F

S

F

(2α360) πR2h2( 1113857 (16)

where centrifugal force F 4076 N α 84deg radius R2

445 mm and thickness h2 26 mm thus the pressureP 24 MPa

In order to select the thickness of the cavity wall of thebalanced terminal the analysis was carried out under theextreme conditions (full-chamber liquid and 30000 rpm)When the pressure is 24MPa different thicknesses of thecavity arc surface force analysis model are built in ANSYS asshown in Figure 10

e relationship between the material thickness and thestress and deformation of the titanium alloy balance ter-minal surface is shown in Figure 11 It can be seen that whenthe material thickness is selected as 2mm under the extremeconditions the stress value and deformation tend to bestable erefore in order to meet the requirement of thebalanced terminal with the spindle acceleration and de-celeration the wall thickness of the force surface should begreater than 2mm We have designed the thickness of thebalanced terminal force surface to be 25mm

e assembled balance terminal of different materialswas trial-produced by us as shown in Figure 12 By analyzingthe requirements of 3D printing technology and strengthtitanium alloy is selected as the manufacturing material forthe integrated balance terminal

So the integrated balanced terminal manufacturedaccording to the design parameters is shown in Figure 13(a)For the nonblocking of the liquid flow and balancing the

h 1l1 l2h 2

(a)α

R 1

R2

(b)

h

Δ1

(c)

Figure 8 Calculation model (a) Section (b) Top view (c) Microelement

6 Shock and Vibration

(a) (b)

Figure 10 Surface force analysis model

P P

Figure 9 Fluid pressure on the cavity surface

0 1 2 3 4 550

100

150

200

250

300

350

400

ickness (mm)

Def

orm

atio

n (μ

m)

(a)

0 1 2 3 4 50

100

200

300

400

500

600

ickness (mm)

Stre

ss (M

pa)

(b)

Figure 11 e relationship between thickness and stress and deformation

(a) (b) (c)

Figure 12 Assembled balance terminal (a) Steel-45 (b) Aluminum alloy-7A04 (c) Titanium alloy-TC4

Shock and Vibration 7

signal test on the terminal surface the flow channel and thesurface were modified as shown in Figure 13(b) e bal-ancing terminal is balanced to eliminate the initial masseccentricity on the balancing machine is will not causenew additional imbalance to the spindle

e 14 model of assembled balanced terminals and in-tegrated balanced terminals is shown in Figure 14 e pa-rameters of the two balanced terminals are shown in Table 1It can be seen that the integrated balanced terminal designedin this paper has obvious advantages in comparison withthe assembled balanced terminal e maximum balancecapacity of balanced terminal increased from 489 gmiddotmm to584 gmiddotmm On the basis of a 194 increase in balance ca-pacity thickness dropped from 38mm to 28mm a drop of263 e volume of balanced terminal dropped from160549 mm3 to 83780 mm3 with 478 drop ratio e massof balanced terminal has dropped from 562 g to 387 g with

a drop ratio of 311 so that the additional mass added by thebalanced terminal to the spindle will be effectively reduced

33 Balanced Procedure and Software e no trial weightbalance method is applied to the spray online balancesystem which can achieve efficient online balance thusmaximizing the protection automation processing link eflow diagram of the no trial weight spray online dynamic isshown in Figure 15 in which three main steps are included(1) acquisition process of rotor structural parameters (2)acquisition process of counterweight scheme and (3) onlinebalance process e balancing procedure is as follows

(1) Establish the physical model and finite elementmodel of the spindle rotor system by separating therotor model

(a) (b)

Figure 13 Titanium alloy integrated balancing terminal (a) Before modification (b) After modification

l2prime = 4mml3prime = 4mm

l1prime = 4mm

(a)

l2 = 25mm

l3 = 2mm

l1 = 2mm

(b)

Figure 14 Comparison of 14 model (a) Assembled terminal (b) Integrated terminal

Table 1 Comparison of two balanced terminal parameters

SpeciesParameters

Balance ability (gmiddotmm) ickness (mm) Volume (mm3) Mass (g) Price (RMB)Assembled terminal 489 38 160549 562 1024000Integrated terminal 584 28 83780 387 580400Variation () 194 263 478 311 433

8 Shock and Vibration

(2) Obtain the structural parameters of the spindle rotorsystem such as mass matrix M stiffness matrix Kand damping matrix C by the finite element simu-lation and simulate the inherent modal character-istics of the spindle

(3) Substitute the correction matrices of structural pa-rameters in Equation (9) to identify the aggravatedinfluence coefficient matrix

(4) Extract the original vibration signal of the spindleand the phase information corresponding to thevibration signal substitute aggravated influencecoefficient matrix into Equation (10) to calculate theunbalance vector

(5) e weighting scheme is searched according to thecavity angle and the unbalance vector

(6) e spray online dynamic balancing system is in-jected into the corresponding cavity to correct theunbalance vector

According to the balance flow chart a measurement andcontrol system based on virtual instruments was developedby our research group e operation interface is shown inFigure 16 e no trial weight online balance measurementand control software consists of vibration signal monitoringtrend of vibration displacement change vibration statusdisplay and monitoring liquid state display and monitoringand online balance information records

Start

Build the spindle physical model

Establish the FEDM of spindlerotor system

Frequency-responsecharacteristic simulation

Obtain the spindle structure parametersM K C G ω

Mode test

Is the inspection errorreasonable

N

Y

Corrected initial value(ndashω2

2M2 + K2 + jω2C2)H2 (ω2) = I

Identify the ETCSRS(α) = (K)ndashω2 (M) + i(C)+(G)ωndash1middotexp (ndashjφ)

Vector synthesis principle for weight matching

Calculate the balance vector (Q)=ndash(α)ndash1middotA

Extract the originalvibration signal A

Get balance scheme

End

Online spray balance

Vibration gt targetvalue

Vibration gt targetvalue

Y

EndN

Y

N

Figure 15 e balance procedure

Shock and Vibration 9

4 Methods and System Validation

41 Experimental Setup In order to verify the application ofno trial weight spray online dynamic on high-speed spindlethe high-speed spindle comprehensive performance test bedwas designed by our research group as shown in Figure 17e test bed comprises a spindle 150 SD40Q7 (Figure 1)whose maximum rotational speed is 40000 rpm and twopairs of SNFA VEX35 bearings lubricated by oil mist aremounted on the spindle with a clearance tolerance to avoidthe preload variation e water cooling system controls thetemperature variation e balancing discs are installed onthe ends of the spindle to suppress vibration A rotatingencoder mounted on the shaft feedbacks the rotationalspeed e spindle rotor is designed with a ratio of 1 7tapered to install the integrated balancing terminal and the

balancing terminal is fixed with a locknut In the front of therotor a HSK C32 tool holder interface is designed emeasuring equipment is composed of a synchronous ac-quisition system developed by our research group A Na-tional Instrument PXI-4472 was used to extract vibrationand phase information German Micro-Epsilon U-05 eddy-current sensor (range 05mm resolution 0025 μm) is usedto measure vibration displacement e NI-PXI6123 is usedto control the hydraulic system A photoelectric sensor wasapplied to measure the phasee layout of the experimentalsetup is shown in Figure 18

42 Measurement Principles e system carries out real-time synchronous acquisition of vibration using eddy-current displacement sensors mounted along the X-axis

Vibrationsignal

Phase signal Vibration state

Data record

Straystate

Figure 16 Measurement and control panel

Diaplacementsensors

Integratedterminal

Sensor supportFixture

Spindle system Phase sensor

Control program

Control card anddata acquisition card

NI-PXI-1003

Nozzle Tool interface

Hydraulic system

24V DC

Control board

Conditioning circuit

Figure 17 Experimental setup

10 Shock and Vibration

X

S1 S2

S3

Z

Y

(a)

90deg

0deg360deg

270deg

180deg

ωY

Xϕθφ

Cavity-2

Cavity-1

Cavity-4

Cavity-3

S2

S1 S3A1

A2

A3

(b)

Figure 18 Measurement principles (a) Sensor layout (b) Angle relation

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

Before balanceAer balance

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

3

25

2

15

1

05

0

286 μm

021 μm

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(c)

287 μm

051 μmAm

plitu

de (μ

m)

3

25

2

15

1

05

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(d)

Figure 19 Continued

Shock and Vibration 11

0 5 10 15 20 25 30Time (seconds)

Am

plitu

de (μ

m)

35

3

25

2

15

Start balance(14286)

Balance end(19021)1

05

0

(e)

Figure 19 Comparison of balance information in the X-Y direction at 12000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

ndash4

0

2

4

ndash2Am

plitu

de (μ

m)

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

(a)

ndash2

ndash3

ndash1

0

1

2

3

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

337 μm

082 μm

(c)

0

05

15

1

2

25

3

Am

plitu

de (μ

m)

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

279 μm

1233 μm

(d)

Figure 20 Continued

12 Shock and Vibration

Am

plitu

de (μ

m)

4

3

2

1

0

Start balance(23337)

Balance end(26082)

0 5 10 15 20 25 3530Time (seconds)

(e)

Figure 20 Comparison of balance information in the X-Y direction at 15000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAfter balance

(b)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

466 μm

052 μm

(c)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

498 μm

131 μm

(d)

Figure 21 Continued

Shock and Vibration 13

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

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elasticity [31] regardless of the tangential displacementcomponent of the inner and outer rings in the rotationprocess the radial centrifugal force of the inner and outerrings is applied as a unit volume force which can beequivalent to the problem of axisymmetric plane stress At anyradius r the deformation displacement ur is [32]

ur 3 + ]8E

ρω2r (1minus ]) B

2+ A

21113872 1113873 +(1 + ])

A2B2

r2minus1minus ]2

3 + ]r2

1113890 1113891

(11)

where ω is angular velocity ρ is material density E is elasticmodulus ] is Poissonrsquos ratio and A and B are the diametersof the inner and outer rings respectively en the mini-mum allowable amount of interference δ is shown as follows

δ uo RF( 1113857minus ui RF( 1113857 3 + v

4Eρw

2RF R

2O minusR

2I1113872 1113873 (12)

where RF is the critical radius of the inner and outer rings(RF 33 mm) RO is the outer radius of the outer ring of thedynamic balance terminal (RO 47 mm) and RI is innerradius of the inner ring of the balancing terminal(RI 28 mm) Balanced terminal material is titanium alloy() When the magnitude of interference is 20 μm the re-lationship between the radial expansion of the inner andouter rings and the rotation speed is shown in Figure 6 Itcan be seen that when the rotational speed reaches the loosecritical speed the interference connection will fail So theassembled balance terminal not only loses the capacity of thechamber seal but also has potential safety hazards Of coursethe magnitude of interference cannot be too large If themagnitude of interference is too large assembly will bedifficult Pressing in force will even cause fatigue damage tothe inner and outer rings

Because of the above problems in the traditional as-sembly balanced terminals an integrated balanced terminalthat combines the needs of spray online dynamic balancingand the technical features of rapid prototyping was proposed

by our research group as shown in Figure 7 e integratedbalancing terminal eliminates the traditional design of theinner and outer rings It is mainly composed of the terminalbody flow path offline weight holes tapered holes andchamberse integrated balanced terminal is formed in onetime using 3D printing technology e tapered holes areused to connect with the spindle e function of offlineweight holes is to eliminate the initial eccentricity of thebalanced terminal in the manufacturing process is willnot cause new additional imbalance to the spindle efunction of the flow path is to direct the fluid to the cor-responding chamber

e capacity of the integrated balanced terminal de-termines its balance capacity In order to calculate thebalance capacity of the balanced terminal the volume ofchamber must be calculated In order to simplify the cal-culation the corner is ignored and the cross section shownin Figure 8(a) is obtained the integral microelement isshown in Figure 8(c)

120

100

80

60

40

20

00 2 4 6

Speed (rpm) (times104)

Radi

al d

ispla

cem

ent (microm

)

Initialinterference

Critical point

Inner ringOuter ring

Figure 6 Relationship between radial displacement and speed

Assembly

Inner ring

Outer ring

Cavity line

Figure 5 Assembled balanced terminal schematic

Flow-path

Offline weighthole

Tapered holeCavity line

Figure 7 Integrated balance terminal diagram

Shock and Vibration 5

en the volume of a single liquid chamber is given by

V1 2πα360

1113946l1

0R1 + l( 1113857 h1 + 2l( 1113857dl

V2 2πα360

1113946l2

0R1 + l1 + l( 1113857 h2 minus 2l( 1113857dl

V V1 + V2

(13)

where h1 145 mm h2 26 mm l1 l2 575 mm R1

33 mm so V asymp 13230 mm3

Assume that the density of the weighted liquid isρ the maximum balance capacity of a single liquidchamber

Q1 2πρα360

1113946l1

0R1 + l( 1113857 h1 + 2l( 1113857(33 + l)dl

Q2 2πρα360

1113946l2

0R1 + l1 + l( 1113857 h2 minus 2l( 1113857 33 + l1 + l( 1113857dl

Q Q1 + Q2

(14)

where ρ 08 gmm3 so Q asymp 413 gmiddotmm It can be knownfrom the balanced terminal cavity distribution structure thatwhen the adjacent two chambers are filled with the balanceliquid and the other chambers are all empty the balancingcapacity of the balanced terminal will reach the maximumvalue at this time e maximum is

2

radicQ 584 gmiddotmm

e balancing terminal rotates with the spindle at a highspeed So balancing the liquid in the terminal chamber willstress the chamber due to centrifugal force Assuming therotational speed is 30000 rpm the cavity of the balanceterminal is filled with the weight fluid with density08 gmm3 and the centrifugal force produced by the weightfluid in a single cavity can be calculated by the followingequation

F mω2r Qω2

(15)

where m is the liquid mass in the cavity r is the radialdistance between the center of gravity of the counterweight

fluid and the axis of rotation ω is the angular velocity andQ m times r hence centrifugal force F 4096 N

When the balanced terminal rotates at a high speed mostof the water pressure generated by the centrifugal force willact on the outer surface of the cavity and the outer surface ofthe cavity is not a regular plane To simplify the calculationthe pressure area on the outer surface of the chamber isequivalent to the projection surface as shown in Figure 9When the balanced terminal is filled with the counterweightfluid the pressure on the outer surface of the chamber iscalculated by the following equation

P F

S

F

(2α360) πR2h2( 1113857 (16)

where centrifugal force F 4076 N α 84deg radius R2

445 mm and thickness h2 26 mm thus the pressureP 24 MPa

In order to select the thickness of the cavity wall of thebalanced terminal the analysis was carried out under theextreme conditions (full-chamber liquid and 30000 rpm)When the pressure is 24MPa different thicknesses of thecavity arc surface force analysis model are built in ANSYS asshown in Figure 10

e relationship between the material thickness and thestress and deformation of the titanium alloy balance ter-minal surface is shown in Figure 11 It can be seen that whenthe material thickness is selected as 2mm under the extremeconditions the stress value and deformation tend to bestable erefore in order to meet the requirement of thebalanced terminal with the spindle acceleration and de-celeration the wall thickness of the force surface should begreater than 2mm We have designed the thickness of thebalanced terminal force surface to be 25mm

e assembled balance terminal of different materialswas trial-produced by us as shown in Figure 12 By analyzingthe requirements of 3D printing technology and strengthtitanium alloy is selected as the manufacturing material forthe integrated balance terminal

So the integrated balanced terminal manufacturedaccording to the design parameters is shown in Figure 13(a)For the nonblocking of the liquid flow and balancing the

h 1l1 l2h 2

(a)α

R 1

R2

(b)

h

Δ1

(c)

Figure 8 Calculation model (a) Section (b) Top view (c) Microelement

6 Shock and Vibration

(a) (b)

Figure 10 Surface force analysis model

P P

Figure 9 Fluid pressure on the cavity surface

0 1 2 3 4 550

100

150

200

250

300

350

400

ickness (mm)

Def

orm

atio

n (μ

m)

(a)

0 1 2 3 4 50

100

200

300

400

500

600

ickness (mm)

Stre

ss (M

pa)

(b)

Figure 11 e relationship between thickness and stress and deformation

(a) (b) (c)

Figure 12 Assembled balance terminal (a) Steel-45 (b) Aluminum alloy-7A04 (c) Titanium alloy-TC4

Shock and Vibration 7

signal test on the terminal surface the flow channel and thesurface were modified as shown in Figure 13(b) e bal-ancing terminal is balanced to eliminate the initial masseccentricity on the balancing machine is will not causenew additional imbalance to the spindle

e 14 model of assembled balanced terminals and in-tegrated balanced terminals is shown in Figure 14 e pa-rameters of the two balanced terminals are shown in Table 1It can be seen that the integrated balanced terminal designedin this paper has obvious advantages in comparison withthe assembled balanced terminal e maximum balancecapacity of balanced terminal increased from 489 gmiddotmm to584 gmiddotmm On the basis of a 194 increase in balance ca-pacity thickness dropped from 38mm to 28mm a drop of263 e volume of balanced terminal dropped from160549 mm3 to 83780 mm3 with 478 drop ratio e massof balanced terminal has dropped from 562 g to 387 g with

a drop ratio of 311 so that the additional mass added by thebalanced terminal to the spindle will be effectively reduced

33 Balanced Procedure and Software e no trial weightbalance method is applied to the spray online balancesystem which can achieve efficient online balance thusmaximizing the protection automation processing link eflow diagram of the no trial weight spray online dynamic isshown in Figure 15 in which three main steps are included(1) acquisition process of rotor structural parameters (2)acquisition process of counterweight scheme and (3) onlinebalance process e balancing procedure is as follows

(1) Establish the physical model and finite elementmodel of the spindle rotor system by separating therotor model

(a) (b)

Figure 13 Titanium alloy integrated balancing terminal (a) Before modification (b) After modification

l2prime = 4mml3prime = 4mm

l1prime = 4mm

(a)

l2 = 25mm

l3 = 2mm

l1 = 2mm

(b)

Figure 14 Comparison of 14 model (a) Assembled terminal (b) Integrated terminal

Table 1 Comparison of two balanced terminal parameters

SpeciesParameters

Balance ability (gmiddotmm) ickness (mm) Volume (mm3) Mass (g) Price (RMB)Assembled terminal 489 38 160549 562 1024000Integrated terminal 584 28 83780 387 580400Variation () 194 263 478 311 433

8 Shock and Vibration

(2) Obtain the structural parameters of the spindle rotorsystem such as mass matrix M stiffness matrix Kand damping matrix C by the finite element simu-lation and simulate the inherent modal character-istics of the spindle

(3) Substitute the correction matrices of structural pa-rameters in Equation (9) to identify the aggravatedinfluence coefficient matrix

(4) Extract the original vibration signal of the spindleand the phase information corresponding to thevibration signal substitute aggravated influencecoefficient matrix into Equation (10) to calculate theunbalance vector

(5) e weighting scheme is searched according to thecavity angle and the unbalance vector

(6) e spray online dynamic balancing system is in-jected into the corresponding cavity to correct theunbalance vector

According to the balance flow chart a measurement andcontrol system based on virtual instruments was developedby our research group e operation interface is shown inFigure 16 e no trial weight online balance measurementand control software consists of vibration signal monitoringtrend of vibration displacement change vibration statusdisplay and monitoring liquid state display and monitoringand online balance information records

Start

Build the spindle physical model

Establish the FEDM of spindlerotor system

Frequency-responsecharacteristic simulation

Obtain the spindle structure parametersM K C G ω

Mode test

Is the inspection errorreasonable

N

Y

Corrected initial value(ndashω2

2M2 + K2 + jω2C2)H2 (ω2) = I

Identify the ETCSRS(α) = (K)ndashω2 (M) + i(C)+(G)ωndash1middotexp (ndashjφ)

Vector synthesis principle for weight matching

Calculate the balance vector (Q)=ndash(α)ndash1middotA

Extract the originalvibration signal A

Get balance scheme

End

Online spray balance

Vibration gt targetvalue

Vibration gt targetvalue

Y

EndN

Y

N

Figure 15 e balance procedure

Shock and Vibration 9

4 Methods and System Validation

41 Experimental Setup In order to verify the application ofno trial weight spray online dynamic on high-speed spindlethe high-speed spindle comprehensive performance test bedwas designed by our research group as shown in Figure 17e test bed comprises a spindle 150 SD40Q7 (Figure 1)whose maximum rotational speed is 40000 rpm and twopairs of SNFA VEX35 bearings lubricated by oil mist aremounted on the spindle with a clearance tolerance to avoidthe preload variation e water cooling system controls thetemperature variation e balancing discs are installed onthe ends of the spindle to suppress vibration A rotatingencoder mounted on the shaft feedbacks the rotationalspeed e spindle rotor is designed with a ratio of 1 7tapered to install the integrated balancing terminal and the

balancing terminal is fixed with a locknut In the front of therotor a HSK C32 tool holder interface is designed emeasuring equipment is composed of a synchronous ac-quisition system developed by our research group A Na-tional Instrument PXI-4472 was used to extract vibrationand phase information German Micro-Epsilon U-05 eddy-current sensor (range 05mm resolution 0025 μm) is usedto measure vibration displacement e NI-PXI6123 is usedto control the hydraulic system A photoelectric sensor wasapplied to measure the phasee layout of the experimentalsetup is shown in Figure 18

42 Measurement Principles e system carries out real-time synchronous acquisition of vibration using eddy-current displacement sensors mounted along the X-axis

Vibrationsignal

Phase signal Vibration state

Data record

Straystate

Figure 16 Measurement and control panel

Diaplacementsensors

Integratedterminal

Sensor supportFixture

Spindle system Phase sensor

Control program

Control card anddata acquisition card

NI-PXI-1003

Nozzle Tool interface

Hydraulic system

24V DC

Control board

Conditioning circuit

Figure 17 Experimental setup

10 Shock and Vibration

X

S1 S2

S3

Z

Y

(a)

90deg

0deg360deg

270deg

180deg

ωY

Xϕθφ

Cavity-2

Cavity-1

Cavity-4

Cavity-3

S2

S1 S3A1

A2

A3

(b)

Figure 18 Measurement principles (a) Sensor layout (b) Angle relation

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

Before balanceAer balance

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

3

25

2

15

1

05

0

286 μm

021 μm

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(c)

287 μm

051 μmAm

plitu

de (μ

m)

3

25

2

15

1

05

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(d)

Figure 19 Continued

Shock and Vibration 11

0 5 10 15 20 25 30Time (seconds)

Am

plitu

de (μ

m)

35

3

25

2

15

Start balance(14286)

Balance end(19021)1

05

0

(e)

Figure 19 Comparison of balance information in the X-Y direction at 12000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

ndash4

0

2

4

ndash2Am

plitu

de (μ

m)

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

(a)

ndash2

ndash3

ndash1

0

1

2

3

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

337 μm

082 μm

(c)

0

05

15

1

2

25

3

Am

plitu

de (μ

m)

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

279 μm

1233 μm

(d)

Figure 20 Continued

12 Shock and Vibration

Am

plitu

de (μ

m)

4

3

2

1

0

Start balance(23337)

Balance end(26082)

0 5 10 15 20 25 3530Time (seconds)

(e)

Figure 20 Comparison of balance information in the X-Y direction at 15000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAfter balance

(b)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

466 μm

052 μm

(c)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

498 μm

131 μm

(d)

Figure 21 Continued

Shock and Vibration 13

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

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Submit your manuscripts atwwwhindawicom

en the volume of a single liquid chamber is given by

V1 2πα360

1113946l1

0R1 + l( 1113857 h1 + 2l( 1113857dl

V2 2πα360

1113946l2

0R1 + l1 + l( 1113857 h2 minus 2l( 1113857dl

V V1 + V2

(13)

where h1 145 mm h2 26 mm l1 l2 575 mm R1

33 mm so V asymp 13230 mm3

Assume that the density of the weighted liquid isρ the maximum balance capacity of a single liquidchamber

Q1 2πρα360

1113946l1

0R1 + l( 1113857 h1 + 2l( 1113857(33 + l)dl

Q2 2πρα360

1113946l2

0R1 + l1 + l( 1113857 h2 minus 2l( 1113857 33 + l1 + l( 1113857dl

Q Q1 + Q2

(14)

where ρ 08 gmm3 so Q asymp 413 gmiddotmm It can be knownfrom the balanced terminal cavity distribution structure thatwhen the adjacent two chambers are filled with the balanceliquid and the other chambers are all empty the balancingcapacity of the balanced terminal will reach the maximumvalue at this time e maximum is

2

radicQ 584 gmiddotmm

e balancing terminal rotates with the spindle at a highspeed So balancing the liquid in the terminal chamber willstress the chamber due to centrifugal force Assuming therotational speed is 30000 rpm the cavity of the balanceterminal is filled with the weight fluid with density08 gmm3 and the centrifugal force produced by the weightfluid in a single cavity can be calculated by the followingequation

F mω2r Qω2

(15)

where m is the liquid mass in the cavity r is the radialdistance between the center of gravity of the counterweight

fluid and the axis of rotation ω is the angular velocity andQ m times r hence centrifugal force F 4096 N

When the balanced terminal rotates at a high speed mostof the water pressure generated by the centrifugal force willact on the outer surface of the cavity and the outer surface ofthe cavity is not a regular plane To simplify the calculationthe pressure area on the outer surface of the chamber isequivalent to the projection surface as shown in Figure 9When the balanced terminal is filled with the counterweightfluid the pressure on the outer surface of the chamber iscalculated by the following equation

P F

S

F

(2α360) πR2h2( 1113857 (16)

where centrifugal force F 4076 N α 84deg radius R2

445 mm and thickness h2 26 mm thus the pressureP 24 MPa

In order to select the thickness of the cavity wall of thebalanced terminal the analysis was carried out under theextreme conditions (full-chamber liquid and 30000 rpm)When the pressure is 24MPa different thicknesses of thecavity arc surface force analysis model are built in ANSYS asshown in Figure 10

e relationship between the material thickness and thestress and deformation of the titanium alloy balance ter-minal surface is shown in Figure 11 It can be seen that whenthe material thickness is selected as 2mm under the extremeconditions the stress value and deformation tend to bestable erefore in order to meet the requirement of thebalanced terminal with the spindle acceleration and de-celeration the wall thickness of the force surface should begreater than 2mm We have designed the thickness of thebalanced terminal force surface to be 25mm

e assembled balance terminal of different materialswas trial-produced by us as shown in Figure 12 By analyzingthe requirements of 3D printing technology and strengthtitanium alloy is selected as the manufacturing material forthe integrated balance terminal

So the integrated balanced terminal manufacturedaccording to the design parameters is shown in Figure 13(a)For the nonblocking of the liquid flow and balancing the

h 1l1 l2h 2

(a)α

R 1

R2

(b)

h

Δ1

(c)

Figure 8 Calculation model (a) Section (b) Top view (c) Microelement

6 Shock and Vibration

(a) (b)

Figure 10 Surface force analysis model

P P

Figure 9 Fluid pressure on the cavity surface

0 1 2 3 4 550

100

150

200

250

300

350

400

ickness (mm)

Def

orm

atio

n (μ

m)

(a)

0 1 2 3 4 50

100

200

300

400

500

600

ickness (mm)

Stre

ss (M

pa)

(b)

Figure 11 e relationship between thickness and stress and deformation

(a) (b) (c)

Figure 12 Assembled balance terminal (a) Steel-45 (b) Aluminum alloy-7A04 (c) Titanium alloy-TC4

Shock and Vibration 7

signal test on the terminal surface the flow channel and thesurface were modified as shown in Figure 13(b) e bal-ancing terminal is balanced to eliminate the initial masseccentricity on the balancing machine is will not causenew additional imbalance to the spindle

e 14 model of assembled balanced terminals and in-tegrated balanced terminals is shown in Figure 14 e pa-rameters of the two balanced terminals are shown in Table 1It can be seen that the integrated balanced terminal designedin this paper has obvious advantages in comparison withthe assembled balanced terminal e maximum balancecapacity of balanced terminal increased from 489 gmiddotmm to584 gmiddotmm On the basis of a 194 increase in balance ca-pacity thickness dropped from 38mm to 28mm a drop of263 e volume of balanced terminal dropped from160549 mm3 to 83780 mm3 with 478 drop ratio e massof balanced terminal has dropped from 562 g to 387 g with

a drop ratio of 311 so that the additional mass added by thebalanced terminal to the spindle will be effectively reduced

33 Balanced Procedure and Software e no trial weightbalance method is applied to the spray online balancesystem which can achieve efficient online balance thusmaximizing the protection automation processing link eflow diagram of the no trial weight spray online dynamic isshown in Figure 15 in which three main steps are included(1) acquisition process of rotor structural parameters (2)acquisition process of counterweight scheme and (3) onlinebalance process e balancing procedure is as follows

(1) Establish the physical model and finite elementmodel of the spindle rotor system by separating therotor model

(a) (b)

Figure 13 Titanium alloy integrated balancing terminal (a) Before modification (b) After modification

l2prime = 4mml3prime = 4mm

l1prime = 4mm

(a)

l2 = 25mm

l3 = 2mm

l1 = 2mm

(b)

Figure 14 Comparison of 14 model (a) Assembled terminal (b) Integrated terminal

Table 1 Comparison of two balanced terminal parameters

SpeciesParameters

Balance ability (gmiddotmm) ickness (mm) Volume (mm3) Mass (g) Price (RMB)Assembled terminal 489 38 160549 562 1024000Integrated terminal 584 28 83780 387 580400Variation () 194 263 478 311 433

8 Shock and Vibration

(2) Obtain the structural parameters of the spindle rotorsystem such as mass matrix M stiffness matrix Kand damping matrix C by the finite element simu-lation and simulate the inherent modal character-istics of the spindle

(3) Substitute the correction matrices of structural pa-rameters in Equation (9) to identify the aggravatedinfluence coefficient matrix

(4) Extract the original vibration signal of the spindleand the phase information corresponding to thevibration signal substitute aggravated influencecoefficient matrix into Equation (10) to calculate theunbalance vector

(5) e weighting scheme is searched according to thecavity angle and the unbalance vector

(6) e spray online dynamic balancing system is in-jected into the corresponding cavity to correct theunbalance vector

According to the balance flow chart a measurement andcontrol system based on virtual instruments was developedby our research group e operation interface is shown inFigure 16 e no trial weight online balance measurementand control software consists of vibration signal monitoringtrend of vibration displacement change vibration statusdisplay and monitoring liquid state display and monitoringand online balance information records

Start

Build the spindle physical model

Establish the FEDM of spindlerotor system

Frequency-responsecharacteristic simulation

Obtain the spindle structure parametersM K C G ω

Mode test

Is the inspection errorreasonable

N

Y

Corrected initial value(ndashω2

2M2 + K2 + jω2C2)H2 (ω2) = I

Identify the ETCSRS(α) = (K)ndashω2 (M) + i(C)+(G)ωndash1middotexp (ndashjφ)

Vector synthesis principle for weight matching

Calculate the balance vector (Q)=ndash(α)ndash1middotA

Extract the originalvibration signal A

Get balance scheme

End

Online spray balance

Vibration gt targetvalue

Vibration gt targetvalue

Y

EndN

Y

N

Figure 15 e balance procedure

Shock and Vibration 9

4 Methods and System Validation

41 Experimental Setup In order to verify the application ofno trial weight spray online dynamic on high-speed spindlethe high-speed spindle comprehensive performance test bedwas designed by our research group as shown in Figure 17e test bed comprises a spindle 150 SD40Q7 (Figure 1)whose maximum rotational speed is 40000 rpm and twopairs of SNFA VEX35 bearings lubricated by oil mist aremounted on the spindle with a clearance tolerance to avoidthe preload variation e water cooling system controls thetemperature variation e balancing discs are installed onthe ends of the spindle to suppress vibration A rotatingencoder mounted on the shaft feedbacks the rotationalspeed e spindle rotor is designed with a ratio of 1 7tapered to install the integrated balancing terminal and the

balancing terminal is fixed with a locknut In the front of therotor a HSK C32 tool holder interface is designed emeasuring equipment is composed of a synchronous ac-quisition system developed by our research group A Na-tional Instrument PXI-4472 was used to extract vibrationand phase information German Micro-Epsilon U-05 eddy-current sensor (range 05mm resolution 0025 μm) is usedto measure vibration displacement e NI-PXI6123 is usedto control the hydraulic system A photoelectric sensor wasapplied to measure the phasee layout of the experimentalsetup is shown in Figure 18

42 Measurement Principles e system carries out real-time synchronous acquisition of vibration using eddy-current displacement sensors mounted along the X-axis

Vibrationsignal

Phase signal Vibration state

Data record

Straystate

Figure 16 Measurement and control panel

Diaplacementsensors

Integratedterminal

Sensor supportFixture

Spindle system Phase sensor

Control program

Control card anddata acquisition card

NI-PXI-1003

Nozzle Tool interface

Hydraulic system

24V DC

Control board

Conditioning circuit

Figure 17 Experimental setup

10 Shock and Vibration

X

S1 S2

S3

Z

Y

(a)

90deg

0deg360deg

270deg

180deg

ωY

Xϕθφ

Cavity-2

Cavity-1

Cavity-4

Cavity-3

S2

S1 S3A1

A2

A3

(b)

Figure 18 Measurement principles (a) Sensor layout (b) Angle relation

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

Before balanceAer balance

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

3

25

2

15

1

05

0

286 μm

021 μm

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(c)

287 μm

051 μmAm

plitu

de (μ

m)

3

25

2

15

1

05

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(d)

Figure 19 Continued

Shock and Vibration 11

0 5 10 15 20 25 30Time (seconds)

Am

plitu

de (μ

m)

35

3

25

2

15

Start balance(14286)

Balance end(19021)1

05

0

(e)

Figure 19 Comparison of balance information in the X-Y direction at 12000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

ndash4

0

2

4

ndash2Am

plitu

de (μ

m)

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

(a)

ndash2

ndash3

ndash1

0

1

2

3

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

337 μm

082 μm

(c)

0

05

15

1

2

25

3

Am

plitu

de (μ

m)

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

279 μm

1233 μm

(d)

Figure 20 Continued

12 Shock and Vibration

Am

plitu

de (μ

m)

4

3

2

1

0

Start balance(23337)

Balance end(26082)

0 5 10 15 20 25 3530Time (seconds)

(e)

Figure 20 Comparison of balance information in the X-Y direction at 15000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAfter balance

(b)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

466 μm

052 μm

(c)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

498 μm

131 μm

(d)

Figure 21 Continued

Shock and Vibration 13

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

(a) (b)

Figure 10 Surface force analysis model

P P

Figure 9 Fluid pressure on the cavity surface

0 1 2 3 4 550

100

150

200

250

300

350

400

ickness (mm)

Def

orm

atio

n (μ

m)

(a)

0 1 2 3 4 50

100

200

300

400

500

600

ickness (mm)

Stre

ss (M

pa)

(b)

Figure 11 e relationship between thickness and stress and deformation

(a) (b) (c)

Figure 12 Assembled balance terminal (a) Steel-45 (b) Aluminum alloy-7A04 (c) Titanium alloy-TC4

Shock and Vibration 7

signal test on the terminal surface the flow channel and thesurface were modified as shown in Figure 13(b) e bal-ancing terminal is balanced to eliminate the initial masseccentricity on the balancing machine is will not causenew additional imbalance to the spindle

e 14 model of assembled balanced terminals and in-tegrated balanced terminals is shown in Figure 14 e pa-rameters of the two balanced terminals are shown in Table 1It can be seen that the integrated balanced terminal designedin this paper has obvious advantages in comparison withthe assembled balanced terminal e maximum balancecapacity of balanced terminal increased from 489 gmiddotmm to584 gmiddotmm On the basis of a 194 increase in balance ca-pacity thickness dropped from 38mm to 28mm a drop of263 e volume of balanced terminal dropped from160549 mm3 to 83780 mm3 with 478 drop ratio e massof balanced terminal has dropped from 562 g to 387 g with

a drop ratio of 311 so that the additional mass added by thebalanced terminal to the spindle will be effectively reduced

33 Balanced Procedure and Software e no trial weightbalance method is applied to the spray online balancesystem which can achieve efficient online balance thusmaximizing the protection automation processing link eflow diagram of the no trial weight spray online dynamic isshown in Figure 15 in which three main steps are included(1) acquisition process of rotor structural parameters (2)acquisition process of counterweight scheme and (3) onlinebalance process e balancing procedure is as follows

(1) Establish the physical model and finite elementmodel of the spindle rotor system by separating therotor model

(a) (b)

Figure 13 Titanium alloy integrated balancing terminal (a) Before modification (b) After modification

l2prime = 4mml3prime = 4mm

l1prime = 4mm

(a)

l2 = 25mm

l3 = 2mm

l1 = 2mm

(b)

Figure 14 Comparison of 14 model (a) Assembled terminal (b) Integrated terminal

Table 1 Comparison of two balanced terminal parameters

SpeciesParameters

Balance ability (gmiddotmm) ickness (mm) Volume (mm3) Mass (g) Price (RMB)Assembled terminal 489 38 160549 562 1024000Integrated terminal 584 28 83780 387 580400Variation () 194 263 478 311 433

8 Shock and Vibration

(2) Obtain the structural parameters of the spindle rotorsystem such as mass matrix M stiffness matrix Kand damping matrix C by the finite element simu-lation and simulate the inherent modal character-istics of the spindle

(3) Substitute the correction matrices of structural pa-rameters in Equation (9) to identify the aggravatedinfluence coefficient matrix

(4) Extract the original vibration signal of the spindleand the phase information corresponding to thevibration signal substitute aggravated influencecoefficient matrix into Equation (10) to calculate theunbalance vector

(5) e weighting scheme is searched according to thecavity angle and the unbalance vector

(6) e spray online dynamic balancing system is in-jected into the corresponding cavity to correct theunbalance vector

According to the balance flow chart a measurement andcontrol system based on virtual instruments was developedby our research group e operation interface is shown inFigure 16 e no trial weight online balance measurementand control software consists of vibration signal monitoringtrend of vibration displacement change vibration statusdisplay and monitoring liquid state display and monitoringand online balance information records

Start

Build the spindle physical model

Establish the FEDM of spindlerotor system

Frequency-responsecharacteristic simulation

Obtain the spindle structure parametersM K C G ω

Mode test

Is the inspection errorreasonable

N

Y

Corrected initial value(ndashω2

2M2 + K2 + jω2C2)H2 (ω2) = I

Identify the ETCSRS(α) = (K)ndashω2 (M) + i(C)+(G)ωndash1middotexp (ndashjφ)

Vector synthesis principle for weight matching

Calculate the balance vector (Q)=ndash(α)ndash1middotA

Extract the originalvibration signal A

Get balance scheme

End

Online spray balance

Vibration gt targetvalue

Vibration gt targetvalue

Y

EndN

Y

N

Figure 15 e balance procedure

Shock and Vibration 9

4 Methods and System Validation

41 Experimental Setup In order to verify the application ofno trial weight spray online dynamic on high-speed spindlethe high-speed spindle comprehensive performance test bedwas designed by our research group as shown in Figure 17e test bed comprises a spindle 150 SD40Q7 (Figure 1)whose maximum rotational speed is 40000 rpm and twopairs of SNFA VEX35 bearings lubricated by oil mist aremounted on the spindle with a clearance tolerance to avoidthe preload variation e water cooling system controls thetemperature variation e balancing discs are installed onthe ends of the spindle to suppress vibration A rotatingencoder mounted on the shaft feedbacks the rotationalspeed e spindle rotor is designed with a ratio of 1 7tapered to install the integrated balancing terminal and the

balancing terminal is fixed with a locknut In the front of therotor a HSK C32 tool holder interface is designed emeasuring equipment is composed of a synchronous ac-quisition system developed by our research group A Na-tional Instrument PXI-4472 was used to extract vibrationand phase information German Micro-Epsilon U-05 eddy-current sensor (range 05mm resolution 0025 μm) is usedto measure vibration displacement e NI-PXI6123 is usedto control the hydraulic system A photoelectric sensor wasapplied to measure the phasee layout of the experimentalsetup is shown in Figure 18

42 Measurement Principles e system carries out real-time synchronous acquisition of vibration using eddy-current displacement sensors mounted along the X-axis

Vibrationsignal

Phase signal Vibration state

Data record

Straystate

Figure 16 Measurement and control panel

Diaplacementsensors

Integratedterminal

Sensor supportFixture

Spindle system Phase sensor

Control program

Control card anddata acquisition card

NI-PXI-1003

Nozzle Tool interface

Hydraulic system

24V DC

Control board

Conditioning circuit

Figure 17 Experimental setup

10 Shock and Vibration

X

S1 S2

S3

Z

Y

(a)

90deg

0deg360deg

270deg

180deg

ωY

Xϕθφ

Cavity-2

Cavity-1

Cavity-4

Cavity-3

S2

S1 S3A1

A2

A3

(b)

Figure 18 Measurement principles (a) Sensor layout (b) Angle relation

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

Before balanceAer balance

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

3

25

2

15

1

05

0

286 μm

021 μm

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(c)

287 μm

051 μmAm

plitu

de (μ

m)

3

25

2

15

1

05

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(d)

Figure 19 Continued

Shock and Vibration 11

0 5 10 15 20 25 30Time (seconds)

Am

plitu

de (μ

m)

35

3

25

2

15

Start balance(14286)

Balance end(19021)1

05

0

(e)

Figure 19 Comparison of balance information in the X-Y direction at 12000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

ndash4

0

2

4

ndash2Am

plitu

de (μ

m)

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

(a)

ndash2

ndash3

ndash1

0

1

2

3

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

337 μm

082 μm

(c)

0

05

15

1

2

25

3

Am

plitu

de (μ

m)

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

279 μm

1233 μm

(d)

Figure 20 Continued

12 Shock and Vibration

Am

plitu

de (μ

m)

4

3

2

1

0

Start balance(23337)

Balance end(26082)

0 5 10 15 20 25 3530Time (seconds)

(e)

Figure 20 Comparison of balance information in the X-Y direction at 15000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAfter balance

(b)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

466 μm

052 μm

(c)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

498 μm

131 μm

(d)

Figure 21 Continued

Shock and Vibration 13

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

signal test on the terminal surface the flow channel and thesurface were modified as shown in Figure 13(b) e bal-ancing terminal is balanced to eliminate the initial masseccentricity on the balancing machine is will not causenew additional imbalance to the spindle

e 14 model of assembled balanced terminals and in-tegrated balanced terminals is shown in Figure 14 e pa-rameters of the two balanced terminals are shown in Table 1It can be seen that the integrated balanced terminal designedin this paper has obvious advantages in comparison withthe assembled balanced terminal e maximum balancecapacity of balanced terminal increased from 489 gmiddotmm to584 gmiddotmm On the basis of a 194 increase in balance ca-pacity thickness dropped from 38mm to 28mm a drop of263 e volume of balanced terminal dropped from160549 mm3 to 83780 mm3 with 478 drop ratio e massof balanced terminal has dropped from 562 g to 387 g with

a drop ratio of 311 so that the additional mass added by thebalanced terminal to the spindle will be effectively reduced

33 Balanced Procedure and Software e no trial weightbalance method is applied to the spray online balancesystem which can achieve efficient online balance thusmaximizing the protection automation processing link eflow diagram of the no trial weight spray online dynamic isshown in Figure 15 in which three main steps are included(1) acquisition process of rotor structural parameters (2)acquisition process of counterweight scheme and (3) onlinebalance process e balancing procedure is as follows

(1) Establish the physical model and finite elementmodel of the spindle rotor system by separating therotor model

(a) (b)

Figure 13 Titanium alloy integrated balancing terminal (a) Before modification (b) After modification

l2prime = 4mml3prime = 4mm

l1prime = 4mm

(a)

l2 = 25mm

l3 = 2mm

l1 = 2mm

(b)

Figure 14 Comparison of 14 model (a) Assembled terminal (b) Integrated terminal

Table 1 Comparison of two balanced terminal parameters

SpeciesParameters

Balance ability (gmiddotmm) ickness (mm) Volume (mm3) Mass (g) Price (RMB)Assembled terminal 489 38 160549 562 1024000Integrated terminal 584 28 83780 387 580400Variation () 194 263 478 311 433

8 Shock and Vibration

(2) Obtain the structural parameters of the spindle rotorsystem such as mass matrix M stiffness matrix Kand damping matrix C by the finite element simu-lation and simulate the inherent modal character-istics of the spindle

(3) Substitute the correction matrices of structural pa-rameters in Equation (9) to identify the aggravatedinfluence coefficient matrix

(4) Extract the original vibration signal of the spindleand the phase information corresponding to thevibration signal substitute aggravated influencecoefficient matrix into Equation (10) to calculate theunbalance vector

(5) e weighting scheme is searched according to thecavity angle and the unbalance vector

(6) e spray online dynamic balancing system is in-jected into the corresponding cavity to correct theunbalance vector

According to the balance flow chart a measurement andcontrol system based on virtual instruments was developedby our research group e operation interface is shown inFigure 16 e no trial weight online balance measurementand control software consists of vibration signal monitoringtrend of vibration displacement change vibration statusdisplay and monitoring liquid state display and monitoringand online balance information records

Start

Build the spindle physical model

Establish the FEDM of spindlerotor system

Frequency-responsecharacteristic simulation

Obtain the spindle structure parametersM K C G ω

Mode test

Is the inspection errorreasonable

N

Y

Corrected initial value(ndashω2

2M2 + K2 + jω2C2)H2 (ω2) = I

Identify the ETCSRS(α) = (K)ndashω2 (M) + i(C)+(G)ωndash1middotexp (ndashjφ)

Vector synthesis principle for weight matching

Calculate the balance vector (Q)=ndash(α)ndash1middotA

Extract the originalvibration signal A

Get balance scheme

End

Online spray balance

Vibration gt targetvalue

Vibration gt targetvalue

Y

EndN

Y

N

Figure 15 e balance procedure

Shock and Vibration 9

4 Methods and System Validation

41 Experimental Setup In order to verify the application ofno trial weight spray online dynamic on high-speed spindlethe high-speed spindle comprehensive performance test bedwas designed by our research group as shown in Figure 17e test bed comprises a spindle 150 SD40Q7 (Figure 1)whose maximum rotational speed is 40000 rpm and twopairs of SNFA VEX35 bearings lubricated by oil mist aremounted on the spindle with a clearance tolerance to avoidthe preload variation e water cooling system controls thetemperature variation e balancing discs are installed onthe ends of the spindle to suppress vibration A rotatingencoder mounted on the shaft feedbacks the rotationalspeed e spindle rotor is designed with a ratio of 1 7tapered to install the integrated balancing terminal and the

balancing terminal is fixed with a locknut In the front of therotor a HSK C32 tool holder interface is designed emeasuring equipment is composed of a synchronous ac-quisition system developed by our research group A Na-tional Instrument PXI-4472 was used to extract vibrationand phase information German Micro-Epsilon U-05 eddy-current sensor (range 05mm resolution 0025 μm) is usedto measure vibration displacement e NI-PXI6123 is usedto control the hydraulic system A photoelectric sensor wasapplied to measure the phasee layout of the experimentalsetup is shown in Figure 18

42 Measurement Principles e system carries out real-time synchronous acquisition of vibration using eddy-current displacement sensors mounted along the X-axis

Vibrationsignal

Phase signal Vibration state

Data record

Straystate

Figure 16 Measurement and control panel

Diaplacementsensors

Integratedterminal

Sensor supportFixture

Spindle system Phase sensor

Control program

Control card anddata acquisition card

NI-PXI-1003

Nozzle Tool interface

Hydraulic system

24V DC

Control board

Conditioning circuit

Figure 17 Experimental setup

10 Shock and Vibration

X

S1 S2

S3

Z

Y

(a)

90deg

0deg360deg

270deg

180deg

ωY

Xϕθφ

Cavity-2

Cavity-1

Cavity-4

Cavity-3

S2

S1 S3A1

A2

A3

(b)

Figure 18 Measurement principles (a) Sensor layout (b) Angle relation

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

Before balanceAer balance

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

3

25

2

15

1

05

0

286 μm

021 μm

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(c)

287 μm

051 μmAm

plitu

de (μ

m)

3

25

2

15

1

05

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(d)

Figure 19 Continued

Shock and Vibration 11

0 5 10 15 20 25 30Time (seconds)

Am

plitu

de (μ

m)

35

3

25

2

15

Start balance(14286)

Balance end(19021)1

05

0

(e)

Figure 19 Comparison of balance information in the X-Y direction at 12000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

ndash4

0

2

4

ndash2Am

plitu

de (μ

m)

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

(a)

ndash2

ndash3

ndash1

0

1

2

3

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

337 μm

082 μm

(c)

0

05

15

1

2

25

3

Am

plitu

de (μ

m)

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

279 μm

1233 μm

(d)

Figure 20 Continued

12 Shock and Vibration

Am

plitu

de (μ

m)

4

3

2

1

0

Start balance(23337)

Balance end(26082)

0 5 10 15 20 25 3530Time (seconds)

(e)

Figure 20 Comparison of balance information in the X-Y direction at 15000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAfter balance

(b)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

466 μm

052 μm

(c)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

498 μm

131 μm

(d)

Figure 21 Continued

Shock and Vibration 13

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

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Submit your manuscripts atwwwhindawicom

(2) Obtain the structural parameters of the spindle rotorsystem such as mass matrix M stiffness matrix Kand damping matrix C by the finite element simu-lation and simulate the inherent modal character-istics of the spindle

(3) Substitute the correction matrices of structural pa-rameters in Equation (9) to identify the aggravatedinfluence coefficient matrix

(4) Extract the original vibration signal of the spindleand the phase information corresponding to thevibration signal substitute aggravated influencecoefficient matrix into Equation (10) to calculate theunbalance vector

(5) e weighting scheme is searched according to thecavity angle and the unbalance vector

(6) e spray online dynamic balancing system is in-jected into the corresponding cavity to correct theunbalance vector

According to the balance flow chart a measurement andcontrol system based on virtual instruments was developedby our research group e operation interface is shown inFigure 16 e no trial weight online balance measurementand control software consists of vibration signal monitoringtrend of vibration displacement change vibration statusdisplay and monitoring liquid state display and monitoringand online balance information records

Start

Build the spindle physical model

Establish the FEDM of spindlerotor system

Frequency-responsecharacteristic simulation

Obtain the spindle structure parametersM K C G ω

Mode test

Is the inspection errorreasonable

N

Y

Corrected initial value(ndashω2

2M2 + K2 + jω2C2)H2 (ω2) = I

Identify the ETCSRS(α) = (K)ndashω2 (M) + i(C)+(G)ωndash1middotexp (ndashjφ)

Vector synthesis principle for weight matching

Calculate the balance vector (Q)=ndash(α)ndash1middotA

Extract the originalvibration signal A

Get balance scheme

End

Online spray balance

Vibration gt targetvalue

Vibration gt targetvalue

Y

EndN

Y

N

Figure 15 e balance procedure

Shock and Vibration 9

4 Methods and System Validation

41 Experimental Setup In order to verify the application ofno trial weight spray online dynamic on high-speed spindlethe high-speed spindle comprehensive performance test bedwas designed by our research group as shown in Figure 17e test bed comprises a spindle 150 SD40Q7 (Figure 1)whose maximum rotational speed is 40000 rpm and twopairs of SNFA VEX35 bearings lubricated by oil mist aremounted on the spindle with a clearance tolerance to avoidthe preload variation e water cooling system controls thetemperature variation e balancing discs are installed onthe ends of the spindle to suppress vibration A rotatingencoder mounted on the shaft feedbacks the rotationalspeed e spindle rotor is designed with a ratio of 1 7tapered to install the integrated balancing terminal and the

balancing terminal is fixed with a locknut In the front of therotor a HSK C32 tool holder interface is designed emeasuring equipment is composed of a synchronous ac-quisition system developed by our research group A Na-tional Instrument PXI-4472 was used to extract vibrationand phase information German Micro-Epsilon U-05 eddy-current sensor (range 05mm resolution 0025 μm) is usedto measure vibration displacement e NI-PXI6123 is usedto control the hydraulic system A photoelectric sensor wasapplied to measure the phasee layout of the experimentalsetup is shown in Figure 18

42 Measurement Principles e system carries out real-time synchronous acquisition of vibration using eddy-current displacement sensors mounted along the X-axis

Vibrationsignal

Phase signal Vibration state

Data record

Straystate

Figure 16 Measurement and control panel

Diaplacementsensors

Integratedterminal

Sensor supportFixture

Spindle system Phase sensor

Control program

Control card anddata acquisition card

NI-PXI-1003

Nozzle Tool interface

Hydraulic system

24V DC

Control board

Conditioning circuit

Figure 17 Experimental setup

10 Shock and Vibration

X

S1 S2

S3

Z

Y

(a)

90deg

0deg360deg

270deg

180deg

ωY

Xϕθφ

Cavity-2

Cavity-1

Cavity-4

Cavity-3

S2

S1 S3A1

A2

A3

(b)

Figure 18 Measurement principles (a) Sensor layout (b) Angle relation

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

Before balanceAer balance

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

3

25

2

15

1

05

0

286 μm

021 μm

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(c)

287 μm

051 μmAm

plitu

de (μ

m)

3

25

2

15

1

05

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(d)

Figure 19 Continued

Shock and Vibration 11

0 5 10 15 20 25 30Time (seconds)

Am

plitu

de (μ

m)

35

3

25

2

15

Start balance(14286)

Balance end(19021)1

05

0

(e)

Figure 19 Comparison of balance information in the X-Y direction at 12000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

ndash4

0

2

4

ndash2Am

plitu

de (μ

m)

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

(a)

ndash2

ndash3

ndash1

0

1

2

3

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

337 μm

082 μm

(c)

0

05

15

1

2

25

3

Am

plitu

de (μ

m)

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

279 μm

1233 μm

(d)

Figure 20 Continued

12 Shock and Vibration

Am

plitu

de (μ

m)

4

3

2

1

0

Start balance(23337)

Balance end(26082)

0 5 10 15 20 25 3530Time (seconds)

(e)

Figure 20 Comparison of balance information in the X-Y direction at 15000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAfter balance

(b)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

466 μm

052 μm

(c)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

498 μm

131 μm

(d)

Figure 21 Continued

Shock and Vibration 13

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

4 Methods and System Validation

41 Experimental Setup In order to verify the application ofno trial weight spray online dynamic on high-speed spindlethe high-speed spindle comprehensive performance test bedwas designed by our research group as shown in Figure 17e test bed comprises a spindle 150 SD40Q7 (Figure 1)whose maximum rotational speed is 40000 rpm and twopairs of SNFA VEX35 bearings lubricated by oil mist aremounted on the spindle with a clearance tolerance to avoidthe preload variation e water cooling system controls thetemperature variation e balancing discs are installed onthe ends of the spindle to suppress vibration A rotatingencoder mounted on the shaft feedbacks the rotationalspeed e spindle rotor is designed with a ratio of 1 7tapered to install the integrated balancing terminal and the

balancing terminal is fixed with a locknut In the front of therotor a HSK C32 tool holder interface is designed emeasuring equipment is composed of a synchronous ac-quisition system developed by our research group A Na-tional Instrument PXI-4472 was used to extract vibrationand phase information German Micro-Epsilon U-05 eddy-current sensor (range 05mm resolution 0025 μm) is usedto measure vibration displacement e NI-PXI6123 is usedto control the hydraulic system A photoelectric sensor wasapplied to measure the phasee layout of the experimentalsetup is shown in Figure 18

42 Measurement Principles e system carries out real-time synchronous acquisition of vibration using eddy-current displacement sensors mounted along the X-axis

Vibrationsignal

Phase signal Vibration state

Data record

Straystate

Figure 16 Measurement and control panel

Diaplacementsensors

Integratedterminal

Sensor supportFixture

Spindle system Phase sensor

Control program

Control card anddata acquisition card

NI-PXI-1003

Nozzle Tool interface

Hydraulic system

24V DC

Control board

Conditioning circuit

Figure 17 Experimental setup

10 Shock and Vibration

X

S1 S2

S3

Z

Y

(a)

90deg

0deg360deg

270deg

180deg

ωY

Xϕθφ

Cavity-2

Cavity-1

Cavity-4

Cavity-3

S2

S1 S3A1

A2

A3

(b)

Figure 18 Measurement principles (a) Sensor layout (b) Angle relation

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

Before balanceAer balance

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

3

25

2

15

1

05

0

286 μm

021 μm

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(c)

287 μm

051 μmAm

plitu

de (μ

m)

3

25

2

15

1

05

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(d)

Figure 19 Continued

Shock and Vibration 11

0 5 10 15 20 25 30Time (seconds)

Am

plitu

de (μ

m)

35

3

25

2

15

Start balance(14286)

Balance end(19021)1

05

0

(e)

Figure 19 Comparison of balance information in the X-Y direction at 12000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

ndash4

0

2

4

ndash2Am

plitu

de (μ

m)

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

(a)

ndash2

ndash3

ndash1

0

1

2

3

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

337 μm

082 μm

(c)

0

05

15

1

2

25

3

Am

plitu

de (μ

m)

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

279 μm

1233 μm

(d)

Figure 20 Continued

12 Shock and Vibration

Am

plitu

de (μ

m)

4

3

2

1

0

Start balance(23337)

Balance end(26082)

0 5 10 15 20 25 3530Time (seconds)

(e)

Figure 20 Comparison of balance information in the X-Y direction at 15000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAfter balance

(b)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

466 μm

052 μm

(c)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

498 μm

131 μm

(d)

Figure 21 Continued

Shock and Vibration 13

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

X

S1 S2

S3

Z

Y

(a)

90deg

0deg360deg

270deg

180deg

ωY

Xϕθφ

Cavity-2

Cavity-1

Cavity-4

Cavity-3

S2

S1 S3A1

A2

A3

(b)

Figure 18 Measurement principles (a) Sensor layout (b) Angle relation

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

Before balanceAer balance

03 032 034 036 038 04minus4

minus2

0

2

4

Time (seconds)

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

3

25

2

15

1

05

0

286 μm

021 μm

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(c)

287 μm

051 μmAm

plitu

de (μ

m)

3

25

2

15

1

05

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

(d)

Figure 19 Continued

Shock and Vibration 11

0 5 10 15 20 25 30Time (seconds)

Am

plitu

de (μ

m)

35

3

25

2

15

Start balance(14286)

Balance end(19021)1

05

0

(e)

Figure 19 Comparison of balance information in the X-Y direction at 12000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

ndash4

0

2

4

ndash2Am

plitu

de (μ

m)

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

(a)

ndash2

ndash3

ndash1

0

1

2

3

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

337 μm

082 μm

(c)

0

05

15

1

2

25

3

Am

plitu

de (μ

m)

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

279 μm

1233 μm

(d)

Figure 20 Continued

12 Shock and Vibration

Am

plitu

de (μ

m)

4

3

2

1

0

Start balance(23337)

Balance end(26082)

0 5 10 15 20 25 3530Time (seconds)

(e)

Figure 20 Comparison of balance information in the X-Y direction at 15000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAfter balance

(b)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

466 μm

052 μm

(c)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

498 μm

131 μm

(d)

Figure 21 Continued

Shock and Vibration 13

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

0 5 10 15 20 25 30Time (seconds)

Am

plitu

de (μ

m)

35

3

25

2

15

Start balance(14286)

Balance end(19021)1

05

0

(e)

Figure 19 Comparison of balance information in the X-Y direction at 12000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

ndash4

0

2

4

ndash2Am

plitu

de (μ

m)

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

(a)

ndash2

ndash3

ndash1

0

1

2

3

03 032 034 036 038 04Time (seconds)

Before balanceAer balance

Am

plitu

de (μ

m)

(b)

Am

plitu

de (μ

m)

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

337 μm

082 μm

(c)

0

05

15

1

2

25

3

Am

plitu

de (μ

m)

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

279 μm

1233 μm

(d)

Figure 20 Continued

12 Shock and Vibration

Am

plitu

de (μ

m)

4

3

2

1

0

Start balance(23337)

Balance end(26082)

0 5 10 15 20 25 3530Time (seconds)

(e)

Figure 20 Comparison of balance information in the X-Y direction at 15000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAfter balance

(b)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

466 μm

052 μm

(c)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

498 μm

131 μm

(d)

Figure 21 Continued

Shock and Vibration 13

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Am

plitu

de (μ

m)

4

3

2

1

0

Start balance(23337)

Balance end(26082)

0 5 10 15 20 25 3530Time (seconds)

(e)

Figure 20 Comparison of balance information in the X-Y direction at 15000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAer balance

(a)

03 032 034 036 038 04ndash6

ndash4

ndash2

0

2

4

6

Time (seconds)

Am

plitu

de (μ

m)

Before balanceAfter balance

(b)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

466 μm

052 μm

(c)

Am

plitu

de (μ

m)

5

4

3

2

1

0

Frequency (Hz)0 200 400 600 800

Before balanceAer balance

498 μm

131 μm

(d)

Figure 21 Continued

Shock and Vibration 13

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

(S1) and the Y-axis (S2) the layout is shown in Figure 18(a)e photoelectric sensor (S3) mounted on the X-axismeasured the phase (Figure 18(b)) e X-direction dis-placement sensor S1 and the phase sensor S3 are aligned withthe zero-hole position to simplify the complex angular re-lationship and eliminate the influence of the initial phaseangleen the lag angle (φ) can be obtained from Equation(17) In Figure 18(b) A1 is vibration peak point A2 is phasepoint and A3 is equivalent point of the rotor unbalanceforce

φ θ + ϕ (17)

43 Experimental Results Analysis e pressure of the hy-draulic system in the experiment is 012MPa and sprayingunit time is 100ms e X-direction vibration displacementis used as the balance reference In order to accurately extractthe imbalance characteristics harmonic wavelet technologywas used to extract the frequency characteristics [33] eonline balance experiments at multiple speeds were verifiedin the experimental system as shown in Figure 17

Comparison of rotation frequency amplitude before andafter balancing at 12000 rpm 15000 rpm and 18000 rpm isshown in Figures 19ndash21 It can be seen from Figure 19 thatthe vibration amplitude in the X direction decreases from286 μm to 021 μm the drop ratio reaches 9266 and thevibration amplitude in the Y direction decreases from287 μm to 051 μm with 8223 drop ratio at 12000 rpm Itcan be seen from the online balance process that the sprayweight is only needed once and the total online balancingtime is 5 seconds

It can be seen from Figure 20 that the vibration am-plitude in the X direction decreases from 337 μm to 082 μmthe drop ratio reaches 7567 and the vibration amplitude inthe Y direction decreases from 279 μm to 123 μm with5591 drop ratio at 15000 rpm It can be seen from theonline balance process that the spray weight is only neededonce and the total online balancing time is 3 seconds

It can be seen from Figure 21 that the vibration am-plitude in the X direction decreases from 466 μm to 052 μmthe drop ratio reaches 8884 and the vibration amplitudein the Y direction decreases from 498 μm to 131 μm with

7369 drop ratio at 18000 rpm It can be seen from theonline balance process that the spray weight is two times andthe total online balancing time is 7 seconds

5 Conclusions and Further Work

Ano trial weight spray online balancemethod is proposed Ahigh-speed spindle finite element analysis model wasestablished and the dynamic influence coefficient matrixwas identified A no trial weight spray online dynamicbalancing system was developed which has the advantagesof no trial weight and high-precision loading A new type ofintegrated balancing terminal that was formed using 3Dprinting technology was first proposed by our researchgroup and its advantages in various aspects are significantlyhigher than traditional assembly balanced terminals Onlinebalance experiments demonstrate the effectiveness of bal-ancing methods and integrated balancing terminals

It is important to note that the shaft is rigid And thefocus is on one end of the bearing which is treated as a single-plane balancing problem But the proposed method in thispaper can be extended to both planes We will further de-velop in-depth research on the method of no trial weightonline dynamic balance considering dynamic stiffness andwe will further improve the loading accuracy of onlinebalance system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

is research was supported by the National Scienceand Technology Major Project of China (Grant Numbers2015ZX04005001 and 2017ZX04013001)

Am

plitu

de (μ

m)

5

4

3

2

1

00 5 10 15 20 25 30

Time (seconds)

Start balance(13466)

Second balance(18178)

Balance end(20052)

(e)

Figure 21 Comparison of balance information in the X-Y direction at 18000 rpm (a) X-time domain (b) Y-time domain (c) X-frequencydomain (d) Y-frequency domain (e) Online balance process

14 Shock and Vibration

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

References

[1] S Zhang and Z Zhang ldquoResearch on the field dynamicbalance technologies for large diesel engine crankshaft sys-temrdquo Shock and Vibration vol 2017 Article ID 715047210 pages 2017

[2] Y Zhang X Mei M Shao and M Xu ldquoAn improvedholospectrum-based balancing method for rotor systems withanisotropic stiffnessrdquo in Proceedings of Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 227 no 2 pp 246ndash260 2013

[3] X Qiao and G Hu ldquoActive control for multinode unbalancedvibration of flexible spindle rotor system with active magneticbearingrdquo Shock and Vibration vol 2017 Article ID 97064939 pages 2017

[4] C Yue X Ren Y Yang and W Deng ldquoUnbalance identi-fication of speed-variant rotary machinery without phaseangle measurementrdquo Shock and Vibration vol 2015 ArticleID 934231 11 pages 2015

[5] H Cao X Zhang and X Chen ldquoe concept and progress ofintelligent spindles a reviewrdquo International Journal of Ma-chine Tools and Manufacture vol 112 pp 21ndash52 2016

[6] P Xin W Hai-Qi G Jin-Ji and M Yu-Zhe ldquoStudy on onlineactive balancing system of rotating machinery and targetcontrol methodrdquo WSEAS Transaction on Systems vol 132014

[7] P Gnielka ldquoModal balancing of flexible rotors without testruns an experimental investigationrdquo Journal of Sound andVibration vol 90 no 2 pp 157ndash172 1983

[8] Y Kang Y P Chang M-H Tseng P-H Tang andY-F Chang ldquoA modified approach based on influence co-efficient method for balancing crank-shaftsrdquo Journal of Soundand Vibration vol 234 no 2 pp 277ndash296 2000

[9] M B Deepthikumar A S Sekhar and M R SrikanthanldquoModal balancing of flexible rotors with bow and distributedunbalancerdquo Journal of Sound and Vibration vol 332 no 24pp 6216ndash6233 2013

[10] L S Qu H Qiu and G H Xu ldquoRotor balancing based onholospectrum analysis principle and practicerdquo ChineseJournal of Mechanical Engineering vol 9 no 1 pp 60ndash631998

[11] G Bin X Li Y Shen andWWang ldquoDevelopment of whole-machine high speed balance approach for turbomachineryshaft system with N + 1 supportsrdquo Measurement vol 122pp 368ndash379 2018

[12] R S Srinivas R Tiwari and C Kannababu ldquoApplication ofactive magnetic bearings in flexible rotordynamic systemsmdashastate-of-the-art reviewrdquo Mechanical Systems and SignalProcessing vol 106 pp 537ndash572 2018

[13] E P Delgado and R H Bannister ldquoBalancing of an exper-imental rotor without trial runsrdquo International Journal ofRotating Machinery vol 8 no 2 pp 99ndash108 2007

[14] S Liu and L Qu A New Field Balancing Method of RotorSystems Based on Holospectrum and Genetic AlgorithmElsevier Science Publishers B V New York NY USA 2008

[15] L Qu X Liu G Peyronne and Y Chen ldquoe holospectruma new method for rotor surveillance and diagnosisrdquo Me-chanical Systems and Signal Processing vol 3 no 3pp 255ndash267 1989

[16] S Liu ldquoA modified low-speed balancing method for flexiblerotors based on holospectrumrdquo Mechanical Systems andSignal Processing vol 21 no 1 pp 348ndash364 2007

[17] J Niebsch and R Ramlau ldquoMathematical imbalance de-termination from vibrational measurements and industrial

applicationsrdquo in Progress in Industrial Mathematics at ECMIpp 293ndash298 Springer Berlin Germany 2008

[18] R Ramlau and J Niebsch ldquoImbalance estimation without testmasses for wind turbinesrdquo Journal of Solar Energy Engi-neering vol 131 no 1 article 011010 2009

[19] Y A Khulief M A Mohiuddin and M El-Gebeily ldquoA newmethod for field-balancing of high-speed flexible rotorswithout trial weightsrdquo International Journal of RotatingMachinery vol 2014 Article ID 603241 11 pages 2014

[20] X Li L Zheng and Z Liu ldquoBalancing of flexible rotorswithout trial weights based on finite element modal analysisrdquoJournal of Vibration and Control vol 19 no 3 pp 461ndash4702012

[21] J V D Vegte and R T Lake ldquoBalancing of rotating systemsduring operationrdquo Journal of Sound and Vibration vol 57no 2 pp 225ndash235 1978

[22] B Hredzak and G Guo ldquoNew electromechanical balancingdevice for active imbalance compensationrdquo Journal of Soundand Vibration vol 294 no 4-5 pp 737ndash751 2006

[23] J D Moon B S Kim and S H Lee ldquoDevelopment of theactive balancing device for high-speed spindle system usinginfluence coefficientsrdquo International Journal of Machine Toolsand Manufacture vol 46 no 9 pp 978ndash987 2006

[24] K Nakamoto K Adachi and K Shirase ldquoProposal of real-time balancing mechanism using magnetic fluid for machinetool spindlerdquo in Manufacturing Systems and Technologies forthe New Frontier pp 387ndash390 Springer London UK 2008

[25] H Fan M Jing R Wang H Liu and J Zhi ldquoNew elec-tromagnetic ring balancer for active imbalance compensationof rotating machineryrdquo Journal of Sound and Vibrationvol 333 no 17 pp 3837ndash3858 2014

[26] H D Nelson and J M McVaugh ldquoe dynamics of rotorbearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 p 593 1976

[27] H D Nelson ldquoA finite rotating shaft element using timo-shenko beam theoryrdquo Journal of Mechanical Design vol 102no 4 p 793 1980

[28] M A Mohiuddin and Y A Khulief ldquoCoupled bendingtorsional vibration of rotors using finite elementrdquo Journal ofSound and Vibration vol 223 no 2 pp 297ndash316 1999

[29] S Lim ldquoFinite element analysis of flexural vibrations in harddisk drive spindle systemsrdquo Journal of Sound and Vibrationvol 233 no 4 pp 597ndash612 2000

[30] J Liu and Y Shao ldquoDynamic modeling for rigid rotor bearingsystems with a localized defect considering additional de-formations at the sharp edgesrdquo Journal of Sound and Vi-bration vol 398 pp 84ndash102 2017

[31] L H Larsson and A S Douglas Elastic-Plastic FractureMechanics MIR Publishers Moscow Russia 1978

[32] M Tsutsumi M Ohya T Aoyama S Shimizu andS Hachiga ldquoDeformation and interface pressure distributionof 110 tapered joints at high rotation speedrdquo InternationalJournal of the Japan Society for Precision Engineering vol 30pp 23ndash28 1996

[33] Y H Heo and K J Kim ldquoDefinitions of non-stationary vi-bration power for timendashfrequency analysis and computationalalgorithms based upon harmonic wavelet transformrdquo Journalof Sound and Vibration vol 336 no 336 pp 275ndash292 2015

Shock and Vibration 15

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

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