Research Article A Study on the Dynamic Performance for ...downloads.hindawi.com/journals/sv/2016/3695950.pdf · uid tracks on the dynamic performances of the HDB. 2. Design of HDB

Research ArticleA Study on the Dynamic Performance for HydraulicallyDamped Rubber Bushings with Multiple Inertia Tracks andOrifices Parameter Identification and Modeling

Chao-Feng Yang1 Zhi-Hong Yin1 Wen-Bin Shangguan12 and Xiao-Cheng Duan12

1School of Mechanical amp Automotive Engineering South China University of Technology Guangzhou 510641 China2Ningbo Tuopu Group Co Ltd Ningbo 315800 China

Correspondence should be addressed to Zhi-Hong Yin mezhyinscuteducn

Received 21 October 2015 Accepted 26 November 2015

Academic Editor Radoslaw Zimroz

Copyright copy 2016 Chao-Feng Yang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Hydraulically damped rubber bushings (HDBs) are important for vehicle noise vibration and harshness (NVH) performance asthey are able to decay the vehiclersquos oscillation induced by engine and roadThe dynamic stiffness and loss angle of anHDB are crucialand it is significant to investigate the relations between the design parameters with the dynamic stiffness and loss angle Thereforethe force-deflection relation of the HDB is measured statically and the dynamic stiffness and loss angle are measured dynamicallyand the test data are analyzed with a view to examine how the measurement results are influenced by the design parameters (thenumber of the fluid tracks) Compared with the results predicted by a nonlinear lumped parameter model whose parameters areextracted by a parameter identification technique using the model the effect of the main rubber and the fluid track on the dynamicstiffness and the loss angle is investigated A unified analytical model of HDB is also developed with the purpose of predictingthe static and dynamic characteristics and the predictions are shown to be well correlated with the measurement data The goodcorrelation suggests the validity of the model and the parameter identification implementation

1 Introduction

Motivation Rubber bushings are commonly used in enginesubsystems vehicle body and vehicle suspension to damptheir oscillations excited by dynamic loads thus improvingdriving safety ride comfort and handling performances [1ndash9] Comparedwith conventional rubber bushings an hydrau-lically damped rubber bushing (HDB) can provide a highviscous damping coefficient in certain frequency range whileexhibiting amplitude dependent properties and frequencydependent properties [5ndash9] which is one of the advantagesof an HDB In designing the frequency characteristics of anHDB designers take care of the value and peak frequencyof the loss angle of the HDB mostly which are dependentupon the performance of the main rubber spring the fluidproperty the shapes of two chambers the fluid tracks and soforth Some effective methods and models have to be invited

to calculate the influences of those factors on the dynamiccharacteristics of an HDB during the design stage Thedynamic performances of an HDB are often characterizedwith two terms dynamic stiffness and loss angle whichare defined as the amplitude ratio of the load response tothe displacement response and the phase difference betweenthe load and displacement responses in frequency domainrespectively [4ndash10]

The lumped parameter (LP) model [10] is often employedto estimate the dynamic performances of an HDB [1 2 4ndash9] Each lumped parameter represents a certain physicalmeaning and the dynamic characteristics of the HDB can bedescribed as a function of lumped parameters explicitly [910] However the traditional method of parameter identifica-tion is a high-cost and time-consuming task for some lumpedparameters such as the HDBrsquos effective piston area and thecompliances of two fluid chambers To obtain these design

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 3695950 16 pageshttpdxdoiorg10115520163695950

2 Shock and Vibration

variables of an HDB a new parameter identification methodis proposed based on the characteristic frequency points ofthe dynamic performances of HDB in this paper

During the design stage of an HDB it is not easy to adjustthe maximum frequency of loss angle to the same frequencyof the most disruptive excitations with one fluid elementdue to geometry limitations of the fluid track [11] Besidesthe hydraulic damping mechanism is relatively sensitive tothe operating environment while multiple damped dynamicvibration absorbers have good robustness against it [12]However the influences of fluid track number on the dynamiccharacteristics of an HDB with multiple inertia tracks andorifices as well as its practical application issues are not fullyinvestigated

Therefore the motivations of this paper are to develop asuperior parameter identificationmethod for theHDB and toreveal the influences of the geometry size and the number offluid tracks on the dynamic characteristics of the HDB by testand simulation approaches

Literature ReviewThe first HDB was introduced in the mid-1980s on MacPherson-type front suspension systems [11]Similar designs are already used as HEM and are adapted forapplication with cylindrical sleeve-type bushing joints simply[11 13] There are also many patents [14ndash20] that claim thedynamic characteristics features of some specific HDBs infrequency domain however no analytical analysis or evenmeasured dynamic properties are provided [4ndash8] Furthervery limited articles have reported the dynamic propertiescharacterization and modeling issues of HDB

Lu and Ari-Gur [1 2] present a linear LP model for anHDB with two inertia tracks and derived the natural fre-quency analytically Then the model is applied to the sim-ulation of an HDB with multiple inertia tracks Howevertheir model is not validated experimentally Sauer and Guy[3] present a fluid system model for an HDB with a bypasstrack and one inertia track in parallel which serves as arelief valve securing theHDB and the adjacent structure fromundesirable shocks but the detailed calculation conclusionsare not carried out Shangguan and Xu [9] propose a nonlin-ear lumped parameter state-space model (by introducing anonlinear track resistance 119877

119894) for an HDB with one inertia

track and the model is verified by experiments while theanalytical study is conducted based on the linearized modelSevensson and Hakansson [21] and Pan et al [22] providean empirical mathematic model including a nonlinear springwith several fluid elements and elastoplastic elements coupledin parallel but the calculation results show that the accuracyof the model is not satisfactory for low amplitude excitationArzanpour andGolnaraghi [23 24] present a linear LPmodelof an HEM employing the same approach as that for anHDB with one inertia track Chai et al [4ndash8] initiate a labo-ratory device with alternative internal configurations andthe dynamic characteristics for different configurations areinvestigated comparatively as well as the steady state andtransient responses of the device However the influence ofthe number of the fluid tracks on the dynamic characteristicsis not discussed The parameters of LP models of HDBwith multiple inertia tracks are identified by methods of

Table 1 Geometric and material parameters of a typical HDB

Parameter ValueLength of inertia track 119897

119894m 019

Cross-sectional area of the track 119860119900119860119894m2 983 times 10minus6

Wet perimeter of the inertia track 120594119894m 127 times 10minus2

Hydraulic diameter of the inertia track 119889119867m 3087 times 10minus3

Length of the orifice 119897119900m 0012

Density of the fluid 120588(kgm3) 111280Poisson ratio of the rubber spring ] 050Density of the rubber spring 120588

119903(kgm3) 105000

curve fitting and Nyquist diagrams [4 8] However the phe-nomenon of the excitation amplitude-invariant fixed pointson the dynamic performance curves of HDBs with differentinternal configurations is not investigated

In summary no uniform LPmodels have been developedto analyze the influences of the number of long fluid tracksand orifices on the dynamic characteristics of HDBs A thor-ough understanding of the mechanism of HDBs is needed

Scope and Objectives As a passive vibration isolation compo-nent in the vehicle suspension and steering wheel assembliesHDBs play an important role in dealing with vehicle systemvibration issues such as brake judder and shimmy [3 11]The configuration of a typical HDB and the runner plate areillustrated in Figures 1 and 2 respectively Its geometric andmaterial parameters are listed in Table 1

The elastomeric rubber spring enclosed by inner andouter metal sleeves has two functions to support the staticload of the vehicle and to provide partial static stiffness andsomewhat damping of the suspension system The typicalpassive hydromechanical bushing usually consists of twoalmost same fluid chambers connected by one ormore inertiatracks andor orifices Typically the fluid chambers filledwith an antifreeze and water mixture which has appropri-ate dynamic and thermal properties required for isolationand control [25] In the event of a dynamically oscillatoryexcitation the chamber pressures change in radical directionand the volume compensation will take place between twofluid chambers Thus the fluid flows back and forth throughthe fluid tracks as an oscillating mass to provide hydraulicdamping [4ndash9 11 13]

Based on the above analysis HDBrsquos engineering designsare conceptually identical to those of HEM but their struc-tures working mechanisms and dynamic properties areessentially different [4ndash8] In particular the lower chamber ofthe HEM is very thin and the pressure in the lower chamberis usually smaller than that in the upper chamber On thecontrary the two chamber pressures of theHDB are relativelyhighTherefore the LPmodels developed forHEM cannot bedirectly applied to HDB [4ndash8]

The disadvantage of the HDB with inertia track is thatthe flow velocity of the liquid column in the inertia trackbecomes very slow due to the viscosity of the fluid once thefrequency excitation reaches its resonant frequency that isit usually can be assumed that the flow ldquoshut offrdquo [25ndash28]

Shock and Vibration 3

(1)(2)

(7)

(6)

(5)

(4)

(3)

(1) Outer metal sleeve(2) Stopper member

(7) Fluid track plate(6) Inertia track(5) Inter metal sleeve

(4) Rubber spring(3) Fluid chamber

Figure 1 The cross-sectional view of HDB

(1)

(2)

(3)

(1) Fluid track plate(2) Inertia fluid track(3) Orifice fluid track

Figure 2 The fluid track plate of hydraulic bushing

which is undesirableHowever theHDBwith large orifice canprovide sufficient damping in a wide frequency range whilethe damping in the low frequency range is deteriorated [11]

The fluid tracks plates of HDBs usually have irregulargeometry and vary from sample to sample significantly Inthis paper the HDBs with different fluid track plates aremanufactured and studied which can provide insights intothe features needed for scientific verification

The objectives of this paper include the following (a)design a controlled experiment to test the static and dynamicperformances of HDB with multiple inertia tracks and ori-fices by change of the fluid track plate and investigate theexcitation amplitude-invariant fixed points on the dynamiccharacteristics of HDBs with different fluid track plates (b)present a newparameter identification technique anddevelopa nonlinear unified LPmodel for theHDBwithmultiple iner-tia tracks and orifices to analyze the influences of the numberof different tracks on the dynamic characteristics of the HDBand estimate the peak frequency of the loss angle (c) validatethe unified analytical model by comparing simulation resultswith test data andpredict the influences of the number of fluidtracks on the dynamic performances of the HDB

2 Design of HDB with MultipleConfigurations and Experimental Study

21 Types and Experimental Procedure A group of HDBtypes with varied configurations are manufactured to inves-tigate the influences of the inertia tracks and orifices on theHDB performances

The basic features of the 10 tested HDBs are summarizedin Table 2 For each type (HB1ndashHB5) two identical samplesare prepared Within these HBDs the type of HB6 is specialbecause it denotes the main rubber of the HDB

Because the fluid of HB1 and HB3 does not providehydraulic damping the LP models are not considered inthis study Different LP models for those HDBs (HB1 HB2HB4 and HB5) are illustrated in Figure 3 119870

119903is the complex

stiffness of the main rubber and 119861119903is the damping ratio of

themain rubber1198621and119862

2are the volumetric compliances of

the two fluid chambers induced by the rubber spring and 1198701

and1198702are the volumetric stiffness of the two fluid chambers

respectively 1198601199011

and 1198601199012

represent the effective pumpingareas of the upper and the lower chambers assuming thatthe behavior of the rubber spring can be approximated by


Table 2 The configurations of HDB samples

Types Configurations Structural featuresHB1 HDB with one inertia track Block one inertia trackHB2 HDB with two inertia tracks Install two fluid track plates symmetrically (one inertia track on each plate)HB3 HDB with no track Block two inertia tracks

HB4 HDB with one inertia and one orifice track Install two fluid track plates symmetrically (one plate with an inertia trackand the other plate with an orifice)

HB5 HDB with two orifice tracks Install two fluid track plates symmetrically (both with an orifice)HB6 Main rubber Empty the chambers of the HDB with multiple configurations

Outer sleeve

Outer sleeve

Inner sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

Qi(t)

IiRi1

Ri2

C1

C2

FT(t)

xi(t)

FT(t)

(a)

Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

Qi(t)

IiRi1

Ri2

C1

C2

FT(t)

xi(t)

FT(t)

IsRs1

Rs2

Qs(t)

xs(t)

(b)

Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

Qi(t)

IiRi1

Ri2

C1

C2

FT(t)

FT(t)

Qo(t) xo(t)

Io Ro1 Ro2

xi(t)

(c)

Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2

FT(t)

FT(t)

Io2 Io1 Ro11Ro12

Ro22 Ro21

Qo1(t)Qo2(t) xo1(t)xo2(t)

(d)

Figure 3 Fluid system model for HDBs with multiple configurations (a) One inertia track (b) two inertia tracks (c) one inertia and anorifice (d) two orifices

that of an artificial cylinder piston 1198751(119905) and 119875

2(119905) denote

the pressures in the upper and the lower chambers Theinertia tracks and orifices are characterized by the fluidinertial coefficients 119868

119894and 119868119900and fluid resistances 119877

119894and 119877

119900

respectivelyThe static preload and a sinusoidal displacementexcitation 119909

119903(119905) = 119883

119903sin120596119905 are applied to the inner metal

sleeve while the outer metal sleeve is fixed 119865119879(119905) is the

dynamic force which is transmitted to the outer metal sleeveby the hydraulic and main rubber paths [4ndash8]

The static and dynamic tests for the above-mentionedHDB with multiple internal configurations are conductedusing elastomer test machine (MTS 831) Detailed descrip-tions of the test method and data processing technique for

obtaining the dynamic properties of an HDB refer to [24] Inaddition it is assumed that the relative displacement betweenthe inner and outer metal sleeves varies in radial directionTherefore only radial damping is considered in this study

22 Experimental Results Figure 4 shows the tests results ofstatic stiffness of the HDBs with different internal configura-tions that is HB1 HB2 HB4 andHB5 It can be seen that theforce-displacement curves for these 4 samples are linear andhave similar gradients under small amplitude excitations (0sim3mm) However they tend to differ with each other and benonlinear for large displacement excitations (gt3mm) Whenthe displacement excitation is greater than 4mm the outer


0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

14000

16000

18000

Forc

e (N

)

Displacement (mm)

HB1HB2

HB4HB5

Figure 4 The tests results static characteristics of the HDBs withdifferent internal configurations

metal sleeve contacts with the stopper member thus theforce-displacement curves exhibit significant nonlinearity

In general the static characteristics of these samplescorrelated well with each other This indicates that thevulcanization process conditions for these samples are stableand consistent

The results of the dynamic tests of the main rubberand the HDBs with multiple internal configurations undersinusoidal excitations are shown in Figure 5

221 The Dynamic Performances of the Main Rubber SpringAs shown in Figures 5(a) and 5(b) a sinusoidal excitation isapplied to the inner sleeve of the main rubber (HB6) witha preload of 387N The changes of the dynamic stiffnessand loss angle of HB6 with excitation frequency are verysmall compared to those of HB1 HB2 HB3 HB4 and HB5Therefore the dynamic stiffness and the loss angle of theHB6 can be reasonably treated as two constants for exampletaking the average of the test data (119870

119903is about 438340Nm

and 119861119903is about 10206Nsm) in the subsequent analysis

222The Dynamic Performances of HDBs with Multiple Con-figurations Figure 5 shows that the dynamic performances oftheHDBswithmultiple configurations are strongly excitationfrequency and amplitude dependent

As the excitation frequency increases the dynamic stiff-ness of bothHB3 andHB6 can be considered as constantThedifference between them can be attributed to the incompress-ibility of the liquid which causes a stiff HDB The dynamicstiffness of HDBs with multiple configurations tends to be ahorizontal line under high frequency band respectively sothey can be considered as amplitude invariant in this scenario

Under the 08mm excitation the frequencies of the peakfor the loss angles of HB1 HB2 HB4 and HB5 are 12 2052 and 88Hz respectively As the frequency increases theloss angles of HB1 HB2 HB4 and HB5 gradually approach

a horizontal line which overlaps the curves of HB3 and HB6This phenomenon indicates that in higher frequency rangethe response of the fluid in tracks is roughly attenuatedand the damping of the HDB is mainly determined by itsmain rubber The bandwidths of the loss angle for the HDBswith different internal configurations are different With theincrease of the number of inertia tracks theHDB can providedamping over a wider frequency band While increasing thenumber of orifices the HDB can provide larger damping in ahigher and wider frequency band

The high frequency dynamic performances of the HDBswith multiple internal configurations can be examined byobserving Figures 5(c) and 5(d) As the number of orificesincreases the notch frequency 119891

119870119889min shifts to a higher valueand the level of the dynamic stiffness 119870

119889decreases simulta-

neously Moreover it is shown that the corresponding lossangles are all 90∘ which means there is a resonance at 119891

119870119889minIt can be found that the loss angle shifts abruptly from

about 0∘ to 180∘ This means the resonance is dominated notonly by the fluid flow resistance but also by the fluid inertialeffect suggesting that HDB with multiple configurationscannot be modelled by a single degree of freedom (SDOF)system under high frequency small amplitude excitation [2527] Since the fluid motion is highly turbulent at higherfrequencies its influence is not modelled here

223 The Fixed Points on Dynamic Performances of HDBswith Multiple Configurations It should be noted that manyexcitation amplitude-invariant fixed points 119872

119894(119894 = 1 sim 4)

and 119873119894(119894 = 1 sim 6) exist on the dynamic test results curves

With the number of inertia tracks increases the dynamicstiffness (119870

119889119872119894) of the fixed point is decreased As the number

of orifices increases 119870119889119872119894

decreases rapidly These fixedpoints contain important information of an HDB and a newparameter identification method can be developed

3 The Nonlinear LP Model foran HDB with One Inertia Track

This section presents the modeling of the HDB with oneinertial track illustrated in Figure 3(a)The following assump-tions are made [29] (i) the influence of gravity is ignoredand the fluid in chamber is incompressible (ii) the inertiaand damping of the fluid in lower and upper chambers areneglected and the fluid pressure in each chamber is uniform(iii) the relative velocity of fluid is constant along the inertiachannel and the properties of inlet and outlet cross sectionsare uniform (iv) the dynamic viscosity and density propertiesof the liquid are constant and the cross sections along inertiatrack have the same shape

With the above assumptions the continuity equations canbe written as

1198701(119860119894119909119894minus 1198601199011119909119903) = 1198751

1198702(1198601199012119909119903minus 119860119894119909119894) = 1198752

(1)


300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)0 20 40 60 80 100

M1

M2

HB1 amplitude 06mm HB1 amplitude 10mmHB2 amplitude 06mm HB2 amplitude 10mmHB3 amplitude 06mm HB3 amplitude 10mmHB4 amplitude 06mm HB5 amplitude 06mmHB6 amplitude 06mm HB6 amplitude 10mm

(a)

0

20

40

60

80

100

Frequency (Hz)0 20 40 60 80 100

N1

N2

Loss

angl

e (∘)


(b)

0500

10001500200025003000350040004500500055006000

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 50 100 150 200 250 300

M1

M2M3

M4

HB1 amplitude = 01mm HB1 amplitude = 03mmHB2 amplitude = 01mm HB2 amplitude = 03mmHB3 amplitude = 01mm HB3 amplitude = 03mmHB4 amplitude = 01mm HB4 amplitude = 03mmHB5 amplitude = 01mm HB5 amplitude = 03mmHB6 amplitude = 01mm HB6 amplitude = 03mm

(c)

0

20

40

60

80

100

120

140

160

Frequency (Hz)0 30 60 90 120 150 180 210 240 270 300

N3

N4

N5

N6

Loss

angl

e (∘)


(d)

Figure 5 The dynamic test results of HDBs with different internal configurations (a) Dynamic stiffness under low frequency band largeamplitude excitations (b) loss angle under low frequency bands large amplitude excitations (c) dynamic stiffness under high frequencyband small amplitude excitations (d) loss angle under high frequency band small amplitude excitations

According to Bernoulli equation for nonstationary flow [30]the momentum equation of the fluid column mass in theinertia track is yielded as follows

Δ119875 = 1198751minus 1198752= 120588119897119894119894+ (120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 (2a)

120578119897=

64120583120574

1205881003816100381610038161003816119894

1003816100381610038161003816 119889119867

Re le 2300

(2b)

120585 = 120585in + 120585out + 120585119897

120585in =1

2 (1 minus 119860out119860 in)

120585out =1

2 (119860out119860 in minus 1)

(2c)

where 120578119897is the major loss coefficient and the calculation

details of 120578119897can be found inWang et al [29] and 120585 is the total

local loss factor including contraction loss coefficient of inlet


120585in expansion loss coefficient of outlet 120585out and the track bendlocal loss factor 120585

119897determined by the curvature radius and

diameter of fluid track according to empirical formulas [29]119860 in and 119860out are the cross-sectional areas of inlet and outletrespectively

The dynamic force transmitted to the outer sleeve can beexpressed as

119865119879 (119905) = 119865

119879119903 (119905) + 119865119879ℎ (119905)

= (119870119903119909119903+ 119861119903119903) + (119860

11990121198752minus 11986011990111198751)

(3)

where 119865119879119903(119905) is the transmitted force of the rubber path and

119865119879ℎ(119905) is the transmitted force of the hydraulic path

31 Analytical Analysis of Dynamic Performance of an HDBwith One Inertia Track Substituting (1) into (2a) (2b) and(2c) the equation of the liquid column flowing in the inertiatrack can be obtained

119894+

1

120588119897119894

(120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 +

1

120588119897119894

(1198702119860119894+ 1198701119860119894) 119909119894

=1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903)

(4)

Then the resonance frequency of liquid column flowingin the inertia track can be derived as follows

120596119899= radic

1198702119860119894+ 1198701119860119894

120588119897119894

= 2120587119891119899 (5)

Equation (5) indicates that the value of the resonantfrequency 119891

119899closely relates not only to the bulk stiffness

values of two chambers but also to 120588 119897119894 119860119894

Since |119894|119894is a quadratic function [31] of the liquid

column velocity the exact solution of (4) cannot be achievedAn alternative is to linearize the nonlinear term as |

119894|119894asymp

81198831198941205961198943120587 utilizing the describing function analysis [31]

Then (4) can be rewritten as

119894+ 2120577120596

119899119894+ 1205962

119899119909119894=

1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903) (6)

where the damping ratio of the system is

120577 =1

2120596119899

(32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) (7)

Substituting 119909119903

= 119883119903119890119895120596119905 119909

119894= 119883119894119890119895(120596119905minus1205931) and 119865

119879=

119879119890119895(120596119905minus120593) into (6) it gives

119883119894

119883119903

119890minus1198951205931 [minus120596

2+ 1205962

119899+ (

32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) 119895120596]

=1

120588119897119894

(11987021198601199012

+ 11987011198601199011)

(8)

Therefore the transfer function of the fluid flowing in aninertia track can be written as

119867119891=

119883119894

119883119903

119890minus1198951205931

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

1

1 minus 1205822 + 119895 (119866120582120596119899+ 119873119883

1198941205822)

(9)

where

119866 =32120583120574

1205881198892

119867

119873 =4120585

3120587119897119894

120582 =120596

120596119899

(10)

Using (1)ndash(3) and (9) the sinusoidal complex stiffness canbe obtained

119870lowast

119889(119895120596) = 119870

119903+ 119895120596119861

119903+ 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(11)

Thus the complex stiffness can be rewritten as

119870lowast

119889(119895120596) =

119865119879(119895120596)

119909119903(119895120596)

=

119865119879119903(119895120596) + 119865

119879119891(119895120596)

119909119903(119895120596)

= 119870lowast

119903+ 119870lowast

119891= 1198701015840+ 11989511987010158401015840

(12)

119870lowast

119903= 119870119903+ 119895120596119861

119903

119870lowast

119891= 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(13)

where 119870lowast

119903is the dynamic stiffness of the rubber spring

and 119870lowast

119891represents dynamic characteristic induced by the

subsystem of chambers and liquid in the inertia track 1198701015840represents the storage dynamic stiffness and 119870

10158401015840 denotesthe loss dynamic stiffness According to (12) and (13) thedynamic performances of HDB with one inertia track can beinvestigated analytically

311 When the Ratio of Excitation Frequency and ResonanceFrequency of Liquid Column Tends to 0 When the excitationfrequency tends to zero that is 120582 rarr 0119867

119891and119870

lowast

119891become

119867119891(120582rarr0)

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

(14)

When 1198601199012

asymp 1198601199011 then

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

asymp 0

119870119889(120582rarr0)

asymp 119870119903(120582rarr0)

(15)

As shown in (14) 119867119891(120582rarr0)

is positive which indicatesthat the displacement excitation 119909

119903and the corresponding

response 119909119894are in-phase and the liquid pumped by the

equivalent piston area of the rubber spring is pumped into


the other chamber in-phaseThus there is almost no pressurefluctuation in both chambers Equation (15) means that thedynamic performance of the HDB is mainly contributed bythe rubber spring

In practice 1198601199012

asymp 1198601199011 119870lowast119891is not equal to zero and the

value of119870119889(120582rarr0)

is slightly bigger than that of119870119903(120582rarr0)

As thetest results shown when the excitation amplitude is 100mm119870119889(119891=0)

= 48547Nmm 119870119903(119891=0)

= 43414Nmm and theratio of119870

119889(119891=0)and119870

119903(119891=0)is 112 that is they are almost the

sameTherefore119870119903can be obtained by testing119870lowast

119889of an HDB

with the liquid under a very low frequency excitation

312 When the Ratio of Excitation Frequency and ResonanceFrequency of Liquid Column Tends to 1 Using (9) whenthe excitation frequency approaches the resonance frequency(119891119899) of the inertia track subsystem that is 120582 rarr 1 119867

119891and

119870lowast

119891become

119867119891(120582rarr1)

=119883119894

119883119903

119890minus1198951205931

= minus119895

11987021198601199012

+ 11987011198601199011

120596119899120588119897119894

1

(119866 + 120596119899119873119883119894)

(16)

119870lowast

119891(120582rarr1)

= 11987021198602

1199012+ 11987011198602

1199011

+ 119895

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

(17a)

119890minus1198951205931 = cos120593

1minus 119895 sin120593

1

119864 =

(11987021198601199012

+ 11987011198601199011)

120588119897119894

(17b)

Substituting (17a) into (11) leads to

119870lowast

119889(120582rarr1)= 119870lowast

119903+ 119870lowast

119891= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011

+ 119895(120596119861119903

+

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

)

(18)

Using (18) the loss angle of the HDB with one inertia trackcan be obtained as120593(120582rarr1)

= arctan(120596119862119903(1198702+ 1198701) + (119870

21198601199012

+ 11987011198601199011)2

120591119891

(1198702+ 1198701) (119870119903+ 11987021198602

1199012+ 11987011198602

1199011)

)

120591119891=

2120596119899

(119866 + radic1198662 + 4119873119883119903119864)

(19)

As shown in (10) (19) many parameters such as 120583 119860119894 120585

119897119894 and 119889

119867can be selected to adjust the loss angleThe smaller

the excitation displacement amplitude 119883119903 the larger the 120591

119891

and 120593(120582=1)

became larger tooThis is the reason why HDB hasamplitude-variant dynamic performances This conclusionmatches well the experimental results shown in Figure 5(b)

313 When the Ratio of Excitation Frequency and ResonanceFrequency of Liquid Column Tends toinfin Using (9) and (13)when the high excitation frequency is applied on an HDBwith one inertia track that is 120582 rarr infin119867

119891and119870

lowast

119891become

119867119891(120582rarrinfin)

997888rarr 0

119870lowast

119891(120582rarr1)= 11987021198602

1199012+ 11987011198602

1199011

(20)

As shown in (20) the displacement response of liquid columnis drastically weakened this effect is sometimes referred toas flow ldquoshut-offrdquo [27] under high excitation frequency Inthis situation the liquid pumped by the piston of the rubberspring is conformed all by two chambers due to their elasticityof bulk which results in pressure fluctuations

Substituting (20) into (11) it can be obtained that

119870lowast

119889(120582rarrinfin)= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011+ 119895120596119861

119903 (21)

As shown in (21) the storage dynamic stiffness under highexcitation frequency is not varied and 119870

119889(120582rarrinfin)is constant

considering that the influence of the damping of the mainrubber is negligible [28] which means that these parameters119870119903 1198701 1198702 1198601199011 and 119860

1199012can be regarded as constants

This analysis result agrees well with the test results shown inFigure 5(a)

32 Parameter IdentificationMethodBased on the Fixed PointsComparing (18) with (21) then

1198701015840

(120582rarr1)= 1198701015840

(120582rarrinfin)asymp 119870119903+ 11987021198602

1199012+ 11987011198602

1199011 (22)

It can be seen that the storage dynamic stiffness tendsto be a constant in higher frequency range and the valuesof 1198701015840

(120582rarr1)and 119870

1015840

(120582rarrinfin)are identical This indicates that

the storage dynamic stiffness curves for different amplitudeexcitations will intersect at 119891

119899and have the value of 119870

119903+

11987021198602

1199012+ 11987011198602

1199011

As shown in Figure 6(a) all the tested curves of storagedynamic stiffness of HB1 pass through the fixed point 119872

5

and all the curves of storage dynamic stiffness of HB3 passthrough the fixed point 119872

5approximately All the curves

almost approach a horizontal line in higher frequency rangeThus the analytical conclusion is verified by test resultswhich implies the validity of the nonlinear LP model

Set the excitation frequency at 1198725as 1198911198985

and thecorresponding storage dynamic stiffness as 1198701015840

(120582=1) Based on

the above analytical analysis the frequency 1198911198985

of the fixedpoint (119872

5) and 119891

119899of the liquid column have the following

relationship

1198911198985

asymp 119891119899=

1

2120587radic1198702119860119894+ 1198701119860119894

120588119897119894

(23)


Frequency (Hz)0 20 40 60 80 100

M5

400

800

1200

1600

2000

2400

Stor

age s

tiffne

ss (N

mm

)

HB1 amplitude = 06mmHB1 amplitude = 08mmHB1 amplitude = 10mmHB1 amplitude = 12mm


(a)

Frequency (Hz)0 20 40 60 80 100

N7

0

200

400

600

800

1000

1200

1400

Loss

stiff

ness

(Nm

m)



(b)Figure 6 The test results of storage and loss dynamic stiffness of the hydraulic bushing (a) Storage stiffness (b) Loss stiffness

Frequency (Hz)0 20 40 60 80 100

300

600

900

1200

1500

1800

2100

2400

2700

Dyn

amic

stiff

ness

(Nm

m)

Tested amplitude = 12mmTested amplitude = 08mmTested amplitude = 06mm

Calculated amplitude = 12mmCalculated amplitude = 08mmCalculated amplitude = 06mm

M1

(a)

Frequency (Hz)0 20 40 60 80 100

N1

0

10

20

30

40

50

60

Loss

angl

e (∘)



(b)

Figure 7 Comparisons between experimental and predicted results of the dynamic characteristics of HB1 (a) Dynamic stiffness (b) Lossangle

Since the curves near the fixed point 1198725are almost

vertical to the frequency axis as shown in Figure 6(a) smallexperiment error of the storage dynamic stiffness will notcause a notable change in 119891

1198985 Therefore the experimental

data of 1198911198985

can be employed to identify 119891119899 Supposing 119870

1

and1198702are proportional the values of119870

1and119870

2can then be

obtained according to (23)Besides the curves of storage dynamic stiffness are almost

perpendicular to the longitudinal axis when 120582 rarr infin so ahigher variation of excitation frequency will only lead to anearly smaller fluctuation of storage dynamic stiffness in highfrequency range On the contrary the slopes of these curvesaround the resonance frequency (when 120582 = 1) are almost

vertical to the frequency axis thus a smaller fluctuation ofexcitation frequency will result in a significant variation ofstorage dynamic stiffness So the parameter identificationresults for 119860

1199011and 119860

1199012based on 119870

1015840

(120582rarrinfin)are more precise

and credibleAccording to the abovemethod and Table 1 these param-

eters identifications are conducted and the results are sum-marized as follows 119870

119903= 486 times 10

5 (Nm) 1198702= 231 times

1011 (Nm5) 119870

1= 208 times 10

11 (Nm5) 1198601199011

= 191 times

10minus3 (m2) and 119860

1199012= 172 times 10

minus3 (m2) Then the dynamicperformances of HB1 are calculated and compared with theexperiment data as shown in Figure 7 It is shown that thecalculation results match well the experiment data which


Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

Q1(t) Q2(t) Q3(t)

I1R1 I3R3I2R2

xi(t)

Figure 8 Fluid system model for an HDB with three nonidenticalinertia tracks

further validates the analytical analysis presented in thisstudy

4 The LP Models for HDB with MultipleInertia Tracks and Orifices

As shown in Figure 5 the dynamic performances of HB2HB4 and HB5 are very similar to that of HB1 This suggeststhat the LP model of HDB with one inertia track is also

applicable for those with multiple inertial tracks Thus it willbe adopted in the following analyses

41TheLPModel for anHDBwithNonidentical Inertia TracksAnLPmodel ofHDBwith three nonidentical inertial tracks isillustrated in Figure 8119860

119894 119897119894(119894 = 1 2 3) denote the lengths and

the cross-sectional areas of the inertia tracks and 119894 representsthe inertial track number 120585

1198941and 1205851198942(119894 = 1 2 3) are the linear

and nonlinear damping coefficients of the fluid motion in theinertia tracks respectively

The inertia 119868119894and the linear and nonlinear resistances 119877

1198941

and 1198771198942(119894 = 1 2 3) of the fluid in the inertia track can be

obtained by

119868119894=120588119860119894

119897119894

1198771198941=

1205851198941

1198602

119894

1198771198942=

1205851198942

1198605

119894

(24)

The fluid volume flow rate in three inertia tracks can bedenoted as 119876

119894(119894 = 1 2 3) respectively The state equation of

the LP model can be written as follows with the state vectordefined as X119879 = (119909

1 1199092 1199093 1199094 1199095) = (119875

1 1198752 1198761 1198762 1198763)

X = AX + B (25)

where

A =

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus (11987711+ 11987712

100381610038161003816100381611990931003816100381610038161003816)

1198681

0 0

1

1198682

minus1

1198682

0minus (11987721+ 11987722

100381610038161003816100381611990941003816100381610038161003816)

1198682

0

1

1198683

minus1

1198683

0 0minus (11987731+ 11987732

100381610038161003816100381611990951003816100381610038161003816)

1198682

]]]]]]]]]]]]]]]]

]

B =

[[[[[[[

[

minus11987011198601199011

11987021198601199012

0

0

0

]]]]]]]

]

(26)

In general the nonlinear resistances 1198771198942(119894 = 1 2 3) are

negligible that is 1198771198942

= 0 (119894 = 1 2 3) Since 119897119894(119894 = 1

2 3) is finite there will be some end effects due to contractionloss of inlet and expansion loss of outlet and the tracksare curvilinear These additional effects (including turbu-lence) are expected to add more damping to the HDB

To compensate these effects an empirical coefficient 120574 isintroduced to better approximate the actual system and toenhance the capillary tube resistance 119877

1198941(119894 = 1 2 3) as

follows [32]

1198771198941= 120574

128120583119897119894

1205871198894

119867

(27)


Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

xi(t)xo(t)

n1Qi(t)n2Qo(t) IiRi

n1IoRon2

Figure 9 Fluid system model of hydraulic bushing with multipleidentical inertia tracks and identical orifices

where 120583 is the dynamic viscosity of the HDB fluid andA thenbecomes a constant matrix A

119871as

A119871=

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus11987711

1198681

0 0

1

1198682

minus1

1198682

0minus11987721

1198682

0

1

1198683

minus1

1198683

0 0minus11987731

1198682

]]]]]]]]]]]]]]]]

]

(28)

Substituting Q119879 = (1198761 1198762 1198763) and 119877

1198942= 0 (119894 = 1 2 3) into

(25) the motion equation of the lumped fluid mass can bewritten as

MQ + CQ + KQ = F (119905) (29)

where M C K and F(119905) are mass matrix damping matrixstiffness matrix and force vector respectively and can beexpressed as

M =[[

[

1198681

1198682

1198683

]]

]

C =[[

[

11987711

11987721

11987731

]]

]

K =[[[

[

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

]]]

]

F (119905) =[[[

[

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

]]]

]

(30)

The characteristic equation of (29) is10038161003816100381610038161003816K minus 120596

2

119899M10038161003816100381610038161003816

= 0 (31)

According to (31) the resonance frequency 119891119899can be

obtained as

119891119899=

120596119899

2120587= radic

(1198701+ 1198702)

120588(1198601

1198971

+1198602

1198972

+1198603

1198973

) (32)

This indicates that there is only one peak frequency forthe loss angle and the system has only one nonzero resonantfrequency

42 The LP Model for an HDB with Multiple Identical InertiaTracks and Identical Orifices An LP model for an HDB withtwo groups of identical inertia tracks and identical orificesis shown in Figure 9 The inertia track group consists of1198991identical inertia tracks with the length of 119897

119894and cross-

sectional area of119860119894for each inertia track Each inertia track is

characterized by the fluid inertances 119868119894 linear fluid resistances

1198771198941 and nonlinear resistances 119877

1198942(119894 = 1 2 119899

1) of the

lumped fluid mass 119868119894can be thought of as the resistance to

change in flow rate 119894to the pressure gradient Δ119875 applied on

the fluid element 119877119894of these elements is fundamentally the

energy dissipation of fluid flow 119876119894over the pressure gradient

Δ119875 of the elementBased on the steady flow assumption the orifice-type

models can also be used to describe the damping of orifice-type tracks for119877

119900concentrated over a short distance Because

the orifice-type track has an aspect ratio of 2119897119900119889119867

asymp 1 itcan be taken as a short-tube orifice for dynamic performanceprediction of HDB The discharge coefficient 119888

119900of such a

short-tube orifice has the following empirical formulation[32] based on the Reynolds number (Re)

119888119900=

(15 + 1374119897119900

119889119867Re

)

minus12119889119867Re119897119900

gt 50

(228 + 64119897119900

119889119867Re

)

minus12119889119867Re119897119900

lt 50

(33)

Using the square-root relation [32 33] and taking intoaccount the impact of the fluid jet the fluid inertances 119868

119900and

the linearized resistance1198771199001(119900 = 1 2 119899

2) of an orifice can

be expressed as

119868119900= 120573

120588119897119900

119860119900

1198771199001

= 120576120588119876119900

11988821199001198602119900

(34)

where 120573 and 120576 are empirical coefficients for adjusting HDBrsquosinertances and damping in nonideal cases 119876

119900denotes the

flow through the orifice [32]Assuming the state variable of the LP model is X and is

defined as X119879 = (1199091 1199092 1199093 1199094) = (119875

1 1198752 119876119894 119876119900) the state

equation then can be obtained as

X = AX + B (35)


Table 3 Comparison of the liquid column resonance frequency

Different HDB HB1 (1198991= 1 1198992= 0) HB2 (119899

1= 2 119899

2= 0) HB4 (119899

1= 1 1198992= 1) HB5 (119899

1= 0 119899

2= 2)

Experiment 12Hz 20Hz 52Hz 82HzSimulation 135Hz 235Hz 585Hz 88Hz

whereA

=

[[[[[[[[[

[

0 0 minus1198701

minus1198701

0 0 1198702

1198702

1

119868119894

minus1

119868119894

minus (1198771198941+ 1198771198942

100381610038161003816100381611990931003816100381610038161003816)

119868119894

0

1

119868119900

minus1

119868119900

0minus (1198771199001+ 1198771199002

100381610038161003816100381611990941003816100381610038161003816)

119868119900

]]]]]]]]]

]

B119879 = (minus11987011198601199011119903 11987021198601199012119903 0 0)

(36)

The dynamic force transmitted to the outer sleeve by therubber and hydraulic paths is

119865119879 (119905) = 119870

119903119909119903 (119905) + 119861

119903119903 (119905) + (119860

11990121198752minus 11986011990111198751) (37)

Accordingly the transfer function can be derived from (34)(35) and (37) in the Laplace domain as follows

119870lowast

119889(119904) =

119871 [119865 (119905)]

119871 [119909119903 (119905)]

= 119870119903+ 119861119903119904 + 1198602

11990121198702+ 1198602

1199011119870

minus [

11987221198852

1199041199042+1198722[(119885119894119904 + 1198991119870) (119885

119904119904 + 1198992119870) minus 119899

21198702] + 119872

21198992119870119885119904119904 minus 119872

21198991119870119885119904119904

(119885119904119904 + 1198992119870)2(119885119894119904 + 1198991119870) minus 119899

2(119885119904119904 + 1198992119870)1198702

]

(38)

where

119872 = 11986011990111198701+ 11986011990121198702

119885119894= 119868119894119904 + 1198771198941

119885119900= 119868119900119904 + 1198771199001

119870 = 1198701+ 1198702

(39)

Using the same technique presented in Section 41 thepeak frequency of the loss angle119891

119899can be explicitly expressed

as

119891119899=

1

2120587

radic(1198701+ 1198702)

120588(1198991119860119894

119897119894

+1198992119860119900

119897119900

) (40)

5 Model Verification and Discussion

51 Model Verification As shown in Figure 9 when 1198992= 0

and 1198991= 1 the LP model is the same as the fluid system

model of anHDBwith one inertia track shown in Figure 3(a)when 119899

2= 0 and 119899

1= 2 the LP model depicts the fluid

system model of an HDB with two inertia tracks shown inFigure 3(b) when 119899

2= 1 and 119899

1= 1 the LPmodel represents

the fluid system model of an HDB with one inertia track andone orifice element shown in Figure 3(c) when 119899

2= 2 and

1198991= 0 the LP model denotes the fluid system model of an

HDB with two orifices shown in Figure 3(d)

The comparisons between the tested and calculateddynamic performances of HDBs with multiple identical iner-tia tracks and identical orifices under excitation amplitude of08mm and lower frequency band (1sim100Hz) are presentedin Figure 10 In general the calculated results agree withthe experiment data reasonably Obvious errors for HB4 andHB5 can be seen around frequencies of 35Hz and 55HzrespectivelyThismainly attributes to the assumption that thefluid-jet length of the short-tube orifice can be regarded asthe length of inertia track which underestimates the impactof liquid turbulence in the orifice However the calculateddynamic performances of HB4 and HB5 can predict thedynamic performances of the HDBs in the whole frequencyrange Therefore the proposed model for an HDB with mul-tiple identical inertia tracks and identical orifices is validated

Based on the geometric and material parameters of theHDB listed in Table 1 the peak frequency of the loss angle 119891

119899

of theHDBwithmultiple identical inertia tracks and identicalorifices can be calculated according to (40) Table 3 showsthe calculated results and the corresponding experiment dataIt can be seen that they match well each other Thus thecalculation method is validated

52 Influences of Key Parameters Based on a lot of parametricanalyses of theHDBwith one inertia track the same dynamicperformances as that of the HDB with two identical inertiatracks can only be achieved by adjusting the length of the fluidtrack


300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)

Tested HB1Tested HB2Tested HB4Tested HB5

Calculated n2 = 0 n1 = 1




0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)

Figure 10 Dynamic performances of HDB (excitation amplitude 08mm) (a) Dynamic stiffness (b) Loss angle

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li

HB1 li = 02liHB1 li = 06liHB1 li = 20liHB1 li = 80li

(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)

Figure 11 Calculated dynamic performances of HDB (excitation amplitude 08mm) (a) Dynamic stiffness (b) Loss angle

As shown in Figure 11 when 04119897119894lt 119897119894lt 06119897

119894 the

curves of loss angle for HB1 approach that of HB2 and when02119897119894lt 119897119894lt 06119897

119894 the curves of dynamic stiffness for HB1 show

consistency with that of HB2 This indicates that the samedynamic performances of an HDB with two identical inertiatracks cannot be achieved simultaneously by only changingthe track length of an HDB with one inertia track

As well known the HDBs have irregular geometry andare constructed with alternate materials from elastomers

to metals It will be a very challenging work to reach anideal dynamic performance by changing those parametersconsidering the huge numbers of experiments Varying thenumber of inertia tracks can be the best choice

In general an HDB with 1198991

ge 2 is more versatile fortuning since there are now many 119897

119894and 119860

119894to change but

as the number of inertia tracks increases the maximumdynamic stiffness the maximum loss angle and the peakfrequency of the loss angle also increase as shown in Figure 12


400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

Tested HB1Tested HB2

0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)

Figure 12 Calculated dynamic performances of HDB with inertia tracks increasing (a) Dynamic stiffness (b) Loss angle

which is good for motion control of say the automotivesuspension bounce mode

53 Prediction of Dynamic Performances of the HDBs Theinfluences of the number of inertia tracks on the dynamicperformances of anHDB can be analytically investigatedwiththe LP model proposed in Section 42 Assume that an HDBconsists of 119899

1(1198992= 0 1 le 119899

1le 7) identical inertia tracks

The influences of the number of inertia tracks on the dynamicperformances of the HDB are shown in Figure 12 It isexpected that an increase in the number of inertia tracks willintroduce more fluid oscillating force contributing to 119865

119879(119905)

Comparing the results for the inertia track cases of1198991

= 2 to 1198991

= 7 shown in Figure 12 it shows that themaximum dynamic stiffness increases from approximately2200Nmm to 3200Nmm Additionally the maximum lossangle increases from approximately 50∘ to 87∘ and the peakfrequency of the loss angle increases from approximately135Hz to 52Hz Thus the sensitivity of the HDBrsquos perfor-mance increases near the maximum loss angle as well as theincrease of the number of inertia tracks Further the numberof inertia tracks can be used to tune the device over a widerrange of frequencies via increasing the peak frequency of theloss angle

The influences of the track type on the performances ofan HDB can be numerically analyzed with the proposed LPmode in Section 42 too Assume that an HDB consists of1198991identical inertia tracks and 119899

2identical orifice tracks The

dynamic performances of theseHDBs are shown in Figure 13When comparing the results for track cases of 119899

1= 1 (119899

2= 2)

and 1198991= 2 (119899

2= 2) to that for the track case of 119899

1= 3

(1198992= 2) it can be seen that the influences of the number of

inertia tracks are not significant since the existence of orifices

Comparing the results for the inertia track cases of 1198991= 2

to 1198991= 7 shown in Figure 12 it is shown that the maximum

dynamic stiffness increases from approximately 2200Nmmto 3200Nmm The maximum loss angle increases fromapproximately 50∘ to 87∘ and the peak frequency of the lossangle increases from approximately 135Hz to 52Hz TheHDBrsquos performances near the maximum loss angle becomemore sensitive as the number of inertia tracks increasesThus increasing the number of inertia tracks can improve thedevice performance over a wider frequency range

Observing the results for the track case of 1198991= 0 (119899

2= 4)

and 1198991

= 0 (1198992

= 6) shown in Figure 13 it can be seenthat themaximum dynamic stiffness decreases from approxi-mately 2300Nmm to 2220Nmm but the dynamic stiffnessincreases from approximately 1050Nmm to 1250Nmm ata very low frequency for example 1 Hz In addition themaximum loss angle decreases from approximately 55∘ to54∘ and the peak frequency of the loss angle increases fromapproximately 154Hz to 196Hz In a word as the number oforifices increases the maximum dynamic stiffness and lossangle decrease slightly and the peak frequency of the lossangle increases significantly Moreover the HDB with moreorifices can provide large damping in a wide frequency rangeTherefore the influences of the number of orifices on theperformances of the HDBs are remarkable

6 Conclusions

Hydraulically damped bushing (HDB) is crucial for thehandling and NVH performances of vehicle In this studythe dynamic stiffness and the loss angle of the HDBs withmultiple internal configurations which are the essentialcharacteristics of the HDBs are tested and the influences of


600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)

Figure 13 Calculated results of the dynamic characteristics of HDB with multiple tracks (a) Dynamic stiffness (b) Loss angle

the HDBrsquos structural features namely the number of inertialtracks and the cross-sectional shape of the track are analyzedexperimentally It is found that (1) inertia tracks can onlyprovide damping in low frequency range and their dampingprovision is almost negligible in high frequency range and (2)fixed points can be observed in test data for all HDB samplesunder all excitation conditions

Lumped parameter (LP) models for the HDB with mul-tiple inertia tracks and orifices are proposed in this paperin order to understand the relations between the HDBrsquosstructure features and its dynamic performance analyticallyThe LPmodels are validated via benchmarking the simulateddynamic stiffness and loss angle against test data Based onthe LP models it is found that increasing the number ofthe inertial tracks andor orifices would benefit the HDBrsquosperformance a great deal since the dynamic stiffness theamplitude and the corresponding frequency of the loss anglewould be significantly improvedHowever it poses a questionlike whether there is a limitation on HDBrsquos performanceimprovement by increasing the number of the inertial tracksThis motivates a future research that aims at finding acompromise between the complexity in manufacturing andassembly induced by the increase of fluid track numberwith the improvement in HDBrsquos performance The proposedmethods of modeling and analysis can be utilized to char-acterize the performances of the HDB with multiple inertiatracks and orifices before prototype

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial supportfrom the Natural Science Foundation of China (Projectnos 51305139 51475171) the Chinese Universities ScientificFund (Project no 2013ZM0016) and the Natural ScienceFoundation of Hubei Province (Project no 2015CFB402)Experiments aremade possible by equipment of nonresonantelastomer testmachine (MTS 831) grants fromNingbo TuopuVibration Isolation System Co Ltd China

References

[1] M Lu and J Ari-Gur ldquoStudy of dynamic properties ofautomotive hydrobushingrdquo ASME Design Engineering DivisionProceedings vol 106 pp 135ndash140 2000

[2] M Lu and J Ari-Gur ldquoStudy of hydromount and hydrobushingwithmultiple inertia tracksrdquo JSAEAnnual Congress Proceedingsvol 68 no 2 pp 5ndash8 2002

[3] W Sauer and Y Guy ldquoHydro bushingsmdashinnovative NVHsolutions in chassis technologyrdquo SAE Paper 2003-01-1475 2003

[4] T Chai R Singh and J Dreyer ldquoDynamic stiffness of hydraulicbushingwithmultiple internal configurationsrdquo SAEPaper 2013-01-1924 SAE International 2013

[5] T Chai J Dreyer andR Singh ldquoTransient response of hydraulicbushing with inertia track and orifice-like elementsrdquo SAE Paper2013-01-1927 2013

[6] T Chai J T Dreyer and R Singh ldquoTime domain responses ofhydraulic bushingwith two flow passagesrdquo Journal of Sound andVibration vol 333 no 3 pp 693ndash710 2014

[7] T Chai J T Dreyer and R Singh ldquoNonlinear dynamicproperties of hydraulic suspension bushing with emphasis onthe flow passage characteristicsrdquo Proceedings of the Institution ofMechanical Engineers Part D Journal of Automobile Engineer-ing vol 229 no 10 pp 1327ndash1344 2014


[8] T Chai J T Dreyer and R Singh ldquoFrequency domainproperties of hydraulic bushing with long and short passagessystem identification using theory and experimentrdquoMechanicalSystems and Signal Processing vol 56ndash57 pp 92ndash108 2015

[9] W B Shangguan and C Xu ldquoExperiment and calculationmethods of the dynamic performances for hydraulic bushingsused in control arms of a suspensionrdquo Journal of Vibration andShock vol 26 no 9 pp 7ndash10 2007

[10] Y-Q Zhang andW-B Shangguan ldquoA novel approach for lowerfrequency performance design of hydraulic engine mountsrdquoComputers amp Structures vol 84 no 8-9 pp 572ndash584 2006

[11] B Heiszliging and M Ersoy Chassis Handbook Vieweg+Teubner2011

[12] K SetoDynamic Vibration Absorber and Its Applications ChinaMachine Press Beijing China 2013

[13] B Piquet C AMaas and F Capou ldquoNext generation of suspen-sion bushings review of current technologies and expansionupon new 3rd generation product datardquo SAE Paper 2007-01-0850 2007

[14] N E Lee ldquoFluid damped resilient mountingrdquo Patent 2582998USA 1952

[15] G L Hipsher ldquoHigh damping resilient bushingrdquo Article ID364226 Patent 364226 8 USA 1972

[16] K Konishi ldquoFluid-filled resilient bushingrdquo Patent 4588174USA 1986

[17] R Kanda ldquoFluid-filled resilient bushing with circumferentialorificerdquo Patent 4693456 USA 1987

[18] R Kanda ldquoFluid-filled cylindrical elastic connector having twoorifice passages with different cross sectional areasrdquo Patent5060918 USA 1991

[19] A Vossel and F Meyerink ldquoHydraulically damping rubberbearing with decoupling elementrdquo Patent 6511058 USA 2003

[20] F Meyerink and C Hoping ldquoHydro-bush bearing with acousticdecouplingrdquo Patent Application 11760846 USA 2007

[21] M Sevensson and M Hakansson Hydrobushing model formulti-body simulation [MS thesis] Lund University LundSweden 2004

[22] X-Y Pan X-X Xie and W-B Shangguan ldquoDynamic proper-ties analysis for hydraulic rubber isolator under excitations withdifferent amplitudesrdquo Chinese Journal of Vibration and Shockvol 31 no 1 pp 144ndash149 2012

[23] S Arzanpour and F Golnaraghi ldquoA novel semi-active mag-netorheological bushing design for variable displacementenginesrdquo Journal of Intelligent Material Systems and Structuresvol 19 no 9 pp 989ndash1003 2008

[24] S Arzanpour and M F Golnaraghi ldquoDevelopment of an activecompliance chamber to enhance the performance of hydraulicbushingsrdquo Journal of Vibration and AcousticsmdashTransactions ofthe ASME vol 132 no 4 pp 1ndash7 2010

[25] R Singh G Kim and P V Ravindra ldquoLinear analysis of auto-motive hydro-mechanical mount with emphasis on decouplercharacteristicsrdquo Journal of Sound and Vibration vol 158 no 2pp 219ndash243 1992

[26] W-B Shangguan and Z-H Lu ldquoExperimental study and sim-ulation of a hydraulic engine mount with fully coupled fluidndashstructure interaction finite element analysis modelrdquo Computersamp Structures vol 82 no 22 pp 1751ndash1771 2004

[27] G Kim and R Singh ldquoA study of passive and adaptive hydraulicenginemount systemswith emphasis onnon-linear characteris-ticsrdquo Journal of Sound and Vibration vol 179 no 3 pp 427ndash4531995

[28] R Fan and Z Lu ldquoFixed points on the nonlinear dynamicproperties of hydraulic engine mounts and parameter identi-fication method experiment and theoryrdquo Journal of Sound andVibration vol 305 no 4-5 pp 703ndash727 2007

[29] L-R Wang Z-H Lu and I Hagiwara ldquoAnalytical analysisapproach to nonlinear dynamic characteristics of hydraulicallydamped rubbermount for vehicle enginerdquoNonlinear Dynamicsvol 61 no 1-2 pp 251ndash264 2010

[30] J Z Lin XD Ruan B G Chen et al FluidMechanics TsinghuaUniversity Press Beijing China 2010

[31] J-H Lee and K-J Kim ldquoAn efficient technique for design ofhydraulic engine mount via design variable-embedded damp-ing modelingrdquo Journal of Vibration and Acoustics vol 127 no 1pp 93ndash99 2005

[32] B Barszcz J T Dreyer and R Singh ldquoExperimental studyof hydraulic engine mounts using multiple inertia tracks andorifices narrow and broad band tuning conceptsrdquo Journal ofSound and Vibration vol 331 no 24 pp 5209ndash5223 2012

[33] E Doebelin System Dynamics Modeling Analysis SimulationDesign CRC Press New York NY USA 1998

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



variables of an HDB a new parameter identification methodis proposed based on the characteristic frequency points ofthe dynamic performances of HDB in this paper

During the design stage of an HDB it is not easy to adjustthe maximum frequency of loss angle to the same frequencyof the most disruptive excitations with one fluid elementdue to geometry limitations of the fluid track [11] Besidesthe hydraulic damping mechanism is relatively sensitive tothe operating environment while multiple damped dynamicvibration absorbers have good robustness against it [12]However the influences of fluid track number on the dynamiccharacteristics of an HDB with multiple inertia tracks andorifices as well as its practical application issues are not fullyinvestigated

Therefore the motivations of this paper are to develop asuperior parameter identificationmethod for theHDB and toreveal the influences of the geometry size and the number offluid tracks on the dynamic characteristics of the HDB by testand simulation approaches

Literature ReviewThe first HDB was introduced in the mid-1980s on MacPherson-type front suspension systems [11]Similar designs are already used as HEM and are adapted forapplication with cylindrical sleeve-type bushing joints simply[11 13] There are also many patents [14ndash20] that claim thedynamic characteristics features of some specific HDBs infrequency domain however no analytical analysis or evenmeasured dynamic properties are provided [4ndash8] Furthervery limited articles have reported the dynamic propertiescharacterization and modeling issues of HDB

Lu and Ari-Gur [1 2] present a linear LP model for anHDB with two inertia tracks and derived the natural fre-quency analytically Then the model is applied to the sim-ulation of an HDB with multiple inertia tracks Howevertheir model is not validated experimentally Sauer and Guy[3] present a fluid system model for an HDB with a bypasstrack and one inertia track in parallel which serves as arelief valve securing theHDB and the adjacent structure fromundesirable shocks but the detailed calculation conclusionsare not carried out Shangguan and Xu [9] propose a nonlin-ear lumped parameter state-space model (by introducing anonlinear track resistance 119877

119894) for an HDB with one inertia

track and the model is verified by experiments while theanalytical study is conducted based on the linearized modelSevensson and Hakansson [21] and Pan et al [22] providean empirical mathematic model including a nonlinear springwith several fluid elements and elastoplastic elements coupledin parallel but the calculation results show that the accuracyof the model is not satisfactory for low amplitude excitationArzanpour andGolnaraghi [23 24] present a linear LPmodelof an HEM employing the same approach as that for anHDB with one inertia track Chai et al [4ndash8] initiate a labo-ratory device with alternative internal configurations andthe dynamic characteristics for different configurations areinvestigated comparatively as well as the steady state andtransient responses of the device However the influence ofthe number of the fluid tracks on the dynamic characteristicsis not discussed The parameters of LP models of HDBwith multiple inertia tracks are identified by methods of

Table 1 Geometric and material parameters of a typical HDB

Parameter ValueLength of inertia track 119897

119894m 019

Cross-sectional area of the track 119860119900119860119894m2 983 times 10minus6

Wet perimeter of the inertia track 120594119894m 127 times 10minus2

Hydraulic diameter of the inertia track 119889119867m 3087 times 10minus3

Length of the orifice 119897119900m 0012

Density of the fluid 120588(kgm3) 111280Poisson ratio of the rubber spring ] 050Density of the rubber spring 120588

119903(kgm3) 105000

curve fitting and Nyquist diagrams [4 8] However the phe-nomenon of the excitation amplitude-invariant fixed pointson the dynamic performance curves of HDBs with differentinternal configurations is not investigated

In summary no uniform LPmodels have been developedto analyze the influences of the number of long fluid tracksand orifices on the dynamic characteristics of HDBs A thor-ough understanding of the mechanism of HDBs is needed

Scope and Objectives As a passive vibration isolation compo-nent in the vehicle suspension and steering wheel assembliesHDBs play an important role in dealing with vehicle systemvibration issues such as brake judder and shimmy [3 11]The configuration of a typical HDB and the runner plate areillustrated in Figures 1 and 2 respectively Its geometric andmaterial parameters are listed in Table 1

The elastomeric rubber spring enclosed by inner andouter metal sleeves has two functions to support the staticload of the vehicle and to provide partial static stiffness andsomewhat damping of the suspension system The typicalpassive hydromechanical bushing usually consists of twoalmost same fluid chambers connected by one ormore inertiatracks andor orifices Typically the fluid chambers filledwith an antifreeze and water mixture which has appropri-ate dynamic and thermal properties required for isolationand control [25] In the event of a dynamically oscillatoryexcitation the chamber pressures change in radical directionand the volume compensation will take place between twofluid chambers Thus the fluid flows back and forth throughthe fluid tracks as an oscillating mass to provide hydraulicdamping [4ndash9 11 13]

Based on the above analysis HDBrsquos engineering designsare conceptually identical to those of HEM but their struc-tures working mechanisms and dynamic properties areessentially different [4ndash8] In particular the lower chamber ofthe HEM is very thin and the pressure in the lower chamberis usually smaller than that in the upper chamber On thecontrary the two chamber pressures of theHDB are relativelyhighTherefore the LPmodels developed forHEM cannot bedirectly applied to HDB [4ndash8]

The disadvantage of the HDB with inertia track is thatthe flow velocity of the liquid column in the inertia trackbecomes very slow due to the viscosity of the fluid once thefrequency excitation reaches its resonant frequency that isit usually can be assumed that the flow ldquoshut offrdquo [25ndash28]


(1)(2)

(7)

(6)

(5)

(4)

(3)





(1)

(2)

(3)

















and 1198601199012







Outer sleeve

Outer sleeve

Inner sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

Qi(t)

IiRi1

Ri2

C1

C2

FT(t)

xi(t)

FT(t)

(a)

Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

Qi(t)

IiRi1

Ri2

C1

C2

FT(t)

xi(t)

FT(t)

IsRs1

Rs2

Qs(t)

xs(t)

(b)

Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

Qi(t)

IiRi1

Ri2

C1

C2

FT(t)

FT(t)

Qo(t) xo(t)

Io Ro1 Ro2

xi(t)

(c)

Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2

FT(t)

FT(t)

Io2 Io1 Ro11Ro12

Ro22 Ro21


(d)



2(119905) denote



119894and 119877

119900


119903(119905) = 119883








0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

14000

16000

18000

Forc

e (N

)

Displacement (mm)

HB1HB2

HB4HB5






119903is about 438340Nm













119894(119894 = 1 sim 4)









1198701(119860119894119909119894minus 1198601199011119909119903) = 1198751

1198702(1198601199012119909119903minus 119860119894119909119894) = 1198752

(1)


300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)0 20 40 60 80 100

M1

M2


(a)

0

20

40

60

80

100

Frequency (Hz)0 20 40 60 80 100

N1

N2

Loss

angl

e (∘)


(b)

0500

10001500200025003000350040004500500055006000

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 50 100 150 200 250 300

M1

M2M3

M4


(c)

0

20

40

60

80

100

120

140

160

Frequency (Hz)0 30 60 90 120 150 180 210 240 270 300

N3

N4

N5

N6

Loss

angl

e (∘)


(d)



Δ119875 = 1198751minus 1198752= 120588119897119894119894+ (120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 (2a)

120578119897=

64120583120574

1205881003816100381610038161003816119894

1003816100381610038161003816 119889119867

Re le 2300

(2b)

120585 = 120585in + 120585out + 120585119897

120585in =1

2 (1 minus 119860out119860 in)

120585out =1

2 (119860out119860 in minus 1)

(2c)









119865119879 (119905) = 119865

119879119903 (119905) + 119865119879ℎ (119905)

= (119870119903119909119903+ 119861119903119903) + (119860

11990121198752minus 11986011990111198751)

(3)




119894+

1

120588119897119894

(120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 +

1

120588119897119894

(1198702119860119894+ 1198701119860119894) 119909119894

=1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903)

(4)


120596119899= radic

1198702119860119894+ 1198701119860119894

120588119897119894

= 2120587119891119899 (5)






119894|119894asymp



119894+ 2120577120596

119899119894+ 1205962

119899119909119894=

1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903) (6)


120577 =1

2120596119899

(32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) (7)


= 119883119903119890119895120596119905 119909

119894= 119883119894119890119895(120596119905minus1205931) and 119865

119879=


119883119894

119883119903

119890minus1198951205931 [minus120596

2+ 1205962

119899+ (

32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) 119895120596]

=1

120588119897119894

(11987021198601199012

+ 11987011198601199011)

(8)


119867119891=

119883119894

119883119903

119890minus1198951205931

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

1

1 minus 1205822 + 119895 (119866120582120596119899+ 119873119883

1198941205822)

(9)

where

119866 =32120583120574

1205881198892

119867

119873 =4120585

3120587119897119894

120582 =120596

120596119899

(10)


119870lowast

119889(119895120596) = 119870

119903+ 119895120596119861

119903+ 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(11)


119870lowast

119889(119895120596) =

119865119879(119895120596)

119909119903(119895120596)

=

119865119879119903(119895120596) + 119865

119879119891(119895120596)

119909119903(119895120596)

= 119870lowast

119903+ 119870lowast

119891= 1198701015840+ 11989511987010158401015840

(12)

119870lowast

119903= 119870119903+ 119895120596119861

119903

119870lowast

119891= 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(13)

where 119870lowast


and 119870lowast





119891and119870

lowast

119891become

119867119891(120582rarr0)

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

(14)

When 1198601199012

asymp 1198601199011 then

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

asymp 0

119870119889(120582rarr0)

asymp 119870119903(120582rarr0)

(15)

As shown in (14) 119867119891(120582rarr0)









value of119870119889(120582rarr0)



= 48547Nmm 119870119903(119891=0)


119889(119891=0)and119870



119889of an HDB



119891and

119870lowast

119891become

119867119891(120582rarr1)

=119883119894

119883119903

119890minus1198951205931

= minus119895

11987021198601199012

+ 11987011198601199011

120596119899120588119897119894

1

(119866 + 120596119899119873119883119894)

(16)

119870lowast

119891(120582rarr1)

= 11987021198602

1199012+ 11987011198602

1199011

+ 119895

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

(17a)

119890minus1198951205931 = cos120593

1minus 119895 sin120593

1

119864 =

(11987021198601199012

+ 11987011198601199011)

120588119897119894

(17b)


119870lowast

119889(120582rarr1)= 119870lowast

119903+ 119870lowast

119891= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011

+ 119895(120596119861119903

+

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

)

(18)


= arctan(120596119862119903(1198702+ 1198701) + (119870

21198601199012

+ 11987011198601199011)2

120591119891

(1198702+ 1198701) (119870119903+ 11987021198602

1199012+ 11987011198602

1199011)

)

120591119891=

2120596119899

(119866 + radic1198662 + 4119873119883119903119864)

(19)


119897119894 and 119889



119891

and 120593(120582=1)



119891and119870

lowast

119891become

119867119891(120582rarrinfin)

997888rarr 0

119870lowast

119891(120582rarr1)= 11987021198602

1199012+ 11987011198602

1199011

(20)



119870lowast

119889(120582rarrinfin)= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011+ 119895120596119861

119903 (21)







1198701015840

(120582rarr1)= 1198701015840

(120582rarrinfin)asymp 119870119903+ 11987021198602

1199012+ 11987011198602

1199011 (22)


(120582rarr1)and 119870

1015840




119903+

11987021198602

1199012+ 11987011198602

1199011


5






(120582=1) Based on



5) and 119891


relationship

1198911198985

asymp 119891119899=

1

2120587radic1198702119860119894+ 1198701119860119894

120588119897119894

(23)


Frequency (Hz)0 20 40 60 80 100

M5

400

800

1200

1600

2000

2400

Stor

age s

tiffne

ss (N

mm

)



(a)

Frequency (Hz)0 20 40 60 80 100

N7

0

200

400

600

800

1000

1200

1400

Loss

stiff

ness

(Nm

m)




Frequency (Hz)0 20 40 60 80 100

300

600

900

1200

1500

1800

2100

2400

2700

Dyn

amic

stiff

ness

(Nm

m)



M1

(a)

Frequency (Hz)0 20 40 60 80 100

N1

0

10

20

30

40

50

60

Loss

angl

e (∘)



(b)





data of 1198911198985


1


1and119870

2can then be




1199011and 119860

1199012based on 119870

1015840




119903= 486 times 10

5 (Nm) 1198702= 231 times

1011 (Nm5) 119870

1= 208 times 10

11 (Nm5) 1198601199011

= 191 times

10minus3 (m2) and 119860

1199012= 172 times 10



Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

Q1(t) Q2(t) Q3(t)

I1R1 I3R3I2R2

xi(t)












1198941


obtained by

119868119894=120588119860119894

119897119894

1198771198941=

1205851198941

1198602

119894

1198771198942=

1205851198942

1198605

119894

(24)




1 1199092 1199093 1199094 1199095) = (119875

1 1198752 1198761 1198762 1198763)

X = AX + B (25)

where

A =

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus (11987711+ 11987712

100381610038161003816100381611990931003816100381610038161003816)

1198681

0 0

1

1198682

minus1

1198682

0minus (11987721+ 11987722

100381610038161003816100381611990941003816100381610038161003816)

1198682

0

1

1198683

minus1

1198683

0 0minus (11987731+ 11987732

100381610038161003816100381611990951003816100381610038161003816)

1198682

]]]]]]]]]]]]]]]]

]

B =

[[[[[[[

[

minus11987011198601199011

11987021198601199012

0

0

0

]]]]]]]

]

(26)



= 0 (119894 = 1 2 3) Since 119897119894(119894 = 1



1198941(119894 = 1 2 3) as

follows [32]

1198771198941= 120574

128120583119897119894

1205871198894

119867

(27)


Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

xi(t)xo(t)

n1Qi(t)n2Qo(t) IiRi

n1IoRon2



119871as

A119871=

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus11987711

1198681

0 0

1

1198682

minus1

1198682

0minus11987721

1198682

0

1

1198683

minus1

1198683

0 0minus11987731

1198682

]]]]]]]]]]]]]]]]

]

(28)


1198942= 0 (119894 = 1 2 3) into


MQ + CQ + KQ = F (119905) (29)


M =[[

[

1198681

1198682

1198683

]]

]

C =[[

[

11987711

11987721

11987731

]]

]

K =[[[

[

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

]]]

]

F (119905) =[[[

[

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

]]]

]

(30)


2

119899M10038161003816100381610038161003816

= 0 (31)


obtained as

119891119899=

120596119899

2120587= radic

(1198701+ 1198702)

120588(1198601

1198971

+1198602

1198972

+1198603

1198973

) (32)



119894and cross-




1198942(119894 = 1 2 119899

1) of the










119900of such a


119888119900=

(15 + 1374119897119900

119889119867Re

)

minus12119889119867Re119897119900

gt 50

(228 + 64119897119900

119889119867Re

)

minus12119889119867Re119897119900

lt 50

(33)


119900and



be expressed as

119868119900= 120573

120588119897119900

119860119900

1198771199001

= 120576120588119876119900

11988821199001198602119900

(34)


119900denotes the


defined as X119879 = (1199091 1199092 1199093 1199094) = (119875

1 1198752 119876119894 119876119900) the state


X = AX + B (35)




1= 2 119899

2= 0) HB4 (119899

1= 1 1198992= 1) HB5 (119899

1= 0 119899

2= 2)


whereA

=

[[[[[[[[[

[

0 0 minus1198701

minus1198701

0 0 1198702

1198702

1

119868119894

minus1

119868119894

minus (1198771198941+ 1198771198942

100381610038161003816100381611990931003816100381610038161003816)

119868119894

0

1

119868119900

minus1

119868119900

0minus (1198771199001+ 1198771199002

100381610038161003816100381611990941003816100381610038161003816)

119868119900

]]]]]]]]]

]

B119879 = (minus11987011198601199011119903 11987021198601199012119903 0 0)

(36)


119865119879 (119905) = 119870

119903119909119903 (119905) + 119861

119903119903 (119905) + (119860

11990121198752minus 11986011990111198751) (37)


119870lowast

119889(119904) =

119871 [119865 (119905)]

119871 [119909119903 (119905)]

= 119870119903+ 119861119903119904 + 1198602

11990121198702+ 1198602

1199011119870

minus [

11987221198852

1199041199042+1198722[(119885119894119904 + 1198991119870) (119885

119904119904 + 1198992119870) minus 119899

21198702] + 119872

21198992119870119885119904119904 minus 119872

21198991119870119885119904119904

(119885119904119904 + 1198992119870)2(119885119894119904 + 1198991119870) minus 119899

2(119885119904119904 + 1198992119870)1198702

]

(38)

where

119872 = 11986011990111198701+ 11986011990121198702

119885119894= 119868119894119904 + 1198771198941

119885119900= 119868119900119904 + 1198771199001

119870 = 1198701+ 1198702

(39)



as

119891119899=

1

2120587

radic(1198701+ 1198702)

120588(1198991119860119894

119897119894

+1198992119860119900

119897119900

) (40)





2= 0 and 119899



2= 1 and 119899



2= 2 and





119899




300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)






0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)


300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)



119894 the








119894and 119860

119894to change but



400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(1)(2)

(7)

(6)

(5)

(4)

(3)





(1)

(2)

(3)

















and 1198601199012







Outer sleeve

Outer sleeve

Inner sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

Qi(t)

IiRi1

Ri2

C1

C2

FT(t)

xi(t)

FT(t)

(a)

Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

Qi(t)

IiRi1

Ri2

C1

C2

FT(t)

xi(t)

FT(t)

IsRs1

Rs2

Qs(t)

xs(t)

(b)

Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

Qi(t)

IiRi1

Ri2

C1

C2

FT(t)

FT(t)

Qo(t) xo(t)

Io Ro1 Ro2

xi(t)

(c)

Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2

FT(t)

FT(t)

Io2 Io1 Ro11Ro12

Ro22 Ro21


(d)



2(119905) denote



119894and 119877

119900


119903(119905) = 119883








0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

14000

16000

18000

Forc

e (N

)

Displacement (mm)

HB1HB2

HB4HB5






119903is about 438340Nm













119894(119894 = 1 sim 4)









1198701(119860119894119909119894minus 1198601199011119909119903) = 1198751

1198702(1198601199012119909119903minus 119860119894119909119894) = 1198752

(1)


300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)0 20 40 60 80 100

M1

M2


(a)

0

20

40

60

80

100

Frequency (Hz)0 20 40 60 80 100

N1

N2

Loss

angl

e (∘)


(b)

0500

10001500200025003000350040004500500055006000

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 50 100 150 200 250 300

M1

M2M3

M4


(c)

0

20

40

60

80

100

120

140

160

Frequency (Hz)0 30 60 90 120 150 180 210 240 270 300

N3

N4

N5

N6

Loss

angl

e (∘)


(d)



Δ119875 = 1198751minus 1198752= 120588119897119894119894+ (120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 (2a)

120578119897=

64120583120574

1205881003816100381610038161003816119894

1003816100381610038161003816 119889119867

Re le 2300

(2b)

120585 = 120585in + 120585out + 120585119897

120585in =1

2 (1 minus 119860out119860 in)

120585out =1

2 (119860out119860 in minus 1)

(2c)









119865119879 (119905) = 119865

119879119903 (119905) + 119865119879ℎ (119905)

= (119870119903119909119903+ 119861119903119903) + (119860

11990121198752minus 11986011990111198751)

(3)




119894+

1

120588119897119894

(120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 +

1

120588119897119894

(1198702119860119894+ 1198701119860119894) 119909119894

=1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903)

(4)


120596119899= radic

1198702119860119894+ 1198701119860119894

120588119897119894

= 2120587119891119899 (5)






119894|119894asymp



119894+ 2120577120596

119899119894+ 1205962

119899119909119894=

1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903) (6)


120577 =1

2120596119899

(32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) (7)


= 119883119903119890119895120596119905 119909

119894= 119883119894119890119895(120596119905minus1205931) and 119865

119879=


119883119894

119883119903

119890minus1198951205931 [minus120596

2+ 1205962

119899+ (

32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) 119895120596]

=1

120588119897119894

(11987021198601199012

+ 11987011198601199011)

(8)


119867119891=

119883119894

119883119903

119890minus1198951205931

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

1

1 minus 1205822 + 119895 (119866120582120596119899+ 119873119883

1198941205822)

(9)

where

119866 =32120583120574

1205881198892

119867

119873 =4120585

3120587119897119894

120582 =120596

120596119899

(10)


119870lowast

119889(119895120596) = 119870

119903+ 119895120596119861

119903+ 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(11)


119870lowast

119889(119895120596) =

119865119879(119895120596)

119909119903(119895120596)

=

119865119879119903(119895120596) + 119865

119879119891(119895120596)

119909119903(119895120596)

= 119870lowast

119903+ 119870lowast

119891= 1198701015840+ 11989511987010158401015840

(12)

119870lowast

119903= 119870119903+ 119895120596119861

119903

119870lowast

119891= 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(13)

where 119870lowast


and 119870lowast





119891and119870

lowast

119891become

119867119891(120582rarr0)

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

(14)

When 1198601199012

asymp 1198601199011 then

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

asymp 0

119870119889(120582rarr0)

asymp 119870119903(120582rarr0)

(15)

As shown in (14) 119867119891(120582rarr0)









value of119870119889(120582rarr0)



= 48547Nmm 119870119903(119891=0)


119889(119891=0)and119870



119889of an HDB



119891and

119870lowast

119891become

119867119891(120582rarr1)

=119883119894

119883119903

119890minus1198951205931

= minus119895

11987021198601199012

+ 11987011198601199011

120596119899120588119897119894

1

(119866 + 120596119899119873119883119894)

(16)

119870lowast

119891(120582rarr1)

= 11987021198602

1199012+ 11987011198602

1199011

+ 119895

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

(17a)

119890minus1198951205931 = cos120593

1minus 119895 sin120593

1

119864 =

(11987021198601199012

+ 11987011198601199011)

120588119897119894

(17b)


119870lowast

119889(120582rarr1)= 119870lowast

119903+ 119870lowast

119891= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011

+ 119895(120596119861119903

+

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

)

(18)


= arctan(120596119862119903(1198702+ 1198701) + (119870

21198601199012

+ 11987011198601199011)2

120591119891

(1198702+ 1198701) (119870119903+ 11987021198602

1199012+ 11987011198602

1199011)

)

120591119891=

2120596119899

(119866 + radic1198662 + 4119873119883119903119864)

(19)


119897119894 and 119889



119891

and 120593(120582=1)



119891and119870

lowast

119891become

119867119891(120582rarrinfin)

997888rarr 0

119870lowast

119891(120582rarr1)= 11987021198602

1199012+ 11987011198602

1199011

(20)



119870lowast

119889(120582rarrinfin)= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011+ 119895120596119861

119903 (21)







1198701015840

(120582rarr1)= 1198701015840

(120582rarrinfin)asymp 119870119903+ 11987021198602

1199012+ 11987011198602

1199011 (22)


(120582rarr1)and 119870

1015840




119903+

11987021198602

1199012+ 11987011198602

1199011


5






(120582=1) Based on



5) and 119891


relationship

1198911198985

asymp 119891119899=

1

2120587radic1198702119860119894+ 1198701119860119894

120588119897119894

(23)


Frequency (Hz)0 20 40 60 80 100

M5

400

800

1200

1600

2000

2400

Stor

age s

tiffne

ss (N

mm

)



(a)

Frequency (Hz)0 20 40 60 80 100

N7

0

200

400

600

800

1000

1200

1400

Loss

stiff

ness

(Nm

m)




Frequency (Hz)0 20 40 60 80 100

300

600

900

1200

1500

1800

2100

2400

2700

Dyn

amic

stiff

ness

(Nm

m)



M1

(a)

Frequency (Hz)0 20 40 60 80 100

N1

0

10

20

30

40

50

60

Loss

angl

e (∘)



(b)





data of 1198911198985


1


1and119870

2can then be




1199011and 119860

1199012based on 119870

1015840




119903= 486 times 10

5 (Nm) 1198702= 231 times

1011 (Nm5) 119870

1= 208 times 10

11 (Nm5) 1198601199011

= 191 times

10minus3 (m2) and 119860

1199012= 172 times 10



Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

Q1(t) Q2(t) Q3(t)

I1R1 I3R3I2R2

xi(t)












1198941


obtained by

119868119894=120588119860119894

119897119894

1198771198941=

1205851198941

1198602

119894

1198771198942=

1205851198942

1198605

119894

(24)




1 1199092 1199093 1199094 1199095) = (119875

1 1198752 1198761 1198762 1198763)

X = AX + B (25)

where

A =

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus (11987711+ 11987712

100381610038161003816100381611990931003816100381610038161003816)

1198681

0 0

1

1198682

minus1

1198682

0minus (11987721+ 11987722

100381610038161003816100381611990941003816100381610038161003816)

1198682

0

1

1198683

minus1

1198683

0 0minus (11987731+ 11987732

100381610038161003816100381611990951003816100381610038161003816)

1198682

]]]]]]]]]]]]]]]]

]

B =

[[[[[[[

[

minus11987011198601199011

11987021198601199012

0

0

0

]]]]]]]

]

(26)



= 0 (119894 = 1 2 3) Since 119897119894(119894 = 1



1198941(119894 = 1 2 3) as

follows [32]

1198771198941= 120574

128120583119897119894

1205871198894

119867

(27)


Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

xi(t)xo(t)

n1Qi(t)n2Qo(t) IiRi

n1IoRon2



119871as

A119871=

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus11987711

1198681

0 0

1

1198682

minus1

1198682

0minus11987721

1198682

0

1

1198683

minus1

1198683

0 0minus11987731

1198682

]]]]]]]]]]]]]]]]

]

(28)


1198942= 0 (119894 = 1 2 3) into


MQ + CQ + KQ = F (119905) (29)


M =[[

[

1198681

1198682

1198683

]]

]

C =[[

[

11987711

11987721

11987731

]]

]

K =[[[

[

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

]]]

]

F (119905) =[[[

[

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

]]]

]

(30)


2

119899M10038161003816100381610038161003816

= 0 (31)


obtained as

119891119899=

120596119899

2120587= radic

(1198701+ 1198702)

120588(1198601

1198971

+1198602

1198972

+1198603

1198973

) (32)



119894and cross-




1198942(119894 = 1 2 119899

1) of the










119900of such a


119888119900=

(15 + 1374119897119900

119889119867Re

)

minus12119889119867Re119897119900

gt 50

(228 + 64119897119900

119889119867Re

)

minus12119889119867Re119897119900

lt 50

(33)


119900and



be expressed as

119868119900= 120573

120588119897119900

119860119900

1198771199001

= 120576120588119876119900

11988821199001198602119900

(34)


119900denotes the


defined as X119879 = (1199091 1199092 1199093 1199094) = (119875

1 1198752 119876119894 119876119900) the state


X = AX + B (35)




1= 2 119899

2= 0) HB4 (119899

1= 1 1198992= 1) HB5 (119899

1= 0 119899

2= 2)


whereA

=

[[[[[[[[[

[

0 0 minus1198701

minus1198701

0 0 1198702

1198702

1

119868119894

minus1

119868119894

minus (1198771198941+ 1198771198942

100381610038161003816100381611990931003816100381610038161003816)

119868119894

0

1

119868119900

minus1

119868119900

0minus (1198771199001+ 1198771199002

100381610038161003816100381611990941003816100381610038161003816)

119868119900

]]]]]]]]]

]

B119879 = (minus11987011198601199011119903 11987021198601199012119903 0 0)

(36)


119865119879 (119905) = 119870

119903119909119903 (119905) + 119861

119903119903 (119905) + (119860

11990121198752minus 11986011990111198751) (37)


119870lowast

119889(119904) =

119871 [119865 (119905)]

119871 [119909119903 (119905)]

= 119870119903+ 119861119903119904 + 1198602

11990121198702+ 1198602

1199011119870

minus [

11987221198852

1199041199042+1198722[(119885119894119904 + 1198991119870) (119885

119904119904 + 1198992119870) minus 119899

21198702] + 119872

21198992119870119885119904119904 minus 119872

21198991119870119885119904119904

(119885119904119904 + 1198992119870)2(119885119894119904 + 1198991119870) minus 119899

2(119885119904119904 + 1198992119870)1198702

]

(38)

where

119872 = 11986011990111198701+ 11986011990121198702

119885119894= 119868119894119904 + 1198771198941

119885119900= 119868119900119904 + 1198771199001

119870 = 1198701+ 1198702

(39)



as

119891119899=

1

2120587

radic(1198701+ 1198702)

120588(1198991119860119894

119897119894

+1198992119860119900

119897119900

) (40)





2= 0 and 119899



2= 1 and 119899



2= 2 and





119899




300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)






0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)


300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)



119894 the








119894and 119860

119894to change but



400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Outer sleeve

Outer sleeve

Inner sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

Qi(t)

IiRi1

Ri2

C1

C2

FT(t)

xi(t)

FT(t)

(a)

Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

Qi(t)

IiRi1

Ri2

C1

C2

FT(t)

xi(t)

FT(t)

IsRs1

Rs2

Qs(t)

xs(t)

(b)

Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

Qi(t)

IiRi1

Ri2

C1

C2

FT(t)

FT(t)

Qo(t) xo(t)

Io Ro1 Ro2

xi(t)

(c)

Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2

FT(t)

FT(t)

Io2 Io1 Ro11Ro12

Ro22 Ro21


(d)



2(119905) denote



119894and 119877

119900


119903(119905) = 119883








0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

14000

16000

18000

Forc

e (N

)

Displacement (mm)

HB1HB2

HB4HB5






119903is about 438340Nm













119894(119894 = 1 sim 4)









1198701(119860119894119909119894minus 1198601199011119909119903) = 1198751

1198702(1198601199012119909119903minus 119860119894119909119894) = 1198752

(1)


300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)0 20 40 60 80 100

M1

M2


(a)

0

20

40

60

80

100

Frequency (Hz)0 20 40 60 80 100

N1

N2

Loss

angl

e (∘)


(b)

0500

10001500200025003000350040004500500055006000

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 50 100 150 200 250 300

M1

M2M3

M4


(c)

0

20

40

60

80

100

120

140

160

Frequency (Hz)0 30 60 90 120 150 180 210 240 270 300

N3

N4

N5

N6

Loss

angl

e (∘)


(d)



Δ119875 = 1198751minus 1198752= 120588119897119894119894+ (120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 (2a)

120578119897=

64120583120574

1205881003816100381610038161003816119894

1003816100381610038161003816 119889119867

Re le 2300

(2b)

120585 = 120585in + 120585out + 120585119897

120585in =1

2 (1 minus 119860out119860 in)

120585out =1

2 (119860out119860 in minus 1)

(2c)









119865119879 (119905) = 119865

119879119903 (119905) + 119865119879ℎ (119905)

= (119870119903119909119903+ 119861119903119903) + (119860

11990121198752minus 11986011990111198751)

(3)




119894+

1

120588119897119894

(120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 +

1

120588119897119894

(1198702119860119894+ 1198701119860119894) 119909119894

=1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903)

(4)


120596119899= radic

1198702119860119894+ 1198701119860119894

120588119897119894

= 2120587119891119899 (5)






119894|119894asymp



119894+ 2120577120596

119899119894+ 1205962

119899119909119894=

1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903) (6)


120577 =1

2120596119899

(32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) (7)


= 119883119903119890119895120596119905 119909

119894= 119883119894119890119895(120596119905minus1205931) and 119865

119879=


119883119894

119883119903

119890minus1198951205931 [minus120596

2+ 1205962

119899+ (

32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) 119895120596]

=1

120588119897119894

(11987021198601199012

+ 11987011198601199011)

(8)


119867119891=

119883119894

119883119903

119890minus1198951205931

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

1

1 minus 1205822 + 119895 (119866120582120596119899+ 119873119883

1198941205822)

(9)

where

119866 =32120583120574

1205881198892

119867

119873 =4120585

3120587119897119894

120582 =120596

120596119899

(10)


119870lowast

119889(119895120596) = 119870

119903+ 119895120596119861

119903+ 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(11)


119870lowast

119889(119895120596) =

119865119879(119895120596)

119909119903(119895120596)

=

119865119879119903(119895120596) + 119865

119879119891(119895120596)

119909119903(119895120596)

= 119870lowast

119903+ 119870lowast

119891= 1198701015840+ 11989511987010158401015840

(12)

119870lowast

119903= 119870119903+ 119895120596119861

119903

119870lowast

119891= 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(13)

where 119870lowast


and 119870lowast





119891and119870

lowast

119891become

119867119891(120582rarr0)

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

(14)

When 1198601199012

asymp 1198601199011 then

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

asymp 0

119870119889(120582rarr0)

asymp 119870119903(120582rarr0)

(15)

As shown in (14) 119867119891(120582rarr0)









value of119870119889(120582rarr0)



= 48547Nmm 119870119903(119891=0)


119889(119891=0)and119870



119889of an HDB



119891and

119870lowast

119891become

119867119891(120582rarr1)

=119883119894

119883119903

119890minus1198951205931

= minus119895

11987021198601199012

+ 11987011198601199011

120596119899120588119897119894

1

(119866 + 120596119899119873119883119894)

(16)

119870lowast

119891(120582rarr1)

= 11987021198602

1199012+ 11987011198602

1199011

+ 119895

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

(17a)

119890minus1198951205931 = cos120593

1minus 119895 sin120593

1

119864 =

(11987021198601199012

+ 11987011198601199011)

120588119897119894

(17b)


119870lowast

119889(120582rarr1)= 119870lowast

119903+ 119870lowast

119891= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011

+ 119895(120596119861119903

+

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

)

(18)


= arctan(120596119862119903(1198702+ 1198701) + (119870

21198601199012

+ 11987011198601199011)2

120591119891

(1198702+ 1198701) (119870119903+ 11987021198602

1199012+ 11987011198602

1199011)

)

120591119891=

2120596119899

(119866 + radic1198662 + 4119873119883119903119864)

(19)


119897119894 and 119889



119891

and 120593(120582=1)



119891and119870

lowast

119891become

119867119891(120582rarrinfin)

997888rarr 0

119870lowast

119891(120582rarr1)= 11987021198602

1199012+ 11987011198602

1199011

(20)



119870lowast

119889(120582rarrinfin)= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011+ 119895120596119861

119903 (21)







1198701015840

(120582rarr1)= 1198701015840

(120582rarrinfin)asymp 119870119903+ 11987021198602

1199012+ 11987011198602

1199011 (22)


(120582rarr1)and 119870

1015840




119903+

11987021198602

1199012+ 11987011198602

1199011


5






(120582=1) Based on



5) and 119891


relationship

1198911198985

asymp 119891119899=

1

2120587radic1198702119860119894+ 1198701119860119894

120588119897119894

(23)


Frequency (Hz)0 20 40 60 80 100

M5

400

800

1200

1600

2000

2400

Stor

age s

tiffne

ss (N

mm

)



(a)

Frequency (Hz)0 20 40 60 80 100

N7

0

200

400

600

800

1000

1200

1400

Loss

stiff

ness

(Nm

m)




Frequency (Hz)0 20 40 60 80 100

300

600

900

1200

1500

1800

2100

2400

2700

Dyn

amic

stiff

ness

(Nm

m)



M1

(a)

Frequency (Hz)0 20 40 60 80 100

N1

0

10

20

30

40

50

60

Loss

angl

e (∘)



(b)





data of 1198911198985


1


1and119870

2can then be




1199011and 119860

1199012based on 119870

1015840




119903= 486 times 10

5 (Nm) 1198702= 231 times

1011 (Nm5) 119870

1= 208 times 10

11 (Nm5) 1198601199011

= 191 times

10minus3 (m2) and 119860

1199012= 172 times 10



Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

Q1(t) Q2(t) Q3(t)

I1R1 I3R3I2R2

xi(t)












1198941


obtained by

119868119894=120588119860119894

119897119894

1198771198941=

1205851198941

1198602

119894

1198771198942=

1205851198942

1198605

119894

(24)




1 1199092 1199093 1199094 1199095) = (119875

1 1198752 1198761 1198762 1198763)

X = AX + B (25)

where

A =

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus (11987711+ 11987712

100381610038161003816100381611990931003816100381610038161003816)

1198681

0 0

1

1198682

minus1

1198682

0minus (11987721+ 11987722

100381610038161003816100381611990941003816100381610038161003816)

1198682

0

1

1198683

minus1

1198683

0 0minus (11987731+ 11987732

100381610038161003816100381611990951003816100381610038161003816)

1198682

]]]]]]]]]]]]]]]]

]

B =

[[[[[[[

[

minus11987011198601199011

11987021198601199012

0

0

0

]]]]]]]

]

(26)



= 0 (119894 = 1 2 3) Since 119897119894(119894 = 1



1198941(119894 = 1 2 3) as

follows [32]

1198771198941= 120574

128120583119897119894

1205871198894

119867

(27)


Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

xi(t)xo(t)

n1Qi(t)n2Qo(t) IiRi

n1IoRon2



119871as

A119871=

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus11987711

1198681

0 0

1

1198682

minus1

1198682

0minus11987721

1198682

0

1

1198683

minus1

1198683

0 0minus11987731

1198682

]]]]]]]]]]]]]]]]

]

(28)


1198942= 0 (119894 = 1 2 3) into


MQ + CQ + KQ = F (119905) (29)


M =[[

[

1198681

1198682

1198683

]]

]

C =[[

[

11987711

11987721

11987731

]]

]

K =[[[

[

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

]]]

]

F (119905) =[[[

[

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

]]]

]

(30)


2

119899M10038161003816100381610038161003816

= 0 (31)


obtained as

119891119899=

120596119899

2120587= radic

(1198701+ 1198702)

120588(1198601

1198971

+1198602

1198972

+1198603

1198973

) (32)



119894and cross-




1198942(119894 = 1 2 119899

1) of the










119900of such a


119888119900=

(15 + 1374119897119900

119889119867Re

)

minus12119889119867Re119897119900

gt 50

(228 + 64119897119900

119889119867Re

)

minus12119889119867Re119897119900

lt 50

(33)


119900and



be expressed as

119868119900= 120573

120588119897119900

119860119900

1198771199001

= 120576120588119876119900

11988821199001198602119900

(34)


119900denotes the


defined as X119879 = (1199091 1199092 1199093 1199094) = (119875

1 1198752 119876119894 119876119900) the state


X = AX + B (35)




1= 2 119899

2= 0) HB4 (119899

1= 1 1198992= 1) HB5 (119899

1= 0 119899

2= 2)


whereA

=

[[[[[[[[[

[

0 0 minus1198701

minus1198701

0 0 1198702

1198702

1

119868119894

minus1

119868119894

minus (1198771198941+ 1198771198942

100381610038161003816100381611990931003816100381610038161003816)

119868119894

0

1

119868119900

minus1

119868119900

0minus (1198771199001+ 1198771199002

100381610038161003816100381611990941003816100381610038161003816)

119868119900

]]]]]]]]]

]

B119879 = (minus11987011198601199011119903 11987021198601199012119903 0 0)

(36)


119865119879 (119905) = 119870

119903119909119903 (119905) + 119861

119903119903 (119905) + (119860

11990121198752minus 11986011990111198751) (37)


119870lowast

119889(119904) =

119871 [119865 (119905)]

119871 [119909119903 (119905)]

= 119870119903+ 119861119903119904 + 1198602

11990121198702+ 1198602

1199011119870

minus [

11987221198852

1199041199042+1198722[(119885119894119904 + 1198991119870) (119885

119904119904 + 1198992119870) minus 119899

21198702] + 119872

21198992119870119885119904119904 minus 119872

21198991119870119885119904119904

(119885119904119904 + 1198992119870)2(119885119894119904 + 1198991119870) minus 119899

2(119885119904119904 + 1198992119870)1198702

]

(38)

where

119872 = 11986011990111198701+ 11986011990121198702

119885119894= 119868119894119904 + 1198771198941

119885119900= 119868119900119904 + 1198771199001

119870 = 1198701+ 1198702

(39)



as

119891119899=

1

2120587

radic(1198701+ 1198702)

120588(1198991119860119894

119897119894

+1198992119860119900

119897119900

) (40)





2= 0 and 119899



2= 1 and 119899



2= 2 and





119899




300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)






0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)


300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)



119894 the








119894and 119860

119894to change but



400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

14000

16000

18000

Forc

e (N

)

Displacement (mm)

HB1HB2

HB4HB5






119903is about 438340Nm













119894(119894 = 1 sim 4)









1198701(119860119894119909119894minus 1198601199011119909119903) = 1198751

1198702(1198601199012119909119903minus 119860119894119909119894) = 1198752

(1)


300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)0 20 40 60 80 100

M1

M2


(a)

0

20

40

60

80

100

Frequency (Hz)0 20 40 60 80 100

N1

N2

Loss

angl

e (∘)


(b)

0500

10001500200025003000350040004500500055006000

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 50 100 150 200 250 300

M1

M2M3

M4


(c)

0

20

40

60

80

100

120

140

160

Frequency (Hz)0 30 60 90 120 150 180 210 240 270 300

N3

N4

N5

N6

Loss

angl

e (∘)


(d)



Δ119875 = 1198751minus 1198752= 120588119897119894119894+ (120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 (2a)

120578119897=

64120583120574

1205881003816100381610038161003816119894

1003816100381610038161003816 119889119867

Re le 2300

(2b)

120585 = 120585in + 120585out + 120585119897

120585in =1

2 (1 minus 119860out119860 in)

120585out =1

2 (119860out119860 in minus 1)

(2c)









119865119879 (119905) = 119865

119879119903 (119905) + 119865119879ℎ (119905)

= (119870119903119909119903+ 119861119903119903) + (119860

11990121198752minus 11986011990111198751)

(3)




119894+

1

120588119897119894

(120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 +

1

120588119897119894

(1198702119860119894+ 1198701119860119894) 119909119894

=1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903)

(4)


120596119899= radic

1198702119860119894+ 1198701119860119894

120588119897119894

= 2120587119891119899 (5)






119894|119894asymp



119894+ 2120577120596

119899119894+ 1205962

119899119909119894=

1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903) (6)


120577 =1

2120596119899

(32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) (7)


= 119883119903119890119895120596119905 119909

119894= 119883119894119890119895(120596119905minus1205931) and 119865

119879=


119883119894

119883119903

119890minus1198951205931 [minus120596

2+ 1205962

119899+ (

32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) 119895120596]

=1

120588119897119894

(11987021198601199012

+ 11987011198601199011)

(8)


119867119891=

119883119894

119883119903

119890minus1198951205931

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

1

1 minus 1205822 + 119895 (119866120582120596119899+ 119873119883

1198941205822)

(9)

where

119866 =32120583120574

1205881198892

119867

119873 =4120585

3120587119897119894

120582 =120596

120596119899

(10)


119870lowast

119889(119895120596) = 119870

119903+ 119895120596119861

119903+ 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(11)


119870lowast

119889(119895120596) =

119865119879(119895120596)

119909119903(119895120596)

=

119865119879119903(119895120596) + 119865

119879119891(119895120596)

119909119903(119895120596)

= 119870lowast

119903+ 119870lowast

119891= 1198701015840+ 11989511987010158401015840

(12)

119870lowast

119903= 119870119903+ 119895120596119861

119903

119870lowast

119891= 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(13)

where 119870lowast


and 119870lowast





119891and119870

lowast

119891become

119867119891(120582rarr0)

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

(14)

When 1198601199012

asymp 1198601199011 then

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

asymp 0

119870119889(120582rarr0)

asymp 119870119903(120582rarr0)

(15)

As shown in (14) 119867119891(120582rarr0)









value of119870119889(120582rarr0)



= 48547Nmm 119870119903(119891=0)


119889(119891=0)and119870



119889of an HDB



119891and

119870lowast

119891become

119867119891(120582rarr1)

=119883119894

119883119903

119890minus1198951205931

= minus119895

11987021198601199012

+ 11987011198601199011

120596119899120588119897119894

1

(119866 + 120596119899119873119883119894)

(16)

119870lowast

119891(120582rarr1)

= 11987021198602

1199012+ 11987011198602

1199011

+ 119895

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

(17a)

119890minus1198951205931 = cos120593

1minus 119895 sin120593

1

119864 =

(11987021198601199012

+ 11987011198601199011)

120588119897119894

(17b)


119870lowast

119889(120582rarr1)= 119870lowast

119903+ 119870lowast

119891= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011

+ 119895(120596119861119903

+

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

)

(18)


= arctan(120596119862119903(1198702+ 1198701) + (119870

21198601199012

+ 11987011198601199011)2

120591119891

(1198702+ 1198701) (119870119903+ 11987021198602

1199012+ 11987011198602

1199011)

)

120591119891=

2120596119899

(119866 + radic1198662 + 4119873119883119903119864)

(19)


119897119894 and 119889



119891

and 120593(120582=1)



119891and119870

lowast

119891become

119867119891(120582rarrinfin)

997888rarr 0

119870lowast

119891(120582rarr1)= 11987021198602

1199012+ 11987011198602

1199011

(20)



119870lowast

119889(120582rarrinfin)= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011+ 119895120596119861

119903 (21)







1198701015840

(120582rarr1)= 1198701015840

(120582rarrinfin)asymp 119870119903+ 11987021198602

1199012+ 11987011198602

1199011 (22)


(120582rarr1)and 119870

1015840




119903+

11987021198602

1199012+ 11987011198602

1199011


5






(120582=1) Based on



5) and 119891


relationship

1198911198985

asymp 119891119899=

1

2120587radic1198702119860119894+ 1198701119860119894

120588119897119894

(23)


Frequency (Hz)0 20 40 60 80 100

M5

400

800

1200

1600

2000

2400

Stor

age s

tiffne

ss (N

mm

)



(a)

Frequency (Hz)0 20 40 60 80 100

N7

0

200

400

600

800

1000

1200

1400

Loss

stiff

ness

(Nm

m)




Frequency (Hz)0 20 40 60 80 100

300

600

900

1200

1500

1800

2100

2400

2700

Dyn

amic

stiff

ness

(Nm

m)



M1

(a)

Frequency (Hz)0 20 40 60 80 100

N1

0

10

20

30

40

50

60

Loss

angl

e (∘)



(b)





data of 1198911198985


1


1and119870

2can then be




1199011and 119860

1199012based on 119870

1015840




119903= 486 times 10

5 (Nm) 1198702= 231 times

1011 (Nm5) 119870

1= 208 times 10

11 (Nm5) 1198601199011

= 191 times

10minus3 (m2) and 119860

1199012= 172 times 10



Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

Q1(t) Q2(t) Q3(t)

I1R1 I3R3I2R2

xi(t)












1198941


obtained by

119868119894=120588119860119894

119897119894

1198771198941=

1205851198941

1198602

119894

1198771198942=

1205851198942

1198605

119894

(24)




1 1199092 1199093 1199094 1199095) = (119875

1 1198752 1198761 1198762 1198763)

X = AX + B (25)

where

A =

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus (11987711+ 11987712

100381610038161003816100381611990931003816100381610038161003816)

1198681

0 0

1

1198682

minus1

1198682

0minus (11987721+ 11987722

100381610038161003816100381611990941003816100381610038161003816)

1198682

0

1

1198683

minus1

1198683

0 0minus (11987731+ 11987732

100381610038161003816100381611990951003816100381610038161003816)

1198682

]]]]]]]]]]]]]]]]

]

B =

[[[[[[[

[

minus11987011198601199011

11987021198601199012

0

0

0

]]]]]]]

]

(26)



= 0 (119894 = 1 2 3) Since 119897119894(119894 = 1



1198941(119894 = 1 2 3) as

follows [32]

1198771198941= 120574

128120583119897119894

1205871198894

119867

(27)


Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

xi(t)xo(t)

n1Qi(t)n2Qo(t) IiRi

n1IoRon2



119871as

A119871=

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus11987711

1198681

0 0

1

1198682

minus1

1198682

0minus11987721

1198682

0

1

1198683

minus1

1198683

0 0minus11987731

1198682

]]]]]]]]]]]]]]]]

]

(28)


1198942= 0 (119894 = 1 2 3) into


MQ + CQ + KQ = F (119905) (29)


M =[[

[

1198681

1198682

1198683

]]

]

C =[[

[

11987711

11987721

11987731

]]

]

K =[[[

[

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

]]]

]

F (119905) =[[[

[

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

]]]

]

(30)


2

119899M10038161003816100381610038161003816

= 0 (31)


obtained as

119891119899=

120596119899

2120587= radic

(1198701+ 1198702)

120588(1198601

1198971

+1198602

1198972

+1198603

1198973

) (32)



119894and cross-




1198942(119894 = 1 2 119899

1) of the










119900of such a


119888119900=

(15 + 1374119897119900

119889119867Re

)

minus12119889119867Re119897119900

gt 50

(228 + 64119897119900

119889119867Re

)

minus12119889119867Re119897119900

lt 50

(33)


119900and



be expressed as

119868119900= 120573

120588119897119900

119860119900

1198771199001

= 120576120588119876119900

11988821199001198602119900

(34)


119900denotes the


defined as X119879 = (1199091 1199092 1199093 1199094) = (119875

1 1198752 119876119894 119876119900) the state


X = AX + B (35)




1= 2 119899

2= 0) HB4 (119899

1= 1 1198992= 1) HB5 (119899

1= 0 119899

2= 2)


whereA

=

[[[[[[[[[

[

0 0 minus1198701

minus1198701

0 0 1198702

1198702

1

119868119894

minus1

119868119894

minus (1198771198941+ 1198771198942

100381610038161003816100381611990931003816100381610038161003816)

119868119894

0

1

119868119900

minus1

119868119900

0minus (1198771199001+ 1198771199002

100381610038161003816100381611990941003816100381610038161003816)

119868119900

]]]]]]]]]

]

B119879 = (minus11987011198601199011119903 11987021198601199012119903 0 0)

(36)


119865119879 (119905) = 119870

119903119909119903 (119905) + 119861

119903119903 (119905) + (119860

11990121198752minus 11986011990111198751) (37)


119870lowast

119889(119904) =

119871 [119865 (119905)]

119871 [119909119903 (119905)]

= 119870119903+ 119861119903119904 + 1198602

11990121198702+ 1198602

1199011119870

minus [

11987221198852

1199041199042+1198722[(119885119894119904 + 1198991119870) (119885

119904119904 + 1198992119870) minus 119899

21198702] + 119872

21198992119870119885119904119904 minus 119872

21198991119870119885119904119904

(119885119904119904 + 1198992119870)2(119885119894119904 + 1198991119870) minus 119899

2(119885119904119904 + 1198992119870)1198702

]

(38)

where

119872 = 11986011990111198701+ 11986011990121198702

119885119894= 119868119894119904 + 1198771198941

119885119900= 119868119900119904 + 1198771199001

119870 = 1198701+ 1198702

(39)



as

119891119899=

1

2120587

radic(1198701+ 1198702)

120588(1198991119860119894

119897119894

+1198992119860119900

119897119900

) (40)





2= 0 and 119899



2= 1 and 119899



2= 2 and





119899




300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)






0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)


300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)



119894 the








119894and 119860

119894to change but



400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)0 20 40 60 80 100

M1

M2


(a)

0

20

40

60

80

100

Frequency (Hz)0 20 40 60 80 100

N1

N2

Loss

angl

e (∘)


(b)

0500

10001500200025003000350040004500500055006000

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 50 100 150 200 250 300

M1

M2M3

M4


(c)

0

20

40

60

80

100

120

140

160

Frequency (Hz)0 30 60 90 120 150 180 210 240 270 300

N3

N4

N5

N6

Loss

angl

e (∘)


(d)



Δ119875 = 1198751minus 1198752= 120588119897119894119894+ (120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 (2a)

120578119897=

64120583120574

1205881003816100381610038161003816119894

1003816100381610038161003816 119889119867

Re le 2300

(2b)

120585 = 120585in + 120585out + 120585119897

120585in =1

2 (1 minus 119860out119860 in)

120585out =1

2 (119860out119860 in minus 1)

(2c)









119865119879 (119905) = 119865

119879119903 (119905) + 119865119879ℎ (119905)

= (119870119903119909119903+ 119861119903119903) + (119860

11990121198752minus 11986011990111198751)

(3)




119894+

1

120588119897119894

(120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 +

1

120588119897119894

(1198702119860119894+ 1198701119860119894) 119909119894

=1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903)

(4)


120596119899= radic

1198702119860119894+ 1198701119860119894

120588119897119894

= 2120587119891119899 (5)






119894|119894asymp



119894+ 2120577120596

119899119894+ 1205962

119899119909119894=

1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903) (6)


120577 =1

2120596119899

(32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) (7)


= 119883119903119890119895120596119905 119909

119894= 119883119894119890119895(120596119905minus1205931) and 119865

119879=


119883119894

119883119903

119890minus1198951205931 [minus120596

2+ 1205962

119899+ (

32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) 119895120596]

=1

120588119897119894

(11987021198601199012

+ 11987011198601199011)

(8)


119867119891=

119883119894

119883119903

119890minus1198951205931

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

1

1 minus 1205822 + 119895 (119866120582120596119899+ 119873119883

1198941205822)

(9)

where

119866 =32120583120574

1205881198892

119867

119873 =4120585

3120587119897119894

120582 =120596

120596119899

(10)


119870lowast

119889(119895120596) = 119870

119903+ 119895120596119861

119903+ 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(11)


119870lowast

119889(119895120596) =

119865119879(119895120596)

119909119903(119895120596)

=

119865119879119903(119895120596) + 119865

119879119891(119895120596)

119909119903(119895120596)

= 119870lowast

119903+ 119870lowast

119891= 1198701015840+ 11989511987010158401015840

(12)

119870lowast

119903= 119870119903+ 119895120596119861

119903

119870lowast

119891= 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(13)

where 119870lowast


and 119870lowast





119891and119870

lowast

119891become

119867119891(120582rarr0)

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

(14)

When 1198601199012

asymp 1198601199011 then

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

asymp 0

119870119889(120582rarr0)

asymp 119870119903(120582rarr0)

(15)

As shown in (14) 119867119891(120582rarr0)









value of119870119889(120582rarr0)



= 48547Nmm 119870119903(119891=0)


119889(119891=0)and119870



119889of an HDB



119891and

119870lowast

119891become

119867119891(120582rarr1)

=119883119894

119883119903

119890minus1198951205931

= minus119895

11987021198601199012

+ 11987011198601199011

120596119899120588119897119894

1

(119866 + 120596119899119873119883119894)

(16)

119870lowast

119891(120582rarr1)

= 11987021198602

1199012+ 11987011198602

1199011

+ 119895

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

(17a)

119890minus1198951205931 = cos120593

1minus 119895 sin120593

1

119864 =

(11987021198601199012

+ 11987011198601199011)

120588119897119894

(17b)


119870lowast

119889(120582rarr1)= 119870lowast

119903+ 119870lowast

119891= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011

+ 119895(120596119861119903

+

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

)

(18)


= arctan(120596119862119903(1198702+ 1198701) + (119870

21198601199012

+ 11987011198601199011)2

120591119891

(1198702+ 1198701) (119870119903+ 11987021198602

1199012+ 11987011198602

1199011)

)

120591119891=

2120596119899

(119866 + radic1198662 + 4119873119883119903119864)

(19)


119897119894 and 119889



119891

and 120593(120582=1)



119891and119870

lowast

119891become

119867119891(120582rarrinfin)

997888rarr 0

119870lowast

119891(120582rarr1)= 11987021198602

1199012+ 11987011198602

1199011

(20)



119870lowast

119889(120582rarrinfin)= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011+ 119895120596119861

119903 (21)







1198701015840

(120582rarr1)= 1198701015840

(120582rarrinfin)asymp 119870119903+ 11987021198602

1199012+ 11987011198602

1199011 (22)


(120582rarr1)and 119870

1015840




119903+

11987021198602

1199012+ 11987011198602

1199011


5






(120582=1) Based on



5) and 119891


relationship

1198911198985

asymp 119891119899=

1

2120587radic1198702119860119894+ 1198701119860119894

120588119897119894

(23)


Frequency (Hz)0 20 40 60 80 100

M5

400

800

1200

1600

2000

2400

Stor

age s

tiffne

ss (N

mm

)



(a)

Frequency (Hz)0 20 40 60 80 100

N7

0

200

400

600

800

1000

1200

1400

Loss

stiff

ness

(Nm

m)




Frequency (Hz)0 20 40 60 80 100

300

600

900

1200

1500

1800

2100

2400

2700

Dyn

amic

stiff

ness

(Nm

m)



M1

(a)

Frequency (Hz)0 20 40 60 80 100

N1

0

10

20

30

40

50

60

Loss

angl

e (∘)



(b)





data of 1198911198985


1


1and119870

2can then be




1199011and 119860

1199012based on 119870

1015840




119903= 486 times 10

5 (Nm) 1198702= 231 times

1011 (Nm5) 119870

1= 208 times 10

11 (Nm5) 1198601199011

= 191 times

10minus3 (m2) and 119860

1199012= 172 times 10



Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

Q1(t) Q2(t) Q3(t)

I1R1 I3R3I2R2

xi(t)












1198941


obtained by

119868119894=120588119860119894

119897119894

1198771198941=

1205851198941

1198602

119894

1198771198942=

1205851198942

1198605

119894

(24)




1 1199092 1199093 1199094 1199095) = (119875

1 1198752 1198761 1198762 1198763)

X = AX + B (25)

where

A =

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus (11987711+ 11987712

100381610038161003816100381611990931003816100381610038161003816)

1198681

0 0

1

1198682

minus1

1198682

0minus (11987721+ 11987722

100381610038161003816100381611990941003816100381610038161003816)

1198682

0

1

1198683

minus1

1198683

0 0minus (11987731+ 11987732

100381610038161003816100381611990951003816100381610038161003816)

1198682

]]]]]]]]]]]]]]]]

]

B =

[[[[[[[

[

minus11987011198601199011

11987021198601199012

0

0

0

]]]]]]]

]

(26)



= 0 (119894 = 1 2 3) Since 119897119894(119894 = 1



1198941(119894 = 1 2 3) as

follows [32]

1198771198941= 120574

128120583119897119894

1205871198894

119867

(27)


Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

xi(t)xo(t)

n1Qi(t)n2Qo(t) IiRi

n1IoRon2



119871as

A119871=

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus11987711

1198681

0 0

1

1198682

minus1

1198682

0minus11987721

1198682

0

1

1198683

minus1

1198683

0 0minus11987731

1198682

]]]]]]]]]]]]]]]]

]

(28)


1198942= 0 (119894 = 1 2 3) into


MQ + CQ + KQ = F (119905) (29)


M =[[

[

1198681

1198682

1198683

]]

]

C =[[

[

11987711

11987721

11987731

]]

]

K =[[[

[

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

]]]

]

F (119905) =[[[

[

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

]]]

]

(30)


2

119899M10038161003816100381610038161003816

= 0 (31)


obtained as

119891119899=

120596119899

2120587= radic

(1198701+ 1198702)

120588(1198601

1198971

+1198602

1198972

+1198603

1198973

) (32)



119894and cross-




1198942(119894 = 1 2 119899

1) of the










119900of such a


119888119900=

(15 + 1374119897119900

119889119867Re

)

minus12119889119867Re119897119900

gt 50

(228 + 64119897119900

119889119867Re

)

minus12119889119867Re119897119900

lt 50

(33)


119900and



be expressed as

119868119900= 120573

120588119897119900

119860119900

1198771199001

= 120576120588119876119900

11988821199001198602119900

(34)


119900denotes the


defined as X119879 = (1199091 1199092 1199093 1199094) = (119875

1 1198752 119876119894 119876119900) the state


X = AX + B (35)




1= 2 119899

2= 0) HB4 (119899

1= 1 1198992= 1) HB5 (119899

1= 0 119899

2= 2)


whereA

=

[[[[[[[[[

[

0 0 minus1198701

minus1198701

0 0 1198702

1198702

1

119868119894

minus1

119868119894

minus (1198771198941+ 1198771198942

100381610038161003816100381611990931003816100381610038161003816)

119868119894

0

1

119868119900

minus1

119868119900

0minus (1198771199001+ 1198771199002

100381610038161003816100381611990941003816100381610038161003816)

119868119900

]]]]]]]]]

]

B119879 = (minus11987011198601199011119903 11987021198601199012119903 0 0)

(36)


119865119879 (119905) = 119870

119903119909119903 (119905) + 119861

119903119903 (119905) + (119860

11990121198752minus 11986011990111198751) (37)


119870lowast

119889(119904) =

119871 [119865 (119905)]

119871 [119909119903 (119905)]

= 119870119903+ 119861119903119904 + 1198602

11990121198702+ 1198602

1199011119870

minus [

11987221198852

1199041199042+1198722[(119885119894119904 + 1198991119870) (119885

119904119904 + 1198992119870) minus 119899

21198702] + 119872

21198992119870119885119904119904 minus 119872

21198991119870119885119904119904

(119885119904119904 + 1198992119870)2(119885119894119904 + 1198991119870) minus 119899

2(119885119904119904 + 1198992119870)1198702

]

(38)

where

119872 = 11986011990111198701+ 11986011990121198702

119885119894= 119868119894119904 + 1198771198941

119885119900= 119868119900119904 + 1198771199001

119870 = 1198701+ 1198702

(39)



as

119891119899=

1

2120587

radic(1198701+ 1198702)

120588(1198991119860119894

119897119894

+1198992119860119900

119897119900

) (40)





2= 0 and 119899



2= 1 and 119899



2= 2 and





119899




300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)






0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)


300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)



119894 the








119894and 119860

119894to change but



400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119865119879 (119905) = 119865

119879119903 (119905) + 119865119879ℎ (119905)

= (119870119903119909119903+ 119861119903119903) + (119860

11990121198752minus 11986011990111198751)

(3)




119894+

1

120588119897119894

(120578119897

119897119894

119889119867

+ 120585)120588

2

10038161003816100381610038161198941003816100381610038161003816 119894 +

1

120588119897119894

(1198702119860119894+ 1198701119860119894) 119909119894

=1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903)

(4)


120596119899= radic

1198702119860119894+ 1198701119860119894

120588119897119894

= 2120587119891119899 (5)






119894|119894asymp



119894+ 2120577120596

119899119894+ 1205962

119899119909119894=

1

120588119897119894

(11987021198601199012119909119903+ 11987011198601199011119909119903) (6)


120577 =1

2120596119899

(32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) (7)


= 119883119903119890119895120596119905 119909

119894= 119883119894119890119895(120596119905minus1205931) and 119865

119879=


119883119894

119883119903

119890minus1198951205931 [minus120596

2+ 1205962

119899+ (

32120583120574

1205881198892

119867

+4120585

3120587119897119894

120596119883119894) 119895120596]

=1

120588119897119894

(11987021198601199012

+ 11987011198601199011)

(8)


119867119891=

119883119894

119883119903

119890minus1198951205931

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

1

1 minus 1205822 + 119895 (119866120582120596119899+ 119873119883

1198941205822)

(9)

where

119866 =32120583120574

1205881198892

119867

119873 =4120585

3120587119897119894

120582 =120596

120596119899

(10)


119870lowast

119889(119895120596) = 119870

119903+ 119895120596119861

119903+ 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(11)


119870lowast

119889(119895120596) =

119865119879(119895120596)

119909119903(119895120596)

=

119865119879119903(119895120596) + 119865

119879119891(119895120596)

119909119903(119895120596)

= 119870lowast

119903+ 119870lowast

119891= 1198701015840+ 11989511987010158401015840

(12)

119870lowast

119903= 119870119903+ 119895120596119861

119903

119870lowast

119891= 11987021198602

1199012+ 11987011198602

1199011

minus (11987021198601199012119860119894+ 11987011198601199011119860119894)119867119891

(13)

where 119870lowast


and 119870lowast





119891and119870

lowast

119891become

119867119891(120582rarr0)

=

11987021198601199012

+ 11987011198601199011

1198702119860119894+ 1198701119860119894

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

(14)

When 1198601199012

asymp 1198601199011 then

119870lowast

119891(120582rarr0)=

11987021198701(1198601199012

minus 1198601199011)2

1198702+ 1198701

asymp 0

119870119889(120582rarr0)

asymp 119870119903(120582rarr0)

(15)

As shown in (14) 119867119891(120582rarr0)









value of119870119889(120582rarr0)



= 48547Nmm 119870119903(119891=0)


119889(119891=0)and119870



119889of an HDB



119891and

119870lowast

119891become

119867119891(120582rarr1)

=119883119894

119883119903

119890minus1198951205931

= minus119895

11987021198601199012

+ 11987011198601199011

120596119899120588119897119894

1

(119866 + 120596119899119873119883119894)

(16)

119870lowast

119891(120582rarr1)

= 11987021198602

1199012+ 11987011198602

1199011

+ 119895

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

(17a)

119890minus1198951205931 = cos120593

1minus 119895 sin120593

1

119864 =

(11987021198601199012

+ 11987011198601199011)

120588119897119894

(17b)


119870lowast

119889(120582rarr1)= 119870lowast

119903+ 119870lowast

119891= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011

+ 119895(120596119861119903

+

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

)

(18)


= arctan(120596119862119903(1198702+ 1198701) + (119870

21198601199012

+ 11987011198601199011)2

120591119891

(1198702+ 1198701) (119870119903+ 11987021198602

1199012+ 11987011198602

1199011)

)

120591119891=

2120596119899

(119866 + radic1198662 + 4119873119883119903119864)

(19)


119897119894 and 119889



119891

and 120593(120582=1)



119891and119870

lowast

119891become

119867119891(120582rarrinfin)

997888rarr 0

119870lowast

119891(120582rarr1)= 11987021198602

1199012+ 11987011198602

1199011

(20)



119870lowast

119889(120582rarrinfin)= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011+ 119895120596119861

119903 (21)







1198701015840

(120582rarr1)= 1198701015840

(120582rarrinfin)asymp 119870119903+ 11987021198602

1199012+ 11987011198602

1199011 (22)


(120582rarr1)and 119870

1015840




119903+

11987021198602

1199012+ 11987011198602

1199011


5






(120582=1) Based on



5) and 119891


relationship

1198911198985

asymp 119891119899=

1

2120587radic1198702119860119894+ 1198701119860119894

120588119897119894

(23)


Frequency (Hz)0 20 40 60 80 100

M5

400

800

1200

1600

2000

2400

Stor

age s

tiffne

ss (N

mm

)



(a)

Frequency (Hz)0 20 40 60 80 100

N7

0

200

400

600

800

1000

1200

1400

Loss

stiff

ness

(Nm

m)




Frequency (Hz)0 20 40 60 80 100

300

600

900

1200

1500

1800

2100

2400

2700

Dyn

amic

stiff

ness

(Nm

m)



M1

(a)

Frequency (Hz)0 20 40 60 80 100

N1

0

10

20

30

40

50

60

Loss

angl

e (∘)



(b)





data of 1198911198985


1


1and119870

2can then be




1199011and 119860

1199012based on 119870

1015840




119903= 486 times 10

5 (Nm) 1198702= 231 times

1011 (Nm5) 119870

1= 208 times 10

11 (Nm5) 1198601199011

= 191 times

10minus3 (m2) and 119860

1199012= 172 times 10



Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

Q1(t) Q2(t) Q3(t)

I1R1 I3R3I2R2

xi(t)












1198941


obtained by

119868119894=120588119860119894

119897119894

1198771198941=

1205851198941

1198602

119894

1198771198942=

1205851198942

1198605

119894

(24)




1 1199092 1199093 1199094 1199095) = (119875

1 1198752 1198761 1198762 1198763)

X = AX + B (25)

where

A =

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus (11987711+ 11987712

100381610038161003816100381611990931003816100381610038161003816)

1198681

0 0

1

1198682

minus1

1198682

0minus (11987721+ 11987722

100381610038161003816100381611990941003816100381610038161003816)

1198682

0

1

1198683

minus1

1198683

0 0minus (11987731+ 11987732

100381610038161003816100381611990951003816100381610038161003816)

1198682

]]]]]]]]]]]]]]]]

]

B =

[[[[[[[

[

minus11987011198601199011

11987021198601199012

0

0

0

]]]]]]]

]

(26)



= 0 (119894 = 1 2 3) Since 119897119894(119894 = 1



1198941(119894 = 1 2 3) as

follows [32]

1198771198941= 120574

128120583119897119894

1205871198894

119867

(27)


Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

xi(t)xo(t)

n1Qi(t)n2Qo(t) IiRi

n1IoRon2



119871as

A119871=

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus11987711

1198681

0 0

1

1198682

minus1

1198682

0minus11987721

1198682

0

1

1198683

minus1

1198683

0 0minus11987731

1198682

]]]]]]]]]]]]]]]]

]

(28)


1198942= 0 (119894 = 1 2 3) into


MQ + CQ + KQ = F (119905) (29)


M =[[

[

1198681

1198682

1198683

]]

]

C =[[

[

11987711

11987721

11987731

]]

]

K =[[[

[

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

]]]

]

F (119905) =[[[

[

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

]]]

]

(30)


2

119899M10038161003816100381610038161003816

= 0 (31)


obtained as

119891119899=

120596119899

2120587= radic

(1198701+ 1198702)

120588(1198601

1198971

+1198602

1198972

+1198603

1198973

) (32)



119894and cross-




1198942(119894 = 1 2 119899

1) of the










119900of such a


119888119900=

(15 + 1374119897119900

119889119867Re

)

minus12119889119867Re119897119900

gt 50

(228 + 64119897119900

119889119867Re

)

minus12119889119867Re119897119900

lt 50

(33)


119900and



be expressed as

119868119900= 120573

120588119897119900

119860119900

1198771199001

= 120576120588119876119900

11988821199001198602119900

(34)


119900denotes the


defined as X119879 = (1199091 1199092 1199093 1199094) = (119875

1 1198752 119876119894 119876119900) the state


X = AX + B (35)




1= 2 119899

2= 0) HB4 (119899

1= 1 1198992= 1) HB5 (119899

1= 0 119899

2= 2)


whereA

=

[[[[[[[[[

[

0 0 minus1198701

minus1198701

0 0 1198702

1198702

1

119868119894

minus1

119868119894

minus (1198771198941+ 1198771198942

100381610038161003816100381611990931003816100381610038161003816)

119868119894

0

1

119868119900

minus1

119868119900

0minus (1198771199001+ 1198771199002

100381610038161003816100381611990941003816100381610038161003816)

119868119900

]]]]]]]]]

]

B119879 = (minus11987011198601199011119903 11987021198601199012119903 0 0)

(36)


119865119879 (119905) = 119870

119903119909119903 (119905) + 119861

119903119903 (119905) + (119860

11990121198752minus 11986011990111198751) (37)


119870lowast

119889(119904) =

119871 [119865 (119905)]

119871 [119909119903 (119905)]

= 119870119903+ 119861119903119904 + 1198602

11990121198702+ 1198602

1199011119870

minus [

11987221198852

1199041199042+1198722[(119885119894119904 + 1198991119870) (119885

119904119904 + 1198992119870) minus 119899

21198702] + 119872

21198992119870119885119904119904 minus 119872

21198991119870119885119904119904

(119885119904119904 + 1198992119870)2(119885119894119904 + 1198991119870) minus 119899

2(119885119904119904 + 1198992119870)1198702

]

(38)

where

119872 = 11986011990111198701+ 11986011990121198702

119885119894= 119868119894119904 + 1198771198941

119885119900= 119868119900119904 + 1198771199001

119870 = 1198701+ 1198702

(39)



as

119891119899=

1

2120587

radic(1198701+ 1198702)

120588(1198991119860119894

119897119894

+1198992119860119900

119897119900

) (40)





2= 0 and 119899



2= 1 and 119899



2= 2 and





119899




300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)






0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)


300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)



119894 the








119894and 119860

119894to change but



400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













value of119870119889(120582rarr0)



= 48547Nmm 119870119903(119891=0)


119889(119891=0)and119870



119889of an HDB



119891and

119870lowast

119891become

119867119891(120582rarr1)

=119883119894

119883119903

119890minus1198951205931

= minus119895

11987021198601199012

+ 11987011198601199011

120596119899120588119897119894

1

(119866 + 120596119899119873119883119894)

(16)

119870lowast

119891(120582rarr1)

= 11987021198602

1199012+ 11987011198602

1199011

+ 119895

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

(17a)

119890minus1198951205931 = cos120593

1minus 119895 sin120593

1

119864 =

(11987021198601199012

+ 11987011198601199011)

120588119897119894

(17b)


119870lowast

119889(120582rarr1)= 119870lowast

119903+ 119870lowast

119891= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011

+ 119895(120596119861119903

+

(11987021198601199012

+ 11987011198601199011)2

(1198702+ 1198701)

2120596119899

119866 + radic1198662 + 4119873119883119903119864

)

(18)


= arctan(120596119862119903(1198702+ 1198701) + (119870

21198601199012

+ 11987011198601199011)2

120591119891

(1198702+ 1198701) (119870119903+ 11987021198602

1199012+ 11987011198602

1199011)

)

120591119891=

2120596119899

(119866 + radic1198662 + 4119873119883119903119864)

(19)


119897119894 and 119889



119891

and 120593(120582=1)



119891and119870

lowast

119891become

119867119891(120582rarrinfin)

997888rarr 0

119870lowast

119891(120582rarr1)= 11987021198602

1199012+ 11987011198602

1199011

(20)



119870lowast

119889(120582rarrinfin)= 119870119903+ 11987021198602

1199012+ 11987011198602

1199011+ 119895120596119861

119903 (21)







1198701015840

(120582rarr1)= 1198701015840

(120582rarrinfin)asymp 119870119903+ 11987021198602

1199012+ 11987011198602

1199011 (22)


(120582rarr1)and 119870

1015840




119903+

11987021198602

1199012+ 11987011198602

1199011


5






(120582=1) Based on



5) and 119891


relationship

1198911198985

asymp 119891119899=

1

2120587radic1198702119860119894+ 1198701119860119894

120588119897119894

(23)


Frequency (Hz)0 20 40 60 80 100

M5

400

800

1200

1600

2000

2400

Stor

age s

tiffne

ss (N

mm

)



(a)

Frequency (Hz)0 20 40 60 80 100

N7

0

200

400

600

800

1000

1200

1400

Loss

stiff

ness

(Nm

m)




Frequency (Hz)0 20 40 60 80 100

300

600

900

1200

1500

1800

2100

2400

2700

Dyn

amic

stiff

ness

(Nm

m)



M1

(a)

Frequency (Hz)0 20 40 60 80 100

N1

0

10

20

30

40

50

60

Loss

angl

e (∘)



(b)





data of 1198911198985


1


1and119870

2can then be




1199011and 119860

1199012based on 119870

1015840




119903= 486 times 10

5 (Nm) 1198702= 231 times

1011 (Nm5) 119870

1= 208 times 10

11 (Nm5) 1198601199011

= 191 times

10minus3 (m2) and 119860

1199012= 172 times 10



Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

Q1(t) Q2(t) Q3(t)

I1R1 I3R3I2R2

xi(t)












1198941


obtained by

119868119894=120588119860119894

119897119894

1198771198941=

1205851198941

1198602

119894

1198771198942=

1205851198942

1198605

119894

(24)




1 1199092 1199093 1199094 1199095) = (119875

1 1198752 1198761 1198762 1198763)

X = AX + B (25)

where

A =

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus (11987711+ 11987712

100381610038161003816100381611990931003816100381610038161003816)

1198681

0 0

1

1198682

minus1

1198682

0minus (11987721+ 11987722

100381610038161003816100381611990941003816100381610038161003816)

1198682

0

1

1198683

minus1

1198683

0 0minus (11987731+ 11987732

100381610038161003816100381611990951003816100381610038161003816)

1198682

]]]]]]]]]]]]]]]]

]

B =

[[[[[[[

[

minus11987011198601199011

11987021198601199012

0

0

0

]]]]]]]

]

(26)



= 0 (119894 = 1 2 3) Since 119897119894(119894 = 1



1198941(119894 = 1 2 3) as

follows [32]

1198771198941= 120574

128120583119897119894

1205871198894

119867

(27)


Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

xi(t)xo(t)

n1Qi(t)n2Qo(t) IiRi

n1IoRon2



119871as

A119871=

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus11987711

1198681

0 0

1

1198682

minus1

1198682

0minus11987721

1198682

0

1

1198683

minus1

1198683

0 0minus11987731

1198682

]]]]]]]]]]]]]]]]

]

(28)


1198942= 0 (119894 = 1 2 3) into


MQ + CQ + KQ = F (119905) (29)


M =[[

[

1198681

1198682

1198683

]]

]

C =[[

[

11987711

11987721

11987731

]]

]

K =[[[

[

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

]]]

]

F (119905) =[[[

[

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

]]]

]

(30)


2

119899M10038161003816100381610038161003816

= 0 (31)


obtained as

119891119899=

120596119899

2120587= radic

(1198701+ 1198702)

120588(1198601

1198971

+1198602

1198972

+1198603

1198973

) (32)



119894and cross-




1198942(119894 = 1 2 119899

1) of the










119900of such a


119888119900=

(15 + 1374119897119900

119889119867Re

)

minus12119889119867Re119897119900

gt 50

(228 + 64119897119900

119889119867Re

)

minus12119889119867Re119897119900

lt 50

(33)


119900and



be expressed as

119868119900= 120573

120588119897119900

119860119900

1198771199001

= 120576120588119876119900

11988821199001198602119900

(34)


119900denotes the


defined as X119879 = (1199091 1199092 1199093 1199094) = (119875

1 1198752 119876119894 119876119900) the state


X = AX + B (35)




1= 2 119899

2= 0) HB4 (119899

1= 1 1198992= 1) HB5 (119899

1= 0 119899

2= 2)


whereA

=

[[[[[[[[[

[

0 0 minus1198701

minus1198701

0 0 1198702

1198702

1

119868119894

minus1

119868119894

minus (1198771198941+ 1198771198942

100381610038161003816100381611990931003816100381610038161003816)

119868119894

0

1

119868119900

minus1

119868119900

0minus (1198771199001+ 1198771199002

100381610038161003816100381611990941003816100381610038161003816)

119868119900

]]]]]]]]]

]

B119879 = (minus11987011198601199011119903 11987021198601199012119903 0 0)

(36)


119865119879 (119905) = 119870

119903119909119903 (119905) + 119861

119903119903 (119905) + (119860

11990121198752minus 11986011990111198751) (37)


119870lowast

119889(119904) =

119871 [119865 (119905)]

119871 [119909119903 (119905)]

= 119870119903+ 119861119903119904 + 1198602

11990121198702+ 1198602

1199011119870

minus [

11987221198852

1199041199042+1198722[(119885119894119904 + 1198991119870) (119885

119904119904 + 1198992119870) minus 119899

21198702] + 119872

21198992119870119885119904119904 minus 119872

21198991119870119885119904119904

(119885119904119904 + 1198992119870)2(119885119894119904 + 1198991119870) minus 119899

2(119885119904119904 + 1198992119870)1198702

]

(38)

where

119872 = 11986011990111198701+ 11986011990121198702

119885119894= 119868119894119904 + 1198771198941

119885119900= 119868119900119904 + 1198771199001

119870 = 1198701+ 1198702

(39)



as

119891119899=

1

2120587

radic(1198701+ 1198702)

120588(1198991119860119894

119897119894

+1198992119860119900

119897119900

) (40)





2= 0 and 119899



2= 1 and 119899



2= 2 and





119899




300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)






0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)


300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)



119894 the








119894and 119860

119894to change but



400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Frequency (Hz)0 20 40 60 80 100

M5

400

800

1200

1600

2000

2400

Stor

age s

tiffne

ss (N

mm

)



(a)

Frequency (Hz)0 20 40 60 80 100

N7

0

200

400

600

800

1000

1200

1400

Loss

stiff

ness

(Nm

m)




Frequency (Hz)0 20 40 60 80 100

300

600

900

1200

1500

1800

2100

2400

2700

Dyn

amic

stiff

ness

(Nm

m)



M1

(a)

Frequency (Hz)0 20 40 60 80 100

N1

0

10

20

30

40

50

60

Loss

angl

e (∘)



(b)





data of 1198911198985


1


1and119870

2can then be




1199011and 119860

1199012based on 119870

1015840




119903= 486 times 10

5 (Nm) 1198702= 231 times

1011 (Nm5) 119870

1= 208 times 10

11 (Nm5) 1198601199011

= 191 times

10minus3 (m2) and 119860

1199012= 172 times 10



Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

Q1(t) Q2(t) Q3(t)

I1R1 I3R3I2R2

xi(t)












1198941


obtained by

119868119894=120588119860119894

119897119894

1198771198941=

1205851198941

1198602

119894

1198771198942=

1205851198942

1198605

119894

(24)




1 1199092 1199093 1199094 1199095) = (119875

1 1198752 1198761 1198762 1198763)

X = AX + B (25)

where

A =

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus (11987711+ 11987712

100381610038161003816100381611990931003816100381610038161003816)

1198681

0 0

1

1198682

minus1

1198682

0minus (11987721+ 11987722

100381610038161003816100381611990941003816100381610038161003816)

1198682

0

1

1198683

minus1

1198683

0 0minus (11987731+ 11987732

100381610038161003816100381611990951003816100381610038161003816)

1198682

]]]]]]]]]]]]]]]]

]

B =

[[[[[[[

[

minus11987011198601199011

11987021198601199012

0

0

0

]]]]]]]

]

(26)



= 0 (119894 = 1 2 3) Since 119897119894(119894 = 1



1198941(119894 = 1 2 3) as

follows [32]

1198771198941= 120574

128120583119897119894

1205871198894

119867

(27)


Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

xi(t)xo(t)

n1Qi(t)n2Qo(t) IiRi

n1IoRon2



119871as

A119871=

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus11987711

1198681

0 0

1

1198682

minus1

1198682

0minus11987721

1198682

0

1

1198683

minus1

1198683

0 0minus11987731

1198682

]]]]]]]]]]]]]]]]

]

(28)


1198942= 0 (119894 = 1 2 3) into


MQ + CQ + KQ = F (119905) (29)


M =[[

[

1198681

1198682

1198683

]]

]

C =[[

[

11987711

11987721

11987731

]]

]

K =[[[

[

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

]]]

]

F (119905) =[[[

[

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

]]]

]

(30)


2

119899M10038161003816100381610038161003816

= 0 (31)


obtained as

119891119899=

120596119899

2120587= radic

(1198701+ 1198702)

120588(1198601

1198971

+1198602

1198972

+1198603

1198973

) (32)



119894and cross-




1198942(119894 = 1 2 119899

1) of the










119900of such a


119888119900=

(15 + 1374119897119900

119889119867Re

)

minus12119889119867Re119897119900

gt 50

(228 + 64119897119900

119889119867Re

)

minus12119889119867Re119897119900

lt 50

(33)


119900and



be expressed as

119868119900= 120573

120588119897119900

119860119900

1198771199001

= 120576120588119876119900

11988821199001198602119900

(34)


119900denotes the


defined as X119879 = (1199091 1199092 1199093 1199094) = (119875

1 1198752 119876119894 119876119900) the state


X = AX + B (35)




1= 2 119899

2= 0) HB4 (119899

1= 1 1198992= 1) HB5 (119899

1= 0 119899

2= 2)


whereA

=

[[[[[[[[[

[

0 0 minus1198701

minus1198701

0 0 1198702

1198702

1

119868119894

minus1

119868119894

minus (1198771198941+ 1198771198942

100381610038161003816100381611990931003816100381610038161003816)

119868119894

0

1

119868119900

minus1

119868119900

0minus (1198771199001+ 1198771199002

100381610038161003816100381611990941003816100381610038161003816)

119868119900

]]]]]]]]]

]

B119879 = (minus11987011198601199011119903 11987021198601199012119903 0 0)

(36)


119865119879 (119905) = 119870

119903119909119903 (119905) + 119861

119903119903 (119905) + (119860

11990121198752minus 11986011990111198751) (37)


119870lowast

119889(119904) =

119871 [119865 (119905)]

119871 [119909119903 (119905)]

= 119870119903+ 119861119903119904 + 1198602

11990121198702+ 1198602

1199011119870

minus [

11987221198852

1199041199042+1198722[(119885119894119904 + 1198991119870) (119885

119904119904 + 1198992119870) minus 119899

21198702] + 119872

21198992119870119885119904119904 minus 119872

21198991119870119885119904119904

(119885119904119904 + 1198992119870)2(119885119894119904 + 1198991119870) minus 119899

2(119885119904119904 + 1198992119870)1198702

]

(38)

where

119872 = 11986011990111198701+ 11986011990121198702

119885119894= 119868119894119904 + 1198771198941

119885119900= 119868119900119904 + 1198771199001

119870 = 1198701+ 1198702

(39)



as

119891119899=

1

2120587

radic(1198701+ 1198702)

120588(1198991119860119894

119897119894

+1198992119860119900

119897119900

) (40)





2= 0 and 119899



2= 1 and 119899



2= 2 and





119899




300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)






0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)


300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)



119894 the








119894and 119860

119894to change but



400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

Q1(t) Q2(t) Q3(t)

I1R1 I3R3I2R2

xi(t)












1198941


obtained by

119868119894=120588119860119894

119897119894

1198771198941=

1205851198941

1198602

119894

1198771198942=

1205851198942

1198605

119894

(24)




1 1199092 1199093 1199094 1199095) = (119875

1 1198752 1198761 1198762 1198763)

X = AX + B (25)

where

A =

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus (11987711+ 11987712

100381610038161003816100381611990931003816100381610038161003816)

1198681

0 0

1

1198682

minus1

1198682

0minus (11987721+ 11987722

100381610038161003816100381611990941003816100381610038161003816)

1198682

0

1

1198683

minus1

1198683

0 0minus (11987731+ 11987732

100381610038161003816100381611990951003816100381610038161003816)

1198682

]]]]]]]]]]]]]]]]

]

B =

[[[[[[[

[

minus11987011198601199011

11987021198601199012

0

0

0

]]]]]]]

]

(26)



= 0 (119894 = 1 2 3) Since 119897119894(119894 = 1



1198941(119894 = 1 2 3) as

follows [32]

1198771198941= 120574

128120583119897119894

1205871198894

119867

(27)


Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

xi(t)xo(t)

n1Qi(t)n2Qo(t) IiRi

n1IoRon2



119871as

A119871=

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus11987711

1198681

0 0

1

1198682

minus1

1198682

0minus11987721

1198682

0

1

1198683

minus1

1198683

0 0minus11987731

1198682

]]]]]]]]]]]]]]]]

]

(28)


1198942= 0 (119894 = 1 2 3) into


MQ + CQ + KQ = F (119905) (29)


M =[[

[

1198681

1198682

1198683

]]

]

C =[[

[

11987711

11987721

11987731

]]

]

K =[[[

[

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

]]]

]

F (119905) =[[[

[

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

]]]

]

(30)


2

119899M10038161003816100381610038161003816

= 0 (31)


obtained as

119891119899=

120596119899

2120587= radic

(1198701+ 1198702)

120588(1198601

1198971

+1198602

1198972

+1198603

1198973

) (32)



119894and cross-




1198942(119894 = 1 2 119899

1) of the










119900of such a


119888119900=

(15 + 1374119897119900

119889119867Re

)

minus12119889119867Re119897119900

gt 50

(228 + 64119897119900

119889119867Re

)

minus12119889119867Re119897119900

lt 50

(33)


119900and



be expressed as

119868119900= 120573

120588119897119900

119860119900

1198771199001

= 120576120588119876119900

11988821199001198602119900

(34)


119900denotes the


defined as X119879 = (1199091 1199092 1199093 1199094) = (119875

1 1198752 119876119894 119876119900) the state


X = AX + B (35)




1= 2 119899

2= 0) HB4 (119899

1= 1 1198992= 1) HB5 (119899

1= 0 119899

2= 2)


whereA

=

[[[[[[[[[

[

0 0 minus1198701

minus1198701

0 0 1198702

1198702

1

119868119894

minus1

119868119894

minus (1198771198941+ 1198771198942

100381610038161003816100381611990931003816100381610038161003816)

119868119894

0

1

119868119900

minus1

119868119900

0minus (1198771199001+ 1198771199002

100381610038161003816100381611990941003816100381610038161003816)

119868119900

]]]]]]]]]

]

B119879 = (minus11987011198601199011119903 11987021198601199012119903 0 0)

(36)


119865119879 (119905) = 119870

119903119909119903 (119905) + 119861

119903119903 (119905) + (119860

11990121198752minus 11986011990111198751) (37)


119870lowast

119889(119904) =

119871 [119865 (119905)]

119871 [119909119903 (119905)]

= 119870119903+ 119861119903119904 + 1198602

11990121198702+ 1198602

1199011119870

minus [

11987221198852

1199041199042+1198722[(119885119894119904 + 1198991119870) (119885

119904119904 + 1198992119870) minus 119899

21198702] + 119872

21198992119870119885119904119904 minus 119872

21198991119870119885119904119904

(119885119904119904 + 1198992119870)2(119885119894119904 + 1198991119870) minus 119899

2(119885119904119904 + 1198992119870)1198702

]

(38)

where

119872 = 11986011990111198701+ 11986011990121198702

119885119894= 119868119894119904 + 1198771198941

119885119900= 119868119900119904 + 1198771199001

119870 = 1198701+ 1198702

(39)



as

119891119899=

1

2120587

radic(1198701+ 1198702)

120588(1198991119860119894

119897119894

+1198992119860119900

119897119900

) (40)





2= 0 and 119899



2= 1 and 119899



2= 2 and





119899




300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)






0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)


300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)



119894 the








119894and 119860

119894to change but



400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Outer sleeve

Inner sleeve

Outer sleeve

Kr1

Kr2

Br1

Br2

xr(t)

P1(t)

P2(t)

Ap1

Ap2

C1

C2 FT(t)

FT(t)

xi(t)xo(t)

n1Qi(t)n2Qo(t) IiRi

n1IoRon2



119871as

A119871=

[[[[[[[[[[[[[[[[

[

0 0 minus1198701

minus1198701

minus1198701

0 0 1198702

1198702

1198702

1

1198681

minus1

1198681

minus11987711

1198681

0 0

1

1198682

minus1

1198682

0minus11987721

1198682

0

1

1198683

minus1

1198683

0 0minus11987731

1198682

]]]]]]]]]]]]]]]]

]

(28)


1198942= 0 (119894 = 1 2 3) into


MQ + CQ + KQ = F (119905) (29)


M =[[

[

1198681

1198682

1198683

]]

]

C =[[

[

11987711

11987721

11987731

]]

]

K =[[[

[

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

1198701+ 11987021198701+ 11987021198701+ 1198702

]]]

]

F (119905) =[[[

[

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

minus11987011198601199011

minus11987021198601199012

0

]]]

]

(30)


2

119899M10038161003816100381610038161003816

= 0 (31)


obtained as

119891119899=

120596119899

2120587= radic

(1198701+ 1198702)

120588(1198601

1198971

+1198602

1198972

+1198603

1198973

) (32)



119894and cross-




1198942(119894 = 1 2 119899

1) of the










119900of such a


119888119900=

(15 + 1374119897119900

119889119867Re

)

minus12119889119867Re119897119900

gt 50

(228 + 64119897119900

119889119867Re

)

minus12119889119867Re119897119900

lt 50

(33)


119900and



be expressed as

119868119900= 120573

120588119897119900

119860119900

1198771199001

= 120576120588119876119900

11988821199001198602119900

(34)


119900denotes the


defined as X119879 = (1199091 1199092 1199093 1199094) = (119875

1 1198752 119876119894 119876119900) the state


X = AX + B (35)




1= 2 119899

2= 0) HB4 (119899

1= 1 1198992= 1) HB5 (119899

1= 0 119899

2= 2)


whereA

=

[[[[[[[[[

[

0 0 minus1198701

minus1198701

0 0 1198702

1198702

1

119868119894

minus1

119868119894

minus (1198771198941+ 1198771198942

100381610038161003816100381611990931003816100381610038161003816)

119868119894

0

1

119868119900

minus1

119868119900

0minus (1198771199001+ 1198771199002

100381610038161003816100381611990941003816100381610038161003816)

119868119900

]]]]]]]]]

]

B119879 = (minus11987011198601199011119903 11987021198601199012119903 0 0)

(36)


119865119879 (119905) = 119870

119903119909119903 (119905) + 119861

119903119903 (119905) + (119860

11990121198752minus 11986011990111198751) (37)


119870lowast

119889(119904) =

119871 [119865 (119905)]

119871 [119909119903 (119905)]

= 119870119903+ 119861119903119904 + 1198602

11990121198702+ 1198602

1199011119870

minus [

11987221198852

1199041199042+1198722[(119885119894119904 + 1198991119870) (119885

119904119904 + 1198992119870) minus 119899

21198702] + 119872

21198992119870119885119904119904 minus 119872

21198991119870119885119904119904

(119885119904119904 + 1198992119870)2(119885119894119904 + 1198991119870) minus 119899

2(119885119904119904 + 1198992119870)1198702

]

(38)

where

119872 = 11986011990111198701+ 11986011990121198702

119885119894= 119868119894119904 + 1198771198941

119885119900= 119868119900119904 + 1198771199001

119870 = 1198701+ 1198702

(39)



as

119891119899=

1

2120587

radic(1198701+ 1198702)

120588(1198991119860119894

119897119894

+1198992119860119900

119897119900

) (40)





2= 0 and 119899



2= 1 and 119899



2= 2 and





119899




300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)






0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)


300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)



119894 the








119894and 119860

119894to change but



400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1= 2 119899

2= 0) HB4 (119899

1= 1 1198992= 1) HB5 (119899

1= 0 119899

2= 2)


whereA

=

[[[[[[[[[

[

0 0 minus1198701

minus1198701

0 0 1198702

1198702

1

119868119894

minus1

119868119894

minus (1198771198941+ 1198771198942

100381610038161003816100381611990931003816100381610038161003816)

119868119894

0

1

119868119900

minus1

119868119900

0minus (1198771199001+ 1198771199002

100381610038161003816100381611990941003816100381610038161003816)

119868119900

]]]]]]]]]

]

B119879 = (minus11987011198601199011119903 11987021198601199012119903 0 0)

(36)


119865119879 (119905) = 119870

119903119909119903 (119905) + 119861

119903119903 (119905) + (119860

11990121198752minus 11986011990111198751) (37)


119870lowast

119889(119904) =

119871 [119865 (119905)]

119871 [119909119903 (119905)]

= 119870119903+ 119861119903119904 + 1198602

11990121198702+ 1198602

1199011119870

minus [

11987221198852

1199041199042+1198722[(119885119894119904 + 1198991119870) (119885

119904119904 + 1198992119870) minus 119899

21198702] + 119872

21198992119870119885119904119904 minus 119872

21198991119870119885119904119904

(119885119904119904 + 1198992119870)2(119885119894119904 + 1198991119870) minus 119899

2(119885119904119904 + 1198992119870)1198702

]

(38)

where

119872 = 11986011990111198701+ 11986011990121198702

119885119894= 119868119894119904 + 1198771198941

119885119900= 119868119900119904 + 1198771199001

119870 = 1198701+ 1198702

(39)



as

119891119899=

1

2120587

radic(1198701+ 1198702)

120588(1198991119860119894

119897119894

+1198992119860119900

119897119900

) (40)





2= 0 and 119899



2= 1 and 119899



2= 2 and





119899




300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)






0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)


300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)



119894 the








119894and 119860

119894to change but



400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










300

600

900

1200

1500

1800

2100

2400

2700

3000D

ynam

ic st

iffne

ss (N

mm

)

Frequency (Hz)






0 20 40 60 80 100

(a)

0

10

20

30

40

50

60

70

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)






(b)


300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Dyn

amic

stiff

ness

(Nm

m)

Frequency (Hz)0 20 40 60 80 100

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(a)

0

20

40

60

80

Frequency (Hz)0 20 40 60 80 100

Loss

angl

e (∘)

Tested HB1

Tested HB2

HB1 li = 04li

HB1 li = 08li

HB1 li = 40li


(b)



119894 the








119894and 119860

119894to change but



400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










400

800

1200

1600

2000

2400

2800

3200

3600

4000

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)


0 20 40 60 80 100

n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(a)

0

20

40

60

80

100

Frequency (Hz)

Loss

angl

e (∘)

0 20 40 60 80 100


n1 = 1

n1 = 2

n1 = 3

n1 = 5

n1 = 7

n1 = 4

n1 = 6

(b)




1(1198992= 0 1 le 119899



119879(119905)


= 2 to 1198991





1= 1 (119899

2= 2)

and 1198991= 2 (119899


1= 3







2= 4)

and 1198991

= 0 (1198992


6 Conclusions



600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










600

800

1000

1200

1400

1600

1800

2000

2200

2400

Frequency (Hz)

Dyn

amic

stiff

ness

(Nm

m)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(a)

0

10

20

30

40

50

60

70

Frequency (Hz)

Loss

angl

e (∘)

0 50 100 150 200 250 300

n1 = 1 n2 = 2

n1 = 3 n2 = 2

n1 = 0 n2 = 4

n1 = 2 n2 = 2

n1 = 3 n2 = 3

n1 = 0 n2 = 6

(b)






Acknowledgments


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article A Study on the Dynamic Performance for ...downloads.hindawi.com/journals/sv/2016/3695950.pdf · uid tracks on the dynamic performances of the HDB. 2. Design of HDB