Meshing Force of Misaligned Spline Coupling and the ...axial-spline coupling used in high-speed turbomachinery with experimental method. Ku et al. [14] developed exper-imental research

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2008, Article ID 321308, 8 pagesdoi:10.1155/2008/321308

Research ArticleMeshing Force of Misaligned Spline Coupling andthe Influence on Rotor System

Guang Zhao, Zhansheng Liu, and Feng Chen

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Guang Zhao, [email protected]

Received 4 January 2008; Accepted 24 March 2008

Recommended by Kazuhiko Kawaike

Meshing force of misaligned spline coupling is derived, dynamic equation of rotor-spline coupling system is established basedon finite element analysis, the influence of meshing force on rotor-spline coupling system is simulated by numerical integralmethod. According to the theoretical analysis, meshing force of spline coupling is related to coupling parameters, misalignment,transmitting torque, static misalignment, dynamic vibration displacement, and so on. The meshing force increases nonlinearlywith increasing the spline thickness and static misalignment or decreasing alignment meshing distance (AMD). Stiffness ofcoupling relates to dynamic vibration displacement, and static misalignment is not a constant. Dynamic behaviors of rotor-splinecoupling system reveal the following: 1X-rotating speed is the main response frequency of system when there is no misalignment;while 2X-rotating speed appears when misalignment is present. Moreover, when misalignment increases, vibration of the systemgets intricate; shaft orbit departs from origin, and magnitudes of all frequencies increase. Research results can provide importantcriterions on both optimization design of spline coupling and trouble shooting of rotor systems.

Copyright © 2008 Guang Zhao et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

The problem of misalignment encountered in rotatingmachinery is of great concern to designers and maintenanceengineers. The need of understanding the phenomena and itsprinciple is important to practical engineers for the purposeof trouble shooting. Most rotating machinery consists of adriver and a driven machine coupled through some type ofcouplings [1]. There are many types of industrial couplingsone of which is spline coupling. Spline coupling allowsshaft to be easily dismantled while providing a high-torquecapacity for minimum size. In addition, it can allow relativeaxial and radial motions between the coupled shafts [2]. Forthese reasons, spline coupling is one of the most commonlyused couplings to link oxidizer pump and reducing agentpump in rocket engine, aviation engine and generator [3],reducer and output shaft in large ship, and so on. However,because of high-rotating speed (such as pump in rocket) orlarge transmitting torque (such as output shaft in ship) ofrotors connected by spline coupling, the support conditionswhich get much rigorous would cause misalignment ofspline coupling unavoidablely. It has been observed by the

author on several occasions that rotating machinery stabilityconditions can change should the alignment state betweenthe driver and the driven machines changes [4]. In oneside, angular misalignment can greatly increase the slip[2], relative slip of small amplitude can occur between thecontacting tooth surfaces. This can give rise to frettingdamage which may limit the life of the coupling and hasthe potential to compromise integrity. This has been aparticular concern in aircraft mechanical systems and hasbeen studied experimentally for many years [5, 6]. In anotherside, misalignment can causes high vibrations with differentsymptoms that sometimes cannot be explained [7–9].

There are many discussions in the industry regardingthe stiffness, damping, and wear of spline coupling. Marmolet al. [10] developed a mathematical model to predict thelateral and angular stiffness and damping coefficients ofspline couplings. Based on these coefficients, they calculatedthe natural frequencies of a rotor system and compared themto the experimental data. With Marmol’s linear stiffness anddamping coefficients, rotor-spline coupling-bearing systemwas studied by Park [11]. In the design charts of flangedand curvic couplings provided by Bannister [12], flexural

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2 International Journal of Rotating Machinery

stiffness values were provided. In 1994, the report of NASA[13] provides the first opportunity to quantify the angularstiffness and equivalent viscous damping coefficients of anaxial-spline coupling used in high-speed turbomachinerywith experimental method. Ku et al. [14] developed exper-imental research on dynamic coefficient of spline coupling.Peng and Li [15] considered the bending, shear, and com-press deformation when calculating the load capability ofinvolute spline, while the deformation of elastic foundationwas neglected. Walton et al. [16] developed three nonlinearanalytical models representing interference fits and axial andcurvic spline couplings. The results show the predictabilityof rotor system instability due to internal fiction, which hasbeen experimentally observed, especially for axial splines.

Summarizing the above discussions, many researcherspay attention to the stiffness of spline coupling, experimentis an effectual and costly way. However, because of mis-alignment, it is difficult to obtain the accurate results. Inaddition, spline coupling, which is one element of rotorsystem essentially, is separated from its rotor system, thereis not enough academic research that integrates the splinestiffness, wear, and misalignment with the rotor system.Consequently, inclusive and principled understanding of theroots of spline wear and misalignment is difficult to achieve.

The present work studied the elastic deformation ofsingle spline referring to the analysis method of toothdeformation; meshing force of misaligned spline couplingwas derived according to the deformation configurationrelationships of all splines in coupling. Graphical simulationswere actualized to obtain the influence of misalignmentand structure parameters on meshing force. To gain agreater insight of the effect of spline coupling on rotorsystem, the dynamic model of rotor-spline coupling systemwas presented with finite element analysis, Newmark-betanumerical integration method was employed to analyzingthe dynamics of the system.

2. MESHING FORCE OF MISALIGNEDSPLINE COUPLING

2.1. Deformation of single spline

Meshing rigidity of gear tooth represents the synthesis effectof all the meshing teeth in the action region [17, 18], incase of sufficient lubrication, the friction of tooth surfacescan be neglected, meshing rigidity is mainly correlated withelastic deformation of single gear tooth [19]. Because singlespline can be considered as typical tooth, the above theoryis also suitable for deriving the spline deformation. Theelastic deformation of rectangle spline can be considered as aplanar problem, generally. Spline is simplified as a cantileverbeam with same section in elastic foundation according tomechanics of materials.

As shown in Figure 1, f is the distribution force actingon the surface of spline, Fi is its equivalent concentrate forceacting on the meshing point i; the deformation caused byFi consists of bending, shear deformation, and deformationof elastic foundation. Because typical spline belongs towidth spline (B/H > 5), deformation of elastic foundation

y

M

f Fi

i

HH

x

LB

A

Lix

z

y

IZc

Zc

Figure 1: Deformation of single spline.

should not be neglected compared to bending and sheardeformation.

The area and inertia of cross-section are A = BHand IZc = BH3/12, respectively; Li is the distance fromi to M; thickness, width, and height of single spline areH, B, and L, respectively; spline number is z; Poisson’sratio and elastic modulus of spline material are v and E,respectively, according to Cornell’s analysis [20], for widthspline, equivalent elastic modulus Ee is E/(1− v2).

(1) Bending deformation is

δBi = FiL3i

3EeI. (1)

(2) Shear deformation is

δSi = kFiLiGA

= 12FiLi(1 + v)5EeA

. (2)

(3) Deformation of elastic foundation, refer to theanalysis of gear tooth [20], is

δMi = FiEe

[5.306BH2

L2i +

2(1− v − 2v2

)(1− v2

)BH

Li +1.534B

]. (3)

Thus, the total elastic deformation is

δi = δBi + δSi + δMi. (4)

Therefore, the rigidity of single spline is

Ki = Fiδ

= Ee(1/3I)L3

i +(5.306/BH2

)L2i + [Q + X]Li + 1.534/B

.

(5)

where Q = 2(1− v − 2v2)/(1− v2)BH , X = 12(1 + v)/5A.Equation (5) shows that the stiffness of single spline is the

function of meshing distance Li.

2.2. Meshing distance of misaligned spline coupling

When there is no misalignment, the centre of male couplingis superposed with the female, as described in Figure 2, themeshing distance for each spline is the same, and we define itas alignment meshing distance (AMD) with the value of L0.

Guang Zhao et al. 3

y

L0

x

Meshing circle Male coupling

Female coupling

Figure 2: AMD of coupling with no misalignment.

y

w

x

1

2

3

z

z − 1

FyFx

Figure 3: Meshing distance of coupling with misalignment in x-axis.

For a real system, there are two kinds of misalignments:static misalignment (e0,ϕ0) and dynamic misalignment(dynamic vibration displacement) (x, y), a general designa-tion of these two is misalignment in the following analysis.So the real misalignment (e,ϕ) is

e =√(x + e0 cosϕ0

)2+(y + e0 sinϕ0

)2,

cosϕ = x + e0 cosϕ0

e,

sinϕ = y + e0 sinϕ0

e.

(6)

For simplifying, assume that the misalignment takes placein x-axis, that is, ϕ0 = 0 and y = 0. The meshing intwo couplings is asymmetric, teeth in right parts of the twocouplings meshing tightly, contrarily, the left parts meshingloosely meshing distance for each spline is different aspresented in Figure 3.

Assuming the spline serial number is 1 in positivedirection of x-axis, then it is 2, 3, . . . , z, anticlockwise, theinterangular from each spline to positive direction of x-axisis

ϕi = 2π(i− 1)z

. (7)

Accordingly, the Li can be presented as

Li = L0 − e cosϕi. (8)

With above analysis, meshing rigidity of single spline dueto misalignment is

Ki = f(Li). (9)

2.3. Meshing force of misaligned spline coupling

In a real rotor-spline coupling system, all splines meshingtightly by transmitting large torque cause a deformation ofeach spline. At the same time, for coupling nodes vibratingwith the system, there is relative displacement betweensplines in two half-couplings, which cause another deforma-tion of each spline; both the deformations generate meshingforce.

(1) Meshing force caused by torque is as follows.When the torque transmitted by coupling is T,

T =z∑i=1

[FTi·Ri

] =z∑i=1

[FTi·

(R + Li

)], (10)

R is radius of the circle in spline root, and

FTi = φLi·Ki, (11)

φ is the torsion angular for each spline caused by deforma-tion, as all splines constitute a whole coupling, φ’s for allsplines are the same when coupling turns around with thestatic torque T. Summarizing the above analysis,

T =z∑i=1

[φLi·Ki·

(R + Li

)]. (12)

So, φ is

φ = T∑zi=1

[LiKi

(R + Li

)] . (13)

Therefore, force transmitted by each spline is

FTi = φLi·Ki. (14)

(2) Meshing force caused by dynamic vibration displace-ment is

F′i =(e′ sinϕi

)·Ki, (15)

and e′ =√x2 + y2.

Accordingly, meshing force caused by misalignment is

Fi = FTi + F′ = (φLi + e′ sinϕi)Ki. (16)

From above equation, we know that, due to dynamicvibration displacement, meshing forces of splines from 1to z/2 get larger, others get smaller, and meshing force foreach spline would not be negative. Therefore, when thetransmitting torque is zero, there would be no meshing


y

w

x

1

2

3

z

z − 1

Fy Fxϕ

Figure 4: Meshing distance of coupling with misalignment in anydirection.

force, even dynamic vibration displacement is still present.Therefore,

Fi =⎧⎨⎩(φLi + e′ sinϕi

)Ki, φLi + e′ sinϕi > 0,

0, φLi + e′ sinϕi ≤ 0.(17)

Two components of the above force are

Fxi = Fi cos θi,

Fyi = Fi sin θi,(18)

θi is angular between meshing force and positive direc-tion of x-axis, it is

θi = ϕi +π

2. (19)

The resultant forces due to misaligned coupling are

Fx =z∑i=1

Fxi =z∑i=1

[(φLi + e′ sinϕi

)Ki cos θi

],

Fy =z∑i=1

Fyi =z∑i=1

[(φLi + e′ sinϕi

)Ki sin θi

].

(20)

2.4. Extending of meshing force due tomisaligned spline coupling

The above analysis is a special misalignment that appearsin x-axis, when the real misalignment is present in anydirection, as shown in Figure 4, which means that there isan angular ϕ from x-axis, anticlockwise. The definition of ϕis the same as (6).

The real meshing forces of misaligned coupling are

fx = Fx cosϕ− Fy sinϕ,

fy = Fx sinϕ + Fy cosϕ.(21)

From above analysis, the meshing force model of mis-aligned coupling is achieved; it is a function of couplingparameters, transmitting torque, static misalignment anddynamic vibration displacement, and so on.

Table 1: Structure parameters of spline coupling.

Name Symbol Unit Value

Number of splines Z 14

Spline length L mm 5.07

Spline width B mm 100

Spline thickness H mm 16

AMD (alignment meshing distance) L0 mm 3.375

Poisson’s ratio v 0.3

Elastic modulus E Pa 2.12× 1011

Radius of spline root R mm 69.5

Transmitting torque T Nm 87000

0

2

4

6

8

10×105

Mes

hin

gfo

rce

(N)

0 1 2 3 4

×10−4Static misalignment (m)

fx(0 deg)fy(0 deg)

fx(30 deg)fy(30 deg)

Figure 5: Meshing forces versus static misalignment.

3. SIMULATIONS OF MESHING FORCE

Structure parameters of spline coupling adopted in thispaper are shown in Table 1.

3.1. Meshing force versus misalignment

(1) Meshing force versus static misalignment is as follows.When the dynamic vibration displacement is x = y =

0.001 mm, the static misalignment e0 changes from 0 to1 mm, and ϕ0 is 0◦ and 30◦, respectively, the curves ofmeshing forces are shown in Figure 5.

According to Figure 5, when the dynamic vibration dis-placement is small and keeps constant, the following hold.

(a) Two meshing forces are equal when there is nomisalignment.

(b) Meshing forces change nonlinearly with smaller staticmisalignment, while they change linearly when themisalignment gets larger.

(2) Meshing force versus dynamic vibration displace-ment is as follows.

Guang Zhao et al. 5

−5

0

5

10

15×105

Mes

hin

gfo

rce

(N)

0 0.5 1 1.5 2×10−5

Dynamic vibration displacement (m)

fxfy

Figure 6: Meshing forces versus dynamic vibration displacement.

−5

0

5×104

Mes

hin

gfo

rce

(N)

3 3.5 4 4.5 5×10−3

Alignment meshing distance (m)

fxfy

Figure 7: Meshing force versus AMD.

When the static misalignment is e0 = 0.5 mm, ϕ0 = 90◦,the dynamic vibration displacement of x and y changes from0 to 0.02 mm, the curves of meshing forces can be obtainedas shown in Figure 6.

Figure 6 shows that when the static misalignmentappears and keeps constant, the following hold.

(a) Meshing force changes linearly with dynamic vibra-tion displacement, approximately, but there is aninflexion.

(b) The slope coefficient of meshing force curve reflectsthe stiffness of coupling, consequently, during thevibration of system, the stiffness of coupling is not aconstant, it relates to dynamic vibration displacementand depends on static misalignment.

−6

−4

−2

0

2

4

6×104

Mes

hin

gfo

rce

(N)

0.01 0.015 0.02 0.025 0.03

Spline thickness (m)

fxfy

Figure 8: Meshing force versus spline thickness.

−4

−2

0

2

4×104

Mes

hin

gfo

rce

(N)

0.04 0.06 0.08 0.1 0.12 0.14 0.16

Spline width (m)

fxfy

Figure 9: Meshing force versus spline width.

3.2. Meshing force versus coupling parameters

Choose the misalignment as x = y = 0, e0 = 0.5 mm, andϕ0 = 45◦, change the alignment meshing distance (AMD) L0

from 3 to 5.0 mm, the spline thickness H from 10 to 30 mm,and the spline width B from 50 to 150 mm, respectively, thecurves of meshing force are represented as shown in Figures7, 8, and 9.

The above simulation results show that meshing forceincreases nonlinearly with the increasing alignment meshingdistance (AMD) and with the decreasing spline thickness.The spline number and width have no influence on meshingforce directly. However, for a spline coupling, the thickness ofspline should be changed when the number of spline alters.


Node no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Bearing no. 1# 2# 3# 4#

Spline coupling

Figure 10: Sketch map of rotor-spline coupling system.

4. DYNAMIC MODEL OF ROTOR-SPLINECOUPLING SYSTEM

2-span symmetry rotor system was used to study theinfluence of spline coupling on dyamics of rotor sytem, thesketch map of system is shown in Figure 10.

Based on finite element analysis, presumption of Euler-Bernoulli beam was employed to derive the dynamic equa-tion of rotor-spline coupling system:

[M]{Z} + [C]{Z} + [K]{Z} = {Q(t)} + {FSP}. (22)

For simplifying, the gravity was neglected and thesupport was assumed as linear spring. In above equation,[M], [C], [K] are system mass, damping (ratio damping),and stiffness matrix, respectively; {Q(t)} is the term ofunbalance force, {Z} = {xj , yj ,−θx j , θy j} (j is the serialnumber of node), {FSP} is the meshing force of splinecoupling, it is

fx = Fx(e, e′) cosϕ− Fy(e, e′) sinϕ,

fy = Fx(e, e′) sinϕ + Fy(e, e′) cosϕ.(23)

In above equation,

e =√(e0 cosϕ0 + dx

)2+(e0 sinϕ0 + dy)

2,

e′ =√

(dx)2 + (dy)2,

cosϕ = e0 cosϕ0 + dx

e,

sinϕ = e0 sinϕ0 + dy

e,

(24)

dx and dy are relative displacements of two half-couplingnodes.

5. DYNAMICS ANALYSIS OF ROTOR-SPLINECOUPLING SYSTEM

Nowadays, the effective method to analyze (22) is numericalmethod. Newmark-beta stepping integration method whichhas high efficiency and good stability was employed to solvethe response of system in this paper.

Choosing the static misalignment (e0,ϕ0) as (0, 0),(0.4 mm, 45◦), and (0.8 mm, 45◦), respectively, the responsesof node 7 (a half-coupling located) in the speed of 5000 rpm

0

0.5

1

1.5×10−5

Vib

rati

onm

agn

itu

deinx

dire

ctio

n(m

)

0 2 4 6 8 10

Integer multiple of rotating speed

(a)

0

2

4×10−5

Vib

rati

onm

agn

itu

deinx

dire

ctio

n(m

)

0 2 4 6 8 10


(b)

Figure 11: Frequency (0, 0).

−2

−1

0

1

2

×10−4

Vib

rati

onm

agn

itu

deiny

dire

ctio

n(m

)

−3 −2 −1 0 1 2 3×10−4

Vibration magnitude in x direction (m)

Figure 12: Shaft orbit (0, 0).

are shown in Figures 11 and 12, 13 and 14, 15, and 16,respectively.

From above analysis, some conclusions can be achievedas follows.

(1) When there is no misalignment, 1X-rotating speedis the main response frequency; shaft orbit enclosesorigin.

(2) When there is misalignment, 1X- and 2X-rotatingspeed are the main response frequencies.

(3) With appearance and increases of misalignment,shaft orbit departs from origin, magnitudes of allfrequencies increase, and frequency of 2X-rotatingspeed increases most rapidly.

Guang Zhao et al. 7

0

2

4

6

×10−6

Vib

rati

onm

agn

itu

deinx

dire

ctio

n(m

)

2 4 6 8


(a)

0

1

2

3

×10−5

Vib

rati

onm

agn

itu

deinx

dire

ctio

n(m

)

2 4 6 8


(b)

Figure 13: Frequency (0.4 mm, 45◦).

−2

−1

0

1

2

×10−4

Vib

rati

onm

agn

itu

deiny

dire

ctio

n(m

)

−2 −1 0 1 2 3×10−4


Figure 14: Shaft orbit (0.4 mm, 45◦).

6. CONCLUSIONS

In this study, meshing force of misaligned spline couplingand its influence on rotor system are studied. The followingsare the outcome of the study.

(1) Meshing force of misaligned spline coupling is notrelated to spline number and width directly, it changeslinearly with transmitting torque while nonlinearly withalignment meshing distance (AMD), spline thickness, andstatic misalignment.

(2) Stiffness of coupling is not a constant; it relatesto dynamic vibration displacement and depends on staticmisalignment.

(3) The dynamics of rotor-spline coupling system showsthe following: 1X-rotating speed is the main responsefrequency when there is no misalignment; while 2X-rotating

0

1

2

×10−5

Vib

rati

onm

agn

itu

deinx

dire

ctio

n(m

)

2 4 6 8


(a)

0

2

4

×10−5

Vib

rati

onm

agn

itu

deinx

dire

ctio

n(m

)

2 4 6 8


(b)

Figure 15: Frequency (0.8 mm, 45◦).

−2

−1

0

1

2

3

×10−4

Vib

rati

onm

agn

itu

deiny

dire

ctio

n(m

)

−2 −1 0 1 2 3 4 5×10−4


Figure 16: Shaft orbit (0.8 mm, 45◦).

speed appears when misalignment is present; with theincrease of misalignment, shaft orbit departs from origin,the magnitudes of all frequencies increase, and 2X-rotatingspeed increases rapidly. Thus, misalignment makes thevibration of system more complicated.

(4) In order to actualize optimization design of splinecoupling with the purpose of reducing the meshing forcecaused by misalignment, reducing the transmitting torqueand misalignment are effectual ways. Moreover, decreasingspline thickness (the number of splines should increasehomologically) or increasing the alignment meshing distance(AMD) is also suitable.

(5) During the trouble shooting to rotor-spline couplingsystem with the aim of decreasing the rotor vibration andreducing frequency components, the effective operationsare aligning the support, monitoring the shaft orbit, and


observing the magnitudes of all frequencies especially the2X-rotating speed which is considered to be a characteristiccommonly observed in the field of misaligned rotatingmachinery.

ACKNOWLEDGMENT

This work was supported by the National Nature ScienceFoundation of China under Grant no. 10632040.

REFERENCES

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[2] S. Medina and A. V. Olver, “Regimes of contact in splinecouplings,” Journal of Tribology, vol. 124, no. 2, pp. 351–357,2002.

[3] Q.-H. Li, “Analysis of distance between keys in involutespline coupling of aviation generates,” Aviation PrecisionManufacturing Technology, vol. 40, no. 2, pp. 40–42, 2004,(Chinese).

[4] K. M. Al-Hussain, “Dynamic stability of two rigid rotorsconnected by a flexible coupling with angular misalignment,”Journal of Sound and Vibration, vol. 266, no. 2, pp. 217–234,2003.

[5] W. D. Weatherford and M. L. Valtierra, “Mechanisms ofwear in misaligned splines,” ASME Journal of LubricationTechnology, vol. 90F, pp. 42–48, 1968.

[6] R. A. Newley, The mechanisms of fretting wear of misalignedsplines in the presence of a lubricant, Ph.D. thesis, University ofLondon, London, UK, 1978.

[7] J. Han and L.-D. Shi, “Study on kinematic mechanismof misalignment fault of rotor system connected by gearcoupling,” Journal of Vibration Engineering, vol. 17, no. 4, pp.416–420, 2004, (Chinese).

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[9] A. S. Sekhar and B. S. Prabhu, “Effects of coupling misalign-ment on vibrations of rotating machinery,” Journal of Soundand Vibration, vol. 185, no. 4, pp. 655–671, 1995.

[10] R. A. Marmol, A. J. Smalley, and J. A. Tecza, “Spline couplinginduced nonsynchronous rotor vibrations,” ASME Journal ofMechanical Design, vol. 102, no. 1, pp. 168–176, 1980.

[11] S. K. Park, Determination of loose spline coupling coefficientsof rotor bearing systems in turbomachinery, Ph.D. thesis, TexasA&M University, College Station, Tex, USA, 1991.

[12] R. H. Bannister, “Methods for modeling flanged and curviccouplings for dynamic analysis of complex rotor construc-tions,” ASME Journal of Mechanical Design, vol. 102, no. 1, pp.130–139, 1980.

[13] C.-P. R. Ku, J. F. Walton Jr., and J. W. Lund, “An investigationof angular stiffness and damping coefficients of an axial splinecoupling in high-speed rotating machinery,” NASA TechnicalReports Server (NTRS) N94-34192, pp. 293–303, NASA Center,Washington, DC, USA, 1994.

[14] C.-P. R. Ku, J. F. Walton, and J. W. Lund, “Dynamic coefficientsof axial spline couplings in high-speed rotating machinery,”ASME Journal of Vibration and Acoustics, vol. 116, no. 3, pp.250–256, 1994.

[15] H.-P. Peng and Z.-M. Li, “A research on the numericalrelations between the load capacity and making precision ofspline on the ends of torsion bar spring,” Mechanism, vol. 31,no. 5, pp. 20–25, 2004, (Chinese).

[16] J. F. Walton Jr., A. Artiles, and J. W. Lund, “Internal rotorfriction instability,” MTI 88TR39, Mechanical TechnologyIncorporated, Latham, NY, USA, 1990.

[17] M. S Tavikoli and D. R. Houser, “Optimum profile modifi-cation for the minimization of static transmission errors ofspur gear,” ASME Journal of Mechanisms, Transmissions, andAutomation in Design, vol. 108, pp. 86–94, 1986.

[18] C.-J. Zhou, J.-Y. Tang, and Y.-X. Wu, “The comparative studyof the bending stress and elastic deformation calculation ofgear tooth,” Journal of Mechanical Transmission, vol. 28, no. 5,pp. 1–6, 2004, (Chinese).

[19] Z.-S. Liu and G. Zhao, “Modeling research on radial force ingear coupling with parallel misalignment,” in Proceedings ofthe 12th World Congress in Mechanism and Machine Science(IFToMM ’07), BesanCon, France, June, 2007, A764.

[20] R. W. Cornell, “Compliance and stress sensitivity of spur gearteeth,” ASME Journal of Mechanical Design, vol. 103, no. 2, pp.447–459, 1981.

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Meshing Force of Misaligned Spline Coupling and the ...axial-spline coupling used in high-speed turbomachinery with experimental method. Ku et al. [14] developed exper-imental research