Modelling and Dynamic Analysis of the Spiral Bevel Gear...

Research ArticleModelling and Dynamic Analysis of the Spiral BevelGear-Shaft-Bearing-Gearbox Coupling System

Haimin Zhu 1 Weifang Chen 1 Rupeng Zhu 1 Jie Gao2 and Meijun Liao2

1National Key Laboratory of Science and Technology on Helicopter TransmissionNanjing University of Aeronautics and Astronautics Nanjing 210016 China2AECC Hunan Aviation Powerplant Research Institute Zhuzhou 412000 China

Correspondence should be addressed to Weifang Chen meewfchennuaaeducn

Received 6 May 2019 Accepted 28 July 2019 Published 8 September 2019

Academic Editor Renato Vidoni

Copyright copy 2019 Haimin Zhu et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To accurately study the dynamic characteristics of the spiral bevel gear transmission system in a helicopter tail transmissionsystem the finite elementmodel of the gear shaft was established by a Timoshenko beam element and themechanical model of thespiral bevel gear was created by the lumped mass method +e substructure method is employed to extract the dynamic pa-rameters from the gearboxrsquos finite element model and the dynamic model of the spiral bevel gear-shaft-bearing-gearbox couplingsystem was built according to the interface coordination conditions In the model the influences of time-varying stiffness a time-varying transmission error gearbox flexibility unbalance excitation and a flexible shaft and bearing support on the systemvibration were taken into account simultaneously On this basis the dynamic differential equations of the full coupling system ofthe spiral bevel gear were derived and the effects of the gearbox flexibility the shaft angle and the unbalance on the dynamicproperties of the system were analysed+e results show that the gearbox flexibility can reduce the gear meshing force and bearingforce in which there is a more significant impact on the bearing force+e shaft angle affects the position size and direction of thesystemrsquos axis trajectory Meanwhile the meshing force and the bearing force of the system are also varied because of the variouspitch angles of the driving and driven gears under different shaft angles +e unbalance of the gear shaft has an effect on thevibration of the spiral bevel gear transmission system in all directions wherein the influence on the torsional vibration is the mostsignificant and the influence increases as the unbalance rises +e unbalance of the gear shaft also affects the meshing force andbearing force which increases as the rotational speed rises +is research provides a theoretical basis to optimize dynamicperformance and reduce the vibration and noise of a spiral bevel gear full coupling system

1 Introduction

+e spiral bevel gear has the characteristics of the firm bearingcapacity a large coincidence degree and stable operation Itsnoise and vibration are relatively small during high-speedoperations and it is widely used in the core transmissionsystems of helicopters and ships +e most common form ofthe spiral bevel gear transmission system is the orthogonal shaftsystem Because of the limitations of space and shape thenonorthogonal spiral bevel gear is widely employed in heli-copter tail transmission systems in which the shaft angle is not90 resulting in the vibration behaviours of the gear pair and therotor system varying accordingly In addition the stiffness ofthe gearbox is relatively high in general so the influence of the

gearbox flexibility can be ignored when studying the dynamiccharacteristics of the gear system However the wall thicknessof the helicopter gearbox is comparatively thin and the flexiblesupport brought by the thin-wall structure of the gearboxcauses a dynamic coupling effect between the gear transmissionsystem and the gearbox +erefore in order to obtain moreaccurate system response and better predict the vibrationalnoise of such systems it is necessary to establish a non-orthogonal spiral bevel gear-shaft-bearing- box coupling me-chanicalmodel to analyse the dynamic properties of the system

Currently the dynamics of the spiral bevel gear trans-mission system has been studied extensively both domes-tically and internationally Wang et al [1] establisheda nonlinear vibration equation for the bending-torsional-

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 9065215 16 pageshttpsdoiorg10115520199065215

axial coupling vibration of a 7-degrees-of-freedom spiralbevel gear +e torsional lateral and axial vibration dis-placements and velocities of the spiral bevel gear systemunder various working conditions are obtained by theoperator algorithm +e results show that with the varietyof the meshing frequency the system enters into chaosthrough period-doubling bifurcation with the variety ofthe support stiffness the system changes into chaosthrough quasi-period bifurcation and in the process of themeshing frequency varying jump phenomena exist in thesystem Feng and Song [2] considered the time-varyingmeshing force of the gear the position of the meshingpoint the transmission error the direction of the frictionforce and the friction coefficient and then establisheda dynamic model of concentrated parameters On thisbasis the effects of the dynamic meshing force and dynamictransmission error on the system response are investigatedWang et al [3] analysed the torsional vibration of the spiralbevel gear transmission system with backlash and time-varying meshing stiffness+e results showed that when thegear system reciprocates the backlash and time-varyingmeshing stiffness will produce a coupling effect causing thegear system to show obvious nonlinear characteristics +einternal excitation frequency the meshing stiffness am-plitude the initial condition and the damping ratio all havean essential impact on the systemrsquos dynamic features +evariation of each parameter makes the systemrsquos responseappear in harmonic period solution and chaotic solutionTang et al [4] considering such influencing factors as thetime-varying meshing stiffness and gear clearance analysedthe effect of the static transmission error on the nonlineardynamic response of a spiral bevel gear system +e dy-namic response bifurcation diagram time domain re-sponse phase diagram and Poincare map are all obtainedby solving the dynamic equations Fan et al [5] examinedthe linear dynamics of the transmission system of a non-orthogonal spiral bevel gear and analysed the systemrsquosinherent characteristics steady-state response and axialtrajectory Cheng and Lim [6] built a three-dimensionalmulti-degrees-of-freedom model of the hypoid gear andanalysed the dynamic coupling and vibrational responsesensitivity using the response function of the meshingforce +en the influences of the system parameters on thedynamic response of the hypoid gear were studied and itwas proven that the design parameters and working con-ditions have significant impacts on the modal and vibrationresponse Yang and Lim [7] considered the nonlinearclearance and time-varying meshing stiffness and solvedthe dynamic meshing force and vibration response by thevariable step size RungendashKutta method +e analysisshowed that damping could dramatically reduce the dy-namic meshing force Wang et al [8] established a hypoidgear dynamics model with a time-varying meshing forceconsidering the effect of the nonlinear clearance andstudied the impact of the different meshing waveforms onthe dynamic response of gear dynamics Xu et al [9 10]constructed the bending-torsion-axial coupling dynamicmodel of the multi-freedom spiral bevel gear through thelumped mass method and then studied the chaotic and

bifurcation properties of the system Yassine et al [11]constructed a dynamic model of a two-stage bevel geartransmission system mainly considering the effect of thetime-varying meshing stiffness +e Newmark numericalalgorithm was utilised to solve the equations and thedynamic features of the system were analysed in the timedomain and the frequency domain Peng [12] createda multibody dynamic model of hypoid gears and studiedthe effects of misalignment torque and eccentricity on thedynamic response of the system Li et al [13] investigatedthe effect of an asymmetric time-varying meshing stiffnesson the spiral bevel gear transmission system It was de-termined that the meshing stiffness has a higher impact onthe driving side than that on the driven side Under a lightload the rise in the meshing stiffness on the drive sidedeteriorates the vibration stability of the transmissionsystem Wu and Yang [14] utilised the extended transfermatrix method to analyse the bending-torsional-coupledvibration of the multi-degrees-of-freedom damped shaftingcaused by external excitation It was shown that the naturalfrequency of bending-torsion coupling is close to thenatural frequency of torsion Choi and Mau [15] analysedthe dynamic behaviours of the gear-coupled rotor systemby using the transfer matrix method +e Timoshenkobeam element was employed to model the rotating shaftand the natural frequency and vibration mode were ac-quired by studying the bending-torsion-coupled vibrationof the system Neriya et al [16] adopted the finite elementmethod to simulate the bending-torsion coupling vibrationof the gear system comparatively early +e finite elementmethod has been widely utilized by scholars both do-mestically and abroad because this method can bettersimulate the influence of a flexible shaft and the bearingparameters on the dynamic response of the gear pair[17ndash20] Li and Hu [21] discussed the linear dynamicproperties of the bending-torsional coupling of the rotor-bearing system with spiral bevel gears via the finite elementmethod after considering the generalized meshing re-lationship between the conical teeth Hua et al [22]employed the finite element method and the lumped pa-rameter method to analyse the dynamics of the spiral bevelgear-rotor system Yavuz et al [23] aking the backlashtime-varying meshing stiffness and the transmission errorinto account built a dynamic model of the spiral bevel gearbased on the finite element method solved differentialequations using the harmonic balance method and studiedthe effects of backlash time-varying meshing stiffness anda bearing stiffness on the system dynamics Choy et al[24 25] examined the effect of the housingrsquos vibration onthe dynamic features of the parallel shaft gear transmissionsystem via the modal synthesis method but the gearboxwas simplified dramatically Ren et al [26] established thegear-shaft-bearing-box coupling dynamic model of theparallel shaft via the impedance synthesis method takingthe dynamical coupling between the transmission systemand the housing into consideration and investigated theinfluence factor of the natural frequency meshing forceand bearing force According to the literature retrieved themodelling methods of the gear transmission systemrsquos

2 Mathematical Problems in Engineering

dynamics mainly contain the lumped mass method thetransfer matrix method and the finite element methodAmong them the finite element method not only considersthe influence of the shaft flexibility and bearing support butalso conveniently couple the forces in the surroundingenvironment which is especially suitable for the complexgeared rotor coupling systems Currently the applicationof the finite element method is very mature for gear systemswith parallel shafts such as spur gears and helical gearsHowever there have been relatively few research works onspiral bevel gear systems with nonparallel shafts and theeffect of thin-walled gearbox flexibility on the dynamics ofthe transmission system has not been taken into account

In this article a nonorthogonal spiral bevel gear trans-mission system supported by a thin-walled gearbox is taken asthe research object First analysis of the transient dynamics iscarried out using finite element software to obtain the time-varying meshing stiffness of the spiral bevel gear pair becausethe tooth surface of the spiral bevel gear is a complex curvedsurface +en the dynamic equation of the nonorthogonalspiral bevel gear transmission system is constructed via thefinite elementmethod and the lumpedmass hybrid modellingmethod +e dynamic parameters of the gearbox system areextracted by the substructure method and are connected bythe nodes of the bearing support thus establishing the dy-namic model of the spiral bevel gear-shaft-bearing-boxcoupling system On this basis the dynamic equations of thewhole coupling system are solved using the Newmark nu-merical algorithm and the effects of the gearbox flexibilitythe shaft angle and the shaftrsquos unbalance value on the dy-namic properties of the coupling system are analysed

2 Mechanical Model of the Spiral Bevel GearFull Coupling System

+e spiral bevel gear full coupling system is a gear-shaft-bearing-box system which is divided into two parts the spiralbevel gear transmission system and the gearbox system Asshown in Figure 1 it contains a hollow input shaft a hollowoutput shaft a pair of nonorthogonal spiral bevel gears fourball bearings and a gearbox connected with the gear trans-mission system +e coordinates O-XYZ in Figure 1 are theglobal coordinate of the entire coupling system

+e three-dimensional model of the system is simplifiedand a mechanical model of the full coupling transmissionsystem of the spiral bevel gear is constructed Because of thedistinct angle between the driven gear shaft and the drivinggear shaft the vibration displacements of the driven gear and itsrotating shaft need to be converted to the global coordinatesystem through a coordinate transform+emechanical modelof the full coupling system of the spiral bevel gear includes theflexible rotating shaft unit the meshing unit the bearingsupport unit and the gearbox unit First the nodes of the beamelements are divided along the axis at the abrupt change of theshaft section the bearing support the gear meshing point thepower point etc +e gearbox system because of its complexstructure is more difficult to model directly using the finiteelement method +e substructure method is employed toconvert the dynamic parameters of the box element to the

connection node of the bearing support and the bearing el-ement is applied to connect the gearbox node and the rotatingshaft node at the bearing position +en the finite elementmethod is utilised to assemble each element matrix and theoverall motion equation of the systemwith the displacement ofeach node as a generalized coordinate can be obtained+rough the above analysis the vibration problem of a fullcoupling system of spiral bevel gears with a continuous massdistribution can be reduced to a vibration problem of a systemwith limited degrees-of-freedom Figure 2 shows the me-chanical model of a full coupling system of the nonorthogonalspiral bevel gear in which nodes 1ndash9 are the nodes corre-sponding to the driving shaftrsquos beam element nodes 10ndash18 arethe nodes corresponding to the beam element of the drivenaxle and nodes 19ndash22 are the nodes of the gearbox unit ac-quired by the substructure method

3 Dynamic Equation of the Spiral Bevel GearFull Coupling System

31 Calculation of the Meshing Stiffness of the Spiral BevelGear +e gearrsquos meshing stiffness refers to the synthetic effectof each pair of gear teeth involved in meshing during themeshing process It is difficult to calculate the meshing stiffnessof the gear teeth using the theoretical analysis method becausethe tooth surface of the spiral bevel gear is a spacemeshing gearMeanwhile the tooth thickness is variable+erefore this studyutilises finite element software to perform dynamic analysisand obtain an accurate meshing stiffness

+e comprehensive elastic deformation δn and contactforce Fn are calculated when multiple pairs of gears mesh atthe same time and then the comprehensive meshingstiffness can be obtained directly +e comprehensiveelastic deformation of the gear teeth includes the contactelastic deformation due to the local Hertz contact δH thedisplacement of the contact position of the gear teethcaused by bending of the gear teeth δb and the deformationof the bearings shafts and supporting structures on thecontact positions δf +e deformation of the bearingsshafts and supporting structures has a relatively smallinfluence on the meshing stiffness +erefore this studyonly takes δH and δb into account so the deformation ofthe gear at a certain moment can be expressed as followsδn 1113936

2i1δHi + 1113936

2i1δbi

First the gearrsquos meshing period is divided into n seg-ments and the meshing stiffness at each time point Ki (i

1 2 3 n) is calculated [27 28] +en a Fourier trans-form is utilized to convert a series of discrete values Ki intoa meshing stiffnessrsquos curve In this article the main pa-rameters of the nonorthogonal spiral bevel gear are shown inTable 1 and then the comprehensive meshing stiffness isanalysed In order to improve the efficiency of calculationnine pairs of gear teeth are selected from the spiral bevel gearpair for the meshing analysis Figure 3 shows the finite el-ement analysis of the meshing stiffness of the spiral bevelgear including the meshing of the gear pair and the contactanalysis of gear pair

Each period T is divided into ten segments on averageand twomeshing periods are taken+en each Ki value at its

Mathematical Problems in Engineering 3

corresponding time ti is calculated By performing a Fouriertransform on a series of discrete stiffness values obtainedfrom the finite element analysis a stiffness curve expressioncan be obtained and the stiffness is transformed into thefollowing form

kh(t) km + 1113944N

i1Ai cos iΩht( 1113857 + Bi sin iΩht( 11138571113858 1113859 (1)

where km is the average value of the time-varying meshingstiffness Ai Bi is the ith-order harmonic amplitude of thetime-varying meshing stiffness Ωh is the excitation

frequency and N represents the order of the Fourier seriesand the series was truncated at N 7 in this article

Figure 4 shows a graph depicting the comprehensivemeshing stiffness of the spiral bevel gear transmissionduring the two meshing cycles

32 Gear Transmission System +e gear transmissionsystem includes the flexible shaft unit spiral bevel gearmeshing unit and the bearing unit which are analysedbelow

X

Y

ZO

Input sha

Output sha Gearbox

(a)

Bearing 1 Bearing 2

Bearing 3

Bearing 4

Gear pair

(b)

(c)

Figure 1 Full coupling system of the spiral bevel gear (a) +e three-dimensional model of the spiral bevel gear fully coupled transmissionsystem (b) Gear transmission system (c) Gearbox system

Meshing element

Box element

Input

Output

Bearing element

Beam elementX

Y

ZO1 2 3 4 75 86

17

9

22

19

11

20

21

12 14 181310 15 16

Figure 2 Mechanical model of the nonorthogonal spiral bevel gear fully coupled transmission system


321 Flexible Shaft Unit +e gear shaftrsquos finite elementmodel is established by the Timoshenko beam elementtaking the shear deformation gyro moment and moment ofinertia into account as shown in Figure 5 which gives thelocal coordinate of the beam element In this article eachnode has six degrees-of-freedom ie the movement alongthe X Y and Z directions and the rotation angle around theX Y and Z directions +e element nodersquos displacement

vector includes 12 degrees-of-freedom which can beexpressed as follows

Xe xi yi zi θxiθyi

θzixi+1 yi+1 zi+1 θx(i+1)

θy(i+1)θz(i+1)

1113960 1113961T

(2)

According to the literature [29 30] the mass matrixMethe stiffness matrix Ke and the gyro matrix Ge of the beamelements can be obtained

It is noted that the directions of the local coordinatesystems of the beam elements are various in the nonparallelshaft gear system +e beam element in the local coordinateneeds to be transformed by the spatial coordinate to theglobal coordinate system +e relationship of the displace-ment of the ith beam element node in the local coordinatesystem and its movement in the global coordinate systemcan be written as follows

xiprime yiprime ziprime θxiprime θyiprime θziprime xi+1prime yi+1prime zi+1prime θxi+1

prime θyi+1prime θzi+1prime1113966 1113967

T

[λ] xi yi zi θxi θyi

θzi xi+1 yi+1 zi+1 θxi+1

θyi+1 θzi+1

1113966 1113967T

(3)

where λ diag[TTTT] and T TcTβTα in which α βand c are the rotation angles of the counterclockwise co-ordinate axis surrounding the global coordinate system

(a)43309 max

38497

33685

28873

24061

19249

14436

96243

48122

000098749 min

(b)

Figure 3 Finite element analysis of the meshing stiffness of thespiral bevel gear pair (a) +e meshing of the gear pairs (b) Contactanalysis of the gear pairs

0 1 2 3Time (s)

4 508

1

12

Mes

hing

stiff

ness

(Nm

)

14

16

18

2 times108

times10ndash4

Fourier curveDiscrete stiffness values

Figure 4 Curve of the comprehensive meshing stiffness of thespiral bevel gear transmission for two meshing cycles

Table 1 Main design parameters of the pinion and gear

Pinion GearYoungrsquos modulus (Pa) E 21e9Poissonrsquos ratio N 03Pressure angle (deg) 225Spiral angle (deg) 35Transverse module (mm) m 56Width of teeth (mm) B 33Shaft angle (deg) 126Damping ratio ξ 005Mean of the transmission error (m) em 2 times 10minus 5

Amplitude of the transmission error (m) er 3 times 10minus 5

Pitch radius (mm) 725 876Number of teeth N1 29 N2 35Hand Right Left

X

Y

Zxi

yi

x

y yi+1xi+1

lsO

θy (i+1)

θz (i+1)

θx (i+1)θxθz

θy

θzi

θyi

θxi

Figure 5 Local coordinate of the beam element


X Y andZ TcTβ andTα are projection vectors along XY and Z respectively and can be given by the followingequations

Tα

1 0 0

0 cos α sin α

0 minus sin α cos α

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tβ

cos β 0 sin β

0 1 0

minus sin β 0 cos β

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tc

cos c sin c 0

minus sin c cos c 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)

Substituting the coordinate transformation matrix (4)into formula (5) the mass matrix the stiffness matrix andthe gyro matrix of the ith beam element node in the globalcoordinate system are calculated which can be described asfollows

Meprime RTMeR

Keprime RTKeR

Geprime RTGeR

(5)

+e matrices of each element of the gear shaft namelyMeprime Keprime and Geprime in the global coordinate system are as-sembled to obtain the mass matrix of the overall gear shaft[Mr] the stiffness matrix of the overall gear shaft [Kr] andthe gyro matrix of the overall gear shaft [Gr] respectively

+e damping of the gear shaft is calculated via Rayleighdamping ie [Cr] α[Mr] + β[Kr] in which α and β are themass damping coefficient and stiffness damping coefficientrespectively which are calculated from the following equations

α 2ω1ω2 ξ1ω2 minus ξ2ω1( 1113857

ω22 minus ω2

1( 1113857

β 2 ξ2ω2 minus ξ1ω1( 1113857

ω22 minus ω2

1

(6)

where ξ1 and ξ2 denote the first and second mode dampingratios respectively generally supposing that ξ1 0005 andξ2 001 ω1 and ω2 are the first- and second-order naturalfrequencies of the rotor system respectively which can beobtained through modal experiments

When each element matrix of the gear shaft is acquiredthe motion differential equation of a single gear shaft canalso be obtained

MreuroX + Cr + ωGr( 1113857 _X + KrX 0 (7)

322 Meshing Unit +e schematic diagram of the dynamicalmodel of the nonorthogonal spiral bevel gear is shown inFigure 6 In this model the local coordinate systems of thepinion and gear are fixed at the center of the two gears namelyOp minus XpYpZp and Og minus XgYgZg respectively +e direction

of the local coordinate system of the driving gear is consistentwith that of the global coordinate system+e dynamicmodel ofthemeshing unit is built based on the lumpedmassmethod andthe lumped mass and the concentrated moment of inertia areutilised to simulate two spiral bevel gears +e model simulatesthe coupling effect during the meshing process through a time-varying meshing stiffness Kh(t) a meshing damping cm anda transmission error e(t) without taking the backlash clearancemeshing friction force and gear meshing position into con-sideration +e bending-torsion-shaft-pendulum coupling vi-bration of the two gears is considered so that the 12-degrees-of-freedom spiral bevel gearmeshing unit can be obtained namelyxp yp zp θpx θpy θpz xq yq zq θqx1113966 θqy θqz

+e spiral bevel gear is a space meshing gear and therelative displacement of the two gears along the meshing linedirection is difficult to obtain directly By rotating the localcoordinate systems of the pinion and gear and performingmultiple coordinate projections the relative displacement ofthe meshing direction of the spiral bevel gear can betransformed into that of the virtual spur gearrsquos meshingdirection +e transformation process of the relative dis-placement of the spiral bevel gear in the direction of themeshing line is shown in Figure 7

+e directional cosine vector of the pinion and gear ni

(i p q) can be calculated with a series of coordinatetransformation as follows

ni nix niy niz1113960 1113961

1

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

TizTiyTix i p g (8)

where Tix Tiy and Tiz indicate the transformation matrixabout the x y and z axes respectively and can be expressedas follows

Tix

1 0 0

0 cos δi sin δi

0 minus sin δi cos δi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tiy

cos minus βi( 1113857 0 sin minus βi( 1113857

0 1 0

minus sin minus βi( 1113857 0 cos minus βi( 1113857

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tiz

cos(pi minus α) sin(pi minus α) 0

minus sin(pi minus α) cos(pi minus α) 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

where δi δ1 and pi minus δ2 (i p q) represent the angles ofthe pinion and gear rotating in the direction of the pitchangles respectively βi β and minus β (i p q) are the midspiralangles of the gears and α denotes the pressure angle

+e directional rotation radii of the meshing forcewhich can be written as follows

λix ni middot uix times ri( 1113857 riyniz minus rizniy

λiy ni middot uiy times ri1113872 1113873 riznix minus rixniz

λiz ni middot uiz times ri( 1113857 rixniy minus riynix

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)


where ri rix riy riz1113960 1113961Trepresents the equivalent posi-

tion vector of the meshing point (this article ignores thevariety of the meshing pointrsquos position) Referring toFigure 6 the equivalent position vectors of the corre-sponding meshing point of the pinion and the gear arerp 0 rp 01113960 1113961

Tand rg 0 rg 01113960 1113961

T respectively in

which rp and rg are the pitch radii of the pinion and gear atthe midpoint of the tooth width respectivelyuxi 1 0 01113858 1113859

T uyi 0 1 01113858 1113859T and uzi 0 0 11113858 1113859

T arethe unit vectors in the X Y and Z directions respectivelyIn the above equation i p g represents the pinion andthe gear respectively

Also the relative displacement of the spiral bevel gear onthe meshing line can be expressed as follows

Xn VXm minus en(t) (11)

where V [npx npy npz λpx λpy λpz minus ngx minus ngy minus ngz λgx

λgy λgz] is the projection vector for the displacement of thespiral bevel gear in all directions to the meshing line Xm

[xp yp zp θpx θpy θpz xg yg zg θgx θgy θgz]T representsthe displacement vector of the spiral bevel gearen(t) em + er sin(Ωt + ϕ) in which em and er represent theaverage value and the amplitude of the transmission errorrespectively Ω indicates the gear meshingrsquos angular

Engagement line

A-A

yg1 A

A

zg1

zgygyp

zpzp1

yp1δ1

δ2

(a)

xp2

xp1

zp1

zp2

β

(b)

Engagementline

θpz2

θgz2

yp2

xp2

yg2

xg2

α

ωp

ωg

(c)

Figure 7 Transformation process of the relative displacement of a spiral bevel gear in the direction of the meshing line (a) Projection alongthe pitch angle direction (b) Projection along the spiral angle direction (c) Projection along the pressure angle direction

Pinion

Gear

θgx

θpx

θgzθgy

θpz

θpy

Yg

Yp

cm

Og

Op

Zg

Zp

Xg

Xp

kh (t)

e (t)

Figure 6 Schematic diagram of the dynamical model of the nonorthogonal spiral bevel gear


frequency and ϕ represents the initial phase angle of theengagement

Combined with the meshing stiffness and relative dis-placement of the spiral bevel gear obtained the meshing forceof the spiral bevel gear pair can be acquired by the followingequation

Fn kh(t)Xn + cm_Xn (12)

where cm represents the meshing damping which can becalculated by the following

cm 2ξ

km1

meqp

+1

meqg

1113888 1113889

1113971

(13)

where ξ is the meshing damping ratio generally taken as003sim017 meqi Izir2i (i p g) denotes the equivalentmass of the gear

According to Newtonrsquos second law the dynamic equationof the spiral bevel gear meshing unit can be written as follows

mp euroxp Fnnpx + mpe1Ω21 cos Ω1t( 1113857

mp euroyp Fnnpy + mpe1Ω21 sin Ω1t( 1113857

mp eurozp Fnnpz

Ipeuroθpx + JpΩ1 _θpy Fnλpx

Ipeuroθpy minus JpΩ1 _θpx Fnλpy

Jpeuroθpz Fnλpz

mg euroxg Fnngx + mge2Ω22 cos Ω2t( 1113857

mg euroyg Fnngy + mge2Ω22 sin Ω2t( 1113857

mg eurozg Fnngz

Igeuroθgx + JgΩ2 _θgy Fnλgx

Igeuroθgy minus JgΩ2 _θgx Fnλgy Jg

euroθgz Fnλgz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

where mp and mg represent the mass of the pinion and gearrespectively Ip and Ig are the diameter rotational inertia ofthe pinion and gear respectively Jp and Jg represent thepolar moment of inertia of the pinion and gear respectivelye1 and e2 denote the pinion and gearrsquos eccentricity re-spectively and Ω1 and Ω2 are the angular velocity of thepinion and gear respectively

Substituting formula (14) into the equations in (15) andarranging the equations into a matrix form the motiondifferential equations of the gear meshing unit can be ob-tained as follows

MmeuroXm + Cm +

Ω1Gm1

Ω2Gm21113890 11138911113888 1113889 _Xm + KmXm

km(t)en(t) + cm _en(t)( 1113857BT

(15)

where _en(t) is the derivative of the comprehensive meshingerror Gm1 and Gm2 represent the gyro matrix of the pinionand the gear respectively and Mm is the mass matrix of themeshing unit which can be expressed as follows

Mm diag mpx mpy mpz Ipx Ipy Ipz mqx mqy mqz Iqx Iqy Iqz1113966 1113967

(16)

+e stiffness matrix Km and the damping matrix Cm ofthe meshing element in the global coordinate system can becalculated as follows

Km kmBTB

Cm cmBTB

(17)

where B [npx npy npz λpx λpy λpz minus npx minus npy minus npz

minus λpx minus λpy minus λpz]T

323 Bearing Unit +e spiral bevel gear transmissionsystem is supported by four angular contact ball bearingsDuring actual operation there is a coupling relationshipbetween the displacements of the bearings and the stiffnessin each direction varies with time +e time-varying stiffnessof the bearings is usually not considered and only the mainstiffness term is retained when calculating the dynamics ofthe gear system +e stiffness matrix of the ball bearing canbe expressed as follows

Kb diag kxx kyy kzz kθx kθy 01113960 1113961 (18)

+e bearing damping takes the same form as the stiffnessand can also be written as follows

Cb diag cxx cyy czz cθx cθy 01113960 1113961 (19)

According to Newtonrsquos second law the dynamic dif-ferential equation of the bearing can be given as follows

MekeuroXek + Cb

_Xek minus _XHk1113872 1113873 + Kb Xek minus XHk( 1113857 0 (20)

where Mek is the mass matrix of the beam element nodesconnected to the kth (k 1 2 3 4) bearingXek _Xek and euroXek

are the displacement velocity and acceleration vectors ofthe beam element nodes connected to the kth (k 1 2 3 4)bearing respectively XHk and _XHk denote the displacementand velocity vectors of the equivalent nodes of the gearboxunit connected to the kth (k 1 2 3 4) bearing respectively

33 Gearbox System In order to establish the dynamicmodel of the gearbox the dynamic parameters of thegearbox need to be obtained including the mass matrix thestiffness matrix and the damping matrix+e gearbox nodesare divided into internal nodes and external nodes when thesubstructure method is employed to extract the gearboxrsquosmatrix information +e external nodes are acquired bycoupling the nodes on the bearing holersquos surface to the holecentre and the remaining finite element nodes are the in-ternal nodes +e gearbox mass stiffness and dampingmatrix can be extracted by a substructure method thatcondenses the internal nodes into external nodes After thatthe mass the stiffness and the damping matrix of theconverted main node of the gearbox are assembled with themass stiffness and damping matrix of the spiral bevel geartransmission system +e substructure method can avoida large-scale finite element matrix and it can be combined


with the dynamic model of the transmission systemconveniently

Figure 8 shows a finite element model diagram of thegearbox using the substructure method+e gearbox is madeof an aluminium alloy with a density of 2700 kgm3 anelastic modulus of 71times 1010 Pa and Poissonrsquos ratio of 033+e model includes 828873 nodes and 585756 elementsTaking the degrees-of-freedom of the translation in eachdirection and rotation around X and Y axes of the bearinghole centre nodes into account meanwhile the screw holesare imposed a fixed constraint at the bottom of the gearbox+erefore a 20times 20 mass matrix MH and a 20times 20 stiffnessmatrix ΚH are formed Rayleigh damping is employed ieCH αHMH + βHKH where αH and βH are the mass co-efficient and the stiffness coefficient respectively

After polycondensation the degrees-of-freedom of thegearbox is significantly reduced and the dynamic equationcan be expressed as follows

MH euroxH + CH _xHk minus _xek( 1113857 + KH xHk minus xek( 1113857 0 (21)

where MH

MH11 MH12 middot middot middot MH1n


⋮ ⋮ ⋱ ⋮MHn1 MHn2 middot middot middot MHnn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is the equivalent

mass matrix of the gearbox substructure in which n is the

number of the bearings KH

KH11 KH12 middot middot middot KH1n


⋮ ⋮ ⋱ ⋮KHn1 KHn2 middot middot middot KHnn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

which is the equivalent stiffness matrix of the gearboxsubstructure CH aHMH + bHKH which is the equivalentdamping matrix of the gearbox substructure XH _XH andeuroXH are the displacement velocity and acceleration vectorsof the gearboxrsquos main node respectively

34 Global Dynamics Equation of the System After obtainingthe dynamic equations of each unit the dynamic equations ofthe whole system can be established Combined with thedifferential equation of the gear shaft unit (7) the differentialequation of the meshing element (15) the differential equationof the bearing element (20) and the motion equation of thegearbox element (21) the full coupling differential equations ofthe entire system can be written as followsMeurox(t) + (C + ΩG)[ _x(t) minus _e(t)] + K(t)[x(t) minus e(t)] Q(t)

(22)where M C G and K are the generalized mass dampinggyroscopic and stiffness matrices respectively X _X and euroXare the generalized displacement velocity and accelerationvectors respectively Q is the excitation force vector of thesystem

4 Solution and Analysis of the DynamicEquation of the Full Coupling System of theSpiral Bevel Gear

From the derivation process and results it can be determinedthat the full coupling system is a complex coupling dynamicsquestion and it is complicated to obtain satisfactory resultsusing the theoretical analysis method Meanwhile the stiffnessmatrix of the entire dynamic equation of the system is a sin-gular matrix ie the system has rigid body displacement Inthis article the rigid body displacement is eliminated by im-posing a fixed constraint on the torsional direction of theoutput end Although the absolute displacement of each de-grees-of-freedom will deviate from the actual one the relativemicrovibration displacement has little effect +en the New-mark method is employed to calculate the numerical solutionof the full coupling system In order to obtain an accuratedynamic response the step size is 1200 of the gear meshingfrequency+e structural parameters of the rotating shaft of thefull coupling system are shown in Table 2 +e relevant pa-rameters of the spiral bevel gear are revealed in Table 1 inwhich the time-varying meshing stiffness of the gear pair isobtained from Part 31 the stiffness and damping of the ballbearings are shown in Table 3 and the structural parameters ofthe gearbox are acquired from Section 33

41 Effect of the Gearbox Flexibility on the DynamicCharacteristics of the Spiral Bevel GearTransmission System

411 Influence of the Gearbox Flexibility on the MeshingForce In order to investigate the gearbox flexibilityrsquos effect

Rigid coupling

(a)

External node

(b)

Figure 8 Finite element model diagram of the gearbox using thesubstructure method


on the meshing force of the gear transmission system themeshing force of the uncoupling model (only consideringthe spiral bevel gear transmission system) and the couplingmodel (the spiral bevel gear-gearbox coupling model con-sidering the gearbox flexibility) are obtained Figure 9 showsthe amplitude-frequency response curve of the meshingforce of the uncoupling and coupling mode at a rotationspeed range of 200sim8000 rmin As seen in Figure 9 thestiffness of the gearbox is introduced into the spiral bevelgear transmission system which reduces the overall stiffnessof the system and absorbs and transmits part of the energyresulting in a reduction in the meshing force at all rotationalspeeds However the magnitude of reduction in the meshingforce is different at various rotational speeds By comparingFigure 9(a) with Figure 9(b) it can be observed that with thevariety of the torque the effect of the gearbox flexibility onmeshing force is also varied Under torques of 300Nm and600Nm the meshing forces of the uncoupling model andthe coupling model are calculated respectively at rotatingspeeds of 3000 rmin 4700 rmin and 7100 rmin +especific values are shown in Table 4 By analysing the data inTable 4 it is demonstrated that the effect of the gearboxflexibility on the meshing force is also larger when the torqueis larger at the same speed

412 Influence of the Gearbox Flexibility on the BearingForce Similarly to study the influence of the gearboxflexibility on the bearing force the differential equations ofthe system are solved in order to obtain the amplitude-frequency response curves of the bearing force of theuncoupling model and coupling model in the speed range of200sim8000 rmin as shown in Figure 10(a) It can be observedthat with the variety of the rotating speed the bearing forcedecreases in the entire rotating speed range while taking thegearbox stiffness into account and the most significantdecrease of the bearing force occurs at the resonance point+e main reason for this is that the vibration of the gearboxis introduced into the entire dynamic system and the variety

of the system stiffness affects the supporting force signifi-cantly and this also means that the force transmitted by thetransmission system to the other structural systems is re-duced considerably +e gearbox flexibility in the trans-mission system helps obtain more accurate dynamic analysisresults which provides a theoretical basis for the optimi-zation design and vibration and noise reduction of thesystem+en the amplitude-frequency response curve of thebearing force of the uncoupling and coupling gearbox in therotating speed range of 200sim8000 rmin is obtained when thetorque increases which is revealed in Figure 10(b) To fa-cilitate the analysis under torques of 300Nm and 600Nmthe bearing forces of uncoupling and coupling the gearbox atrotating speeds of 3000 rmin 4700 rmin and 7100 rminare calculated respectively as shown in Table 5 By ob-serving the data in the table it can be determined that thechange rate of the bearing force is large when the torque islarge at the same speed

According to the above analysis the flexibility of thegearbox has an impact on the meshing force and the bearingforce of which the influence on the bearing force isprominent In the analysis of the overall dynamic response ofthe coupling system supported by the thin-walled gearboxthe effect of the gearbox flexibility needs to be taken intoaccount

42 Influence of the Shaft Angle on the Dynamic Character-istics of the Spiral Bevel Gear Full Coupling System Based onthe research in the previous section the influence of theflexibility of the thin-walled gearbox on the spiral bevel geartransmission system cannot be ignored +us this sectiondirectly studies the effect of the shaft angle on the dynamiccharacteristics of the spiral bevel gear-shaft-bearing-gearboxfull coupling system

421 Effect of the Shaft Angle on the Axis Trajectory of theGear Studying the axis locus of the critical nodes canpredict whether the geared rotor transmission system willencounter rub-impact during operation or not and theinvestigation of the gearrsquos axis locus is especially helpful inunderstanding the meshing status of the gear It is assumedthat the speed of the driving spiral bevel gear is 4200 rminthe torque is 600Nm and the other parameters keep in-variant +e variation of the axial trajectory of the drivenspiral bevel gear is analysed when the shaft angles of thespiral bevel gear transmission system are 70 90 110 and126 as shown in Figure 11+e centre of the trajectory of thedriven spiral bevel gear is varied when the shaft angle hasdiverse values as shown in Figure 11(a) +e centres of theaxis locus under each shaft angle are set to the same positionto observe the variety of the direction and size of the axislocus It can be seen in Figure 11(b) that the variation in theshaft angle not only affects the direction of the axis locus butalso alters the amplitude of the axis locus +e calculationshows that the axis locus of the other nodes will also changewith the variety of the shaft angle +erefore the impact ofthe shaft angle must be taken into account when studying the

Table 2 Shaft element parameters

Driving and driven shaftNode to node 1-2 2-3 3-4 4ndash9 10ndash15 15-16 16-17 17-18Outer diameter(mm) 48 57 60 75 75 60 57 48

Inner diameter(mm) 44 45 45 52 52 45 45 44

Shaft length(mm) 20 26 28 108 140 28 26 20

Table 3 Bearing parameters

Stiffness Bearing 1-2 Bearing 3-4kxx kyy (Nm) 2times108 2times108

kzz (Nm) 1times 108 1times 108

kθxθx kθyθy (Nmrad) 1times 106 1times 106

Damping Bearing 1-2 Bearing 3-4c (Nsm) 1times 103 1times 103


vibration features of the nonorthogonal spiral bevel gear-shaft-bearing-gearbox full coupling system

422 Effect of the Shaft Angle on the Meshing Force andBearing Force Supposing that the driving gearrsquos rotation

speed is 4200 rmin the torque is 600 Nm and otherparameters kept invariant then the meshing force andthe bearing force variation in each direction are obtainedwhen the shaft angle is 70 90 110 and 126 respectivelyas shown in Figure 12 It can be demonstrated from the

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

BoxRigid

(a)

times104

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

65

7

75

8

85

9

95

10

105

Mes

hing

forc

e (N

)

(b)

Figure 9 Amplitude-frequency response curve of the meshing forces of the uncoupling and coupling model under different torques (a)T 300Nm (b) T 600Nm

Table 4 Change rate of the meshing force under different models

Torque (Nm) Speed (rmin) Meshing force of coupling model (N) Meshing force of uncoupled model (N) Rate of change ()

3003000 46570 49240 544700 66190 69000 417100 42130 44920 62

6003000 71170 76010 644700 95210 100100 487100 70070 75880 76

Speed (rmin)

0100020003000400050006000700080009000

10000

Bear

ing

forc

e (N

)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000

(a)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

0

2000

4000

6000

8000

10000

12000

Bear

ing

forc

e (N

)

(b)

Figure 10 Amplitude-frequency response curves of the bearing forces of the uncoupling and coupling model under different torques(a) T 300Nm (b) T 600Nm


figure that the meshing force and the bearing force of thespiral bevel gear pair are various with the variety of the shaftangles +e meshing force varies because the pitch angles ofthe pinion and the gear are various under different shaftangles and the projection results of the vibration displace-ment of the gear pair in various directions to the meshing linevary At the same time the forcersquos components decomposedinto diverse directions are also different resulting in varia-tions in the bearing force for each direction In conclusion thevibration behaviours of the spiral bevel gear pair and the fullcoupling system will vary with the variety of the shaft angle

43 Influence of Unbalance on the Dynamic Characteristics ofa Spiral Bevel Gear Full Coupling System

431 Ce Influence of the Unbalance on Frequency ResponseCharacteristics of the Coupling System An unbalanceamount which is equal to 5 times 10minus 4 kgmiddotm is applied to thedrive shaftrsquos node 4 with no imbalance given to the othernodes +e driving shaft speed n1 is in the range of200sim800 rmin and the three-dimensional spectrum of thefull coupling system at the driving and driven gears areobtained respectively as shown in Figure 13 As seen fromFigure 13(a) because of the coupling effect between therotor and the gear the X-direction displacement vibrationspectrum of the driving gear includes the rotation fre-quency of the driving shaft f1 the gear meshing frequencyfm and its harmonic frequency 2fm Meanwhile it can beseen from Figure 13(b) that the torsional vibration spec-trum of the driving gear also includes the rotation fre-quency of the driving shaft f1 the meshing frequency fmand its harmonic frequency 2fm As seen fromFigure 13(c) the X-direction displacement vibrationspectrum of the driven gear includes the meshing fre-quency fm and its harmonic frequency 2fm+e imbalanceof the driving shaft has little effect on the vibration dis-placement of the driven gear in the X direction where therotation frequency amplitude of the driving shaft is smallwhich is not revealed in the figure As seen fromFigure 13(d) because of the coupling effect of bending andtorsion of the spiral bevel gear pair the driven gearrsquostorsional vibration not only contains the meshing fre-quency fm and its harmonic frequency 2fm but also thedriving shaftrsquos rotation frequency f1 where the rotationalfrequency amplitude increases with the rise of the speed+us the effect of the imbalance of the rotating shaft on thegearrsquos vibration in the torsional direction is more prom-inent than that on the lateral vibration

+e unbalance amount applied on the driving shaftrsquosnode 4 is changed to 10 times 10minus 4kgmiddotm and the other condi-tions remain unchanged +e speed of the driving shaft n1varies from 200 rmin to 8000 rmin and the three-di-mensional spectrum of the full coupling system at the pinionand gear are acquired as shown in Figure 14 ComparingFigures 13(a) and 13(b) and 14(a) and 14(b) it can be

ndash58 ndash56 ndash54 ndash52 ndash5 ndash48 ndash46 ndash44Gear2 X

ndash3

ndash2

ndash1

0

1

2

Gea

r2 Y

70deg90deg

110deg126deg

times10ndash5

times10ndash5

(a)

70deg90deg

110deg126deg

ndash4 ndash2 0 2 4Gear2 X

ndash1

ndash05

0

05

1

15

Gea

r2 Y

times10ndash6

times10ndash6

(b)

Figure 11 Axis trajectory diagram of a driven gear under variousshaft angles

Table 5 Change rate of the bearing force under different models

Torque (Nm) Speed (rmin) Bearing force of coupling model (N) Bearing force of uncoupling model (N) Rate of change ()

3003000 2213 3307 334700 6393 9808 357100 1278 1893 32

6003000 2741 4292 374700 7964 11700 407100 1545 2431 36


demonstrated that the amplitude corresponding to the ro-tating frequency of the driving gear in the lateral and tor-sional vibration spectrograms increases significantly whenthe imbalance on the driving shaft is rising while the am-plitude corresponding to the meshing frequency hardlychanges It can be seen from Figure 14(c) that the drivengearrsquos vibration spectrum exhibits the driving shaftrsquos rotatingfrequency f1 at a high rotational speed indicating that thecoupling effect of the driving shaftrsquos imbalance on the drivengearrsquos vibration displacement is also risen with an increase inthe unbalance of the driving shaft Comparing Figures 13(d)and 14(d) it can be revealed that the amplitude of the ro-tation frequency f1 rises in the driven gearrsquos torsional vi-bration spectrum which also indicates that the increase ofthe driving shaftrsquos unbalance increases the effect of thedriven gearrsquos torsional vibration and the increased ampli-tude is more obvious than that of the transverse vibration

432 Influence of the Unbalance on the Meshing Force andBearing Force In order to analyse the influence of theunbalance of the rotating shaft on the meshing force and thebearing force the unbalance amount on the driving shaftrsquosnode 4 is taken as 0 5 times 10minus 4 kgmiddotm and 10 times 10minus 4 kgmiddotmrespectively +e amplitude-frequency response curves of themeshing force and the bearing force within a rotation speedrange of 200sim8000 rmin are calculated as shown in Figure 15It can be revealed from the figure that the meshing force andthe bearing force have little influence at low rotational speedswith the increase of the eccentricity however the eccentricityhas a great influence on the meshing force and bearing forcewhen speed continuously increase +erefore when the speedof the full coupling system is high the eccentric amount of therotating shaft should be reduced to diminish its impact on themeshing force and bearing force so as to ensure the stable andreliable operation of the system

0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)

Figure 12 Meshing force and bearing force under various shaft angles (a) Meshing force (b) Bearing force in the X direction (c) Bearingforce in the Y direction (d) Bearing force in the Z direction


Frequency (Hz) Rotatio

nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)

Figure 13 +ree-dimensional spectrum of the coupling system as the unbalance is 5 times 10minus 4 kgmiddotm (a) Lateral vibration of the driving gear in the Xdirection (b) Torsional vibration of the driving gear (c) Lateral vibration of the driven gear in theXdirection (d) Torsional vibration of the driven gear

1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm

Frequency (Hz) Rotational s

peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion

A full coupling dynamic model of the spiral bevel gear wasbuilt taking the flexibility of the shaft the bearing sup-porting the unbalance amount a time-varying meshingstiffness a comprehensive transmission error and thegearbox flexibility into account +e Timoshenko beam el-ement was employed to simulate the flexible shaft in themodel and the gears and bearings were treated according tothe concentrated mass Because of the complex structure ofthe gearbox system the substructure method was applied toextract its mass stiffness and damping matrix +e fullcoupling dynamic equations of the spiral bevel gear-shaft-bearing-gearbox were constructed and the coupling dy-namic equations were solved via the Newmark numericalalgorithm +e following conclusions were obtained

+e stiffness of the gearbox affects the meshing force andbearing force of the spiral bevel gear transmission system+e meshing force and bearing forces are smaller than thoseof the rigid support when considering the gearbox flexibilitywherein the gearbox flexibility has a more significant effecton the bearing force

+e shaft angle of the full coupling system of thenonorthogonal spiral bevel gear affects the axis locusmeshing force and bearing force simultaneously For var-ious shaft angles the position size and direction of thegearrsquos axis locus vary Because the pitch angles of the pinionand gear are different under various shaft angles themeshing force and bearing force of the system also vary Asa result the vibrational characteristics of the gear pair andthe entire coupling system also change

Because of the coupling of bending and torsion of thesystem the unbalance of the rotating shaft exerts an effect onthe vibration of the spiral bevel gear transmission system inall directions where the torsional vibration has the greatestinfluence and the effect increase with the variety of

unbalance +e unbalance of the rotating shaft also affectsthe meshing force and the bearing force +e influence islittle at a low rotational speed However the unbalanceexerts a significant effect on the meshing force and bearingforce when speed continuously increases

Data Availability

All the data studied in this article are derived from a specifictype of helicopter tail drive system ensuring that the re-search content is close to the actual work situation

Conflicts of Interest

+e authors declare that they have no potential conflicts ofinterest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (nos 51775265 and 51775277)

References

[1] S Wang Y W Shen and H J Dong ldquoNonlinear dynamicalcharacteristics of a spiral bevel gear system with backlash andtime-varying stiffnessrdquo Chinese Journal of Mechanical Engi-neering vol 39 no 02 pp 28ndash31 2003 in Chinese

[2] Z H Feng and C S Song ldquoEffects of geometry design pa-rameters on the static strength and dynamics for spiral bevelgearrdquo International Journal of Rotating Machinery vol 2017Article ID 6842938 8 pages 2017

[3] L H Wang Y Y Huang R F Li and T J Lin ldquoStudy onnonlinear vibration characteristics of spiral bevel trans-mission systemrdquo ChinaMechanical Engineering vol 18 no 3pp 260ndash264 2007 in Chinese

[4] J-Y Tang Z-H Hu L-J Wu and S-Y Chen ldquoEffect of statictransmission error on dynamic responses of spiral bevel

Rotational speed (rmin)

0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)

05 times 10ndash4 kgm10 times 10ndash4 kgm

0 2000 4000 6000 8000

(a)


0 2000 4000 6000 8000Rotational speed (rmin)

4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)

Figure 15 Amplitude-frequency response curves of themeshing force and the bearing force (a) Amplitude-frequency response curve of thebearing force (b) Amplitude-frequency response curve of the meshing force


gearsrdquo Journal of Central South University vol 20 no 3pp 640ndash647 2013

[5] Y S Fan S M Wang Z Yang and H X Liu ldquoVibrationanalysis method for multi-rotor system geared by angularspiral bevel gearsrdquo Journal of Harbin Institute of Technologyvol 43 no 3 pp 111ndash116 2011 in Chinese

[6] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[7] J Yang and T C Lim ldquoInfluence of propeller shaft bendingvibration on drivetrain gear dynamicsrdquo International Journalof Automotive Technology vol 16 no 1 pp 57ndash65 2015

[8] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 1-2 pp 302ndash329 2007

[9] J L Xu L Wan andW X Luo ldquoInfluence of bearing stiffnesson the nonlinear dynamics of a shaft-final drive systemrdquoShock and Vibration vol 2016 Article ID 3524609 14 pages2016

[10] J L Xu F C Zeng and X G Su ldquoCoupled bending-torsionalnonlinear vibration and bifurcation characteristics of spiralbevel gear systemrdquo Shock and Vibration vol 2017 Article ID6835301 14 pages 2017

[11] D Yassine H AhmedW Lassaad and HMohamed ldquoEffectsof gear mesh fluctuation and defaults on the dynamic be-havior of two-stage straight bevel systemrdquo Mechanism andMachine Ceory vol 82 no 24 pp 71ndash86 2014

[12] T Peng CoupledMulti-Body Dynamic and Vibration Analysisof Hypoid and Bevel Geared Rotor System University ofCincinnati Cincinnati OH USA 2010

[13] Y N Li G Y Li and L Zheng ldquoInfluence of asymmetricmesh stiffness on dynamics of spiral bevel gear transmissionsystemrdquo Mathematical Problems in Engineering vol 2010Article ID 124148 13 pages 2010

[14] J-S Wu and I-H Yang ldquoComputer method for torsion-and-flexure-coupled forced vibration of shafting system withdampingrdquo Journal of Sound and Vibration vol 180 no 3pp 417ndash435 1995

[15] S-T Choi and S-Y Mau ldquoDynamic analysis of geared rotor-bearing systems by the transfer matrix methodrdquo Journal ofMechanical Design vol 123 no 4 pp 562ndash568 2001

[16] S V Neriya R B Bhat and T S Sankar ldquo+e coupledtorsional-flexural vibration of a geared shaft system usingfinite element methodrdquo Ce Shock and Vibration Bulletinvol 55 no 3 pp 13ndash251 1985

[17] A Kahraman H Nevzat Ozguven D R Houser andJ J Zakrajsek Dynamic Analysis of Geared Rotors by FiniteElements NASA Washington DC USA 2015

[18] M Kubur A Kahraman D M Zini and K Kienzle ldquoDy-namic analysis of a multi-shaft helical gear transmission byfinite elements model and experimentrdquo Journal of Vibrationand Acoustics vol 126 no 7 pp 398ndash406 2004

[19] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science andTechnology vol 27 no 11 pp 3231ndash3242 2013

[20] S Chen J Tang Y Li and Z Hu ldquoRotordynamics analysis ofa double-helical gear transmission systemrdquo Meccanicavol 51 no 1 pp 251ndash268 2016

[21] M Li andH Y Hu ldquoDynamic analysis of a spiral bevel-gearedrotor-bearing systemrdquo Journal of Sound and Vibrationvol 259 no 3 pp 605ndash624 2003

[22] X Hua T C Lim T Peng andW EWali ldquoDynamic analysisof spiral bevel geared rotor systems applying finite elementsand enhanced lumped parametersrdquo International Journal ofAutomotive Technology vol 13 no 1 pp 97ndash107 2012

[23] S D Yavuz Z B Saribay and E Cigeroglu ldquoNonlinear time-varying dynamic analysis of a spiral bevel geared systemrdquoNonlinear Dynamics vol 92 no 4 pp 1901ndash1919 2018

[24] F K Choy Y K Tu J J Zakrajsek and D P TownsendldquoEffects of gear box vibration and mass imbalance on thedynamics of multi-stage gear transmissionsrdquo NASA Wash-ington DC USA 1991

[25] F K Choy Y K Ruan R K Tu J J Zakrajsek andD P Townsend ldquoModal analysis of multistage gear systemscoupled with gearbox vibrationsrdquo NASA Washington DCUSA 1991

[26] Y Ren S Chang G Liu L Wu and T C Lim ldquoImpedancesynthesis based vibration analysis of geared transmissionsystemrdquo Shock and Vibration vol 2017 Article ID 484653214 pages 2017

[27] J Wei A Zhang D Qin et al ldquoA coupling dynamics analysismethod for a multistage planetary gear systemrdquo Mechanismand Machine Ceory vol 110 pp 27ndash49 2017

[28] F Concli L Cortese R Vidoni F Nalli and G Carabin ldquoAmixed FEM and lumped-parameter dynamic model forevaluating the modal properties of planetary gearboxesrdquoJournal of Mechanical Science and Technology vol 32 no 7pp 3047ndash3056 2018

[29] J R Hutchinson ldquoShear coefficients for Timoshenko beamtheoryrdquo Journal of Applied Mechanics vol 68 no 1pp 87ndash92 2001

[30] M I Friswell J E T Penny S D Garvey and A W LeesDynamics of Rotating Machines Cambridge University PressLondon UK 2010


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axial coupling vibration of a 7-degrees-of-freedom spiralbevel gear +e torsional lateral and axial vibration dis-placements and velocities of the spiral bevel gear systemunder various working conditions are obtained by theoperator algorithm +e results show that with the varietyof the meshing frequency the system enters into chaosthrough period-doubling bifurcation with the variety ofthe support stiffness the system changes into chaosthrough quasi-period bifurcation and in the process of themeshing frequency varying jump phenomena exist in thesystem Feng and Song [2] considered the time-varyingmeshing force of the gear the position of the meshingpoint the transmission error the direction of the frictionforce and the friction coefficient and then establisheda dynamic model of concentrated parameters On thisbasis the effects of the dynamic meshing force and dynamictransmission error on the system response are investigatedWang et al [3] analysed the torsional vibration of the spiralbevel gear transmission system with backlash and time-varying meshing stiffness+e results showed that when thegear system reciprocates the backlash and time-varyingmeshing stiffness will produce a coupling effect causing thegear system to show obvious nonlinear characteristics +einternal excitation frequency the meshing stiffness am-plitude the initial condition and the damping ratio all havean essential impact on the systemrsquos dynamic features +evariation of each parameter makes the systemrsquos responseappear in harmonic period solution and chaotic solutionTang et al [4] considering such influencing factors as thetime-varying meshing stiffness and gear clearance analysedthe effect of the static transmission error on the nonlineardynamic response of a spiral bevel gear system +e dy-namic response bifurcation diagram time domain re-sponse phase diagram and Poincare map are all obtainedby solving the dynamic equations Fan et al [5] examinedthe linear dynamics of the transmission system of a non-orthogonal spiral bevel gear and analysed the systemrsquosinherent characteristics steady-state response and axialtrajectory Cheng and Lim [6] built a three-dimensionalmulti-degrees-of-freedom model of the hypoid gear andanalysed the dynamic coupling and vibrational responsesensitivity using the response function of the meshingforce +en the influences of the system parameters on thedynamic response of the hypoid gear were studied and itwas proven that the design parameters and working con-ditions have significant impacts on the modal and vibrationresponse Yang and Lim [7] considered the nonlinearclearance and time-varying meshing stiffness and solvedthe dynamic meshing force and vibration response by thevariable step size RungendashKutta method +e analysisshowed that damping could dramatically reduce the dy-namic meshing force Wang et al [8] established a hypoidgear dynamics model with a time-varying meshing forceconsidering the effect of the nonlinear clearance andstudied the impact of the different meshing waveforms onthe dynamic response of gear dynamics Xu et al [9 10]constructed the bending-torsion-axial coupling dynamicmodel of the multi-freedom spiral bevel gear through thelumped mass method and then studied the chaotic and

bifurcation properties of the system Yassine et al [11]constructed a dynamic model of a two-stage bevel geartransmission system mainly considering the effect of thetime-varying meshing stiffness +e Newmark numericalalgorithm was utilised to solve the equations and thedynamic features of the system were analysed in the timedomain and the frequency domain Peng [12] createda multibody dynamic model of hypoid gears and studiedthe effects of misalignment torque and eccentricity on thedynamic response of the system Li et al [13] investigatedthe effect of an asymmetric time-varying meshing stiffnesson the spiral bevel gear transmission system It was de-termined that the meshing stiffness has a higher impact onthe driving side than that on the driven side Under a lightload the rise in the meshing stiffness on the drive sidedeteriorates the vibration stability of the transmissionsystem Wu and Yang [14] utilised the extended transfermatrix method to analyse the bending-torsional-coupledvibration of the multi-degrees-of-freedom damped shaftingcaused by external excitation It was shown that the naturalfrequency of bending-torsion coupling is close to thenatural frequency of torsion Choi and Mau [15] analysedthe dynamic behaviours of the gear-coupled rotor systemby using the transfer matrix method +e Timoshenkobeam element was employed to model the rotating shaftand the natural frequency and vibration mode were ac-quired by studying the bending-torsion-coupled vibrationof the system Neriya et al [16] adopted the finite elementmethod to simulate the bending-torsion coupling vibrationof the gear system comparatively early +e finite elementmethod has been widely utilized by scholars both do-mestically and abroad because this method can bettersimulate the influence of a flexible shaft and the bearingparameters on the dynamic response of the gear pair[17ndash20] Li and Hu [21] discussed the linear dynamicproperties of the bending-torsional coupling of the rotor-bearing system with spiral bevel gears via the finite elementmethod after considering the generalized meshing re-lationship between the conical teeth Hua et al [22]employed the finite element method and the lumped pa-rameter method to analyse the dynamics of the spiral bevelgear-rotor system Yavuz et al [23] aking the backlashtime-varying meshing stiffness and the transmission errorinto account built a dynamic model of the spiral bevel gearbased on the finite element method solved differentialequations using the harmonic balance method and studiedthe effects of backlash time-varying meshing stiffness anda bearing stiffness on the system dynamics Choy et al[24 25] examined the effect of the housingrsquos vibration onthe dynamic features of the parallel shaft gear transmissionsystem via the modal synthesis method but the gearboxwas simplified dramatically Ren et al [26] established thegear-shaft-bearing-box coupling dynamic model of theparallel shaft via the impedance synthesis method takingthe dynamical coupling between the transmission systemand the housing into consideration and investigated theinfluence factor of the natural frequency meshing forceand bearing force According to the literature retrieved themodelling methods of the gear transmission systemrsquos











2i1δHi + 1113936

2i1δbi






kh(t) km + 1113944N






X

Y

ZO

Input sha

Output sha Gearbox

(a)

Bearing 1 Bearing 2

Bearing 3

Bearing 4

Gear pair

(b)

(c)


Meshing element

Box element

Input

Output

Bearing element

Beam elementX

Y

ZO1 2 3 4 75 86

17

9

22

19

11

20

21

12 14 181310 15 16







θy(i+1)θz(i+1)

1113960 1113961T

(2)





T



θyi+1 θzi+1

1113966 1113967T

(3)


(a)43309 max

38497

33685

28873

24061

19249

14436

96243

48122

000098749 min

(b)


0 1 2 3Time (s)

4 508

1

12

Mes

hing

stiff

ness

(Nm

)

14

16

18

2 times108

times10ndash4







X

Y

Zxi

yi

x

y yi+1xi+1

lsO

θy (i+1)

θz (i+1)

θx (i+1)θxθz

θy

θzi

θyi

θxi




Tα

1 0 0

0 cos α sin α


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tβ

cos β 0 sin β

0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tc

cos c sin c 0

minus sin c cos c 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)


Meprime RTMeR

Keprime RTKeR

Geprime RTGeR

(5)



α 2ω1ω2 ξ1ω2 minus ξ2ω1( 1113857

ω22 minus ω2

1( 1113857

β 2 ξ2ω2 minus ξ1ω1( 1113857

ω22 minus ω2

1

(6)









ni nix niy niz1113960 1113961

1

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

TizTiyTix i p g (8)


Tix

1 0 0

0 cos δi sin δi


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tiy


0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tiz



0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)






⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)




Tand rg 0 rg 01113960 1113961

T respectively in


T uyi 0 1 01113858 1113859T and uzi 0 0 11113858 1113859







Engagement line

A-A

yg1 A

A

zg1

zgygyp

zpzp1

yp1δ1

δ2

(a)

xp2

xp1

zp1

zp2

β

(b)

Engagementline

θpz2

θgz2

yp2

xp2

yg2

xg2

α

ωp

ωg

(c)


Pinion

Gear

θgx

θpx

θgzθgy

θpz

θpy

Yg

Yp

cm

Og

Op

Zg

Zp

Xg

Xp

kh (t)

e (t)







cm 2ξ

km1

meqp

+1

meqg

1113888 1113889

1113971

(13)





mp eurozp Fnnpz



Jpeuroθpz Fnλpz



mg eurozg Fnngz



euroθgz Fnλgz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



MmeuroXm + Cm +

Ω1Gm1

Ω2Gm21113890 11138911113888 1113889 _Xm + KmXm

km(t)en(t) + cm _en(t)( 1113857BT

(15)



(16)


Km kmBTB

Cm cmBTB

(17)








MekeuroXek + Cb










where MH




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦








Rigid coupling

(a)

External node

(b)












Shaft length(mm) 20 26 28 108 140 28 26 20










0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

BoxRigid

(a)

times104

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

65

7

75

8

85

9

95

10

105

Mes

hing

forc

e (N

)

(b)




3003000 46570 49240 544700 66190 69000 417100 42130 44920 62

6003000 71170 76010 644700 95210 100100 487100 70070 75880 76

Speed (rmin)

0100020003000400050006000700080009000

10000

Bear

ing

forc

e (N

)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000

(a)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

0

2000

4000

6000

8000

10000

12000

Bear

ing

forc

e (N

)

(b)








ndash3

ndash2

ndash1

0

1

2

Gea

r2 Y

70deg90deg

110deg126deg

times10ndash5

times10ndash5

(a)

70deg90deg

110deg126deg


ndash1

ndash05

0

05

1

15

Gea

r2 Y

times10ndash6

times10ndash6

(b)




3003000 2213 3307 334700 6393 9808 357100 1278 1893 32

6003000 2741 4292 374700 7964 11700 407100 1545 2431 36




0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)




nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018

















2i1δHi + 1113936

2i1δbi






kh(t) km + 1113944N






X

Y

ZO

Input sha

Output sha Gearbox

(a)

Bearing 1 Bearing 2

Bearing 3

Bearing 4

Gear pair

(b)

(c)


Meshing element

Box element

Input

Output

Bearing element

Beam elementX

Y

ZO1 2 3 4 75 86

17

9

22

19

11

20

21

12 14 181310 15 16







θy(i+1)θz(i+1)

1113960 1113961T

(2)





T



θyi+1 θzi+1

1113966 1113967T

(3)


(a)43309 max

38497

33685

28873

24061

19249

14436

96243

48122

000098749 min

(b)


0 1 2 3Time (s)

4 508

1

12

Mes

hing

stiff

ness

(Nm

)

14

16

18

2 times108

times10ndash4







X

Y

Zxi

yi

x

y yi+1xi+1

lsO

θy (i+1)

θz (i+1)

θx (i+1)θxθz

θy

θzi

θyi

θxi




Tα

1 0 0

0 cos α sin α


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tβ

cos β 0 sin β

0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tc

cos c sin c 0

minus sin c cos c 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)


Meprime RTMeR

Keprime RTKeR

Geprime RTGeR

(5)



α 2ω1ω2 ξ1ω2 minus ξ2ω1( 1113857

ω22 minus ω2

1( 1113857

β 2 ξ2ω2 minus ξ1ω1( 1113857

ω22 minus ω2

1

(6)









ni nix niy niz1113960 1113961

1

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

TizTiyTix i p g (8)


Tix

1 0 0

0 cos δi sin δi


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tiy


0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tiz



0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)






⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)




Tand rg 0 rg 01113960 1113961

T respectively in


T uyi 0 1 01113858 1113859T and uzi 0 0 11113858 1113859







Engagement line

A-A

yg1 A

A

zg1

zgygyp

zpzp1

yp1δ1

δ2

(a)

xp2

xp1

zp1

zp2

β

(b)

Engagementline

θpz2

θgz2

yp2

xp2

yg2

xg2

α

ωp

ωg

(c)


Pinion

Gear

θgx

θpx

θgzθgy

θpz

θpy

Yg

Yp

cm

Og

Op

Zg

Zp

Xg

Xp

kh (t)

e (t)







cm 2ξ

km1

meqp

+1

meqg

1113888 1113889

1113971

(13)





mp eurozp Fnnpz



Jpeuroθpz Fnλpz



mg eurozg Fnngz



euroθgz Fnλgz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



MmeuroXm + Cm +

Ω1Gm1

Ω2Gm21113890 11138911113888 1113889 _Xm + KmXm

km(t)en(t) + cm _en(t)( 1113857BT

(15)



(16)


Km kmBTB

Cm cmBTB

(17)








MekeuroXek + Cb










where MH




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦








Rigid coupling

(a)

External node

(b)












Shaft length(mm) 20 26 28 108 140 28 26 20










0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

BoxRigid

(a)

times104

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

65

7

75

8

85

9

95

10

105

Mes

hing

forc

e (N

)

(b)




3003000 46570 49240 544700 66190 69000 417100 42130 44920 62

6003000 71170 76010 644700 95210 100100 487100 70070 75880 76

Speed (rmin)

0100020003000400050006000700080009000

10000

Bear

ing

forc

e (N

)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000

(a)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

0

2000

4000

6000

8000

10000

12000

Bear

ing

forc

e (N

)

(b)








ndash3

ndash2

ndash1

0

1

2

Gea

r2 Y

70deg90deg

110deg126deg

times10ndash5

times10ndash5

(a)

70deg90deg

110deg126deg


ndash1

ndash05

0

05

1

15

Gea

r2 Y

times10ndash6

times10ndash6

(b)




3003000 2213 3307 334700 6393 9808 357100 1278 1893 32

6003000 2741 4292 374700 7964 11700 407100 1545 2431 36




0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)




nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018









kh(t) km + 1113944N






X

Y

ZO

Input sha

Output sha Gearbox

(a)

Bearing 1 Bearing 2

Bearing 3

Bearing 4

Gear pair

(b)

(c)


Meshing element

Box element

Input

Output

Bearing element

Beam elementX

Y

ZO1 2 3 4 75 86

17

9

22

19

11

20

21

12 14 181310 15 16







θy(i+1)θz(i+1)

1113960 1113961T

(2)





T



θyi+1 θzi+1

1113966 1113967T

(3)


(a)43309 max

38497

33685

28873

24061

19249

14436

96243

48122

000098749 min

(b)


0 1 2 3Time (s)

4 508

1

12

Mes

hing

stiff

ness

(Nm

)

14

16

18

2 times108

times10ndash4







X

Y

Zxi

yi

x

y yi+1xi+1

lsO

θy (i+1)

θz (i+1)

θx (i+1)θxθz

θy

θzi

θyi

θxi




Tα

1 0 0

0 cos α sin α


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tβ

cos β 0 sin β

0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tc

cos c sin c 0

minus sin c cos c 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)


Meprime RTMeR

Keprime RTKeR

Geprime RTGeR

(5)



α 2ω1ω2 ξ1ω2 minus ξ2ω1( 1113857

ω22 minus ω2

1( 1113857

β 2 ξ2ω2 minus ξ1ω1( 1113857

ω22 minus ω2

1

(6)









ni nix niy niz1113960 1113961

1

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

TizTiyTix i p g (8)


Tix

1 0 0

0 cos δi sin δi


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tiy


0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tiz



0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)






⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)




Tand rg 0 rg 01113960 1113961

T respectively in


T uyi 0 1 01113858 1113859T and uzi 0 0 11113858 1113859







Engagement line

A-A

yg1 A

A

zg1

zgygyp

zpzp1

yp1δ1

δ2

(a)

xp2

xp1

zp1

zp2

β

(b)

Engagementline

θpz2

θgz2

yp2

xp2

yg2

xg2

α

ωp

ωg

(c)


Pinion

Gear

θgx

θpx

θgzθgy

θpz

θpy

Yg

Yp

cm

Og

Op

Zg

Zp

Xg

Xp

kh (t)

e (t)







cm 2ξ

km1

meqp

+1

meqg

1113888 1113889

1113971

(13)





mp eurozp Fnnpz



Jpeuroθpz Fnλpz



mg eurozg Fnngz



euroθgz Fnλgz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



MmeuroXm + Cm +

Ω1Gm1

Ω2Gm21113890 11138911113888 1113889 _Xm + KmXm

km(t)en(t) + cm _en(t)( 1113857BT

(15)



(16)


Km kmBTB

Cm cmBTB

(17)








MekeuroXek + Cb










where MH




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦








Rigid coupling

(a)

External node

(b)












Shaft length(mm) 20 26 28 108 140 28 26 20










0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

BoxRigid

(a)

times104

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

65

7

75

8

85

9

95

10

105

Mes

hing

forc

e (N

)

(b)




3003000 46570 49240 544700 66190 69000 417100 42130 44920 62

6003000 71170 76010 644700 95210 100100 487100 70070 75880 76

Speed (rmin)

0100020003000400050006000700080009000

10000

Bear

ing

forc

e (N

)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000

(a)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

0

2000

4000

6000

8000

10000

12000

Bear

ing

forc

e (N

)

(b)








ndash3

ndash2

ndash1

0

1

2

Gea

r2 Y

70deg90deg

110deg126deg

times10ndash5

times10ndash5

(a)

70deg90deg

110deg126deg


ndash1

ndash05

0

05

1

15

Gea

r2 Y

times10ndash6

times10ndash6

(b)




3003000 2213 3307 334700 6393 9808 357100 1278 1893 32

6003000 2741 4292 374700 7964 11700 407100 1545 2431 36




0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)




nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018












θy(i+1)θz(i+1)

1113960 1113961T

(2)





T



θyi+1 θzi+1

1113966 1113967T

(3)


(a)43309 max

38497

33685

28873

24061

19249

14436

96243

48122

000098749 min

(b)


0 1 2 3Time (s)

4 508

1

12

Mes

hing

stiff

ness

(Nm

)

14

16

18

2 times108

times10ndash4







X

Y

Zxi

yi

x

y yi+1xi+1

lsO

θy (i+1)

θz (i+1)

θx (i+1)θxθz

θy

θzi

θyi

θxi




Tα

1 0 0

0 cos α sin α


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tβ

cos β 0 sin β

0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tc

cos c sin c 0

minus sin c cos c 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)


Meprime RTMeR

Keprime RTKeR

Geprime RTGeR

(5)



α 2ω1ω2 ξ1ω2 minus ξ2ω1( 1113857

ω22 minus ω2

1( 1113857

β 2 ξ2ω2 minus ξ1ω1( 1113857

ω22 minus ω2

1

(6)









ni nix niy niz1113960 1113961

1

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

TizTiyTix i p g (8)


Tix

1 0 0

0 cos δi sin δi


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tiy


0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tiz



0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)






⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)




Tand rg 0 rg 01113960 1113961

T respectively in


T uyi 0 1 01113858 1113859T and uzi 0 0 11113858 1113859







Engagement line

A-A

yg1 A

A

zg1

zgygyp

zpzp1

yp1δ1

δ2

(a)

xp2

xp1

zp1

zp2

β

(b)

Engagementline

θpz2

θgz2

yp2

xp2

yg2

xg2

α

ωp

ωg

(c)


Pinion

Gear

θgx

θpx

θgzθgy

θpz

θpy

Yg

Yp

cm

Og

Op

Zg

Zp

Xg

Xp

kh (t)

e (t)







cm 2ξ

km1

meqp

+1

meqg

1113888 1113889

1113971

(13)





mp eurozp Fnnpz



Jpeuroθpz Fnλpz



mg eurozg Fnngz



euroθgz Fnλgz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



MmeuroXm + Cm +

Ω1Gm1

Ω2Gm21113890 11138911113888 1113889 _Xm + KmXm

km(t)en(t) + cm _en(t)( 1113857BT

(15)



(16)


Km kmBTB

Cm cmBTB

(17)








MekeuroXek + Cb










where MH




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦








Rigid coupling

(a)

External node

(b)












Shaft length(mm) 20 26 28 108 140 28 26 20










0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

BoxRigid

(a)

times104

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

65

7

75

8

85

9

95

10

105

Mes

hing

forc

e (N

)

(b)




3003000 46570 49240 544700 66190 69000 417100 42130 44920 62

6003000 71170 76010 644700 95210 100100 487100 70070 75880 76

Speed (rmin)

0100020003000400050006000700080009000

10000

Bear

ing

forc

e (N

)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000

(a)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

0

2000

4000

6000

8000

10000

12000

Bear

ing

forc

e (N

)

(b)








ndash3

ndash2

ndash1

0

1

2

Gea

r2 Y

70deg90deg

110deg126deg

times10ndash5

times10ndash5

(a)

70deg90deg

110deg126deg


ndash1

ndash05

0

05

1

15

Gea

r2 Y

times10ndash6

times10ndash6

(b)




3003000 2213 3307 334700 6393 9808 357100 1278 1893 32

6003000 2741 4292 374700 7964 11700 407100 1545 2431 36




0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)




nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018









Tα

1 0 0

0 cos α sin α


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tβ

cos β 0 sin β

0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tc

cos c sin c 0

minus sin c cos c 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)


Meprime RTMeR

Keprime RTKeR

Geprime RTGeR

(5)



α 2ω1ω2 ξ1ω2 minus ξ2ω1( 1113857

ω22 minus ω2

1( 1113857

β 2 ξ2ω2 minus ξ1ω1( 1113857

ω22 minus ω2

1

(6)









ni nix niy niz1113960 1113961

1

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

TizTiyTix i p g (8)


Tix

1 0 0

0 cos δi sin δi


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tiy


0 1 0


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Tiz



0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)






⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)




Tand rg 0 rg 01113960 1113961

T respectively in


T uyi 0 1 01113858 1113859T and uzi 0 0 11113858 1113859







Engagement line

A-A

yg1 A

A

zg1

zgygyp

zpzp1

yp1δ1

δ2

(a)

xp2

xp1

zp1

zp2

β

(b)

Engagementline

θpz2

θgz2

yp2

xp2

yg2

xg2

α

ωp

ωg

(c)


Pinion

Gear

θgx

θpx

θgzθgy

θpz

θpy

Yg

Yp

cm

Og

Op

Zg

Zp

Xg

Xp

kh (t)

e (t)







cm 2ξ

km1

meqp

+1

meqg

1113888 1113889

1113971

(13)





mp eurozp Fnnpz



Jpeuroθpz Fnλpz



mg eurozg Fnngz



euroθgz Fnλgz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



MmeuroXm + Cm +

Ω1Gm1

Ω2Gm21113890 11138911113888 1113889 _Xm + KmXm

km(t)en(t) + cm _en(t)( 1113857BT

(15)



(16)


Km kmBTB

Cm cmBTB

(17)








MekeuroXek + Cb










where MH




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦








Rigid coupling

(a)

External node

(b)












Shaft length(mm) 20 26 28 108 140 28 26 20










0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

BoxRigid

(a)

times104

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

65

7

75

8

85

9

95

10

105

Mes

hing

forc

e (N

)

(b)




3003000 46570 49240 544700 66190 69000 417100 42130 44920 62

6003000 71170 76010 644700 95210 100100 487100 70070 75880 76

Speed (rmin)

0100020003000400050006000700080009000

10000

Bear

ing

forc

e (N

)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000

(a)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

0

2000

4000

6000

8000

10000

12000

Bear

ing

forc

e (N

)

(b)








ndash3

ndash2

ndash1

0

1

2

Gea

r2 Y

70deg90deg

110deg126deg

times10ndash5

times10ndash5

(a)

70deg90deg

110deg126deg


ndash1

ndash05

0

05

1

15

Gea

r2 Y

times10ndash6

times10ndash6

(b)




3003000 2213 3307 334700 6393 9808 357100 1278 1893 32

6003000 2741 4292 374700 7964 11700 407100 1545 2431 36




0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)




nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018










Tand rg 0 rg 01113960 1113961

T respectively in


T uyi 0 1 01113858 1113859T and uzi 0 0 11113858 1113859







Engagement line

A-A

yg1 A

A

zg1

zgygyp

zpzp1

yp1δ1

δ2

(a)

xp2

xp1

zp1

zp2

β

(b)

Engagementline

θpz2

θgz2

yp2

xp2

yg2

xg2

α

ωp

ωg

(c)


Pinion

Gear

θgx

θpx

θgzθgy

θpz

θpy

Yg

Yp

cm

Og

Op

Zg

Zp

Xg

Xp

kh (t)

e (t)







cm 2ξ

km1

meqp

+1

meqg

1113888 1113889

1113971

(13)





mp eurozp Fnnpz



Jpeuroθpz Fnλpz



mg eurozg Fnngz



euroθgz Fnλgz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



MmeuroXm + Cm +

Ω1Gm1

Ω2Gm21113890 11138911113888 1113889 _Xm + KmXm

km(t)en(t) + cm _en(t)( 1113857BT

(15)



(16)


Km kmBTB

Cm cmBTB

(17)








MekeuroXek + Cb










where MH




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦








Rigid coupling

(a)

External node

(b)












Shaft length(mm) 20 26 28 108 140 28 26 20










0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

BoxRigid

(a)

times104

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

65

7

75

8

85

9

95

10

105

Mes

hing

forc

e (N

)

(b)




3003000 46570 49240 544700 66190 69000 417100 42130 44920 62

6003000 71170 76010 644700 95210 100100 487100 70070 75880 76

Speed (rmin)

0100020003000400050006000700080009000

10000

Bear

ing

forc

e (N

)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000

(a)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

0

2000

4000

6000

8000

10000

12000

Bear

ing

forc

e (N

)

(b)








ndash3

ndash2

ndash1

0

1

2

Gea

r2 Y

70deg90deg

110deg126deg

times10ndash5

times10ndash5

(a)

70deg90deg

110deg126deg


ndash1

ndash05

0

05

1

15

Gea

r2 Y

times10ndash6

times10ndash6

(b)




3003000 2213 3307 334700 6393 9808 357100 1278 1893 32

6003000 2741 4292 374700 7964 11700 407100 1545 2431 36




0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)




nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018












cm 2ξ

km1

meqp

+1

meqg

1113888 1113889

1113971

(13)





mp eurozp Fnnpz



Jpeuroθpz Fnλpz



mg eurozg Fnngz



euroθgz Fnλgz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)



MmeuroXm + Cm +

Ω1Gm1

Ω2Gm21113890 11138911113888 1113889 _Xm + KmXm

km(t)en(t) + cm _en(t)( 1113857BT

(15)



(16)


Km kmBTB

Cm cmBTB

(17)








MekeuroXek + Cb










where MH




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦








Rigid coupling

(a)

External node

(b)












Shaft length(mm) 20 26 28 108 140 28 26 20










0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

BoxRigid

(a)

times104

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

65

7

75

8

85

9

95

10

105

Mes

hing

forc

e (N

)

(b)




3003000 46570 49240 544700 66190 69000 417100 42130 44920 62

6003000 71170 76010 644700 95210 100100 487100 70070 75880 76

Speed (rmin)

0100020003000400050006000700080009000

10000

Bear

ing

forc

e (N

)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000

(a)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

0

2000

4000

6000

8000

10000

12000

Bear

ing

forc

e (N

)

(b)








ndash3

ndash2

ndash1

0

1

2

Gea

r2 Y

70deg90deg

110deg126deg

times10ndash5

times10ndash5

(a)

70deg90deg

110deg126deg


ndash1

ndash05

0

05

1

15

Gea

r2 Y

times10ndash6

times10ndash6

(b)




3003000 2213 3307 334700 6393 9808 357100 1278 1893 32

6003000 2741 4292 374700 7964 11700 407100 1545 2431 36




0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)




nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018












where MH




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣







⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦








Rigid coupling

(a)

External node

(b)












Shaft length(mm) 20 26 28 108 140 28 26 20










0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

BoxRigid

(a)

times104

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

65

7

75

8

85

9

95

10

105

Mes

hing

forc

e (N

)

(b)




3003000 46570 49240 544700 66190 69000 417100 42130 44920 62

6003000 71170 76010 644700 95210 100100 487100 70070 75880 76

Speed (rmin)

0100020003000400050006000700080009000

10000

Bear

ing

forc

e (N

)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000

(a)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

0

2000

4000

6000

8000

10000

12000

Bear

ing

forc

e (N

)

(b)








ndash3

ndash2

ndash1

0

1

2

Gea

r2 Y

70deg90deg

110deg126deg

times10ndash5

times10ndash5

(a)

70deg90deg

110deg126deg


ndash1

ndash05

0

05

1

15

Gea

r2 Y

times10ndash6

times10ndash6

(b)




3003000 2213 3307 334700 6393 9808 357100 1278 1893 32

6003000 2741 4292 374700 7964 11700 407100 1545 2431 36




0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)




nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018

















Shaft length(mm) 20 26 28 108 140 28 26 20










0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

BoxRigid

(a)

times104

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

65

7

75

8

85

9

95

10

105

Mes

hing

forc

e (N

)

(b)




3003000 46570 49240 544700 66190 69000 417100 42130 44920 62

6003000 71170 76010 644700 95210 100100 487100 70070 75880 76

Speed (rmin)

0100020003000400050006000700080009000

10000

Bear

ing

forc

e (N

)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000

(a)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

0

2000

4000

6000

8000

10000

12000

Bear

ing

forc

e (N

)

(b)








ndash3

ndash2

ndash1

0

1

2

Gea

r2 Y

70deg90deg

110deg126deg

times10ndash5

times10ndash5

(a)

70deg90deg

110deg126deg


ndash1

ndash05

0

05

1

15

Gea

r2 Y

times10ndash6

times10ndash6

(b)




3003000 2213 3307 334700 6393 9808 357100 1278 1893 32

6003000 2741 4292 374700 7964 11700 407100 1545 2431 36




0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)




nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018











0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

BoxRigid

(a)

times104

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

65

7

75

8

85

9

95

10

105

Mes

hing

forc

e (N

)

(b)




3003000 46570 49240 544700 66190 69000 417100 42130 44920 62

6003000 71170 76010 644700 95210 100100 487100 70070 75880 76

Speed (rmin)

0100020003000400050006000700080009000

10000

Bear

ing

forc

e (N

)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000

(a)

BoxRigid

0 1000 2000 3000 4000 5000 6000 7000 8000Speed (rmin)

0

2000

4000

6000

8000

10000

12000

Bear

ing

forc

e (N

)

(b)








ndash3

ndash2

ndash1

0

1

2

Gea

r2 Y

70deg90deg

110deg126deg

times10ndash5

times10ndash5

(a)

70deg90deg

110deg126deg


ndash1

ndash05

0

05

1

15

Gea

r2 Y

times10ndash6

times10ndash6

(b)




3003000 2213 3307 334700 6393 9808 357100 1278 1893 32

6003000 2741 4292 374700 7964 11700 407100 1545 2431 36




0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)




nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018













ndash3

ndash2

ndash1

0

1

2

Gea

r2 Y

70deg90deg

110deg126deg

times10ndash5

times10ndash5

(a)

70deg90deg

110deg126deg


ndash1

ndash05

0

05

1

15

Gea

r2 Y

times10ndash6

times10ndash6

(b)




3003000 2213 3307 334700 6393 9808 357100 1278 1893 32

6003000 2741 4292 374700 7964 11700 407100 1545 2431 36




0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)




nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018










0064 00645 0065 006552

3

4

5

6

7M

eshi

ng fo

rce

70deg90deg

110deg126deg

t (s)

times104

(a)

70deg90deg

110deg126deg

00625 0063 00635 0064 00645 0065t (s)

6000

6500

7000

7500

8000

8500

9000

9500

Fx

(b)

70deg90deg

110deg126deg

t (s)0062 00625 0063 00635 0064 00645 0065 00655 0066

ndash1000

0

1000

2000

3000

4000

5000

Fy

(c)

70deg90deg

110deg126deg

t (s)00625 0063 00635 0064 00645 0065

ndash4800

ndash4600

ndash4400

ndash4200

ndash4000

ndash3800

ndash3600

ndash3400

ndash3200

ndash3000

ndash2800

Fz

(d)




nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018









nal sp

eed (r

min)

Am

plitu

de

80007000

60000 5000

times10ndash6

400010002000 3000

3000 20004000 10005000

6000

5

f1

fm

2fm

(a)

Am

plitu

de

times10ndash5

f1

fm

2fm

Rotatio

nal sp

eed (r

min)

80007000

60000 5000

400010002000 3000

Frequency (Hz)

3000 20004000 100050006000

12

(b)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

Am

plitu

de

times10ndash6

012

fm

2fm

(c)

Rotatio

nal sp

eed (r

min)

80007000

60005000

400010002000 3000

Frequency (Hz)

3000 20004000 10005000

6000

2fm

3fmAm

plitu

de0

times10ndash6

5

f1

fm

(d)


1000

10002000

30004000

50006000

70008000

20003000

40005000

6000

0

5

Am

plitu

de

times10ndash6

fm

fm

2fm


peed (r

min)

(a)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash5

210

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

(b)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

Am

plitu

de

times10ndash6

Frequency (Hz)

210

Rotational s

peed (r

min)

f1

fm

2fm

(c)

10002000

30004000

50006000

70008000

10002000

30004000

50006000

0

5

Am

plitu

de

times10ndash6

Frequency (Hz)

Rotational s

peed (r

min)

f1

fm

2fm

3fm

(d)



5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018








5 Conclusion






Data Availability




Acknowledgments


References






0

2000

4000

6000

8000

10000

Bear

ing

forc

e (N

)


0 2000 4000 6000 8000

(a)



4

45

5

55

6

65

7

75

Mes

hing

forc

e (N

)

times104

(b)






































Journal of












Journal of







Volume 2018






Volume 2018











































Journal of












Journal of







Volume 2018






Volume 2018








Modelling and Dynamic Analysis of the Spiral Bevel Gear...

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Transcript of Modelling and Dynamic Analysis of the Spiral Bevel Gear...