mathematicalmodelinganddynamicanalysisofplanetarygears...

Research ArticleMathematical Modeling andDynamic Analysis of Planetary GearsSystem with Time-Varying Parameters

Zhengming Xiao Jinxin Cao and Yinxin Yu

Faculty of Mechanical and Electrical Engineering Kunming University of Science and Technology Kunming Yunnan China

Correspondence should be addressed to Zhengming Xiao suzemsinacom

Received 14 January 2020 Accepted 20 February 2020 Published 16 March 2020

Guest Editor Jian Huang

Copyright copy 2020 Zhengming Xiao et al (is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Planetary gears are widely used in automobiles helicopters heavy machinery etc due to the high speed reductions in compactspaces however the gear fault and early damage induced by the vibration of planetary gears remains a key concern (e time-varying parameters have a vital influence on dynamic performance and reliability of the gearbox An analytical model is proposedto investigate the effect of gear tooth crack on the gear mesh stiffness and then the dynamical model of the planetary gears withtime-varying parameters is established (e natural characteristics of the transmission system are calculated and the dynamicresponses of transmission components as well as dynamic meshing force of each pair of gear are investigated based on varyinginternal excitations induced by time-varying parameters and tooth root crack (e effects of gear tooth root crack size on theplanetary gear dynamics are simulated and the mapping rules between damage degree and gear dynamics are revealed In order toverify the theoretical model and simulation results the planetary gear test rig was built by assembling faulty and healthy gearseparately (e failure mechanism and dynamic characteristics of the planetary gears with tooth root crack are clarified bycomparing the analytical results and experimental data

1 Introduction

Planetary transmissions are an important form of me-chanical transmission Because of its advantages of hightransmission ratio high bearing capacity and compactstructure it is widely used in complex mechanicalequipment such as aerospace wind power generationequipment and mining machinery Due to the influenceof time-varying parameters the planetary gear systemalways has the problem of nonlinear dynamics and ex-cessive vibration (e planetary transmission system hasa high failure rate and the root crack is one of the mostimportant forms of gear failure due to complicatedinternal structure and large load during operation Whenthe root of the transmission system is cracked the vi-bration of the system will be intensified and in seriouscases the equipment will be damaged (erefore thedynamic study of the planetary gear transmission systemwith tooth root crack failure and analysis of the fault

mechanism has an important theoretical significance andengineering application value for improving the reli-ability and service life of the gear Mathematical mod-eling and analysis are important methods to solve thenonlinear dynamics of planetary gear transmissionsystems (e control of nonlinear dynamic systems is awidely recognized challenging issue It is promising todevelop vibration reduction design of the planetary gearsystem based on U-model because of the unique ad-vantages of U-model in nonlinear control [1 2] Somescholars have carried out a lot of work on the dynamics ofgear systems by mathematical modeling Bonori andPellicano [3] presented a dynamic model of a single pairof gear transmission systems and analyzed the effects ofrandom manufacturing errors on the dynamic responseof the system Chaari et al [4] established an analyticalmodel of time-varying gear meshing stiffness based onthe analytical method and analyzed the variation of crackto gear stiffness under two different parameters Guo and

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 3185624 9 pageshttpsdoiorg10115520203185624

Parker [5] established a dynamic model of the planetarygear train based on the lumped parameter method andsolved the dynamic model to obtain the dynamic re-sponse of the transmission system Chen and Shao [6]studied the effects of internal ring cracks on the time-varying mesh stiffness and dynamic response of plane-tary gear trains Some scholars have also carried out a lotof research on the dynamics of planetary transmissionsystems Zhou [7] studied the influence of crack pa-rameters on dynamic characteristics by establishing afinite element model Wan [8] proposed a meshingstiffness correction method based on the potential energymethod (e dynamic model of a gear transmissionsystem with tooth root crack was established by using thelumped parameter method and the dynamic equationwas solved to obtain the dynamic response In previousplanetary transmission dynamics analysis the bearingwas simplified to the ideal constraint boundary and thebearing was simplified to a constant stiffness coefficientspring also ignoring the influence of the bearing time-varying stiffness on the dynamic characteristics of thetransmission system Bearings and gears are the keycomponents of the planetary transmission system (edynamic characteristics of the bearings have an impor-tant influence on the stability and service life of thetransmission system Walters [9] proposed the dynamicanalysis model of the rolling bearing (e dynamic modelof the rolling bearing was established and the drag forcebetween the rolling element and the ferrule was obtainedby considering the four degrees of freedom of the rollingelement and the six degrees of freedom of the cageHarris and Kotzalas [10] considered the effect of elas-tohydrodynamics on the basis of quasi-static analysis andimproved the pseudostatic analysis method of rollingbearings In establishing the balance equations of therolling element the cage and the inner ring the rollingelementrsquos revolution and bearing deformation parame-ters are obtained

Zhou [11] considered the time-varying stiffness of rollingbearings established the coupled dynamics model of MW-class wind turbine gear transmission system by using thelumped parameter method obtained the inherent character-istics of the transmission system and solved the dynamics ofeach bearing contact stress but it does not consider the in-fluence of the centrifugal force of the roller on the bearingstiffnessMohammed et al [12] presents an investigation of theperformance of crack propagation scenarios to compare thesescenarios from a fault diagnostics point of view Park et al [13]proposes a variance of energy residual (VER) method forplanetary gear fault detection under variable-speed conditions

In the planetary gear transmission system the dynamicmeshing excitation of the gear teeth is transmitted to thecasing through the bearing causing vibration and noise ofthe casing (e dynamic characteristics of the bearing havean important influence on the performance of the entiretransmission system In the past the scholars neglected theinfluence of the mass of the rolling element and the cen-trifugal force of the roller on the time-varying stiffness of the

bearing when studying the gear-bearing coupling dynamicsystem At the same time the dynamic characteristics of thefaulty planetary transmission system were not studied Inthis paper the planetary transmission systemwith tooth rootcrack is taken as the research object According to the Hertzcontact theory considering the mass of the rolling elementand the effect of centrifugal force the time-varying bearingstiffness model is established At the same time the time-varying bearing stiffness the transmission error and thetime-varying meshing stiffness of the cracked gear are in-troduced (e gear-bearing coupling dynamics model withcrack is established (e influence of time-varying bearingstiffness on the dynamic characteristics of the gear trans-mission system is analyzed (e influence of crack failure onthe dynamic characteristics of the gear is explored whichprovides a more detailed mathematical method for the faultdiagnosis and vibration control of the planetary geartransmission system

2 Dynamic Model of Planetary Gear System

(e dynamic characteristics of the planetary gear trans-mission system are affected by many factors (e me-chanical model is simplified as shown in Figure 1 It isassumed that the three planet gears in the planetarytransmission system are evenly distributed along thecircumference and the geometric parameters and phys-ical parameters of the three planet gears are the same (eengagement between the gear pairs is simplified to aspring with time-varying stiffness Considering the gearmeshing error the time-varying bearing stiffness of thebearing and the time-varying meshing factor of the gearthe planetary gear translation-torsion dynamics model isestablished by the lumped parameter method Eachcomponent has 2 translational degrees of freedom and 1torsional degree of freedom and the model has 18 degreesof freedom

erp2crp2

krp2

erp3

ksp3

esp3

csp3

esp2

ksp2

csp2ksu

csuesp1 ksp1

csp1

cpy kpy

cpx

kpxerp1

crp1

krp1

crp3

krp3

ccu

cru kru

csy ksyt

ksxt

csx

kcu

Y

X0

x

y

ζn

ψk

ηn

Figure 1 Mechanical model of planetary gear system

2 Mathematical Problems in Engineering

Ring

mr euroxr minus 2ωc _yr minus ω2cxr( 1113857 minus 1113944

N

n1crn

_δrn + krnδrn1113872 1113873sinψrn + crx _xr + krxxr 0

mr euroyr + 2ωc _xr minus ω2cyr( 1113857 + 1113944

N

n1crn

_δrn + krnδrn1113872 1113873cosψrn + cry _yr + kryyr 0

Ir

r2r1113888 1113889 eurour + 1113944

N

n1crn

_δrn + krnδrn1113872 1113873 + cru _uc + kruuc Tr

rr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Sun

ms euroxs minus 2ωc _ys minus ω2cxs( 1113857 minus 1113944

N

n1csn

_δsn + ksnδsn1113872 1113873sinψsn + csx _xs + ksxtxs 0

ms euroys + 2ωc _xs minus ω2cys( 1113857 + 1113944

N

n1csn

_δsn + ksnδsn1113872 1113873cosψsn + csy _ys + ksytys 0

Is

r2s1113888 1113889 eurous + 1113944

N

n1ksnδsn + csn

_δsn1113872 1113873 + csu _us + ksuus Ts

rs

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Carrier

1113944N

n1kpn δnr cosψn minus δnt sinψn( 1113857 + cpn

_δnr cosψn minus _δnt sinψn1113872 11138731113960 1113961 + mc euroxc minus 2ωc _yc minus ω2cxc1113872 1113873 + ccx _xc

+kcxxc 0

1113944

N

n1kpn δnr sinψn + δnt cosψn( 1113857 + cpn

_δnr sinψn + _δnt cosψn1113872 11138731113960 1113961 + mc euroyc + 2ωc _xc minus ω2cyc1113872 1113873

+ccy _yc + kcyyc 0

Ic

r2c1113888 1113889 eurouc + 1113944

N

n1kpnδnt + cpn

_δnt1113872 1113873 + ccu _uc + kcuuc Tc

rc

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Planet

mpneuroζn minus 2ωc _ηn minus ω2

cζn1113872 1113873 minus csn_δsn + ksnδsn1113872 1113873cos αs minus crn

_δrn + krnδrn1113872 1113873cos αr minus cpn_δnr minus kpnδnr 0

mpn euroηn + 2ωc_ζn minus ω2

cηn1113872 1113873 minus csn_δsn + ksnδsn1113872 1113873sin αs + crn

_δrn + krnδrn1113872 1113873sin αr minus cpn_δnt minus kpnδnt 0

Ipn

r2pn

⎛⎝ ⎞⎠euroun + csn_δsn + ksnδsn minus crn

_δrn minus krnδrn 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where δsn is the elastic deformation of the sun gear and thenth planet gear on the meshing line

δsn minusxs sinψsn + ys cosψsn minus ζn sin αs minus ηn cos αs

+ us + un + esn(2)

where δrn is the elastic deformation of the ring and the nthplanet gear on the meshing line

δrn yr cosψrn minus xr sinψrn minus ζn sin αr minus ηn cos αr

+ ur minus un + ern(3)

where δnr is the projection of the planet carrier in the ζdirection relative to the nth planet gear

δnr xc cosψn + yc sinψn minus ζn (4)

Mathematical Problems in Engineering 3

where δnt is the projection of the planet carrier in the ηdirection with respect to the nth planet gear

δnt minusxc sinψn + yc cosψn + uc minus ηn (5)

where ψsn ψn minus αs ψrn ψn+ αr ψn is the position angle ofthe nth planet gear αr is the meshing angle between the ringand the planet gear and αs is the meshing angle between thesun gear and the planet gear

(e dynamic equations of the various componentsestablished above are collated (e dynamic equations of theplanetary transmission system can be expressed in the formof a matrix (e dynamic equation of the transmissionsystem can be expressed as follows

M eurox + ωcG _x + Kb + Km minus ω2cKω1113960 1113961x F + T (6)

where G is the gyromatrix Km is the meshing stiffnessmatrix Kω is the centripetal stiffness matrix x is the gen-eralized displacement matrix of the system M is the gen-eralized mass matrix of the system Kb is the bearing supportstiffness matrix Km is the meshing stiffness matrix T is theexternal load of the system and F is the internal excitation

3 Time-Varying Stiffness Model of Bearing

In order to simplify the calculation it is assumed that thesteel ball rotates at a constant speed and the movementbetween the inner and outer rings of the bearing is purerolling Regardless of the oil film stiffness under bearinglubrication the bearing stiffness of the bearing can be an-alyzed according to Hertz contact theory

As shown in Figure 2 during the movement of thebearing the inner and outer rings of the bearing are sub-jected to the weight of the roller and the centrifugal force(erefore the forces of the contact between the roller andthe inner and outer rings of the bearing are different (econtact force Qo between the roller and the outer ring is asfollows

Qo Q minus mg sinφk + mRpω2c

ωc π 1 minus DbDp1113872 1113873cos β1113872 1113873ns

60

(7)

where ωc is the angular velocity of the cage Q is the load onthe inner ring of the bearing m is the mass of the steel ball(e radius of the circle where the geometric center of theroller is located is RpDb is the diameter of the roller ns is therotational speed of the shaft and β is the contact angle

According to Hertz contact theory the relationshipbetween the contact force Q of the rolling bearing steel balland the elastic approaching amount δ can be expressed asfollows

Q Kcδ32

(8)

where Q is the contact load of the steel ball and Kc is theHertz contact stiffness

According to the force balance condition the compo-nent forces fx and fy of the bearing x and y directions areequal to the sum of the component forces of the load Qk of

each roller in the x and y directions and fx and fy can beexpressed by the following formulas

fx 1113944N

k1Qk cos φk( 1113857

fy 1113944N

k1Qk sin φk( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(9)

(e radial stiffness kb of the bearing is

kb 1113944

N

k1Kk cosφk1113858 1113859 (10)

where Kk is the contact stiffness of the kth roller(e relevant parameters of the rolling bearing in the

planetary transmission system are shown in Table 1 (ebearingmaterial is bearing steel which is solved according tothe time-varying bearing stiffness model of the rollingbearing Figure 3 shows the variation of the radial stiffness ofthe bearing with the angle of the steel ball as the positionangle changes

4 Effect of Crack on Gear Meshing Stiffness

During the movement of the gear the root position willproduce a bending stress greater than other positions dueto the periodicity of the load which will easily lead to theoccurrence of fatigue cracks in the gear teeth When cal-culating the gear meshing stiffness the gear teeth aregenerally considered be cantilever beams (ere are fivekinds of stiffness in the gear meshing pair Hertz contactstiffness kh bending stiffness kb shear stiffness ka andradial compression stiffness ks [12] In addition to the abovestiffness there is also a matrix flexible stiffness kf of thegear When the gear is cracked the surface contact area andradial force of the gear are constant so the Hertz contactstiffness radial compression stiffness and matrix flexibilityof the gear are unchanged (e gear meshing stiffness of thecrack mainly considers the bending stiffness and shearsstiffness

In this paper the planetary gearbox with tooth root crackfault is taken as the research object (e relevant parametersof the planetary box are shown in Table 2 When the cracktooth enters the meshing state the meshing stiffness of thecracked gear is calculated Figure 4 shows the comparison ofthe stiffness of a simulated crack occurrence (the tooth crack

0

y

x

Q

Ball

Innerrace

Outerrace

ki

ko

mgωc

Figure 2 Bearing rolling element-ring model


has an expansion angle αc of 60deg and a crack depth q of18mm) when engaged with a healthy gear pair

5 Dynamic Response and Analysis

51ModalAnalysis of PlanetaryGears (emodal analysis ofthe gear-bearing coupling dynamics model of the abovementioned planetary transmission system is carried out

Regardless of the effect of damping the average stiffness isused instead of the time-varying stiffness and the dynamicequation is expressed as follows

M eurox + ωcG _x + Kb + Km minus ω2cKω1113960 1113961x 0 (11)

(us the n positive real roots ωi which are the naturalfrequencies of the system could be determined (e mainmode equation is

ω2i Mϕi Kb + Km1113858 1113859ϕi (12)

(e planetary gears system has 18 degrees of freedom(e natural frequencies of the system are calculated asshown in Table 3 and the corresponding vibration modesare shown in Figure 5 (ere are three kinds of vibrationmode rigid mode rotational modes (no repetitive fre-quency no translation only torsion) and translationalmodes (there is a repetitive frequency no torsion onlytranslation) (e phenomenon of 5 equivalent natural fre-quencies (repetitive frequency) occurs in the system becausethe geometrical and physical parameters of the planetarygears in the transmission system are assumed to be the sameresulting in a symmetrical type of the planetary transmissionstage (e occurrence of repetitive frequencies causes dif-ferent types of vibration to be induced at one frequency

52DynamicResponseAnalysis of Planetary System In orderto deeply study the influence of sun gear root crack failure onthe dynamic characteristics of the planetary gear trans-mission system (is paper selects the fault model and fault-free model of the sun gear crack 18mm Considering thetime-varying meshing stiffness of healthy and tooth crackedgears the directional parameter method is used to establishthe translation-torsion dynamics modeling of the planetarytransmission system (e dynamic equation is solved byRungendashKutta method (e input torque of the system is30Nm and the dynamic response of the planetary trans-mission system at 1000 rmin is analyzed (e meshingfrequency of the transmission system is 33468Hz and thebearing ball revolution frequency is 1181Hz

Figures 6ndash8 shows the vibration velocity response of ahealthy and root-cracked gear transmission component Asshown in Figure 6 the peak velocity of the sun gear of thehealthy system is 525mms and the peak velocity of the sungear with root crack fault transmission system is 877mmsAs shown in Figure 7 the peak velocity of the planet gear ofthe healthy system is 283mms and the peak velocity of thesun gear with root crack fault transmission system is395mms As shown in Figure 8 the peak velocity of thering gear of the healthy system is 0295mms and the peakvelocity of the ring gear with the root crack fault trans-mission system is 0362mms

(rough the comparison of the time domain waveformof the healthy and faulty transmission systems it can be seen

Table 1 Geometric parameters of the bearing

Outer diameter D (mm) Inner diameter d (mm) Bearing width B (mm) Ball diameter Db (mm) Number of rolling elements nb80 40 18 12 9

0 40 80 120 160

Bear

ing

radi

al st

iffne

ss(k

sxtN

middotmndash1

)

Bearing cage rotation angle (deg)

351

349

347

345

343

times108

Figure 3 Time-varying bearing stiffness

Table 2 Parameters of planetary gearbox

Parameter Ring Carrier Planetgear Sun gear

Mass (kg) 1012 27 045 08Number of teeth 71 4229 28Base circle diameter(mm) 15012 112 20 592

Modulus (mm) 225Engagement angle (deg) 20deg

83

5

7

9

0 2 4 106 12

Mes

hing

stiff

ness

(ksp

Nmiddotm

ndash1)

Engagement angle (deg)

10times108

NormalCrack

Figure 4 Time-varying meshing stiffness


Table 3 Natural frequencies of the system

Vibration mode Natural frequencies (Hz)Rigid mode f1 0Rotational modes f2 f3 1104 f5 f6 4595 f10 f11 9286 f13 f14 15111 f16 f17 21544Translational modes f4 1207 f7 4930 f8 5967 f9 6467 f12 10421 f15 17898 f18 23060

(a) (b) (c)

Figure 5 Vibration modes of planetary transmission system (a) rigid mode (b) rotational mode and (c) translational mode

ndash5

0

5

0 005004003002001

Velo

city

(vsy

mm

middotsndash1)

Time (ts)

(a)

Time (ts)

Velo

city

(vsy

mm

middotsndash1)

ndash10

0

10

0 002 004 006 008 01

(b)

Figure 6 Vibration velocity response of the sun gear (a) Healthy planetary transmission system (b) Faulty planetary transmission system

0 005004003002001Time (ts)

Velo

city

(vpy

mm

middotsndash1)

ndash4ndash2

024

(a)

Time (ts)

Velo

city

(vpy

mm

middotsndash1)

ndash5

0

5

0 002 004 006 008 01

(b)

Figure 7 Vibration velocity response of planet gear (a) Healthy planetary transmission system (b) Faulty planetary transmission system

0 005004003002001Time (ts)

Velo

city

(vry

mm

middotsndash1)

020

ndash02ndash04

(a)

Velo

city

(vry

mm

middotsndash1)

Time (ts)0 002 004 006 008 01

0204

0ndash02ndash04

(b)

Figure 8 Vibration velocity of ring (a) Healthy planetary transmission system (b) Faulty planetary transmission system


that a periodic shock signal appears in the time domainwaveform diagram of the faulty system compared to thehealthy transmission system (e occurrence of sun gearcracks causes the gear teeth of the cracks to reduce themeshing stiffness whichmakes the vibration velocity the sungear the planet gears and the ring in the system increasesignificantly which aggravates the vibration of the planetarytransmission system

(e spectrum analysis of the vibration velocity responseof the sun gear in the y direction of the transmission systemis performed Figures 9 and 10 are the spectrum diagrams ofthe vibration velocity response of the healthy and the toothroot crack faulty transmission system As shown in Figure 9in the spectrum diagram of the healthy transmission systemthe meshing frequency of the transmission system is3347Hz and the bearing ball revolution frequency is1184Hz and the meshing frequency fm is mainly dominantand the peak of the first frequency is relatively high At thesame time a sideband appears on both sides of the meshingfrequency and the distance of the sideband from themeshing frequency is the frequency at which the ballrevolves

Compared with healthy gears sidebands appear aroundthe meshing frequency in the spectrum of root crack failure

as shown in Figure 10 (is is because the sun gear shaftfrequency fs produces a frequency modulation effect on themeshing frequency (e distance between the sideband andthe meshing frequency is nfs (n 1 2 3 ) At the sametime the amplitude of the meshing frequency of the systemincreases

6 Experimental Analysis

Taking the planetary transmission gearbox as the ex-perimental object the test was carried out on theplanetary gearbox with the sun gear root crack failure(e test bench and measurement points are shown inFigure 11 Measuring points 1ndash3 are set on the outputbearing ring gear and input bearing respectively (evibration acceleration signals of the healthy and faultytransmission systems are acquired and the vibrationsignal in the z direction of the measuring point 1 areshown in Figure 12

(e spectrum of the vibration acceleration signals areshown in Figure 13 In the spectrum diagram of the healthytransmission system the meshing frequency fm and itsmultiplication are mainly dominant the meshing frequencyof the system is 3346Hz and the frequency of the bearing

times103

3347Hz

0

fm ndash fb fm + fb

Vel

ocity

(mm

middotsndash1)

Vel

ocity

(mm

middotsndash1)

0

04

08

12

0 2 3 4 51Frequency (Hz)

002

006

01fm

2fm

3fm

200 400 6000Frequency (Hz)

(a)

times10ndash3

X 1184

Vel

ocity

(mm

middotsndash1)

130 150 170 190110Frequency (Hz)

0

2

4

6

(b)

Figure 9 Spectrum diagram of sun gear (a) Spectrum of healthy signal (b) Spectrum refinement of healthy signals


times103

15

1

05

00

Velo

city

(mm

middotsndash1)

(a)

100 200 300 400 500Frequency (Hz)

012

008

004

0

Velo

city

(mm

middotsndash1)

fm ndash 2fs

fm ndash 3fs fm + 3fs

fm

(b)

Figure 10 Spectrum diagram of sun gear with root crack (a) Spectrum of fault signal (b) Spectrum refinement of fault signal


steel ball is 1189Hz At the same time a sideband having acarrier frequency fc as a modulation signal appears aroundthe meshing frequency

Because the influence of the vibration signal transmis-sion path is not considered there is no sideband caused bythe frequency fc modulation of the carrier around themeshing frequency and the spectrum of the actual vibrationsignal is slightly different

As shown in Figure 14 the spectrum of the vibrationresponse of the fault-containing system is obtained (eamplitude of the faulty system spectrum is larger than that ofthe healthy system (e meshing frequency of the system is3346Hz (e sideband of the sun frequency fs is modulatedaround the meshing frequency (e correctness of thesimulation model is verified

02 04 06 08 10Time (s)

ndash50

0

50

Acc

eler

atio

n(m

middotsndash2)

(a)

ndash100

0

100

Acc

eler

atio

n(m

middotsndash2)

02 04 06 08 10Time (s)

(b)

Figure 12 Vibration acceleration of test point 1 (a) Healthy system (b) Fault system

310 320 330300Frequency (Hz)

0

02

04

06

08

Acc

eler

atio

n (m

middotsndash2)

fm ndash 2fc

fm ndash fc

fm

(a)

120 122118Frequency (Hz)

0

1

2

Acc

eler

atio

n (m

middotsndash2)

fb

times10ndash3

(b)

Figure 13 Spectrum diagram of the healthy system (a) System meshing frequency (b) Frequency conversion bearing steel ball

Testpoint 1

Testpoint 2

Testpoint 3

z

x y

Figure 11 Test setup

fm ndash 2fs

fm ndash 2fc

fm ndash fs

fm ndash fcfm

fm ndash fs + 2fc

0

02

04

06

08

1

Acc

eler

atio

n (m

middotsndash2)

310 320 330300Frequency (Hz)

Figure 14 Spectrum diagram of the faulty system


7 Conclusions

(e gear-bearing coupling dynamics model of the planetarytransmission system is established by using the lumpedparameter method(e natural characteristics analysis of thesystem is carried out and the influencing factors such as thetime-varying parameters of the meshing stiffness and thebearing stiffness are considered (e dynamic characteristicsof the planetary transmission system are calculated andsimulated (e system dynamic response and frequencydomain characteristics are obtained by theoretical analysisand bench test (e next work is to analyze the influence ofdesign and operation parameters on system dynamics so asto propose effective methods for controlling system vibra-tion (e specific conclusions are as follows

(1) Based on the analytical method the natural fre-quency and modal characteristics of the geartransmission system are obtained (e natural fre-quencies of the system are mainly concentrated inthe 1104 to 23060Hz

(2) By solving the dynamic model and comparing andanalyzing the vibration response of the transmissionsystem it is shown that the vibration of the trans-mission system increases due to the influence of theroot crack In the frequency domain in the middle ofthe spectrum of the faulty transmission system asideband occurs around the meshing frequency (isis because the frequency shift fs of the sun gear has amodulation effect on the meshing frequency fm

(3) (rough the bench experiment we can find thesidebands generated by the modulation of the carrierfrequency fc in the spectrogram of the vibrationacceleration of the faulty system and also there aresidebands generated by the frequency fs modulationof the sun gear (e experimental results verify thecorrectness of the mathematical model It also guidesthe future direction to control nonlinear dynamics ofthe planetary gear system

Data Availability

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

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is work was supported by the National Natural ScienceFoundation of China (Grant no 51965025)

References

[1] Q Zhu L Liu S Li and W Zhang ldquoControl of complexnonlinear dynamic rational systemsrdquo Complexity vol 2018Article ID 8953035 12 pages 2018

[2] Q M Zhu D Y Zhao and J Zhang ldquoA general U-blockmodel-based design procedure for nonlinear polynomialcontrol systemsrdquo International Journal of Systems Sciencevol 47 no 14 pp 1ndash10 2016

[3] G Bonori and F Pellicano ldquoNon-smooth dynamics of spurgears with manufacturing errorsrdquo Journal of Sound and Vi-bration vol 306 no 1-2 pp 271ndash283 2007

[4] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of MechanicsmdashASolids vol 28 no 3pp 461ndash468 2009

[5] Y Guo and R G Parker ldquoDynamic modeling and analysis of aspur planetary gear involving tooth wedging and bearingclearance nonlinearityrdquo European Journal of MechanicsmdashASolids vol 29 no 6 pp 1022ndash1033 2010

[6] Z Chen and Y Shao ldquoDynamic simulation of planetary gearwith tooth root crack in ring gearrdquo Engineering FailureAnalysis vol 31 no 9 pp 8ndash18 2013

[7] Q Zhou A Dynamic Analysis and Numerical Simulation ofCylindrical Spur Gear and Crack Fault PhD thesis TaiyuanUniversity of Technology Taiyuan China 2008

[8] Z Wan ldquoTime-varying mesh stiffness algorithm correctionand tooth crack dynamic modelingrdquo Journal of MechanicalEngineering vol 49 no 11 p 153 2013

[9] C T Walters ldquo(e dynamics of ball bearingsrdquo Journal ofTribology vol 93 no 1 pp 1ndash10 1970

[10] T A Harris andM N KotzalasAdvanced Concepts of BearingTechnology Rolling Bearing Analysis CRC Press Boca RatonFL USA 5th edition 2007

[11] Z G Zhou Study on Dynamics and Time-dependent Reli-ability of Gear Transmission System of Wind Turbine underRandom Wind Conditions PhD thesis Chongqing Univer-sity Chongqing China 2012

[12] O D Mohammed M Rantatalo J O Aidanpaa andU Kumar ldquoVibration signal analysis for gear fault diagnosiswith various crack progression scenariosrdquo Mechanical Sys-tems amp Signal Processing vol 41 no 1-2 pp 176ndash195 2013

[13] J Park Y Kim K Na and B D Youn ldquoVariance of energyresidual (VER) an efficient method for planetary gear faultdetection under variable-speed conditionsrdquo Journal of Soundand Vibration vol 453 no 4 pp 253ndash267 2019


Parker [5] established a dynamic model of the planetarygear train based on the lumped parameter method andsolved the dynamic model to obtain the dynamic re-sponse of the transmission system Chen and Shao [6]studied the effects of internal ring cracks on the time-varying mesh stiffness and dynamic response of plane-tary gear trains Some scholars have also carried out a lotof research on the dynamics of planetary transmissionsystems Zhou [7] studied the influence of crack pa-rameters on dynamic characteristics by establishing afinite element model Wan [8] proposed a meshingstiffness correction method based on the potential energymethod (e dynamic model of a gear transmissionsystem with tooth root crack was established by using thelumped parameter method and the dynamic equationwas solved to obtain the dynamic response In previousplanetary transmission dynamics analysis the bearingwas simplified to the ideal constraint boundary and thebearing was simplified to a constant stiffness coefficientspring also ignoring the influence of the bearing time-varying stiffness on the dynamic characteristics of thetransmission system Bearings and gears are the keycomponents of the planetary transmission system (edynamic characteristics of the bearings have an impor-tant influence on the stability and service life of thetransmission system Walters [9] proposed the dynamicanalysis model of the rolling bearing (e dynamic modelof the rolling bearing was established and the drag forcebetween the rolling element and the ferrule was obtainedby considering the four degrees of freedom of the rollingelement and the six degrees of freedom of the cageHarris and Kotzalas [10] considered the effect of elas-tohydrodynamics on the basis of quasi-static analysis andimproved the pseudostatic analysis method of rollingbearings In establishing the balance equations of therolling element the cage and the inner ring the rollingelementrsquos revolution and bearing deformation parame-ters are obtained

Zhou [11] considered the time-varying stiffness of rollingbearings established the coupled dynamics model of MW-class wind turbine gear transmission system by using thelumped parameter method obtained the inherent character-istics of the transmission system and solved the dynamics ofeach bearing contact stress but it does not consider the in-fluence of the centrifugal force of the roller on the bearingstiffnessMohammed et al [12] presents an investigation of theperformance of crack propagation scenarios to compare thesescenarios from a fault diagnostics point of view Park et al [13]proposes a variance of energy residual (VER) method forplanetary gear fault detection under variable-speed conditions

In the planetary gear transmission system the dynamicmeshing excitation of the gear teeth is transmitted to thecasing through the bearing causing vibration and noise ofthe casing (e dynamic characteristics of the bearing havean important influence on the performance of the entiretransmission system In the past the scholars neglected theinfluence of the mass of the rolling element and the cen-trifugal force of the roller on the time-varying stiffness of the

bearing when studying the gear-bearing coupling dynamicsystem At the same time the dynamic characteristics of thefaulty planetary transmission system were not studied Inthis paper the planetary transmission systemwith tooth rootcrack is taken as the research object According to the Hertzcontact theory considering the mass of the rolling elementand the effect of centrifugal force the time-varying bearingstiffness model is established At the same time the time-varying bearing stiffness the transmission error and thetime-varying meshing stiffness of the cracked gear are in-troduced (e gear-bearing coupling dynamics model withcrack is established (e influence of time-varying bearingstiffness on the dynamic characteristics of the gear trans-mission system is analyzed (e influence of crack failure onthe dynamic characteristics of the gear is explored whichprovides a more detailed mathematical method for the faultdiagnosis and vibration control of the planetary geartransmission system

2 Dynamic Model of Planetary Gear System

(e dynamic characteristics of the planetary gear trans-mission system are affected by many factors (e me-chanical model is simplified as shown in Figure 1 It isassumed that the three planet gears in the planetarytransmission system are evenly distributed along thecircumference and the geometric parameters and phys-ical parameters of the three planet gears are the same (eengagement between the gear pairs is simplified to aspring with time-varying stiffness Considering the gearmeshing error the time-varying bearing stiffness of thebearing and the time-varying meshing factor of the gearthe planetary gear translation-torsion dynamics model isestablished by the lumped parameter method Eachcomponent has 2 translational degrees of freedom and 1torsional degree of freedom and the model has 18 degreesof freedom

erp2crp2

krp2

erp3

ksp3

esp3

csp3

esp2

ksp2

csp2ksu

csuesp1 ksp1

csp1

cpy kpy

cpx

kpxerp1

crp1

krp1

crp3

krp3

ccu

cru kru

csy ksyt

ksxt

csx

kcu

Y

X0

x

y

ζn

ψk

ηn

Figure 1 Mechanical model of planetary gear system


Ring


N

n1crn



N

n1crn


Ir

r2r1113888 1113889 eurour + 1113944

N

n1crn


rr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Sun


N

n1csn



N

n1csn


Is

r2s1113888 1113889 eurous + 1113944

N

n1ksnδsn + csn


rs

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Carrier

1113944N



+kcxxc 0

1113944

N



+ccy _yc + kcyyc 0

Ic

r2c1113888 1113889 eurouc + 1113944

N

n1kpnδnt + cpn


rc

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Planet







Ipn

r2pn



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



+ us + un + esn(2)


















60

(7)



Q Kcδ32

(8)




fx 1113944N


fy 1113944N


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(9)


kb 1113944

N

k1Kk cosφk1113858 1113859 (10)






0

y

x

Q

Ball

Innerrace

Outerrace

ki

ko

mgωc









ω2i Mϕi Kb + Km1113858 1113859ϕi (12)







0 40 80 120 160

Bear

ing

radi

al st

iffne

ss(k

sxtN

middotmndash1

)


351

349

347

345

343

times108






83

5

7

9

0 2 4 106 12

Mes

hing

stiff

ness

(ksp

Nmiddotm

ndash1)


10times108

NormalCrack





(a) (b) (c)


ndash5

0

5

0 005004003002001

Velo

city

(vsy

mm

middotsndash1)

Time (ts)

(a)

Time (ts)

Velo

city

(vsy

mm

middotsndash1)

ndash10

0

10

0 002 004 006 008 01

(b)


0 005004003002001Time (ts)

Velo

city

(vpy

mm

middotsndash1)

ndash4ndash2

024

(a)

Time (ts)

Velo

city

(vpy

mm

middotsndash1)

ndash5

0

5

0 002 004 006 008 01

(b)


0 005004003002001Time (ts)

Velo

city

(vry

mm

middotsndash1)

020

ndash02ndash04

(a)

Velo

city

(vry

mm

middotsndash1)

Time (ts)0 002 004 006 008 01

0204

0ndash02ndash04

(b)










times103

3347Hz

0

fm ndash fb fm + fb

Vel

ocity

(mm

middotsndash1)

Vel

ocity

(mm

middotsndash1)

0

04

08

12


002

006

01fm

2fm

3fm


(a)

times10ndash3

X 1184

Vel

ocity

(mm

middotsndash1)

130 150 170 190110Frequency (Hz)

0

2

4

6

(b)



times103

15

1

05

00

Velo

city

(mm

middotsndash1)

(a)

100 200 300 400 500Frequency (Hz)

012

008

004

0

Velo

city

(mm

middotsndash1)

fm ndash 2fs


fm

(b)






02 04 06 08 10Time (s)

ndash50

0

50

Acc

eler

atio

n(m

middotsndash2)

(a)

ndash100

0

100

Acc

eler

atio

n(m

middotsndash2)

02 04 06 08 10Time (s)

(b)


310 320 330300Frequency (Hz)

0

02

04

06

08

Acc

eler

atio

n (m

middotsndash2)

fm ndash 2fc

fm ndash fc

fm

(a)


0

1

2

Acc

eler

atio

n (m

middotsndash2)

fb

times10ndash3

(b)


Testpoint 1

Testpoint 2

Testpoint 3

z

x y


fm ndash 2fs

fm ndash 2fc

fm ndash fs

fm ndash fcfm

fm ndash fs + 2fc

0

02

04

06

08

1

Acc

eler

atio

n (m

middotsndash2)

310 320 330300Frequency (Hz)



7 Conclusions





Data Availability




Acknowledgments


References















Ring


N

n1crn



N

n1crn


Ir

r2r1113888 1113889 eurour + 1113944

N

n1crn


rr

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Sun


N

n1csn



N

n1csn


Is

r2s1113888 1113889 eurous + 1113944

N

n1ksnδsn + csn


rs

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Carrier

1113944N



+kcxxc 0

1113944

N



+ccy _yc + kcyyc 0

Ic

r2c1113888 1113889 eurouc + 1113944

N

n1kpnδnt + cpn


rc

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Planet







Ipn

r2pn



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)



+ us + un + esn(2)


















60

(7)



Q Kcδ32

(8)




fx 1113944N


fy 1113944N


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(9)


kb 1113944

N

k1Kk cosφk1113858 1113859 (10)






0

y

x

Q

Ball

Innerrace

Outerrace

ki

ko

mgωc









ω2i Mϕi Kb + Km1113858 1113859ϕi (12)







0 40 80 120 160

Bear

ing

radi

al st

iffne

ss(k

sxtN

middotmndash1

)


351

349

347

345

343

times108






83

5

7

9

0 2 4 106 12

Mes

hing

stiff

ness

(ksp

Nmiddotm

ndash1)


10times108

NormalCrack





(a) (b) (c)


ndash5

0

5

0 005004003002001

Velo

city

(vsy

mm

middotsndash1)

Time (ts)

(a)

Time (ts)

Velo

city

(vsy

mm

middotsndash1)

ndash10

0

10

0 002 004 006 008 01

(b)


0 005004003002001Time (ts)

Velo

city

(vpy

mm

middotsndash1)

ndash4ndash2

024

(a)

Time (ts)

Velo

city

(vpy

mm

middotsndash1)

ndash5

0

5

0 002 004 006 008 01

(b)


0 005004003002001Time (ts)

Velo

city

(vry

mm

middotsndash1)

020

ndash02ndash04

(a)

Velo

city

(vry

mm

middotsndash1)

Time (ts)0 002 004 006 008 01

0204

0ndash02ndash04

(b)










times103

3347Hz

0

fm ndash fb fm + fb

Vel

ocity

(mm

middotsndash1)

Vel

ocity

(mm

middotsndash1)

0

04

08

12


002

006

01fm

2fm

3fm


(a)

times10ndash3

X 1184

Vel

ocity

(mm

middotsndash1)

130 150 170 190110Frequency (Hz)

0

2

4

6

(b)



times103

15

1

05

00

Velo

city

(mm

middotsndash1)

(a)

100 200 300 400 500Frequency (Hz)

012

008

004

0

Velo

city

(mm

middotsndash1)

fm ndash 2fs


fm

(b)






02 04 06 08 10Time (s)

ndash50

0

50

Acc

eler

atio

n(m

middotsndash2)

(a)

ndash100

0

100

Acc

eler

atio

n(m

middotsndash2)

02 04 06 08 10Time (s)

(b)


310 320 330300Frequency (Hz)

0

02

04

06

08

Acc

eler

atio

n (m

middotsndash2)

fm ndash 2fc

fm ndash fc

fm

(a)


0

1

2

Acc

eler

atio

n (m

middotsndash2)

fb

times10ndash3

(b)


Testpoint 1

Testpoint 2

Testpoint 3

z

x y


fm ndash 2fs

fm ndash 2fc

fm ndash fs

fm ndash fcfm

fm ndash fs + 2fc

0

02

04

06

08

1

Acc

eler

atio

n (m

middotsndash2)

310 320 330300Frequency (Hz)



7 Conclusions





Data Availability




Acknowledgments


References


























60

(7)



Q Kcδ32

(8)




fx 1113944N


fy 1113944N


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(9)


kb 1113944

N

k1Kk cosφk1113858 1113859 (10)






0

y

x

Q

Ball

Innerrace

Outerrace

ki

ko

mgωc









ω2i Mϕi Kb + Km1113858 1113859ϕi (12)







0 40 80 120 160

Bear

ing

radi

al st

iffne

ss(k

sxtN

middotmndash1

)


351

349

347

345

343

times108






83

5

7

9

0 2 4 106 12

Mes

hing

stiff

ness

(ksp

Nmiddotm

ndash1)


10times108

NormalCrack





(a) (b) (c)


ndash5

0

5

0 005004003002001

Velo

city

(vsy

mm

middotsndash1)

Time (ts)

(a)

Time (ts)

Velo

city

(vsy

mm

middotsndash1)

ndash10

0

10

0 002 004 006 008 01

(b)


0 005004003002001Time (ts)

Velo

city

(vpy

mm

middotsndash1)

ndash4ndash2

024

(a)

Time (ts)

Velo

city

(vpy

mm

middotsndash1)

ndash5

0

5

0 002 004 006 008 01

(b)


0 005004003002001Time (ts)

Velo

city

(vry

mm

middotsndash1)

020

ndash02ndash04

(a)

Velo

city

(vry

mm

middotsndash1)

Time (ts)0 002 004 006 008 01

0204

0ndash02ndash04

(b)










times103

3347Hz

0

fm ndash fb fm + fb

Vel

ocity

(mm

middotsndash1)

Vel

ocity

(mm

middotsndash1)

0

04

08

12


002

006

01fm

2fm

3fm


(a)

times10ndash3

X 1184

Vel

ocity

(mm

middotsndash1)

130 150 170 190110Frequency (Hz)

0

2

4

6

(b)



times103

15

1

05

00

Velo

city

(mm

middotsndash1)

(a)

100 200 300 400 500Frequency (Hz)

012

008

004

0

Velo

city

(mm

middotsndash1)

fm ndash 2fs


fm

(b)






02 04 06 08 10Time (s)

ndash50

0

50

Acc

eler

atio

n(m

middotsndash2)

(a)

ndash100

0

100

Acc

eler

atio

n(m

middotsndash2)

02 04 06 08 10Time (s)

(b)


310 320 330300Frequency (Hz)

0

02

04

06

08

Acc

eler

atio

n (m

middotsndash2)

fm ndash 2fc

fm ndash fc

fm

(a)


0

1

2

Acc

eler

atio

n (m

middotsndash2)

fb

times10ndash3

(b)


Testpoint 1

Testpoint 2

Testpoint 3

z

x y


fm ndash 2fs

fm ndash 2fc

fm ndash fs

fm ndash fcfm

fm ndash fs + 2fc

0

02

04

06

08

1

Acc

eler

atio

n (m

middotsndash2)

310 320 330300Frequency (Hz)



7 Conclusions





Data Availability




Acknowledgments


References





















ω2i Mϕi Kb + Km1113858 1113859ϕi (12)







0 40 80 120 160

Bear

ing

radi

al st

iffne

ss(k

sxtN

middotmndash1

)


351

349

347

345

343

times108






83

5

7

9

0 2 4 106 12

Mes

hing

stiff

ness

(ksp

Nmiddotm

ndash1)


10times108

NormalCrack





(a) (b) (c)


ndash5

0

5

0 005004003002001

Velo

city

(vsy

mm

middotsndash1)

Time (ts)

(a)

Time (ts)

Velo

city

(vsy

mm

middotsndash1)

ndash10

0

10

0 002 004 006 008 01

(b)


0 005004003002001Time (ts)

Velo

city

(vpy

mm

middotsndash1)

ndash4ndash2

024

(a)

Time (ts)

Velo

city

(vpy

mm

middotsndash1)

ndash5

0

5

0 002 004 006 008 01

(b)


0 005004003002001Time (ts)

Velo

city

(vry

mm

middotsndash1)

020

ndash02ndash04

(a)

Velo

city

(vry

mm

middotsndash1)

Time (ts)0 002 004 006 008 01

0204

0ndash02ndash04

(b)










times103

3347Hz

0

fm ndash fb fm + fb

Vel

ocity

(mm

middotsndash1)

Vel

ocity

(mm

middotsndash1)

0

04

08

12


002

006

01fm

2fm

3fm


(a)

times10ndash3

X 1184

Vel

ocity

(mm

middotsndash1)

130 150 170 190110Frequency (Hz)

0

2

4

6

(b)



times103

15

1

05

00

Velo

city

(mm

middotsndash1)

(a)

100 200 300 400 500Frequency (Hz)

012

008

004

0

Velo

city

(mm

middotsndash1)

fm ndash 2fs


fm

(b)






02 04 06 08 10Time (s)

ndash50

0

50

Acc

eler

atio

n(m

middotsndash2)

(a)

ndash100

0

100

Acc

eler

atio

n(m

middotsndash2)

02 04 06 08 10Time (s)

(b)


310 320 330300Frequency (Hz)

0

02

04

06

08

Acc

eler

atio

n (m

middotsndash2)

fm ndash 2fc

fm ndash fc

fm

(a)


0

1

2

Acc

eler

atio

n (m

middotsndash2)

fb

times10ndash3

(b)


Testpoint 1

Testpoint 2

Testpoint 3

z

x y


fm ndash 2fs

fm ndash 2fc

fm ndash fs

fm ndash fcfm

fm ndash fs + 2fc

0

02

04

06

08

1

Acc

eler

atio

n (m

middotsndash2)

310 320 330300Frequency (Hz)



7 Conclusions





Data Availability




Acknowledgments


References

















(a) (b) (c)


ndash5

0

5

0 005004003002001

Velo

city

(vsy

mm

middotsndash1)

Time (ts)

(a)

Time (ts)

Velo

city

(vsy

mm

middotsndash1)

ndash10

0

10

0 002 004 006 008 01

(b)


0 005004003002001Time (ts)

Velo

city

(vpy

mm

middotsndash1)

ndash4ndash2

024

(a)

Time (ts)

Velo

city

(vpy

mm

middotsndash1)

ndash5

0

5

0 002 004 006 008 01

(b)


0 005004003002001Time (ts)

Velo

city

(vry

mm

middotsndash1)

020

ndash02ndash04

(a)

Velo

city

(vry

mm

middotsndash1)

Time (ts)0 002 004 006 008 01

0204

0ndash02ndash04

(b)










times103

3347Hz

0

fm ndash fb fm + fb

Vel

ocity

(mm

middotsndash1)

Vel

ocity

(mm

middotsndash1)

0

04

08

12


002

006

01fm

2fm

3fm


(a)

times10ndash3

X 1184

Vel

ocity

(mm

middotsndash1)

130 150 170 190110Frequency (Hz)

0

2

4

6

(b)



times103

15

1

05

00

Velo

city

(mm

middotsndash1)

(a)

100 200 300 400 500Frequency (Hz)

012

008

004

0

Velo

city

(mm

middotsndash1)

fm ndash 2fs


fm

(b)






02 04 06 08 10Time (s)

ndash50

0

50

Acc

eler

atio

n(m

middotsndash2)

(a)

ndash100

0

100

Acc

eler

atio

n(m

middotsndash2)

02 04 06 08 10Time (s)

(b)


310 320 330300Frequency (Hz)

0

02

04

06

08

Acc

eler

atio

n (m

middotsndash2)

fm ndash 2fc

fm ndash fc

fm

(a)


0

1

2

Acc

eler

atio

n (m

middotsndash2)

fb

times10ndash3

(b)


Testpoint 1

Testpoint 2

Testpoint 3

z

x y


fm ndash 2fs

fm ndash 2fc

fm ndash fs

fm ndash fcfm

fm ndash fs + 2fc

0

02

04

06

08

1

Acc

eler

atio

n (m

middotsndash2)

310 320 330300Frequency (Hz)



7 Conclusions





Data Availability




Acknowledgments


References






















times103

3347Hz

0

fm ndash fb fm + fb

Vel

ocity

(mm

middotsndash1)

Vel

ocity

(mm

middotsndash1)

0

04

08

12


002

006

01fm

2fm

3fm


(a)

times10ndash3

X 1184

Vel

ocity

(mm

middotsndash1)

130 150 170 190110Frequency (Hz)

0

2

4

6

(b)



times103

15

1

05

00

Velo

city

(mm

middotsndash1)

(a)

100 200 300 400 500Frequency (Hz)

012

008

004

0

Velo

city

(mm

middotsndash1)

fm ndash 2fs


fm

(b)






02 04 06 08 10Time (s)

ndash50

0

50

Acc

eler

atio

n(m

middotsndash2)

(a)

ndash100

0

100

Acc

eler

atio

n(m

middotsndash2)

02 04 06 08 10Time (s)

(b)


310 320 330300Frequency (Hz)

0

02

04

06

08

Acc

eler

atio

n (m

middotsndash2)

fm ndash 2fc

fm ndash fc

fm

(a)


0

1

2

Acc

eler

atio

n (m

middotsndash2)

fb

times10ndash3

(b)


Testpoint 1

Testpoint 2

Testpoint 3

z

x y


fm ndash 2fs

fm ndash 2fc

fm ndash fs

fm ndash fcfm

fm ndash fs + 2fc

0

02

04

06

08

1

Acc

eler

atio

n (m

middotsndash2)

310 320 330300Frequency (Hz)



7 Conclusions





Data Availability




Acknowledgments


References


















02 04 06 08 10Time (s)

ndash50

0

50

Acc

eler

atio

n(m

middotsndash2)

(a)

ndash100

0

100

Acc

eler

atio

n(m

middotsndash2)

02 04 06 08 10Time (s)

(b)


310 320 330300Frequency (Hz)

0

02

04

06

08

Acc

eler

atio

n (m

middotsndash2)

fm ndash 2fc

fm ndash fc

fm

(a)


0

1

2

Acc

eler

atio

n (m

middotsndash2)

fb

times10ndash3

(b)


Testpoint 1

Testpoint 2

Testpoint 3

z

x y


fm ndash 2fs

fm ndash 2fc

fm ndash fs

fm ndash fcfm

fm ndash fs + 2fc

0

02

04

06

08

1

Acc

eler

atio

n (m

middotsndash2)

310 320 330300Frequency (Hz)



7 Conclusions





Data Availability




Acknowledgments


References















7 Conclusions





Data Availability




Acknowledgments


References















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