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Research ArticleThe Influence of the Motor Traction Vibration on Fatigue Life ofthe Bogie Frame of the Metro Vehicle

Qiushi Wang Jinsong Zhou Dao Gong Tengfei Wang Jiangxue Chen Taiwen Youand Zhanfei Zhang

Institute of Rail Transit Tongji University Shanghai 201814 China

Correspondence should be addressed to Jinsong Zhou jinsongzhoutongjieducn

Received 6 January 2020 Revised 25 June 2020 Accepted 4 October 2020 Published 23 October 2020

Academic Editor F Viadero

Copyright copy 2020 Qiushi Wang 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

During the service life of metro vehicles the cracks frequently appear at the root of motor seats +is indicates the previousantifatigue designs are unable to cover the actual operating environment Some individual loads such as motor vibration havebeen ignored or wrongly understood which leads to the occurrence of local insufficient fatigue life of the frame To illustrate theinfluence of the motor vibration on the fatigue life of the bogie frame a metro vehicle was taken as an example first the precisefinite element model of the frame was established and its correctness was verified then the vibration characteristics of the framewere analyzed by sweep frequency calculation and finally considering the vibration acceleration signal of the motor measured ona metro line as the excitation the influence of the random vibration on the fatigue life of the frame under traction and idle runningconditions was compared and analyzed by solving the power spectral density of the dynamic stress response at the weak fatiguenodes of the structure +e results show that the energy of the vibration in the frame is mainly concentrated in modes 6 and 7which are excited by the motor transverse and vertical vibration respectively and contribute a lot to the fatigue damage of theframe the fatigue life of the vital positions of the frame under the traction condition significantly reduces compared to the idlerunning condition and the contribution of the low-amplitude vibration above 300Hz to the fatigue damage can be ignored Atlast the importance of the influence of the motor traction vibration on the fatigue life should be fully considered in the metrovehicle design was proposed

1 Introduction

During the operation metro vehicles were excited by variousloads +ese loads not only come from track irregularitiesbut also the inside as well such as traction motor vibrationwithin a wide frequency range [1ndash3] Some studies men-tioned that many fatigue cracks appeared around the areanear to the traction motor seats during their service lifewhich was thought to be related to the local elastic vibration[4 5] +ere are only a few research studies about motorvibration fatigue on the metro bogie frame Some studiesshow that ① the vertical vibration of the frame suspensiontraction motor has an evident impact on the frame fatiguestrength [6] ② decoupling optimization of the motorsuspension system can effectively improve the vibrationisolation to reduce the fatigue damage of the motor seat

[7 8] +e above studies have confirmed the importance ofthe influence of the motor vibration on the fatigue strengthof the frame from the obverse and reverse but have not beenstudied in detail

Figure 1 shows a complete time-domain signal of themotor acceleration vibration from station A to station B in acity metro line It can be seen that the metro vehicle hasexperienced ldquotractionmdashidle runningmdashbrakerdquo conditionssuccessively and the traction condition includes the startingcondition and rated condition It is worth noting that in alarge number of previous fatigue designs and calculationsthe starting condition was often considered as an extraor-dinary condition which leads to the influence of the startingcondition on the fatigue of the frame which is omitted[9ndash11] For this question EN13749-2005 (Appendix D in[12]) has stated that it is necessary to consider the number of

HindawiShock and VibrationVolume 2020 Article ID 7385861 11 pageshttpsdoiorg10115520207385861

applications of these loads in any fatigue damage analysis[12] For such frequently started railway vehicles as metrothe ratio of the starting condition to their entire service life issignificantly different from the high-speed EMU in the sameservice mileage Besides according to the existing researchhigh-amplitude loads often contribute a lot to the fatiguedamage [13] However the starting conditions often causelarge-amplitude loads +erefore the research on the in-fluence of the motor traction vibration (including startingcondition) on the fatigue life of the bogie frame in metrovehicles has practical significance

+e vibration under the braking condition is not causedby motor traction so it will not be discussed in this researchSome studies show that the longitudinal vibration of themotor has contributed a little to the fatigue damage of theframe so the study on the influence of the longitudinalmotor vibration will not be performed in this paper [6]

+e outline of this paper is as follows first the modalshape and the main frequency of the frame were calculatedby the finite element method (FEM) and the FEM resultswere compared with the experimental modal analysis (EMA)results to verify the correctness of the model +en thedynamic characteristics of the bogie frame were analyzedthrough sweep frequency calculation Finally by fullyconsidering the contribution of the high-frequency vibrationto the fatigue damage of the frame the fatigue life of thebogie frame was calculated separately with the motor vi-bration under traction and idle running conditions within1300Hz

2 Establishment and Verification of the Model

21 Finite Element Calculation +e finite element model ofthe metro bogie frame was built by HyperMesh 190 asshown in Figure 2 +e motor seats gearbox seats and partsof the midbeam were meshed as three-dimensional solidelements +e side beams ribs etc were meshed as two-dimensional shell elements +e numbers of solid and shellelements are 286492 and 102839 and the numbers of nodesfor the solid and shell are 63996 and 101933+e coordinatesystem is defined as follows the forward direction of thevehicle is the X-axis the Y-axis is parallel with the track

plane and perpendicular to the X-axis and the Z-axis isperpendicular to the track plane Rigid connections wereestablished between the motor and the motor mountingbase and the completely free modes of the frame without theunconstrained condition were calculated

According to the modal analysis theory of structurevibration before the modal verification the bogie systemrsquosnatural modal characteristics need to be solved +edamping ratio is generally 001sim01 for the bogie framestructure +erefore the influence of the damping ratio onthe natural frequency can be ignored +e fundamentalequation of an undamped free vibration system can beexpressed as

[M] eurou +[K] u 0 (1)

where [M] represents the mass matrix [K] represents thestiffness matrix and u represents the displacement vectorAssume that the special solution of formula (1) is as follows

u φ1113864 1113865sinω t minus t0( 1113857 (2)

where φ represents an n-order eigenvector ω representsthe eigenvalue corresponding to φ and t0 represents aconstant determined by the initial conditions

Solve (1) by substituting (2) and get analytical solution(3)+e calculation results are (ω2

1 φ1) (ω22 φ2) (ω2

n φn)+e parameters ω1 ω2 ωn represent the eigenvalues ofthe system +e eigenvectors φ1 φ2 φn representthe natural modes of the system

[K] φ1113864 1113865 minus ω2[M] φ1113864 1113865 0 (3)

22 Experimental Modal Test +e EMA (experimentalmodal analysis) of the frame was carried out to verify thesimulation modelrsquos correctness +e hammering methodand SISO test method were used to perform the EMA onthe bogie frame +e frame was not equipped withwheelsets gearboxes and other components With sus-pending by a cranersquos hoisting rope it can be considered as acompletely free and unconstrained condition as shown inFigure 3

100

50

0

ndash50

ndash100

Acce

lera

tion

(ms

2 )

290 300 310 320 330 340 350 360Time (s)

Idle running conditionBraking condition

Traction condition

Figure 1 Vertical acceleration vibration signal of a metro motorfrom station A to station B

Brake seat

Gearbox seat

Motor

Secondary springholder

Primary spring holder

Z X

Y

Figure 2 Finite element model of the bogie frame

2 Shock and Vibration

As shown in Figure 4 positions 1 and 7 with the three-direction piezoelectric acceleration sensors stuck here arethe vibration response test points +e hammering pointswere arranged at the other 30 vital geometric positions of theframe +e signals caused by the medium force hammerwere collected through the CPCI box

23 Analysis and Verification +e FEM model (Figure 2)was imported into commercial software +e block Lanczosalgorithm was used to calculate the modal shape and themain frequency of the bogie frame with the frequency below100Hz +e modal shapes are shown in Figure 5 with thetheoretical calculation results on the left and the modal testresults on the right

+eMAC function (modal assurance criterion) was usedto describe the correlation between theoretical andmeasuredmodal shapes as shown in formula (4)+e closer theMACrsquoscalculated value is to 1 the higher the two modal shapesrsquocorrelation degree is +e calculation results of the corre-lation are shown in Table 1

MAC ϕe1113864 1113865j ϕf1113966 1113967k

1113872 11138731113960 1113961 ϕe1113864 1113865

T

j ϕf1113966 1113967k

1113872 11138732

ϕe1113864 1113865T

j ϕe1113864 1113865j ϕf1113966 1113967T

kϕf1113966 1113967

k

(4)

where φej represents the j-th modal shape vector of the finiteelement analysis results and φfk represents the k-th modalshape vector of the experimental modal analysis results

All the relative errors between the FEM results and EMAresults for natural frequency corresponding to each modeare less than 750 (themaximum error is 737) which canbe accepted in engineering as shown in Table 1 +ereforethe accuracy of the finite element model has been verified

3 Random Vibration Analysis

Stress (strain) is always essential data for fatigue assessmentand life prediction +ere are two kinds of methods tocalculate the stress in the bogie frame static stress methodand dynamic stress method For example UIC615-4 [14] andEN13749 the standards for fatigue strength design of thebogie frame are the static stress analysis methods Althoughthe methods are practical and straightforward the effect of

1213 15

11

14 16 1718

222 25

2423

27

29 31

30 32

2120

1019

9

8

26 28

4

5

3

7 (test point)

6

MidframeSide frame

1 (test point)

Figure 4 Layout of test pointsMotor Test point 1

Test point 7

Figure 3 Experimental modal test for the bogie frame

Frequency = 64972Hz Frequency = 60293Hz

Frequency = 52680HzFrequency = 56066Hz

Frequency = 68662Hz

Frequency = 71291Hz

Frequency = 78885Hz


Frequency = 73072Hz

Frequency = 69687Hz

Frequency = 65694Hz

Figure 5 Comparison of modal shapes of FEM and EMA results

Shock and Vibration 3

resonance fatigue has been ignored +e random vibrationanalysis is the method based on the dynamic stress mainlyincluding the quasi-static method time-domain methodand frequency-domain method

① +e quasi-static method is generally used to analyzethe engineering problem that the loading frequency isfar away from the main frequency of the structure+e dynamic stress response of each key point isobtained by using the quasi-static response coeffi-cient and random load spectrum

② +e time-domain method is based on the input of thetime-domain signal of the random load and obtainsthe dynamic stress response of each point throughthe mode superposition method or direct integrationmethod While dealing with a long time-domainsignal the method will spend a lot of running timeand cost a lot of storage space Sometimes it cannotbe even solved out

③ +e frequency-domain method is a widely usedanalysis method +e time-domain signal is trans-formed into the power spectral density (PSD) in thefrequency domain with the Fourier transform As theinput the PSD for displacement velocity or accel-eration is used to calculate the dynamic stress re-sponse based on the established transfer function

Because of the long time-domain signal of the whole metroline about 245 kilometres the time-domain method cannot beapplied to complete the analysis +erefore the frequency-domain method was used in this analysis to solve the dynamicstress response of several nodes of the bogie frame

31 Power Spectral Density (PSD) Function +e vibrationgenerated by the operation of metro vehicles can be regardedas a wide-sense stationary random process (WSRP) +eautocorrelation function Rqq(τ) of the time-domain signalq(t) depends on the time difference τ t2 minus t1 +e auto-correlation function Rqq(τ) is defined as equation (5) Be-sides Rqq(τ) and power spectrum density Sqq(ω) are aFourier-transform pair defined as equation (6)

Rqq(τ) 1113946+infin

minus infinq1 t1( 1113857q2 t2( 1113857p q1 q2 t1 t2( 1113857dq1dq2 (5)

Sqq(ω) 12π

1113946+infin

minus infinRqq(τ)e

minus jωτdτ (6)

+rough the piezoelectric acceleration sensor(Figure 6(a)) and CPCI box (Figure 6(b)) the vertical andlateral acceleration vibration signals of the motor aremeasured on a metro line Since the vibration of the motormeasured on a metro operation line comes from the tractionand idle running conditions together (Figure 1) the motorrsquosvibration acceleration signals in the traction condition andthe idle running condition were selected respectively Andthen the power spectrum density (PSD) of accelerationobtained by Fourier transform is shown in Figure 7

Power spectrum density (PSD) function represents thedistribution of energy in the frequency domain in a randomprocess [15] As shown in Figure 7 the main vibration of themotor is the vertical vibration because the amplitude ofvertical vibrations is more than ten times of the transversevibration +e maximum value of the vertical accelerationpeak is 584 (ms2)2 under the traction condition +emaximum value of the vertical acceleration peak is 322 (ms2)2 under the idle running condition It shows that thevibration of the motor under the traction condition isenhanced

Besides except for the energy distribution within 500Hzunder the traction condition there are still visible peaksignals at the frequency domain of 1000Hzsim2500Hz asshown in Figure 7(a) +erefore the influence of the high-frequency vibration of the motor on the fatigue strength ofthe frame was considered in this analysis Because of thelimited storage capacity and calculation capacity of thecomputer the signals with frequency less than 1300Hz wereintercepted for analysis

32 eoretical Analysis of the Random VibrationAccording to the mechanical vibration theory there are twomethods which are used to take acceleration as the input inrandom vibration analysis fixed-point excitation methodand relative motion method +e fixed-point excitationneeds to convert the acceleration into force or torque andthen apply it to the structure However there is no bettermethod yet to identify the absolute excitation force of arigidly connected structure

+e relative motion method requires applying a fullconstraint on the load excitation point and applying anacceleration field in the whole system which is equivalent tothe load excitation point which vibrates at the same accel-eration of the applied acceleration field in the system

Table 1 Modal analysis of FEM and EMA results

FEA modal result Test modal resultError () MAC Modal shape

Modal order Frequency (Hz) Modal order Frequency (Hz)1 49370 1 47010 478 096 Whole shearing deformation2 56066 2 52680 604 094 Whole twisting deformation3 64972 3 60293 720 094 Side beam bending as ldquoVrdquo type4 68662 4 65694 432 095 Side beam bending as ldquodiamondrdquo type5 71291 5 69687 225 098 Side beam lateral bending on the reverse direction6 78885 6 73072 737 096 Side beam lateral bending in the same direction7 84279 7 79868 523 095 Side beam lateral bending in the same direction


In this paper the relative motion method (Figure 8) wasused to analyze the influence of the motor vibration on theframersquos fatigue life +e measured motor acceleration vi-bration signal was directly used as the input condition andX Y and Z direction constraints were applied on the centerof mass of the motor +e other constraints were released atthe same time +e theoretical derivation process is asfollows

+e single-degree-of-freedom system is shown in Fig-ure 8 Its dynamic differential equation can be written as

meuroy + c( _y minus t _x) + k(y minus x) 0 (7)

where y represents the absolute displacement of thesystem response x represents the displacement of thereference position and z y-x represents the relativedisplacement between them Equation (7) can be writtenas equation (8) +erefore the dynamic differentialequation of the multiple degrees of freedom system can bewritten as equation (9)

meuroz + c _z + kz minus m eurox (8)

[M] euroz +[C] _z +[K] z minus [M] E eurox (t) (9)

According to the mechanical vibration theory the cal-culation results of the response of the MDOF system can beexpressed as

Szz(ω)1113858 1113859 [H]lowast

Sxx(ω)1113858 1113859[H]T (10)

And the spectral matrix relationship between the inputand the response can be expressed as equations (11) and(12)

Sxz(ω)1113858 1113859 Sxx(ω)1113858 1113859[H]T (11)

Szx(ω)1113858 1113859 [H]lowast

Sxx(ω)1113858 1113859 (12)

However the solution z of equation (8) is the relativedisplacement +e absolute displacement y is the

Acceleration sensorMotor

(a)

CPCI box

Computer

(b)

Figure 6 (a) Motor test point (b) CPCI box and computer

Acc

eler

atio

n ((

ms

2 )2 ∙Hzndash1

)

0 500 1000 1500 2000Frequency (Hz)

6

5

4

3

2

1

0

6

5

4

3

2

1

00 50 100 150 200 250 350300 400 450 500

High frequency with low amplitude

Traction conditionIdle running condition

(a)

040

035

030

025

020

015

010

005

Acc

eler

atio

n ((

ms

2 )2 ∙Hzndash1

)

0000 500 1000 1500 2000 2500

Frequency (Hz)


(b)

Figure 7 (a) Acceleration PSD of the motor (vertical) (b) Acceleration PSD of the motor (horizontal)


superposition of the relative displacement z and the refer-ence position displacement x y(t) z(t) + x(t) +e auto-correlation function of absolute displacement y(t) can beeasily derived as

Ryy(τ) E[Y(t) middot Y(t + τ)] E[(Z(t) + X(t))

middot t(Z(t + τ) + X(t + τ))]

Rzz(τ) + Rzx(τ) + Rxz(τ) + Rxx(τ)

(13)

According to equation (6) the absolute displacementpower spectral density of the system can be expressed as

Syy(ω)1113960 1113961 Szz(ω)1113858 1113859 + Szx(ω)1113858 1113859 + Sxz(ω)1113858 1113859 + Sxx(ω)1113858 1113859

(14)

where [M] [C] and [K] represent the mass damping andstiffness matrices of the system structure E represents theindicating vector of the inertial force and [H] represents thefrequency response function matrix

33 Analysis of Vibration Characteristics Based on SweepFrequency Calculation +e random vibration analysismodule of finite element commercial software was used toanalyze the framersquos structural vibration characteristicsAccording to the theory of the relative motion method thecenter of mass of the motor is constrained in X Y and Zdirections and the unit-amplitude-sine acceleration field in+Y and +Z directions was applied to the system to performthe sweep frequency calculation +e range of the sweepfrequency is set at 0sim1300Hz

Figure 9 shows the calculation results of the root meansquare (RMS) cloud diagram for the dynamic stress +ethree nodes in the most concentrated area of σRMS are se-lected as the focus nodes P1 (number 152331) P2 (number164533) and P3 (number 208674) as shown in Figure 9(b)P1 is located at the motor seat body σRMS 1369MPa P2 islocated at the root of the motor seat base (at the fillet of themidbeam) σRMS 1081MPa P3 is located in the middle ofthe bottom plate of the side beam (at the fillet with themidbeam) σRMS 797MPa +e above three positions arealso consistent with where the actual fatigue cracks haveappeared +erefore the three nodes are taken as the laterresearch object

+e dynamic stress responses of P1 P2 and P3 within250Hz are extracted for analysis as shown in Figure 10Mode 2 (56066Hz) and mode 7 (84279Hz) of the framegenerate large resonance due to the vertical vibration of themotor Mode 6 (78885Hz) of the frame generates largeresonance due to the motorrsquos transverse vibration Besidesthe closer the node is to the motor the more resonant modeswill occur (number of resonance frequencies P1gtP2gtP3)+is proved that the influence of the motor traction vi-bration on the frame is local elastic resonance

34 Analysis of the RandomVibration Based on theMeasuredSignal Considering the lateral and vertical vibration ac-celeration signal of the motor measured on a metro line asthe excitation (Figure 7) the relative motion method wasused to solve the dynamic stress power spectral density of therandom vibration response of the frame in actual operation+e technical route is as follows

(1) +e center of mass of the motor is constrained in XY and Z directions in the frame finite elementmodel

(2) +e measured acceleration vibration power spectraldensity of the motor is decomposed +e frequencyrange is set at 0sim1300Hz and the frequency step isset as Δω 01Hz ωj ωmin + j∙Δω (j 0 1 2 n)

(3) Several frequency-domain PSD signals which comefrom Figure 7 are taken as the acceleration field in +Yand +Z directions to apply to the frame system at thesame time

(4) +e dynamic stress response power spectral densityof the nodes is calculated after the modal expansion

+e dynamic stress responses of P1 P2 and P3 areextracted and the double logarithmic spectrum is drawn foranalysis as shown in Figure 11 +e calculation resultsrsquo dataare organized as shown in Table 2 +e main frequencies ofthe key nodes (P1 P2 and P3) under the traction and idlerunning conditions are the same +e dynamic stress re-sponse amplitude of mode 7 and mode 6 of the model ismore significant than others (Table 2) +e results show thatthe vibration energy of the frame is mainly concentrated inmode 7 (84279Hz) and mode 6 (78885Hz) which con-tribute a lot to the fatigue damage of the frame

According to the analysis of vibration characteristicsgiven in Section 33 mode 7 (84279Hz) is excited by themotorrsquos vertical vibration mode 6 (78885Hz) is excited bythe motorrsquos transverse vibration Even though the transversevibration amplitude of the motor is much smaller than thevertical vibration (Figure 7(b)) it still has a high impact onthe framersquos dynamic stress response+erefore in the fatiguestrength design and assessment of the frame the influence ofthe transverse vibration of the motor on the fatigue life of theframe should not be ignored

4 Fatigue Assessment in the FrequencyDomain

+e fatigue assessment method based on frequency-domainstatistics was used to calculate the vibration fatigue life Some

Reference position

k c

+x

+ym

Figure 8 Dynamic model of the relative motion method


50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

84279

7888556066

6866249370

160470173660

71291

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)

Z directionY direction

(a)

50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

56066

49370 6866271291

84279 173660

160470

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(b)

0 50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

0

84279

78885

7129156066

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(c)

Figure 10 (a) Dynamic stress response of P1 (b) Dynamic stress response of P2 (c) Dynamic stress response of P3

P2 (num 164533)

P1 (num 152331)

P3 (num 208674)+1112e + 02+1019e + 02+9260e + 01+8334e + 01+7208e + 01+6282e + 01+5556e + 01+4630e + 01+3704e + 01+2778e + 01+1852e + 01+9260e + 00+0000e + 00

RS RS11SNEG (traction = ndash10)(Avg 75)

(a)

P2 (σRMS = 1081MPa)P1(σRMS = 1369MPa)P3 (σRMS = 797MPa)

+1112e + 02+1019e + 02+9260e + 01+8334e + 01+7208e + 01+6282e + 01+5556e + 01+4630e + 01+3704e + 01+2778e + 01+1852e + 01+9260e + 00+0000e + 00


(b)

Figure 9 (a) RMS stress cloud diagram for the whole bogie frame (MPa) (b) RMS stress cloud diagram for the local part (MPa)


studies show that Dirlikrsquos formulation is more reliable thanother fatigue calculation methods with broadband signals[16 17] Dirlikrsquos formulation is written as

D E[P] middot T

C1113946infin

0S

mp(S)dS (15)

where D represents the cumulative fatigue damage E[P]represents the expected number of peaks T (second) rep-resents the test time C and M represent the parameters ofthe S-N curve S represents the stress range level and p(S)represents the probability density of S (MPa) which can beexpressed as follows

1000100

101

01001

1E ndash 31E ndash 41E ndash 5

1E ndash 8

1E ndash 61E ndash 7

Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)

1 10 100 1000Frequency (Hz)


84279Hz78885Hz68710Hz56066Hz


(a)

1000100

101

01001


1E ndash 8


1 10 100 1000

Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)

Frequency (Hz)


8427978885

6871056066

(b)

1000100

101

01001

1E ndash 31E ndash 41E ndash 51E ndash 61E ndash 71E ndash 8

Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



8427978885

56066

(c)

Figure 11 (a) PSD of P1 dynamic stress response (b) PSD of P2 dynamic stress response (c) PSD of P3 dynamic stress response

Table 2 +e peak and frequency of the dynamic stress response at the key nodes

FrequencyP1stress (MPa2∙Hzminus 1) P2stress (MPa2∙Hzminus 1) P3stress (MPa2∙Hzminus 1)

Traction Idle running Traction Idle running Traction Idle running84279 2157150 1479530 578429 396752 3024340 207914078885 699092 501669 80046 57410 770571 55241956066 366335 128741 12440 6830 62182 2112168710 153482 65530 38997 12132 2761 086349225 133416 47614 8123 2918 2397 0926261520 87341 16146 1029 0399 2614 0702


p(S) D1Q( 1113857e

minus ZQ+ D2 middot ZR2

1113872 1113873eminus z22middotR2

+ D3 middot Zeminus Z22

1113874 1113875

2 m0

radic

D1 2 Xm minus c

21113872 1113873

1 + c2

D2 1 minus c minus D1 + D

21

1 minus R

D3 1 minus D1 minus D2

Z S

2 m0

radic

Q 125 c minus D3 minus D2R( 1113857

D1

R c minus Xm minus D

21

1 minus c minus D1 + D21

c m2m0m4

radic

Xm m1

m0

m2

m4

1113970

E[P]

m2

m4

1113970

mn 1113946 fn

middot G(f)df

(16)

where mn (n 0 1 2 3 4) represents the moment of area ofthe PSD and G(f) 2S(f ) represents the value of the single-side PSD at frequency f (Hz)

Based on the PalmgrenndashMiner rule assume the sum ofdamage at fatigue failure is 1 and total fatigue life Ttot can besolved as follows

Ttot C

E[P] middot 1113938infin0 S

mp(S)dS

(17)

41 Comparison of Fatigue Life between Traction and IdleRunningConditions Extract the dynamic stress responses ofP1 P2 and P3 calculated in Section 24 (Figure 11) andimport them into MSC fatigue software to calculate thefatigue life of the bogie frame By fully considering thecontribution of low-amplitude loads to the fatigue damagethe S-N curve of nominal stress associated with 16MnR steelwas extended with a slope of minus 1(2m minus 1) starting from thepoint associated with 5times107 times cycles

According to the BS7608-2015 welding structure fatigueanalysis standard the S-N curve level was selected as Class-Bto set the parameters as shown in Figure 12 [18 19]

As the calculation results are shown in Table 3 byconsidering the vibration frequency within 1300Hz thefatigue life of key node P1 is 109523 (seconds) under the idlerunning condition the fatigue life of key node P1 is 107134(seconds) under the traction condition Comparing with theidle running condition the fatigue life of P1 P2 and P3under the traction condition is lost about 9931((109523ndash107134)109523 9959) 9966 and 9919 re-spectively+erefore the influence of the motor vibration onthe fatigue strength of the metro bogie frame should be fullyconsidered especially the motor start conditions

lm

lm + 25 times 107

101Fatigue life N (cycles)

Stre

ss ra

nge S

a (M

Pa)

10104

102

103

104

106 108 1010 1012

Figure 12 S-N curve of 16MnR steel (95 reliability)

Table 3 Comparison of fatigue life under the traction condition and idle running condition

Condition P1time (s) P2time (s) P3time (s)Traction 107134 107898 1010012

Idle running 109523 1010365 1012102

Life loss () 9959 9966 9919


42 Influence of the Vibration with High Frequency and LowAmplitude onFatigueDamage It can be seen from Figure 7that there are some vibrations of high frequency with lowamplitude around 1000 Hz which leads to the low am-plitude in the dynamic stress power spectral density of P1P2 and P3 as shown in Figure 11 To investigate theinfluence of dynamic stress of high frequency and lowamplitude on the fatigue damage on the bogie frame themotor vibration frequency within 300 Hz and 1300 Hz wasselected for fatigue life calculation under the tractioncondition

Finally the results show that the bogie frame can endure13614398 (s) within 300Hz and 13614444 (s) within1300Hz which is about 46(s) in difference Because thecontribution of the vibration above 300Hz to the fatiguedamage is too low its effect can be ignored+e fundamentalreason is that the stress response amplitude is too small whenthe frequency is above 300Hz

5 Conclusion

According to the random vibration signal of a motor of thevehicle measured on a metro line the influence of thetractionmotor vibration on the fatigue life of the bogie framewas studied

(1) With the motorrsquos traction vibration the weakpositions of fatigue strength are located at themotor seat body the root of the motor seat base (atthe fillet of the midbeam) and the middle of thebottom plate of the side beam (from the fillet withthe midbeam) +e above three positions areconsistent with where the actual fatigue damageappeared

(2) Mode 7 (84279Hz) andmode 6 (78885Hz) with thehigher corresponding stress amplitude contribute alot to the fatigue damage Besides mode 7 is excitedby themotorrsquos vertical vibration mode 6 is excited bythe motorrsquos transverse vibration

(3) Compared with the idle running condition the fa-tigue life under the traction condition is lost a lot+erefore the influence of the motor vibration onthe fatigue strength of the metro bogie frame cannotbe ignored especially the motor start conditions Inthis case the contribution of the vibration above300Hz to the fatigue damage is too small and itseffect can be ignored

Data Availability

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

Conflicts of Interest

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

Acknowledgments

+is research was supported by the National Natural ScienceFoundation of China (Grant no 51805373)

References

[1] Y Yao Y Yan Z Hu and K Chen ldquo+e motor active flexiblesuspension and its dynamic effect on the high-speed trainbogierdquo Journal of Dynamic Systems Measurement andControl vol 140 no 5 pp 209ndash212 2018

[2] Y Yao X Zhang and X Liu ldquo+e active control of the lateralmovement of a motor suspended under a high-speed loco-motiverdquo Proceedings of the Institution of Mechanical Engi-neers Part F Journal of Rail and Rapid Transit vol 230 no 6pp 1509ndash1520 2015

[3] G Yang YWei G Zhao et al ldquoResearch on key mechanics ofhigh-speed trainrdquo Advances in Mechanics vol 45 no 1pp 217ndash460 2015

[4] L Zhang Z S Ren S G Sun and G Yang ldquoResearch on theinfluence of railway bogie elastic vibration to fatigue liferdquoRailway Locomotive amp Car vol 35 no 2 pp 115ndash119 2018

[5] B J Wang S G Sun Xi Wang Li Zhang L Dong andC Y Jiang ldquoResearch on characteristics of operation loadsand fatigue damage of metro train bogie framerdquo Journal of theChina Railway Society vol 41 no 5 pp 53ndash60 2019

[6] Q An F Li and M H Fu ldquoInfluences of traction motorvibration on bogie frame fatigue strengthrdquo Journal ofSouthwest Jiaotong University vol 45 no 5 pp 209ndash2122010

[7] Z Xia D Gong and J Zhou ldquoDecoupling the optimumdesign of motor suspension system for metro vehiclesrdquoJournal of Tongji University vol 45 no 3 pp 1675ndash16802017

[8] M Zigo E Arslan W Mack and G Kepplinger ldquoEfficientfrequency-domain based fatigue life estimation of spot weldsin vehicle componentsrdquo Forschung im Ingenieurwesen vol 83no 3 pp 921ndash931 2019

[9] M Kassner ldquoFatigue strength analysis of a welded railwayvehicle structure by different methodsrdquo International Journalof Fatigue vol 34 no 2 pp 103ndash111 2012

[10] W Tang WWang Y Wang and Q Li ldquoFatigue strength andmodal analysis of bogie frame for DMUs exported to TunisiardquoJournal of Applied Mathematics and Physics vol 2 no 2pp 342ndash348 2014

[11] B Chen P Zhi and Y Li ldquoFatigue strength analysis of bogieframe in consideration of parameter uncertaintyrdquo Frattura EdIntegrita Strutturale vol 13 no 48 pp 385ndash399 2019

[12] EN13749-2005 Railway Applications Methods of SpecifyingStructural Requirements of Bogie Frames British StandardsInstitution London UK 2005

[13] S Li P Wu and Z Zeng ldquoExtrapolation of load historiesbased on kernel density methodrdquo Journal of the China RailwaySociety vol 39 no 7 pp 25ndash31 2017

[14] UIC615-4-2003Motive Power Units-Bogie and Running Gear-Bogie Frame Structure Strength Tests International Union ofRailways France 2003

[15] M Aykan andM Ccedilelik ldquoVibration fatigue analysis and multi-axial effect in testing of aerospace structuresrdquo MechanicalSystems and Signal Processing vol 23 no 2 pp 897ndash9072009

[16] T Dirlik Application of computers in fatigue analysis PhDthesis +e University of Warwick Coventry UK 1985


[17] M Mrsnik J Slavic and M Boltezar ldquoFrequency-domainmethods for a vibration-fatigue-life estimation- Applicationto real datardquo International Journal of Fatigue vol 47 no 2pp 8ndash17 2013

[18] British Standards Institution BS7608 B S Code of Practice forFatigue Design and Assessment of Steel Structures BritishStandards Institution UK 2015

[19] P Dong and J Hong ldquo+e master S-N curve approach tofatigue of piping and vessel weldsrdquo Weld vol 48 no 2pp 28ndash36 2004


applications of these loads in any fatigue damage analysis[12] For such frequently started railway vehicles as metrothe ratio of the starting condition to their entire service life issignificantly different from the high-speed EMU in the sameservice mileage Besides according to the existing researchhigh-amplitude loads often contribute a lot to the fatiguedamage [13] However the starting conditions often causelarge-amplitude loads +erefore the research on the in-fluence of the motor traction vibration (including startingcondition) on the fatigue life of the bogie frame in metrovehicles has practical significance

+e vibration under the braking condition is not causedby motor traction so it will not be discussed in this researchSome studies show that the longitudinal vibration of themotor has contributed a little to the fatigue damage of theframe so the study on the influence of the longitudinalmotor vibration will not be performed in this paper [6]

+e outline of this paper is as follows first the modalshape and the main frequency of the frame were calculatedby the finite element method (FEM) and the FEM resultswere compared with the experimental modal analysis (EMA)results to verify the correctness of the model +en thedynamic characteristics of the bogie frame were analyzedthrough sweep frequency calculation Finally by fullyconsidering the contribution of the high-frequency vibrationto the fatigue damage of the frame the fatigue life of thebogie frame was calculated separately with the motor vi-bration under traction and idle running conditions within1300Hz

2 Establishment and Verification of the Model

21 Finite Element Calculation +e finite element model ofthe metro bogie frame was built by HyperMesh 190 asshown in Figure 2 +e motor seats gearbox seats and partsof the midbeam were meshed as three-dimensional solidelements +e side beams ribs etc were meshed as two-dimensional shell elements +e numbers of solid and shellelements are 286492 and 102839 and the numbers of nodesfor the solid and shell are 63996 and 101933+e coordinatesystem is defined as follows the forward direction of thevehicle is the X-axis the Y-axis is parallel with the track

plane and perpendicular to the X-axis and the Z-axis isperpendicular to the track plane Rigid connections wereestablished between the motor and the motor mountingbase and the completely free modes of the frame without theunconstrained condition were calculated

According to the modal analysis theory of structurevibration before the modal verification the bogie systemrsquosnatural modal characteristics need to be solved +edamping ratio is generally 001sim01 for the bogie framestructure +erefore the influence of the damping ratio onthe natural frequency can be ignored +e fundamentalequation of an undamped free vibration system can beexpressed as

[M] eurou +[K] u 0 (1)

where [M] represents the mass matrix [K] represents thestiffness matrix and u represents the displacement vectorAssume that the special solution of formula (1) is as follows

u φ1113864 1113865sinω t minus t0( 1113857 (2)

where φ represents an n-order eigenvector ω representsthe eigenvalue corresponding to φ and t0 represents aconstant determined by the initial conditions

Solve (1) by substituting (2) and get analytical solution(3)+e calculation results are (ω2

1 φ1) (ω22 φ2) (ω2

n φn)+e parameters ω1 ω2 ωn represent the eigenvalues ofthe system +e eigenvectors φ1 φ2 φn representthe natural modes of the system

[K] φ1113864 1113865 minus ω2[M] φ1113864 1113865 0 (3)

22 Experimental Modal Test +e EMA (experimentalmodal analysis) of the frame was carried out to verify thesimulation modelrsquos correctness +e hammering methodand SISO test method were used to perform the EMA onthe bogie frame +e frame was not equipped withwheelsets gearboxes and other components With sus-pending by a cranersquos hoisting rope it can be considered as acompletely free and unconstrained condition as shown inFigure 3

100

50

0

ndash50

ndash100

Acce

lera

tion

(ms

2 )

290 300 310 320 330 340 350 360Time (s)

Idle running conditionBraking condition

Traction condition

Figure 1 Vertical acceleration vibration signal of a metro motorfrom station A to station B

Brake seat

Gearbox seat

Motor

Secondary springholder

Primary spring holder

Z X

Y

Figure 2 Finite element model of the bogie frame





MAC ϕe1113864 1113865j ϕf1113966 1113967k

1113872 11138731113960 1113961 ϕe1113864 1113865

T

j ϕf1113966 1113967k

1113872 11138732

ϕe1113864 1113865T

j ϕe1113864 1113865j ϕf1113966 1113967T

kϕf1113966 1113967

k

(4)





1213 15

11

14 16 1718

222 25

2423

27

29 31

30 32

2120

1019

9

8

26 28

4

5

3

7 (test point)

6

MidframeSide frame

1 (test point)


Test point 7




Frequency = 68662Hz

Frequency = 71291Hz

Frequency = 78885Hz


Frequency = 73072Hz

Frequency = 69687Hz

Frequency = 65694Hz











Sqq(ω) 12π

1113946+infin

minus infinRqq(τ)e

minus jωτdτ (6)

















Szz(ω)1113858 1113859 [H]lowast

Sxx(ω)1113858 1113859[H]T (10)


Sxz(ω)1113858 1113859 Sxx(ω)1113858 1113859[H]T (11)

Szx(ω)1113858 1113859 [H]lowast

Sxx(ω)1113858 1113859 (12)



(a)

CPCI box

Computer

(b)


Acc

eler

atio

n ((

ms

2 )2 ∙Hzndash1

)

0 500 1000 1500 2000Frequency (Hz)

6

5

4

3

2

1

0

6

5

4

3

2

1

00 50 100 150 200 250 350300 400 450 500



(a)

040

035

030

025

020

015

010

005

Acc

eler

atio

n ((

ms

2 )2 ∙Hzndash1

)

0000 500 1000 1500 2000 2500

Frequency (Hz)


(b)







(13)



(14)














Reference position

k c

+x

+ym



50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

84279

7888556066

6866249370

160470173660

71291

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(a)

50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

56066

49370 6866271291

84279 173660

160470

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(b)

0 50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

0

84279

78885

7129156066

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(c)


P2 (num 164533)

P1 (num 152331)



(a)


+1112e + 02+1019e + 02+9260e + 01+8334e + 01+7208e + 01+6282e + 01+5556e + 01+4630e + 01+3704e + 01+2778e + 01+1852e + 01+9260e + 00+0000e + 00


(b)




D E[P] middot T

C1113946infin

0S

mp(S)dS (15)


1000100

101

01001


1E ndash 8


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



84279Hz78885Hz68710Hz56066Hz


(a)

1000100

101

01001


1E ndash 8


1 10 100 1000

Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)

Frequency (Hz)


8427978885

6871056066

(b)

1000100

101

01001


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



8427978885

56066

(c)






p(S) D1Q( 1113857e




1113874 1113875

2 m0

radic

D1 2 Xm minus c

21113872 1113873

1 + c2


21

1 minus R


Z S

2 m0

radic


D1


21


c m2m0m4

radic

Xm m1

m0

m2

m4

1113970

E[P]

m2

m4

1113970

mn 1113946 fn

middot G(f)df

(16)



Ttot C


mp(S)dS

(17)




lm

lm + 25 times 107


Stre

ss ra

nge S

a (M

Pa)

10104

102

103

104

106 108 1010 1012




Idle running 109523 1010365 1012102

Life loss () 9959 9966 9919




5 Conclusion





Data Availability




Acknowledgments


References

























MAC ϕe1113864 1113865j ϕf1113966 1113967k

1113872 11138731113960 1113961 ϕe1113864 1113865

T

j ϕf1113966 1113967k

1113872 11138732

ϕe1113864 1113865T

j ϕe1113864 1113865j ϕf1113966 1113967T

kϕf1113966 1113967

k

(4)





1213 15

11

14 16 1718

222 25

2423

27

29 31

30 32

2120

1019

9

8

26 28

4

5

3

7 (test point)

6

MidframeSide frame

1 (test point)


Test point 7




Frequency = 68662Hz

Frequency = 71291Hz

Frequency = 78885Hz


Frequency = 73072Hz

Frequency = 69687Hz

Frequency = 65694Hz











Sqq(ω) 12π

1113946+infin

minus infinRqq(τ)e

minus jωτdτ (6)

















Szz(ω)1113858 1113859 [H]lowast

Sxx(ω)1113858 1113859[H]T (10)


Sxz(ω)1113858 1113859 Sxx(ω)1113858 1113859[H]T (11)

Szx(ω)1113858 1113859 [H]lowast

Sxx(ω)1113858 1113859 (12)



(a)

CPCI box

Computer

(b)


Acc

eler

atio

n ((

ms

2 )2 ∙Hzndash1

)

0 500 1000 1500 2000Frequency (Hz)

6

5

4

3

2

1

0

6

5

4

3

2

1

00 50 100 150 200 250 350300 400 450 500



(a)

040

035

030

025

020

015

010

005

Acc

eler

atio

n ((

ms

2 )2 ∙Hzndash1

)

0000 500 1000 1500 2000 2500

Frequency (Hz)


(b)







(13)



(14)














Reference position

k c

+x

+ym



50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

84279

7888556066

6866249370

160470173660

71291

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(a)

50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

56066

49370 6866271291

84279 173660

160470

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(b)

0 50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

0

84279

78885

7129156066

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(c)


P2 (num 164533)

P1 (num 152331)



(a)


+1112e + 02+1019e + 02+9260e + 01+8334e + 01+7208e + 01+6282e + 01+5556e + 01+4630e + 01+3704e + 01+2778e + 01+1852e + 01+9260e + 00+0000e + 00


(b)




D E[P] middot T

C1113946infin

0S

mp(S)dS (15)


1000100

101

01001


1E ndash 8


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



84279Hz78885Hz68710Hz56066Hz


(a)

1000100

101

01001


1E ndash 8


1 10 100 1000

Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)

Frequency (Hz)


8427978885

6871056066

(b)

1000100

101

01001


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



8427978885

56066

(c)






p(S) D1Q( 1113857e




1113874 1113875

2 m0

radic

D1 2 Xm minus c

21113872 1113873

1 + c2


21

1 minus R


Z S

2 m0

radic


D1


21


c m2m0m4

radic

Xm m1

m0

m2

m4

1113970

E[P]

m2

m4

1113970

mn 1113946 fn

middot G(f)df

(16)



Ttot C


mp(S)dS

(17)




lm

lm + 25 times 107


Stre

ss ra

nge S

a (M

Pa)

10104

102

103

104

106 108 1010 1012




Idle running 109523 1010365 1012102

Life loss () 9959 9966 9919




5 Conclusion





Data Availability




Acknowledgments


References






























Sqq(ω) 12π

1113946+infin

minus infinRqq(τ)e

minus jωτdτ (6)

















Szz(ω)1113858 1113859 [H]lowast

Sxx(ω)1113858 1113859[H]T (10)


Sxz(ω)1113858 1113859 Sxx(ω)1113858 1113859[H]T (11)

Szx(ω)1113858 1113859 [H]lowast

Sxx(ω)1113858 1113859 (12)



(a)

CPCI box

Computer

(b)


Acc

eler

atio

n ((

ms

2 )2 ∙Hzndash1

)

0 500 1000 1500 2000Frequency (Hz)

6

5

4

3

2

1

0

6

5

4

3

2

1

00 50 100 150 200 250 350300 400 450 500



(a)

040

035

030

025

020

015

010

005

Acc

eler

atio

n ((

ms

2 )2 ∙Hzndash1

)

0000 500 1000 1500 2000 2500

Frequency (Hz)


(b)







(13)



(14)














Reference position

k c

+x

+ym



50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

84279

7888556066

6866249370

160470173660

71291

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(a)

50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

56066

49370 6866271291

84279 173660

160470

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(b)

0 50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

0

84279

78885

7129156066

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(c)


P2 (num 164533)

P1 (num 152331)



(a)


+1112e + 02+1019e + 02+9260e + 01+8334e + 01+7208e + 01+6282e + 01+5556e + 01+4630e + 01+3704e + 01+2778e + 01+1852e + 01+9260e + 00+0000e + 00


(b)




D E[P] middot T

C1113946infin

0S

mp(S)dS (15)


1000100

101

01001


1E ndash 8


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



84279Hz78885Hz68710Hz56066Hz


(a)

1000100

101

01001


1E ndash 8


1 10 100 1000

Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)

Frequency (Hz)


8427978885

6871056066

(b)

1000100

101

01001


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



8427978885

56066

(c)






p(S) D1Q( 1113857e




1113874 1113875

2 m0

radic

D1 2 Xm minus c

21113872 1113873

1 + c2


21

1 minus R


Z S

2 m0

radic


D1


21


c m2m0m4

radic

Xm m1

m0

m2

m4

1113970

E[P]

m2

m4

1113970

mn 1113946 fn

middot G(f)df

(16)



Ttot C


mp(S)dS

(17)




lm

lm + 25 times 107


Stre

ss ra

nge S

a (M

Pa)

10104

102

103

104

106 108 1010 1012




Idle running 109523 1010365 1012102

Life loss () 9959 9966 9919




5 Conclusion





Data Availability




Acknowledgments


References





























Szz(ω)1113858 1113859 [H]lowast

Sxx(ω)1113858 1113859[H]T (10)


Sxz(ω)1113858 1113859 Sxx(ω)1113858 1113859[H]T (11)

Szx(ω)1113858 1113859 [H]lowast

Sxx(ω)1113858 1113859 (12)



(a)

CPCI box

Computer

(b)


Acc

eler

atio

n ((

ms

2 )2 ∙Hzndash1

)

0 500 1000 1500 2000Frequency (Hz)

6

5

4

3

2

1

0

6

5

4

3

2

1

00 50 100 150 200 250 350300 400 450 500



(a)

040

035

030

025

020

015

010

005

Acc

eler

atio

n ((

ms

2 )2 ∙Hzndash1

)

0000 500 1000 1500 2000 2500

Frequency (Hz)


(b)







(13)



(14)














Reference position

k c

+x

+ym



50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

84279

7888556066

6866249370

160470173660

71291

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(a)

50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

56066

49370 6866271291

84279 173660

160470

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(b)

0 50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

0

84279

78885

7129156066

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(c)


P2 (num 164533)

P1 (num 152331)



(a)


+1112e + 02+1019e + 02+9260e + 01+8334e + 01+7208e + 01+6282e + 01+5556e + 01+4630e + 01+3704e + 01+2778e + 01+1852e + 01+9260e + 00+0000e + 00


(b)




D E[P] middot T

C1113946infin

0S

mp(S)dS (15)


1000100

101

01001


1E ndash 8


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



84279Hz78885Hz68710Hz56066Hz


(a)

1000100

101

01001


1E ndash 8


1 10 100 1000

Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)

Frequency (Hz)


8427978885

6871056066

(b)

1000100

101

01001


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



8427978885

56066

(c)






p(S) D1Q( 1113857e




1113874 1113875

2 m0

radic

D1 2 Xm minus c

21113872 1113873

1 + c2


21

1 minus R


Z S

2 m0

radic


D1


21


c m2m0m4

radic

Xm m1

m0

m2

m4

1113970

E[P]

m2

m4

1113970

mn 1113946 fn

middot G(f)df

(16)



Ttot C


mp(S)dS

(17)




lm

lm + 25 times 107


Stre

ss ra

nge S

a (M

Pa)

10104

102

103

104

106 108 1010 1012




Idle running 109523 1010365 1012102

Life loss () 9959 9966 9919




5 Conclusion





Data Availability




Acknowledgments


References


























(13)



(14)














Reference position

k c

+x

+ym



50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

84279

7888556066

6866249370

160470173660

71291

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(a)

50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

56066

49370 6866271291

84279 173660

160470

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(b)

0 50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

0

84279

78885

7129156066

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(c)


P2 (num 164533)

P1 (num 152331)



(a)


+1112e + 02+1019e + 02+9260e + 01+8334e + 01+7208e + 01+6282e + 01+5556e + 01+4630e + 01+3704e + 01+2778e + 01+1852e + 01+9260e + 00+0000e + 00


(b)




D E[P] middot T

C1113946infin

0S

mp(S)dS (15)


1000100

101

01001


1E ndash 8


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



84279Hz78885Hz68710Hz56066Hz


(a)

1000100

101

01001


1E ndash 8


1 10 100 1000

Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)

Frequency (Hz)


8427978885

6871056066

(b)

1000100

101

01001


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



8427978885

56066

(c)






p(S) D1Q( 1113857e




1113874 1113875

2 m0

radic

D1 2 Xm minus c

21113872 1113873

1 + c2


21

1 minus R


Z S

2 m0

radic


D1


21


c m2m0m4

radic

Xm m1

m0

m2

m4

1113970

E[P]

m2

m4

1113970

mn 1113946 fn

middot G(f)df

(16)



Ttot C


mp(S)dS

(17)




lm

lm + 25 times 107


Stre

ss ra

nge S

a (M

Pa)

10104

102

103

104

106 108 1010 1012




Idle running 109523 1010365 1012102

Life loss () 9959 9966 9919




5 Conclusion





Data Availability




Acknowledgments


References






















50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

84279

7888556066

6866249370

160470173660

71291

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(a)

50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

00

56066

49370 6866271291

84279 173660

160470

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(b)

0 50 100 150 200 250Frequency (Hz)

50000

40000

30000

20000

10000

0

84279

78885

7129156066

Stre

ss P

SD ((

MPa

)2 ∙Hzndash1

)


(c)


P2 (num 164533)

P1 (num 152331)



(a)


+1112e + 02+1019e + 02+9260e + 01+8334e + 01+7208e + 01+6282e + 01+5556e + 01+4630e + 01+3704e + 01+2778e + 01+1852e + 01+9260e + 00+0000e + 00


(b)




D E[P] middot T

C1113946infin

0S

mp(S)dS (15)


1000100

101

01001


1E ndash 8


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



84279Hz78885Hz68710Hz56066Hz


(a)

1000100

101

01001


1E ndash 8


1 10 100 1000

Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)

Frequency (Hz)


8427978885

6871056066

(b)

1000100

101

01001


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



8427978885

56066

(c)






p(S) D1Q( 1113857e




1113874 1113875

2 m0

radic

D1 2 Xm minus c

21113872 1113873

1 + c2


21

1 minus R


Z S

2 m0

radic


D1


21


c m2m0m4

radic

Xm m1

m0

m2

m4

1113970

E[P]

m2

m4

1113970

mn 1113946 fn

middot G(f)df

(16)



Ttot C


mp(S)dS

(17)




lm

lm + 25 times 107


Stre

ss ra

nge S

a (M

Pa)

10104

102

103

104

106 108 1010 1012




Idle running 109523 1010365 1012102

Life loss () 9959 9966 9919




5 Conclusion





Data Availability




Acknowledgments


References























D E[P] middot T

C1113946infin

0S

mp(S)dS (15)


1000100

101

01001


1E ndash 8


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



84279Hz78885Hz68710Hz56066Hz


(a)

1000100

101

01001


1E ndash 8


1 10 100 1000

Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)

Frequency (Hz)


8427978885

6871056066

(b)

1000100

101

01001


Stre

ss P

SD ((

ms

)2 ∙Hzndash1

)



8427978885

56066

(c)






p(S) D1Q( 1113857e




1113874 1113875

2 m0

radic

D1 2 Xm minus c

21113872 1113873

1 + c2


21

1 minus R


Z S

2 m0

radic


D1


21


c m2m0m4

radic

Xm m1

m0

m2

m4

1113970

E[P]

m2

m4

1113970

mn 1113946 fn

middot G(f)df

(16)



Ttot C


mp(S)dS

(17)




lm

lm + 25 times 107


Stre

ss ra

nge S

a (M

Pa)

10104

102

103

104

106 108 1010 1012




Idle running 109523 1010365 1012102

Life loss () 9959 9966 9919




5 Conclusion





Data Availability




Acknowledgments


References






















p(S) D1Q( 1113857e




1113874 1113875

2 m0

radic

D1 2 Xm minus c

21113872 1113873

1 + c2


21

1 minus R


Z S

2 m0

radic


D1


21


c m2m0m4

radic

Xm m1

m0

m2

m4

1113970

E[P]

m2

m4

1113970

mn 1113946 fn

middot G(f)df

(16)



Ttot C


mp(S)dS

(17)




lm

lm + 25 times 107


Stre

ss ra

nge S

a (M

Pa)

10104

102

103

104

106 108 1010 1012




Idle running 109523 1010365 1012102

Life loss () 9959 9966 9919




5 Conclusion





Data Availability




Acknowledgments


References
























5 Conclusion





Data Availability




Acknowledgments


References






















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