Fatigue Life Calculation Method and Experimental Study of

Research ArticleFatigue Life Calculation Method and Experimental Study of theMultiple Corrugated Diaphragm Coupling

Angang Cao 1 Chunhua Ding2 Wei Li1 and Zhiyong Zhang1

1School of Mechanical Engineering Zhengzhou University of Science and Technology Zhengzhou 450064 China2e 703 Research Institute of CSSC Harbin 150078 China

Correspondence should be addressed to Angang Cao caoangang88163com

Received 16 August 2021 Revised 18 October 2021 Accepted 15 November 2021 Published 27 November 2021

Academic Editor Nur Izzi Md Yusoff

Copyright copy 2021 Angang Cao 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

+emultiple corrugated diaphragm (MCD) coupling is a new flexible coupling developed based on the diaphragm and diaphragmdisc coupling Compared to traditional couplings the MCD coupling has the advantages of high torque diameter high torqueweight and high compensation capability It is more suitable for high power speed and high power density working conditions+eMCD coupling is subjected to axial angular torque and centrifugal force loads+e fatigue failure caused by alternating stressis the primary failure mode of the coupling +e fatigue life of the MCD coupling cannot be accurately calculated because of thecomplexity of the force in operation Some theoretical simplifications can only obtain the approximate result In this paper aparameterized finite element model of the MCDs is established A method for calculating the fatigue safety factor of the MCDcoupling is proposed based on a modified Goodman curve to know the design of theMCD coupling+e feasibility of this methodis verified by the fatigue life test of the coupling

1 Introduction

+e multiple corrugated diaphragm (MCD) coupling is anew flexible coupling developed based on the diaphragmand diaphragm disc coupling Its core component is the highstrength and high toughness stainless steel diaphragm diskgroup Multiple separated diaphragm discs in the disc grouphave an internal fault protection function making thecoupling have greater flexibility and obtain greater torquetransmission capacity without increasing the diameter of thecoupling In power transfer and deformation compensationthe parallel corrugated diaphragm increases the profile byexpanding the wave +erefore compared with the tradi-tional coupling the MCD coupling has the advantages ofhigh torque diameter ratio high torque weight ratio andlarge compensation capacity which is more suitable forincreased power and speed conditions [1ndash4]

Previous researchers conducted several studies on thecoupling applied in different fields Nagesh et al conductedfatigue analysis on metal diaphragm couplings of variousmaterials through test data obtained from the S-N curve

+ey analyzed the influence of angle offset on the fatigue lifeof metal diaphragm couplings [5] Verucchi et al carried outan experimental study on the structure of the diaphragmcoupling of the turbine +ey proved that the diaphragmstructure with shot-peening treatment could improve thefatigue life of the design by at least 10 [6] Li et al in-troduced the coupling and other coupling mechanisms intheir work [7] Herbstritt et al introduced an economicalcoupling [8] Duong and Kazerounian designed an im-proved diaphragm coupling for aerospace applications [9]Daniel discussed the virtual entity method of creating elasticcoupling with computer-aided technology and completedthe modeling design of the flexible coupling model [10]Dobre et al carried out finite element analysis on the elasticcoupling and demonstrated the influence of axial deviationon the diaphragm coupling in the process of transmittingtorque that is the maximum equivalent stress increases withthe increase of axial deviation [11] Wang and Chang studiedthe axial vibration characteristics of the coupling anddemonstrated the shock absorption characteristics of thecoupling as an elastic coupling mechanism [12] Farias et al

HindawiShock and VibrationVolume 2021 Article ID 6674636 9 pageshttpsdoiorg10115520216674636

discussed the metal components of the coupling andsummarized the categories and suitable types of variousmetal components in the coupling [13] Lun et al analyzedthe strength and vibration characteristics of the diaphragmcoupling +ey showed the influence of vibration charac-teristics of the diaphragm coupling on the strength under theworking state [14] Mancuso introduced several couplingsfor gas turbines and mentioned that the multiple corrugateddiaphragm couplings are more suitable for gas turbines Stillthe design and calculation methods of corrugated filmcouplings were not introduced [15] Li studied the corru-gated diaphragmrsquos structural strength fatigue life and vi-bration characteristics by establishing the parametric finiteelement model of the corrugated diaphragm [16] Yang et alused the MSC Patran module in finite element analysissoftware to check the nonlinear strength of the diaphragmcoupling [17] Li and Li analyzed the dynamic characteristicsand vibration mechanism of parallel misalignment andangular misalignment middle rotors by building a multispanrotor-bearing system testbed [18] Feng et al established themodel of trunnion joint to study the influence of rotatingspeed and the ratio of inner radius to the outer radius on theaxial stiffness which provides a reference for designers tochoose the proper ratio of inner radius to the outer radius ofthe rotating circular plate [19]

According to the existing references many research in-stitutions on diaphragm couplings membrane disc couplingsand laminated couplings have been more in depth Manyscholars carried out extensive research from the theoreticalcalculation and finite element analysis to the stress stiffnessand fatigue life test +e diaphragm coupling membrane disccoupling laminated coupling stress analysis fatigue lifeanalysis design method processing technology test methodand installation application have an emotional experienceHowever research studies on theMCD coupling are relativelyfew +e research on the fatigue stress and fatigue life of theMCD coupling is even less For the design of the couplingfatigue life is a problem that must be considered and paidattention to +erefore it is necessary to study the fatiguestress and fatigue life of the MCDs to promote the applicationand development of the MCD coupling

In this paper the finite element model of the MCDs isestablished by Ansys finite element analysis software Amethod for calculating the fatigue safety factor of the MCDcoupling is presented based on the third strength theory andmodified Goodman curve +en the fatigue life test methodof the MCD coupling is designed +e axial fatigue life testand angular fatigue life test are carried out which verify thefeasibility of the calculation method of the fatigue safetyfactor of the MCDs to a certain extent

2 Geometric Parameters and WorkingParameters of the MCDs

+e shape of the MCDs is shown in Figure 1 And thespecific geometric parameters of the MCDs in this paper areshown in Table 1 +e influence of the bolt hole and splinewas ignored in the subsequent analysis [4] +e load pa-rameters of the MCDs in this paper are shown in Table 2

3 Fatigue Safety Factor of the MCDs

Disk couplings require very high processing technology andunique materials mainly because they adapt to moderateflexibility so fatigue strength is their main performanceindex To determine fatigue strength the method proposedby NB Rothfuss is to draw the average composite stress andalternating composite stress on the fatigue diagram of equallife and set 107 cycle lifelines as the fatigue limit parameterwith infinite life limit For fatigue diagrams three averagestress correction methods have been invented to avoid thetrouble of fatigue tests under different average stressesnamely Goodman method Gerber method and Soderbergmethod +e three modified fatigue diagrams are shown inFigure 2 Considering the dispersion of fatigue data theGerber parabola is not safe and the calculation is compli-cated +e Soderberg line can make safe predictions and inmany cases it seems overly safe Although the Goodman lineis not 100 safe it is much safer than the Gerber parabolaand it is easy to calculate so it is widely used in the fatiguedesign of mechanical parts In this paper the Goodmancurve is simplified into a straight line and the modifiedGoodman curve is used as the infinite lifeline to calculate thefatigue safety factor [20]

According to the Ansys help file we can see that Plane42is defined by four nodes having two degrees of freedom ateach node translations in the nodal x- and y-directionsPlane25 is defined by four nodes having threedegrees of freedom per node translations in the nodal x- y-and z-direction Plane25 is a generalization of the axisym-metric version of Plane42 Obviously for the MCDs onlythe x- and y-direction is not enough and Plane25 is moresuitable for the analysis of MCDs In addition the FE modelis characterized by using a simple two-dimensional model toanalyze three-dimensional forces +erefore this paperadopts the Plane25 element for FE modeling

In Ansys software APDL (Ansys Parametric DesignLanguage) was used for parametric modeling and Plane25element was used to mesh the membrane disc with quad-rilateral ring elements [1 2 4] +e finite element model ofthe MCDs was obtained [21ndash26] By applying an axial loadangular load centrifugal force load and torque load to thefinite element model of the MCDs the radial stress tan-gential stress and torsional shear stress of the MCDs underdifferent loads can be obtained [27ndash29]

+e stress generated by the axial load and the centrifugalload of theMCDs is in the same plane so the two stresses canbe combined linearly

σr σrx + σrl

σt σtx + σtl(1)

In the formula σr is the synthetic radial stress σt is thesynthetic tangential stress σrx is the radial stress due to axialload σtx is the tangential stress due to axial load σrl is theradial stress generated under centrifugal force loading σtl isthe tangential stress generated under centrifugal forceloading

+erefore the maximum and minimum principalstresses of the static stress of the MCDs are

2 Shock and Vibration

σstaticmax σr + σt

2+

σr minus σt( 11138572

4+ τ2

1113971

σstaticmin σr + σt

2minus


4+ τ2

1113971

(2)

In the formula τ is the shear stress generated by torqueload

+e dynamic stress of the MCDs is the alternating stresscaused by angular deformation +e MCDs change once perrotation which also belongs to the plane stress state

+erefore the maximum and minimum principal stresses ofthe dynamic stress are

σdynamicmax σrf

σdynamicmin σtf(3)

In the formula σrϕ is the radial stress under angular loadand σtϕ is the tangential stress under angular load

According to the third strength theory the static anddynamic working stresses of the MCD can be obtained asfollows [30]

When σmaxσmin lt 0 the working stress is

Outer ringbolt hole

Sinusoidalwave form profile

inside ringspline

Figure 1 +e shape of the MCDs

Table 1 Geometric parameters of the MCDs

Serial number Name of the parameter Symbol (unit) Value1 +e inner radius of the profile a (mm) 1042 +e outside radius of the profile b (mm) 1623 +e thickness of the profile t (mm) 054 +e amplitude of the profile c (mm) 355 +e thickness of the hub h1 (mm) 126 +e thickness of the flange h2 (mm) 127 +e radius of the transition fillet r2 (mm) 28 +e radius of the inner circle r0 (mm) 899 +e radius of the outer circle r1 (mm) 18210 +e number of diaphragms in an assembly m (piece) 15


Axial load (mm) Angle load (deg) Torque load (Nmiddotm) Centrifugal load (rmin)25 025 55148 5200

Shock and Vibration 3

σ σmax1113868111386811138681113868

1113868111386811138681113868 + σmin1113868111386811138681113868

1113868111386811138681113868 (4)

When σmaxσmin gt 0 the working stress is

σ max σmax1113868111386811138681113868

1113868111386811138681113868 σmin1113868111386811138681113868

11138681113868111386811138681113966 1113967 (5)

If only symmetrical cyclic alternating stress is assumed theworking stress is required to be σdynamic le [σdynamic]When onlystatic stress is applied the operational stress is needed to beσstatic le [σstatic ] In the formula [σdynamic] is the allowabledynamic stress and [σstatic ] is the allowable static stress In thegeneral case under this cyclic characteristic the line betweenthe composite stress P of static and dynamic stresses and theorigin O and the intersection point of the modified Goodmancurve isM It is safe when the ratio ofA to B is more significantthan one +e modified Goodman curve is assumed to be astraight line for simplicity as shown in Figure 3

+en the safety factor of the MCD coupling is

n OM

OP (6)

According to the above derivation fatigue safety factorsat each position of the profile line of the MCD coupling canbe obtained by using Ansys software as shown in Figure 4+e safety factor at each position of the MCD profile line isgreater than 1 and the minimum value is 134 which meetsthe design requirements

4 Fatigue Life Test of the MCD Coupling

During the coupling operation the external axial or angularexcitation will inevitably lead to the accelerated destructionof the flexible components which will reduce the life of thecoupling Under the rated compensation capacity and tor-que the stress level on the flexible components of thecoupling is far below the materialrsquos ultimate strength and

the fatigue stress is the direction that the coupling designneeds to focus +e fatigue failure caused by alternatingstress is the primary failure mode of the coupling In thispaper through the fatigue life test of the coupling thefeasibility of calculating the fatigue safety factor of the MCDcoupling is verified

41 Accelerated Fatigue Test Principles +e coupling isgenerally designed according to the infinite life or the samelife as the unit but the life design is usually calculated by 20years +erefore the accelerated equivalent fatigue test mustbe carried out because the cycle time of regular fatigue testsis too large and time-consuming According to the identicalfatigue test method the equivalent fatigue test of the cou-pling can be carried out by increasing the excitation fre-quency and vibration amplitude According to the sinusoidal

σe

σy σu0

Mean stress (σm)

Stre

ss am

plitu

de (σ

m)

Soderberg

Gerber

Goodman

Figure 2 +ree average stress correction methods

O

M

P

05σ-1

σdynamic

σdynamic

σsσstatic σstatic

Figure 3 Goodman curve

1

2

3

4

5

Fatig

ue sa

fety

fact

or

110 120 130 140 150 160 170100radius (mm)

Figure 4 +e fatigue safety factor of the MCD coupling


vibration fatigue equivalent test method there is a rela-tionship between the test time before and after the equivalenttest and the excitation amplitude as follows

t1

t2

x2

x11113888 1113889

k

s2

s1times

t21t22

1113888 1113889

k

(7)

In the formula t1 and t2 are the time before and after theequivalent test and x1

and x2

are the accelerationmagnitudes

before and after the equivalent test K value depends onmaterial damping characteristics geometric shape loadingform and temperature environmental conditions K 6 isrecommended in similar trials in the United States in ChinaKvalue is determined to be between 5 and 7 through many tests+is paper prefers to be conservative K 5

Because of the limitation of the flexible coupling com-pensation performance test of vibration amplitude adjustmentdamaged coupling compensation is too significant and toomuch To accelerate the fatigue test this test will be set to10mm axial vibration amplitude and the angular amplitude isset as 574mm In this case the axial vibration amplitude is twotimes of the rated axial compensation of the coupling Also themiddle length of the coupling is 658 mm so the angularamplitude is about 2 times of the rated angular compensation

For the axial vibration the excitation comes from theaxial vibration of the main engine +e primary engineexcitation frequency is taken at 20 and the unit works 24hours a day then the coupling fatigue alternating timeswithin 20 years are

N1 ttotal times f 20 times 365 times 24 times 3600 times 20 126 times 1010(8)

Set the excitation frequency of the equivalent test as35Hz and the excitation amplitude as 10mm thenaccording to equation (7) the number of cycles in theidentical test is 146times106 times about 116 hours

For the angular vibration the excitation comes from thecombined action of angular deviation and circular rotation+e coupling speed is 5200 rmin the unit works 24 hours aday then the coupling fatigue alternating times in 20 yearsare

N2 ttotal times f 20 times 365 times 24 times 3600 times520060

546 times 1010

(9)


42 Accelerated Fatigue Life Test Schemes Fatigue damage ofthe flexible coupling refers to that in the use process underthe condition of repeated stress the performance of flexiblecomponents will be reduced or damaged so that they cannotcontinue to use

In this paper the accelerated fatigue test of the MCDcoupling was carried out on the vibration testbed of thecoupling (as shown in Figure 5) +e influence of fatiguedeformation was judged by the axial stiffness changed beforeand after the test

+e test scheme is as follows

(1) +e axial stiffness of two groups of the MCD as-semblies (numbered 1 and 2 respectively) in the

(a) (b)

Figure 5 Coupling vibration test bench (a) Angular vibration test (b) Axial vibration test


prototype was measured and the results wererecorded

(2) +e prototype of the MCD coupling was installed inthe axial vibration test position of the coupling vi-bration testbed +e vibration frequency of thetestbed was adjusted to 35Hz the vibration ampli-tude was adjusted to 10mm and the 12-houraccelerated fatigue test was carried out

(3) Remove the prototype check whether there is anydamage in the appearance of the MCDs and then

measure the axial stiffness of the two groups of theMCD assemblies and record the results

(4) Install the prototype of the MCD coupling in theangular vibration test position of the coupling vi-bration test bench (as shown in Figure 6) adjust thevibration frequency of the test bench to 150Hz andthe vibration amplitude to 574mm and conduct a14-hour accelerated fatigue test


Figure 6 Angular vibration test of the MCD coupling

0 25 50 75 100 125 150 175 200 225 250 275Axial deformation (001 mm)

0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)

Theoretical minimumTheoretical maximum

Before fatigue testAfter fatigue test

(a)

25 50 75 100 125 150 175 200 225 250 2750Axial deformation (001 mm)

0

5

10

15

20

25

30

35A

xial

forc

e (KN

)



(b)

Figure 7 Axial stiffness curve of the MCD assemblies before and after the axial fatigue life test (a)+eMCD assemblies of 1 (b)+eMCDassemblies of 2



5 Results and Discussion

After inspection no damage was found in the appearance ofthe two MCD assemblies after the axial fatigue life test Aftersorting the axial stiffness test data of the MCD assembliesbefore and after the axial fatigue life test are shown inFigure 7

After inspection no damage was found in the appear-ance of the twoMCD assemblies after the angular fatigue lifetest After sorting the axial stiffness test data of the MCDassemblies before and after the angular fatigue life test areshown in Figure 8

+e average axial stiffness of the two groups of MCDassemblies before and after the fatigue test is shown inTable 3

After the fatigue test no damage was found in the twoMCD assemblies It can be seen from the stiffness curvesbefore and after the fatigue life test of the MCDs in Figures 7and 8 and the stiffness values of MCD assemblies before andafter the fatigue test in Table 3 that after the axial fatigue testthe axial stiffness of 1 MCD assemblies decreased by 246and the axial stiffness of 2 MCD assemblies decreased by310 after the angular fatigue test the axial stiffness of 1MCD assemblies and 2 MCD assemblies decreased by

417 and 451 respectively compared with that beforethe fatigue test After the two fatigue tests the stiffness of 1MCD assemblies decreased by 663 and the axial stiffnessof 2 MCD assemblies decreased by 761 compared withthat before the tests +e results show that the axial stiffnessof the MCDs has little change after the fatigue test and theperformance of the MCD coupling in the fatigue life canmeet the design requirements

6 Conclusions

In this paper the parameterized finite element model of theMCDs was established by Ansys software and the followingresults were obtained

(1) A method for calculating the fatigue safety factor ofthe MCDs is presented Based on this the MCDs canbe optimized to improve the performance of theMCD coupling

(2) +e axial and angular accelerated fatigue life tests ofthe MCD coupling are carried out using the cou-plingsrsquo vibration testbed After the fatigue life testthe axial stiffness of the MCD assemblies is reducedcompared with that before the test However it is stillwithin the design range which proves that theperformance of the MCDs designed in this papermeets the design requirements during the fatigue life

0

5

10

15

20

25

30

35A

xial

forc

e (KN

)


eoretical minimumeoretical maximum

Before fatigue testAer fatigue test

(a)

0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)




(b)

Figure 8 Axial stiffness curve of the MCD assemblies before and after the angular fatigue life test (a) +e MCD assemblies of 1 (b) +eMCD assemblies of 2

Table 3 Stiffness value of the MCD assemblies before and after the fatigue test

Partnumber

Axial stiffnessbefore test (kN

mm)

Axial stiffness after theaxial fatigue test (kN

mm)

Relative to the changebefore the experiment

()

Axial stiffness after theangular fatigue test (kN

mm)


()1 10198 9948 246 9522 6632 901 873 310 8323 761


period and verifies the feasibility of the calculationmethod of the fatigue safety factor of the MCDcoupling proposed in this paper

Data Availability

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

Conflicts of Interest

Angang Cao received a PhD degree from Harbin Ship-building Boiler Turbine Research Institute Harbin China in2020 Now he is an Associate Professor at the ZhengzhouUniversity of Science and Technology His current researchinterests include mechanical transmission machinery dy-namics and mechanical structure Chunhua Ding received amasterrsquos degree from Harbin Shipbuilding Boiler TurbineResearch Institute Harbin China in 2011 Now he is aprofessor of engineering in the CSIC No 703 ResearchInstitute His current research interests include mechanicaltransmission machinery dynamics and mechanical struc-tureWei Li received a masterrsquos degree fromXirsquoan Universityof Technology Xirsquoan China in 2010 Now he is an AssociateProfessor at the Zhengzhou University of Science andTechnology His current research interests include me-chanical transmission machinery dynamics andmechanicalstructure Zhiyong Zhang received a masterrsquos degree fromZhengzhou University Zhengzhou China in 2014 Now heis a lecturer at the Zhengzhou University of Science andTechnology His current research interests include me-chanical transmission machinery dynamics andmechanicalstructure

Acknowledgments

+is work was supported by the Key Scientific ResearchProject of Colleges and Universities of Henan ProvincialDepartment of Education (nos 20B460016 21B460020 and22B460027)

References

[1] A Cao C Ding and W Xu ldquoStudy on optimization ofwaveform diaphragm molded linesrdquo Journal of Engineeringfor ermal Energy and Power vol 34 no 1 pp 120ndash1242019 (in Chinese)

[2] A Cao S Chang C Ding and S Li ldquoDesign and optimizationof transition fillet for waveform diaphragmrdquo Ship Science andTechnology vol 40 no 4 pp 139ndash142 2018 (in Chinese)

[3] A Cao L Xing and X Liu ldquoTorsional stiffness of waveformdiaphragm with sinusoidal molded lines on the basis ofmembrane theoryrdquo Journal of Engineering for ermal Energyand Power vol 35 no 8 pp 158ndash164 2020 (in Chinese)

[4] A Cao L Xing X Zheng and Y Zhu ldquoAnalytical and ex-perimental study on the axial stiffness of multiple corrugateddiaphragm coupling based on the corrugated circular platetheoryrdquo Proceedings of the Institution of Mechanical Engineers- Part C Journal of Mechanical Engineering Science Article ID095440622110145 2021

[5] S Nagesh A M Junaid Basha and T D Singh ldquoDynamicperformance analysis of high speed flexible coupling of gasturbine engine transmission systemrdquo Journal of MechanicalScience and Technology vol 29 no 1 pp 173ndash179 2015

[6] C Verucchi J Bossio G Bossio and G Acosta ldquoMisalign-ment detection in induction motors with flexible coupling bymeans of estimated torque analysis and MCSArdquo MechanicalSystems and Signal Processing vol 80 pp 570ndash581 2016

[7] J J Li E C Yang W J Liu and Y Li ldquoCoupling modelingand analysis of a wind energy converterrdquo Advances in Me-chanical Engineering vol 8 no 6 2016

[8] W R Herbstritt D J Konopka and T L Slagle ldquoIntro-duction of an economical diaphragm coupling for mid-rangeapplicationsrdquo in Proceedings of the 2000 ASME designengingeering technical conference p 13 Baltimore MD USASeptember 2000

[9] L Duong and K Kazerounian ldquoDesign improvement of themechanical coupling diaphragms for aerospace applica-tionsrdquo Mechanics Based Design of Structures and Machinesvol 35 no 4 pp 467ndash479 2007

[10] D Daniel ldquoComputer aided design of elastic coupling withmetallic membranes using solid modelingrdquo JIDEG vol 3no 4 pp 21ndash24 2008

[11] D Dobre I Simion V G Adir G M Adir and N PasculdquoFinite element analysis of the flexible coupling with metallicmembranesrdquo Annals of DAAAM amp Proceedings vol 1pp 421ndash423 2010

[12] X F Wang and D G Chang ldquoKinematic and dynamic an-alyses of tripod sliding universal jointsrdquo Journal of MechanicalDesign vol 131 no 6 Article ID 061011 2009

[13] R C Farias J E Cohen C Jutten and P Comon ldquoJointdecompositions with flexible couplingsrdquo in Proceedings of theLatent Variable Analysis and Signal Separation LiberecCzech Republic August 2015

[14] L Lun C Dengqing and W Jin ldquoRigid flexible couplingdynamic modeling and vibration control for flexible space-craft based on its global analytical modesrdquo Science China(Technological Sciences) vol 62 no 4 pp 92ndash102 2019

[15] J R Mancuso ldquoFlexible couplings for gas turbine applica-tionsrdquo in Proceedings of the ASME 1989 International GasTurbine and Aeroengine Congress and Exposition TorontoCanada June 1989

[16] X Li ldquoStructure analysis and optimization of marine cor-rugated diaphragm couplingrdquo Masterrsquos +esis (in Chinese)Harbin Engineering University Harbin China 2018

[17] L Yang X Xu H Qiao and C Zeng ldquoFinite element analysisof elastic membrane coupling based on MSC Patranrdquo Ma-chinery vol 42 no 1 pp 50ndash52 2015 (in Chinese)

[18] M Li and Z Li ldquo+eoretical and experimental study ondynamics of rotor-bearing system with the faults of couplingmisalignmentrdquo Journal of Vibration Measurement amp Diag-nosis vol 35 no 2 pp 345ndash351 2015 (in Chinese)

[19] S Feng H Geng B Zhang L Yang and L Yu ldquoAxial stiffnessof a rotating trunnion jointrdquo Proceedings of the Institution ofMechanical Engineers - Part C Journal of Mechanical Engi-neering Science vol 229 no 16 pp 2846ndash2853 2014

[20] C Yue S Guo and M Li ldquoAnsys fluent based modeling andhydrodynamic analysis for a spherical under water robotrdquo inProceedings of the IEEE International Conference on Mecha-tronics amp Automation IEEE Takamatsu Japan August 2013

[21] ANSYS Inc ANSYS Modeling and Meshing Guide ANSYSInc Canonsburg PA USA 2004

[22] K Bao Q Zhang Y Liu and D Jin ldquoFatigue life of thewelding seam of a tracked vehicle body structure evaluated


using the structural stress methodrdquo Engineering FailureAnalysis vol 120 2021 (in Chinese)

[23] R Unarine D Desai S Glen and J Tamba ldquoSignificance ofresidual stresses in fatigue life prediction of micro gas turbinebladesrdquo Engineering Failure Analysis vol 120 2021

[24] W Jiang X Xie T Wang et al ldquoFatigue life prediction of316L stainless steel weld joint including the role of residualstress and its evolution experimental and modelingrdquo Inter-national Journal of Fatigue vol 143 2021

[25] H Feng Y Wang and X Jiang ldquoA maximum likelihoodmethod for estimating probabilistic strain amplitude-fatiguelife curvesrdquo Acta Mechanica Solida Sinica vol 31 no 1pp 80ndash93 2018

[26] S R Prasad and A S Sekhar ldquoLife estimation of shafts usingvibration based fatigue analysisrdquo Journal of Mechanical Scienceand Technology vol 32 no 9 pp 4071ndash4078 2018

[27] Q B Liu W K Shi and Z Y Chen ldquoRubber fatigue lifeprediction using a random forest method and nonlinearcumulative fatigue damage modelrdquo Journal of Applied Poly-mer Science vol 137 no 14 2020

[28] X H Deng ldquoAnalysis and prediction of gear fatigue liferdquo inProceedings of the 2018 4th International Conference on En-vironmental Science and Material Application Xirsquoan ChinaDecember 2018

[29] P M Wavish I R Mccoll and S B Leen ldquoDesign of a multi-axial test specimen for fretting fatigue in splined couplingsrdquoAdvances in Experimental Mechanics IV vol 8 pp 267ndash2722005

[30] X F Yang X U Wei and G P Guo ldquoExperimental study ofvibration fatigue of coupling diaphragm based on nonlinearvibrationrdquo Journal of Experimental Mechanics vol 31 no 6pp 763ndash768 2016


discussed the metal components of the coupling andsummarized the categories and suitable types of variousmetal components in the coupling [13] Lun et al analyzedthe strength and vibration characteristics of the diaphragmcoupling +ey showed the influence of vibration charac-teristics of the diaphragm coupling on the strength under theworking state [14] Mancuso introduced several couplingsfor gas turbines and mentioned that the multiple corrugateddiaphragm couplings are more suitable for gas turbines Stillthe design and calculation methods of corrugated filmcouplings were not introduced [15] Li studied the corru-gated diaphragmrsquos structural strength fatigue life and vi-bration characteristics by establishing the parametric finiteelement model of the corrugated diaphragm [16] Yang et alused the MSC Patran module in finite element analysissoftware to check the nonlinear strength of the diaphragmcoupling [17] Li and Li analyzed the dynamic characteristicsand vibration mechanism of parallel misalignment andangular misalignment middle rotors by building a multispanrotor-bearing system testbed [18] Feng et al established themodel of trunnion joint to study the influence of rotatingspeed and the ratio of inner radius to the outer radius on theaxial stiffness which provides a reference for designers tochoose the proper ratio of inner radius to the outer radius ofthe rotating circular plate [19]

According to the existing references many research in-stitutions on diaphragm couplings membrane disc couplingsand laminated couplings have been more in depth Manyscholars carried out extensive research from the theoreticalcalculation and finite element analysis to the stress stiffnessand fatigue life test +e diaphragm coupling membrane disccoupling laminated coupling stress analysis fatigue lifeanalysis design method processing technology test methodand installation application have an emotional experienceHowever research studies on theMCD coupling are relativelyfew +e research on the fatigue stress and fatigue life of theMCD coupling is even less For the design of the couplingfatigue life is a problem that must be considered and paidattention to +erefore it is necessary to study the fatiguestress and fatigue life of the MCDs to promote the applicationand development of the MCD coupling

In this paper the finite element model of the MCDs isestablished by Ansys finite element analysis software Amethod for calculating the fatigue safety factor of the MCDcoupling is presented based on the third strength theory andmodified Goodman curve +en the fatigue life test methodof the MCD coupling is designed +e axial fatigue life testand angular fatigue life test are carried out which verify thefeasibility of the calculation method of the fatigue safetyfactor of the MCDs to a certain extent

2 Geometric Parameters and WorkingParameters of the MCDs

+e shape of the MCDs is shown in Figure 1 And thespecific geometric parameters of the MCDs in this paper areshown in Table 1 +e influence of the bolt hole and splinewas ignored in the subsequent analysis [4] +e load pa-rameters of the MCDs in this paper are shown in Table 2

3 Fatigue Safety Factor of the MCDs

Disk couplings require very high processing technology andunique materials mainly because they adapt to moderateflexibility so fatigue strength is their main performanceindex To determine fatigue strength the method proposedby NB Rothfuss is to draw the average composite stress andalternating composite stress on the fatigue diagram of equallife and set 107 cycle lifelines as the fatigue limit parameterwith infinite life limit For fatigue diagrams three averagestress correction methods have been invented to avoid thetrouble of fatigue tests under different average stressesnamely Goodman method Gerber method and Soderbergmethod +e three modified fatigue diagrams are shown inFigure 2 Considering the dispersion of fatigue data theGerber parabola is not safe and the calculation is compli-cated +e Soderberg line can make safe predictions and inmany cases it seems overly safe Although the Goodman lineis not 100 safe it is much safer than the Gerber parabolaand it is easy to calculate so it is widely used in the fatiguedesign of mechanical parts In this paper the Goodmancurve is simplified into a straight line and the modifiedGoodman curve is used as the infinite lifeline to calculate thefatigue safety factor [20]

According to the Ansys help file we can see that Plane42is defined by four nodes having two degrees of freedom ateach node translations in the nodal x- and y-directionsPlane25 is defined by four nodes having threedegrees of freedom per node translations in the nodal x- y-and z-direction Plane25 is a generalization of the axisym-metric version of Plane42 Obviously for the MCDs onlythe x- and y-direction is not enough and Plane25 is moresuitable for the analysis of MCDs In addition the FE modelis characterized by using a simple two-dimensional model toanalyze three-dimensional forces +erefore this paperadopts the Plane25 element for FE modeling

In Ansys software APDL (Ansys Parametric DesignLanguage) was used for parametric modeling and Plane25element was used to mesh the membrane disc with quad-rilateral ring elements [1 2 4] +e finite element model ofthe MCDs was obtained [21ndash26] By applying an axial loadangular load centrifugal force load and torque load to thefinite element model of the MCDs the radial stress tan-gential stress and torsional shear stress of the MCDs underdifferent loads can be obtained [27ndash29]

+e stress generated by the axial load and the centrifugalload of theMCDs is in the same plane so the two stresses canbe combined linearly

σr σrx + σrl

σt σtx + σtl(1)

In the formula σr is the synthetic radial stress σt is thesynthetic tangential stress σrx is the radial stress due to axialload σtx is the tangential stress due to axial load σrl is theradial stress generated under centrifugal force loading σtl isthe tangential stress generated under centrifugal forceloading

+erefore the maximum and minimum principalstresses of the static stress of the MCDs are



2+


4+ τ2

1113971


2minus


4+ τ2

1113971

(2)




σdynamicmax σrf





Outer ringbolt hole


inside ringspline







σ σmax1113868111386811138681113868

1113868111386811138681113868 + σmin1113868111386811138681113868

1113868111386811138681113868 (4)


σ max σmax1113868111386811138681113868

1113868111386811138681113868 σmin1113868111386811138681113868

11138681113868111386811138681113966 1113967 (5)



n OM

OP (6)






σe

σy σu0

Mean stress (σm)

Stre

ss am

plitu

de (σ

m)

Soderberg

Gerber

Goodman


O

M

P

05σ-1

σdynamic

σdynamic



1

2

3

4

5

Fatig

ue sa

fety

fact

or

110 120 130 140 150 160 170100radius (mm)




t1

t2

x2

x11113888 1113889

k

s2

s1times

t21t22

1113888 1113889

k

(7)


and x2









546 times 1010

(9)






(a) (b)











0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)



(a)


0

5

10

15

20

25

30

35A

xial

forc

e (KN

)



(b)










6 Conclusions




0

5

10

15

20

25

30

35A

xial

forc

e (KN

)




(a)

0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)




(b)



Partnumber


mm)


mm)


()


mm)


()1 10198 9948 246 9522 6632 901 873 310 8323 761



Data Availability




Acknowledgments


References



































2+


4+ τ2

1113971


2minus


4+ τ2

1113971

(2)




σdynamicmax σrf





Outer ringbolt hole


inside ringspline







σ σmax1113868111386811138681113868

1113868111386811138681113868 + σmin1113868111386811138681113868

1113868111386811138681113868 (4)


σ max σmax1113868111386811138681113868

1113868111386811138681113868 σmin1113868111386811138681113868

11138681113868111386811138681113966 1113967 (5)



n OM

OP (6)






σe

σy σu0

Mean stress (σm)

Stre

ss am

plitu

de (σ

m)

Soderberg

Gerber

Goodman


O

M

P

05σ-1

σdynamic

σdynamic



1

2

3

4

5

Fatig

ue sa

fety

fact

or

110 120 130 140 150 160 170100radius (mm)




t1

t2

x2

x11113888 1113889

k

s2

s1times

t21t22

1113888 1113889

k

(7)


and x2









546 times 1010

(9)






(a) (b)











0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)



(a)


0

5

10

15

20

25

30

35A

xial

forc

e (KN

)



(b)










6 Conclusions




0

5

10

15

20

25

30

35A

xial

forc

e (KN

)




(a)

0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)




(b)



Partnumber


mm)


mm)


()


mm)


()1 10198 9948 246 9522 6632 901 873 310 8323 761



Data Availability




Acknowledgments


References


































σ σmax1113868111386811138681113868

1113868111386811138681113868 + σmin1113868111386811138681113868

1113868111386811138681113868 (4)


σ max σmax1113868111386811138681113868

1113868111386811138681113868 σmin1113868111386811138681113868

11138681113868111386811138681113966 1113967 (5)



n OM

OP (6)






σe

σy σu0

Mean stress (σm)

Stre

ss am

plitu

de (σ

m)

Soderberg

Gerber

Goodman


O

M

P

05σ-1

σdynamic

σdynamic



1

2

3

4

5

Fatig

ue sa

fety

fact

or

110 120 130 140 150 160 170100radius (mm)




t1

t2

x2

x11113888 1113889

k

s2

s1times

t21t22

1113888 1113889

k

(7)


and x2









546 times 1010

(9)






(a) (b)











0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)



(a)


0

5

10

15

20

25

30

35A

xial

forc

e (KN

)



(b)










6 Conclusions




0

5

10

15

20

25

30

35A

xial

forc

e (KN

)




(a)

0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)




(b)



Partnumber


mm)


mm)


()


mm)


()1 10198 9948 246 9522 6632 901 873 310 8323 761



Data Availability




Acknowledgments


References



































t1

t2

x2

x11113888 1113889

k

s2

s1times

t21t22

1113888 1113889

k

(7)


and x2









546 times 1010

(9)






(a) (b)











0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)



(a)


0

5

10

15

20

25

30

35A

xial

forc

e (KN

)



(b)










6 Conclusions




0

5

10

15

20

25

30

35A

xial

forc

e (KN

)




(a)

0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)




(b)



Partnumber


mm)


mm)


()


mm)


()1 10198 9948 246 9522 6632 901 873 310 8323 761



Data Availability




Acknowledgments


References










































0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)



(a)


0

5

10

15

20

25

30

35A

xial

forc

e (KN

)



(b)










6 Conclusions




0

5

10

15

20

25

30

35A

xial

forc

e (KN

)




(a)

0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)




(b)



Partnumber


mm)


mm)


()


mm)


()1 10198 9948 246 9522 6632 901 873 310 8323 761



Data Availability




Acknowledgments


References









































6 Conclusions




0

5

10

15

20

25

30

35A

xial

forc

e (KN

)




(a)

0

5

10

15

20

25

30

35

Axi

al fo

rce (

KN)




(b)



Partnumber


mm)


mm)


()


mm)


()1 10198 9948 246 9522 6632 901 873 310 8323 761



Data Availability




Acknowledgments


References



































Data Availability




Acknowledgments


References












































Documents

Fatigue Life Calculation Method and Experimental Study of