diagnosis of the misaligned faults of the vertical test

Research ArticleDiagnosis of the Misaligned Faults of the Vertical TestInstrument of High-Precision Industrial Robot Reducer

Zhen Yu 1 and Yuan Zhang 2

1State Key Laboratory of Precision Measuring Technology and Instrument Tianjin University Tianjin 300072 China2School of Management China University of Mining amp Technology Beijing China

Correspondence should be addressed to Yuan Zhang zhangyuan_yuan999163com

Received 20 April 2021 Revised 23 June 2021 Accepted 16 July 2021 Published 27 July 2021

Academic Editor Francisco Beltran-Carbajal

Copyright copy 2021 Zhen Yu and Yuan Zhang is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

High-precision reducer is the core component of industrial robots In order to achieve the comprehensive performance testing ofprecision reducers an instrument with a vertical layout and a cylindrical structure is designed As a rotating machine theinevitable coupling misalignment of the instrument can lead to vibration faults which lead to errors in the test So it is prettynecessary to diagnose and monitor the coupling misalignment while the instrument is working e causes of the couplingmisaligned fault of the instrument and the relationship between the misalignment fault and torque ripple are analyzed in thispaper A method of using the torque transducer in the measurement chain of the instrument to diagnose the coupling mis-alignment is proposed in this paper Many experiments have been done to test the capability of detecting the coupling mis-alignment using this method Experimental results show that the amplitude of torque ripple of the shaft is linearly related to thecoupling misalignment and is quadratically related to the rotation speed of the shaft when the misalignment exists in the shaftecombination of components at the rotation frequency (fr) and the additional components can be used to diagnose faults due tocoupling misalignment

1 Introduction

As a core component of the actuators in robots the robotreducer directly affects their performance factors such as thekinematic accuracy load capacity and fatigue life [1]erefore the comprehensive performance test of precisionreducers is required for developing high-precision industrialrobots and it has become the focus of attention of re-searchers [2] Some progress has been achieved concerningthe performance testing of precision reducers [3 4] ereducer detector often shows vibration faults caused by shaftmisalignment [5] An instrument designed by Dhaouadiand Ghorbel is designed to be a vertical structure to ensurean excellent shaft misalignment [6ndash8] However themethods to reduce the shaft misalignment caused by themanufacturing tolerances incur high costs and require mucheffort [9] Even if an accurate alignment is secured it cannotbe continued for a long duration due to many effects such as

thermal deformation and load imbalance [10 11] Manystudies [12ndash14] indicated that the misalignment wouldundoubtedly lead to deterioration of whole lubricationcharacteristics of bearing and even the failure of the system

Flexible couplings are usually used to overcome slightmisalignment of the rotating machinery [15ndash17] Howeveras is described above many factors can increase misalignedfaults erefore a feasible method for misaligned faultdiagnosis is required Recently many efforts have been madeto diagnose misaligned faults of rotating machinery throughthe combination of signal acquisition data processing andfault classification [18ndash22] Vibration analysis is the mostcommon technique for monitoring and diagnosing me-chanical systems [23ndash25] e signal acquisition for certainmechanical faults is accomplished by installing an acceler-ometer or proximity probe [26 27] However due to theconfusing spectral characteristics of vibration vibration-based misalignment detection led to less reliability [10]

HindawiShock and VibrationVolume 2021 Article ID 5516025 17 pageshttpsdoiorg10115520215516025

Furthermore especially for the vertical reducer detector witha torque sensor in the measuring chain the additionalsensors mounted on the mechanism can cause a series ofproblems such as signals acquired to vary according to thepositioning of sensors [28 29] onefold information ac-quisition [30] and additional cost

Some scholars and experts have made some progress onthe research of detection and monitoring of coupling mis-alignment based on torque signal [31] However their workonly shows the possibilities of torque measurements as avaluable technique for detecting misalignment and moretheoretical studies are needed to show the relationshipbetween the coupling misalignment and the ripple of torqueIn other works techniques for estimating torque from thesensing of the electrical variables of the motors have beenproposed and tested using the frequency spectrum of thecurrent electrical motor for diagnosis [16] However theestimated torque which includes the component due toeither misalignment of stator and rotor or rotor eccentricityof the motor cannot represent the torque transmitted by theshaft us it cannot show the effect of coupling mis-alignment of the shaft clearly Besides the experiment iscarried out on the horizontal instrument which cannotrepresent the effect of the method on the vertical instrument

In order to solve the problem of diagnosis couplingmisalignment in the vertical robot reducer detector amethod of using the torque transducer to diagnose thecoupling misaligned fault of performance testing instru-ments of precision reducers is proposed in this paper toovercome the shortcoming of diagnosis of misaligned faultusing an accelerometer or proximity probe e paper ismainly organized as follows Firstly the structure of avertical performance testing instrument of precision re-ducers including guide rail mechanism measurementmodule on the input side (MMIS) tested assemblies (TA)measurement module on the output side (MMOS) andworkbench is introduced in Section 2 combined withanalysis of the causes of the misaligned fault Next therelationships between the misaligned fault and the torqueripple are analyzed in Section 3 Moreover the experimentrsquoseffects using the Fast Fourier Transform and power spectrumanalysis of the torque signal to diagnose the misaligned faultare revealed in Section 4 Finally some conclusions are givenin the fifth section

2 Structure of the Vertical Instrument andAnalysis of the Causes of theMisaligned Fault

e vertical instrument is designed to measure the com-prehensive performance of the reducer as shown in Figure 1is instrument consists of four assistance modules whichare guide rail mechanism tested assemblies (TA) powersupply system and workbench and two measurementmodules namely the measurement module at the input sideof the reducer (MMIS) and that at the output side of thereducer (MMOS) Each measurement module is mainlycomposed of the five following subsystems motor torquetransducer test mode conversion part angle encoder andother connecting parts e red component in Figure 1

represents the TA e robot reducer was installed insidethe TA which is used to achieve a rapid installation ofdifferent reducer types e connections with the mea-surement systems adopt a uniform and standard structure

e instrument is of vertical type in which functionalcomponents are sequentially connected in series in the verticaldirection MMIS MMOS and TA are the core componentse causes of the misaligned fault are as follows

(1) Although the disk supports inside the hollow cyl-inder are designed to ensure that the shafts arecentered the manufacturing tolerances and assemblyerror of the disk supports can still lead to shaftmisalignment

(2) e flexible coupling is used in the connecting partsto overcome slight misalignment between the shaftsHowever many factors can increase misalignedfaults and flexible coupling can never solve theproblem caused by increasing misalignment

(3) Moreover the test mode conversion componentsallow the input or output shaft of the tested reducerto be under unconstrained driven or locked con-ditions thereby satisfying the functional require-ments of different dynamic and static performancetests However there is no guarantee that the con-version components move along the axis echanging of the test mode can increase the mis-aligned faults because of the motion of the con-version components

Since all the three factors above can lead to the mis-aligned fault and the misalignment mainly between theshafts connected by the flexible coupling and spline cou-pling a feasible method is required to realize the misalignedfault diagnosis In this paper the torque transducer is used todiagnose the misaligned fault of performance testing in-struments of precision reducers through testing the torqueripple e relationships between the misaligned fault andthe torque ripple are analyzed in the next section

3 Relationships between the Misaligned Faultand the Torque Ripple

ere are three common types of shaft misalignment (asshown in Figure 2) parallel misalignment (a) angle mis-alignment (b) and comprehensive misalignment (c) Due tothe inevitable manufacturing tolerances and assembly errorthe three kinds of misalignment faults shown in Figure 2 canexist in the flexible coupling and spline coupling at each ofthe two shafts of MMIS and MMOS Especially for therectangular spline sleeve which connects two spline shafts atthe same time the misalignment fault can exist Due to theexcessive spline clearance both spline shafts can be con-sidered unconstrained at one end and the other end issupported in the state of high-precision bearing In thetransmission process it is likely that the misalignment be-tween the two spline shafts and the spline sleeve changeswith the rotation of the shaft at is the three states inFigure 2 may appear alternately

2 Shock and Vibration

e rotating shaft of MMIS which is the same as that ofMMOS is shown in Figure 3 When the parallel or angularoffsets exist in the shaft the mass center of the shaft does notcoincide with the rotation center e shaft will be affectedby the electromagnetic torque of the motor and the eccentrictorque Under the effect of the cogging effect and mis-alignments the electromagnetic torque of the motor and theeccentric torque change with the rotation of the shaft edriving torque which is defined as the sum of the elec-tromagnetic torque of the motor and the eccentric torquecan be expressed as the sum of averaged torque and torqueripple [32]

T T + 1113944n

Tn cos nθout +emptyn( 1113857 (1)

where T and Tn are the averaged torque and amplitude forthe nth driving torque respectively n is the harmonic ordersof the output shaft and θout and emptyn are the position of the

output shaft and phase offset for the nth driving torquerespectively

e ripple of the driving torque can cause the frictiontorque of the shaft to change with the shaft rotation at isbecause the friction torque is caused by the friction gen-erated in the bearing because of rotation e ripple of thedriving torque leads to the ripple of the acceleration of theshaft e ripple of the acceleration of the shaft can cause theripple of the rotation speed and finally leads to the ripple ofthe friction torque of bearings [33] en the friction torquecan be expressed as follows

Tf μNr + μrmΔr 1113944n

ω2n cos nθout +emptyn( 1113857 (2)

where Tf and μ are the friction torque and friction coeffi-cient respectively N is the supporting force of the bearing ris the radius of the shaftm is the eccentric mass of the shaftΔr is the misalignment of the shaft n is the harmonic orders

Measurement module atthe input side of the reducer

Tested assemblies

Work bench

Torquetransducer

Measurement module at theoutput side of the reducer

Robotreducer

Guide railmechanism

Power supply system

Servomotor

Connecting partHollow cylinder

Test modeconversion part


Connecting part

Torque motor

Torque transducerAngle encoderSpline sleeve

Hollow cylinderAngle encoderSpline sleeve

Figure 1 e vertical precision robot reducer detector

(a) (b) (c)

Figure 2 ree states of shaft misalignment (a) e parallel misalignment (b) the angle misalignment (c) the comprehensivemisalignment

Shock and Vibration 3

of the output shaft ωn is the amplitude for the nth rotationspeed of the shaft and θout and emptyn are the position of theoutput shaft and phase offset for the nth driving torquerespectively

e driving torque and the friction torque which arechanging with the shaft rotation eventually lead to thetorque ripple of the shaft As is analyzed above the drivingtorque and the friction torque are affected by the mis-alignment e torque ripple of the shaft is also affected bythe misalignment According to the analysis in the secondsection the misalignment is mainly between the shaftsconnected by the flexible coupling and spline coupling eimpact of misalignment of various shaft parts on torqueripple especially about the misalignment between the twospline shafts and the spline sleeve is analyzed in detail in thefollowing

31 Torque Ripple Caused by the Misalignment of FlexibleCoupling When the parallel misalignment Δx exists be-tween the motor shaft and the transmission shaft the flexiblecoupling connecting the two shafts will be deformedAccording to Hookersquos law the deformed flexible couplingwill produce reaction forces F1prime and F2prime on the shafts at bothends of the flexible coupling Besides the mass center of theshaft does not coincide with the rotation center e shaftwill be affected by the eccentric force At this time thetransmission shaft is not only affected by the driving torqueT transmitted by the flexible coupling but also affected by thereaction force and the eccentric force caused by the parallelmisalignmente reaction force and the eccentric force will

eventually be balanced by the supporting forces F1 and F2provided by the bearing Finally the friction torque Tf1 willbe brought due to the reaction force and the eccentric forceacting on the bearing erefore the parallel misalignmentΔx exists between the motor shaft and drive shaft producingfriction torque which affects the torque transmitted throughthe shaft Because the driving torque is changing with theshaft rotation it will cause the rotation speed to change withthe shaft rotatione ripple of the rotation speed will lead tothe friction torque change with the shaft rotation Figure 4shows the force and friction torque caused by the parallelmisalignment Δx between the motor shaft and the trans-mission shaft

e torque transmitted by the transmission shaft T1 isthe drive torque minus the friction torqueWhen the parallelmisalignment Δx exists between the motor shaft and thetransmission shaft the torque transmitted by shaft Tprime can beexpressed as follows [34]

T1 T + 1113944n

Tn cos nθout +emptyn( 1113857 minus μ1k1r1x

minus μ1m1r1x 1113944n

ω2n cos nθout +emptyn( 1113857

(3)

where T and Tn are the averaged torque and amplitude forthe nth driving torque respectively n is the harmonic ordersof the output shaft θout andemptyn are the position of the outputshaft and phase offset for the nth driving torque respec-tively μ1 is the friction coefficient of the transmission shaftk1 is the coefficient of elasticity of the transmission shaft m1is the mass of the transmission shaft r1 is the radius of thetransmission shaftΔx is the parallel misalignment and ωn isthe amplitude for the nth rotation speed of the shaft

e misalignment errors comprise radial and angulardisplacements When the angle misalignment Δθ existsbetween the motor shaft and the transmission shaft theflexible coupling connecting the two shafts will be deformede deformed coupling will produce a reaction force on theshafts at both ends of the flexible coupling According toHookersquos law the reaction force on the transmission shaftcaused by the angle misalignment can be expressed asfollows

F k1L1 sin(Δθ) (4)

where L1 is the length of the transmission shaft Δθ is theangular misalignment and k1 is the coefficient of elasticity ofthe transmission shaft

Besides when the angular offsets exist in the shaft themass center of the shaft does not coincide with the rotationcenter e shaft will be affected by the eccentric force eeccentric force on the shaft caused by the angle misalign-ment can be expressed as follows

Fe m1L1 sin(Δθ) 1113944n


where Fe is the eccentric force m1 is the mass of thetransmission shaft L1 is the length of the transmission shaftΔθ is the angular misalignment n is the harmonic orders ofthe output shaft ωn is the amplitude for the nth rotation

Servo motor

Coupling

Bearing

Bearing

Bearing

Spline coupling

Torque transducer

Angle encoder

Figure 3 e rotating shaft of MMIS


speed of the shaft and θout and emptyn are the position of theoutput shaft and phase offset for the nth driving torquerespectively

At this time the transmission shaft is not only affected bythe driving torque T transmitted by the flexible coupling butalso affected by the reaction force and the eccentric forcecaused by the angular misalignment e reaction force andthe eccentric force will eventually be balanced by the sup-porting force provided by the bearing Finally the frictiontorque Tf1prime will be brought due to the reaction force and theeccentric force acting on the bearing erefore the anglemisalignment Δθ exists between the motor shaft and driveshaft producing friction torque which affects the torquetransmitted through the shaft Because the driving torque ischanging with the shaft rotation it will cause the rotationspeed to change with the shaft rotation e ripple of therotation speed will lead to the friction torque change with theshaft rotation Figure 5 shows the force and friction torquecaused by the angle misalignment Δθ between the motorshaft and the transmission shaft

When an angular misalignment error exists the torquetransmitted by shaft T1prime is the drive torque minus the frictiontorque When the angle misalignment Δθ exists between themotor shaft and the transmission shaft the torque trans-mitted by shaft T1prime can be expressed as follows [35]

T1prime T + 1113944n

Tn cos nθout +emptyn( 1113857 minus μ1k1L1 sin(Δθ)

minus μ1m1r1L1 tan(Δθ) 1113944n


(6)

where T and Tn are the averaged torque and amplitude forthe nth driving torque respectively L1 is the length of thetransmitted shaft Δθ is the angular misalignment k1 is thecoefficient of elasticity of the transmission shaft μ1 is thefriction coefficient of the transmission shaft m1 is the massof the transmission shaft r1 is the radius of the transmissionshaft n is the harmonic orders of the output shaft ωn is theamplitude for the nth rotation speed of the shaft and θout

and emptyn are the position of the output shaft and phase offsetfor the nth driving torque respectively

32 Torque Ripple Caused by the Misalignment of SplineCoupling e spline coupling is a complex structure espline structure in our instrument is equivalent to two sets ofspline coupling One spline sleeve connects two spline shafts atthe same time In general the torque is transmitted to the splinesleeve through the transmission shaft and then is transferred tothe torque transducer shaft by the spline sleeve Due to theexcessive spline clearance both spline shafts can be consideredunconstrained at one end and supported by a high-precisionbearing at the other end In the transmission process it is likelythat the misalignment between the two spline shafts and thespline sleeve changes with the rotation of the shaft Besides aslong as there is a misalignment between the spline shaft andspline sleeve only 25sim50 of the splinersquos teeth participate inmeshing simultaneously in actual work Besides themeshing ofthe spline shaft and spline sleeve is uneven some of the splinersquosteethmeshing is tight and some of the splinersquos teethmeshing isloose As a result the engagement force of each splinersquos teeth isdifferent and a resultant force is generated at the teeth of thespline shaft offsete resultant force produces sliding betweenthe spline sleeve and the spline shaft So the friction torquewhich affects the torque transmitted through the shaft isgenerated e resultant force is balanced by the friction forceand support force provided by the bearing Due to misalign-ment the spline sleeve pointmeshed with the spline shaft at thespline shaft offset is changed with the shaft rotation

When the parallel misalignment Δxprime exists between thetransmission shaft and the torque transducer shaft themeshing of the spline shaft and spline sleeve is uneven efriction torque Tf2 caused by the uneven meshing of thespline shaft and spline sleeve and the sliding between thespline sleeve and the spline shaft is generated At this timethe torque transducer shaft is not only affected by the drivingtorque T transmitted by the transmission shaft but also

Coupling

BearingTprimef1

TprimeR1

∆θ

FFprime

Servo motor

Figure 5 e force and the friction torque caused by the angularmisalignment of the flexible coupling

Coupling

Bearing

Tf1

TR1

F2

F1

Fprime

Fprime

Servo motor

∆X

Figure 4 e force and the friction torque caused by the parallelmisalignment of the flexible coupling


affected by the friction torque Tf2 caused by the parallelmisalignment erefore the parallel misalignment Δxprime thatexists between the two spline shafts produces friction torquewhich affects the torque transmitted through the shaftBecause the driving torque is changing with the shaft ro-tation it will cause the rotation speed to change with theshaft rotatione ripple of the rotation speed will lead to thefriction torque change with the shaft rotation Figure 6shows the force and friction torque Tf2 caused by the par-allel misalignment Δxprime between the two spline shafts

e torque transmitted by the torque transducer shaft T2 isthe drive torque minus the friction torque When the parallelmisalignment Δx1 exists between the transmission shaft andthe torque transducer shaft the torque transmitted by thetorque transducer shaft T2 can be expressed as follows [34]

T2 T + 1113944n

Tn cos nθout +emptyn( 1113857 minus μ2k2r2Δxprime

minus μ2m2r2 1113944n

ω2n L0 minusxprime cos nθout +emptyn( 1113857( 1113857cos nθout +emptyn( 1113857

(7)

where T and Tn are the averaged torque and amplitude forthe nth driving torque respectively n is the harmonic ordersof the output shaft θout andemptyn are the position of the outputshaft and phase offset for the nth driving torque respec-tively μ2 is the friction coefficient of the torque transducershaft m2 is the mass of the torque transducer shaft r2 is theradius of the torque transducer shaft k2 is the coefficient ofelasticity of the torque transducer shaft Δxprime is the parallelmisalignment L0 is the equivalent meshing distance of eachsplinersquos teeth in good alignment condition and ωn is theamplitude for the nth rotation speed of the shaft

When an angular misalignment Δθprime exists between thetransmission shaft and the torque transducer shaft themeshing of the spline shaft and spline sleeve is uneven efriction torque Tf2prime caused by the uneven meshing of the splineshaft and spline sleeve and the sliding between the spline sleeveand the spline shaft is generated erefore the angularmisalignment Δθprime that exists between the two spline shaftsproduces friction torque which affects the torque transmittedthrough the shaft Because the driving torque is changed withthe shaft rotation it will cause the rotation speed to change withthe shaft rotation e ripple of the rotation speed will lead tothe friction torque change with the shaft rotation Figure 7shows the force and friction torque Tf2prime caused by the angularmisalignment Δθprime between the two spline shafts

e torque transmitted by the torque transducer shaft T2prime isthe drive torque minus the friction torque When the angularmisalignment Δθprime exists between the transmission shaft andthe torque transducer shaft the torque transmitted by thetorque transducer shaft T2prime can be expressed as follows [35]


Tn cos nθout +emptyn( 1113857 minus μ2k2L2 sin θprime( 1113857

minus μ2k2L2 sin θprime( 1113857


ω2n L0 minus L2 tan θprime cos nθout +emptyn( 1113857( 1113857cos nθout +emptyn( 1113857

(8)

Transmission sha

Spline sleeve

BearingBearing

Torque transducer shaTorque transducer sha

FT1

FT2

TR2

Tf2 ∆Xprime

Figure 6 e force and the friction torque caused by the parallelmisalignment of spline coupling

Transmission shaft

Spline sleeve

BearingBearing

Torque transducer shaftTorque transducer shaft

FprimeT1

FprimeT2

TprimeR2

Tprimef2 ∆θprime

Figure 7 e force and the friction torque caused by the angularmisalignment of spline coupling


where T and Tn are the averaged torque and amplitude forthe nth driving torque respectively n is the harmonicorders of the output shaft θout and emptyn are the position ofthe output shaft and phase offset for the nth drivingtorque respectively μ2 is the friction coefficient of thetorque transducer shaft m2 is the eccentric mass of thetorque transducer shaft r2 is the radius of the torquetransducer shaft k2 is the coefficient of elasticity of thetorque transducer shaft Δθprime is the angular misalignmentL0 is the equivalent meshing distance of each splinersquos teethin good alignment condition ωn is the amplitude for thenth rotation speed of the shaft and L2 is the length of thetorque transducer shaft

4 Experimental Results of Diagnosing theMisalignment Using the Fast FourierTransform and Power Spectrum Analysis ofthe Torque Signal

According to the analysis proposed in the above section itcan be seen that the amplitude of torque ripple of theshaft is linearly related to the misalignment of the flexi-ble coupling and the spline coupling and is quadraticallyrelated to the rotation speed of the shaft when the parallelor angular misalignment exists in the shaft Moreover thefrequency of the torque ripple is the same as the frequencyof the rotation speed So the amplitude and frequency oftorque ripple can be used to diagnose the misalignment ofthe MMIS and MMOS Experiments have been done totest the capability of detecting the misalignment of theflexible coupling and the spline coupling of the MMIS andMMOS using the measurement and analysis of torqueripple e Fast Fourier Transform and power spectrumanalysis of torque are used to obtain the amplitude andfrequency of torque ripple e singular value decomposi-tion (SVD) filtering stage is included in the data processingto obtain the amplitude and frequency of torque rippleSVD filtering algorithm is to decompose the signal into aseries of pure signal subspace and noise subspace whichare corresponding to the singular matrix In the singularmatrix the larger singular value corresponds to the puresignal and the smaller singular value corresponds to thenoise signal erefore after determining the reasonableeffective rank of the singular matrix the singular valueswith the same number of effective ranks are taken fromthe large to the small and the remaining values are set tozero en the matrix estimation of pure signal is obtainedby inverse operation of SVD and the final signal aftersingular value denoising is obtained by inverse recon-struction of phase space In signal processing SVD fil-tering stage which has excellent invariance and stabilityis mainly used for the extraction of periodic componentsand the denoising of signals By using the maximum salientpoint of singular value difference spectrum to accuratelyjudge the order of noise reduction the interference ofnoise components in the signal can be effectively elimi-nated and the main components of fault information canbe retained

e structure of the MMIS and MMOS is introduced inthe second section as shown in Figure 1 In the experimentthe motor drives the transmission shaft coupled to thetorque transducer shaft at means the test mode con-version components make the shaft of the MMIS or MMOSunder driven conditions e angular misalignment con-ditions and radial misalignment conditions are controlled byadjusting the radial and angular displacements of the con-version components and the transmission shaft when in-stalling the shaft of MMIS or MMOS as shown in Figure 8e torque transducers in MMIS andMMOS are used to testthe torque ripple e diagrammatic sketch of the experi-mental device (MMIS) is shown in Figure 9

As an example the details of adjusting the radial mis-alignment in the experiment are shown as follows (1) Putthe hollow cylinder in the center of the turntable (2) Put thedisk support on the hollow cylinder (3) Adjust the radialmisalignment between the torque transducer shaft and thedisk support using the dial gauge (4) Adjust the radialmisalignment between the disk supports and the hollowcylinder using the dial gauge and install the disk support onthe hollow cylinder (5) Adjust the radial misalignmentbetween the transmission shaft and the hollow cylinderusing the dial gauge and install the transmission shaft on thedisk support

e torque transducers in the instrument are producedby the HBM companyemodel of the torque transducer isT40B e measuring range of the torque transducer in theMMIS is 0ndash50Nm and its measuring accuracy is 01 in fullscale e measuring range of the torque transducer inMMOS is 0ndash2000Nm and its measuring accuracy is 005in full scale As the best product of torque transducer of theworld the maximum allowable radial misalignment ofT40B can reach about 2mm But when the radial mis-alignments reach more than 100 μm the radial misalign-ments will lead to torque oscillation e servo motor andtorque motor used in the MMIS and MMOS are made bythe Modrol Electric Co Ltd e model of the servo motoris SMS15-42P2C and the model of the torque motor isMDD310 e loading accuracies of the servo motor andtorque motor are 01 in full scale e load ranges of theservo motor and torque motor are 50 Nm and 2000Nmrespectively A PXIe acquisition system produced by the NIcompany is used to collect the torque transducerrsquos signalse accuracy of torque signal acquisition of the PXIeacquisition system is 01 in full scale e torque signaldata is collected using LabVIEWtrade software at a rate of 25 ksamples per second

First of all experiments of diagnosing the misalignmentfault of the shaft are carried out for alignment a 05mmradial misalignment and a 10mm radial misalignmente experiments are carried out under no load Using theMMIS as an example the rotation speed is increased by200 rpm every time from 0 rpm to 800 rpm e torquetransducer tests the transmitted torque of the MMIS underdifferent rotation speeds e ripple of the transmittedtorque under different rotation speeds is analyzed using theFast Fourier Transform and power spectrum analysis afterSVD filtering


Flexible coupling

Servo motor

Transmission sha

Hollow cylinder

Disk supports

Disk supports

Disk supports

Torque transducersha

Torque transducer

Bearing

Bearing

Bearing

Spline coupling

Spline sleeve

Test modeconversioncomponents

Figure 9 e diagrammatic sketch of the experimental device (MMIS)

(a) (b) (c)

Figure 8 Adjusting the angular misalignment conditions and radial misalignment conditions of the shaft (a) Adjusting the radial andangular misalignment of the torque transducer shaft (b) adjusting the misalignment between the disk supports and the hollow cylinder(c) adjusting the radial and angular misalignment of the transmission shaft


10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20

Frequency (Hz)25 30 35

2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (200Hz) 4fr (400Hz) 8fr (800Hz)

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)

Figure 10 e frequency spectrum of the transmitted torque for shaft alignment under different rotation speeds (a) 200 rpm (b) 400 rpm(c) 600 rpm (d) 800 rpm

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

Figure 11 Continued


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz) 4fr (400Hz) 8fr (800Hz)

0 15 30 45 60Frequency (Hz)

75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)

Figure 11 e power spectrum of the transmitted torque for shaft alignment under different rotation speeds (a) 200 rpm (b) 400 rpm(c) 600 rpm (d) 800 rpm

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)

Figure 12 e frequency spectrum of the transmitted torque for 05mm radial misalignment under different rotation speeds (a) 200 rpm(b) 400 rpm (c) 600 rpm (d) 800 rpm


Figure 10 shows the frequency spectrum of the trans-mitted torque under different rotation speeds for the case ofshaft alignment For such conditions the rotational fre-quency when the rotation speed is increased by 200 rpmevery time from 0 rpm to 800 rpm is increased by 333Hzevery time from 0Hz to 1333Hz e figure showing theabsence of components at the rotation frequency (fr)which is shown in the third section is able to detect faultmisalignment e additional components of twice therotational frequency (2fr) fourth of the rotational fre-quency (4fr) and even eighth of the rotational frequency(8fr) are also observed in Figure 10 is component maybe due to the cogging torque and electromagnetic torqueripple Figure 11 shows the power spectrum of the trans-mitted torque under different rotation speeds for the case ofshaft alignment e powers of the additional componentsof twice the rotational frequency (2fr) fourth of the ro-tational frequency (4fr) and even eighth of the rotationalfrequency (8fr) are all the same is phenomenon meansthat all the torque ripple causes such as the cogging torque

and electromagnetic torque ripple have the same effect onthe torque ripple

Figure 12 shows the frequency spectrum for 05mmradial misalignment Figure 13 shows the frequency spec-trum for 10mm radial misalignment In this particular casea component at the rotation frequency (fr) of greater am-plitude can be observed ese components clearly showmisalignment between the transmission shaft and the torquetransducer shaft Furthermore it can be observed that thecomponents of twice the rotational frequency (2fr) fourth ofthe rotational frequency (4fr) and eighth of the rotationalfrequency (8fr) are higher than those for aligned shaftsFigure 14 shows the power spectrum of the transmittedtorque under different rotation speeds for 05mm radialmisalignment Figure 15 shows the power spectrum of thetransmitted torque under different rotation speeds for10mm radial misalignment e powers of the additionalcomponents of the rotational frequency (fr) twice the ro-tational frequency (2fr) fourth of the rotational frequency(4fr) and even eighth of the rotational frequency (8fr) are

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

150 30 45 60Frequency (Hz)

75 90 105

(d)



decreased with the increase of frequency is phenomenoncan be explained by the friction torque due to misalignmenton shaft extension Such torque increases the eccentricitywhich appears in the components at fr After comparison wecan come to the conclusion that the component fr and thepower of the additional components both allow diagnosingfaults due to misalignment

From Figures 10ndash15 it can be seen that the robustnessand effects of the SVD filtering are so good that thoseunuseful signals of high frequency and low amplitudes havebeen filtered at means only the main components oftorque ripple which can be used to diagnose the mis-alignment fault left after SVD filtering

In order to verify the diagnostic accuracy and robustnessof the proposed method and the SVD filtering methodcomparison experiments of diagnosing the misalignmentfault of the shaft are carried out for alignment a 0125mmradial misalignment a 025mm radial misalignment a0375mm radial misalignment a 05mm radial misalign-ment a 0625mm radial misalignment a 075mm radial

misalignment a 0875mm radial misalignment and a10mm radial misalignment Figure 16 shows the amplitudeof the components of torque ripple at fr 2fr and 4fr foralignment 0125mm radial misalignment 025mm radialmisalignment 0375mm radial misalignment 05mm radialmisalignment 0625mm radial misalignment 075mm ra-dial misalignment 0875mm radial misalignment and10mm radial misalignment under the same rotation speedsFigure 17 shows the amplitude of torque ripple at fr underdifferent rotation speeds for alignment 05mm radialmisalignment and 10mm radial misalignment respec-tively From the two figures we can see that the amplitude oftorque ripple of the shaft is linearly related to the radiusmisalignment and is quadratically related to the rotationspeed of the shaft

en comparison experiments of diagnosing themisalignment fault of the shaft are carried out for align-ment a 0125deg angular misalignment a 025deg angularmisalignment a 0375deg angular misalignment a 05deg angularmisalignment a 0625deg angular misalignment a 075deg

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)

Figure 14 e power spectrum of the transmitted torque for 05mm radial misalignment under different rotation speeds (a) 200 rpm(b) 400 rpm (c) 600 rpm (d) 800 rpm


020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 10 20 30 40


2fr (133Hz)

3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)

(b)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)

(c)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)


105

09

075

06

045

03

015

Radial misalignment (mm)

00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)

Radial misalignment (mm)0 0125 025 0375 05 0625 075 0875 1

105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)

Figure 16 Continued


angular misalignment a 0875deg angular misalignmentand 1deg angular misalignment e experiments are car-ried out under no load Using the MMIS as an examplethe rotation speed is increased by 200 rpm every timefrom 0 rpm to 800 rpm e torque transducer tests thetransmitted torque of the MMIS under different rotationspeeds Figure 18 shows the amplitudes of the componentof torque ripple at fr 2fr and 4fr for alignment 0125degangular misalignment 025deg angular misalignment0375deg angular misalignment 05deg angular misalignment

0625deg angular misalignment 075deg angular misalign-ment 0875deg angular misalignment and 1deg angularmisalignment under the same rotation speeds Figure 19shows the amplitude of torque ripple at fr under differentrotation speeds for alignment 05deg angular misalignmentand 1deg angular misalignment respectively From the twofigures we can see that the amplitude of torque ripple ofthe shaft is linearly related to the angular misalignmentand is quadratically related to the rotation speed of theshaft


175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)

Figure 16 e amplitude of the torque ripple component at fr 2fr and 4fr for alignment 0125mm radial misalignment 025mm radialmisalignment 0375mm radial misalignment 05mm radial misalignment 0625mm radial misalignment 075mm radial misalignment0875mm radial misalignment and 10mm radial misalignment under different rotation speeds Red fr blue 2fr green 4fr (a) 200 rpm(b) 400 rpm (c) 600 rpm (d) 800 rpm

21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400

Rotation speed (rpm)

10mm radial misalignment


600 800

Alignment

Figure 17 e amplitude of torque ripple at fr under different rotation speeds for alignment 05mm radial misalignment and 10mmradial misalignment Red alignment blue 05mm radial misalignment and green 10mm radial misalignment


105

09

075

06

045

03

015

Angular misalignment (deg)

00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)

Angular misalignment (deg)0 0125 025 0375 05 0625 075 0875 1

105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)

Figure 18 e amplitude of the torque ripple components at fr 2fr and 4fr for alignment 0125deg angular misalignment 025deg angularmisalignment 0375deg angular misalignment 05deg angular misalignment 0625deg angular misalignment 075deg angular misalignment 0875degangular misalignment and 1deg angular misalignment under different rotation speeds Red fr blue 2fr green 4fr (a) 200 rpm (b) 400 rpm(c) 600 rpm (d) 800 rpm

21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400


1deg angular misalignment


600 800

Alignment

Figure 19 e amplitude of torque ripple at fr under different rotation speeds for alignment 05deg angular misalignment and 1deg angularmisalignment Red alignment blue 05deg angular misalignment green 1deg angular misalignment


5 Conclusions

A method of using the torque transducer to diagnose thecoupling misaligned fault of performance testing instru-ments of precision reducers is proposed in this paper ecauses of the coupling misaligned fault of the instrument andthe relationship between the misalignment fault and torqueripple are analyzed Experiments have been done to test thecapability of detecting the coupling misalignment of theinstrument using the measurement and analysis of torqueripple From the analysis and experimental results presentedin the third and fourth sections we can see that the am-plitude of torque ripple of the shaft is linearly related to themisalignment of the flexible coupling and the spline cou-pling and is quadratically related to the rotation speed of theshaft when the parallel or angular misalignment exists in theshaft e amplitude and frequency of torque ripple can beused to diagnose themisalignment of theMMIS andMMOSe frequency components of the torque ripple test by thetorque transducer in the MMIS and MMOS allow identi-fying radial and angular misalignment when the increasingmisalignment exists in the shaft connected by the flexiblecoupling and spline coupling

Such a characteristic can be demonstrated with smalllevels of angular and radial misalignment faults ecomponents of the torque at fr and 2fr frequencies aresuitable for detecting misalignment erefore it is con-sidered a more suitable fault indicator than that of thecomponent at fr frequency Moreover the powers of theadditional components of the rotational frequency (fr)twice the rotational frequency (2fr) fourth of the rotationalfrequency (4fr) and even eighth of the rotational frequency(8fr) are decreased with the increase of frequency isphenomenon can be explained by the friction torque due tomisalignment on shaft extension Such torque increases theeccentricity which appears in the components at fr Aftercomparison we can come to the conclusion that thecombination of components at the rotation frequency (fr)and the additional components can be used to diagnosefaults due to coupling misalignment

e method proposed in this paper is mainly about thediagnosis of static misalignment caused by installation andthe manufacturing tolerances ose misaligned faults canlead to continuing periodical torque ripple e frequencyand the amplitude of torque ripple can be used to diagnosethe misaligned fault But when talking about the torqueoscillation caused by the joint flexibility or backlash at jointson one side it is not periodical signal and on the other sidethe amplitude of the torque oscillation will change with therotation of shaft So the torque oscillation caused by thejoint flexibility or backlash at joints will not influence theeffect of the method proposed in this paper e diagnosis ofthe joint flexibility or backlash at joints will be our researchwork in the future

Data Availability

e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was financially supported by the National KeyResearch and Development Program of China (NKRDPC)(no 2017YFF0108100)

References

[1] A-D Pham and H-J Ahn ldquoHigh precision reducers forindustrial robots driving 4th industrial revolution state ofarts analysis design performance evaluation and perspec-tiverdquo International Journal of Precision Engineering andManufacturing-Green Technology vol 5 no 4 pp 519ndash5332018

[2] X Y Chu H Xu X Wu J Tao and G Shao ldquoe method ofselective assembly for the RV reducer based on genetic al-gorithmrdquo ProceedingsmdashInstitution of Mechanical EngineersPart C Journal of Mechanical Engineering Science vol 232no 6 pp 921ndash929 2019

[3] I Makoto and N Hiroyuki ldquo3rd international conference onadvances in control and optimization of dynamical systemsrdquoIFAC Proceedings Volumes vol 47 no 3 pp 6831ndash6836 2014

[4] P Peng and J Wang ldquoNOSCNN a robust method for faultdiagnosis of RV reducerrdquo Measurement vol 138 pp 652ndash658 2019

[5] J Piotrowski Shaft Alignment Handbook CRC Press BocaRaton FL USA 3rd edition 2006

[6] R Dhaouadi F H Ghorbel and P S Gandhi ldquoA new dy-namic model of hysteresis in harmonic drivesrdquo IEEETransactions on Industrial Electronics vol 50 no 6pp 1165ndash1171 2003

[7] F H Ghorbel P S Gandhi and F Alpeter ldquoOn the kinematicerror in harmonic drive gearsrdquo Journal of Mechanical Designvol 123 no 1 pp 90ndash97 2001

[8] P S Gandhi and F H Ghorbel ldquoClosed-loop compensationof kinematic error in harmonic drives for precision controlapplicationsrdquo IEEE Transactions on Control Systems Tech-nology vol 10 no 6 pp 759ndash768 2002

[9] S H Park J C Park S W Hwang J H Kim H J Park andM S Lim ldquoSuppression of torque ripple caused by mis-alignment of the gearbox by using harmonic current injectionmethodrdquo IEEEASME Transactions on Mechatronicsvol 99pp 1ndash10 2020

[10] J Piotrowski Shaft Alignment Handbook Dekker New YorkNY USA 1986

[11] M Xu and R D Marangoni ldquoVibration analysis of a motor-flexible coupling-rotor system subject to misalignment andunbalance part I theoretical model and analysisrdquo Journal ofSound and Vibration vol 176 no 5 pp 663ndash679 1994

[12] G Xiang Y Han J Wang K Xiao and J Li ldquoA transienthydrodynamic lubrication comparative analysis for misalignedmicro-grooved bearing considering axial reciprocating move-ment of shaftrdquo Tribology International vol 132 pp 11ndash23 2019

[13] W Litwin ldquoExperimental research on water lubricated threelayer sliding bearing with lubrication grooves in the upperpart of the bush and its comparison with a rubber bearingrdquoTribology International vol 82 pp 153ndash161 2015

[14] S B Shenoy and R Pai ldquoeoretical investigations on theperformance of an externally adjustable fluid-film bearing


including misalignment and turbulence effectsrdquo TribologyInternational vol 42 no 7 pp 1088ndash1100 2009

[15] J M Bossio G Bossio and C DeAngelo ldquoAngular mis-alignment in induction motors with flexible couplingrdquo inProceedings of IEEE Industrial Electronics Conferencepp 1033ndash1038 IEEE Porto Portugal November 2009

[16] 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

[17] K Lu Y Jin P Huang F Zhang H Zhang and C Fu ldquoeapplications of pod method in dual rotor-bearing systemswith coupling misalignmentrdquo Mechanical Systems and SignalProcessing vol 150 Article ID 107236 2021

[18] J M Bossio C H De Angelo and G R Bossio ldquoSelf-or-ganizing map approach for classification of mechanical androtor faults on induction motorsrdquo Neural Computing ampApplications vol 23 no 1 pp 41ndash51 2013

[19] Y Lei Z He Y Zi and X Chen ldquoNew clustering algorithm-basedfault diagnosis using compensation distance evaluation tech-niquerdquo Mechanical Systems and Signal Processing vol 22 no 2pp 419ndash435 2008

[20] W L Du J F Tao Y M Li and C L Liu ldquoWavelet leadersmultifractal features based fault diagnosis of rotating mecha-nismrdquoMechanical Systems and Signal Processing vol 43 no 1-2 pp 57ndash75 2014

[21] O Janssens V Slavkovikj B Vervisch K StockmanM Loccufier and S Verstockt ldquoConvolutional neural net-work based fault detection for rotating machineryrdquo Journal ofSound and Vibration vol 377 pp 331ndash345 2016

[22] W Zhou B Lu T G Habetler and R G Harley ldquoIncipientbearing fault detection via motor stator current noise can-cellation using wiener filterrdquo IEEE Transactions on IndustryApplications vol 45 no 4 pp 1309ndash1317 2009

[23] A D Nembhard J K Sinha and A Yunusa-KaltungoldquoExperimental observations in the shaft orbits of relativelyflexible machines with different rotor related faultsrdquo Mea-surement vol 75 pp 320ndash337 2015

[24] W X Yang and P J Tavner ldquoEmpirical mode decompositionan adaptive approach for interpreting shaft vibratory signalsof large rotating machineryrdquo Journal of Sound and Vibrationvol 321 no 3ndash5 pp 1144ndash1170 2009

[25] A K Jalan and A R Mohanty ldquoModel-based fault diagnosisof a rotor-bearing system for misalignment and unbalanceunder steady-state conditionrdquo Journal of Sound and Vibra-tion vol 327 no 3ndash5 pp 604ndash622 2009

[26] M A Khan M A Shahid S A Ahmed S Z KhanK A Khan and S A Ali ldquoGear misalignment diagnosis usingstatistical features of vibration and airborne sound spec-trumsrdquo Measurement vol 145 pp 419ndash435 2019

[27] L Wang F Yun S Yao and J Liu ldquoMeasurement methodand pipe wall misalignment adjustment algorithm of the pipebutting machinerdquo Measurement vol 94 pp 873ndash882 2016

[28] T H Patel and A K Darpe ldquoExperimental investigations onvibration response of misaligned rotorsrdquo Mechanical Systemsand Signal Processing vol 23 no 7 pp 2236ndash2252 2009

[29] J H Jung B C Jeon B D Youn M Kim D Kim andY Kim ldquoOmnidirectional regeneration (ODR) of proximitysensor signals for robust diagnosis of journal bearing sys-temsrdquo Mechanical Systems and Signal Processing vol 90pp 189ndash207 2017

[30] J Wang Y Peng andW Qiao ldquoCurrent-aided order trackingof vibration signals for bearing fault diagnosis of direct-drive

wind turbinesrdquo IEEE Transactions on Industrial Electronicsvol 63 no 10 pp 6336ndash6346 2016

[31] M Chandra Sekhar Reddy and A S Sekhar ldquoDetection andmonitoring of coupling misalignment in rotors using torquemeasurementsrdquo Measurement vol 61 pp 111ndash122 2015

[32] S-H Park J-C Park S-W Hwang J-H Kim H-J Parkand M-S Lim ldquoSuppression of torque ripple caused bymisalignment of the gearbox by using harmonic current in-jection methodrdquo IEEE vol 25 no 4 pp 1990ndash1999 2020

[33] X Liu D Liang J Du Y Yu X Yang and Z Luo ldquoEffectsanalysis of misalignments on dynamic characteristics test forpermanent magnet synchronous motorrdquo in Proceedings of the2014 17th International Conference on Electrical Machines andSystems (ICEMS) vol 10 Hangzhou China 2014

[34] J Piotrowski ldquoDetecting misalignment on rotating machin-eryrdquo in Shaft Alignment Handbook pp 35ndash87 CRC PressBoca Raton FL USA 3rd edition 2007

[35] C Salomon W Santana L DaSilva E Bonaldi L DeOliveiraand J DaSilva ldquoA stator flux synthesis approach for torqueestimation of induction motors using a modified stator re-sistance considering the losses effectrdquo in Proceedings of IEEEInternational Electric Machines amp Drives Conferencepp 1369ndash1375 IEMDC Chicago IL USA May 2013


Furthermore especially for the vertical reducer detector witha torque sensor in the measuring chain the additionalsensors mounted on the mechanism can cause a series ofproblems such as signals acquired to vary according to thepositioning of sensors [28 29] onefold information ac-quisition [30] and additional cost

Some scholars and experts have made some progress onthe research of detection and monitoring of coupling mis-alignment based on torque signal [31] However their workonly shows the possibilities of torque measurements as avaluable technique for detecting misalignment and moretheoretical studies are needed to show the relationshipbetween the coupling misalignment and the ripple of torqueIn other works techniques for estimating torque from thesensing of the electrical variables of the motors have beenproposed and tested using the frequency spectrum of thecurrent electrical motor for diagnosis [16] However theestimated torque which includes the component due toeither misalignment of stator and rotor or rotor eccentricityof the motor cannot represent the torque transmitted by theshaft us it cannot show the effect of coupling mis-alignment of the shaft clearly Besides the experiment iscarried out on the horizontal instrument which cannotrepresent the effect of the method on the vertical instrument

In order to solve the problem of diagnosis couplingmisalignment in the vertical robot reducer detector amethod of using the torque transducer to diagnose thecoupling misaligned fault of performance testing instru-ments of precision reducers is proposed in this paper toovercome the shortcoming of diagnosis of misaligned faultusing an accelerometer or proximity probe e paper ismainly organized as follows Firstly the structure of avertical performance testing instrument of precision re-ducers including guide rail mechanism measurementmodule on the input side (MMIS) tested assemblies (TA)measurement module on the output side (MMOS) andworkbench is introduced in Section 2 combined withanalysis of the causes of the misaligned fault Next therelationships between the misaligned fault and the torqueripple are analyzed in Section 3 Moreover the experimentrsquoseffects using the Fast Fourier Transform and power spectrumanalysis of the torque signal to diagnose the misaligned faultare revealed in Section 4 Finally some conclusions are givenin the fifth section

2 Structure of the Vertical Instrument andAnalysis of the Causes of theMisaligned Fault

e vertical instrument is designed to measure the com-prehensive performance of the reducer as shown in Figure 1is instrument consists of four assistance modules whichare guide rail mechanism tested assemblies (TA) powersupply system and workbench and two measurementmodules namely the measurement module at the input sideof the reducer (MMIS) and that at the output side of thereducer (MMOS) Each measurement module is mainlycomposed of the five following subsystems motor torquetransducer test mode conversion part angle encoder andother connecting parts e red component in Figure 1

represents the TA e robot reducer was installed insidethe TA which is used to achieve a rapid installation ofdifferent reducer types e connections with the mea-surement systems adopt a uniform and standard structure

e instrument is of vertical type in which functionalcomponents are sequentially connected in series in the verticaldirection MMIS MMOS and TA are the core componentse causes of the misaligned fault are as follows

(1) Although the disk supports inside the hollow cyl-inder are designed to ensure that the shafts arecentered the manufacturing tolerances and assemblyerror of the disk supports can still lead to shaftmisalignment

(2) e flexible coupling is used in the connecting partsto overcome slight misalignment between the shaftsHowever many factors can increase misalignedfaults and flexible coupling can never solve theproblem caused by increasing misalignment

(3) Moreover the test mode conversion componentsallow the input or output shaft of the tested reducerto be under unconstrained driven or locked con-ditions thereby satisfying the functional require-ments of different dynamic and static performancetests However there is no guarantee that the con-version components move along the axis echanging of the test mode can increase the mis-aligned faults because of the motion of the con-version components

Since all the three factors above can lead to the mis-aligned fault and the misalignment mainly between theshafts connected by the flexible coupling and spline cou-pling a feasible method is required to realize the misalignedfault diagnosis In this paper the torque transducer is used todiagnose the misaligned fault of performance testing in-struments of precision reducers through testing the torqueripple e relationships between the misaligned fault andthe torque ripple are analyzed in the next section

3 Relationships between the Misaligned Faultand the Torque Ripple

ere are three common types of shaft misalignment (asshown in Figure 2) parallel misalignment (a) angle mis-alignment (b) and comprehensive misalignment (c) Due tothe inevitable manufacturing tolerances and assembly errorthe three kinds of misalignment faults shown in Figure 2 canexist in the flexible coupling and spline coupling at each ofthe two shafts of MMIS and MMOS Especially for therectangular spline sleeve which connects two spline shafts atthe same time the misalignment fault can exist Due to theexcessive spline clearance both spline shafts can be con-sidered unconstrained at one end and the other end issupported in the state of high-precision bearing In thetransmission process it is likely that the misalignment be-tween the two spline shafts and the spline sleeve changeswith the rotation of the shaft at is the three states inFigure 2 may appear alternately



T T + 1113944n









Tested assemblies

Work bench

Torquetransducer


Robotreducer

Guide railmechanism

Power supply system

Servomotor




Connecting part

Torque motor




(a) (b) (c)








T1 T + 1113944n




(3)









Servo motor

Coupling

Bearing

Bearing

Bearing

Spline coupling

Torque transducer

Angle encoder










(6)





Coupling

BearingTprimef1

TprimeR1

∆θ

FFprime

Servo motor


Coupling

Bearing

Tf1

TR1

F2

F1

Fprime

Fprime

Servo motor

∆X





T2 T + 1113944n




(7)









(8)

Transmission sha

Spline sleeve

BearingBearing


FT1

FT2

TR2

Tf2 ∆Xprime


Transmission shaft

Spline sleeve

BearingBearing


FprimeT1

FprimeT2

TprimeR2

Tprimef2 ∆θprime











Flexible coupling

Servo motor

Transmission sha

Hollow cylinder

Disk supports

Disk supports

Disk supports


Torque transducer

Bearing

Bearing

Bearing

Spline coupling

Spline sleeve



(a) (b) (c)



10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (200Hz) 4fr (400Hz) 8fr (800Hz)

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

Figure 11 Continued


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz) 4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)






20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)








01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)



020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 10 20 30 40


2fr (133Hz)

3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)

(b)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)

(c)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)


105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)

Figure 16 Continued





175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References









































T T + 1113944n









Tested assemblies

Work bench

Torquetransducer


Robotreducer

Guide railmechanism

Power supply system

Servomotor




Connecting part

Torque motor




(a) (b) (c)








T1 T + 1113944n




(3)









Servo motor

Coupling

Bearing

Bearing

Bearing

Spline coupling

Torque transducer

Angle encoder










(6)





Coupling

BearingTprimef1

TprimeR1

∆θ

FFprime

Servo motor


Coupling

Bearing

Tf1

TR1

F2

F1

Fprime

Fprime

Servo motor

∆X





T2 T + 1113944n




(7)









(8)

Transmission sha

Spline sleeve

BearingBearing


FT1

FT2

TR2

Tf2 ∆Xprime


Transmission shaft

Spline sleeve

BearingBearing


FprimeT1

FprimeT2

TprimeR2

Tprimef2 ∆θprime











Flexible coupling

Servo motor

Transmission sha

Hollow cylinder

Disk supports

Disk supports

Disk supports


Torque transducer

Bearing

Bearing

Bearing

Spline coupling

Spline sleeve



(a) (b) (c)



10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (200Hz) 4fr (400Hz) 8fr (800Hz)

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

Figure 11 Continued


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz) 4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)






20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)








01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)



020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 10 20 30 40


2fr (133Hz)

3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)

(b)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)

(c)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)


105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)

Figure 16 Continued





175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References













































T1 T + 1113944n




(3)









Servo motor

Coupling

Bearing

Bearing

Bearing

Spline coupling

Torque transducer

Angle encoder










(6)





Coupling

BearingTprimef1

TprimeR1

∆θ

FFprime

Servo motor


Coupling

Bearing

Tf1

TR1

F2

F1

Fprime

Fprime

Servo motor

∆X





T2 T + 1113944n




(7)









(8)

Transmission sha

Spline sleeve

BearingBearing


FT1

FT2

TR2

Tf2 ∆Xprime


Transmission shaft

Spline sleeve

BearingBearing


FprimeT1

FprimeT2

TprimeR2

Tprimef2 ∆θprime











Flexible coupling

Servo motor

Transmission sha

Hollow cylinder

Disk supports

Disk supports

Disk supports


Torque transducer

Bearing

Bearing

Bearing

Spline coupling

Spline sleeve



(a) (b) (c)



10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (200Hz) 4fr (400Hz) 8fr (800Hz)

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

Figure 11 Continued


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz) 4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)






20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)








01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)



020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 10 20 30 40


2fr (133Hz)

3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)

(b)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)

(c)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)


105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)

Figure 16 Continued





175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References















































(6)





Coupling

BearingTprimef1

TprimeR1

∆θ

FFprime

Servo motor


Coupling

Bearing

Tf1

TR1

F2

F1

Fprime

Fprime

Servo motor

∆X





T2 T + 1113944n




(7)









(8)

Transmission sha

Spline sleeve

BearingBearing


FT1

FT2

TR2

Tf2 ∆Xprime


Transmission shaft

Spline sleeve

BearingBearing


FprimeT1

FprimeT2

TprimeR2

Tprimef2 ∆θprime











Flexible coupling

Servo motor

Transmission sha

Hollow cylinder

Disk supports

Disk supports

Disk supports


Torque transducer

Bearing

Bearing

Bearing

Spline coupling

Spline sleeve



(a) (b) (c)



10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (200Hz) 4fr (400Hz) 8fr (800Hz)

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

Figure 11 Continued


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz) 4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)






20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)








01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)



020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 10 20 30 40


2fr (133Hz)

3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)

(b)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)

(c)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)


105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)

Figure 16 Continued





175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References










































T2 T + 1113944n




(7)









(8)

Transmission sha

Spline sleeve

BearingBearing


FT1

FT2

TR2

Tf2 ∆Xprime


Transmission shaft

Spline sleeve

BearingBearing


FprimeT1

FprimeT2

TprimeR2

Tprimef2 ∆θprime











Flexible coupling

Servo motor

Transmission sha

Hollow cylinder

Disk supports

Disk supports

Disk supports


Torque transducer

Bearing

Bearing

Bearing

Spline coupling

Spline sleeve



(a) (b) (c)



10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (200Hz) 4fr (400Hz) 8fr (800Hz)

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

Figure 11 Continued


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz) 4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)






20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)








01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)



020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 10 20 30 40


2fr (133Hz)

3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)

(b)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)

(c)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)


105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)

Figure 16 Continued





175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References
















































Flexible coupling

Servo motor

Transmission sha

Hollow cylinder

Disk supports

Disk supports

Disk supports


Torque transducer

Bearing

Bearing

Bearing

Spline coupling

Spline sleeve



(a) (b) (c)



10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (200Hz) 4fr (400Hz) 8fr (800Hz)

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

Figure 11 Continued


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz) 4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)






20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)








01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)



020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 10 20 30 40


2fr (133Hz)

3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)

(b)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)

(c)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)


105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)

Figure 16 Continued





175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References








































10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (200Hz) 4fr (400Hz) 8fr (800Hz)

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz) 4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 10 20 30 40


2fr (133Hz) 4fr (267Hz) 8fr (534Hz)

(b)

Figure 11 Continued


01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz) 4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)






20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)








01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)



020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 10 20 30 40


2fr (133Hz)

3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)

(b)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)

(c)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)


105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)

Figure 16 Continued





175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References








































01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz) 4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz) 4fr (533Hz) 8fr (1067Hz)

(d)


10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

10

09

08

07

06

05

Torq

ue (N

middotm)

04

03

02

01

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)






20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)








01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)



020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 10 20 30 40


2fr (133Hz)

3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)

(b)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)

(c)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)


105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)

Figure 16 Continued





175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References











































20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

00 5 10 15 20


2fr (667Hz)3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (200Hz)3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

20

18

16

14

12

10

Torq

ue (N

middotm)

08

06

04

02

0

2fr (267Hz)3fr (40Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)


75 90 105

(d)








01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)



020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 10 20 30 40


2fr (133Hz)

3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)

(b)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)

(c)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)


105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)

Figure 16 Continued





175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References













































01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (133Hz)3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)


50 60 70

(b)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

0

2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)


75 90 105

(c)

01

009

008

007

006

005

Pow

er (J

)

004

003

002

001

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)



020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 10 20 30 40


2fr (133Hz)

3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)

(b)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)

(c)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)


105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)

Figure 16 Continued





175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References








































020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 5 10 15 20


2fr (667Hz)

3fr (10Hz)

fr (333Hz)

4fr (133Hz) 8fr (267Hz)

(a)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 10 20 30 40


2fr (133Hz)

3fr (20Hz)

fr (667Hz)

4fr (267Hz) 8fr (534Hz)

(b)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (200Hz)

3fr (30Hz)

fr (10Hz)

4fr (400Hz) 8fr (800Hz)

(c)

020

018

016

014

012

010

Pow

er (J

)

008

006

004

002

00 15 30 45 60


2fr (267Hz)

3fr (30Hz)

fr (133Hz)

4fr (533Hz) 8fr (1067Hz)

(d)


105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)

Figure 16 Continued





175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References











































175

15

125

10

075

05

025

0

Torq

ue (N

middotm)

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References








































105

09

075

06

045

03

015


00 0125 025 0375 05 0625 075 0875 1

Torq

ue (N

middotm)

fr 2fr

4fr

(a)


105

09

075

06

045

03

015

0

Torq

ue (N

middotm) fr

2fr

4fr

(b)


Torq

ue (N

middotm)

175

15

125

10

075

05

025

0

fr

2fr

4fr

(c)


Torq

ue (N

middotm)

21

18

15

12

09

06

03

0

fr

2fr

4fr

(d)


21

18

15

12

Torq

ue (N

middotm)

09

06

03

00 200 400




600 800

Alignment



5 Conclusions




Data Availability




Acknowledgments


References








































5 Conclusions




Data Availability




Acknowledgments


References








































diagnosis of the misaligned faults of the vertical test

Documents