Global Sensitivity Analysis of High Speed Shaft Subsystem ...

Research ArticleGlobal Sensitivity Analysis of High Speed Shaft Subsystem of aWind Turbine Drive Train

Saeed Asadi Viktor Berbyuk and Haringkan Johansson

Department of Mechanics and Maritime Sciences Chalmers University of Technology 412 96 Goteborg Sweden

Correspondence should be addressed to Saeed Asadi saeedasadichalmersse

Received 21 September 2017 Revised 21 December 2017 Accepted 25 December 2017 Published 1 February 2018

Academic Editor Ryoichi Samuel Amano

Copyright copy 2018 Saeed Asadi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The wind turbine dynamics are complex and critical area of study for the wind industry Quantification of the effective factors towind turbine performance is valuable for making improvements to both power performance and turbine health In this paper theglobal sensitivity analysis of validated mathematical model for high speed shaft drive train test rig has been developed in order toevaluate the contribution of systems input parameters to the specified objective functionsThedrive train in this study consists of a 3-phase induction motor flexible shafts shaftsrsquo coupling bearing housing and disk with an eccentric massThe governing equationswere derived by using the Lagrangian formalism and were solved numerically by Newmark method The variance based globalsensitivity indices are introduced to evaluate the contribution of input structural parameters correlated to the objective functionsThe conclusion from the current research provides informative beneficial data in terms of design and optimization of a drive trainsetup and also can provide better understanding of wind turbine drive train system dynamics with respect to different structuralparameters ultimately designing more efficient drive trains Finally the proposed global sensitivity analysis (GSA) methodologydemonstrates the detectability of faults in different components

1 Introduction

Nowadays wind turbine industry is growing very fast andwind power has become one of the most cost effectiveof all renewable energy sources because of environmentalconcerns and finite resource of fossil fuels [1] Therefore ithas attracted political and business interest and extensiveresearch is carried out at different universities and institutes[2]

It is very important to reduce the costs of designmanufacturing and maintenance and optimize the powerperformance of wind turbines in order to be cost competitivein the current market A better understanding and conse-quently improvement of wind turbine drive train dynamicsperformance could lead to overall cost efficiency in windturbine operations This is done by advanced system mod-elling identifying the reasons for premature failures and faultdiagnoses [3] in different components in drive train of windturbines Examples of multibody dynamic analysis to predictthe componentsrsquo loads in wind turbine drive trains are in

[4ndash6] Generally the dynamic performance of wind turbinedrive train can be investigated based on different pointsof view such as power performance operational speedsloads in specific components like coupling and bearingsin both healthy and faulty situations and misalignmentparameters [7ndash14] Different structural dynamic parameterscan directly influence a wind turbine ultimate performanceand in terms of loading high speed shaft components arethe most important since the components experience majorextreme loads during different regimes especially in transientone

The current research deals with mathematical modellingof high speed shaft drive train test rig A typical wind turbinedrive train is composed of a generator as an electrical partwith a control set flexible couplings a speed-up gearbox amain shaft and a wind rotor as an air-operated system [15]The power from the rotor is transferred through main shaftgearbox and high speed shaft to the generator

We propose the global sensitivity analysis (GSA) as a toolto better investigate wind turbine drive train dynamics more

HindawiInternational Journal of Rotating MachineryVolume 2018 Article ID 9674364 20 pageshttpsdoiorg10115520189674364

2 International Journal of Rotating Machinery

specifically high speed shaft drive train modelsThe aim is toquantify the input structural parametersrsquo effect to objectivefunctions of the system dynamics (interaction quantificationset [16]) From GSA the most influential design parameterson objective functions of wind turbines can be determinedwhich could ease the optimization process by narrowingdown the number of input design parameters

Methodology of sensitivity analysis is various and it hasbeen used for different applications Reviews on differentGSA methods can be found in [16ndash20] Some examplesof sensitivity analysis are derivative-based Morris one-at-a-time (OAT) local sensitivity method [21] and SobolSaltellimethod providing the variance based sensitivity [22 23]whereas the SobolSaltelli variance based sensitivity methodprovides indices that quantify the relative contribution ofeach input structural parameter to set of objective functions[22 23] In the variance based GSA input variables arechosen randomly in a proper rangeThen the contribution ofinput parameter to objective functions is investigated via thesensitivity indices These sensitivity indices are quantifyingthe contribution of a random variable to the total varianceof the response which is basis of variance based globalsensitivity analysis in [23]

Computational effort when design analysis requires alarge set of loading conditions to be evaluated such aswind turbine drive trains is very important As one of themost common methods for GSA Monte Carlo simulationfor these systems is not efficient since it is computation-ally expensive On the other hand GSA within analysisof variance decomposition (ANOVA) or high dimensionalmodel representation (HDMR) theory is suitable method interms of being cost effective It decomposes the responsefunction of a high dimensional system into combination oflow dimensional system set This reduces the computationaleffort A multiplicative dimensional reduction method (M-DRM) for GSA was proposed by Zhang and Pandey [24]entailing simple representation of GSA indices The methodhas proved to be as accurate as Monte Carlo with much lesscomputational effort [22]

Several works have been done within sensitivity analysisof rigid and flexible multibody systems with respect tostructural parameters such as mass moment of inertiastiffness and damping values [25ndash28] In [29] extendedFourier amplitude sensitivity test method was used and theresults of a single turbine and a set of large group of turbineswas compared by GSA Sensitivity analysis to perform a largeset of data analysis for a wind farm was reported in [30]

Considering that the faults occur in high speed shaft drivetrains this could be a subject of study [31 32] The test rigfor high speed shaft drive train as subsystem of wind turbinehas been constructed in order to do experimental study andacquire data for validation of developedmathematicalmodelAnother purpose of the test rig is to investigate strategies todetect faults Early fault detection is important since it canreduce the cost of maintenance drastically

Ultimately by introducing faults and defects and differentcomponents of high speed shaft drive train we investigatethe detectability of faults and failures within GSA Such an

analysis could lead to fault diagnose in components of windturbine drive train and prevent failures in early stages

In the following sections the proposed mathematicalmodel of high speed shaft drive train test rig with the exper-imental setup at the lab is presented Then input variablesand different operational scenarios and objective functionsare introduced Different sources of faults and failures suchas defect in the bearing backlash in the coupling andtorque ripple in the motor are introduced and modelledUltimately the GSA indices for the validated model havebeen investigated Results of GSA allow one to understandthe priorities in contribution of system parameters for anyspecific objective function and lead one to focus on the areasof operation that show greatest opportunities for optimiza-tion The outcome of the current research leads to usefulinformation regarding wind turbine drive train dynamicsbehaviour on different operational scenarios which not onlyprovide possibility to reduce the number of input parametersfor optimization and speed up the process but also giveinsight to design functional components in proper ranges andin a smarter way In particular the main aim of the paperis to apply the GSA for the proposed mathematical modelof high speed shaft drive train including aforementionedfault sources Quantifying the faults effects on the drive trainobjective functions could be applied as a methodology tocontribute to virtual condition monitoring in order to detectand predict faults and prevent failures in early stages indifferent components of drive trains

2 High Speed Shaft Drive TrainSystem Test Rig

The most common wind turbine is of lateral (horizontal)axis type which entails main supporting tower upon whichsits a nacelle and rotor Inside the nacelle sits a main shaftsupported by bearings on which the rotor hangs The mainshaft is connected to a gearbox which in turn is connected tothe generator with control system (Figure 1) The high speedshaft subsystem is one part where faults in bearing are oftenobserved

To support experimental and numerical studies of windturbine drive train dynamics a scaled down test rig has beenbuilt and instrumented as reported in [33 34] It entails amotor rotor disk with shaft where the rotor shaft is passingthrough the bearing housing and connected to the motor bya coupling A small eccentric mass can be added on the rotordisk to give a varying loadThe setup of the drive train test rigis represented in Figure 2

The test rig is driven by ABB motor model M3BP160MLA 6 (6 pole 75 kW) with a variable frequency controller(ACS355 from ABB providing speed up to 1000 rpm) Inaddition the test rig entails motor shaft adapter shaftrsquoscoupling bearing housing with flexible support a disk onwhich an eccentricmass can be added and bedplate onwhichall the components are mounted

The data acquisition system is SKF ptitude ObserverSKF offers under its Windcon 30 condition monitoringsystemThe set of sensors comprise accelerometers and Eddy

International Journal of Rotating Machinery 3

12

3

4

5

6 7 8 9 10 11 12

High speed sha subsystem

1

2

3

Main bearingMain shaBrakes

7

8

9

GearboxFantailFantail motor

10

11

12

TowerRotor hubRotor blade

4

5

6

Measuring

GeneratorClutch

instruments

Figure 1 Overview of a high speed shaft subsystem as part of wind turbine drive train [52]

Displacement sensorsDisplacement sensors (bearing housing)

(coupling torque)Telemeter

(tip deflection)

Figure 2 The test rig with sensors location

probes displacement sensors which detect vibrations in thetest rig and deflection at both tip and bearing housingAdditionally in order to measure the torque in coupling fullbridge strain gauges are put on themotor shaft and the signalis acquired using a telemeter system (TEL 1-PCM)

In previousworks [33 34] the coupling torque deflectionfields in both lateral and vertical directions in tip and bearing

housing have been measured and used for model validationTo support future studies of defect detection the objectivefunctions derived in this paper are based on these sensors

21 Mathematical Model of Drive Train System Test RigAn engineering abstract of the test rig is shown in Fig-ure 3 The mathematical model has been developed based


Test bed

Table top

Adjustablemotor mount

Shacoupling

Motor amp shaadapterM3BP-160-MLA-6

6-pole 3-phaseinduction machine

ACS-355 frequencycontroller

Test

clamps4x hold down

housing mountAdjustable bearing

displacement sensorsBearing housing with

Eccentric test load

farme

(a)

Rigid shaFlexible sha

Bedplate

Cb

kb

lbl

C

X

Y

Z

kC

k

r

kb

Jr

mD

WD

l

g

B

(b)

Figure 3 A sketch of the test rig (a) and an engineering abstract (b)

on both rigid and flexible shaft assumption in bendingmodes considering that the torsional flexibility is concen-trated at coupling The motor is rigidly attached to thebedplate

211 Governing Equations The kinetic potential and dissi-pative energies and nonconservative forces are derived con-sidering that the shaft is rotating and its bending deflection119908(119909) is assumed to occur in the radial direction only The


disk is seen as a rigid body and the shaft torsional flexibilityis concentrated in coupling as well as linear along the shaftlength A linear spring model is adopted to simplify theflexibility of structure of the bearings [35]

119879119863 ≐ 12119898119863 [[1198631205931015840 (119897)]2 + [119908119863120593 (119897)]2 1206012119904 ]+ 12119898119890 [ 11990810158402 (119897) + (119908 (119897) + 119903119890)2 1206012119904 ] + 12119869119863 1206012119863+ 12119868119863 [1198631205931015840 (119897)]2

119879 = 119879119863 + 12119869119892 1206012119892 + 12 [119869119888 + 119869sh2 ] 1206012119904 + 12 [119869sh2 ] 1206012119863119881 = (119898119863 + 119898119890) 119892119908 (119897) cos120601119904 + 12119896sh120601 (120601119863 minus 120601119904)2

+ 12119896119862120601 (120601119904 minus 120601119892)2 + 12119896120595119887 [119908119863120593 (119897119887)]2 + 119881sh119877 = 12119862sh

120601 ( 120601119863 minus 120601119904)2 + 12119862119862120601 ( 120601119904 minus 120601119892)2 + 12119862119887119891 10038161003816100381610038161003816 12060111990410038161003816100381610038161003816+ 12119862119887 [119863120593 (119897119887)]2

(1)

Here 119881sh = int1198970119864119868(11990810158401015840(119909))2119889119909 is the contribution of the

shaft bending to potential energy and the indices sh and 119863denotes for shaft and disk respectively The complete list ofstructural parameters is given in Table 1 The shaft torsion isparametrized by 120601119863 120601119904 and 120601119892 as an angle of rotation in thedisk (rotor tip) shaft and motor respectively

Applying Lagrange formalism the nonlinear set of equa-tions of motion can be written as matrix equation with timevariant coefficients as follows

119872(119902) 119902 + 119861 (119902 119902 119905) = 119865 (119902 119902 119905) (2)

where 119872 is the inertia matrix 119861 is the vector related tointernal forces acting on the test rig and 119902 is the vector ofgeneralized coordinate to be defined in Section 212119865 is an external load vector and it is a combination of119872∙BH(119902(119905))119879BL

119862 (119902(119905)) and119872rip119892 (119902(119905))which applies to specific

degree of freedom119872∙BH corresponds to external forces due to the distur-bance of the bearing defects imposed to the system (describedin Section 223) The defect may impose the disturbancetorque into bending and torsional degrees of freedom (∙ =120595 120601119903)119879BL119862 refers to the backlash of the coupling which appears

as an internal load in the governing equations The backlashhas a contribution in the potential energy since it is modelledas a spring with gap119872rip

119892 is torque ripples imposed by the motor a source offaults in drive train dynamics representing electromechanicalinteraction between components

212 Parametrization of the Model Since the coupling isassumed to be very stiff in torsion (apart from its backlash)

the shaft torsional flexibility and coupling play are here addedand included in 119896120601 and 119879BL

119862 We introduce 120601119903 as the rotor(disk plus shaft) angle such that the shaft twist is representedby Δ def= 120601119892 minus 120601119903 where 120601119892 is the motor shaft angleHence 120601119904 and 120601119863 are both replaced by 120601119903 in the governingequations in Section 211We assume that the shaft inertia andcoupling are abstracted as 119869119903 = 119869sh + 119869119862 Coupling torsionalstiffness is represented in a way that includes both shaft andcoupling torsional stiffnessrsquos Similarly the shaft and couplingdamping 119862sh

120601 and 119862119862120601 are represented by 119862120601The shaft deflection is parametrized in terms of rigid and

flexible contribution to total deflection as 119908(119909) = 120595119909 +119908119863120593(119909) where 120595 is the angular deflection at the coupling119908119863 is the rotor shaft tip deflection due to shaft flexibilityand 120593(119909) is taken as the shape of the fundamental bendingeigenmode as described in [33] In [33] we developed andvalidated the mathematical model of high speed shaft drivetrain test rig considering the shaft flexibility in terms ofbending To obtain a suitable shape function 120593(119909) the Euler-Bernoulli beam theory is applied [36] Only radial vibrationamplitudes of the beam were considered The fundamentalbending eigenmode based on structural parameters in Table 1was found at 120573 = 08734 and its shape function 120593 for a shaftrotating at speed Ω = 1000 rpm gives the eigenfrequencyat 120596⋆ = 1094608 rads The next eigenmode was foundat 120596⋆ = 6151 rads which is much higher than relevantexcitation frequency The deflection fields in two locationsnamely tip (119909 = 119897) and bearing hub (119909 = 119897119887) are assessedwithin the aforementioned assumptions Thus the tip radialdeflection would be 120595119897 + 119908119863120593(119897) and the bearing housingradial deflection becomes120595119897119887+119908119863120593(119897119887)The displacements invertical and lateral directions become mirrored by rotationalangle 120601119903 Note that we simplify the deflection field in thebearing hub For more realistic model shaft prebending andrelative motion of inner ring with respect to the bearing hubframe must be considered

To conclude the vector of generalized coordinate isdefined as 119902 = [120595 120601119903 120601119892 119908119863]22 Faults Modelling in the Drive Train

221 Electromechanical Modelling of the Motor Motor Torquewith Ripple Considering that wind turbine drive train con-tains both mechanical and electrical components it is ofimportant concern to have a model which studies electrome-chanical interactions In induction motors many kinds ofexcitation occur such as electromagnetic forces exist thatis unbalanced magnetic pull Torsional vibration of drivetrain system could lead to significant fluctuation of motorspeed Such oscillation of the angular speed superimposed onthe average motor angular speed causes severe oscillation ofelectric currents in motor windings [37] In this section wemodel torque ripple as source of this fluctuations in motorspeed

In the current mathematical model we assume the motoras a 6-pole induction machine with rated power 75 kW atspeed 975 rpm The speed is set from a frequency convertersuch that the torque is given by the difference from set speed


Table 1 Test rig nominal and randomized input parameters

Parameter Symbol Nominal value(119883⋆) 119883min119883⋆ 119883max119883⋆

Motor inertia with respect to 119909 axis 119869119892 0087 kgm2 05 2Rotor inertia with respect to 119909 axis 119869119903 0095 kgm2 05 2Rotor shaft inertia with respect to 119909 axis 119869sh 00003 kgm2

Rotor inertia with respect to 120595 (minor axis) 119868119903 19083 kgm2 05 2Disk inertia with respect to 119910 axis 119868119863 18 kgm2

Shaft inertia with respect to 119910 axis 119868119884sh 05 kgm2

Disk mass 119898119863 1452 kg 075 15Rotor shaft mass 119898sh 385 kgEccentric mass value mounted in the rotor disk 119898119890 744 gr 0 4Distance of eccentric mass from center 119903119890 8 cmRotor shaft and disk mass 119898119903 1835 kgLength from coupling to the tip 119897 085m 08 12Length from coupling to bearing housing 119897119887 072m 08 12Length from coupling to CG of disk and rotor shaft 1198970 075mDamping factor 120577119889 001 sCoupling torsional stiffness 119896120601 2600Nmrad 075 5Coupling bending stiffness 119896120595 56 kNmrad 075 5Bearing mount lateral and vertical stiffness coefficients 1198960119887 4308 kNm 075 5Bearing hub bending stiffness 119896120595

11988701119896120595 075 5

Coupling torsional damping 119862120601 26 kNmrad 05 10Coupling bending damping 119862120595 56 kNm srad 05 10Bearing mount lateral and vertical damping coefficients 119862119887 4308 kN sm 05 10Bearing friction damping coefficient 119862119887119891 00023Nsm 08 3Rotor torsional damping coefficient plus bearing friction 119862119891

1198871199030006Nsm 08 3

Motor torsional damping coefficient 119862119887119892 0006Nsm 05 10Shaft bending stiffness 119864119868 76699 kNm2 05 10Angle dependency factor of 119896119887 120572119887 1 08 15Torque ripples parameters 119872(119894)rip 10Nm 0 3Backlash gap BL 0002m 0 3Backlash regularization parameter 120590 40Bearing torque disturbance zone for defect 120576 01Maximum value of rotational bearing defect torque 119872120601119903BH 30Nm 0 5Maximum value of radial bearing defect torque 119872120595BH 30Nm 0 5

120596set119892 and actual speed 120601119892 A cycle of ramp startup-steady state-

shutdown can thus be described by

119872mech = 119872markΔ120596mark(120596set119892 minus 120601119892) (3)

where

120596set119892 = 1199051199050120596

max119892 119905 lt 1199050

120596set119892 = 120596max

119892 1199050 le 119905 lt 1199051119872mark = 0 1199051 le 119905

(4)

where Δ120596mark = 2560 times 2120587 rads 119872mark = 734561Nm 1199050is the startup-time and 1199051 is the instance of time when motor

is turned off (without braking) From electrical measurementof the voltage from the frequency converter it was found thatin addition to119872mech there is a small periodic torque (ripple)acting on the generator rotor at frequencies 120596set

119892 3120596set119892 and6120596set

119892 Hence the following electrical torque is considered

119872119892 (119905) = 119872markΔ120596mark(120596set119892 minus 120601119892) +119872(1)rip sin (120601119892)

+119872(2)rip sin (3120601119892) +119872(3)rip sin (6120601119892) (5)

where 119872(119894)rip (119894 = 1 2 3) are coefficients that contribute totorque ripple with corresponding frequencies and have beenobtained based on comparison of the torque ripple range insteady state in simulation and experiments


minus01

minus005

0

005

01

Torq

ue (N

m)

minus008 minus006 minus004 minus002 0 002 004 006 008 01minus01

Δ (rad)

Figure 4 Backlash effect as a torque to the coupling

222 Backlash Modelling of the Coupling Here we proposebacklash as nonlinear stiffness properties to the systemstructure which is common source of faults in transmissionsystem dynamics (Figure 4) This phenomenon affects thegoverning equations Properly functioning mechanical sys-tems need to have a certain clearance (gap play) betweenthe components transmitting motion under load As part ofinternal forces 119879int is defined as follows [38]

119879int (Δ) = 119896120601 (Δ minus BL) + (119896120601 minus BL)times [119892 (Δ minus BL) minus 119892 (Δ + BL)]2

119892 (119909) = |119909| tanh (120590119909) (6)

223 Modelling of Defects in Rolling Elements (Bearing) Dis-turbance Torque from Bearing Inner Ring Bearing fault diag-nosis is important in condition monitoring of any rotatingmachine Early fault detection is one of the concerns withinthe wind turbine system modelling To this end we need tomodel source of faults in the bearingDynamicmodelling andvibration response simulation of rolling bearings with surfacedefects is a common assumption within fault modellingWhen bearing balls roll over defects shock pulses with highfrequency are generated with excessive vibration leading thewhole system to failure This characteristic impulse-responsesignal is detected by the vibration monitor An inner racewayrotates with shaftrsquos rotation and goes through the loaded zoneof bearing every rotation This mechanism shows that thevibration of inner raceway with defect is modulated by theshaft rotation [39ndash41]

Here we propose a simple modelling of surface defect forbearing which imposes a disturbance torque in certain anglesin the inner raceway connected to the rotor shaft (similar towhen a ball passes from the defects and causes a excitationto the system) Figure 5 shows the external torque imposedin radial and rotational directions As part of external forces119872∙BH is defined based on impulse factor 120576 and maximumtorque119872∙BH as shown in Figure 5(a)

Proper value for maximum disturbance is 119872maxBH =30Nm 120576(=01) is the angular length of active contact zone

where the defect is assumed (Figure 5)Depending on the rolling elements number included in

the bearing structure the geometric relationship among itscomponents and shaft rotational speed characteristic rota-tional frequencies initiating from respective bearing elements

could be obtained In this regard the characteristic passingfrequency in the inner ring is defined as follows

119891119894 = 1198911199031198731198872 (1 + 119863119887119863119888 cos120593) (7)

where 119891119903 is the rotor rotational frequency (= 1206011199032120587) 119873119887is number of rolling balls (119873119887 = 9) 119863119887(=72mm) and119863119888(=535mm) are the balls diameter and bearing pitch ((119863119900+119863119894)2) diameters respectively 120593 is the contact angle (120593 =0 rad) The bearing defect location (here the inner race) willdetermine that 119891119894(asymp1055119891119903) will be exhibited in the machinevibration signal

224 Uneven Mounting Stiffness For subsequent study ofexcitation of uneven mounting stiffness the bearing mountstiffness has been defined as function of rotor shaft angle120601119903(119905)as follows

119896119887 ≐ 1198960119887radiccos2120601119903 + 1120572119887 sin2120601119903 (8)

where 120572119887 is a parameter corresponding to angle dependencyof 119896119887 and defines different loads in vertical and lateraldirections imposed to the bearing springs

23 Numerical Values of Parameters in the High Speed ShaftDrive Train Test Rig Table 1 represents the parameters andtheir numerical values assumed in the mathematical model

The damping coefficients are chosen as proportional torespective stiffness coefficient 119862∙ ≐ 120577119889119896∙ where ∙ = 120601 120595 119887

The value of 119862119887119891 which represents the bearing frictiondamping has been defined from SKF manual handbook [42]

24 Operational Scenarios In order to study the GSA indifferent conditions we introduce the following set of opera-tional scenarios in wind turbine drive train dynamic analysissuch as transient steady state and shutdown

(i) Transient a torque is applied to reach the presetspeed

(ii) Steady state it is where the structure is under almostconstant speed control

(iii) Shutdown in this regime there is no torque imposedto the system and it is left to slow down to stop

Note that the corresponding motor speeds([1000 800 600] rpm) correspond to the specific frequencyin the frequency controller ([50 40 30]Hz resp)

25 System Model Response for the Nominal Values Here wepresent the simulation response of the mathematical modelbased on structural parameters nominal values representedin Table 1 The displacement fields in both lateral and verticaldirection with their FFT have been presented

The deflection fields include three different operationalscenarios based on Figure 6 with steady state of 119905 = [1 6] secand shutdown starting at 119905 = 6 secThe tip deflection fields inboth vertical and lateral directions are presented with zoom


100

80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

Torq

ue (N

m)

Time (s)

109876543210

M∙（

(a)

O

z

y

k

k

c

c

Ball

defect

Inner amp Outerraceway

(b)

Figure 5 Typical disturbance torque (119872∙BH) with zoom (a) and inner race defect modelling sketch with rolling bearing elements (b)

Transient Steady state Shutdown

t0 t1

600

800

1000

Mot

or sp

eed

(rpm

)

8 9 104 5 6 721 30

Time (s)

ＧＲg = 1000 rpm

ＧＲg = 800 rpm

ＧＲg = 600 rpm

t0 t1

Figure 6 Motor speed for operational scenarios transient steadystate and shutdown for 3 different maximum speeds

in steady state (119905 = [5 6] sec) and shutdown (119905 = [7 8] sec)in Figure 7

The bearing housing deflection fields in both vertical andlateral directions are presented with zoom in steady state(119905 = [5 6] sec) and shutdown (119905 = [7 8] sec) are shown inFigure 8

The frequency response functions of the tip deflection inboth vertical and lateral directions are presented with zoomon the dominant frequency in Figure 9

The coupling torque with its frequency response is pre-sented here in Figure 10

The equations of motion have been solved using New-mark numerical procedure method implemented in Matlab

software In order to satisfy the convergence criteria theproper time step has been chosen as Δ119905 = 10minus43 Objective Functions

In this section we introduce the set of objective functionsThe introduced objective functions are assessed in order toevaluate the dynamic performance of the developed highspeed shaft models

(i) Tip Deflection in Both Vertical and Lateral Directionsin Steady State and Shutdown Regimes (1198741198651ndash4119861 ) The RMSvalues of the tip deflection in both lateral and verticalin two operational scenarios (steady state and shutdown)(RMS(119884std) RMS(119885std) and RMS(119884SD) RMS(119885SD) resp) areobtained within the mathematical model and the sensitivityindices based on the structural inputs have been defined forthese objective functions

(ii) Bearing Hub Deflection in Both Vertical and LateralDirections in Steady State and Shutdown Regimes (1198741198655ndash8119861 )RMS value of the bearing housing bending in both lateral andvertical directions in two operational scenarios (steady stateand shutdown) (RMS(BHstd

119884 ) RMS(BHstd119885 ) and RMS(BHSD

119884 )RMS(BHSD

119885 ) resp) is obtained within the mathematicalmodel and the sensitivity indices based on the structuralinputs have been defined for these objective functions Amodel of gearbox is required to consist of shaft and bearinghousing stiffness which allows interaction between housingand the internal components to be investigated


times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

2

0

minus2

minus45 655

2 4 6 8 100Time (s)

(a)

7 75 8

times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

2

1

0

minus1

(b)

Figure 7 Tip deflection with zoom in steady state in vertical (a) and with zoom in shutdown regime in lateral (b) directions for the totalmodel

5 55 6

times10minus4

times10minus4

2 4 6 8 100Time (s)

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

minus2

minus1

0

1

2

(a)

7 75 8

times10minus4

times10minus5

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

15

10

5

0

minus5

(b)

Figure 8 Bearing housing deflection with zoom in steady state in vertical (a) and with zoom in shutdown regime in lateral (b) directions forthe total model

(iii) Frequency Response of the Tip Deflection in Lateral andVertical Directions in Steady State and Shutdown (1198741198659ndash12119861 )These are the desired objective function representationsFFT(119884std) FFT(119885std) and FFT(119884SD) FFT(119885SD) respectivelyThe modal sensitivity analysis allows one to recognize theeigenfrequency response to design parameters changes It canalso help for the better identification of important dynamicproperties

(iv) Coupling Torque in Different Operational Scenarios Tran-sient Steady State and Shutdown Cases (1198741198651ndash3119879 ) In orderto determine the influence of the design parameters in

torsional vibrationmodel RMS values of the coupling torquein different operational scenarios transient steady stateand shutdown cases are investigated as objective functions(RMS(CTtr) RMS(CTstd) and RMS(CTSD) resp) The cou-pling torque is defines as119872120601 = 119896120601Δ

Coupling torque in transient regime is very importantto measure and quantify the parameters which have themost effect since the component experiences themost severeloadings due to this regime nature By shutting down themotor there is not any load imposed by motor side and therange of the coupling torque is decreased from steady statevalue to zero in short period of time


50 100 1500Frequency (Hz)

times10minus5

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

umtimes10minus5

0

1

2

3

4

1510 20 3025

(a)

times10minus5 times10minus5

50 100 150 2000Frequency (Hz)

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

um

0

1

2

3

4

1510 20 3025

(b)

Figure 9 FFT of tip deflection in steady state regime in vertical (a) and lateral (b) directions for the total model

minus10

minus5

0

5

10

15

20

Cou

plin

g to

rque

(Nm

)

2 4 6 8 100Time (s)

3minus20

minus10

0

10

20

435

minus15

(a)

0

10

20

30

40

50

60

70

80

90

100

Am

plitu

de sp

ectr

um

50 100 150 2000Frequency (Hz)

0

10

20

30

40

40 50 6010 20 30

(b)

Figure 10 Coupling torque with zoom in the steady state (a) and its FFT with zoom in the peaks (b) for the total model

(v) The First and Second Peaks in FFT Diagram of CouplingTorque in Steady State (1198741198654-5119879 ) There are several picks inthe coupling torque frequency response of which the mostdominant one is of interest to study and investigating themost influential parameter would give valuable conclusionform the vibrational point of view Studying the torsionalmodel we came up with a conclusion that some peaks inFFT are related to the ripple due to motor Quantifying thosepeaks would give the contribution of the fault source to theobjective function which is very valuable information interms of optimization

(vi) The Peak in FFT Diagram of Coupling Torque in Shut-down Regime (1198741198656119879) During shutdown we expect that onlytorsional eignefrequency of the structure plays the role inFFT

4 Global Sensitivity Analysis ofHigh Speed Shaft Test Rig

The efficiency and applicability of global sensitivity analysisis already proven by some mathematical and mechanicalexamples [24] The aim here is to apply this method to thehigh speed shaft drive train dynamics with respect to thestructural parameters

41Theoretical Concept of Variance Based Sensitivity AnalysisEach of the objective functions in Section 3 can be seen asa function of the test rig structural parameters representedas vector of 119899 variables X = [1198831 1198832 119883119899]119879 We haveadopted the variance based sensitivity indices developed bySaltelli et al [43] and mainly founded on the previous workof the Russian mathematician Zhang and Pandey [22] These


methods are defined as global sensitivity analysis (GSA)since the set of sensitivity indices are calculated consideringthe entire range of variations in parameters compared toOAT (once at a time) or differential approaches [44]

Upon considering the structural parametersX as randomthe objectives OF = ℎ(X) are also random The expectationoperation is denoted by119864[∙] such that themean and varianceof ℎ are calculated as follows

120583OF = 119864 [ℎ (X)] 119881OF = 119864 [(ℎ (X) minus 120583OF)2] (9)

Under certain assumptions of independence of 119883119894 s theANOVA decomposition of ℎ(X) can be expressed as follows[45 46]

ℎ (X) = ℎ0 + 119899sum119894=1

ℎ119894 (119883119894) +sum119894lt119895

ℎ119894119895 (119883119894 119883119895) + sdot sdot sdot+ ℎ12sdotsdotsdot119899 (1198831 1198832 119883119899)

(10)

Upon taking the variance of (10) the total variance of OF canbe decomposed in the same manner [47]

119881OF = 119899sum119894=1

119881119894 +sum119894lt119895

119881119894119895 + sdot sdot sdot + 11988112sdotsdotsdot119899 (11)

where it was assumed that the components of the function(11) are orthogonal and can be expressed in terms of ℎ(X)integrals as

1198811198941 sdotsdotsdot119894119904 = int (ℎ1198941 sdotsdotsdot119894119904)2 1198891199091198941 sdot sdot sdot 119889119909119894119904 (12)

119881OF = int (ℎ)2 119889X minus (ℎ0)2 (13)

In case 119883119894rsquos are not uniformly distributed it is possi-ble to define a probability function for random variables1198831 1198832 119883119904 and adjust (12) and (13) accordingly [22]

The constants 119881OF and 1198811198941 sdotsdotsdot119894119904 are called variances of ℎand ℎ1198941 sdotsdotsdot119894119904 respectively The first-order terms 119881119894 representthe partial variance in the response due to the individualeffect of a random variable 119883119894 while the higher order termsshow the interaction effects between two or more randomvariables This decomposition puts forward two conceptsnamely the primary effect referring to the effect of a termassociated with only one random variable [48] and the totaleffect referring to both the individual effect of a randomvariable as well as its interactionwith other random variablesThe primary effect index of a random variable119883119894 is obtainedby the normalization of themain-effect variance over the totalvariance in OF

119878119894 = 119881119894119881OF(14)

and the total sensitivity index is defined as

119878119879119894 = 119881119879119894119881OF (15)

where 119881119879119894 = 119881OF minus 119881minus119894[119864119894(OF |sim 119883119894)] Note that 119881minus119894[∙]refers to the variance of quantity with randomized set of Xexcept 119883119894 taken as nominal value More elaborate derivationof equations is presented in [22]

411 Multiplicative Dimensional Reduction Method (M-DRM) To Compute the Simplified Sensitivity Indices Inorder to reduce the computational effort associated with thenumerical evaluation of (13) [22] we propose to utilize amultiplicative form of dimensional reduction method (M-DRM) for sensitivity analysis of which the correspondingdeterministic model OF = ℎ(119883) is written with reference toa fixed input point that is X = c referred to as the cut pointwith coordinates c To this end the 119894th invariance functionsdefined by fixing all input variables but119883119894 to their respectivecut point coordinates are derived as

ℎ119894 (119883119894) = ℎ (1198881 119888119894minus1 119883119894 119888119894+1 119888119899) (16)

Based on this concept amultiplicative dimensional reductionmethod approximates the deterministic function ℎ(X) asfollows [24]

ℎ (X) asymp [ℎ (c)]1minus119899 sdot 119899prod119894=1

ℎ119894 (119883119894 cminus119894) (17)

where ℎ(c) = ℎ(1198881 1198882 119888119899) is a constant value and ℎ(119883119894 cminus119894)represents the function value for the case that all inputs except119883119894 are fixed at their respective cut point coordinates M-DRM is particularly useful for approximating the integralsrequired for evaluation of the sensitivity indices described inthe previous section

In order to write compact mathematical expression 120588119896and 120579119896 denoted as the 119896mean andmean square values respec-tively associate with input parameters 119883119896 The Gaussianquadrature is computationally efficient for one-dimensionalintegrals calculation The definitions are as follows

120588119896 = 119864 [ℎ119896 (119883119896)]asymp 119873sum119897=1

119908119896119897ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899) 120579k = 119864 [[ℎ119896 (119883119896)]2]

asymp 119873sum119897=1

119908119896119897 [ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899)]2

(18)

where 119873 is the total number of integration points and119883119896119897 and 119908119896119897 are the 119897th Gaussian integration abscissa andweight function respectively The total number of functionevaluations required to calculate the sensitivity indices usingthis approach is 119873X times 119873 with 119873X as number of designparameters


For computed 120579119894 and 120588119894 (119894 = 1 119899) the primary andtotal sensitivity indices are approximated as shown as follows

119878119894 = 119864119894 (119881 (OF | 119883119894))119881 (OF (X)) asymp 1205791198941205882119894 minus 1prod119899119896=11205791198961205882119896 minus 1

119878119879119894 = 1 minus 119864119894 (119881 (OF |sim 119883119894))119881 (OF (X)) asymp 1 minus 1205882119894 1205791198941 minus prod119899119896=11205882119896120579119896 (19)

The primary sensitivity index 119878119894 gives an indication ofhow strong the direct influence of 119883119894 is on OF without anyinteractions with other variables The total sensitivity index119878119879119894 gives an indication of the total influence of119883119894 onOF fromits own direct effects along with its interaction with othervariables [49 50]

The differences between the primary effect and thetotal effect then give an indication of how important theinteractions of119883119894 are with other variables in influencing OFIf themain effect is small whereas the total effect is large then119883119894 does influenceOF but only through interactive effects withother system inputs

The introduced sensitivity indices accuracy depends onnumber of integration points used that can be determined viaa convergence study [24 51]

42 System Input Parameters Used in GSA Assuming thebending model with torsional parameters imposed to thesystem we will have a model which contains bending andtorsional response of the system In order to apply GSA forthis model we propose the following set of input parametersas follows

XTOT = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (20)

The nominal values of the input parameters as well asthe range of parameters variation are given in Table 1 119883⋆ ischosen as cut point (see Section 411)

5 Results from GSA of High SpeedShaft (HSS) Model

In this section the global sensitivity analysis of the developedmathematicalmodels corresponding to high speed shaftdrivetrain system has been presented Both torsional and bend-ing models are examined under three different operationalscenarios with different motor speeds in steady state (OS119894 =120596max119892 = [50 40 30]Hz)The analysis presented here focuses on the response

characteristics particular to chosen objective functions It isexpected that the operational scenarios will influence thedynamics and structural response to different input variables

501 Convergence Study The number of function evalua-tions using M-DRM method is 119873X times 119873 For specific setof input variables and on each operational scenario allobjective functions are evaluated at the same time by asingle time integration of the model In order to achievean acceptable approximation of the sensitivity analysisindices using the M-DRM method it is important to assessthe convergence of the results by increasing number ofGaussian quadrature integration abscissas denoted by 119873in (18)

The convergence study shows that for a nonlinear systemsuch as high speed shaft drive train with system input param-eters prescribed in Section 5 sufficient level of accuracy forthe sensitivity analysis can be obtained with 21 times 20 = 420(119873X = 21 design parameters and 119873 = 20 integrationabscissas) function evaluations This is one of the mostprominent aspects of the global sensitivity analysis using

this approach which could reduce the computational effortdrastically

51 GSA of Drive Train High Speed (HSS) Shaft SubsystemModel In this sectionwe apply GSA to the totalmodel whichcomprises bending and torsional parameters all togetherTheinput parameters are defined in (20)

For the model containing all bending and torsionalparameters the primary and total sensitivity indices are asshown in Figure 11

It can be concluded from this that the torsional inputparameters (coupling torsional stiffness 119896120601 and bearingdamping due to friction 119862119887119891 and motor torque ripples 119872(119894)ripbacklash parameter BL and bearing impact disturbance119872120601119903BH) do not have significant influence to the bendingobjective functions (OF1minus11119861 ) The same conclusion couldbe made based on the bending input parameters (couplingbending stiffness 119896120595 and bearing mounting stiffness 119896119887 andshaft bending stiffness 119864119868) which do not seem to have majoreffect on torsional objective function (OF2119879)

Based on this analysis we conclude that the torsional andbending response of the model can be analysed separatelywhich allows for substantially simpler analysis which willdiscussed next

52 HSS Bending Model After separating the two modelswe analyse the sensitivity of HSS within the frame ofbending model based on bending objective functions suchas deflection field in bearing hub and tip in both lateraland vertical directions and the corresponding frequencydomains

Considering the separation of model the simulationresults of the bending model have been obtained based onsemi-inverse dynamics approach (ldquoperfectrdquo speed control)by prescribing the motor angular speed where it constitutes


025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(b)

Figure 11 Primary sensitivity 119878119894 (a) and total sensitivity ST119894 (b) indices for combined bending and torsional model for OS1

the input to the system and is prescribed as a ramp loadingexpressed by Heaviside function

120596119892 (119905) = 120596max1198921199050 119905119867 (1199050 minus 119905) + 120596max

119892 119867(119905 minus 1199050)119867 (1199051 minus 119905)+ 120596max119892 ( 119905end minus 119905119905end minus 1199051)119867(119905 minus 1199051)

(21)

where 1199050 and 1199051 denote the instant of time of the end oftransient and instant of time of the start of the shutdownoperations respectively 119905end is duration of the operationalscenarios (see Figure 6)

In order to investigate the bending model sensitivitythe input parameter appearing in the governing equationscorresponding to the model is chosen as follows

X119861 = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872120595BH] (22)

Here the sensitivity analysis has been performed for bendingmodel for 3 different operational scenarios with differentmotor speeds in steady state regime

It was shown that within these 3 different motor frequen-cies the contribution of the input parameters is not changingdrastically The obtained results of GSA are presented inFigure 12 The results indicates that the most influentialstructural parameters which have the major effect to theobjective functions are coupling bending stiffness 119896120595 andbearing mounting stiffness 119896119887 shaft bending stiffness 119864119868and bearing hub and tip locations 119897119887 119897 Also the dampingparameters do not have significant effect on both primary andtotal sensitivity indices

More detailed analysis of the results based on eachobjective functions and influence of the input parameters ispresented as follows

(i) Tip Deflection in Vertical and Lateral Directions in SteadyState (1198741198651-2119861 ) and Shutdown Scenarios (1198741198653-4119861 ) It is illustratedthat the coupling bending stiffness 119896120595 and bearing hubmounting stiffness 119896119887 are the most dominant influentialparameters respectively

It could be seen fromFigure 12 that the shaft bending stiff-ness 119864119868 hasmore influence in tip deflection field compared to

the other objective functionThis is somehow logical since thefirst bending mode shape is higher in tip (119909 = 119897) compared tobearing hub (119909 = 119897119887) [33]

Moreover the geometric locations of bearing hub and tip(119897119887 119897) are influential parameters in terms of deflection field inboth bearing hub and tip

(ii) Bearing Hub Deflection in Vertical and Lateral Directionsin Steady State (1198741198655-6119861 ) and Shutdown Scenarios (1198741198657-8119861 ) Inbearing deflection field the influence of 119897119887 is more than 119897 Intip deflection field the influence of 119897 is more than 119897119887

The bearing bending stiffness 119896120595119887does not have a major

effect to the system objective functionsIt could be seen that the angle dependency of the bearing

stiffness 120572119887 is not having a major effect to all of objectivefunctions since the realistic range for 120572119887 is small

(iii) Frequency Response of the Tip Deflection in Vertical andLateral Directions in Steady State (1198741198659-10119861 ) and ShutdownScenarios (11987411986511-12119861 ) The influences of stiffness parameters 119896120595and 119896119887 and 119864119868 and 119897119887 and also inertia parameters 119868119903 119898119863 and119898119890 are the most dominant parameters respectively This tellsthat aforementioned parameters are the most important onesin terms of frequency response of the tip deflection


025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)

Figure 12 Primary sensitivity 119878119894 (a) and total sensitivity ST119894 (b) indices for bending model for OS123

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)

Figure 13 Primary sensitivity 119878119894 (a) and total sensitivity ST119894 (b) indices for bending model with bearing defect assumption for OS1

Among themass inertia parameters 119868119903119898119890 and119898119863 affectmost of the bending objective functions

It could be also seen form Figure 12 that the dampinginput parameters have the least influences on both primaryand total sensitivity indices

To sum up the conclusion related to the GSA of HSSsubsystem within the frame of bending model 119896119887 and 119896120595 and119864119868 and 119868119903 and 119898119863 and 119898119890 have the major effects on bothprimary and total sensitivity indices

However according to the sensitivity analysis results restof parameters stated in the system modelling do not have aremarkable effect on thementioned objective functions Suchinformation would be particularly useful in attenuating thenumber of design parameters for optimization applications ofwind turbine drive trains setups which leads to cost efficientsimulations

53 Bending Model with Bearing Defect Detection of Faults inInner Ring Here we study the bearing defect influence to the

objective functions The aim is to investigate the sensitivityindices with respect to the bearing defect parameter 119872120595BHimposing a radial load to the bearing housing and seek itseffect to the bearing deflection and loads

It is hypothesized that if the sensitivity index with respectto parameter 119872120595BH is comparable to other sensitivities incertain OF we can expect that a sensor measuring this OFis able to detect defects of size related to the magnitude of119872120595BH The obtained results on GSA with bearing defect arepresented in Figure 13 and summarized below

The minimum value of119872120595BH which makes the sensitivityindices to be detectable has been investigated

(i) Deflection fields in vertical and lateral directionsin steady state and shutdown scenarios for the tip(OF1ndash4119861 ) and bearing hub (OF5ndash8119861 ) respectively inFigure 13 comparing 119872120595BH sensitivity indices forobjective functions tip and bearing hub deflections(OF1ndash4119861 versus OF5ndash8119861 ) the influence of the bearing


defect119872120595BH in bearing hub is more than the tip Thisis due to direct influence of the defect in the bearingby imposing the defect torque which means that thebearing may experience more damage by introducinga defect in the inner ring

(ii) Frequency response of the tip deflection in steadystate (OF9-10119861 ) and shutdown (OF11-12119861 ) scenarios inFFT of tip deflection bearing defect has showndifferent influence compared to the healthy case (nodefect) This means that the frequency response ofstructure in both healthy and faulty conditions isdifferent and by studying FFT of system response andinvestigating their sensitivity indices we could detectsome faults sources in specific components

54 HSS Torsional Model The torsional model has beenstudied based on the faults sources introduced in the previoussections Backlash effects on coupling creating nonlinearitiesto the system structure motor torque ripple with specificpeaks in the frequency domain and bearing defect distur-bance torque imposing on the rotor angular direction areapplied within system modelling In order to optimize thedesign and upgrade the structural performance the systemresponse must be studied based on these assumptions andcompared with the healthy conditions This gives a betterinsight to understand the faults sources and could facilitatethe way to prevent the failures in drive train by creatingstructures which are less sensitive with respect to these faultssources This could be studied within GSA by investigatingand optimizing the sensitivity indices

The sensitivity analysis for torsional model has beenperformed based on 3 different operational scenarios (seeFigure 6) for motor speeds in steady state regime In orderto investigate the torsional model sensitivity analysis theinput parameters appearing in the governing equations cor-responding to the model are chosen as followsX119879

= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)

541 Torsion Vibration Model of Drive Train Test Rig withBacklash Upon considering only the torsional response ofthe test rig the complete model in Section 21 is reduced alumped torsion vibration model An engineering abstract ofthe model is depicted in Figure 14 and entails two inertiasdisk 119869119903 and generator rotor 119869119892 respectively Each of theseinertias is supported by bearings The shaft and couplingflexibility is lumped together and represented by a torsionspring Due to gravity and eccentric mass there will be anexternal torque 119872119903(=119898119890119903119890119892 cos120601119903) acting on the disk andthe induced electrical torque from the motor119872119892 acts on thegenerator rotor

Hence we have the following set of equations to describethe test rig torsional vibration response

[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg

Figure 14 Engineering abstract of the test rig in torsional vibration

Here we define 119862119891119887119903(=119862119887119903 + 119862119887119891) as combination structural

damping in rotor and the bearing damping due to the frictionBased on the results (see Figure 15) it could be demon-

strated that the most influential input parameters to thetorsional objective functions are coupling torsional stiffness119896120601 and backlash in coupling BL motor torque ripples 119872(119894)ripand bearing defect parameter 119872120601119903BH Also the structuraldamping of the rotor and bearing friction coefficient 119862119891

119887119903has

some effects to both primary and total sensitivity indicesTheeffects of damping coefficient (except the bearing damping)and rotor and motor mass inertia values (119869119903 and 119869119892) are notsignificant compared to the other aforementioned parame-ters In total sensitivity 119878119879119894 also eccentric mass has shownsome impact which is expectable since it imposes an externaltorque to the structure

More detailed analysis of the results based on eachobjective functions and influence of the input parameters isstated as follows

(i) Coupling Torque in Transient and Steady State Regimes(1198741198651-2119879 )The coupling torque has strong sensitivity in terms oftorque ripple parameters119872(119894)rip (119894 = 1 2 3) arising from themotor This tells the significance of motor ripple assumptionin drive trains as one type of fault source

(ii) Coupling Torque and Its FFT in Shutdown Regime (11987411986536119879 )It is clear that in shutdown regime there is no any torqueripple imposed by the motor of which objective functionsOF3119879 and OF6119879 sensitivity indices for torque ripples areidentically zero

In FFT coupling torque influence is more than steadystate case since there is no ripple imposed by the motorAlso eccentric mass has some contribution especially in totalsensitivity index

(iii) Frequency Response of the Coupling Torque in SteadyState Regime (1198741198654-5119879 ) In FFT of the coupling torque insteady state regime the most dominant parameter is 119896120601Then the backlash and motor torque ripples have significantcontribution This tells that in vibration analysis it is veryimportant to identify the vibration faults sources namely herebacklash and ripple which may affect the system structure ina severe way


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)

Figure 15 Primary sensitivity 119878119894 (a) and total sensitivity ST119894 (b) indices for torsional model for OS1 with backlash

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)

Figure 16 Primary sensitivity 119878119894 (a) and total sensitivity ST119894 (b) indices for torsional model for OS1 with no backlash

(iv) Bearing Defect Maximum Value 119872120601119903119861119867 By increasingthe bearing defect maximum torque (Figure 5) we couldinvestigate its contribution to the sensitivity indices The aimis to define the minimum value of defect parameter 119872max

BHsuch that the sensitivity indices related to this parameterwould be larger than a specific value (119872max

BH gt 100Nm) Thismethodology contributes to detect predict and prevent thefaults and failures in bearings

Summing up the results of GSA for torsional model wecould state that the most influential parameters are 119896120601 andfaults sources BL and119872(119894)rip and also 119862119887119891 respectively

For disturbance torque coming from bearing housing119872120601119903BH we notice that its influence is more in total sensitivityindex rather than primary one This means that 119872(119894)rip influ-ence is more when the other parameters are changed Thesame conclusion could be made for ripple parameters119872(119894)rip

542 Torsional Model with No Backlash In order to empha-size into the bearing fault detection we study the systemstructure response based on no backlash (Figure 16) Thenwe investigate the influence of the input parameters to theobjective functions in case when the shaft coupling has abacklash

The obtained results have led to the following conclu-sions

(i) The bearing defect (quantified in terms of 119872120601119903BH)could be detected easier in case where no backlashcompared to with backlash (more contribution sensi-tivity indices especially the primary one) This is veryimportant conclusion and means that in the systemswith a couple of sources of faults the detectability ofeach fault may not be easy and it get more complexby introducing more complex system and assumingmore faults


025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)

Figure 17 Primary sensitivity 119878119894 (a) and total sensitivity ST119894 (b) indices for bending model for ℎres

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)

Figure 18 Primary sensitivity 119878119894 (a) and total sensitivity ST119894 (b) indices for torsional model for ℎres

(ii) The influence of bearing defect in the FFT for firstpeak and the second peaks is small though recogniz-able This draws the same conclusion for the bendingmodel with defect which tells that some specificpeaks in the FFT correspond to the system structuralfaults

55 Special Cases Resonance Investigation The operatingspeed ranges in wind turbines and system excitation fre-quency have a wide range The system may encounterresonances leading to severe vibration and affect the normaloperation of the machine Thus the drive train dynamicsmust be studied with more focus in excitation ranges

To focus on the excitation ranges we investigate theprimary and secondary sensitivity indices within the case inwhich the input parameters leads to resonance It means thatwe only take the randomized input parameters which are nearto the resonance frequency of the system in terms of bendingand torsion

Xres = X such that norm (eig (SYS (X))minus eig (SYS (c))) le TOL (25)

where SYS(X) and SYS(c) denote for the system eigenvaluebased on randomized set of X and nominal structural

parameters presented in Table 1 respectively Also TOL =1Hz is considered as a tolerance for frequency fluctuationSome conclusions could be made based on aforemen-

tioned cases and the results are presented in Figures 17 and18

(i) By selecting those randomized variables which basedon them the eigenfrequency of the structure is nearto specific value (50Hz) the sensitivity of eigenfre-quency variables takes the most contribution This isexpectable since in the resonance points the systeminfluential parameters are only the ones which theeigenfrequency of the system is based on them

(ii) In optimum case where the randomized values areidentical to the bending and torsional eigenfrequen-cies we expect that only the input parameters haveinfluence that they are in the eigenfrequency defini-tion

6 Conclusion and Outlook

The global sensitivity analysis for both torsional and bendingmodels of high speed shaft drive train test rig with respect toits structural parameters has been carried out using M-DRMmethod


The current study showed the feasibility and efficiencyof the M-DRM for global sensitivity analysis of nonlineardynamical system (high speed shaft drive train) whichprovided the results in a computationally efficient frame-workTheM-DRM applied here revealed practical significantresults which can reduce the number of input design param-eters for optimization of wind turbine drive train systems

By comparing the sensitivity of objective functions withrespect to different input structural parameters a betterunderstanding of drive train system dynamics is obtainedVertical and lateral deflections in bearing housing and tipcoupling torque in 3 different operational scenarios FFTof these functions have been introduced as the objectivefunctions to represent the test rig dynamic performance Toreflect the effect of operational scenarios the aforementionedoutputs have been evaluated within transient steady stateand shutdown cases in 3 different motor speeds The effectof the fundamental structural parameters on the dynamicresponse of the HSS drive train test rig has been recognizedBy comparing the sensitivity indices of corresponding objec-tive functions a better understanding of drive train systemdynamics is obtained

Based on the present analysis the following conclusionscould be stated

(i) GSA for complex systems is computationally expen-sive by using ordinary algorithms like Monte CarloNevertheless M-DRM method has been successfullyapplied to high speed shaft drive train model andprovided physically reasonable results in an efficientcomputational framework

(ii) For considered HSS drive train system model theestimated primary and total sensitivity indices coin-cide revealing no variance interaction effects betweenthe input parameters within different operational sce-narios Also there is a direct correspondence between119878119894 and 119878119879119894 which means that the higher order effectsare negligible

(iii) Deflection fields in both tip and bearing hub aremostly sensitive with respect to bearing mountingstiffness 119896119887 and coupling bending stiffness 119896120595 andgeometric parameters 119897 119897119887 and shaft bending stiffness119864119868

(iv) Coupling torque mostly in all regimes is influencedmainly by coupling torsional stiffness 119896120601 couplingbacklash BL and motor torque ripples 119872(119894)rip respec-tively This means that the proposed faults sourcesmay lead to drastic contributions in componentsloads In this research we have quantified thesecontributions

(v) Dynamic modelling and vibration responses simu-lation are important for fault mechanism studies toprovide proofs for defect detection and fault diagnoseThe most significant demonstration in the presentedresearch is the applicability of GSA towards defectdetection and faults diagnose in functional compo-nents of drive train The results suggest that from

sensitivity indices detectability of some faults suchas bearing defect could be analysed This analysisillustrates the applicability of the GSA method foradvancing modern condition monitoring systems forbearings

The global sensitivity analysis provided a useful methodto deeply understand some aspects of the initial structuralparameters to the dynamic outputs At this stage a prelim-inary assessment of the key parameters has been performedMoreover fault modelling in functional components suchas motor ripples bearing defects coupling backlash andtheir detectability within GSA has been investigated Thisfacilitates understanding the faults sources and their effectto other functional components which leads to better designand optimize drive trains more efficiently with less failure

Finally the items for future work could be outlined asfollows

(i) Apply GSA for real wind turbine structural param-eters such that we first propose to apply the sameanalysis forNREL 5MWwind turbine and investigateGSA sensitivity indices

(ii) Carry out dimensional analysis and upscaling of thetest rig and apply GSA for Pi Buckingham theory theaim is to prescribe system structure based onminimalnumber of dimensionless parameters which couldlead to upscale the structure and predict the real windturbine behaviours

(iii) Apply wind turbine aerodynamic loads as an externalload to the rotor (119872119903) in order to get more realisticresponse of the system considering the stochasticnature of wind loads it would be interesting tostudy the current model based on realistic calculatedaerodynamic loads extracted from FAST software

(iv) Study the bearing modelling in more advanced anddetailed approach the bearing defects here havebeen modelled by introducing an radial excitation tocertain DOFs imposed on inner ring of the bearingFor future work it is important to consider also ballmass inertia and outer ring dynamics [40 41]

(v) Model the electromechanical interaction of windturbine drive train since wind turbine contains bothmechanical and electrical components to have betterunderstanding of drive train dynamics the studymust be done within this interaction

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This project is financed through the Swedish Wind PowerTechnology Centre (SWPTC) SWPTC is a research centrefor design of wind turbines The purpose of the centreis to support Swedish industry with knowledge of designtechniques as well as maintenance in the field of wind power


The centre is funded by the Swedish Energy Agency andChalmers University of Technology as well as academic andindustrial partners In particular a donation of motors andfrequency converters by ABB CR WindCon30 conditionmonitoring system and set of sensors by SKF is gratefullyacknowledged Besides the authors would like to thank JanMoller for his contribution in the laboratory

References

[1] International Renewable Energy Agency Wind Power vol 1Power sector Issue 55 of Renewable energy technologies Costanalysis series

[2] F Dincer ldquoThe analysis on wind energy electricity generationstatus potential and policies in the worldrdquo Renewable ampSustainable Energy Reviews vol 15 no 9 pp 5135ndash5142 2011

[3] J Ribrant and L Bertling ldquoSurvey of failures in wind powersystems with focus on Swedish wind power plants during 1997-2005rdquo in Proceedings of the 2007 IEEE Power Engineering SocietyGeneral Meeting PES USA June 2007

[4] J L M Peeters D Vandepitte and P Sas ldquoAnalysis of internaldrive train dynamics in a wind turbinerdquoWind Energy vol 9 no1-2 pp 141ndash161 2006

[5] F Oyague ldquoGearbox Modeling and Load Simulation of a Base-line 750-kW Wind Turbine Using State-of-the-Art SimulationCodesrdquo Tech Rep NREL41160 2009

[6] W Shi H-C Park S Na J Song S Ma and C-W KimldquoDynamic analysis of three-dimensional drivetrain system ofwind turbinerdquo International Journal of Precision Engineering andManufacturing vol 15 no 7 pp 1351ndash1357 2014

[7] S Asadi Drive train system dynamic analysis application towind turbines Licentiate Thesis Chalmers University of Tech-nology 2016

[8] F Oyague Fault Identification in Drive Train ComponentsUsing Vibration Signature Analysis TP-500-41160 NationalRenewable Energy Laboratory 2009

[9] K G Scott D Infield N Barltrop J Coultate and A ShahajldquoEffects of extreme and transient loads on wind turbine drivetrainsrdquo in Proceedings of the 50th AIAA Aerospace SciencesMeeting Including the New Horizons Forum and AerospaceExposition January 2012

[10] G Semrau S Rimkus and T Das ldquoNonlinear Systems Analysisand Control of Variable Speed Wind Turbines for MultiregimeOperationrdquo Journal of Dynamic Systems Measurement andControl vol 137 no 4 Article ID 041007 2015

[11] T M Ericson and R G Parker ldquoNatural frequency clusters inplanetary gear vibrationrdquo Journal of Vibration andAcoustics vol135 no 6 Article ID 061002 2013

[12] M Singh and S Santoso ldquoDynamic Models for Wind Turbinesand Wind Power Plantsrdquo Tech Rep NRELSR-5500-527802011

[13] J Wang D Qin and Y Ding ldquoDynamic behavior of windturbine by a mixed flexible-rigid multi-body modelrdquo Journal ofSystem Design and Dynamics vol 3 no 3 pp 403ndash419 2009

[14] S Struggl V Berbyuk and H Johansson ldquoReview on windturbines with focus on drive train system dynamicsrdquo WindEnergy vol 18 no 4 pp 567ndash590 2015

[15] C Zhu Z Huang Q Tang and Y Tan ldquoAnalysis of non-linear coupling dynamic characteristics of gearbox systemabout wind-driven generatorrdquo Chinese Journal of MechanicalEngineering vol 41 no 8 pp 203ndash207 2005

[16] B Iooss and L Lemaitre ldquoA review on global sensitivity analysismethodsrdquo in Uncertainty management in Simulationoptimiza-tion of Complex Systems Algorithms andApplications CMeloniand G Dellino Eds pp 101ndash122 Springer US 2015

[17] C I Reedijk Sensitivity Analysis of Model Output Performanceof various local and global sensitivity measures on reliabilityproblems [Master thesis] Delft University of Technology 2000

[18] F Pianosi K Beven J Freer et al ldquoSensitivity analysis ofenvironmental models A systematic review with practicalworkflowrdquo Environmental Modeling and Software vol 79 pp214ndash232 2016

[19] E Borgonovo and E Plischke ldquoSensitivity analysis a review ofrecent advancesrdquo European Journal of Operational Research vol248 no 3 pp 869ndash887 2016

[20] X-Y Zhang M N Trame L J Lesko and S Schmidt ldquoSobolsensitivity analysis A tool to guide the development and evalu-ation of systems pharmacologymodelsrdquoCPT Pharmacometricsamp Systems Pharmacology vol 4 no 2 pp 69ndash79 2015

[21] H M Wainwright S Finsterle Y Jung Q Zhou and JT Birkholzer ldquoMaking sense of global sensitivity analysesrdquoComputers amp Geosciences vol 65 pp 84ndash94 2014

[22] X Zhang and M D Pandey ldquoAn effective approximation forvariance-based global sensitivity analysisrdquo Reliability Engineer-ing amp System Safety vol 121 pp 164ndash174 2014

[23] A Saltelli ldquoMaking best use of model evaluations to computesensitivity indicesrdquoComputer Physics Communications vol 145no 2 pp 280ndash297 2002

[24] X Zhang and M D Pandey ldquoStructural reliability analysisbased on the concepts of entropy fractional moment anddimensional reduction methodrdquo Structural Safety vol 43 pp28ndash40 2013

[25] J M P Dias and M S Pereira ldquoSensitivity Analysis of Rigid-Flexible Multibody Systemsrdquo Multibody System Dynamics vol1 no 3 pp 303ndash322 1997

[26] K D Bhalerao M Poursina and K Anderson ldquoAn efficientdirect differentiation approach for sensitivity analysis of flexiblemultibody systemsrdquoMultibody System Dynamics vol 23 no 2pp 121ndash140 2010

[27] D Bestle and J Seybold ldquoSensitivity analysis of constrainedmultibody systemsrdquo Archive of Applied Mechanics vol 62 no3 pp 181ndash190 1992

[28] S MMousavi Bideleh and V Berbyuk ldquoGlobal sensitivity anal-ysis of bogie dynamics with respect to suspension componentsrdquoMultibody System Dynamics vol 37 no 2 pp 145ndash174 2016

[29] P M McKay R Carriveau D S-K Ting and J L JohrendtldquoGlobal sensitivity analysis ofwind turbine power outputrdquoWindEnergy vol 17 no 7 pp 983ndash995 2014

[30] K Dykes A Ning R King P Graf G Scott and P SVeers ldquoSensitivity Analysis of Wind Plant Performance to KeyTurbine Design Parameters A Systems Engineering Approachrdquoin Conference Paper NRELCP-5000-60920 2014 Contract NoDE-AC36-08GO28308 National Harbor Maryland USA 2014

[31] J Helsen P Peeters K Vanslambrouck F Vanhollebeke andW Desmet ldquoThe dynamic behavior induced by different windturbine gearbox suspension methods assessed by means of theflexible multibody techniquerdquo Journal of Renewable Energy vol69 pp 336ndash346 2014

[32] B Marrant The validation of MBS multi-megawatt gearboxmodels on a 132 MW test rig Simpack User Meeting 2012 3

[33] S Asadi V Berbyuk andH Johansson ldquoVibrationDynamics ofa Wind Turbine Drive Train High Speed Subsystem Modelling


and Validatiordquo in Proceedings of the ASME International DesignEngineering Technical Conferences and Computers and Infor-mation in Engineering Conference pp C2015ndash46016 AugustBoston Massachusetts USA 2015

[34] S Asadi V Berbyuk andH Johansson ldquoStructural dynamics ofa wind turbine drive train high speed subsystem Mathematicalmodelling and validationrdquo in Proc of the International Con-ference on Engineering Vibration Ljubljana 7 - 10 September[editors Miha Boltezar Janko Slavic Marian Wiercigroch] -EBook - Ljubljana pp 553ndash562 2015

[35] M Todorov I Dobrev and F Massouh ldquoAnalysis of torsionaloscillation of the drive train in horizontal-axis wind turbinerdquoin Proceedings of the 2009 8th International Symposium onAdvanced Electromechanical Motion Systems and Electric DrivesJoint Symposium ELECTROMOTION 2009 France July 2009

[36] T A Silva and N M Maia ldquoElastically restrained Bernoulli-Euler beams applied to rotary machinery modellingrdquo ActaMechanica Sinica vol 27 no 1 pp 56ndash62 2011

[37] T Iwatsubo Y Yamamoto and R Kawai ldquoStart-up torsionalvibration of rotating machine driven by synchronous motorrdquo inProc of the Int Conference on Rotordynamics IFToMM

[38] T C Kim T E Rook and R Singh ldquoEffect of smootheningfunctions on the frequency response of an oscillator withclearance non-linearityrdquo Journal of Sound and Vibration vol263 no 3 pp 665ndash678 2003

[39] B Ghalamchi J Sopanen andAMikkola ldquoSimple and versatiledynamic model of spherical roller bearingrdquo International Jour-nal of RotatingMachinery vol 2013 Article ID 567542 13 pages2013

[40] S P Harsha ldquoNonlinear dynamic analysis of an unbalancedrotor supported by roller bearingrdquo Chaos Solitons amp Fractalsvol 26 no 1 pp 47ndash66 2005

[41] L Niu H Cao Z He and Y Li ldquoDynamic modeling andvibration response simulation for high speed rolling ball bear-ings with localized surface defects in racewaysrdquo Journal ofManufacturing Science and EngineeringmdashTransactions of theASME vol 136 no 4 Article ID 041015 2014

[42] Rolling bearings (general catalogue published by SKF group)2012

[43] A Saltelli P Annoni I Azzini F Campolongo M Ratto and STarantola ldquoVariance based sensitivity analysis of model outputDesign and estimator for the total sensitivity indexrdquo ComputerPhysics Communications vol 181 no 2 pp 259ndash270 2010

[44] A Saltelli and P Annoni ldquoHow to avoid a perfunctory sensitiv-ity analysisrdquo Environmental Modeling and Software vol 25 no12 pp 1508ndash1517 2010

[45] I M Sobol ldquoTheorems and examples on high dimensionalmodel representationrdquo Reliability Engineering amp System Safetyvol 79 no 2 pp 187ndash193 2003

[46] H Rabitz and O F Alis ldquoGeneral foundations of high-dimensional model representationsrdquo Journal of MathematicalChemistry vol 25 no 2-3 pp 197ndash233 1999

[47] I M Sobol ldquoGlobal sensitivity indices for nonlinear mathemat-ical models and their Monte Carlo estimatesrdquoMathematics andComputers in Simulation vol 55 no 1-3 pp 271ndash280 2001

[48] H Liu andWChen ldquoProbabilistic SensitivityAnalysisMethodsfor Design under Uncertainty Tech B224rdquo Tech Rep Inte-gratedDesignAutomation Laboratory Department ofMechan-ical Engineering Northwestern University

[49] A Saltelli and IM Sobolrsquo ldquoAbout the use of rank transformationin sensitivity analysis of model outputrdquo Reliability Engineeringamp System Safety vol 50 no 3 pp 225ndash239 1995

[50] T Homma and A Saltelli ldquoImportance measures in globalsensitivity analysis of nonlinear modelsrdquo Reliability Engineeringamp System Safety vol 52 no 1 pp 1ndash17 1996

[51] B Sudret ldquoGlobal sensitivity analysis using polynomial chaosexpansionsrdquo Reliability Engineering amp System Safety vol 93 no7 pp 964ndash979 2008

[52] httpwwwclimatetechwikiorgtechnologyoffshore-wind

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom


specifically high speed shaft drive train modelsThe aim is toquantify the input structural parametersrsquo effect to objectivefunctions of the system dynamics (interaction quantificationset [16]) From GSA the most influential design parameterson objective functions of wind turbines can be determinedwhich could ease the optimization process by narrowingdown the number of input design parameters

Methodology of sensitivity analysis is various and it hasbeen used for different applications Reviews on differentGSA methods can be found in [16ndash20] Some examplesof sensitivity analysis are derivative-based Morris one-at-a-time (OAT) local sensitivity method [21] and SobolSaltellimethod providing the variance based sensitivity [22 23]whereas the SobolSaltelli variance based sensitivity methodprovides indices that quantify the relative contribution ofeach input structural parameter to set of objective functions[22 23] In the variance based GSA input variables arechosen randomly in a proper rangeThen the contribution ofinput parameter to objective functions is investigated via thesensitivity indices These sensitivity indices are quantifyingthe contribution of a random variable to the total varianceof the response which is basis of variance based globalsensitivity analysis in [23]

Computational effort when design analysis requires alarge set of loading conditions to be evaluated such aswind turbine drive trains is very important As one of themost common methods for GSA Monte Carlo simulationfor these systems is not efficient since it is computation-ally expensive On the other hand GSA within analysisof variance decomposition (ANOVA) or high dimensionalmodel representation (HDMR) theory is suitable method interms of being cost effective It decomposes the responsefunction of a high dimensional system into combination oflow dimensional system set This reduces the computationaleffort A multiplicative dimensional reduction method (M-DRM) for GSA was proposed by Zhang and Pandey [24]entailing simple representation of GSA indices The methodhas proved to be as accurate as Monte Carlo with much lesscomputational effort [22]

Several works have been done within sensitivity analysisof rigid and flexible multibody systems with respect tostructural parameters such as mass moment of inertiastiffness and damping values [25ndash28] In [29] extendedFourier amplitude sensitivity test method was used and theresults of a single turbine and a set of large group of turbineswas compared by GSA Sensitivity analysis to perform a largeset of data analysis for a wind farm was reported in [30]

Considering that the faults occur in high speed shaft drivetrains this could be a subject of study [31 32] The test rigfor high speed shaft drive train as subsystem of wind turbinehas been constructed in order to do experimental study andacquire data for validation of developedmathematicalmodelAnother purpose of the test rig is to investigate strategies todetect faults Early fault detection is important since it canreduce the cost of maintenance drastically

Ultimately by introducing faults and defects and differentcomponents of high speed shaft drive train we investigatethe detectability of faults and failures within GSA Such an

analysis could lead to fault diagnose in components of windturbine drive train and prevent failures in early stages

In the following sections the proposed mathematicalmodel of high speed shaft drive train test rig with the exper-imental setup at the lab is presented Then input variablesand different operational scenarios and objective functionsare introduced Different sources of faults and failures suchas defect in the bearing backlash in the coupling andtorque ripple in the motor are introduced and modelledUltimately the GSA indices for the validated model havebeen investigated Results of GSA allow one to understandthe priorities in contribution of system parameters for anyspecific objective function and lead one to focus on the areasof operation that show greatest opportunities for optimiza-tion The outcome of the current research leads to usefulinformation regarding wind turbine drive train dynamicsbehaviour on different operational scenarios which not onlyprovide possibility to reduce the number of input parametersfor optimization and speed up the process but also giveinsight to design functional components in proper ranges andin a smarter way In particular the main aim of the paperis to apply the GSA for the proposed mathematical modelof high speed shaft drive train including aforementionedfault sources Quantifying the faults effects on the drive trainobjective functions could be applied as a methodology tocontribute to virtual condition monitoring in order to detectand predict faults and prevent failures in early stages indifferent components of drive trains

2 High Speed Shaft Drive TrainSystem Test Rig

The most common wind turbine is of lateral (horizontal)axis type which entails main supporting tower upon whichsits a nacelle and rotor Inside the nacelle sits a main shaftsupported by bearings on which the rotor hangs The mainshaft is connected to a gearbox which in turn is connected tothe generator with control system (Figure 1) The high speedshaft subsystem is one part where faults in bearing are oftenobserved

To support experimental and numerical studies of windturbine drive train dynamics a scaled down test rig has beenbuilt and instrumented as reported in [33 34] It entails amotor rotor disk with shaft where the rotor shaft is passingthrough the bearing housing and connected to the motor bya coupling A small eccentric mass can be added on the rotordisk to give a varying loadThe setup of the drive train test rigis represented in Figure 2

The test rig is driven by ABB motor model M3BP160MLA 6 (6 pole 75 kW) with a variable frequency controller(ACS355 from ABB providing speed up to 1000 rpm) Inaddition the test rig entails motor shaft adapter shaftrsquoscoupling bearing housing with flexible support a disk onwhich an eccentricmass can be added and bedplate onwhichall the components are mounted

The data acquisition system is SKF ptitude ObserverSKF offers under its Windcon 30 condition monitoringsystemThe set of sensors comprise accelerometers and Eddy


12

3

4

5

6 7 8 9 10 11 12


1

2

3


7

8

9


10

11

12


4

5

6

Measuring

GeneratorClutch

instruments




(tip deflection)







Test bed

Table top


Shacoupling




Test

clamps4x hold down



Eccentric test load

farme

(a)


Bedplate

Cb

kb

lbl

C

X

Y

Z

kC

k

r

kb

Jr

mD

WD

l

g

B

(b)






119879119863 ≐ 12119898119863 [[1198631205931015840 (119897)]2 + [119908119863120593 (119897)]2 1206012119904 ]+ 12119898119890 [ 11990810158402 (119897) + (119908 (119897) + 119903119890)2 1206012119904 ] + 12119869119863 1206012119863+ 12119868119863 [1198631205931015840 (119897)]2

119879 = 119879119863 + 12119869119892 1206012119892 + 12 [119869119888 + 119869sh2 ] 1206012119904 + 12 [119869sh2 ] 1206012119863119881 = (119898119863 + 119898119890) 119892119908 (119897) cos120601119904 + 12119896sh120601 (120601119863 minus 120601119904)2

+ 12119896119862120601 (120601119904 minus 120601119892)2 + 12119896120595119887 [119908119863120593 (119897119887)]2 + 119881sh119877 = 12119862sh

120601 ( 120601119863 minus 120601119904)2 + 12119862119862120601 ( 120601119904 minus 120601119892)2 + 12119862119887119891 10038161003816100381610038161003816 12060111990410038161003816100381610038161003816+ 12119862119887 [119863120593 (119897119887)]2

(1)




119872(119902) 119902 + 119861 (119902 119902 119905) = 119865 (119902 119902 119905) (2)





















11988701119896120595 075 5


1198871199030006Nsm 08 3





where

120596set119892 = 1199051199050120596

max119892 119905 lt 1199050

120596set119892 = 120596max

119892 1199050 le 119905 lt 1199051119872mark = 0 1199051 le 119905

(4)



119892 3120596set119892 and6120596set



+119872(2)rip sin (3120601119892) +119872(3)rip sin (6120601119892) (5)



minus01

minus005

0

005

01

Torq

ue (N

m)


Δ (rad)




119892 (119909) = |119909| tanh (120590119909) (6)







119891119894 = 1198911199031198731198872 (1 + 119863119887119863119888 cos120593) (7)



119896119887 ≐ 1198960119887radiccos2120601119903 + 1120572119887 sin2120601119903 (8)













100

80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

Torq

ue (N

m)

Time (s)

109876543210

M∙（

(a)

O

z

y

k

k

c

c

Ball

defect


(b)



t0 t1

600

800

1000

Mot

or sp

eed

(rpm

)

8 9 104 5 6 721 30

Time (s)

ＧＲg = 1000 rpm

ＧＲg = 800 rpm

ＧＲg = 600 rpm

t0 t1












119884 )RMS(BHSD



times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

2

0

minus2

minus45 655

2 4 6 8 100Time (s)

(a)

7 75 8

times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

2

1

0

minus1

(b)


5 55 6

times10minus4

times10minus4

2 4 6 8 100Time (s)

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

minus2

minus1

0

1

2

(a)

7 75 8

times10minus4

times10minus5

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

15

10

5

0

minus5

(b)








times10minus5

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

umtimes10minus5

0

1

2

3

4

1510 20 3025

(a)


50 100 150 2000Frequency (Hz)

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

um

0

1

2

3

4

1510 20 3025

(b)


minus10

minus5

0

5

10

15

20

Cou

plin

g to

rque

(Nm

)

2 4 6 8 100Time (s)

3minus20

minus10

0

10

20

435

minus15

(a)

0

10

20

30

40

50

60

70

80

90

100

Am

plitu

de sp

ectr

um

50 100 150 2000Frequency (Hz)

0

10

20

30

40

40 50 6010 20 30

(b)












ℎ (X) = ℎ0 + 119899sum119894=1

ℎ119894 (119883119894) +sum119894lt119895


(10)


119881OF = 119899sum119894=1

119881119894 +sum119894lt119895




119881OF = int (ℎ)2 119889X minus (ℎ0)2 (13)



119878119894 = 119881119894119881OF(14)


119878119879119894 = 119881119879119894119881OF (15)



ℎ119894 (119883119894) = ℎ (1198881 119888119894minus1 119883119894 119888119894+1 119888119899) (16)



ℎ119894 (119883119894 cminus119894) (17)



120588119896 = 119864 [ℎ119896 (119883119896)]asymp 119873sum119897=1

119908119896119897ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899) 120579k = 119864 [[ℎ119896 (119883119896)]2]

asymp 119873sum119897=1

119908119896119897 [ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899)]2

(18)

























025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(b)



120596119892 (119905) = 120596max1198921199050 119905119867 (1199050 minus 119905) + 120596max


(21)



X119861 = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872120595BH] (22)














025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)


025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)
















= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



12

3

4

5

6 7 8 9 10 11 12


1

2

3


7

8

9


10

11

12


4

5

6

Measuring

GeneratorClutch

instruments




(tip deflection)







Test bed

Table top


Shacoupling




Test

clamps4x hold down



Eccentric test load

farme

(a)


Bedplate

Cb

kb

lbl

C

X

Y

Z

kC

k

r

kb

Jr

mD

WD

l

g

B

(b)






119879119863 ≐ 12119898119863 [[1198631205931015840 (119897)]2 + [119908119863120593 (119897)]2 1206012119904 ]+ 12119898119890 [ 11990810158402 (119897) + (119908 (119897) + 119903119890)2 1206012119904 ] + 12119869119863 1206012119863+ 12119868119863 [1198631205931015840 (119897)]2

119879 = 119879119863 + 12119869119892 1206012119892 + 12 [119869119888 + 119869sh2 ] 1206012119904 + 12 [119869sh2 ] 1206012119863119881 = (119898119863 + 119898119890) 119892119908 (119897) cos120601119904 + 12119896sh120601 (120601119863 minus 120601119904)2

+ 12119896119862120601 (120601119904 minus 120601119892)2 + 12119896120595119887 [119908119863120593 (119897119887)]2 + 119881sh119877 = 12119862sh

120601 ( 120601119863 minus 120601119904)2 + 12119862119862120601 ( 120601119904 minus 120601119892)2 + 12119862119887119891 10038161003816100381610038161003816 12060111990410038161003816100381610038161003816+ 12119862119887 [119863120593 (119897119887)]2

(1)




119872(119902) 119902 + 119861 (119902 119902 119905) = 119865 (119902 119902 119905) (2)





















11988701119896120595 075 5


1198871199030006Nsm 08 3





where

120596set119892 = 1199051199050120596

max119892 119905 lt 1199050

120596set119892 = 120596max

119892 1199050 le 119905 lt 1199051119872mark = 0 1199051 le 119905

(4)



119892 3120596set119892 and6120596set



+119872(2)rip sin (3120601119892) +119872(3)rip sin (6120601119892) (5)



minus01

minus005

0

005

01

Torq

ue (N

m)


Δ (rad)




119892 (119909) = |119909| tanh (120590119909) (6)







119891119894 = 1198911199031198731198872 (1 + 119863119887119863119888 cos120593) (7)



119896119887 ≐ 1198960119887radiccos2120601119903 + 1120572119887 sin2120601119903 (8)













100

80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

Torq

ue (N

m)

Time (s)

109876543210

M∙（

(a)

O

z

y

k

k

c

c

Ball

defect


(b)



t0 t1

600

800

1000

Mot

or sp

eed

(rpm

)

8 9 104 5 6 721 30

Time (s)

ＧＲg = 1000 rpm

ＧＲg = 800 rpm

ＧＲg = 600 rpm

t0 t1












119884 )RMS(BHSD



times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

2

0

minus2

minus45 655

2 4 6 8 100Time (s)

(a)

7 75 8

times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

2

1

0

minus1

(b)


5 55 6

times10minus4

times10minus4

2 4 6 8 100Time (s)

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

minus2

minus1

0

1

2

(a)

7 75 8

times10minus4

times10minus5

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

15

10

5

0

minus5

(b)








times10minus5

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

umtimes10minus5

0

1

2

3

4

1510 20 3025

(a)


50 100 150 2000Frequency (Hz)

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

um

0

1

2

3

4

1510 20 3025

(b)


minus10

minus5

0

5

10

15

20

Cou

plin

g to

rque

(Nm

)

2 4 6 8 100Time (s)

3minus20

minus10

0

10

20

435

minus15

(a)

0

10

20

30

40

50

60

70

80

90

100

Am

plitu

de sp

ectr

um

50 100 150 2000Frequency (Hz)

0

10

20

30

40

40 50 6010 20 30

(b)












ℎ (X) = ℎ0 + 119899sum119894=1

ℎ119894 (119883119894) +sum119894lt119895


(10)


119881OF = 119899sum119894=1

119881119894 +sum119894lt119895




119881OF = int (ℎ)2 119889X minus (ℎ0)2 (13)



119878119894 = 119881119894119881OF(14)


119878119879119894 = 119881119879119894119881OF (15)



ℎ119894 (119883119894) = ℎ (1198881 119888119894minus1 119883119894 119888119894+1 119888119899) (16)



ℎ119894 (119883119894 cminus119894) (17)



120588119896 = 119864 [ℎ119896 (119883119896)]asymp 119873sum119897=1

119908119896119897ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899) 120579k = 119864 [[ℎ119896 (119883119896)]2]

asymp 119873sum119897=1

119908119896119897 [ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899)]2

(18)

























025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(b)



120596119892 (119905) = 120596max1198921199050 119905119867 (1199050 minus 119905) + 120596max


(21)



X119861 = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872120595BH] (22)














025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)


025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)
















= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Test bed

Table top


Shacoupling




Test

clamps4x hold down



Eccentric test load

farme

(a)


Bedplate

Cb

kb

lbl

C

X

Y

Z

kC

k

r

kb

Jr

mD

WD

l

g

B

(b)






119879119863 ≐ 12119898119863 [[1198631205931015840 (119897)]2 + [119908119863120593 (119897)]2 1206012119904 ]+ 12119898119890 [ 11990810158402 (119897) + (119908 (119897) + 119903119890)2 1206012119904 ] + 12119869119863 1206012119863+ 12119868119863 [1198631205931015840 (119897)]2

119879 = 119879119863 + 12119869119892 1206012119892 + 12 [119869119888 + 119869sh2 ] 1206012119904 + 12 [119869sh2 ] 1206012119863119881 = (119898119863 + 119898119890) 119892119908 (119897) cos120601119904 + 12119896sh120601 (120601119863 minus 120601119904)2

+ 12119896119862120601 (120601119904 minus 120601119892)2 + 12119896120595119887 [119908119863120593 (119897119887)]2 + 119881sh119877 = 12119862sh

120601 ( 120601119863 minus 120601119904)2 + 12119862119862120601 ( 120601119904 minus 120601119892)2 + 12119862119887119891 10038161003816100381610038161003816 12060111990410038161003816100381610038161003816+ 12119862119887 [119863120593 (119897119887)]2

(1)




119872(119902) 119902 + 119861 (119902 119902 119905) = 119865 (119902 119902 119905) (2)





















11988701119896120595 075 5


1198871199030006Nsm 08 3





where

120596set119892 = 1199051199050120596

max119892 119905 lt 1199050

120596set119892 = 120596max

119892 1199050 le 119905 lt 1199051119872mark = 0 1199051 le 119905

(4)



119892 3120596set119892 and6120596set



+119872(2)rip sin (3120601119892) +119872(3)rip sin (6120601119892) (5)



minus01

minus005

0

005

01

Torq

ue (N

m)


Δ (rad)




119892 (119909) = |119909| tanh (120590119909) (6)







119891119894 = 1198911199031198731198872 (1 + 119863119887119863119888 cos120593) (7)



119896119887 ≐ 1198960119887radiccos2120601119903 + 1120572119887 sin2120601119903 (8)













100

80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

Torq

ue (N

m)

Time (s)

109876543210

M∙（

(a)

O

z

y

k

k

c

c

Ball

defect


(b)



t0 t1

600

800

1000

Mot

or sp

eed

(rpm

)

8 9 104 5 6 721 30

Time (s)

ＧＲg = 1000 rpm

ＧＲg = 800 rpm

ＧＲg = 600 rpm

t0 t1












119884 )RMS(BHSD



times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

2

0

minus2

minus45 655

2 4 6 8 100Time (s)

(a)

7 75 8

times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

2

1

0

minus1

(b)


5 55 6

times10minus4

times10minus4

2 4 6 8 100Time (s)

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

minus2

minus1

0

1

2

(a)

7 75 8

times10minus4

times10minus5

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

15

10

5

0

minus5

(b)








times10minus5

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

umtimes10minus5

0

1

2

3

4

1510 20 3025

(a)


50 100 150 2000Frequency (Hz)

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

um

0

1

2

3

4

1510 20 3025

(b)


minus10

minus5

0

5

10

15

20

Cou

plin

g to

rque

(Nm

)

2 4 6 8 100Time (s)

3minus20

minus10

0

10

20

435

minus15

(a)

0

10

20

30

40

50

60

70

80

90

100

Am

plitu

de sp

ectr

um

50 100 150 2000Frequency (Hz)

0

10

20

30

40

40 50 6010 20 30

(b)












ℎ (X) = ℎ0 + 119899sum119894=1

ℎ119894 (119883119894) +sum119894lt119895


(10)


119881OF = 119899sum119894=1

119881119894 +sum119894lt119895




119881OF = int (ℎ)2 119889X minus (ℎ0)2 (13)



119878119894 = 119881119894119881OF(14)


119878119879119894 = 119881119879119894119881OF (15)



ℎ119894 (119883119894) = ℎ (1198881 119888119894minus1 119883119894 119888119894+1 119888119899) (16)



ℎ119894 (119883119894 cminus119894) (17)



120588119896 = 119864 [ℎ119896 (119883119896)]asymp 119873sum119897=1

119908119896119897ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899) 120579k = 119864 [[ℎ119896 (119883119896)]2]

asymp 119873sum119897=1

119908119896119897 [ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899)]2

(18)

























025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(b)



120596119892 (119905) = 120596max1198921199050 119905119867 (1199050 minus 119905) + 120596max


(21)



X119861 = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872120595BH] (22)














025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)


025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)
















= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




119879119863 ≐ 12119898119863 [[1198631205931015840 (119897)]2 + [119908119863120593 (119897)]2 1206012119904 ]+ 12119898119890 [ 11990810158402 (119897) + (119908 (119897) + 119903119890)2 1206012119904 ] + 12119869119863 1206012119863+ 12119868119863 [1198631205931015840 (119897)]2

119879 = 119879119863 + 12119869119892 1206012119892 + 12 [119869119888 + 119869sh2 ] 1206012119904 + 12 [119869sh2 ] 1206012119863119881 = (119898119863 + 119898119890) 119892119908 (119897) cos120601119904 + 12119896sh120601 (120601119863 minus 120601119904)2

+ 12119896119862120601 (120601119904 minus 120601119892)2 + 12119896120595119887 [119908119863120593 (119897119887)]2 + 119881sh119877 = 12119862sh

120601 ( 120601119863 minus 120601119904)2 + 12119862119862120601 ( 120601119904 minus 120601119892)2 + 12119862119887119891 10038161003816100381610038161003816 12060111990410038161003816100381610038161003816+ 12119862119887 [119863120593 (119897119887)]2

(1)




119872(119902) 119902 + 119861 (119902 119902 119905) = 119865 (119902 119902 119905) (2)





















11988701119896120595 075 5


1198871199030006Nsm 08 3





where

120596set119892 = 1199051199050120596

max119892 119905 lt 1199050

120596set119892 = 120596max

119892 1199050 le 119905 lt 1199051119872mark = 0 1199051 le 119905

(4)



119892 3120596set119892 and6120596set



+119872(2)rip sin (3120601119892) +119872(3)rip sin (6120601119892) (5)



minus01

minus005

0

005

01

Torq

ue (N

m)


Δ (rad)




119892 (119909) = |119909| tanh (120590119909) (6)







119891119894 = 1198911199031198731198872 (1 + 119863119887119863119888 cos120593) (7)



119896119887 ≐ 1198960119887radiccos2120601119903 + 1120572119887 sin2120601119903 (8)













100

80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

Torq

ue (N

m)

Time (s)

109876543210

M∙（

(a)

O

z

y

k

k

c

c

Ball

defect


(b)



t0 t1

600

800

1000

Mot

or sp

eed

(rpm

)

8 9 104 5 6 721 30

Time (s)

ＧＲg = 1000 rpm

ＧＲg = 800 rpm

ＧＲg = 600 rpm

t0 t1












119884 )RMS(BHSD



times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

2

0

minus2

minus45 655

2 4 6 8 100Time (s)

(a)

7 75 8

times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

2

1

0

minus1

(b)


5 55 6

times10minus4

times10minus4

2 4 6 8 100Time (s)

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

minus2

minus1

0

1

2

(a)

7 75 8

times10minus4

times10minus5

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

15

10

5

0

minus5

(b)








times10minus5

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

umtimes10minus5

0

1

2

3

4

1510 20 3025

(a)


50 100 150 2000Frequency (Hz)

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

um

0

1

2

3

4

1510 20 3025

(b)


minus10

minus5

0

5

10

15

20

Cou

plin

g to

rque

(Nm

)

2 4 6 8 100Time (s)

3minus20

minus10

0

10

20

435

minus15

(a)

0

10

20

30

40

50

60

70

80

90

100

Am

plitu

de sp

ectr

um

50 100 150 2000Frequency (Hz)

0

10

20

30

40

40 50 6010 20 30

(b)












ℎ (X) = ℎ0 + 119899sum119894=1

ℎ119894 (119883119894) +sum119894lt119895


(10)


119881OF = 119899sum119894=1

119881119894 +sum119894lt119895




119881OF = int (ℎ)2 119889X minus (ℎ0)2 (13)



119878119894 = 119881119894119881OF(14)


119878119879119894 = 119881119879119894119881OF (15)



ℎ119894 (119883119894) = ℎ (1198881 119888119894minus1 119883119894 119888119894+1 119888119899) (16)



ℎ119894 (119883119894 cminus119894) (17)



120588119896 = 119864 [ℎ119896 (119883119896)]asymp 119873sum119897=1

119908119896119897ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899) 120579k = 119864 [[ℎ119896 (119883119896)]2]

asymp 119873sum119897=1

119908119896119897 [ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899)]2

(18)

























025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(b)



120596119892 (119905) = 120596max1198921199050 119905119867 (1199050 minus 119905) + 120596max


(21)



X119861 = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872120595BH] (22)














025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)


025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)
















= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









11988701119896120595 075 5


1198871199030006Nsm 08 3





where

120596set119892 = 1199051199050120596

max119892 119905 lt 1199050

120596set119892 = 120596max

119892 1199050 le 119905 lt 1199051119872mark = 0 1199051 le 119905

(4)



119892 3120596set119892 and6120596set



+119872(2)rip sin (3120601119892) +119872(3)rip sin (6120601119892) (5)



minus01

minus005

0

005

01

Torq

ue (N

m)


Δ (rad)




119892 (119909) = |119909| tanh (120590119909) (6)







119891119894 = 1198911199031198731198872 (1 + 119863119887119863119888 cos120593) (7)



119896119887 ≐ 1198960119887radiccos2120601119903 + 1120572119887 sin2120601119903 (8)













100

80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

Torq

ue (N

m)

Time (s)

109876543210

M∙（

(a)

O

z

y

k

k

c

c

Ball

defect


(b)



t0 t1

600

800

1000

Mot

or sp

eed

(rpm

)

8 9 104 5 6 721 30

Time (s)

ＧＲg = 1000 rpm

ＧＲg = 800 rpm

ＧＲg = 600 rpm

t0 t1












119884 )RMS(BHSD



times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

2

0

minus2

minus45 655

2 4 6 8 100Time (s)

(a)

7 75 8

times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

2

1

0

minus1

(b)


5 55 6

times10minus4

times10minus4

2 4 6 8 100Time (s)

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

minus2

minus1

0

1

2

(a)

7 75 8

times10minus4

times10minus5

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

15

10

5

0

minus5

(b)








times10minus5

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

umtimes10minus5

0

1

2

3

4

1510 20 3025

(a)


50 100 150 2000Frequency (Hz)

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

um

0

1

2

3

4

1510 20 3025

(b)


minus10

minus5

0

5

10

15

20

Cou

plin

g to

rque

(Nm

)

2 4 6 8 100Time (s)

3minus20

minus10

0

10

20

435

minus15

(a)

0

10

20

30

40

50

60

70

80

90

100

Am

plitu

de sp

ectr

um

50 100 150 2000Frequency (Hz)

0

10

20

30

40

40 50 6010 20 30

(b)












ℎ (X) = ℎ0 + 119899sum119894=1

ℎ119894 (119883119894) +sum119894lt119895


(10)


119881OF = 119899sum119894=1

119881119894 +sum119894lt119895




119881OF = int (ℎ)2 119889X minus (ℎ0)2 (13)



119878119894 = 119881119894119881OF(14)


119878119879119894 = 119881119879119894119881OF (15)



ℎ119894 (119883119894) = ℎ (1198881 119888119894minus1 119883119894 119888119894+1 119888119899) (16)



ℎ119894 (119883119894 cminus119894) (17)



120588119896 = 119864 [ℎ119896 (119883119896)]asymp 119873sum119897=1

119908119896119897ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899) 120579k = 119864 [[ℎ119896 (119883119896)]2]

asymp 119873sum119897=1

119908119896119897 [ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899)]2

(18)

























025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(b)



120596119892 (119905) = 120596max1198921199050 119905119867 (1199050 minus 119905) + 120596max


(21)



X119861 = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872120595BH] (22)














025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)


025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)
















= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



minus01

minus005

0

005

01

Torq

ue (N

m)


Δ (rad)




119892 (119909) = |119909| tanh (120590119909) (6)







119891119894 = 1198911199031198731198872 (1 + 119863119887119863119888 cos120593) (7)



119896119887 ≐ 1198960119887radiccos2120601119903 + 1120572119887 sin2120601119903 (8)













100

80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

Torq

ue (N

m)

Time (s)

109876543210

M∙（

(a)

O

z

y

k

k

c

c

Ball

defect


(b)



t0 t1

600

800

1000

Mot

or sp

eed

(rpm

)

8 9 104 5 6 721 30

Time (s)

ＧＲg = 1000 rpm

ＧＲg = 800 rpm

ＧＲg = 600 rpm

t0 t1












119884 )RMS(BHSD



times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

2

0

minus2

minus45 655

2 4 6 8 100Time (s)

(a)

7 75 8

times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

2

1

0

minus1

(b)


5 55 6

times10minus4

times10minus4

2 4 6 8 100Time (s)

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

minus2

minus1

0

1

2

(a)

7 75 8

times10minus4

times10minus5

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

15

10

5

0

minus5

(b)








times10minus5

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

umtimes10minus5

0

1

2

3

4

1510 20 3025

(a)


50 100 150 2000Frequency (Hz)

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

um

0

1

2

3

4

1510 20 3025

(b)


minus10

minus5

0

5

10

15

20

Cou

plin

g to

rque

(Nm

)

2 4 6 8 100Time (s)

3minus20

minus10

0

10

20

435

minus15

(a)

0

10

20

30

40

50

60

70

80

90

100

Am

plitu

de sp

ectr

um

50 100 150 2000Frequency (Hz)

0

10

20

30

40

40 50 6010 20 30

(b)












ℎ (X) = ℎ0 + 119899sum119894=1

ℎ119894 (119883119894) +sum119894lt119895


(10)


119881OF = 119899sum119894=1

119881119894 +sum119894lt119895




119881OF = int (ℎ)2 119889X minus (ℎ0)2 (13)



119878119894 = 119881119894119881OF(14)


119878119879119894 = 119881119879119894119881OF (15)



ℎ119894 (119883119894) = ℎ (1198881 119888119894minus1 119883119894 119888119894+1 119888119899) (16)



ℎ119894 (119883119894 cminus119894) (17)



120588119896 = 119864 [ℎ119896 (119883119896)]asymp 119873sum119897=1

119908119896119897ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899) 120579k = 119864 [[ℎ119896 (119883119896)]2]

asymp 119873sum119897=1

119908119896119897 [ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899)]2

(18)

























025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(b)



120596119892 (119905) = 120596max1198921199050 119905119867 (1199050 minus 119905) + 120596max


(21)



X119861 = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872120595BH] (22)














025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)


025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)
















= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



100

80

60

40

20

0

minus20

minus40

minus60

minus80

minus100

Torq

ue (N

m)

Time (s)

109876543210

M∙（

(a)

O

z

y

k

k

c

c

Ball

defect


(b)



t0 t1

600

800

1000

Mot

or sp

eed

(rpm

)

8 9 104 5 6 721 30

Time (s)

ＧＲg = 1000 rpm

ＧＲg = 800 rpm

ＧＲg = 600 rpm

t0 t1












119884 )RMS(BHSD



times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

2

0

minus2

minus45 655

2 4 6 8 100Time (s)

(a)

7 75 8

times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

2

1

0

minus1

(b)


5 55 6

times10minus4

times10minus4

2 4 6 8 100Time (s)

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

minus2

minus1

0

1

2

(a)

7 75 8

times10minus4

times10minus5

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

15

10

5

0

minus5

(b)








times10minus5

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

umtimes10minus5

0

1

2

3

4

1510 20 3025

(a)


50 100 150 2000Frequency (Hz)

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

um

0

1

2

3

4

1510 20 3025

(b)


minus10

minus5

0

5

10

15

20

Cou

plin

g to

rque

(Nm

)

2 4 6 8 100Time (s)

3minus20

minus10

0

10

20

435

minus15

(a)

0

10

20

30

40

50

60

70

80

90

100

Am

plitu

de sp

ectr

um

50 100 150 2000Frequency (Hz)

0

10

20

30

40

40 50 6010 20 30

(b)












ℎ (X) = ℎ0 + 119899sum119894=1

ℎ119894 (119883119894) +sum119894lt119895


(10)


119881OF = 119899sum119894=1

119881119894 +sum119894lt119895




119881OF = int (ℎ)2 119889X minus (ℎ0)2 (13)



119878119894 = 119881119894119881OF(14)


119878119879119894 = 119881119879119894119881OF (15)



ℎ119894 (119883119894) = ℎ (1198881 119888119894minus1 119883119894 119888119894+1 119888119899) (16)



ℎ119894 (119883119894 cminus119894) (17)



120588119896 = 119864 [ℎ119896 (119883119896)]asymp 119873sum119897=1

119908119896119897ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899) 120579k = 119864 [[ℎ119896 (119883119896)]2]

asymp 119873sum119897=1

119908119896119897 [ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899)]2

(18)

























025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(b)



120596119892 (119905) = 120596max1198921199050 119905119867 (1199050 minus 119905) + 120596max


(21)



X119861 = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872120595BH] (22)














025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)


025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)
















= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

2

0

minus2

minus45 655

2 4 6 8 100Time (s)

(a)

7 75 8

times10minus4

times10minus4

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

2

1

0

minus1

(b)


5 55 6

times10minus4

times10minus4

2 4 6 8 100Time (s)

minus3

minus2

minus1

0

1

2

Vert

ical

defl

ectio

n (m

)

minus2

minus1

0

1

2

(a)

7 75 8

times10minus4

times10minus5

minus3

minus2

minus1

0

1

2

Late

ral d

eflec

tion

(m)

2 4 6 8 100Time (s)

15

10

5

0

minus5

(b)








times10minus5

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

umtimes10minus5

0

1

2

3

4

1510 20 3025

(a)


50 100 150 2000Frequency (Hz)

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

um

0

1

2

3

4

1510 20 3025

(b)


minus10

minus5

0

5

10

15

20

Cou

plin

g to

rque

(Nm

)

2 4 6 8 100Time (s)

3minus20

minus10

0

10

20

435

minus15

(a)

0

10

20

30

40

50

60

70

80

90

100

Am

plitu

de sp

ectr

um

50 100 150 2000Frequency (Hz)

0

10

20

30

40

40 50 6010 20 30

(b)












ℎ (X) = ℎ0 + 119899sum119894=1

ℎ119894 (119883119894) +sum119894lt119895


(10)


119881OF = 119899sum119894=1

119881119894 +sum119894lt119895




119881OF = int (ℎ)2 119889X minus (ℎ0)2 (13)



119878119894 = 119881119894119881OF(14)


119878119879119894 = 119881119879119894119881OF (15)



ℎ119894 (119883119894) = ℎ (1198881 119888119894minus1 119883119894 119888119894+1 119888119899) (16)



ℎ119894 (119883119894 cminus119894) (17)



120588119896 = 119864 [ℎ119896 (119883119896)]asymp 119873sum119897=1

119908119896119897ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899) 120579k = 119864 [[ℎ119896 (119883119896)]2]

asymp 119873sum119897=1

119908119896119897 [ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899)]2

(18)

























025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(b)



120596119892 (119905) = 120596max1198921199050 119905119867 (1199050 minus 119905) + 120596max


(21)



X119861 = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872120595BH] (22)














025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)


025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)
















= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




times10minus5

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

umtimes10minus5

0

1

2

3

4

1510 20 3025

(a)


50 100 150 2000Frequency (Hz)

0

05

1

15

2

25

3

35

Am

plitu

de sp

ectr

um

0

1

2

3

4

1510 20 3025

(b)


minus10

minus5

0

5

10

15

20

Cou

plin

g to

rque

(Nm

)

2 4 6 8 100Time (s)

3minus20

minus10

0

10

20

435

minus15

(a)

0

10

20

30

40

50

60

70

80

90

100

Am

plitu

de sp

ectr

um

50 100 150 2000Frequency (Hz)

0

10

20

30

40

40 50 6010 20 30

(b)












ℎ (X) = ℎ0 + 119899sum119894=1

ℎ119894 (119883119894) +sum119894lt119895


(10)


119881OF = 119899sum119894=1

119881119894 +sum119894lt119895




119881OF = int (ℎ)2 119889X minus (ℎ0)2 (13)



119878119894 = 119881119894119881OF(14)


119878119879119894 = 119881119879119894119881OF (15)



ℎ119894 (119883119894) = ℎ (1198881 119888119894minus1 119883119894 119888119894+1 119888119899) (16)



ℎ119894 (119883119894 cminus119894) (17)



120588119896 = 119864 [ℎ119896 (119883119896)]asymp 119873sum119897=1

119908119896119897ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899) 120579k = 119864 [[ℎ119896 (119883119896)]2]

asymp 119873sum119897=1

119908119896119897 [ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899)]2

(18)

























025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(b)



120596119892 (119905) = 120596max1198921199050 119905119867 (1199050 minus 119905) + 120596max


(21)



X119861 = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872120595BH] (22)














025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)


025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)
















= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







ℎ (X) = ℎ0 + 119899sum119894=1

ℎ119894 (119883119894) +sum119894lt119895


(10)


119881OF = 119899sum119894=1

119881119894 +sum119894lt119895




119881OF = int (ℎ)2 119889X minus (ℎ0)2 (13)



119878119894 = 119881119894119881OF(14)


119878119879119894 = 119881119879119894119881OF (15)



ℎ119894 (119883119894) = ℎ (1198881 119888119894minus1 119883119894 119888119894+1 119888119899) (16)



ℎ119894 (119883119894 cminus119894) (17)



120588119896 = 119864 [ℎ119896 (119883119896)]asymp 119873sum119897=1

119908119896119897ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899) 120579k = 119864 [[ℎ119896 (119883119896)]2]

asymp 119873sum119897=1

119908119896119897 [ℎ (1198881 119888119896minus1 119883119896119897 119888119896+1 119888119899)]2

(18)

























025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(b)



120596119892 (119905) = 120596max1198921199050 119905119867 (1199050 minus 119905) + 120596max


(21)



X119861 = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872120595BH] (22)














025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)


025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)
















= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia

























025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

OF2T

MBHBLM(i)

ripbEICbfCCbCk

bkkbklblmemDIrJgJr

(b)



120596119892 (119905) = 120596max1198921199050 119905119867 (1199050 minus 119905) + 120596max


(21)



X119861 = [119869119903 119869119892 119868119903 119898119863 119898119890 119897 119897119887 119896120601 119896120595 119896119887 119896120595119887 119862120601 119862120595 119862119887 119862119887119891 119864119868 120572119887 119872120595BH] (22)














025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)


025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)
















= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

bEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)


025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(a)

025

02

015

01

005

MBHbEICbfCCbCk

bkkbklblmemDIrJgJr

OF1B

OF2B

OF3B

OF4B

OF5B

OF6B

OF7B

OF8B

OF9B

OF10B

OF11B

OF12B

(b)
















= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







= [119869119903 119869119892 119898119890 119896120601 119862119891119887119903

119862119887119892 119872(1)rip 119872(2)rip 119872(3)rip BL 119872120601119903BH] (23)



[119869119903 + 1198981198901199032119890 ] 120601119903 + 119862119891119887119903 120601119903 + 119896120601 (120601119903 minus 120601119892) = 119872119903 (119905) 119869119892 120601119892 + 119862119887119892 120601119892 + 119896120601 (120601119892 minus 120601119903) = 119872119892 (119905)

(24)

Xre

me

r

Jr

k

g

Jg Mg

Cf

br Cbg





119887119903has








05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)


05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(a)

05

06

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHC

f

brCbgmeJgJr M(i)

ripk

(b)











025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(a)

025

02

015

01

005

OF1B

OF2B

OF5B

OF6B

OF9B

OF10B

bEICbfCCbCk

bkkbklblmemDIrJgJr

(b)


05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(a)

05

04

03

02

01

0

OF1T

OF2T

OF3T

OF4T

OF5T

OF6T

MBHBLC

f

brCbgmeJgJr M(i)

ripk

(b)
































Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





















Acknowledgments




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




References

























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia

























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Global Sensitivity Analysis of High Speed Shaft Subsystem ...

Documents

Transcript of Global Sensitivity Analysis of High Speed Shaft Subsystem ...