DevelopmentofaFive-Degree-of-FreedomSeatedHumanModel ...downloads.hindawi.com/journals/sv/2018/1649180.pdf · 2.2. Derivation of Nonlinear Equations of Motion. še nonlinear equations

Research ArticleDevelopment of a Five-Degree-of-Freedom Seated Human Modeland Parametric Studies for Its Vibrational Characteristics

Jong-Jin Bae and Namcheol Kang

School of Mechanical Engineering Kyungpook National University Daegu Republic of Korea

Correspondence should be addressed to Namcheol Kang nckangknuackr

Received 28 June 2018 Revised 22 September 2018 Accepted 30 September 2018 Published 23 October 2018

Academic Editor Giuseppe Ruta

Copyright copy 2018 Jong-Jin Bae andNamcheol Kangis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is study focuses on the biodynamic responses of a seated human model to whole-body vibrations in a vehicle Five-degree-of-freedom nonlinear equations of motion for a human model were derived and human parameters such as spring constants anddamping coefficients were extracted using a three-step optimization processes that applied the experimental data to themathematical human model e natural frequencies and mode shapes of the linearized model were also calculated In order toexamine the effects of the human parameters parametric studies involving initial segment angles and stiffness values wereperformed Interestingly mode veering was observed between the fourth and fifth human modes when combining two differentspring stiffness values Finally through the frequency responses of the human model nonlinear characteristics such as frequencyshift and jump phenomena were clearly observed

1 Introduction

Vibrational characteristics of seated humans are an im-portant consideration in the automotive industry becausethey play amajor role in riding comfort Furthermore recentsignificant advances in electric and autonomous vehiclesmay affect the perception and emotions of occupants ina vehicle For example in conventional vehicles combustionengine-induced vibration and noise act in concert to maskroad-induced vibration and noise generally this does notoccur with electric vehicles us we can expect that driversand passengers will be more sensitive to road-induced vi-brations and that the importance of analyzing vibrationalperception in vehicles will increase

e characteristics of the dynamic responses of a seatedhuman body aremainly affected by low-frequency vibrations(below 50Hz) A number of studies have also found that thefundamental frequency of a seated human exposed to whole-body vibration is lower than 10Hz [1ndash5] e natural fre-quencies and frequency responses of a seated human aredecided depending on body weight length of human seg-ments and sitting posture (e g slouched or erect) [6ndash10] Inaddition dynamic characteristics of a seated human involvenonlinearities such as the frequency softening phenomenon

in which the first natural frequency decreases as the mag-nitude of the excited vibrations increases [11ndash13] Mansfieldand Griffin [11] inferred that complex human factors such asmuscle forces and the bending or buckling of a spine couldgive rise to these nonlinearities erefore it is challengingto quantitatively analyze the dynamic responses of a seatedhuman body owing to its complexity and nonlinearity

Various finite element models of the human have beendesigned to investigate the complex characteristics of a seatedhuman Vavalle et al [14] introduced the whole body finiteelement model for conducting dynamic simulations underimpact loading conditions Siefert et al [15] developeda human-seat interaction system and extracted the seattransfer functions and contact pressure between a humanbody and a seat Butz et al [16] proposed the finite elementmodel of the lower extremity based on anatomy data andcalculated the dynamic responses of the human model in themilitary vehicles Although the finite element method is ca-pable of sophisticated modelling of the entire human bodyand local segments its computational time increases as thenumber of elements increases in nature In addition it is notstraightforward to develop finite element models of a humanbody that can accommodate variations in geometric pa-rameters such as body segment lengths and sitting postures

HindawiShock and VibrationVolume 2018 Article ID 1649180 15 pageshttpsdoiorg10115520181649180

To overcome the disadvantages of finite element modelslumped parameter models consisting of masses springs anddampers have been widely emphasized in various studies Weiand Griffin [17] utilized single-degree- and two-degree-of-freedom human models to mathematically predict seattransmissibility ey also revealed that the two-degree-of-freedom provided the better fitness to the experimental resultsthan the single-degree-of-freedom Choi andHan [18] used thevertical seat-human model that calculated the vibrationalperformance of semiactive seat suspensions quantitatively withSEAT values and vibration dose value (VDV) Bai et al [19]analyzed the dynamic characteristics of a four-degree-of-freedom biodynamic model with various structural configu-rations to identify the best configuration An improved genericalgorithmwas considered to determine the human parametersAlthough these human models that take only translationalmotions into account can be used to describe frequency re-sponses of occupants exposed to whole-body vibrations theycannot represent the swingmotions of human segments owingto the absence of rotational degrees-of-freedom erefore inorder to more accurately grasp and analyze the dynamicmovements of human segments it is essential to developa biodynamic human model that considers rotational motions

Lumped human models with rotational degrees-of-free-dom have been studied extensively to interpret fore-and-aftand pitch movements in the sagittal plane Matsumoto andGriffin [20] developed four- and five-degree-of-freedomhuman models including the viscera with rotational springsand dampers and they showed that the models having therotational degree-of-freedom represented the more reason-able descriptions for the dynamic characteristics of the seatedhuman body Cho and Yoon [21] presented nine-degree-of-freedom human model consisting of three rigid bodies andcompared with the single- two- and three-degree-of-free-dom models to verify the performance of the proposed nine-degree-of-freedom human model Further they also in-vestigated the effect of the seat backrest Kim et al [22]proposed a seven-degree-of-freedom seat-mannequin modelwhose parameters were obtained using indentation and swingexperimental tests the frequency response functions of theseat-mannequin model were presented and compared tothe experimental results Zheng et al [23] proposed seven-degree-of-freedom human model and showed analyticaland experimental apparent masses with and withouta backrest in the vertical and fore-and-aft directions Alsothe sensitive parameters were presented according to thepresence or absence of backrest Mohajer et al [24] pre-sented a human biomechanical model that consisted of 15rigid bodies e ride comfort was also estimated withrespect to road roughness vehicle speed and the weights ofthe subjects using the proposed human model

To develop the human model we used a Lagrangianformulation to derive the nonlinear equations of motion fora five-degree-of-freedom model e spring constants anddamping coefficients were extracted from experimental datain the literature using an optimization process e naturalfrequencies and mode shapes were also calculated from thelinearized human model In addition several parametricstudies were performed Finally we calculated the frequency

response curves of the nonlinear human model and thencompared those of the linear model nonlinear model andthe experiment

2 Nonlinear and LinearFive-Degree-of-Freedom Models

21 Model Description In order to investigate dynamiccharacteristics of a seated human we developed a lumpedparameter model consisting of masses dampers and springsIn the proposed model we considered three rigid bodiesmdashahead trunk and lower body (including the thighs and pelvis)e trunk and lower body are the suggested measurementpoints of vibrations in BS 6841 which is the standard forassessing human vibrations [25] Further the motions of thehead-neck segment are a prominent source of discomfort [10]because they may affect the vision of vehicle occupants andmay cause motion sickness Foot support was not consideredbecause the vibrations transmitted through foot supports arerelatively small compared to the vibrations affecting the trunkand lower body [26]

Based on these assumptions we developed a five-degree-of-freedom human model consisting of xh zh θ1 θ2 and θ3as shown in Figure 1 xh and zh respectively represent thehorizontal and vertical displacements of the hip joint withrespect to the basi-centric coordinate system while θ1 θ2and θ3 respectively represent the angular displacements ofthe lower body trunk and head around the y-axis Becausewe focus our interpretation on analyzing motions in therotational and vertical directions at the x-z plane (sagittalplane) dynamic behavior in the lateral direction was notconsidered e base motion zb is the input vertical dis-placement and velocity of the seat floor It was assumed thatno components overlapped and each segment was con-nected to a torsional spring and damper to represent therotational motion e translational springs were connectedto the surface of each component It is also assumed thateach human segment has uniform mass distribution ejoints of the human model were modelled using the pin jointcondition because the elongation and compression at therotational joints of an actual human body are very small Inother words the independent translational movements ofeach segment were not considered in our model

e combination of seat foam and human skin wasconsidered as the translational springs and dampers Allsprings assumed to be massless and frictionless were innonstretched and nonrotated conditions at the initial con-figuration Although seat foam has nonlinear viscoelasticproperties [27ndash29] it has been reported that such propertiesdo not significantly affect the transmissibility of human bodyvibrations in practice [13] It was also assumed that becausedeformations of the human back and buttock tissue aftersitting on a seat are sufficiently small their nonlinearity couldbe disregarded erefore considering the combination ofseat foam and human tissue as the linear translational springand damper provides a convenient and reasonable approachto developing this lumped parameter human model ebackrest was set to be parallel to the trunk of the humanmodel at the initial configuration

2 Shock and Vibration

It is necessary to dene translational spring deformationvalues to reect actual deformation characteristics of the seatfoam and tissue Unlike vertical human models comprisingone-dimensional motion in vertical directions each segment ofthe proposed ve-degree-of-freedommodel is able to representhorizontal and rotational motions When the deformationvalue of the translational spring is dened by the distancebetween two connected points the spring forces are generatedin the diagonal direction under this denition In the case ofactual foam and tissue deformation when the contact points aremoved the foam of the seat located at the moved positiongenerates the force to support a human body in the normaldirection To this end the tangential force of the translationalspringwas disregarded and the deformation of the translationalspring in the normal direction of the seat oor and seatback wasused for calculating the dynamic responses (Figure 2)

22 Derivation of Nonlinear Equations of Motion enonlinear equations of motion for the ve-degree-of-freedommathematical model were derived using the Lagrange equa-tion e locations of the center of mass at each segment are

x1 xh +12Lth cos θ1

x2 xh +12Ltr cos θ2

x3 xh + Ltr cos θ2 +12Lh cos θ3

z1 zh +12Lth sin θ1

z2 zh +12Ltr sin θ2

z3 zh + Ltr sin θ2 +12Lh sin θ3

(1)

where xh and zh are the horizontal and vertical displace-ments of the hip joint xi and zi are the locations of thecenters of mass of the lower body trunk and head re-spectively and Lth Ltr and Lh dene the lengths of the lowerbody trunk and head respectivelye kinetic energy of thehuman model is derived as follows

T 12sum3

i1mi _x2i + _z2i( ) +

12sum3

i1Ji _θ

2i (2)

where mi and Ji respectively indicate the mass and massmoment of inertia of the corresponding segment It can beeasily seen that translational and rotational kinetic energyare considered for three rigid bodies e derived po-tential energy and Rayleighrsquos dissipation function arewritten as

V 12sum4

i1kiδ

2i +

12sum2

i1ktiδ

2ti +sum

3

i1migΔi (3)

D 12sum4

i1ci _δ

2i +

12sum2

i1cti _δ

2ti (4)

where V is the potential energy associated with the trans-lational and torsional spring forces and gravity D is theRayleighrsquos dissipation function representing linear functionsof velocities and ki ci and g respectively denote trans-lational spring constants damping coecients and gravi-tational acceleration kti and cti denote torsional stinessvalues and damping coecients respectively δi and δti arethe displacements of the translational springs and torsionalsprings respectively Now applying the Lagrange equationto Equations (1)ndash(4) yields

δ3

δ2δ1

Figure 2 Directional denition of the translational springdeections

g

Lt2

m3rsquo

k4

k3

c4

c3L3

(xh zh)

L4

Ltr

Lh

J3rsquo

kt2 ct2

03

02

01

m2

m1 J1 Lt1

c2c1z

xO

k2k1

L1

L2

Lth

kt1 ct1

zb

J2

Figure 1 e proposed ve-degree-of-freedom seated biodynamichuman mathematical model (red colored symbols generalizedcoordinates)

Shock and Vibration 3

A11 0 A13 A14 A15

A22 A23 A24 A25

A33 0 0

A44 A45

Sym A55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

euroxh

eurozh

euroθ1euroθ2euroθ3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

B1

B2

B3

B4

B5

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(5)

e elements of each matrix are given in Appendix APrior to examining the dynamic characteristics using

nonlinear equations of motion linearization was performedIt is not a simple task to simulate the nonlinear humanmodel to obtain dynamic characteristics such as the naturalfrequency mode shape and parametric sensitivity andexcessive computational time would be required ereforewe carried out the linearization by expending the Taylorseries at the initial configurations which makes it possible torepresent the equations of motion with a set of generalizedcoordinates In the linearization of nonlinear equations ofmotion the following approximations were considered

sin θ sin θlowast + θ0( 1113857 asymp sin θ0 + cos θ0 middot θlowast

cos θ cos θlowast + θ0( 1113857 asymp cos θ0 minus sin θ0 middot θlowast(6)

where θlowast denotes a small perturbation around the initial con-figuratione linearized equations of motion can be written as

Meurox + C _x + Kx _zbCb + zbKb (7)

where xT is the generalized coordinate vector(xlowasth zlowasth θlowast1 θlowast2 θlowast3 )T and each matrix and vector is config-ured as follows

M

M11 0 M13 M14 M15

M22 M23 M24 M25

M33 0 0M44 M45

Sym M55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C

C11 C12 0 C14 0C22 C23 C24 0

C33 C34 0C44 C45

Sym C55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K

K11 K12 0 K14 0K22 K23 K24 0

K33 K34 0K44 K45

Sym K55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cb 0 Cb2 Cb3 0 01113858 1113859T

Kb 0 Kb2 Kb2 0 01113858 1113859T (8)

e elements of each matrix are given in Appendix BTaking the Fourier transform of Equation (7) the linearequations of motion in the frequency domain are given by

X(jω) minusω2M + jωC + K1113872 1113873minus1

jωCb + Kb( 1113857Zb(jω) (9)

us we can calculate the natural frequency and modeshape of the human model using the mass and stiffnessmatrix e seat-to-head (STH) transmissibility of the linearmodel can be computed from Equation (9) it will be used todetermine the human parameters by a comparison with theexperimental results

23 Identification of Human Parameters e inertial andgeometric properties stiffness and damping coefficients arecrucial parameters that determine the dynamic responses ofthe proposed human model In this study we determinedthe human parameters through the following three steps

(1) Determination of inertial and geometric properties(2) Extraction of stiffness values(3) Extraction of damping coefficients

e inertial and geometric parameters were measurableusing the measurement tools further we chose the massmoment of inertia and human segment length data from thereported literature related to anthropometry By contrast thestiffness and damping coefficients are not easy to measureexperimentally and thus we extracted these parameters byusing the optimization process from the experimental STHtransmissibility results [30] In the second and third steps wedetermined the stiffness values and damping coefficients

231 Determination of Inertial and Geometric ParametersIn the initial step we determined the mass moment of inertiaand segment length values In the second and third step toobtain the stiffness and damping coefficients the experimentalsubjects in the whole-body experiments were Korean males intheir late 20s [21 30] and thus we tried to select the mass andgeometric parameters from Korean anthropometric data asclose as possible to these experimental subjects [31 32]However we used themassmoment of inertia from alternativeanthropometric data which is the similar configuration of theproposed human model because of a lack of validated officialdata for Korean males [22] e human mathematical modelin this study had a backrest angle of 111deg a seat pan angle of 12deg[12] and a head angle of 100deg (to the horizontal line) [22] emass mass moment of inertia and geometric length valuesused in the human mathematical model are listed in Table 1

232 Extraction of the Stiffness and Damping CoefficientsIn this study the springs and dampers underneath thelower body and the springs and dampers connected betweenthe backrest and trunk were set to have different stiffnessvalues and damping coefficients e foam of the seatbackshows the different force-displacement relationship fromthe seat pad [33] However in the sitting condition on a seatit is assumed that the stiffness of a human body in the vi-cinity of the buttock and thigh has approximately identicalvalue in the case of small penetration based on load-deflection measurement [34] Furthermore the deformation


characteristics of the foam in the seatback or seat pad wereassumed to be similar therefore we assumed that thestiffness and damping coefficients of the translationalsprings and dampers connected to the same component ofthe human model have the equal values respectively (ie k1 k2 k3 k4 c1 c2 and c3 c4)

To identify the stiffness and damping coefficients theexperimental STH transmissibility was obtained from thereported literature [30] In the optimization to obtain humanparameters we used the ldquofminconrdquo function in MATLAB(2015b) which is known to be effective for convex opti-mization Firstly the stiffness values were extracted based onthe natural frequency of the experimental results Becausethe estimation of the natural frequency tends to be morerepeatable than the estimation of amplitude in general vi-bration experiments we considered that the natural fre-quency values reported in the literature are more reliablethan those for amplitude we subsequently prioritized de-termining the natural frequency of the human model Wethen determined the damping coefficients by minimizing thedifference between the amplitude of the human mathe-matical model and the experiment

e stiffness of the five-degree-of-freedom humanmodelwas determined based on the first and second natural fre-quencies (42Hz and 75Hz) from the experimental resultse following objective function was used in the process fordetermining the stiffness values

E1 min w(2)1 f

ex1 minusf

cal1

11138681113868111386811138681113868

11138681113868111386811138681113868 + w(2)2 f

ex2 minusf

cal2

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875 (10)

where fex1 fex

2 fcal1 fcal

2 denote the first and second naturalfrequency of the experiment and the proposed humanmodel respectively w

(2)1 w

(2)1 are the weighting factors (20

and 10 were applied respectively) e superscripts indicatethe second step We calculated the natural frequencies of thehuman model using a mass and stiffness matrix whose el-ements are optimized spring constants and both resultsindicate the same natural frequency as listed in Table 2 emode shape was also computed as shown in Figure 3 it willbe explained in detail in Section 32

e objective function for the extraction of the dampingcoefficient is as follows

E2 min w(3)1 D1 + w

(3)2 D21113872 1113873 (11)

where

D1 1113938

fb

faTex fi( 1113857

11138681113868111386811138681113868111386811138681113868minus Tcak fi( 1113857

11138681113868111386811138681113868111386811138681113868df

1113938fb

faTex fi( 1113857

11138681113868111386811138681113868111386811138681113868df

(12)

D2 max Tex

( 1113857minusmax Tcal

1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868 (13)

where Tex is the experimental STH transmissibility and Tcal

is the STH transmissibility of the human mathematicalmodel fa and fb are the first and final frequencies of theexperimental data (0Hz and 20Hz respectively) In order tomaximize the correlation between our calculated amplitudesand the experimental results near the first natural frequencythe weighting factors w

(3)1 w

(3)1 are set to 10 and 20 re-

spectively e initial and optimized stiffness and dampingcoefficients are listed in Table 3

It can be seen that the first natural frequency and themaximum amplitude of the STH transmissibility of thehuman mathematical model calculated from the optimizedparameters are consistent with those of the experimentalSTH transmissibility as shown in Figure 4 e frequencyresponse curve of the humanmodel around the fundamentalfrequency is in good agreement with that of the experimentalSTH transmissibility in the range of 0ndash7Hz However bothresults differ at frequencies ranging from 7ndash15Hz Inpractice it is not easy to simultaneously represent the firstand second peaks in the frequency response curves using ourmodel owing to the limited configurations of the proposedhuman model us we focused on fitting to the amplitudeof the first natural frequency in the frequency domainTypically in the frequency response curves of a seatedhuman the first resonance frequency is clearly distinguishedfrom the higher mode Moreover similar frequency bandsand amplitudes are reported in many studies [1 6 15 21]However the second natural frequency of a seated human isrelatively undetectable and is significantly affected by in-dividual subject characteristics such as sitting posture bodyweight and length of human segments For exampleRakheja et al [6] reported the apparent masses of seatedsubjects with respect to their sitting postures In this liter-ature the clarity of the second peak is dependent on the bodymass measurement point and magnitude of the excitedvibrations In addition the amplitude of the fundamental

Table 1 Inertial properties and length data of the five-degree-of-freedom human model

Property Value

Mass (kg)m1 1049m2 3398m3 667

Mass moment of inertia (kgmiddotm2)J1 023J2 205J3 003

Length (mm)

Ltr 59860Lth 57170Lh 21710L1 8800L2 45980L3 10000L4 47890Lt1 15620Lt2 22400

Table 2 Natural frequencies of the proposed human model andexperiment

ModeNatural frequency (Hz)

Proposed model Experiment1 420 4202 750 7503 999 mdash4 1970 mdash5 2135 mdash


frequency is much greater than that of the higher-ordernatural frequency and it can be inferred that tting to therst natural frequency represents the dynamic characteris-tics more accurately In this sense it can be concluded thatthe optimized parameters of the proposed human model canadequately represent the dynamic characteristics of thehuman body

3 Study of Parameters in Human Linear Model

Understanding how parameters aect the dynamic char-acteristics of the human model is an important issue in themechanical approach to explaining the dynamic responses ofthe human body Such understanding also requires para-metric analysis of the human mathematical model which isachieved by varying the human parameters us we ana-lyzed the variations in the natural frequencies of the humanmodel according to changes in the parameters which mainlyaect the dynamic responses of the proposed model Inorder to conduct the parameter study we selected the in-clination angles of the human model including seat panbackrest and head angles and the translational and torsionalstiness values

31 Variations in Rotational Angles Parameter studies fo-cused on the angle of the seat pan backrest and head wereperformed in order to analyze the variations in naturalfrequencies caused by varying the sitting posture In thelinear humanmodel the angles of the seat pan backrest andhead were dened as θ10 θ20 and θ30 respectively eoptimized stiness and damping coecients were applied tothe parametric study and the natural frequencies of thehuman mathematical model were calculated when θ10 θ20and θ30 were changed in turn

Figure 5 shows the variations in natural frequency thatresulted from variations in the inclination angles In thisgure the rst second and third columns respectivelyindicate the resonance frequency with respect to the seatpan backrest and head angle and the rst to fth rowsrepresent the order of the natural frequency e y-axis scalewith respect to the inclination angle was determined basedon the sensitivity of the corresponding parameter

e changes in natural frequencies with respect to theseat pan angle are shown in the rst column of Figure 5e angle of the seat pan was changed in the range of 0deg to20deg at 2deg intervals We set the y-axis scale to plusmn5 of thecorresponding natural frequency because the variation inall natural frequencies is relatively small e black dottedline indicates the baseline value of the seat pan angle

1st mode 2nd mode 3rd mode

5th mode4th mode

Figure 3 Mode shapes of the ve-degree-of-freedom humanmodel (solid line initial conguration dashed line mode shape congurationblack color head blue color trunk red color lower body)

Table 3 Initial and optimized values of the stiness and dampingcoecients

Parameter Parameter Initial value Optimized value

Stiness(kNm kNmrad)

k1 k2 8000 6660k3 k4 10000 9554kt1 200 142kt2 100 112

Damping coecient(kNsm kNmsrad)

c1 c2 150 089c3 c4 150 094ct1 030 030ct2 020 020

2000

1

2

STH

tran

smiss

ibili

ty 3

4

5 10Frequency (Hz)

15

ExperimentProposed human model

Figure 4 STH transmissibility of experimental results [30] and theve-degree-of-freedom human linear model calculated using theoptimized stiness and damping coecients


As can be seen the natural frequencies of the humanmathematical model are not significantly affected by thechange in the seat pan angle As the seat pan angle in-creases the natural frequencies excluding the fourth modetend to decrease as a whole but the variation values areless than 5 ese tendencies were also observed in theexperimental results Wang et al reported no statisticalsignificance between the variations in the seat pan angleand the variations in the first natural frequency of thehuman body [8]

e second column of Figure 5 shows the variations innatural frequencies caused by varying the backrest angle from80deg to 130deg at 10deg intervals e y-axis scale was set to plusmn30 ofthe corresponding order of the natural frequency To verify the

human mathematical model we performed a comparisonbetween the first natural frequencies of the human model andthe experiment e experimental natural frequency resultingfrom varying the backrest angle is shown in Figure 5 in thesecond column of the first row as the solid blue line in thiscase the foam thickness is 150mm [35] e fundamentalfrequency of the human model was reduced by approximately20 as the backrest angle increased from 80deg to 130deg and theexperimental results show a similar tendency From thistendency it is inferred that the human back muscles becomerelaxed as the backrest angle increases and that the firstnatural frequency decreases owing to the reduction in thestiffness of the human back [35] In addition the secondnatural frequency increases as the backrest angle increases and

72

76

74

78

f 2 (H

z)

104102

109896

f 3 (H

z)

205

20

195

19

f 4 (H

z)

22

215

21

2050 10Seat pan angle (degree)

20

f 5 (H

z)4

41

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

42

43

44 44

43

42

41

4

78

76

74

72

104102

109896

Head angle (degree)70 90 110 130

22

215

21

205

205

195

19

20

5

4

3

10

8

6

12

10

8

6

25

20

15

Backrest angle (degree)80 90 100 110 120 130


20Head angle (degree)

70 90 110 130Backrest angle (degree)

80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130

25

20

15

Figure 5 Natural frequencies of the five-degree-of-freedom linear model with respect to the seat pan backrest and head angle (dottedvertical green line baseline angle blue line in the graph in the second column of the first row experimental results [35] the top through thebottom rows represent the first through the fifth modes respectively)


the third natural frequency slightly decreases and then in-creasesere was no significant change in the fourth and fifthnatural frequencies according to the backrest angle changeerefore it can be expected that our human mathematicalmodel enables us to represent variations in the first naturalfrequency through comparisons with experimental data

e variations in the natural frequencies with respect tothe head angle were also calculated as shown in the thirdcolumn of Figure 5 e head angles were varied between 70degand 130deg at intervals of 10deg e y-axis scale was set to plusmn5 ofthe corresponding natural frequency As the head angle in-creased all natural frequencies had different tendencies butdid not change substantially Similar to the results of thechange in the seat pan angle the head angle seems to have onlya minor role in the natural frequencies of the human model

32 Effect of Translational and Torsional Stiffness To in-vestigate the effect of the translational and torsional springconstants we analyzed the change in the natural frequenciesaccording to the change in stiffness (Figure 6) e naturalfrequency of the human model was calculated by changing thebaseline stiffness values from 11000 to two times the baselinevaluee black dotted lines indicate the baseline stiffness values

e first natural frequency increases with an increase ink1 and the change in k2 does not have a significant effectAlso the first natural frequency slightly increases with anincrease in k3 and k4 when k3 and k4 are greater than thebaseline e variations in kt1 and kt2 which respectivelyrepresent the hip joint and headneck joint do not signif-icantly affect the first natural frequency ese tendenciescan also be confirmed from the configurations of the cor-responding mode shape It can be expected that the varia-tions in the k2 stiffness spring which is connected relativelyfar from the hip joint would not have a significant effect onthe first mode because the first mode shape is comprisedmainly of the vertical motion of the hip joint In addition therotational motions of the head and trunk are relatively smallcompared to the vertical motion of the hip joint thus thetorsional spring has a very small effect on variations in thefirst natural frequency In summary the fundamental fre-quency of the human model was sensitive to the changes ink1 k3 and k4 stiffness in particular the variations in k1stiffness play a major role in the first mode

e second natural frequency was sensitive to k3 k4 kt1and kt2 stiffness In particular the effects of varying k3 and k4are more significant e change in k3 stiffness has a minoreffect when the k4 value varies below the baseline value butthe second natural frequency is varied considerably owing toan increase in the k3 value when the k4 value is greater thanthe baseline Moreover it can be seen that the second naturalfrequency indicates a minor change with respect to thevariations in k1e dominant motion of the second mode isthe rotation of the trunk us the translational motion ofthe hip joint and the rotation of the head and thigh are notsignificant erefore the springs that affect the rotation ofthe trunk are k3 k4 and kt1 owing to its mode shape ethird mode is particularly influenced by the changes in thetranslational stiffness values k3 and k4 e stiffness values of

the k3 and k4 springs which connect the backrest and trunkplay an important role because the hip joint moves in thehorizontal direction e other stiffness values also haverelatively little influence on the third-order natural fre-quency It can be seen that kt2 and k2 are the most influentialparameters in the fourth and fifth modese configurationsof corresponding mode shapes support these findings andthe rotational motions of the head and lower body occursimultaneously in the fourth and fifth modes ese modeshave no impact on the variations in the hip joint torsionalspring kt1 or the translational springs k3 and k4

Interestingly the veering phenomenon was observedfor variations in k2 and kt2 in the fourth and fifth modes[36] Further the veering regions are changed owing to thevariations in k2 and kt2 Figure 7 shows the natural fre-quency loci of the fourth and fifth modes when k2 and kt2vary from 11000 to two times the baseline stiffness valueWe illustrate the natural frequency loci using a three-di-mensional graph in order to more efficiently describe howdifferent parameters cause modes to be influenced by thepresence of the veering phenomenon associated with multiparameters k2 and kt2 [37] Here the upper and lowerplanes indicate the fourth and fifth natural frequenciesrespectively It can be seen that the two planes approacheach other according to the variations in each parameterand then they veer away abruptly In the veering region aninterchange between the fourth and fifth eigenvectors alsooccurs and their mode shape subsequently changes epresence of the eigenvalue veering phenomenon in thefourth and fifth modes is of prime importance becausesmall variations in k2 and kt2 stiffness values can give rise tolarge changes in the eigenvectors within the veering regionIn this human mathematical model such veering phe-nomena play a key role in the design of seat and sus-pensions for a vehicle

In order to investigate the veering quantitativelya veering index value was calculated from the combinationof the modal dependence factor (MDF) and the cross-sensitivity quotient (CSQ) [38] as follows

VIij MDFij times CSQij times MDFji (14)

e MDF was also defined as

MDFij zϕiza( 1113857

TMϕj1113966 11139672

zϕiza( 1113857TM zϕiza( 1113857

(15)

where ϕj and ϕj are mass-normalized eigenvectors andM isthe stiffness matrix e eigenvector sensitivity with respectto parameter a was approximated by

zϕi

zaasymp minus

ϕTi (zMzδ)ϕi

2+

αij

Δλij

ϕi (16)

where Δλij λj minus λi and λi denotes the eigenvalue associ-ated with i-th mode e modal coupling αij was defined as

αij ϕTj

zKzaminus λi

zMza

1113888 1113889ϕi (17)

where K is the stiffness matrix e CSQ is also given by


CSQij α2ij

α2ij + Δσji21113872 11138732 (18)

where Δσji σj minus σi and σi indicates the eigenvalue sen-sitivity It can be written as

σi zλi

za ϕTi

zMzaminus λi

zKza

1113888 1113889ϕi (19)

Figure 8 shows the veering indices with respect to k2 andkt2 stiffness A veering index approaching 1 indicates that theeffect of the veering phenomenon is relatively large us itcan be expected that the folded area of the eigenvalue planein Figure 7 represents the larger veering indices in Figure 8

Further the veering intensity band changes with respect tok2 and kt2 stiffness this demonstrates that the eigenvaluecurvature of the fourth and fifth modes is changed whenstiffness values vary erefore the natural frequency lociand veering index are considered simultaneously for betterunderstanding of this veering phenomenon

4 Dynamic Simulation of NonlinearFive-Degree-of-Freedom Human Model

We carried out dynamic analysis of a five-degree-of-freedomnonlinear human model in the frequency domain to in-vestigate its nonlinearity Because nonlinear dynamic sys-tems may have multiple responses at the same excitation

51st mode

Unit Hz

3rd mode

4th mode

5th mode

2nd mode

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

210210210k1(k1)ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k3(k3)ref kt1(kt1)ref

210210210k1(k1)ref k3(k3)ref kt1(kt1)ref




0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

Figure 6 Natural frequency of the human model with respect to the translational and torsional stiffness values (dotted black vertical andhorizontal line baseline values of the stiffness the top through the bottom rows respectively represent the first through the fifth modes)


frequency steady-state responses are needed to reach thesolution for the frequency response function We also usedthe optimized parameters of the human model in theanalysis of the linear model For the nonlinear humanmodel we applied a nonlinear force-displacement re-lationship to the translational springs to more accuratelyreect the actual deformation behavior of seat foam andtissue When the deection of the translational spring islarger than the initial length of the corresponding spring itmeans that the connection between the human body and theseat would be physically removed erefore we consideredthat the tension force of the translational spring was set tozero when δi ge δi0 Here the linear spring model was appliedto k3 and k4 owing to horizontal constraints

In order to compute the frequency response function ofthe ve-degree-of-freedom nonlinear model near the fun-damental frequency the base oor was excited by harmonicexcitation within a frequency range of 01Hz to 5Hz atintervals of 01Hz To extract the steady-state responses ofthe human model a sucient excitation time must be taken

into account to ensure steady-state responses we set thesimulation time at each frequency step to 50 seconds eexcitation frequency was also increased and then decreaseduniformly using 01Hz intervals e displacement excita-tion was applied to the base of the nonlinear human modeland the amplitude of the excitation was changed in a rangefrom 3mm to 12mm in 3mm increments to analyze thedynamic characteristics of the human model with respect tothe amplitude of the excitation displacement

Figure 9 displays the translational and angular dis-placements with respect to the excitation frequency andamplitude in the frequency domain e amplitudes of thetranslational and angular displacements increase as theexcitation amplitude increases It can be also conrmed thatthe rst natural frequencies of each generalized coordinatehave slightly dierent values For example the locations ofthe maximum amplitude for the vertical displacements ofthe hip joint zh and the angular displacements of the headθ3 occur at 40Hz and 43Hz respectively ere is a minordierence between the rst natural frequency of the non-linear model and that of the linear model (f1 of linear model 42Hz) and it may occur owing to its nonlinearity

Importantly it can be observed that a frequency soft-ening phenomenon occurs in which the rst natural fre-quency decreases with the increase in the excitationamplitude e fundamental frequency of the vertical dis-placements of the hip joint and the angular displacements ofthe head was respectively reduced by approximately 15and 14 while the excitation amplitude increased from3mm to 12mm e rst natural frequency exhibiteda minor variation as the excitation amplitude increased from3mm to 6mm e fundamental frequencies in the trans-lational and angular displacements were rapidly decreasedwhen the excitation amplitude exceeded 6mm In additionthe jump phenomenon (in which the amplitude dramaticallychanges) becomes clear in the high excitation amplitude Asan example study Manseld et al reported that the fun-damental frequency of experimental subjects decreased by222 from 54Hz to 42Hz based on median data andpresumed that this frequency shifting would be caused byvarious complex causes such as muscle and tissue responses[11] is phenomenon is more clearly observed in thefrequency response functions of the rotational coordinatesIn order to describe the multiple amplitudes at the sameexcitation frequency the frequency response functions inthe overlapped range are given using a zoom-in graph asshown in Figure 10 the amplitudes were calculated using001Hz intervals

Figure 10 shows STH transmissibility data from thelinear and nonlinear humanmodels and the experimenteresults of the linear model and experimental data are thesame as in Figure 4 and are replotted in Figure 10 forcomparison with the frequency response of the nonlinearmodel We also calculated the STH transmissibility of thenonlinear human model from the vertical displacements ofthe head at the center of gravity and provide the nonlinearhuman model results associated with an excitation ampli-tude of 3mm for comparison with those from the linearmodel Further the STH transmissibility of the nonlinear

1k2(k2)ref

2

15

k t2

(kt2

) ref

1

05

05 15 20

02

04

06

08

1

Veer

ing

inde

x

Figure 8 Veering index of the fourth and fth modes with respectto k2 and kt2

2

1

0 0

1

2

Nat

ural

freq

uenc

y (H

z)

kt2 (kt2 )ref k2(k2)ref

5

10

15

20

25

30

Figure 7 Mode veering between the fourth and fth modes forvariations in k2 and kt2 (lower plane fourth natural frequenciesupper plane fth natural frequencies red line natural frequencyloci when kt2 varies with baseline value of k2 blue line naturalfrequency loci with respect to k2 with baseline value of kt2)


human model was calculated in a range of 0 to 20Hz atfrequency intervals of 01Hz

It can be seen that the STH transmissibility of thenonlinear model is in good agreement with that of the linearmodel e maximum amplitude of the nonlinear model isslightly lower than the amplitude of the experimental dataand linear human model is illustrates the fact that if the

amplitude of the vertical excitation is relatively small thelinear human model could sufficiently represent frequencyresponse function However it is more difficult to expressthe frequency response characteristics of the human bodyusing the linear human model alone for large excitationamplitudes that cause shifting of the first natural frequencyand thus the necessity of the nonlinear model becomes moreimportant Further the nonlinear model enables one todescribe rapid changes in amplitude such as those caused bythe jump phenomenon erefore it could be expected thatthe nonlinear model is more suitable for representing thevibrational characteristics of a seated human subjected towhole-body vibrations generated from the road in which theprofiles are changed from a smooth to a rough surface andvice versa

5 Conclusions

In this study we first derived the equations of motion fora human model e determination of the human param-eters was performed in three steps e inertial and geo-metrical parameters were selected on the basis ofanthropometry reference data comprised of measurableproperties and the stiffness and damping coefficients wereextracted from the experimental STH results using theoptimization process e mode shapes were also obtainedusing the mass and stiffness matrix of the linearized model

Am

plitu

de o

f zh (

mm

)

0

10

20

30

Am

plitu

de o

f θ3 (

degr

ee)

0

1

2

3

Excitation frequency (Hz)0 1 2 3 4 5

3 325 350

1

2

3

3 325 3518

20

22

24

26

Figure 9 Amplitude of the vertical displacement of the hip joint and the angular displacement of the head with four different excitationamplitudes using the nonlinear human model (the circle and asterisk symbols indicate the amplitude values with increasing and decreasingexcitation frequency respectively)

Excitation frequency (Hz)

4

3

STH

tran

smiss

ibili

ty

2

1

00 5 10 15 20

ExperimentLinear modelNonlinear model

Figure 10 STH transmissibility of the linear and nonlinear pro-posed mathematical model and the experimental results [30](nonlinear model excitation amplitude of 3mm)


In addition we analyzed the variations in the naturalfrequencies of the linear human model when human pa-rameters were varied For this parametric study the in-clination angle and stiffness values were considered as theprominent parameters in the dynamic characteristics of thehuman model As a result the first natural frequency is mostsensitive to the backrest angle represented by spring stiff-ness value k1e variations in natural frequencies accordingto the change in stiffness values were compared with theconfigurations of the corresponding mode shape Here themode veering occurs between the loci of the fourth and fifthnatural frequencies for the variations in k2 and kt2

e frequency response functions of the nonlinear hu-man model were presented using the steady-state amplitudeof translational and angular displacements caused by har-monic base excitation We note that frequency shifting wasobserved in the first mode and various studies have reportedsimilar phenomenon us the proposed human modelcould be reasonably expected to accurately exhibit the dy-namic response characteristics of a seated human

Appendix

Appendix A

e elements of the matrix in Equation (5) are

A11 m1 + m2 + m3

A13 minus12m1Lth sin θ1

A14 minus12Ltr m2 + 2m3( 1113857sin θ2

A15 minus12m3Lh

euroθ3 sin θ3

A22 m1 + m2 + m3

A23 12m1Lth cos θ1

A24 12Ltr m2 + 2m3( 1113857cos θ2

A25 12m3Lh cos θ3

A33 J1 +14m1L

2th

A44 J2 +14m2L

2tr + m3L

2tr

A45 12m3LtrLh cos θ3 minus θ2( 1113857

A55 J3 +14m3L

2h

B1 12m1Lth

_θ21 cos θ11113874 1113875 +

12Ltr m2 + 2m3( 1113857 _θ

22 cos θ21113874 1113875

+12m3Lh

_θ23 cos θ31113874 1113875minus c3

_δ3 + c4_δ4 + k3δ3 + k4δ41113872 1113873

middot sin θ20

B2 12m1Lth

_θ21 sin θ1 +

12Ltr m2 + 2m3( 1113857 _θ

22 sin θ2

+12m3Lh

_θ23 sin θ3 minus c1

_δ1 minus c2_δ2 minus k1δ1 minus k2δ2

+ c3_δ3 + c4

_δ4 + k3δ3 + k4δ41113872 1113873cos θ20 minus m1 + m2 + m3( 1113857g

B3 minus c1_δ1 + k1δ11113872 1113873

12Lt1 sin θ1 + L1 cos θ11113874 1113875

minus c2_δ2 + k2δ21113872 1113873

12Lt1 sin θ1 + L2 cos θ11113874 1113875 + ct1

_δt1

+ kt1δt1 minus12m1gLth cos θ1

B4 12m3LtrLh

_θ23 sin θ3 minus θ2( 1113857minus c3

_δ3 + k3δ31113872 1113873

middot12Lt2 sin θ2 minus θ20( 1113857minusL3 cos θ2 minus θ20( 11138571113882 1113883minus c4

_δ4 + k4δ41113872 1113873

middot12Lt2 sin θ2 minus θ20( 1113857minusL4 cos θ2 minus θ20( 11138571113882 1113883minus ct1

_δt1

+ ct2_δt2 minus kt1δt1 + kt2δt2 minus

12m2 + m31113874 1113875gLtr cos θ2

B5 minus12m3LtrLh

_θ22 sin θ3 minus θ2( 1113857minus ct2

_δt2 minus kt2δt2

minus12m3gLh cos θ3 (A1)

where Li is the distance from the hip joint to the positionwhere the i-th translational spring and damper are con-nected Lt1 and Lt2 denote the thickness values of the lowerbody and trunk respectively e subscript 0 of θi0 indicatesthe initial inclination angle Moreover the translational andtorsional spring displacements are given below

δ1 zh minusLt1

2cos θ1 + L1 sin θ1 minus zb minus δ10

δ2 zh minusLt1


δ3 xh sin θ20 minus zh cos θ20 minus L3 sin θ2 minus θ20( 1113857

minusLt2

2cos θ2 minus θ20( 1113857minusxs3 sin θ20 + zs3 cos θ20 minus δ30


minusLt2


δt1 θ2 minus θ1 minus δt10


(A2)


where xs3 zs3 xs4 and zs4 are the connected locationsbetween the fixed backrest of the seat and the trunk of thehuman mathematical model δi0 and δti0 denote the initiallength of the translational springs and the initial angle of thetorsional springs respectively

Appendix B

e elements of the mass damping and stiffness matrix ofthe five-degree-of-freedom linear model are given byM11 m1 + m2 + m3

M13 minus12m1Lth sin θ10

M14 minus12

m2 + 2m3( 1113857Ltr sin θ20

M15 minus12m3Lh sin θ30

M22 m1 + m2 + m3

M23 12m1Lth cos θ10

M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr

M45 12m3LhLtr cos θ20 minus θ30( 1113857

M55 J3 +14m3L

2h

C11 c3 + c4( 1113857sin2 θ20 C12 minus c3 + c4( 1113857sin θ20 cos θ20

C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12

2 c1L1 + c2L2( 1113857cos θ10 + c1 + c2( 1113857Lt1 sin θ101113864 1113865

C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L

221113872 1113873cos2 θ10 + c1L1 + c2L2( 1113857Lt1 sin θ10 cos θ10

+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20

K12 minus k3 + k4( 1113857sin θ20 cos θ20

K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12

2 k1L1 + k2L2( 1113857cos θ10 + k1 + k2( 1113857Lt1 sin θ101113864 1113865

K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L

221113872 1113873cos2 θ10 + k1L1 + k2L2( 1113857Lt1 sin θ10 cos θ10

+14

4kt1 minus 2m1gLth sin θ10 + k1 + k2( 1113857L2t1sin

2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2

K55 kt2 minus12m3gLh sin θ30

Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability

e length and thickness data of the Korean male are ac-cessible at httpssizekoreakrin the section of 5th mea-surement survey

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National ResearchFoundation of Korea (NRF) grant funded by the Koreagovernment (Ministry of Education) (No 2015R1D1A1A01060582)

References

[1] T E Fairley and M J Griffin ldquoe apparent mass of theseated human body vertical vibrationrdquo Journal of Bio-mechanics vol 22 no 2 pp 81ndash94 1989

[2] Z Zhou and M J Griffin ldquoResponse of the seated humanbody to whole-body vertical vibration biodynamic responsesto sinusoidal and random vibrationrdquo Ergonomics vol 57no 5 pp 693ndash713 2014

[3] E Kim M Fard and K Kato ldquoCharacterisation of the hu-man-seat coupling in response to vibrationrdquo Ergonomicsvol 60 no 8 pp 1085ndash1100 2017


[4] M Tarabini S Solbiati G Moschioni B Saggin andD Scaccabarozzi ldquoAnalysis of non-linear response of thehuman body to vertical whole-body vibrationrdquo Ergonomicsvol 57 no 11 pp 1711ndash1723 2014

[5] S D Smith J A Smith and D R Bowden ldquoTransmissioncharacteristics of suspension seats in multi-axis vibrationenvironmentsrdquo International Journal of Industrial Ergo-nomics vol 38 no 5-6 pp 434ndash446 2008

[6] S Rakheja I Stiharu H Zhang and P E Boileau ldquoSeatedoccupant interactions with seat backrest and pan and bio-dynamic responses under vertical vibrationrdquo Journal of Soundand Vibration vol 298 no 3 pp 651ndash671 2006

[7] J L Coyte D Stirling H Du and M Ros ldquoSeated whole-body vibration analysis technologies and modeling a sur-veyrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 46 no 6 pp 725ndash739 2016

[8] W Wang S Rakheja and P E Boileau ldquoEffects of sittingpostures on biodynamic response of seated occupants undervertical vibrationrdquo International Journal of Industrial Ergo-nomics vol 34 no 4 pp 289ndash306 2004

[9] I Hermanns N Raffler R P Ellegast S Fischer andB Gores ldquoSimultaneous field measuring method of vibrationand body posture for assessment of seated occupationaldriving tasksrdquo International Journal of Industrial Ergonomicsvol 38 no 3 pp 255ndash263 2008

[10] S Rahmatalla and J DeShaw ldquoPredictive discomfort of non-neutral headndashneck postures in forendashaft whole-body vibra-tionrdquo Ergonomics vol 54 no 3 pp 263ndash272 2011

[11] N J Mansfield and M J Griffin ldquoNon-linearities in apparentmass and transmissibility during exposure to whole-bodyvertical vibrationrdquo Journal of Biomechanics vol 33 no 8pp 933ndash941 2000

[12] Y Qiu andM J Griffin ldquoTransmission of forendashaft vibration toa car seat using field tests and laboratory simulationrdquo Journalof Sound and Vibration vol 264 no 1 pp 135ndash155 2003

[13] S Tufano and M J Griffin ldquoNonlinearity in the verticaltransmissibility of seating the role of the human body ap-parent mass and seat dynamic stiffnessrdquo Vehicle SystemDynamics vol 51 no 1 pp 122ndash138 2013

[14] N A Vavalle D P Moreno A C Rhyne J D Stitzel andF S Gayzik ldquoLateral impact validation of a geometricallyaccurate full body finite element model for blunt injurypredictionrdquo Annals of Biomedical Engineering vol 41 no 3pp 497ndash512 2013

[15] A Siefert S Pankoke and H P Wolfel ldquoVirtual optimisationof car passenger seats simulation of static and dynamic effectson driversrsquo seating comfortrdquo International Journal of In-dustrial Ergonomics vol 38 no 5 pp 410ndash424 2008

[16] K Butz C Spurlock R Roy et al ldquoDevelopment of theCAVEMAN human body model validation of lower ex-tremity sub-injurious response to vertical accelerative load-ingrdquo Stapp Car Crash Journal vol 61 pp 175ndash209 2017

[17] L Wei and J Griffin ldquoe prediction of seat transmissibilityfrom measures of seat impedancerdquo Journal of Sound andVibration vol 214 no 1 pp 121ndash137 1998

[18] S B Choi and Y M Han ldquoVibration control of electro-rheological seat suspension with human-body model usingsliding mode controlrdquo Journal of Sound and Vibrationvol 303 no 1-2 pp 391ndash404 2007

[19] X X Bai S X XuW Cheng and L J Qian ldquoOn 4-degree-of-freedom biodynamic models of seated occupants lumped-parameter modelingrdquo Journal of Sound and Vibrationvol 402 pp 122ndash141 2017

[20] Y Matsumoto and M J Griffin ldquoModelling the dynamicmechanisms associated with the principal resonance of theseated human bodyrdquo Clinical Biomechanics vol 16pp S31ndashS44 2001

[21] Y Cho and Y S Yoon ldquoBiomechanical model of human on seatwith backrest for evaluating ride qualityrdquo International Journalof Industrial Ergonomics vol 27 no 5 pp 331ndash345 2001

[22] S K Kim S W White A K Bajaj and P Davies ldquoSimplifiedmodels of the vibration of mannequins in car seatsrdquo Journal ofSound and Vibration vol 264 no 1 pp 49ndash90 2003

[23] G Zheng Y Qiu and M J Griffin ldquoAn analytic model of thein-line and cross-axis apparent mass of the seated humanbody exposed to vertical vibration with and without a back-restrdquo Journal of Sound and Vibration vol 330 no 26pp 6509ndash6525 2011

[24] N Mohajer H Abdi S Nahavandi and K Nelson ldquoDi-rectional and sectional ride comfort estimation using anintegrated human biomechanical-seat foam modelrdquo Journalof Sound and Vibration vol 403 pp 38ndash58 2017

[25] BS 6841Measurement and Evaluation of Human Exposure toWhole-Body Mechanical Vibration and Repeated ShockBritish Standard London 1987

[26] M J GriffinHandbook of Human Vibration Academic PressCambridge MA USA 1990

[27] R K Ippili P Davies A K Bajaj and L HagenmeyerldquoNonlinear multi-body dynamic modeling of seatndashoccupantsystem with polyurethane seat and H-point predictionrdquo In-ternational Journal of Industrial Ergonomics vol 38 no 5pp 368ndash383 2008

[28] G Joshi A K Bajaj and P Davies ldquoWhole-body vibratoryresponse study using a nonlinear multi-body model of seat-occupant system with viscoelastic flexible polyurethanefoamrdquo Industrial Health vol 48 no 5 pp 663ndash674 2010

[29] M L Ju H Jmal R Dupuis and E Aubry ldquoVisco-hypere-lastic constitutive model for modeling the quasi-static be-havior of polyurethane foam in large deformationrdquo PolymerEngineering and Science vol 55 no 8 pp 1795ndash1804 2015

[30] C C Liang and C F Chiang ldquoModeling of a seated humanbody exposed to vertical vibrations in various automotiveposturesrdquo Industrial Health vol 46 no 2 pp 125ndash137 2008

[31] S J Park S C Park J H Kim and C B Kim ldquoBiomechanicalparameters on body segments of Korean adultsrdquo InternationalJournal of Industrial Ergonomics vol 23 no 1 pp 23ndash311999

[32] Online database of the size of Korean body httpsizekoreakr

[33] M M Verver R De Lange J F A M van Hoof andJ S Wismans ldquoAspects of seat modelling for seating comfortanalysisrdquo Applied Ergonomics vol 36 no 1 pp 33ndash42 2005

[34] D M Brienza P E Karg and C E Brubaker ldquoSeat cushiondesign for elderly wheelchair users based on minimization ofsoft tissue deformation using stiffness and pressure mea-surementsrdquo IEEE Transactions on Rehabilitation Engineeringvol 4 no 2 pp 320ndash327 1996

[35] M G Toward andM J Griffin ldquoApparent mass of the humanbody in the vertical direction effect of seat backrestrdquo Journalof Sound and Vibration vol 327 no 3 pp 657ndash669 2009

[36] A W Leissa ldquoOn a curve veering aberrationrdquo Zeitschrift furAngewandte Mathematik und Physik ZAMP vol 25 no 1pp 99ndash111 1974

[37] A Gallina L Pichler and T Uhl ldquoEnhanced meta-modellingtechnique for analysis of mode crossing mode veering andmode coalescence in structural dynamicsrdquo Mechanical Sys-tems and Signal Processing vol 26 no 7 pp 2297ndash2312 2011


[38] J L du Bois S Adhikari and N A Lieven ldquoOn the quan-tification of eigenvalue curve veering a veering indexrdquoJournal of Applied Mechanics vol 78 no 4 article 0410072011


International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

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To overcome the disadvantages of finite element modelslumped parameter models consisting of masses springs anddampers have been widely emphasized in various studies Weiand Griffin [17] utilized single-degree- and two-degree-of-freedom human models to mathematically predict seattransmissibility ey also revealed that the two-degree-of-freedom provided the better fitness to the experimental resultsthan the single-degree-of-freedom Choi andHan [18] used thevertical seat-human model that calculated the vibrationalperformance of semiactive seat suspensions quantitatively withSEAT values and vibration dose value (VDV) Bai et al [19]analyzed the dynamic characteristics of a four-degree-of-freedom biodynamic model with various structural configu-rations to identify the best configuration An improved genericalgorithmwas considered to determine the human parametersAlthough these human models that take only translationalmotions into account can be used to describe frequency re-sponses of occupants exposed to whole-body vibrations theycannot represent the swingmotions of human segments owingto the absence of rotational degrees-of-freedom erefore inorder to more accurately grasp and analyze the dynamicmovements of human segments it is essential to developa biodynamic human model that considers rotational motions

Lumped human models with rotational degrees-of-free-dom have been studied extensively to interpret fore-and-aftand pitch movements in the sagittal plane Matsumoto andGriffin [20] developed four- and five-degree-of-freedomhuman models including the viscera with rotational springsand dampers and they showed that the models having therotational degree-of-freedom represented the more reason-able descriptions for the dynamic characteristics of the seatedhuman body Cho and Yoon [21] presented nine-degree-of-freedom human model consisting of three rigid bodies andcompared with the single- two- and three-degree-of-free-dom models to verify the performance of the proposed nine-degree-of-freedom human model Further they also in-vestigated the effect of the seat backrest Kim et al [22]proposed a seven-degree-of-freedom seat-mannequin modelwhose parameters were obtained using indentation and swingexperimental tests the frequency response functions of theseat-mannequin model were presented and compared tothe experimental results Zheng et al [23] proposed seven-degree-of-freedom human model and showed analyticaland experimental apparent masses with and withouta backrest in the vertical and fore-and-aft directions Alsothe sensitive parameters were presented according to thepresence or absence of backrest Mohajer et al [24] pre-sented a human biomechanical model that consisted of 15rigid bodies e ride comfort was also estimated withrespect to road roughness vehicle speed and the weights ofthe subjects using the proposed human model

To develop the human model we used a Lagrangianformulation to derive the nonlinear equations of motion fora five-degree-of-freedom model e spring constants anddamping coefficients were extracted from experimental datain the literature using an optimization process e naturalfrequencies and mode shapes were also calculated from thelinearized human model In addition several parametricstudies were performed Finally we calculated the frequency

response curves of the nonlinear human model and thencompared those of the linear model nonlinear model andthe experiment

2 Nonlinear and LinearFive-Degree-of-Freedom Models

21 Model Description In order to investigate dynamiccharacteristics of a seated human we developed a lumpedparameter model consisting of masses dampers and springsIn the proposed model we considered three rigid bodiesmdashahead trunk and lower body (including the thighs and pelvis)e trunk and lower body are the suggested measurementpoints of vibrations in BS 6841 which is the standard forassessing human vibrations [25] Further the motions of thehead-neck segment are a prominent source of discomfort [10]because they may affect the vision of vehicle occupants andmay cause motion sickness Foot support was not consideredbecause the vibrations transmitted through foot supports arerelatively small compared to the vibrations affecting the trunkand lower body [26]

Based on these assumptions we developed a five-degree-of-freedom human model consisting of xh zh θ1 θ2 and θ3as shown in Figure 1 xh and zh respectively represent thehorizontal and vertical displacements of the hip joint withrespect to the basi-centric coordinate system while θ1 θ2and θ3 respectively represent the angular displacements ofthe lower body trunk and head around the y-axis Becausewe focus our interpretation on analyzing motions in therotational and vertical directions at the x-z plane (sagittalplane) dynamic behavior in the lateral direction was notconsidered e base motion zb is the input vertical dis-placement and velocity of the seat floor It was assumed thatno components overlapped and each segment was con-nected to a torsional spring and damper to represent therotational motion e translational springs were connectedto the surface of each component It is also assumed thateach human segment has uniform mass distribution ejoints of the human model were modelled using the pin jointcondition because the elongation and compression at therotational joints of an actual human body are very small Inother words the independent translational movements ofeach segment were not considered in our model

e combination of seat foam and human skin wasconsidered as the translational springs and dampers Allsprings assumed to be massless and frictionless were innonstretched and nonrotated conditions at the initial con-figuration Although seat foam has nonlinear viscoelasticproperties [27ndash29] it has been reported that such propertiesdo not significantly affect the transmissibility of human bodyvibrations in practice [13] It was also assumed that becausedeformations of the human back and buttock tissue aftersitting on a seat are sufficiently small their nonlinearity couldbe disregarded erefore considering the combination ofseat foam and human tissue as the linear translational springand damper provides a convenient and reasonable approachto developing this lumped parameter human model ebackrest was set to be parallel to the trunk of the humanmodel at the initial configuration










(1)


T 12sum3

i1mi _x2i + _z2i( ) +

12sum3

i1Ji _θ

2i (2)


V 12sum4

i1kiδ

2i +

12sum2

i1ktiδ

2ti +sum

3

i1migΔi (3)

D 12sum4

i1ci _δ

2i +

12sum2

i1cti _δ

2ti (4)


δ3

δ2δ1


g

Lt2

m3rsquo

k4

k3

c4

c3L3

(xh zh)

L4

Ltr

Lh

J3rsquo

kt2 ct2

03

02

01

m2

m1 J1 Lt1

c2c1z

xO

k2k1

L1

L2

Lth

kt1 ct1

zb

J2



A11 0 A13 A14 A15

A22 A23 A24 A25

A33 0 0

A44 A45

Sym A55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

euroxh

eurozh


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

B1

B2

B3

B4

B5

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(5)








M

M11 0 M13 M14 M15

M22 M23 M24 M25

M33 0 0M44 M45

Sym M55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C

C11 C12 0 C14 0C22 C23 C24 0

C33 C34 0C44 C45

Sym C55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K

K11 K12 0 K14 0K22 K23 K24 0

K33 K34 0K44 K45

Sym K55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cb 0 Cb2 Cb3 0 01113858 1113859T

Kb 0 Kb2 Kb2 0 01113858 1113859T (8)



jωCb + Kb( 1113857Zb(jω) (9)











E1 min w(2)1 f

ex1 minusf

cal1

11138681113868111386811138681113868

11138681113868111386811138681113868 + w(2)2 f

ex2 minusf

cal2

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875 (10)

where fex1 fex

2 fcal1 fcal


(2)1 w




E2 min w(3)1 D1 + w

(3)2 D21113872 1113873 (11)

where

D1 1113938

fb

faTex fi( 1113857

11138681113868111386811138681113868111386811138681113868minus Tcak fi( 1113857

11138681113868111386811138681113868111386811138681113868df

1113938fb

faTex fi( 1113857

11138681113868111386811138681113868111386811138681113868df

(12)

D2 max Tex


1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868 (13)



(3)1 w





Property Value

Mass (kg)m1 1049m2 3398m3 667


Length (mm)













5th mode4th mode




Stiness(kNm kNmrad)

k1 k2 8000 6660k3 k4 10000 9554kt1 200 142kt2 100 112


c1 c2 150 089c3 c4 150 094ct1 030 030ct2 020 020

2000

1

2

STH

tran

smiss

ibili

ty 3

4

5 10Frequency (Hz)

15







72

76

74

78

f 2 (H

z)

104102

109896

f 3 (H

z)

205

20

195

19

f 4 (H

z)

22

215

21


20

f 5 (H

z)4

41

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

42

43

44 44

43

42

41

4

78

76

74

72

104102

109896


22

215

21

205

205

195

19

20

5

4

3

10

8

6

12

10

8

6

25

20

15





80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130

25

20

15














TMϕj1113966 11139672

zϕiza( 1113857TM zϕiza( 1113857

(15)


zϕi

zaasymp minus

ϕTi (zMzδ)ϕi

2+

αij

Δλij

ϕi (16)


αij ϕTj

zKzaminus λi

zMza

1113888 1113889ϕi (17)



CSQij α2ij

α2ij + Δσji21113872 11138732 (18)


σi zλi

za ϕTi

zMzaminus λi

zKza

1113888 1113889ϕi (19)





51st mode

Unit Hz

3rd mode

4th mode

5th mode

2nd mode

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

210210210k1(k1)ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref






0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2









1k2(k2)ref

2

15

k t2

(kt2

) ref

1

05

05 15 20

02

04

06

08

1

Veer

ing

inde

x


2

1

0 0

1

2

Nat

ural

freq

uenc

y (H

z)


5

10

15

20

25

30






5 Conclusions


Am

plitu

de o

f zh (

mm

)

0

10

20

30

Am

plitu

de o

f θ3 (

degr

ee)

0

1

2

3


3 325 350

1

2

3

3 325 3518

20

22

24

26



4

3

STH

tran

smiss

ibili

ty

2

1

00 5 10 15 20






Appendix

Appendix A


A11 m1 + m2 + m3



A15 minus12m3Lh

euroθ3 sin θ3

A22 m1 + m2 + m3

A23 12m1Lth cos θ1

A24 12Ltr m2 + 2m3( 1113857cos θ2

A25 12m3Lh cos θ3

A33 J1 +14m1L

2th

A44 J2 +14m2L

2tr + m3L

2tr


A55 J3 +14m3L

2h

B1 12m1Lth

_θ21 cos θ11113874 1113875 +

12Ltr m2 + 2m3( 1113857 _θ

22 cos θ21113874 1113875

+12m3Lh

_θ23 cos θ31113874 1113875minus c3

_δ3 + c4_δ4 + k3δ3 + k4δ41113872 1113873

middot sin θ20

B2 12m1Lth

_θ21 sin θ1 +

12Ltr m2 + 2m3( 1113857 _θ

22 sin θ2

+12m3Lh



+ c3_δ3 + c4


B3 minus c1_δ1 + k1δ11113872 1113873

12Lt1 sin θ1 + L1 cos θ11113874 1113875

minus c2_δ2 + k2δ21113872 1113873

12Lt1 sin θ1 + L2 cos θ11113874 1113875 + ct1

_δt1


B4 12m3LtrLh


_δ3 + k3δ31113872 1113873


_δ4 + k4δ41113872 1113873


_δt1


12m2 + m31113874 1113875gLtr cos θ2

B5 minus12m3LtrLh


_δt2 minus kt2δt2



δ1 zh minusLt1


δ2 zh minusLt1



minusLt2



minusLt2




(A2)



Appendix B



M14 minus12

m2 + 2m3( 1113857Ltr sin θ20


M22 m1 + m2 + m3


M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr


M55 J3 +14m3L

2h


C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12


C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L


+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20


K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12


K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L


+14


2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2


Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability




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










(1)


T 12sum3

i1mi _x2i + _z2i( ) +

12sum3

i1Ji _θ

2i (2)


V 12sum4

i1kiδ

2i +

12sum2

i1ktiδ

2ti +sum

3

i1migΔi (3)

D 12sum4

i1ci _δ

2i +

12sum2

i1cti _δ

2ti (4)


δ3

δ2δ1


g

Lt2

m3rsquo

k4

k3

c4

c3L3

(xh zh)

L4

Ltr

Lh

J3rsquo

kt2 ct2

03

02

01

m2

m1 J1 Lt1

c2c1z

xO

k2k1

L1

L2

Lth

kt1 ct1

zb

J2



A11 0 A13 A14 A15

A22 A23 A24 A25

A33 0 0

A44 A45

Sym A55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

euroxh

eurozh


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

B1

B2

B3

B4

B5

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(5)








M

M11 0 M13 M14 M15

M22 M23 M24 M25

M33 0 0M44 M45

Sym M55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C

C11 C12 0 C14 0C22 C23 C24 0

C33 C34 0C44 C45

Sym C55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K

K11 K12 0 K14 0K22 K23 K24 0

K33 K34 0K44 K45

Sym K55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cb 0 Cb2 Cb3 0 01113858 1113859T

Kb 0 Kb2 Kb2 0 01113858 1113859T (8)



jωCb + Kb( 1113857Zb(jω) (9)











E1 min w(2)1 f

ex1 minusf

cal1

11138681113868111386811138681113868

11138681113868111386811138681113868 + w(2)2 f

ex2 minusf

cal2

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875 (10)

where fex1 fex

2 fcal1 fcal


(2)1 w




E2 min w(3)1 D1 + w

(3)2 D21113872 1113873 (11)

where

D1 1113938

fb

faTex fi( 1113857

11138681113868111386811138681113868111386811138681113868minus Tcak fi( 1113857

11138681113868111386811138681113868111386811138681113868df

1113938fb

faTex fi( 1113857

11138681113868111386811138681113868111386811138681113868df

(12)

D2 max Tex


1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868 (13)



(3)1 w





Property Value

Mass (kg)m1 1049m2 3398m3 667


Length (mm)













5th mode4th mode




Stiness(kNm kNmrad)

k1 k2 8000 6660k3 k4 10000 9554kt1 200 142kt2 100 112


c1 c2 150 089c3 c4 150 094ct1 030 030ct2 020 020

2000

1

2

STH

tran

smiss

ibili

ty 3

4

5 10Frequency (Hz)

15







72

76

74

78

f 2 (H

z)

104102

109896

f 3 (H

z)

205

20

195

19

f 4 (H

z)

22

215

21


20

f 5 (H

z)4

41

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

42

43

44 44

43

42

41

4

78

76

74

72

104102

109896


22

215

21

205

205

195

19

20

5

4

3

10

8

6

12

10

8

6

25

20

15





80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130

25

20

15














TMϕj1113966 11139672

zϕiza( 1113857TM zϕiza( 1113857

(15)


zϕi

zaasymp minus

ϕTi (zMzδ)ϕi

2+

αij

Δλij

ϕi (16)


αij ϕTj

zKzaminus λi

zMza

1113888 1113889ϕi (17)



CSQij α2ij

α2ij + Δσji21113872 11138732 (18)


σi zλi

za ϕTi

zMzaminus λi

zKza

1113888 1113889ϕi (19)





51st mode

Unit Hz

3rd mode

4th mode

5th mode

2nd mode

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

210210210k1(k1)ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref






0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2









1k2(k2)ref

2

15

k t2

(kt2

) ref

1

05

05 15 20

02

04

06

08

1

Veer

ing

inde

x


2

1

0 0

1

2

Nat

ural

freq

uenc

y (H

z)


5

10

15

20

25

30






5 Conclusions


Am

plitu

de o

f zh (

mm

)

0

10

20

30

Am

plitu

de o

f θ3 (

degr

ee)

0

1

2

3


3 325 350

1

2

3

3 325 3518

20

22

24

26



4

3

STH

tran

smiss

ibili

ty

2

1

00 5 10 15 20






Appendix

Appendix A


A11 m1 + m2 + m3



A15 minus12m3Lh

euroθ3 sin θ3

A22 m1 + m2 + m3

A23 12m1Lth cos θ1

A24 12Ltr m2 + 2m3( 1113857cos θ2

A25 12m3Lh cos θ3

A33 J1 +14m1L

2th

A44 J2 +14m2L

2tr + m3L

2tr


A55 J3 +14m3L

2h

B1 12m1Lth

_θ21 cos θ11113874 1113875 +

12Ltr m2 + 2m3( 1113857 _θ

22 cos θ21113874 1113875

+12m3Lh

_θ23 cos θ31113874 1113875minus c3

_δ3 + c4_δ4 + k3δ3 + k4δ41113872 1113873

middot sin θ20

B2 12m1Lth

_θ21 sin θ1 +

12Ltr m2 + 2m3( 1113857 _θ

22 sin θ2

+12m3Lh



+ c3_δ3 + c4


B3 minus c1_δ1 + k1δ11113872 1113873

12Lt1 sin θ1 + L1 cos θ11113874 1113875

minus c2_δ2 + k2δ21113872 1113873

12Lt1 sin θ1 + L2 cos θ11113874 1113875 + ct1

_δt1


B4 12m3LtrLh


_δ3 + k3δ31113872 1113873


_δ4 + k4δ41113872 1113873


_δt1


12m2 + m31113874 1113875gLtr cos θ2

B5 minus12m3LtrLh


_δt2 minus kt2δt2



δ1 zh minusLt1


δ2 zh minusLt1



minusLt2



minusLt2




(A2)



Appendix B



M14 minus12

m2 + 2m3( 1113857Ltr sin θ20


M22 m1 + m2 + m3


M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr


M55 J3 +14m3L

2h


C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12


C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L


+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20


K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12


K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L


+14


2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2


Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability




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


A11 0 A13 A14 A15

A22 A23 A24 A25

A33 0 0

A44 A45

Sym A55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

euroxh

eurozh


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

B1

B2

B3

B4

B5

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(5)








M

M11 0 M13 M14 M15

M22 M23 M24 M25

M33 0 0M44 M45

Sym M55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C

C11 C12 0 C14 0C22 C23 C24 0

C33 C34 0C44 C45

Sym C55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K

K11 K12 0 K14 0K22 K23 K24 0

K33 K34 0K44 K45

Sym K55

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cb 0 Cb2 Cb3 0 01113858 1113859T

Kb 0 Kb2 Kb2 0 01113858 1113859T (8)



jωCb + Kb( 1113857Zb(jω) (9)











E1 min w(2)1 f

ex1 minusf

cal1

11138681113868111386811138681113868

11138681113868111386811138681113868 + w(2)2 f

ex2 minusf

cal2

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875 (10)

where fex1 fex

2 fcal1 fcal


(2)1 w




E2 min w(3)1 D1 + w

(3)2 D21113872 1113873 (11)

where

D1 1113938

fb

faTex fi( 1113857

11138681113868111386811138681113868111386811138681113868minus Tcak fi( 1113857

11138681113868111386811138681113868111386811138681113868df

1113938fb

faTex fi( 1113857

11138681113868111386811138681113868111386811138681113868df

(12)

D2 max Tex


1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868 (13)



(3)1 w





Property Value

Mass (kg)m1 1049m2 3398m3 667


Length (mm)













5th mode4th mode




Stiness(kNm kNmrad)

k1 k2 8000 6660k3 k4 10000 9554kt1 200 142kt2 100 112


c1 c2 150 089c3 c4 150 094ct1 030 030ct2 020 020

2000

1

2

STH

tran

smiss

ibili

ty 3

4

5 10Frequency (Hz)

15







72

76

74

78

f 2 (H

z)

104102

109896

f 3 (H

z)

205

20

195

19

f 4 (H

z)

22

215

21


20

f 5 (H

z)4

41

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

42

43

44 44

43

42

41

4

78

76

74

72

104102

109896


22

215

21

205

205

195

19

20

5

4

3

10

8

6

12

10

8

6

25

20

15





80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130

25

20

15














TMϕj1113966 11139672

zϕiza( 1113857TM zϕiza( 1113857

(15)


zϕi

zaasymp minus

ϕTi (zMzδ)ϕi

2+

αij

Δλij

ϕi (16)


αij ϕTj

zKzaminus λi

zMza

1113888 1113889ϕi (17)



CSQij α2ij

α2ij + Δσji21113872 11138732 (18)


σi zλi

za ϕTi

zMzaminus λi

zKza

1113888 1113889ϕi (19)





51st mode

Unit Hz

3rd mode

4th mode

5th mode

2nd mode

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

210210210k1(k1)ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref






0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2









1k2(k2)ref

2

15

k t2

(kt2

) ref

1

05

05 15 20

02

04

06

08

1

Veer

ing

inde

x


2

1

0 0

1

2

Nat

ural

freq

uenc

y (H

z)


5

10

15

20

25

30






5 Conclusions


Am

plitu

de o

f zh (

mm

)

0

10

20

30

Am

plitu

de o

f θ3 (

degr

ee)

0

1

2

3


3 325 350

1

2

3

3 325 3518

20

22

24

26



4

3

STH

tran

smiss

ibili

ty

2

1

00 5 10 15 20






Appendix

Appendix A


A11 m1 + m2 + m3



A15 minus12m3Lh

euroθ3 sin θ3

A22 m1 + m2 + m3

A23 12m1Lth cos θ1

A24 12Ltr m2 + 2m3( 1113857cos θ2

A25 12m3Lh cos θ3

A33 J1 +14m1L

2th

A44 J2 +14m2L

2tr + m3L

2tr


A55 J3 +14m3L

2h

B1 12m1Lth

_θ21 cos θ11113874 1113875 +

12Ltr m2 + 2m3( 1113857 _θ

22 cos θ21113874 1113875

+12m3Lh

_θ23 cos θ31113874 1113875minus c3

_δ3 + c4_δ4 + k3δ3 + k4δ41113872 1113873

middot sin θ20

B2 12m1Lth

_θ21 sin θ1 +

12Ltr m2 + 2m3( 1113857 _θ

22 sin θ2

+12m3Lh



+ c3_δ3 + c4


B3 minus c1_δ1 + k1δ11113872 1113873

12Lt1 sin θ1 + L1 cos θ11113874 1113875

minus c2_δ2 + k2δ21113872 1113873

12Lt1 sin θ1 + L2 cos θ11113874 1113875 + ct1

_δt1


B4 12m3LtrLh


_δ3 + k3δ31113872 1113873


_δ4 + k4δ41113872 1113873


_δt1


12m2 + m31113874 1113875gLtr cos θ2

B5 minus12m3LtrLh


_δt2 minus kt2δt2



δ1 zh minusLt1


δ2 zh minusLt1



minusLt2



minusLt2




(A2)



Appendix B



M14 minus12

m2 + 2m3( 1113857Ltr sin θ20


M22 m1 + m2 + m3


M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr


M55 J3 +14m3L

2h


C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12


C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L


+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20


K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12


K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L


+14


2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2


Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability




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





E1 min w(2)1 f

ex1 minusf

cal1

11138681113868111386811138681113868

11138681113868111386811138681113868 + w(2)2 f

ex2 minusf

cal2

11138681113868111386811138681113868

111386811138681113868111386811138681113874 1113875 (10)

where fex1 fex

2 fcal1 fcal


(2)1 w




E2 min w(3)1 D1 + w

(3)2 D21113872 1113873 (11)

where

D1 1113938

fb

faTex fi( 1113857

11138681113868111386811138681113868111386811138681113868minus Tcak fi( 1113857

11138681113868111386811138681113868111386811138681113868df

1113938fb

faTex fi( 1113857

11138681113868111386811138681113868111386811138681113868df

(12)

D2 max Tex


1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868 (13)



(3)1 w





Property Value

Mass (kg)m1 1049m2 3398m3 667


Length (mm)













5th mode4th mode




Stiness(kNm kNmrad)

k1 k2 8000 6660k3 k4 10000 9554kt1 200 142kt2 100 112


c1 c2 150 089c3 c4 150 094ct1 030 030ct2 020 020

2000

1

2

STH

tran

smiss

ibili

ty 3

4

5 10Frequency (Hz)

15







72

76

74

78

f 2 (H

z)

104102

109896

f 3 (H

z)

205

20

195

19

f 4 (H

z)

22

215

21


20

f 5 (H

z)4

41

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

42

43

44 44

43

42

41

4

78

76

74

72

104102

109896


22

215

21

205

205

195

19

20

5

4

3

10

8

6

12

10

8

6

25

20

15





80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130

25

20

15














TMϕj1113966 11139672

zϕiza( 1113857TM zϕiza( 1113857

(15)


zϕi

zaasymp minus

ϕTi (zMzδ)ϕi

2+

αij

Δλij

ϕi (16)


αij ϕTj

zKzaminus λi

zMza

1113888 1113889ϕi (17)



CSQij α2ij

α2ij + Δσji21113872 11138732 (18)


σi zλi

za ϕTi

zMzaminus λi

zKza

1113888 1113889ϕi (19)





51st mode

Unit Hz

3rd mode

4th mode

5th mode

2nd mode

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

210210210k1(k1)ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref






0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2









1k2(k2)ref

2

15

k t2

(kt2

) ref

1

05

05 15 20

02

04

06

08

1

Veer

ing

inde

x


2

1

0 0

1

2

Nat

ural

freq

uenc

y (H

z)


5

10

15

20

25

30






5 Conclusions


Am

plitu

de o

f zh (

mm

)

0

10

20

30

Am

plitu

de o

f θ3 (

degr

ee)

0

1

2

3


3 325 350

1

2

3

3 325 3518

20

22

24

26



4

3

STH

tran

smiss

ibili

ty

2

1

00 5 10 15 20






Appendix

Appendix A


A11 m1 + m2 + m3



A15 minus12m3Lh

euroθ3 sin θ3

A22 m1 + m2 + m3

A23 12m1Lth cos θ1

A24 12Ltr m2 + 2m3( 1113857cos θ2

A25 12m3Lh cos θ3

A33 J1 +14m1L

2th

A44 J2 +14m2L

2tr + m3L

2tr


A55 J3 +14m3L

2h

B1 12m1Lth

_θ21 cos θ11113874 1113875 +

12Ltr m2 + 2m3( 1113857 _θ

22 cos θ21113874 1113875

+12m3Lh

_θ23 cos θ31113874 1113875minus c3

_δ3 + c4_δ4 + k3δ3 + k4δ41113872 1113873

middot sin θ20

B2 12m1Lth

_θ21 sin θ1 +

12Ltr m2 + 2m3( 1113857 _θ

22 sin θ2

+12m3Lh



+ c3_δ3 + c4


B3 minus c1_δ1 + k1δ11113872 1113873

12Lt1 sin θ1 + L1 cos θ11113874 1113875

minus c2_δ2 + k2δ21113872 1113873

12Lt1 sin θ1 + L2 cos θ11113874 1113875 + ct1

_δt1


B4 12m3LtrLh


_δ3 + k3δ31113872 1113873


_δ4 + k4δ41113872 1113873


_δt1


12m2 + m31113874 1113875gLtr cos θ2

B5 minus12m3LtrLh


_δt2 minus kt2δt2



δ1 zh minusLt1


δ2 zh minusLt1



minusLt2



minusLt2




(A2)



Appendix B



M14 minus12

m2 + 2m3( 1113857Ltr sin θ20


M22 m1 + m2 + m3


M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr


M55 J3 +14m3L

2h


C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12


C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L


+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20


K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12


K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L


+14


2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2


Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability




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









5th mode4th mode




Stiness(kNm kNmrad)

k1 k2 8000 6660k3 k4 10000 9554kt1 200 142kt2 100 112


c1 c2 150 089c3 c4 150 094ct1 030 030ct2 020 020

2000

1

2

STH

tran

smiss

ibili

ty 3

4

5 10Frequency (Hz)

15







72

76

74

78

f 2 (H

z)

104102

109896

f 3 (H

z)

205

20

195

19

f 4 (H

z)

22

215

21


20

f 5 (H

z)4

41

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

42

43

44 44

43

42

41

4

78

76

74

72

104102

109896


22

215

21

205

205

195

19

20

5

4

3

10

8

6

12

10

8

6

25

20

15





80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130

25

20

15














TMϕj1113966 11139672

zϕiza( 1113857TM zϕiza( 1113857

(15)


zϕi

zaasymp minus

ϕTi (zMzδ)ϕi

2+

αij

Δλij

ϕi (16)


αij ϕTj

zKzaminus λi

zMza

1113888 1113889ϕi (17)



CSQij α2ij

α2ij + Δσji21113872 11138732 (18)


σi zλi

za ϕTi

zMzaminus λi

zKza

1113888 1113889ϕi (19)





51st mode

Unit Hz

3rd mode

4th mode

5th mode

2nd mode

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

210210210k1(k1)ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref






0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2









1k2(k2)ref

2

15

k t2

(kt2

) ref

1

05

05 15 20

02

04

06

08

1

Veer

ing

inde

x


2

1

0 0

1

2

Nat

ural

freq

uenc

y (H

z)


5

10

15

20

25

30






5 Conclusions


Am

plitu

de o

f zh (

mm

)

0

10

20

30

Am

plitu

de o

f θ3 (

degr

ee)

0

1

2

3


3 325 350

1

2

3

3 325 3518

20

22

24

26



4

3

STH

tran

smiss

ibili

ty

2

1

00 5 10 15 20






Appendix

Appendix A


A11 m1 + m2 + m3



A15 minus12m3Lh

euroθ3 sin θ3

A22 m1 + m2 + m3

A23 12m1Lth cos θ1

A24 12Ltr m2 + 2m3( 1113857cos θ2

A25 12m3Lh cos θ3

A33 J1 +14m1L

2th

A44 J2 +14m2L

2tr + m3L

2tr


A55 J3 +14m3L

2h

B1 12m1Lth

_θ21 cos θ11113874 1113875 +

12Ltr m2 + 2m3( 1113857 _θ

22 cos θ21113874 1113875

+12m3Lh

_θ23 cos θ31113874 1113875minus c3

_δ3 + c4_δ4 + k3δ3 + k4δ41113872 1113873

middot sin θ20

B2 12m1Lth

_θ21 sin θ1 +

12Ltr m2 + 2m3( 1113857 _θ

22 sin θ2

+12m3Lh



+ c3_δ3 + c4


B3 minus c1_δ1 + k1δ11113872 1113873

12Lt1 sin θ1 + L1 cos θ11113874 1113875

minus c2_δ2 + k2δ21113872 1113873

12Lt1 sin θ1 + L2 cos θ11113874 1113875 + ct1

_δt1


B4 12m3LtrLh


_δ3 + k3δ31113872 1113873


_δ4 + k4δ41113872 1113873


_δt1


12m2 + m31113874 1113875gLtr cos θ2

B5 minus12m3LtrLh


_δt2 minus kt2δt2



δ1 zh minusLt1


δ2 zh minusLt1



minusLt2



minusLt2




(A2)



Appendix B



M14 minus12

m2 + 2m3( 1113857Ltr sin θ20


M22 m1 + m2 + m3


M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr


M55 J3 +14m3L

2h


C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12


C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L


+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20


K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12


K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L


+14


2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2


Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability




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





72

76

74

78

f 2 (H

z)

104102

109896

f 3 (H

z)

205

20

195

19

f 4 (H

z)

22

215

21


20

f 5 (H

z)4

41

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

f 2 (H

z)f 3

(Hz)

f 4 (H

z)f 5

(Hz)

f 1 (H

z)

42

43

44 44

43

42

41

4

78

76

74

72

104102

109896


22

215

21

205

205

195

19

20

5

4

3

10

8

6

12

10

8

6

25

20

15





80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130




80 90 100 110 120 130

25

20

15














TMϕj1113966 11139672

zϕiza( 1113857TM zϕiza( 1113857

(15)


zϕi

zaasymp minus

ϕTi (zMzδ)ϕi

2+

αij

Δλij

ϕi (16)


αij ϕTj

zKzaminus λi

zMza

1113888 1113889ϕi (17)



CSQij α2ij

α2ij + Δσji21113872 11138732 (18)


σi zλi

za ϕTi

zMzaminus λi

zKza

1113888 1113889ϕi (19)





51st mode

Unit Hz

3rd mode

4th mode

5th mode

2nd mode

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

210210210k1(k1)ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref






0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2









1k2(k2)ref

2

15

k t2

(kt2

) ref

1

05

05 15 20

02

04

06

08

1

Veer

ing

inde

x


2

1

0 0

1

2

Nat

ural

freq

uenc

y (H

z)


5

10

15

20

25

30






5 Conclusions


Am

plitu

de o

f zh (

mm

)

0

10

20

30

Am

plitu

de o

f θ3 (

degr

ee)

0

1

2

3


3 325 350

1

2

3

3 325 3518

20

22

24

26



4

3

STH

tran

smiss

ibili

ty

2

1

00 5 10 15 20






Appendix

Appendix A


A11 m1 + m2 + m3



A15 minus12m3Lh

euroθ3 sin θ3

A22 m1 + m2 + m3

A23 12m1Lth cos θ1

A24 12Ltr m2 + 2m3( 1113857cos θ2

A25 12m3Lh cos θ3

A33 J1 +14m1L

2th

A44 J2 +14m2L

2tr + m3L

2tr


A55 J3 +14m3L

2h

B1 12m1Lth

_θ21 cos θ11113874 1113875 +

12Ltr m2 + 2m3( 1113857 _θ

22 cos θ21113874 1113875

+12m3Lh

_θ23 cos θ31113874 1113875minus c3

_δ3 + c4_δ4 + k3δ3 + k4δ41113872 1113873

middot sin θ20

B2 12m1Lth

_θ21 sin θ1 +

12Ltr m2 + 2m3( 1113857 _θ

22 sin θ2

+12m3Lh



+ c3_δ3 + c4


B3 minus c1_δ1 + k1δ11113872 1113873

12Lt1 sin θ1 + L1 cos θ11113874 1113875

minus c2_δ2 + k2δ21113872 1113873

12Lt1 sin θ1 + L2 cos θ11113874 1113875 + ct1

_δt1


B4 12m3LtrLh


_δ3 + k3δ31113872 1113873


_δ4 + k4δ41113872 1113873


_δt1


12m2 + m31113874 1113875gLtr cos θ2

B5 minus12m3LtrLh


_δt2 minus kt2δt2



δ1 zh minusLt1


δ2 zh minusLt1



minusLt2



minusLt2




(A2)



Appendix B



M14 minus12

m2 + 2m3( 1113857Ltr sin θ20


M22 m1 + m2 + m3


M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr


M55 J3 +14m3L

2h


C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12


C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L


+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20


K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12


K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L


+14


2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2


Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability




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













TMϕj1113966 11139672

zϕiza( 1113857TM zϕiza( 1113857

(15)


zϕi

zaasymp minus

ϕTi (zMzδ)ϕi

2+

αij

Δλij

ϕi (16)


αij ϕTj

zKzaminus λi

zMza

1113888 1113889ϕi (17)



CSQij α2ij

α2ij + Δσji21113872 11138732 (18)


σi zλi

za ϕTi

zMzaminus λi

zKza

1113888 1113889ϕi (19)





51st mode

Unit Hz

3rd mode

4th mode

5th mode

2nd mode

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

210210210k1(k1)ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref






0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2









1k2(k2)ref

2

15

k t2

(kt2

) ref

1

05

05 15 20

02

04

06

08

1

Veer

ing

inde

x


2

1

0 0

1

2

Nat

ural

freq

uenc

y (H

z)


5

10

15

20

25

30






5 Conclusions


Am

plitu

de o

f zh (

mm

)

0

10

20

30

Am

plitu

de o

f θ3 (

degr

ee)

0

1

2

3


3 325 350

1

2

3

3 325 3518

20

22

24

26



4

3

STH

tran

smiss

ibili

ty

2

1

00 5 10 15 20






Appendix

Appendix A


A11 m1 + m2 + m3



A15 minus12m3Lh

euroθ3 sin θ3

A22 m1 + m2 + m3

A23 12m1Lth cos θ1

A24 12Ltr m2 + 2m3( 1113857cos θ2

A25 12m3Lh cos θ3

A33 J1 +14m1L

2th

A44 J2 +14m2L

2tr + m3L

2tr


A55 J3 +14m3L

2h

B1 12m1Lth

_θ21 cos θ11113874 1113875 +

12Ltr m2 + 2m3( 1113857 _θ

22 cos θ21113874 1113875

+12m3Lh

_θ23 cos θ31113874 1113875minus c3

_δ3 + c4_δ4 + k3δ3 + k4δ41113872 1113873

middot sin θ20

B2 12m1Lth

_θ21 sin θ1 +

12Ltr m2 + 2m3( 1113857 _θ

22 sin θ2

+12m3Lh



+ c3_δ3 + c4


B3 minus c1_δ1 + k1δ11113872 1113873

12Lt1 sin θ1 + L1 cos θ11113874 1113875

minus c2_δ2 + k2δ21113872 1113873

12Lt1 sin θ1 + L2 cos θ11113874 1113875 + ct1

_δt1


B4 12m3LtrLh


_δ3 + k3δ31113872 1113873


_δ4 + k4δ41113872 1113873


_δt1


12m2 + m31113874 1113875gLtr cos θ2

B5 minus12m3LtrLh


_δt2 minus kt2δt2



δ1 zh minusLt1


δ2 zh minusLt1



minusLt2



minusLt2




(A2)



Appendix B



M14 minus12

m2 + 2m3( 1113857Ltr sin θ20


M22 m1 + m2 + m3


M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr


M55 J3 +14m3L

2h


C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12


C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L


+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20


K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12


K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L


+14


2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2


Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability




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


CSQij α2ij

α2ij + Δσji21113872 11138732 (18)


σi zλi

za ϕTi

zMzaminus λi

zKza

1113888 1113889ϕi (19)





51st mode

Unit Hz

3rd mode

4th mode

5th mode

2nd mode

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

5

4

3

9876

12

10

8

24

20

16

26

22

18

210210210k1(k1)ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 2(k 2

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k 4(k 4

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref

k t2

(kt2

) ref






0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2









1k2(k2)ref

2

15

k t2

(kt2

) ref

1

05

05 15 20

02

04

06

08

1

Veer

ing

inde

x


2

1

0 0

1

2

Nat

ural

freq

uenc

y (H

z)


5

10

15

20

25

30






5 Conclusions


Am

plitu

de o

f zh (

mm

)

0

10

20

30

Am

plitu

de o

f θ3 (

degr

ee)

0

1

2

3


3 325 350

1

2

3

3 325 3518

20

22

24

26



4

3

STH

tran

smiss

ibili

ty

2

1

00 5 10 15 20






Appendix

Appendix A


A11 m1 + m2 + m3



A15 minus12m3Lh

euroθ3 sin θ3

A22 m1 + m2 + m3

A23 12m1Lth cos θ1

A24 12Ltr m2 + 2m3( 1113857cos θ2

A25 12m3Lh cos θ3

A33 J1 +14m1L

2th

A44 J2 +14m2L

2tr + m3L

2tr


A55 J3 +14m3L

2h

B1 12m1Lth

_θ21 cos θ11113874 1113875 +

12Ltr m2 + 2m3( 1113857 _θ

22 cos θ21113874 1113875

+12m3Lh

_θ23 cos θ31113874 1113875minus c3

_δ3 + c4_δ4 + k3δ3 + k4δ41113872 1113873

middot sin θ20

B2 12m1Lth

_θ21 sin θ1 +

12Ltr m2 + 2m3( 1113857 _θ

22 sin θ2

+12m3Lh



+ c3_δ3 + c4


B3 minus c1_δ1 + k1δ11113872 1113873

12Lt1 sin θ1 + L1 cos θ11113874 1113875

minus c2_δ2 + k2δ21113872 1113873

12Lt1 sin θ1 + L2 cos θ11113874 1113875 + ct1

_δt1


B4 12m3LtrLh


_δ3 + k3δ31113872 1113873


_δ4 + k4δ41113872 1113873


_δt1


12m2 + m31113874 1113875gLtr cos θ2

B5 minus12m3LtrLh


_δt2 minus kt2δt2



δ1 zh minusLt1


δ2 zh minusLt1



minusLt2



minusLt2




(A2)



Appendix B



M14 minus12

m2 + 2m3( 1113857Ltr sin θ20


M22 m1 + m2 + m3


M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr


M55 J3 +14m3L

2h


C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12


C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L


+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20


K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12


K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L


+14


2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2


Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability




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








1k2(k2)ref

2

15

k t2

(kt2

) ref

1

05

05 15 20

02

04

06

08

1

Veer

ing

inde

x


2

1

0 0

1

2

Nat

ural

freq

uenc

y (H

z)


5

10

15

20

25

30






5 Conclusions


Am

plitu

de o

f zh (

mm

)

0

10

20

30

Am

plitu

de o

f θ3 (

degr

ee)

0

1

2

3


3 325 350

1

2

3

3 325 3518

20

22

24

26



4

3

STH

tran

smiss

ibili

ty

2

1

00 5 10 15 20






Appendix

Appendix A


A11 m1 + m2 + m3



A15 minus12m3Lh

euroθ3 sin θ3

A22 m1 + m2 + m3

A23 12m1Lth cos θ1

A24 12Ltr m2 + 2m3( 1113857cos θ2

A25 12m3Lh cos θ3

A33 J1 +14m1L

2th

A44 J2 +14m2L

2tr + m3L

2tr


A55 J3 +14m3L

2h

B1 12m1Lth

_θ21 cos θ11113874 1113875 +

12Ltr m2 + 2m3( 1113857 _θ

22 cos θ21113874 1113875

+12m3Lh

_θ23 cos θ31113874 1113875minus c3

_δ3 + c4_δ4 + k3δ3 + k4δ41113872 1113873

middot sin θ20

B2 12m1Lth

_θ21 sin θ1 +

12Ltr m2 + 2m3( 1113857 _θ

22 sin θ2

+12m3Lh



+ c3_δ3 + c4


B3 minus c1_δ1 + k1δ11113872 1113873

12Lt1 sin θ1 + L1 cos θ11113874 1113875

minus c2_δ2 + k2δ21113872 1113873

12Lt1 sin θ1 + L2 cos θ11113874 1113875 + ct1

_δt1


B4 12m3LtrLh


_δ3 + k3δ31113872 1113873


_δ4 + k4δ41113872 1113873


_δt1


12m2 + m31113874 1113875gLtr cos θ2

B5 minus12m3LtrLh


_δt2 minus kt2δt2



δ1 zh minusLt1


δ2 zh minusLt1



minusLt2



minusLt2




(A2)



Appendix B



M14 minus12

m2 + 2m3( 1113857Ltr sin θ20


M22 m1 + m2 + m3


M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr


M55 J3 +14m3L

2h


C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12


C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L


+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20


K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12


K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L


+14


2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2


Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability




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





5 Conclusions


Am

plitu

de o

f zh (

mm

)

0

10

20

30

Am

plitu

de o

f θ3 (

degr

ee)

0

1

2

3


3 325 350

1

2

3

3 325 3518

20

22

24

26



4

3

STH

tran

smiss

ibili

ty

2

1

00 5 10 15 20






Appendix

Appendix A


A11 m1 + m2 + m3



A15 minus12m3Lh

euroθ3 sin θ3

A22 m1 + m2 + m3

A23 12m1Lth cos θ1

A24 12Ltr m2 + 2m3( 1113857cos θ2

A25 12m3Lh cos θ3

A33 J1 +14m1L

2th

A44 J2 +14m2L

2tr + m3L

2tr


A55 J3 +14m3L

2h

B1 12m1Lth

_θ21 cos θ11113874 1113875 +

12Ltr m2 + 2m3( 1113857 _θ

22 cos θ21113874 1113875

+12m3Lh

_θ23 cos θ31113874 1113875minus c3

_δ3 + c4_δ4 + k3δ3 + k4δ41113872 1113873

middot sin θ20

B2 12m1Lth

_θ21 sin θ1 +

12Ltr m2 + 2m3( 1113857 _θ

22 sin θ2

+12m3Lh



+ c3_δ3 + c4


B3 minus c1_δ1 + k1δ11113872 1113873

12Lt1 sin θ1 + L1 cos θ11113874 1113875

minus c2_δ2 + k2δ21113872 1113873

12Lt1 sin θ1 + L2 cos θ11113874 1113875 + ct1

_δt1


B4 12m3LtrLh


_δ3 + k3δ31113872 1113873


_δ4 + k4δ41113872 1113873


_δt1


12m2 + m31113874 1113875gLtr cos θ2

B5 minus12m3LtrLh


_δt2 minus kt2δt2



δ1 zh minusLt1


δ2 zh minusLt1



minusLt2



minusLt2




(A2)



Appendix B



M14 minus12

m2 + 2m3( 1113857Ltr sin θ20


M22 m1 + m2 + m3


M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr


M55 J3 +14m3L

2h


C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12


C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L


+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20


K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12


K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L


+14


2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2


Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability




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




Appendix

Appendix A


A11 m1 + m2 + m3



A15 minus12m3Lh

euroθ3 sin θ3

A22 m1 + m2 + m3

A23 12m1Lth cos θ1

A24 12Ltr m2 + 2m3( 1113857cos θ2

A25 12m3Lh cos θ3

A33 J1 +14m1L

2th

A44 J2 +14m2L

2tr + m3L

2tr


A55 J3 +14m3L

2h

B1 12m1Lth

_θ21 cos θ11113874 1113875 +

12Ltr m2 + 2m3( 1113857 _θ

22 cos θ21113874 1113875

+12m3Lh

_θ23 cos θ31113874 1113875minus c3

_δ3 + c4_δ4 + k3δ3 + k4δ41113872 1113873

middot sin θ20

B2 12m1Lth

_θ21 sin θ1 +

12Ltr m2 + 2m3( 1113857 _θ

22 sin θ2

+12m3Lh



+ c3_δ3 + c4


B3 minus c1_δ1 + k1δ11113872 1113873

12Lt1 sin θ1 + L1 cos θ11113874 1113875

minus c2_δ2 + k2δ21113872 1113873

12Lt1 sin θ1 + L2 cos θ11113874 1113875 + ct1

_δt1


B4 12m3LtrLh


_δ3 + k3δ31113872 1113873


_δ4 + k4δ41113872 1113873


_δt1


12m2 + m31113874 1113875gLtr cos θ2

B5 minus12m3LtrLh


_δt2 minus kt2δt2



δ1 zh minusLt1


δ2 zh minusLt1



minusLt2



minusLt2




(A2)



Appendix B



M14 minus12

m2 + 2m3( 1113857Ltr sin θ20


M22 m1 + m2 + m3


M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr


M55 J3 +14m3L

2h


C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12


C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L


+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20


K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12


K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L


+14


2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2


Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability




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



Appendix B



M14 minus12

m2 + 2m3( 1113857Ltr sin θ20


M22 m1 + m2 + m3


M24 12

m2 + 2m3( 1113857Ltr cos θ20

M25 12m3Lh cos θ30

M33 J1 +14m1L

2th

M44 J2 +14m2L

2tr + m3L

2tr


M55 J3 +14m3L

2h


C14 minus c3L3 + c4L3( 1113857sin θ20

C22 c1 + c2 + c3 + c4( 1113857cos2 θ20

C23 12


C24 c3L3 + c4L3( 1113857cos θ20

C33 c1L21 + c2L


+14

4ct1 + c1 + c2( 1113857L2t1sin

2 θ101113966 1113967

C34 minusct1

C44 ct1 + ct2 + c3L23 + c4L

24

C45 minusct2

C55 ct2

K11 k3 + k4( 1113857sin2 θ20


K14 minus k3L3 + k4L3( 1113857sin θ20

K22 k1 + k2 + k3 + k4( 1113857cos2 θ20

K23 12


K24 k3L3 + k4L3( 1113857cos θ20

K33 k1L21 + k2L


+14


2 θ101113966 1113967

K34 minuskt1

K44 kt1 + kt2 + k3L23 + k4L

24 minus

12

m2 + 2m3( 1113857gLtr sin θ20

K45 minuskt2


Cb2 minus c1 + c2( 1113857

Cb3 minus12


Kb2 minus k1 + k2( 1113857

Kb3 minus12


(B1)

Data Availability




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









































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




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

DevelopmentofaFive-Degree-of-FreedomSeatedHumanModel ...downloads.hindawi.com/journals/sv/2018/1649180.pdf · 2.2. Derivation of Nonlinear Equations of Motion. še nonlinear equations