Low-frequency vibration analysis of human body in semi-supine posture exposed to vertical excitation

European Journal of Mechanics / A Solids xxx (xxxx) xxx

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Low-frequency vibration analysis of human body in semi-supine posture exposed to vertical excitation Rajesh Govindan *, V.H. Saran, S.P. Harsha Vibration & Noise Control Lab, Mechanical and Industrial Engineering Department, IIT Roorkee, India

A R T I C L E I N F O

A B S T R A C T

Keywords: Semi-supine Apparent mass Thoracic cage Soft abdomen tissue Tornado diagram

In the new space vehicle architecture, the astronauts in semi-supine posture would be exposed to elevated vi­ bration level. The present study is focused to develop a multi-body model of human in a semi-supine posture to analyze their biodynamic responses due to vertical excitation. The study proposes a linear 15-DOF model of a semi-supine human in the sagittal plane. The human body model is represented by five body segments as headneck, torso-arm, pelvis, thigh, and legs. Each segment is interconnected with the translational and rotational spring-damper elements to simulate the relative motion. The tissues in contact with the rigid vibrating support are modeled using translational spring-damper elements. The model parameters were identified using optimi­ zation scheme by minimizing the least square error between normalized magnitude and phase of predicted and target vertical apparent mass in the frequency domain. The parameter sensitivity analysis shows that vertical stiffness and damping of underneath tissue of torso has substantial influence on peak modulus of apparent mass and corresponding resonance frequency. The modal analysis reveals that the principal resonance frequency for human body in semi-supine posture is greater as compared to seated and standing posture.

1. Introduction Humans are often exposed to vibration and shock while operating on land, sea, air and space vehicles. The transmission of these mechanical energy has a detrimental effect on the person’s health and on their physical and psychological performance (Adelstein et al., 2008; Johan­ ning, 2015; Narayanamoorthy and Huzur Saran, 2011). The human body is highly sensitive to low-frequency vibration. At a frequency below 2 Hz, the human body shows no significant responses as it moves like a simple mass with no relative internal motion while protective means can be used to reduce the vibration transmission at a frequency above 20 Hz (Clark et al., 1962). In order to protect the person from the harmful effect of low-frequency vibration, it is necessary to understand the dynamic behavior of the human body under such environment. Though experimental studies provide a great insight into human biodynamic, they are expensive and limited to the availability of experimental setup. Since, the last many decades, numerous biodynamic model, mostly for seated and standing posture have been developed to investigate the dynamic responses of the human body subjected to whole-body vibra­ tion. In these models, the human body is either idealized by discrete

spring-mass-dampers (i.e. lumped parameter model) (Gan et al., 2015; Matsumoto and Griffin, 2003; Muksian and Nash, 1976) or by rigid bodies connected by viscoelastic elements and muscle groups (multi-­ body model) (Kumbhar et al., 2013; Wang and Rahmatalla, 2013; Zheng et al., 2011) or using detailed finite-element-based schemes (Currie-­ Gregg and Carney, 2019; Govindan and Harsha, 2018; Zhang et al., 2015). Literature related to the effect of whole-body vibration on the human body in supine posture is scarce. In this framework, the effect of unidi­ rectional forced vibration on humans in sitting erect, sitting relaxed and semi-supine posture in form of a circumferential strain of chest, abdomen, pelvis, and thigh is determined (Clark et al., 1962). The maximum strain for semi-supine and sitting posture occurred at 6.7 Hz and between 4 and 6 Hz respectively. The mean strain of chest and abdomen at resonance frequency were quite higher for semi-supine posture whereas strain value was more than double for pelvis in sitting posture. Vogt et al. (1973) proposed a lumped parameter model comprising of three mass-spring-damper systems connected to an un-sprung mass to predict the impedance and phase value of the supine human body under sustained acceleration. The results revealed that the effective masses near driving point increases under sustained

* Corresponding author. E-mail addresses: [email protected] (R. Govindan), [email protected] (V.H. Saran), [email protected] (S.P. Harsha). https://doi.org/10.1016/j.euromechsol.2019.103906 Received 14 May 2019; Received in revised form 28 September 2019; Accepted 13 November 2019 Available online 18 November 2019 0997-7538/© 2019 Elsevier Masson SAS. All rights reserved.

Please cite this article as: Rajesh Govindan, European Journal of Mechanics / A Solids, https://doi.org/10.1016/j.euromechsol.2019.103906

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European Journal of Mechanics / A Solids xxx (xxxx) xxx

acceleration at the expense of upper masses. That is the tissues of the human body in contact with the support acts like pure mass under sus­ tained acceleration. The spring constant increases near driving point with þGx and damping coefficients depend on the mass and sustained acceleration. In another study, Vogt et al. (1978) proposed a multi-degree-of-freedom (DOF) model to predict the measured vibration transmissibility and mechanical impedance of supine humans exposed to a sinusoidal vibration and constant acceleration. The results exhibited that thorax, due to its anatomical configuration, responds differently from the rest of the body segments. The thorax showed lower vibration transmissibility compared to the abdomen and the impedance curve for the thorax showed one principal resonance peak compared to the two prominent resonances of the abdomen. A three dimensional multi-body model of supine human with its underlying transport system was developed to predict the response under multidirectional whole-body vibration (Wang and Rahmatalla, 2013). The model consists of three segments representing the head, torso-arm, and pelvis-leg, interconnected via translational and rota­ tional spring-damper components representing tissue and muscles. The contact interface between the supine human and transport system was modeled using the translational spring-dampers system. The model pa­ rameters were identified using an optimization scheme by minimizing the error between modulus and phase of the predicted and experimental transmissibility. In a series of studies, the biodynamic response of semi-supine humans under vertical (Huang and Griffin, 2008a) and longitudinal horizontal (Huang and Griffin, 2008b) whole-body vibration was investigated. The subjects were exposed to two types of vibration, continuous and intermittent random vibration. As the vibration magnitude increased, the median resonance frequency of the apparent mass decreases with both intermittent random vibration and continuous random vibration. However, the median change in resonance frequency was greater for continuous vibration than for intermittent vibration. A parallel two degree of freedom lumped parameter model were used to fit the vertical in-line apparent mass and phases to obtain primary reso­ nance frequency. The 2-DOF LP model of semi-supine human doesn’t represent any anatomical body segment of the human body. It lacks the coupling between the sub-systems and predicts response only in the direction corresponding to input excitation. It doesn’t consider the relative angular motion between adjacent body segments. The present understanding of the impact of whole-body vibration on the human body in semi-supine posture is very limited. The major reason being the body posture not common in regular modes of transportation or military vehicles. The primary use of this posture is during space flight. In the new space vehicle architecture, i.e. stack launch and capsule re-entry, the astronauts in semi-supine posture would be exposed to elevated vibration level (Adelstein et al., 2009, 2008). To the best of the authors’ knowledge, other than the 2-DOF LP model, no biodynamic model is available to predict and simulate human biody­ namic response in semi-supine posture. The present study intends to develop a multi-body model of the human body in a semi-supine posture to simulate the response under vertical whole-body vibration. The human body has been modeled in five rigid body segments as; the head-neck, torso-arms, pelvis, thighs, and legs, which are connected by translational and rotational springdamper components. The interface contact between the body and un­ derlying vibrating rigid support is modeled using the translational spring-damper element. The optimization scheme has been used for the identification of model parameters by minimizing the least square error between normalized magnitude and phase of predicted and synthesized experimental target data of vertical apparent mass in the frequency domain. Further, a parameter sensitivity analysis was performed to determine the impact of the various model parameter on the response of a semi-supine human. Also, modal analysis was carried out to obtain natural frequencies and associated mode shapes. In addition, the effec­ tive modal mass was calculated to determine the contribution of each

body segment in each vibration mode. 2. Multibody biodynamic modeling 2.1. Model description The study proposes a linear 15-DOF model of a semi-supine human in the sagittal plane. The human body model is represented by five rigid body segments as shown in Fig. 1. Segment-1 represents the head with neck, segment-2 represents the torso and combined the thoracic and lumbar spine, abdomen and arms, segment-3 represents the pelvis and extends from L5 vertebrae to the ischial tuberosities, segment-4 repre­ sents the thighs, and segment-5 represents the legs with feet of the body. Each body segment has two translational motion; one horizontal (x-axis) and one vertical (z-axis) in the sagittal plane; and in addition one rotational motion about normal (y-axis) to the sagittal plane. The geo­ metric parameters defining the location of the center of gravity of each segment, joints between adjacent segments and contact points between segment and vibrating rigid support are derived from the literature (NASA Reference Publication, 1978; Armstrong, 1988; Dempster and Gaughran, 1889) and are summarized in Table 1. The location of the center of gravity of each body segment is expressed as � � x Gi ¼ i (1) zi The adjacent body segments are interconnected by bushing joints Jk (k ¼ 1 to 4) having both translational and rotational degree of freedom between the segments. The bushing joints are represented using linear and torsional sets of springs (kk, krk) and dampers (ck, crk). Each joint connected two segments includes two positions. Thus two subscripts (Jik ) are used to represent the displacement of joints, where the first subscript indicates the body index (i ¼ 1 to 5) and the second subscript indicates the joint index (k ¼ 1 to 4). For example, the Joint J1 con­ necting head-neck and torso includes two positions, J1 ¼ ½J11 J21 �: J11 representing the position of J1 on head-neck side (segment 1) and J21 representing the position of J1 on the torso side (segment 2). The loca­ tion of connecting joint Jik between two body segments is expressed as: � � x Jik ¼ Jik ​ (2) zJik The relationship between the displacement of the joint and center of gravity of each body segment can be expressed as: Jik ¼ Gi þ R dJik

(3)

where dJik ¼ ½dJik x ; dJik z �T , is the vector from the center of gravity Gi to the connecting joint Jk between two body segments.and ℝ is a rotation matrix given as: � � 0 θi R¼ (4) θi 0 The deformation of body tissue at the interface of body and rigid support is simulated through a set of linear springs (kvi) and dampers (cvi). These spring-damper elements represent the stiffness or flexibility of tissue of the respective segment. The contact points of the body with rigid support is given as: � � x C i ¼ ci ​ (5) zc i The relationship between the displacement of the contact point and center of gravity of each body segment can be expressed as: C i ¼ Gi þ R

d Ci

(6)

where, dCi ¼ ½dCi x ; dCi z �T , is the vector from the center of gravity Gi to 2

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Fig. 1. 15 DOF Semi-supine human body model. Table 1 Geometric parameters. Center of Gravity

Joint

Contact

Parameter

Coordinate (x, z)

Parameter

Coordinate (x, z)

Parameter

Coordinate (x, z)

G1 G2 G3 G4 G5

610, 108 391, 64 21, 49 67, 217 241, 429

J1 J2 J3 J4 -

567, 97 0, 0 55, 67 87, 413 –

C1 C2 C3 C4 C5

610, 18 391, 26 21, 66 149, 217 241, 384

NOTE: All dimensions are in mm.

the contact point Ci at the body-support interface of each segment. To reduce the complexity, the contact points are assumed to be located below the center of gravity at a distance depending upon the circumference of the body segment, which are based on anthropometric data (Armstrong, 1988). The entire body is on rigid vibrating support, which moves in the vertical direction (z0) due to external excitation.

fI; i ¼ kvi ðzCi

where q ¼ fx1 ; z1 ; θ1 ; x2 ; z2 ; θ2 ; x3 ; z3 ; θ3 ; x4 ; z4 ; θ4 ; x5 ; z5 ; θ5 g are the in­ dependent coordinate corresponding to the horizontal, vertical and rotational displacement of the center of gravity of each body segment of the model. The kinetic energy, T of the system is given as:

In the present model, each body segment has 3 DOF: xi, the hori­ zontal displacement of ith body segment; zi, the vertical displacement of ith body segment; θi, the absolute rotational displacement of ith body segment normal to sagittal plane; The coupled human body system has a total of 15 DOF with xi ¼ {x1, x2, x3, x4, x5} and zi ¼ {z1, z2, z3, z4, z5} are the horizontal and vertical displacement of head-neck, torso, pelvis, thighs and legs, respectively, and θi ¼ {θ1, θ2, θ3, θ4, θ5} are the rota­ tional displacement of each body segment. The forces at the joints are expressed as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ft; Jk ¼ kk ðxJik xJðiþ1Þk Þ2 þ ðzJik zJðiþ1Þk Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ck ðz_Jik z_Jðiþ1Þk Þ2 þ ðz_Jik z_Jðiþ1Þk Þ2 (7) θkþ1 Þ þ crk ðθ_ k

θ_ kþ1 Þ

(9)

z_0 Þ þ chi x_Ci

where z0 is the input displacement. The equations of motion for the model were derived through Lagrangian dynamics approach using Eq. (10). � � d ∂T ∂T ∂V ∂D ¼0 (10) þ þ ∂q ∂q ∂q_ dt ∂q_

2.2. Equations of motion

fr;Jk ¼ krk ðθk

z0 Þ þ khi xCi þ cvi ðz_Ci

T¼

5 5 � 1X 1X 2 mi x_2i þ z_2i þ Ii θ_ 2 i¼1 2 i¼1 i

(11)

where mi ; is the mass of each body segment and Ii , is the moment of inertia of each body segment. The potential energy, V of the system is given as: V¼

(8)

The forces at the interface between each body segment and the support are expressed as:

þ

3

5 � 1X khi x2Ci þ kvi ðzCi 2 i¼1 4 1X kk ðxJik 2 i;k¼1

z0 Þ2

�

xJðiþ1Þk Þ2 þ kk ðzJik

zJðiþ1Þk Þ2 þ krk ðθJik

θJðiþ1Þk Þ2

�

(12)

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The dissipation energy, D of the system is given as: D¼ þ

1 2

5 � X

� chi x_2Ci þ cvi z_Ci

z_0

interface. It indicates the extents to which vibration is transmitted through the body to the distant point (e.g. forehead, sternum, upper abdomen, etc.) from the driving point (i.e. vibrating support). The transfer function for vertical transmissibility due to vertical excitation is defined as

�2 �

i¼1

4 1X ck ðx_Jik 2 i;k¼1

x_Jðiþ1Þk Þ2 þ ck ðz_Jik

z_Jðiþ1Þk Þ2 þ crk ðθ_ Jik

θ_ Jðiþ1Þk Þ2

�

(13)

TRV ðjωÞ ¼

The equations of motion in terms of independent coordinates q are obtained by substituting Eqs.(11)–(13) in Eq. (10) and performing the partial differentiation with respect to q. For the proposed 15 DOF system there are 15 coupled differential equations and in matrix form can be expressed as:

TRH ðjωÞ ¼

TRR ðjωÞ ¼

θi ðjωÞ z0 ðωÞ

(20)

The identification of unknown model parameters was performed by optimization using the nonlinear constrained multivariable function. The objective function is designed to minimize the least square error between the predicted and measured target value of median normalized modulus and phase of vertical apparent mass in the frequency domain. The measured apparent mass data is derived through a synthesis of experimental data of semi-supine human subjects exposed to vertical whole-body vibration obtained by Huang and Griffin (2008a). The overall objective function E is given as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� 8 E ¼ min W1 � E1 þ W2 � E2 > > > > > > N > �2 1 X > < E1 ¼ Mp ðfi Þ Ms ðfi Þ N i¼1 (21) > > > > N > X > �2 1 > > ∅p ðfi Þ ∅s ðfi Þ : E2 ¼ N i¼1 where, E1 and E2 are sums of square error for modulus and phase of vertical apparent mass respectively. W1 and W2 are weighting factors assigned to the error in the modulus and phase of vertical apparent mass achieve the best fit. Mp ðfi Þ and ∅p ðfi Þ are the predicted normalized modulus and phase of vertical apparent mass at a frequency fi , Ms ðfi Þ and ∅s ðfi Þ are the synthesized normalized modulus and phase of vertical apparent mass at a frequency fi . N is the size of the sample of synthesized target data of apparent mass in the frequency range 0.1–20 Hz. The unknown model parameter vector (Γ) includes translational and rotational mass of each body segments, translational and rotational spring and damper coefficients at the segment joints and translational spring and damper coefficients at the contact interface between the body and support, and is given by

(16)

where FV ðjωÞ is the vertical force on the support surface and A0 ðjωÞ is the support acceleration. The cross-axis apparent mass of the model was defined as: 5 P ðkhi þ jωchi ÞxCi FH ðjωÞ i¼1 McV ðjωÞ ¼ ¼ A0 ðjωÞ ω2 z0

(19)

3. Identification of model parameters

To assess the response of the human body to excitation biodynamic transfer function such as apparent mass and transmissibility are employed. Apparent Mass is the complex ratio of driving periodic force to the resultant acceleration measured at the input point (i.e. body-support interface). Under the dynamic condition, the mass of the system de­ pends on the frequency, therefore normalized apparent mass of human body during excitation varies from 1 at the static condition to nearly 1.5 at resonance. The vertical apparent mass of the model was defined as: z0 Þ

xi ðjωÞ z0 ðωÞ

The transfer function for rotational transmissibility due to vertical excitation is defined as

where M, C, and K are 15 � 15 mass, damping coefficient, and spring stiffness matrices respectively; €z, z_ and z is 15 � 1 acceleration, velocity, and displacement vectors respectively. fk and fc are 15 � 1 coefficient vectors for the support motion; z_0 and z0 are velocity and displacement due to external excitation of support. The details of M, C, K, fk, and fc are given in Appendix A. For the linear system, the frequency-domain method is more suit­ able, therefore Laplace transformations are performed to obtain equa­ tions of motion in s-domain and then to convert them into the frequency domain, s ¼ jω is substituted, where ω is excitation frequency. The ratio of transformation for the vertical and rotational displacement of the center of gravity to the rigid support displacement with zero initial condition corresponding to the independent coordinates of the model is expressed as: � � 1 QðjωÞ ¼ Mω2 þ jCω þ K ðf k þ jf c ωÞ (15)

5 P ðkvi þ jωcvi ÞðzCi FV ðjωÞ i¼1 MV ðjωÞ ¼ ¼ a0 ðjωÞ ω2 z0

(18)

The transfer function for horizontal transmissibility due to vertical excitation is defined as

(14)

M€z þ Cz_ þ Kz ¼ f k z0 þ f c z_0

zi ðjωÞ z0 ðωÞ

(17)

Γ ¼ ½m1 ;m2 ;m3 ;m4 ;m5 ;I1 ;I2 ;I3 ;I4 ;I5 ;kh1 ;kv1 ;k1 ;kh2 ;kv2 ;k2 ;kh3 ;kv3 ;k3 ;kh4 ;kv4 ;k4 ;kh5 ;kv5 ;kr1 ;kr2 ;kr3 ;kr4 ;ch1 ;cv1 ;c1 ;ch2 ;cv2 ;c2 ;ch3 ;cv3 ;c3 ;ch4 ;cv4 ;c4 ;ch5 ;cv5 ;cr1 ;cr2 ;cr3 ;cr4 �T (22)

where FH ðjωÞ is the horizontal force on the support surface. Transmissibility (TR) is a complex ratio of the response motion of the individual body segment to the support motion at the body-support

To quantify how well the results of parameter identification would predict the synthesized target value, the adjusted R-squared value was calculated using the equation: 4

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R2adj ¼ 1

SSE ðn 1Þ SSTðn p 1Þ

European Journal of Mechanics / A Solids xxx (xxxx) xxx

Table 2 Translational and rotational mass.

(23)

Rotational Mass (kg-m2)

Translational Mass (kg)

where n ¼ Sample size; p ¼ No. of unknown parameters; SSE¼ Sum of square error; SST ¼ sum square mean. The multivariable optimization problem is solved using ‘fmincon’ function available in MATLAB. The variable tolerance and objective function toleration was set as 10 6. The optimization procedure is started with a set of initial, upper and lower bounds values for the un­ known parameters chosen by referring to published studies (Bai et al., 2017; Dempster and Gaughran, 1889; Zheng et al., 2011) and updated after each optimization run. The constrained condition is imposed on the mass parameters so that their sum equals whole body mass (median mass ¼ 66.4 kg based on experimental study (Huang and Griffin, 2008a)). The variation of objective function value for the final optimi­ zation run to acquire the highest adjusted R-squared value is shown in Fig. 2. The model parameters obtained after parameter identification are shown in Tables 2–4. The predicted normalized modulus and phase of vertical apparent mass are compared with the synthesized target value in Fig. 3(a). The primary resonance peak in the predicted normalized modulus of vertical apparent mass is obtained at 7.85 Hz as compared to 7.43 Hz in the synthesized target data. Although, the peak modulus of vertical apparent mass is slightly underpredicted by the model and predicted phase showed a slight deviation from the target, the modulus, and phase of vertical apparent mass fits quite well with the synthesized target data. The adjusted R-squared values obtained for the modulus and phase of vertical apparent mass are 0.9836 and 0.9556, respectively. The performance of the model is evaluated by comparing the hori­ zontal cross-axis apparent mass with the experimental study on semisupine human exposed to vertical excitation by Huang and Griffin (2008a) (Fig. 3(b)). The experimental response shows three distin­ guishable peaks; the first peak below 4.0–8.0 Hz, the second peak be­ tween 4.0 or 8.0 Hz and 12.0 Hz and third peak around 14.0–16.0 Hz. The model has precisely captured the first two peaks within the range; primary peak at 9.11 Hz and a secondary peak at 4.42 Hz. The third peak in the experimental response was reported due to non-rigidity of the platform in the horizontal direction and hence not acquired in the simulated response. The trend of cross-axis apparent mass matches quite

Parameter

Initial

Optimized

Parameter

Initial

Optimized

m1 m2 m3 m4 m5

5.639 27.760 9.557 18.194 5.248

5.693 26.858 11.302 18.193 4.352

I1 I2 I3 I4 I5

0.330 0.690 0.533 0.847 0.942

1.352 1.030 0.977 0.926 0.980

well with the experimental response. Though the model has slightly under-predicted the peak amplitude of horizontal cross-axis apparent mass, peak frequency was predicted precisely within the reported fre­ quency range. The model performance was further validated by comparing the vertical transmissibility ratio with the experimental response reported by Huang and Griffin (2009). They measured apparent mass and transmissibility at three body locations, viz., sternum, upper abdomen and lower abdomen for relaxed (P1) and constrained (P3) semi-supine human subjects exposed to vertical vibration. They observed the insig­ nificant difference in vertical apparent mass on postural change but the substantial effect was noted in transmissibility to the upper abdomen. In the present study, segment 2 i.e. torso comprises of thorax and abdomen, thus, its transmissibility ratio is compared with the experimental response to validate model performance (Fig. 4). The response of torso found in agreement with that of the sternum in both posture below 6.0 Hz, thereafter it follows a similar trend as upper abdomen in relaxed semi-supine posture. The peak frequency in both posture at the sternum and upper abdomen was reported in the range 6.0–10.0 Hz. The model predicts primary peak frequency at torso within this range at around 8.0 Hz. The transmissibility ratio at torso is observed to lie between the response of sternum and upper abdomen in both postures. This is because in the present study objective function for parameter identifi­ cation was based on apparent mass and torso comprise both measured locations. Thus, it can be ascertained that the model is capable to map the response of the complete upper body in both postures. The predicted transmissibility responses to vertical excitation of each body segment shown in Fig. 5. The horizontal transmissibility (Fig. 5(a))

Fig. 2. Variation of objective function value with the number of iteration. 5

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Table 3 Translational stiffness and damping. Translational Stiffness (N/m)

Translational Damping (N-s/m)

Parameter

Initial

Optimized

Parameter

Initial

Optimized

kh1 kv1 k1 kh2 kv2 k2 kh3 kv3 k3 kh4 kv4 k4 kh5 kv5

8985.677 36,022.006 25,595.456 8585.138 66,842.669 12,768.952 1964.515 158,947.759 11,810.939 1695.674 179,451.137 31,930.147 5236.908 12,650.985

6790.025 36,381.671 23,859.866 7863.712 66,774.301 12,332.309 1864.758 158,837.653 11,863.965 1619.151 179,134.918 32,093.861 5223.895 12,597.343

ch1 cv1 c1 ch2 cv2 c2 ch3 cv3 c3 ch4 cv4 c4 ch5 cv5

100.194 381.214 100.054 527.808 828.788 166.726 200.943 2714.255 100.025 104.415 1219.836 767.984 272.667 1675.201

100.047 3996.853 102.387 100.257 676.217 215.739 3567.244 1515.299 2362.078 688.992 1723.840 3999.625 780.596 3998.378

13.0–15.0 Hz. Head has an amplifying effect below around 7.0 Hz and then attenuate the vertical excitation. The rotational transmissibility response (Fig. 5(c)) for head showed the highest peak around 10.0 Hz while other body segment showed a meager response to vertical excitation. The vibration transmission path in semi-supine posture involves less soft tissues of the lower back and more skeletal structure of the entire back. Moreover, the response of each body segment depends on the tendency of its content to displaced under whole-body vibration (Vogt et al., 1973). The previous study on supine human showed lower vertical transmissibility of thorax relative to abdomen due to its anatomical configuration (Vogt et al., 1978). The torso in the proposed semi-supine model comprised both stiff thoracic cage and soft abdomen tissues. The higher transmissibility of torso indicates that soft abdomen tissues have a higher impact than a thoracic cage.

Table 4 Translational stiffness and damping. Rotational Stiffness (N-m/rad)

Rotational Damping (N-m-s/rad)

Parameter

Initial

Optimized

Parameter

Initial

Optimized

kr1 kr2 kr3 kr4

207.758 208.6698 240.5937 230.6265

101.8995 107.4917 378.457 130.7559

cr1 cr2 cr3 cr4

357.263 410.814 114.957 981.728

431.852 305.533 675.375 369.707

is observed to be highest for the head with two unique peaks, one below 5.0 Hz and second around 10.0 Hz. The peak transmissibility in the horizontal direction for the torso is obtained at 10.0 Hz. The primary peak in the vertical transmissibility response (Fig. 5(b)) to vertical excitation is obtained for torso around 8.0 Hz. The pelvis and thigh represented quite similar vertical transmissibility with a peak around

Fig. 3. Performance comparison of the model with the synthesized experimental data (Huang and Griffin, 2008a) (a) Normalized vertical apparent mass. (b) Normalized cross-axis apparent mass of the semi-supine human body. 6

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Fig. 4. Performance comparison of the model with synthesized experimental data (Huang and Griffin, 2009) in terms of vertical transmissibility for semi-supine posture (P1- relaxed semi-supine, P3 - constrained semi-supine)).

Fig. 5. (a) Horizontal transmissibility response due to vertical excitation (b) Vertical transmissibility response due to vertical excitation (c) Rotational trans­ missibility response due to vertical excitation.

4. Parameter sensitivity analysis

� � 0 gðχ0 Þ g χþ i ; χei ∅i ¼ 0 gðχ Þ

The sensitivity analysis is performed to interrogate the model re­ sponses to the change in the input parameters. The sensitivity analysis is carried out through a finite difference approach, where the effect of a change in the input parameter(s) on the model response is expressed in terms of finite difference and residuals (Borgonovo and Plischke, 2016). Sensitiveness or total effect of the model parameter (χ i) on model response can be expressed as:

(25)

(24)

∅i;j : represents the residual interaction effect due to the simultaneous variation of χ i and χ j . � � þ 0 gðχ0 Þ g χþ i ; χj ; χei;j ∅i;j ¼ (26) ∅i ∅j 0 gðχ Þ

where ∅i : individual effect of varying one model parameter at a time from the nominal case to the sensitivity case.

0 gðχþ i ; χei Þ : model response obtained by an increment in the nominal input parameter χ0i while other parameters retain their nominal value.

∅Ti ¼

n X

∅i þ i¼1

X ∅i;j þ … þ ∅1;2;…;n j¼1

χ þi ¼ ð1 þ δÞχ 0i χ: vector of the mass, stiffness and damping parameter 7

(27)

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χ ¼ ½m1 ; m2 ; m3 ; m4 ; m5 ; kh1 ; kv1 ; k1 ; kh2 ; kv2 ; k2 ; kh3 ; kv3 ; k3 ; kh4 ; kv4 ; k4 ; kh5 ; kv5 ; kr1 ; kr2 ; kr3 ; kr4 ; ch1 ; cv1 ; c1 ; ch2 ; cv2 ; c2 ; ch3 ; cv3 ; c3 ; ch4 ; cv4 ; c4 ; ch5 ; cv5 ; cr1 ; cr2 ; cr3 ; cr4 �

(28)

interactive effect, a fall in any mass parameter would increase the apparent mass and decrease otherwise. This suggests the effect of mass parameters is overshadowed by other parameters on simultaneous interaction. The primary resonance frequency (Fig. 6(d)) corresponding to peak apparent mass is most affected by m2. Also, the interaction effect would magnify the peak resonance frequency response with both in­ crease or decrease in m2 mass. The influence of torso mass (m2) on peak vertical apparent mass and corresponding resonance frequency is reasonably consistent as it comprised a major proportion of total body mass.

The interaction effect of model parameter χ i with all other parame­ ters are combined into one indicator and given as: ∅Ii ¼ ∅Ti

(29)

∅i

In the present study, the residual interactions are considered only up to second-order to examine the model response. The sensitiveness of two model responses (Rk, k ¼ 1,2) is examined by perturbation of the input model parameters (χ) with perturbation factor δ ¼ �10% of its nominal value. Here R1, represents peak modulus of vertical apparent mass and R2, represents resonance frequency are corresponding to peak modulus of vertical apparent mass. The information of the sensitivity analysis is represented with the aid of generalized tornado diagram. In the diagram, the sensitivity of each parameter on the response is displayed with triplet bars, indicating in­ dividual effect (∅i ), total effect (∅Ti ) and interaction effect (∅Ii ). The sensitiveness of mass, stiffness and damping parameters on the model responses, viz., peak modulus of apparent mass (R1) and corresponding resonance frequency (R2), are presented in Fig. 6. Out of 18 stiffness and damping parameter each, only five most sensitive parameters based on total effect have been displayed to preserve visual clarity.

4.2. Sensitiveness of stiffness parameters The peak magnitude of vertical apparent mass (Fig. 6(b)) is highly responsive to perturbation in kv2. The vertical apparent mass magnitude increases with the increase in kv2, kh1, and k2, while it decreases with the increase in kv4 and kv3. The interaction effect has a positive sign with kv2, kh1, kv3 and k2 and negative sign with kv4. Therefore, with a rise in kv2, kh1, kv3, and k2 parameters, the individual effect amplifies while atten­ uating otherwise. A similar effect is observed with the rise/drop in the kv4. The resonance frequency (Fig. 6(e)) corresponding to peak modulus of apparent mass is most sensitive to kv2 and k1. The interaction effect diminishes the individual effect with a shift in kv2 on either side of its nominal value whereas a shift in k1 results in amplification. This in­ dicates the interaction effect causes peak resonance frequency to reduce as kv2 increases while with the rise in k1, it increases. The vertical stiffness parameters have a considerable impact on peak modulus of vertical apparent mass and corresponding resonance frequency. The influence of horizontal and rotational stiffness observed marginal.

4.1. Sensitiveness of mass parameters The perturbation in the mass parameters (Fig. 6(a)) shows that the modulus of peak apparent mass is most sensitive to m2 and m4 and least sensitive to m1. The individual effect represents that the vertical apparent mass increases as any mass parameter increases and vice-versa. The interaction effect of m2 was found to be most significant. It is also observed that both positive and negative perturbation in the mass parameter results in negative interaction effect. It means that due to

Fig. 6. Generalized Tornado Diagram for the sensitiveness of mass, stiffness and damping parameters on peak modulus of apparent mass (R1) and corresponding resonance frequency (R2): Effect (

þ

∅Ii ).

Individual Effect ( ∅i ),

Total Effect ( ∅Ti ),

Interaction Effect ( ∅Ii ),

8

Individual Effect (þ ∅i ),

Total Effect ( ∅Ti ), þ

Interaction

R. Govindan et al.

European Journal of Mechanics / A Solids xxx (xxxx) xxx

corresponding resonance frequency (Fig. 6(f)). The vertical apparent mass reduces with the rise in damping parameters and vice-versa. The peak apparent mass response shows the interaction effect of cv2 is most significant. Due to interaction, a positive shift in damping parameters

4.3. Sensitiveness of damping parameters The sensitivity of damping parameter cv2 has a significant influence on peak modulus of the vertical apparent mass (Fig. 6(c)) and

Fig. 7. Mode shapes of semi-supine human body model in the low frequency (0.1–20 Hz) range. The scale of the mode vector is enlarged for clear representation. 9

R. Govindan et al.

European Journal of Mechanics / A Solids xxx (xxxx) xxx

would result in an increase of peak apparent mass and vice-versa. With an increase in cv1 and c4 and decrease in cv4, cv3 and c4, the resonance fre­ quency remains unaffected. Compared to individual effect, the interac­ tion effect has a dominating influence on the total effect in the resonance frequency response.

fDg is the influence vector which represents the displacements of the masses due to the static application of a unit ground displacement and rotation. For free vibration analysis, unit displacement is considered for each degree of freedom (Govindan and Harsha, 2018; Irvine, 2013). The undamped natural frequencies acquired through modal analysis show ten low frequencies (0.1–20 Hz) vibration modes. The mode shapes for low-frequency vibration modes for the semi-supine model are illustrated in Fig. 7. The undeformed or initial posture of the body is represented by blue color and deformed posture at the specified fre­ quency is depicted by red color. The effectiveness of each body seg­ ment’s DOF during specified vibration mode is characterized by effective modal mass (Fig. 8). For the given mode, body segment with larger effective modal mass has higher energy content and thus would be the major contributor to the overall body motion (Govindan and Harsha, 2018). Mode shapes for first four natural frequencies, the body segments were dominated by longitudinal motion along the horizontal direction. In the later modes, with the exception of the seventh mode, the vertical motion has a greater influence. A similar observation is noted in the effective modal mass too. The horizontal DOF (xi) has higher contribu­ tion up to the fourth mode and later vertical DOF (zi) contributed effectively in the effective modal mass. Though the contribution of rotational DOF to effective modal mass are marginal at low frequencies, translational motion is always coupled with rotation. The effect of rotational DOF is more significant at higher frequency range. The first mode at 1.586 Hz has horizontal out-of-phase motion of thigh, pelvis, and torso. The second mode is obtained at frequency 3.347 Hz, where the torso and head have in-phase motion whereas thigh moves out-of-phase. The third horizontal mode at 5.327 Hz is observed to have out-of-phase shear deformation of calf tissues of legs while the upper body remained almost stationary. The fourth mode comprised out-ofphase shear and axial deformation of pelvis and thigh tissues in contact with the rigid base support. In the first four horizontal modes, thigh, torso, leg, and pelvis, respectively have a major contribution to the effective modal mass and thus they would significantly influence the response of the system under horizontal excitation.

5. Modal analysis The modal analysis of the proposed model for the semi-supine human body is performed to determine the natural frequencies and associated modes. The frequencies and modes are examined to analyze vibration mode and DOF that contribute to the primary resonance observed in the response of apparent mass. The equation of motion for free undamped vibration is given as ½M�f€xg þ ½K�fxg ¼ f0g

(30)

where M and K are the mass and stiffness matrix obtained through the € are the displacement and acceleration optimized model parameters. x; x vectors, respectively, with 15 DOF for the proposed model. The solution to Eq. (30). was determined in terms of eigenvalues and eigenvectors. The eigenvalues represent the natural frequencies and eigenvectors represent vibration modes. To identify modes that will contribute to excitation in a particular direction, the effective modal mass was calculated. A mode with a large effective mass would be readily excited by base excitation and thus usually a significant contributor to the response of the system. The effective modal mass for the ith mode is given as Meff ;i ¼

γ2i T f∅gi ½M�f∅gi

(31)

where γ i is the coefficient vector, expressed as: γ i ¼ f∅gTi ½M�fDg

(32)

f∅gi is the eigenvector matrix for the ith mode.

Fig. 8. Undamped natural frequency and effective modal mass of semi-supine human body model: H- Horizontal DOF (xi); V- Vertical DOF (zi) and, R- Rotational DOF (θi). 10

European Journal of Mechanics / A Solids xxx (xxxx) xxx

R. Govindan et al.

The first vertical mode is witnessed at 8.611 Hz with an in-phase motion of the torso and head. The second resonance mode is obtained at 8.822 Hz involve vertical motion of leg along with pitch motion about the knee joint. The seventh mode is a sought of combined mode has both horizontal and vertical motion along with pitch motion of torso and head. While the head and torso move in-phase vertically, they move outof-phase in a horizontal direction causing longitudinal deformation of tissues connecting them. The eighth mode involves shear deformation of thigh tissues in contact with the rigid support. The out-of-phase vertical motion is observed for head, torso, and pelvis in the ninth vibration mode whereas the tenth mode show in-phase motion. Due to the effective proportion of torso and thigh in total body mass, these have a major contribution in the effective modal mass in the vertical direction. The higher modes have a dominating pitch motion of head, pelvis, leg, torso, and thigh, respectively. While the head has counter-clockwise motion others have clockwise motion in the respective modes. Hopkin’s study on semi-supine human body reported resonance at 7 Hz and 10.5 Hz in mechanical impedance response (Hopkins, 1961) as cited in (Clark et al., 1962). Vykukal reported primary resonance fre­ quency in the impedance response with semi-supine space crew between 7 and 11 Hz exposed to sinusoidal excitation of 2.8 m/s2 rms over the frequency range 1–70 Hz (Vykukal, 1968) as cited in (Huang and Griffin, 2008a). Huang and Griffin study on the human body in semi-supine posture found the range of resonance frequencies of apparent mass be­ tween 7.81 Hz and 10.01 Hz when exposed to five different magnitudes of continuous random vibration (Huang and Griffin, 2008a). The reso­ nance frequencies obtained for the proposed model are consistent with the results of previous studies on the human body in a semi-supine posture.

body and the rigid vibrating support were modeled with translational spring-damper elements at the interface. Based on the analysis, the following conclusions are drawn: � The optimization scheme based on minimization of least squared error between predicted and the target value of normalized modulus and phase of vertical apparent mass is employed to identify the 46 unknown model parameters. The model developed depicted an adjusted R-squared value of 0.9836 and 0.9556, respectively for normalized modulus and phase of vertical apparent mass. � The model is capable of capturing the overall response of the semisupine human body exposed to vertical excitation in terms of apparent mass and transmissibility. Moreover, it can describe the head motion due to vertical excitation, which would influence the visual acuity and stereopsis. � The vibration transmissibility in the horizontal and vertical direction due to vertical excitation was found higher for head and torso respectively. The vertical degree of freedom has substantial influence while the rotational degree of freedom has a marginal impact on vertical apparent mass at primary resonance. � The peak of vertical apparent mass and corresponding resonance frequency were highly sensitive to vertical stiffness and damping parameter of the torso and also to its mass. Among horizontal model parameters, the effect of head stiffness and damping was more evident to influence the apparent mass. The effect of rotational pa­ rameters was meager in the sensitivity analysis. � The modal analysis showed the principal and secondary resonance frequency at 8.611 Hz and 12.99 Hz respectively for the semi-supine human, which are higher as compared to other postures like sitting and standing (5–6 Hz). � The response of the torso can be refined by modeling thorax and abdomen separately. Moreover, pelvis and thigh can be considered as a single segment to reduce complexity since their individual re­ sponses show no significant difference. Further, nonlinearity in the apparent mass and transmissibility response could be modeled using nonlinear spring-damper elements.

6. Conclusions A 15 DOF multi-body model having five rigid bodies is developed to predict the biodynamic response of the semi-supine human exposed to vertical excitation. The segments were interconnected with translational and rotational spring-damper elements. The interaction between the

Appendix A A1.1. Mass matrix for 15 DOF Semi-supine model M : diagonal ​ matrix of ðm1 ; m1 ; I1 ; m2 ; m2 ; I2 ; m3 ; m3 ; I3 ; m4 ; m4 ; I4 ; m5 ; m5 ; I5 Þ

A1.2. Stiffness matrix for 15 DOF Semi-supine model 0

K1;1 K¼K ¼@ ⋮ K15;1 T

1 … K1;15 ⋱ ⋮ A ⋯ K15;15

K1;1 K1;6 K2;2 K2;6

¼ k1 þ kh1 ¼ k1 dJ21 z ¼ k1 þ kv1 ¼ k1 dJ21 x

K3;6 K4;4 K4;9 K5;5 K5;9

¼ k1 ðdJ11 x dJ21 x þ dJ11 z dJ21 z Þ ¼ k1 þ k2 þ kh2 ¼ k2 dJ32 z ¼ k1 þ k2 þ kv2 ¼ k2 dJ32 x

K3;3 ¼ k1 ðd2J11 x þ d2J11 z Þ þ kv1 d2c1 x þ kh1 d2c1 z þ kr1 kr1

K1;3 K1;2 K2;3 K2;6 K3;4

¼ k1 dJ11 z kh1 dC1 z ¼ K1;5 ¼ K1;7 ¼ … ¼ K1;15 ¼ 0 ¼ k1 dJ11 x kv1 dC1 x ¼ k1 dJ21 x ¼ k1 dJ11 z

K3;7 K4;6 K4;5 K5;6 K5;7 K6;7 K6;9

¼ ¼ ¼ ¼ ¼ ¼ ¼

… ¼ K3;15 ¼ 0 k1 dJ21 z þ k2 dJ22 z þ kh2 dC2 z K4;8 ¼ K4;10 ¼ … ¼ K4;15 ¼ 0 k1 dJ21 x k2 dJ22 x kv2 dC2 x K5;10 ¼ … ¼ K5;15 ¼ 0 k2 dJ22 z k2 ðdJ22 x dJ32 x þ dJ22 z dJ32 z Þ kr2

K1;4 ¼

k1

K2;5 ¼

k1

K3;5 ¼ k1 dJ11 x K4;7 ¼

k2

K5;8 ¼

k2

K6;8 ¼ k2 dJ22 x K6;10 ¼ … ¼ K6;15 ¼ 0 (continued on next page)

11

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R. Govindan et al.

(continued ) K6;6 ¼ k1 ðd2J21 x þ d2J21 z Þ þ k2 ðd2J22 x þ d2J22 z Þþ kv2 d2c2 x þ kh2 d2c2 z þ kr1 þ kr2

K7;7 ¼ k2 þ k3 þ kh3 K7;12 ¼ k3 dJ43 z K8;8 ¼ k2 þ k3 þ kv3 K8;12 ¼ k3 dJ43 x

K9;9 ¼ k2 ðd2J32 x þ d2J32 z Þ þ k3 ðd2J33 x þ d2J33 z Þþ

kv3 d2c3 x þ kh3 d2c3 z þ kr2 þ kr3

K10;10 K10;15 K11;11 K11;14

¼ k3 þ k4 þ kh4 ¼ k4 dJ54 z ¼ k3 þ k4 þ kv4 ¼ k4

K12;12 ¼ k3 ðd2J43 x þ d2J43 z Þ þ k4 ðd2J44 x þ d2J44 z Þþ kv4 d2c4 x þ kh4 d2c4 z þ kr3 þ kr4

K13;13 ¼ k4 þ kh5 K14;14 ¼ k4 þ kv5

K7;9 ¼ k2 dJ32 z þ k3 dJ33 z þ kh3 dC3 z K7;8 ¼ K7;11 ¼ K7;13 ¼ … ¼ K7;15 ¼ 0 K8;9 ¼ k2 dJ32 x k3 dJ33 x kv3 dC3 x K8;10 ¼ K8;13 ¼ … ¼ K8;15 ¼ 0 K9;10 ¼ k3 dJ33 z K9;12 ¼ k3 ðdJ33 x dJ43 x þ dJ33 z dJ43 z Þ kr3

K7;10 ¼

k3

K8;11 ¼

k3

K10;12 K10;11 K11;12 K11;15 K12;13 K12;15

K10;13 ¼

¼ k3 dJ43 z þ k4 dJ44 z þ kh4 dC4 z ¼ K10;14 ¼ 0 ¼ k3 dJ43 x k4 dJ44 x kv4 dC4 x ¼ k4 dJ54 x ¼ k4 dJ44 z ¼ k4 ðdJ44 x dJ54 x þ dJ44 z dJ54 z Þ kr4

K13;14 ¼ 0 K14;15 ¼ k4 dJ54 x

K15;15 ¼ k4 ðd2J54 x þ d2J54 z Þ þ kv5 d2c5 x þ kh5 d2c5 z þ kr4

K9;11 ¼ k3 dJ33 x K9;13 ¼ … ¼ K9;15 ¼ 0 k4

K11;13 ¼ 0 K12;14 ¼ k4 dJ44 x K13;15 ¼ k4 dJ54 z þ kh5 dC5 z

kv5 dC5 x

A1.3. Damping matrix for 15 DOF Semi-supine model 0

C1;1 C¼C ¼@ ⋮ C15;1 T

1 … C1;15 ⋱ ⋮ A ⋯ C15;15

C1;1 C1;6 C2;2 C2;6

¼ c1 þ ch1 ¼ c1 dJ21 z ¼ c1 þ cv1 ¼ c1 dJ21 x

C3;6 C4;4 C4;9 C5;5 C5;9

¼ c1 ðdJ11 x dJ21 x þ dJ11 z dJ21 z Þ ¼ c1 þ c2 þ ch2 ¼ c2 dJ32 z ¼ c1 þ c2 þ cv2 ¼ c2 dJ32 x

C3;3 ¼ c1 ðd2J11 x þ d2J11 z Þ þ cv1 d2c1 x þ ch1 d2c1 z þ cr1 cr1

C6;6 ¼ c1 ðd2J21 x þ d2J21 z Þ þ c2 ðd2J22 x þ d2J22 z Þþ

cv2 d2c2 x þ ch2 d2c2 z þ cr1 þ cr2

C7;7 ¼ c2 þ c3 þ ch3 C7;12 ¼ c3 dJ43 z C8;8 ¼ c2 þ c3 þ cv3 C8;12 ¼ c3 dJ43 x

C9;9 ¼ c2 ðd2J32 x þ d2J32 z Þ þ c3 ðd2J33 x þ d2J33 z Þþ

cv3 d2c3 x þ ch3 d2c3 z þ cr2 þ cr3

C10;10 C10;15 C11;11 C11;14

¼ c3 þ c4 þ ch4 ¼ c4 dJ54 z ¼ c3 þ c4 þ cv4 ¼ c4

C12;12 ¼ c3 ðd2J43 x þ d2J43 z Þ þ c4 ðd2J44 x þ d2J44 z Þþ cv4 d2c4 x þ ch4 d2c4 z þ cr3 þ cr4

C13;13 ¼ c4 þ ch5 C14;14 ¼ c4 þ cv5

C1;3 C1;2 C2;3 C2;4 C3;4

¼ c1 dJ11 z ch1 dC1 z ¼ C1;5 ¼ C1;7 ¼ … ¼ C1;15 ¼ 0 ¼ c1 dJ11 x cv1 dC1 x ¼ C2;7 ¼ … ¼ C2;15 ¼ 0 ¼ c1 dJ11 z

C3;7 C4;6 C4;5 C5;6 C5;7 C6;7 C6;9

¼ … ¼ C3;15 ¼ 0 ¼ c1 dJ21 z þ c2 dJ22 z þ ch2 dC2 z ¼ C4;8 ¼ C4;10 ¼ … ¼ C4;15 ¼ 0 ¼ c1 dJ21 x c2 dJ22 x cv2 dC2 x ¼ C5;10 ¼ … ¼ C5;15 ¼ 0 ¼ c2 dJ22 z ¼ c2 ðdJ22 x dJ32 x þ dJ22 z dJ32 z Þ cr2

c1

C2;5 ¼

c1

C3;5 ¼ c1 dJ11 x C4;7 ¼

c2

C5;8 ¼

c2

C6;8 ¼ c2 dJ22 x C6;10 ¼ … ¼ C6;15 ¼ 0

C7;9 ¼ c2 dJ32 z þ c3 dJ33 z þ ch3 dC3 z C7;8 ¼ C7;11 ¼ C7;13 ¼ … ¼ C7;15 ¼ 0 C8;9 ¼ c2 dJ32 x c3 dJ33 x cv3 dC3 x C8;10 ¼ C8;13 ¼ … ¼ C8;15 ¼ 0 C9;10 ¼ c3 dJ33 z C9;12 ¼ c3 ðdJ33 x dJ43 x þ dJ33 z dJ43 z Þ cr3

C7;10 ¼

c3

C8;11 ¼

c3

C10;12 C10;11 C11;12 C11;15 C12;13 C12;15

C10;13 ¼

¼ c3 dJ43 z þ c4 dJ44 z þ ch4 dC4 z ¼ C10;14 ¼ 0 ¼ c3 dJ43 x c4 dJ44 x cv4 dC4 x ¼ c4 dJ54 x ¼ c4 dJ44 z ¼ c4 ðdJ44 x dJ54 x þ dJ44 z dJ54 z Þ cr4

C13;14 ¼ 0 C14;15 ¼ c4 dJ54 x

C15;15 ¼ c4 ðd2J54 x þ d2J54 z Þ þ cv5 d2c5 x þ ch5 d2c5 z þ cr4

C1;4 ¼

C9;11 ¼ c3 dJ33 x C9;13 ¼ … ¼ C9;15 ¼ 0 c4

C11;13 ¼ 0 C12;14 ¼ c4 dJ44 x C13;15 ¼ c4 dJ54 z þ ch5 dC5 z

cv5 dC5 x

A1.4. Coefficient vectors for the support motion for 15 DOF Semi-supine model � fk ¼ 0

kv1

kv1 dc1 x

0 kv2

kv2 dc2 x

0

kv3

kv3 dc3 x

0

kv4

kv4 dc4 x

0

kv5

kv5 dc5 x

� f c ¼ 0 cv1

cv1 dc1 x

0

cv2 dc2 x

0

cv3

cv3 dc3 x

0

cv4

cv4 dc4 x

0

cv5

cv5 dc5 x

cv2

References

�T �T

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Adelstein, B.D., Beutter, B.R., Kaiser, M.K., Mccann, R.S., Stone, L.S., Anderson, M.R., 2009. Effects of Transverse Seat Vibration on Near-Viewing Readability of

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