Modeling of Human Reactions to Whole-Body Vibration

Farid M. L. Amirouche Assistant Professor,

Department of Mechanical Engineering, University of Illinois at Chicago,

Chicago, Illinois 60680

Modeling of Human Reactions to Whole-Body Vibration A computer-automated approach for studying the human body vibration is presented. This includes vertical, horizontal, and torsional vibration. The procedure used is based on Finite Segment Modeling (FSM) of the human body, thus treating it as a mechanical structure. Kane's equations as developed by Huston et al. are used to formulate the governing equations of motion. The connective tissues are modeled by springs and dampers. In addition, the paper presents the transient response of different parts of the body due to a sinusoidal forcing function as well as an impulse function applied to the lower torso in the vertical direction.

Introduction Human body vibration is a phenomenon affecting about 8

million workers in the United States (see reference [2]). Among these are light and heavy equipment operators, machine tool operators, and truck drivers. Many harmful side effects of the vibration can be both physiological and neurological, which in many cases lead to permanent injury. Some results [1,2] suggest that muscular weakness movements resulting from vibration could be affected. This is especially the case in precision work. For example, performance in an aiming type of task for an individual who is under vibration could be affected by an increase in hand tremor.

Studies by Patil and Palanichamy [3] and Muskian and Nash [4, 5] have shown the human body could be modeled as a series of lumped masses connected by springs and dampers (muscles, ligaments, disks vertebrae). This approach has proven to be effective when the body is undergoing vertical motion. In most of the research reported in the literature [3-8] on human body vibration (HBV), sinusoidal motion has always been used to approximate the stimulus to the body. However, the vibration conditions experienced in practice are random with frequent impulses to account for bumps and irregularities as is the case for the tractor ride.

Most of the research on the effects of mechanical vibration on humans is based upon statistical analysis of experimental data. References [6-10] show some of the work and methods used to measure the body response to different types of vibration. The seriousness of the problem caused by vibration has led some researchers to extend their research in looking at the effect of longitudinal as well as transverse and torsional vibration. In references [11, 12] Anderson, Pope, and Wilder have suggested that low back pain could be a result of exposure to vibration due to both vertical and bending motions. It has also been suggested by Sjoflot and Suggs [13] that man is more affected and shows more degraded performance under vertical vibration alone. Worst is a combination of transverse and vertical motion (see reference [13]).

Most recently, Huston and Amirouche [14] have shown that

Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAI ENGINEERING. Manuscript received by the Bioengineering Division May 13, 1986; revised manuscript received April 24, 1987.

the human body dynamic response for low accelerations could be studied by new techniques used in the analysis of multibody systems. This paper presents a computer automated approach used in monitoring the human body vibration response in the vertical, horizontal, and torsional directions. The procedures developed herein are based on Kane's equations as formulated by Huston and Passerello [15]. Since the frequency range of concern for whole body vibration is between 0.5 and 12 Hz, the equations of motions are formulated allowing the body to undergo small oscillations. The connective tissues, muscles, ligaments, and disks vertabra are modeled by linear and nonlinear spring and damping forces (see references [3, 5, 6, 7]). The geometrical data used are based on the 572 and Hybrid III dummies used in the automobile crash accidents (see reference [18]). The procedures developed could be useful in studying the effects of whole body vibration when some parts of the body come in contact with a vibrating medium. Indeed, this unique approach could be used to help establish minimal standards of vibration safety. Using these standards, designers would be able to seek means of reducing the deleterious vibration effects.

The paper is divided into five sections. Part one presents the modeling of the human body. The second and third parts discuss the formulation of the kinematics and the equations of motion. Part four presents results due to vertical vibration of a seated man under swept sinusoidal function and impulse functions. The final part consists of the conclusion.

I Modeling of the Human Body

Consider a typical model of a human body as depicted by Fig. 1. It consists of a series of connected cylinders and frustums of elliptical cones. However, there might be bodies of arbitrary shapes. Let these bodies be numbered as shown in Fig. 1. For analysis purposes the model described is looked upon as a "tree-like" structure based upon the body numbering. These are: 1) a body array L0 (k); 2) a lower body array Lx (k); 3) a connecting body array L2(k). For the model of Fig. 1, these arrays are:

210/Vol. 109, AUGUST 1987 Transactions of the ASME

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Table 1 Configuration of the structure given by Fig. 1

Fig. 1 A seated human body model

L0(Ar) = { l , 2 , 3 , 4, 5, 6, 7, 8,9, 10, 11, 12, 13] (1)

L{ (k) = (1, 1, 1, 1, 1, 6, 7, 8, 9, 10, 10, 12, 12] (2)

L2(k) = {0, 0, 0, 0, 0, 1,6, 1,8, 3, 3, 3, 3] (3)

Based on the three arrays described above the topology of the mechanical structure representing the human model is given by a block array Af described in Table 1 (see reference [16]). The exponent element P is equal to E(k) and determines the number of iterations to go from segment Bk to a reference frame R. The configuration of the human model represented by Table 1, where the main body is the trunk, and the legs and arms the branches, is further illustrated in Appendix b.

(a) Transformation Matrices. Consider a typical pair of adjoining segments of the model. Let these be called Bj and Bk. Let n;7 and nki (i = 1, 2, 3) be mutually perpendicular vector sets fixed in Bj and Bk. Then a transformation matrix SJK defining the relative orientation of Bj and Bk may be introduced (see Appendix a). Especially, the elements of SJK are:

SJKim= ii:, >nk (4) "ji' llkm

If R is an inertial reference frame, the transformation matrix between Bk and R may be obtained by multiplying the

X i

2

3

4

5

6

1

1

O

2

1

O

3

1

0

4

1

0

5

1 0

6

6

1

1

0

7

7

6

6

1

1

O

a 8

I

I

0

9

9

8

8

1

1

O

10

10

3

1

0

1

0

1 1

10

3

1

O

12

(\1

3

1

O

13

12

3

1

0

. p /\ Tree Array

K

H

1

1

2

1

3

1

4

1

5

1

6

3

7

3 a 3

9

5

10

5

1 1

3

12

3 13 3

E(K) Exponent A r r a y

transformation matrices between adjacent bodies in the chain of bodies between R and Bk. The multiplication could be simplified if the bodies (segments) in a particular branch (in this case it could be a leg or an arm) are aligned therefore the transformation shifter matrix will reduce to identity. Reference [16] shows how we can cut down in computation by using such a technique.

II Kinematics

(a) Coordinates. Let each body of the model have 6 degrees of freedom. Hence, for the N bodies of the system there are 6Ndegrees of freedom. Let these be characterized by the coordinates Xj 0 = 1 , . . . ,67V). Further, let these coordinates be divided into 2N sets of triplets describing first the rotation of the bodies, and second the translation of the bodies.

(Jb) Angular Velocity. Consider two connecting segments from the structure given in Fig. 1. The angular velocity of segment k with respect to its adjacent lower segment j is given by

J—k o = (cos/3cos7a - sin7/3)n,v + (cos/3sin7a

+ C0S7)3) nJ2 + ( - sin/3a 4- 7) nj3

where a, 13, and 7 represent the orientations of segment Bk

with respect to Bj. In the analysis of the human body vibration the oscillations between body elements are small hence the angular velocity in equation (5) reduces to

J—k (6)

From reference [15], Huston and Passerello have shown that

N o m e n c l a t u r e

hPk = E(k) =

R = L0{k) = L,(k) = L2(k) =

SJK =

nkm -X, =

J-k =

CO

uklm -

tree array exponent array inertia frame body array lower-body array connecting-body array shifter matrix between B}

and Bk

unit vectors fixed in Bk

generalized coordinates angular velocity of Bk with respect to Bj partial angular velocity array

angular acceleration of Bk

with respect to R Gk

t* Tic

g*

Vklm a*

alp

mk

*kmn

= center mass point of Bk

= translation vector of Bk

relative to Bj = position vector of Gk

relative to 0^ = position vector of Qk

relative to 0y = partial velocity array = mass center acceleration of

Bk relative to R = generalized mass matrix = mass of Bk

= mass center based inertia dyadic of Bk

I'km

Tkm

Kip

c,P k (/••/)

C< r >"

F*(t) CO

/

generalized active forces active forces components acting on Bk

active torques components acting on Bk

global stiffness matrix global damping matrix local stiffness due to rotation (/•) or translation (f) local damping due to rotation (/•) or translation (t) velocity excitation force circular frequency frequency of oscillation

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the angular velocity of any body k with respect to an inertial frame R could be expressed as

R-k_ CO - 0)kln X,nn

(7)

where the n0,„ are unit vectors fixed in an inertial reference frame R and the coefficients wklm are components of the so-called "partial angular velocity vectors" (see Kane and Levin-son [17]). (Regarding notation, there is a sum on repeated indices over the range of the index.) Interestingly, the partial angular velocity vector is given by:

wkln; {d^kjdX^ .n0 (8)

( /=1,2 ,— 6/V)

where coWm is analogous to a base vector in the 67V-dimensional space. For the human body model of Fig. 1, the wk!m are listed in Table 2.

X 1

z

3

4

5

6

7

a

9

10

11

1Z

13

0

I

I

I

I

0

0

0

0

0

0

0

0

7 , a . 9

0

D

I

I

I

Table 2

,„.„ ° 0

0

I

I

a

13.14 15

0

0

0

0

I

0

0

0

0

D

0

0

0

ufc/m f o r Fi9-16,17

IB

0

0

0

0

0

S I B

5 1 6

0

D

0

0

0

0

19.SO 21

0

0

0

0

0

0

S 0 7

0

0

0

0

0

0

ZZ.Z3 Z4

0

0

0

0

0

0

0

S I S

S I S

0

0

D

0

1

^ o

0 SQ9

D

33 ,29 30

0

0

0

0

0

0

0

0

0

saiio)

S3dO)

0

0

31,32 33 0

0

0

0

0

0

0

0

0

0

SOfil l

53112)

" i -

S3I1ZIS0I13I

37,33 39

0

0

0

0

0

0

0

0

a

0

o

0

0

(c) Angular Acceleration. The angular acceleration ak of a typical body Bk may be obtained by differentiation of equation (7):

ak = ( V>klmX-l + UklmXl) n0m (9)

for small oscillations, typical of linear response to industrial vibration, the first term is neglected. Hence, ak is:

«*="Hm^; n 0m ( 1 0)

Again we see that the angular acceleration is a function of the block array cow„, given in Table 2.

(d) Mass Center Velocities and Accelerations. Consider again the "Human Body" in Fig. 1. From (reference [16]) it is seen that the velocity and acceleration of the mass center Gk of a typical segment Bk may be obtained as follows: let 0k (k = 1, . . . ,N) be the origin or "reference point" of Bk. Then 0^ is also the connection point of Bk with Bj the adjacent lower numbered body. Let Qk be the point of Bj which is the connecting point with Bk. Let £k measure the translation of Bk

relative to Bj. Then 0̂ . and Qk are coincident when %k = 0. Finally, let gk locate Qk relative to 0,-, and let rk locate Gk

(center of mass) relative 0 .̂. Then, the position vector of Gk

relative to 0, a point fixed in inertial frame R is given by:

P* = £ i+g 2 + ?2 + g2+ • • • +tj + ik + tk + rk (11) By differentiating, we obtain the velocity of Gk, which can be expressed in the form:

\k=VklmX,n0m (12)

where, as before, there is a sum over the range of the repeated indices. From (14) it is seen that the partial velocity vectors' Vklm for / < 37V is given by:

Van, = WKmMrkh + £ WSmM (zsh + g,„) (13) s

where the indicated sum on 5 is over the bodies connecting Bk

to fi[, and where WKmM is defined as:

WKnM=empiuk/iS0Kph (14)

The rkn are the n0,„ components of tk, the qkn are the n0„ components of qk and e,ml are the permutation coefficients defined as:

1 -(m—s) (s — t) (t — m)

for />37V, the Vklm are:

Vklm ="*:(/-3/>/)m

(15)

(16)

By differentiation of equation (10), the acceleration of Gk

may be written as:

'klmX) + Vkim XI ?/)n<> (17)

Again for small oscillations, the first term may be neglected so that ak

m a y t>e written as:

»k=ykimX,n0m (18)

where Vklm array is given by equation (16) and (13).

Il l Equations of Motion

Using Kane's equations [see Kane and Levinson (1985)], together with procedures of Huston and Passerello [15], the equations of motion of the human body model may be written in the form:

alpXp=Fp(I,p, = l, . . . ,6/V) (19)

where alp is known as the "generalized mass matrix" and is given by

--mkVH r Vkpm +Iknmljlklm<J>kpn (20)

mk is the mass of Bk and Ikmn are the n0m and n0„ components of the mass center based inertia dyadic of Bk. Fp is the generalized active force associated with Xp and is discussed in the next section. As before, there is a sum over the range of the repeated indices.

Equation (19) form a set of 67V simultaneous ordinary differential equations for the 6/V generalized coordinates X, of the system.

(a) Generalized Forces. If the system of forces acting on typical body Bk is equivalent to a single force F^ passing through Gk together with a couple with torque Tk, then the

to the generalized active force Ft contribution F, of ¥k

is:

F,=

and Tk t

y klmr km + wklmTkm (21)

where Fkm and Tkm are the n0m components of ¥k and Tk. For two typical adjoining bodies Bj and Bk let the forces exerted by Bj on Bk equivalent to a single force ¥jk passing through 0^ together with a couple with torque MJk. Then, by the law of action and reaction, the forces exerted by Bk on Bj are equivalent to a single force - F ^ passing through a point 0k of Bj which coincides with 0k together with a couple with torque - Mjk. By using the definition of equation (21), it is found [see reference (16)] that the contributions of the components of FyVt

— VJk, Mjk, and - M ^ to F, are uncoupled. That is each Ft

contains only one of the components. This result is extremely useful in modeling the effects of muscles, ligaments, and other connective tissue of the model. Indeed, a general representation of the connective tissues and the external applied forces, may be written in the form:



Table 3 Model dataset

t rans I a t i on -

Fig. 2 Two typical adjoining segments By and Bk

FP=- CIPXP ~KIPXP +fp ( 0

(/ ,P=1 ,6N)

where the Clp and the Klp form the elements of the damping and stiffness matrices for modeling the connective tissues at the joints. The forcing functions or impact forces are represented by / * (/).

(b) Vibration Analysis. The equations of motion of the model presented in Fig. 1 is given by equation (19) and equation (22).

alpXp + ClpXp +KlpXp = / * ( / ) (/,/?= 1,6W) (23)

Equation (23) forms a set of 6N ordinary differential equations. Each segment of the human model is allowed three translations and three rotations as illustrated by Fig. 2. If we only restrict our study to the vertical vibration, then the equations of motion reduce to a set of N ordinary differential equations where each segment will only be allowed to translate along the vertical axis. Since the moments and the forces at the connecting joints appear singly that is one per equation, then a linear modeling of the connective tissues and muscles could be represented by diagonal stiffness and damping matrices.

K= ' Kr

3

0 "

K, . C =

~cr

0

0 "

c, . (24)

Kr and K, are the stiffness matrices corresponding to the rotation and translation, respectively. Whereas Cr and C, correspond to the damping coefficient. Both Kr and K, are matrices where the diagonal terms are given by

K r -

kr, 0

(37V X 37V)

(25)

Kt =

k'i 0

k<

where k\r,,) (/' = be expressed as

(3N.3N)

1 ,--37V) are local stiffness matrices that could

(r,t) k, •

(r,t) = k, SOI

(/ = D

(27)

k/''" is a local stiffness due to rotation or translation for segment /, and SOI denotes the shifter matrix between segment I and the fixed reference frame R. Following the same analogy the local damping matrix with respect to R could be expressed

N a .e

Lower torso

Center torso

Upper torso

Neck

Head

Upper Arm

Lower Arm

Upper Leg

Lower Leg

Mass kg

11.76

2.69

17.36

0,88

4.42

2.22

2.15

9.68

4.42

Moment of Inertia (lx, ly, Iz) kgzn2

(0.1297, 0.0817, 0.1393)

(0.0140, 0.01S9, 0.0186)

(0.2352, 0.1896, 0.1508)

(0.0014, 0.0014, 0.0006)

(0.248), 0.0307, 0.0184)

{0.016S, 0.0156, 0.0016)

(0.0311, 0.301, 0.0014)

(0.13, 0.1387, 0.0017)

(0.1315, 0.1271, 0.0046)

Location Center of Gravity (ra)

(0.026, 0., -0.079)

(0.033, 0., 0.072)

{0.029, 0 . , 0.162)

{0., 0., 0.063)

(0.006, 0., 0028)

(0. , 0., -0.122)

(0. , 0 . , -0.167)

(0 . , 0.006, -0.207)

(0.016, 0., -0.272)

Location Origin of Joints (m)

(0.0, 0., 0.)

{0., 0., 0.132)

{0.065, 0., 0.318)

(0. , 0., 0.124)

(0.03, 0.189, 0.260)

{0.08, ±0.2, -0.261}

(0.042, 0.087, -0.07)

(0. , +0.008, -0.405)

(0.46, 0., 0.)

Table 4 Characteristics of the linear coefficients of the spring and dampers after Muskian (1974)

Dampinq Gonstant (c)

(kqf/cm/sec)

c n = 0.373

c 2 ? = 0.298

c 3 3 = 0.298

c . . = 3.651 44

c 5 5 = 3.651

Spring Constant (k)

(kqf/cm)

k l l = 2 "

k2 2 = 0.8941

k J 3 = 0.8941

k,„ = 53.64 44

k 5 s = 53.64

(r,t) C. =

r,t C, SOI (28)

The elements of both k/r '" and C/r'() represent the coefficients for modeling the forces and moments due to the connective tissues, muscles, disks, and ligaments at the joints. The diagonal elements of those matrices represent the forces between adjacent segments while the off-diagonal terms correspond to nonadjacent segment forces.

(c) Model Data and Results. The dataset used here includes masses, moment of inertia, center of gravity location and distance between joints. These data are based on a 50 percentile sized part 572 dummy design. Table 3 presents the proposed model data collected from [18]. The connective tissues are modeled by springs and dampers. The values of the tissue springs and damping parameters are obtained from studies on the characteristics of specific subsystem (Muskian, [4]). Table 4 presents a listing of those values. Since the data for modeling the connective tissues at the joints were not available for all individual parts of the human body, some of the coefficients, especially the legs and the arms, were approximated based on their respective cross-section areas. The stiffness and damping matrices are both diagonal matrices with coefficients given by Table 4. For the model presented in Fig. 1, the oscillation of the chair could be represented by a sinusoidal function applied to the lower torso. This function is given by:

7 7 ( 0 =/o sin «f (29)

where the chair characteristics are given by:

a>: circular frequency f0: max. amplitude of the force F*

the circular frequency is a function of the frequency of oscillation of the chair (/) that is

CO = 2 T T / (30)



Simulatioi

Experiment - Sitting Relaxed

Frequency, Hz

Fig. 3 The transmission of vibrations from the seat to the head at varied postures obtained from Coermann [7] compared with the analytical results of the model presented. (The transmission factor is defined as the ratio of acceleration of the head divided by the acceleration of the seat (RMS).)

If we restrict our analysis to vertical vibration, the chair vibration is imposed to the lower torso along the z direction. Equations (23) along with equations (28) and (29) are simulated in computer program called Human Body Vibration or HBV (currently in its final stage at the University of Illinois at Chicago), and integrated to solve for the displacement X,, the velocity Xt an acceleration X, of individual parts of the body. For the swept sinusoidal force applied to the lower torso the maximum amplitude of acceleration was maintained at (1) g and the frequency of excitation varied from 1 to 20 Hz. For each frequency used in the forcing function the maximum displacements, velocities, and accelerations were recorded. The root mean square method (RMS) (see reference [2]) was used and the ratio of the acceleration and displacement of different parts of the body to the seat were computed.

To validate our results, experimental data from Coermann [7] is used. Coermann performed experiments to measure the response of the human body to sinusoidal vibrations of the seat at low frequencies. The subject sat on a stiff plate which was connected over force transducers to the vertically vibrating excitor producing sinusoidal motions at low accelerations. The natural frequency of the system consisting of the mass of the subject and the elasticity of the plate must be at least 6 times higher than the highest frequency studied (in our case 20 c/s).

Eight subjects at ages 29-47 with different weights and heights were employed in the measurements. The amplitude of the shake table was kept at a level comfortable to the subject. The legs of the subject hung down freely over the side of the shake table. The hands lay on the upper thigh and all belting was removed from the waist. Records were taken at erect and relaxed postures of the sitting subject and at an erect posture standing with stiff knees. For each condition about fifty records were taken to cover the frequency range of 1 to 20 Hz. The exposure time to vibration was an average of one minute for each record.

For measuring the transmitted forces and the displacements, oval-shaped ring force transducers and variable reluctance displacement transducers were used.

The transmission of vibrations through the body to the head, defined as the ratio between the acceleration on the head to the acceleration on the seat, were calculated and plotted for low accelerations at the frequency range studied. Light weight

Table 5 Stiffness and damping coefficients used in the simulation of the model subjected to a sinusoidal forcing function

Segment

Lower Torso

Middle Torso

Upper Torso

NecV.

Haad

Stiffness Coefficient Vertical (Nt/m)

50000

105000

105000

120000

120000

Damping Coefficient Vertical Nt/m/sec

1000

2000

2000

2500

2500

(1) The coefficient refers to the joint at the origin of the local

axis of the segment

accelerometers were mounted on the top of the head with an elastic bandage to measure the acceleration of the head.

For measuring relative displacements a flexible aluminum girdle with two molds covering the hip bone was tightened around the pelvis, and another flexible aluminum plate was mounted by elastic belts over the area of the first thoracic vertebra. The relative displacements between the shake table and the hip girdle and between the hip girdle and the neckplace were measured by elastic capillary tubes filled with mercury and connected to a resistance bridge.

A simulation of the human model presented in this paper for the same conditions as the experiment conducted by Coermann [7] was sought. The connective springs and dampers values given by reference [4] had to be modified in order to achieve better results. Those values are given by Table 5.

In Fig. 3, the transmission of vibration to the head as obtained from the experiment and simulation are compared. The simulation closely fits the experimental findings for the sitting relaxed posture, with the major peak at 4.85 Hz having an acceleration ratio of slightly over 2. Simulation also reveals another peak at 2.2 Hz. At higher natural frequencies of the model in the range, peaks are not observed due to the high damping coefficients used.

In Figs. 4 and 5, relative body displacements between the shake table and pelvis, and between the pelvis and neck, relative to 1 g on the shake table are compared with the experimental findings. For each frequency these are the peaks of the relative displacement waves. In conformance with the experiment, at each frequency the relative displacement between



Simulation

Experiment

12 14 Frequency, Hz

Fig. 4 Relative body displacements between shake table and pelvis relative to 1g on the shake table for a sitting human body swept by sinusoidal forcing function. Experimental values are from Coermann [7]

7.5 10.0 .12.5 15.0 17.5

Frequency (HZI

Fig. 7 Acceleration transfer of the head for a sitting human model with an impulse function of 15 N and 0.25 s duration applied to the chair. The acceleration transfer is computed using the fast fourier transform (FFT).

12 14 Frequency, Hz

Fig. 5 Relative body displacements between pelvis and neck relative to 1g on the shake table for a sitting human body swept by a sinusoidal forcing function. Experimental values are from Coermann [7]

max

D.O 2.5 5.0 7.5 10.0 12 5 15.0 17.5

Frequency IHZ)

Fig. 8 Acceleration transfer of the upper torso for a sitting human model subjected to an impulse function of 15 N and a 0.25 s duration. The acceleration transfer is computed using the fast fourier transfer (FFT).

Fig. 6 Square impulse function

the pelvis and neck is smaller than the relative displacement between the shake table and pelvis. Second, a square impulse function was used as an impact force applied to the lower torso replacing the sinusoidal function. The impulse function is given by Fig. 6. Using fourier transforms the frequency response of the accelerations of the head, upper torso and the lower torso are given by Figs. 7-9. Figure 7 shows a resonance peak at 3 Hz. The peak of resonance for the upper torso and the lower torso occur at 2.0 Hz (see Figs. 8 and 9).

Frequency (HZ J

Fig. 9 Acceleration transfer of the lower torso for a sitting human model subjected to an impulse function ot 15 N and a 0.25 s duration. The acceleration transfer is computed using the fast fourier transform (FFT).



IV Discussion References Two cases were investigated. The first one involves the

sinusoidal forcing function applied to the lower torso. In this analysis the procedures developed in this paper show a good agreement with previous work done under the same conditions. However, the amplitude ratio between different parts of the body responses and the input accelerations were different. Actually if higher acceleration is allowed to the chair oscillation; the neck showed an unrestrained motion, and more than one peak of resonance in the range of frequency prescribed occurred. The second case where the sinusoidal function was replaced by the square impulse functions, the procedure used was consistant and proved to be effective. The individual parts of the body motion responses to different input of excitation reported here show the general approach to predicting the skeletal system response. This is important due to the fact that the experimental data collected from measuring the responses between surface mounted transducers and those mounted on pins rigidly attached to the skeletal system have shown a substantial difference [see reference (11)]. The analysis presented could be enhanced by dividing each segment of the model presented into more segments. The forces and damping coefficients used in this paper do not account for the forces due to the muscles that interconnect with other segments of the body other than the adjacent segments.

In our analysis a number of questions concerning the modeling of the muscles were raised. The results obtained in the simulation lead to the magnitudes of springs and dampers forces (assumed linear). Those forces represent the equivalent forces acting at the joints for the interconnected body segments. It is apparent from the model presented that for low accelerations type of stimulus to the body, some muscles are passive and, therefore, do not contribute to the generalized active forces. This is especially the case for muscles that interconnect the non-adjacent body segments. Those forces could become a factor when the acceleration increases. They play an interesting role in constraining the body from undergoing large displacements. Their contribution to the equations of motion could easily be achieved by using the off diagonal terms in the stiffness and damping matrices. It is further believed that a detailed investigation of the spine motion could be useful in understanding the role of the muscles, intervertebral disks and other connective tissues.

V Conclusion

The procedures developed in this paper are very useful in extracting the natural frequency and mode shapes of such models of the human body as presented in Fig. 1. It is also an effective and computer oriented method that can be used to monitor the transient response of each individual part of the body. The external forces are easily incorporated in the equations of motion due to the fact that the generalized forces are uncoupled. Further, the procedures developed can be used to study not only the human body response due to vertical vibration, but its response to translation and rotation in the three directions X, Y, Z.

It is suggested that more experimental data is needed for better formulation of the springs and dampers for the model presented. Of course it is quite difficult to evaluate those constants because individuals react differently to various vibration media. True mathematical relationships may become apparent from additional empirical data, therefore, reducing the amount of testing necessary to establish thresholds for safety standards.

1 Kelsey, J. L., and Hardy, R. J., "Driving of Motor Vehicles as a Risk Factor for Acute Herniated Lumbar Intervertebral Disc," Am. J. Epidem., Vol. 102, No. 1, 1975, pp. 63-73.

2 ISO 2631-1978 (E) Guide for the Evaluation of Human Exposure to Whole Body Vibration, International Organization for Standardization, Ref. No. ISO 2631-1978 (E).

3 Patil, M. K., Palanichamy, M. S., and Ghista, D. N., "Man-Tractor System Dynamics: Towards a Better Suspension System for Human Ride Comfort," J. Biomechanics, Vol. II, 1978, pp. 397-406.

4 Muskian, R., and Nash, C. D., "A Model for the Response of Seated Humans to Sinusoidal Displacement of the Seat," J. Biomechanics, Vol. 7, 1974, pp. 209-205.

5 Muskian, R., "A Nonlinear Model of the Human Body in the Sitting Position Subjected to Sinusoidal Displacements of the Seat," Ph.D. Dissertation, University of Rhode Island Kingston, 1970.

6 Pradko, F., Lee, R., and Greene, J. D., "Human Vibration Response Theory," ASME Biomechanics Monograph, 1967, pp. 205-222.

7 Coermann, R. R., "The Mechanical Impedance of the Human Body in Sitting and Standing Positions at Low Frequencies," Human Factors, Vol. 4, 1962, pp. 227-253.

8 Panjabi, M. M., Andersson, G. B. J., Jorneus, L., Hult, E., andMattson, L., "In Vivo Measurements of Spinal Column Vibrations," / . Bone Joint Surgery, Vol. 68A, 1984, pp. 695-702.

9 Pope, M. H., Wilder, D. G., and Frymoyer, J. W., "Vibration as an Etiologic Factor in Low Back Pain," Proc. Confer. Engineering Aspects of the Spine, Engineering Aspects of the Spine, Meeting Brit. Orthop, Assoc, and Inst. Mech. Eng., U.K., 1980.

10 Quandieu, P. , and Pellieux, L., "Study in Site et in Vivo of the Acceleration of Lumbar Vertebrae of a Primate Exposed to Vibration in the Z-Axis," Journal of Biomechanics, Vol. 15, 1982, pp. 985-1002.

11 Pope, M. H., Svensson, M., Broman, H., Andersson, G. B. J., "Mounting of the Transducers in Measurement of Segmental Motion of the Spine," Journal of Biomechanics, Vol. 19, No. 8, 1986, pp. 675-677.

12 Wilder, D. G., Woodworth, B. B., Frymoyer, J. W., and Pope, M. H., "Vibration and the Human Spine," Spine, Vol. 7, 1982, pp. 243-254.

13 Sjoflot, L., and Suggs, C. W., "Human Reactions to Whole Body Transverse Angular Vibrations Compared to Linear Vertical Vibrations," Ergonomics, Vol. 16, 1973, pp. 455-468.

14 Amirouche, M. L., and Huston, R. L., Modeling of Human Body Vibration, Trends in Ergonomics/Human Factors I, Elsevier Science Publishers, B. V. (North Holland), 1985, pp. 65-70.

15 Huston, R. L., and Passerello, C , "On the Dynamics of Chain Systems," ASME Paper No. 74 WA/Aug. 11, 1974.

16 Amirouche, M. L., "Flexibility and Transient Effects in Multibody Systems," Ph.D. Thesis, University of Cincinnati, Ohio, 1984.

17 Kane, T. R., and Levinson, D. A., Dynamics Theory and Application McGraw-Hill, 1985.

18 Wismans, J., "Comparison of Mass Distribution Data of the Part 572 Dummy," S.A.E., Warrendale, PA, Mathematical Simulation Subcommittee, 1983.

19 Goldman, B. E., and Von Gierke, H. E., "The Effects of Shock and Vibration on Man, Naval Research Institute Report No. 60-3, Bethesda, Md., 1960, (or Chapter 44 of Shock and Vibration Handbook, edited by Harris and Crede, 2nd ed., 1976, McGraw-Hill, New York).

20 Huston, J. C , Passerello, C. E., and Huston, R. L., "Numerical Prediction of Head/Neck Response to Shock-Impact, Measurement and Prediction of Structural and Biodynamic Crash-Impact Response," ASME, 1976, pp. 137-150.

21 Radke, A. O., "Vehicle Vibration. . .Man's Environment," ASME Paper 57-A, 1957, p. 54.

22 Zagorski, 3., Jakubowski, R., Solecki, L., Sadlo, A., and Kasperek, W., "Studies on the Transmission of Vibrations in Human Organism Exposed to Low-Frequency Whole-Body Vibration," Acta Physiol. Polonica, Vol. 27, No. 4, pp. 347-354.

23 Broderson, A. N., and Von Gierke, H. E., "Mechanical Impedance and Its Variation in the Restrained Primate during Prolonged Vibration," ASME Publ. No. 71-WA-/BHF-8, (10 pp.) (AMRL-TR-71-67).

24 Garg, D. P., and Ross, M. A., "Vertical Mode Human Body Vibration Transmissibility," IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-6, pp. 102-112.

25 Mertens, H., "Nonlinear Behavior of Sitting Humans Under Increasing Gravity," Aviat. Space Environ. Med., Vol. 49, 1978, pp. 287-298.

26 Griffin, M. J., Lewis, C. H., Parsons, K. C , and Whitham, E. M., "The Biodynamic Response of the Human Body and Its Application to Standards," AGARD Conf. Proc. No. 253, A28-1-A28-18.

27 Linder, G. S., "Mechanical Vibration Effects on Human Being," Aerospace Medicine, Vol. 33, pp. 939-950.

216/ Vol. 109, AUGUST 1987 Transactions of the ASME


A P P E N D I X

(a). The transformation matrix between two typical adjoin- the transformation between any body k and fixed reference ing bodies is frame R for small oscillations is

S0K--

1 - ( 7 l + 7 2 + " - + 7*) ( f t + f t + ""- + ft)

(?1 +72+--7*) 1 - (« !+« ; , +-- + «,</)

- ( / 3 , + | 3 2 + ~ + /3*) ( a , + a 2 + -- + «*) 1

SJK=

CpkCyk

C$kSyk

. -«ft

SakSpkCy/c — CakS

SakSPkSyk+

CakCyk

SakCPk

for small oscillations

SJK=

' 1

7*

. ~ft

yk

~Jk

1

<*k

CakSpkClk + SakSyk

CakSI3kSyk-SakCyk

CakC0k

ft -ctk

1

(b) Description of Table 1

The configuration of the human body model in Fig. 1 could be obtained by knowing the total number of segments of the body, the lower body array, and the connecting body array which represent row 2 and row 3, respectively. For instance any column describes the chain of segments needed to go from segment K to the reference frame R (0). Tracking segment 11: from Table 1 Column 11 gives the following: Segment 11 has lower body 10, then 10 is connected to segment 3, then 3 has lower body 1 which is connected to 0. This translates into 11, 10, 3, 2, 1, to be the chain of segments connecting 11 to 1.

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Modeling of Human Reactions to Whole-Body Vibration