1
Chapter 8 Human Vibration -- A Model for Human Vibration Studies and for Predicting
Response to Jolting and Jarring
There has long been an interest in modeling human body vibration and specifically, the response
of the human frame exposed to vibratory environments. Long term exposure to vibration is
believed to be the cause of numerous maladies including low back disorders and vascular and
neurological disorders.
Recently analysts have been interested in studying human response to repeated jarring and
jolting superposed upon a background of random vibration, as occurs with operators of large
motor vehicles and off-highway equipment. Of particular interest is the long-term effect of this
jarring and jolting upon spinal stability.
To study this phenomena it is useful to have a human body model which can mimic whole body
vibration and the response to jarring and jolting, while at the same time overcoming the
complexity of the human system with its varying geometries and nonhomogeneous material
properties. That is, models are sought which can accurately represent the vibration and yet be
tractable for quantitative (numerical) analysis.
In here we aim to present such a model with the objective of being able to study both whole body
vibration (WBV) and the jolting and jarring (JAJ) phenomena. Our model is a finite-segment
(lumped mass) system as in Figures 1 and 2. Such models have been successfully employed for
the past two decades and more for studying human response to high acceleration (and
deceleration) environments as in motor-vehicle accidents.
As noted, research in human vibration has been ongoing for many years. The volume of Griffin
[1] summarizes much of the work up to 1990. More recently Bovenzi and Hulshof [2] provided
a comprehensive review of literature on low back pain -- an update of their earlier review of
1987. Attempts to measure, monitor, and assess the harmful effects of WBV in general have
been documented by Donati [3], Griffin [4], Lewis and Griffin [5], and Moeda and Morioka [6].
Low back pain, spinal degeneration and instability, and herniated discs, appear to be the most
frequent of the adverse affects of WBV and thus presumably of JAJ as well. Other excellent
reviews have been prepared by Wilder and Pope [7], Seidel and Heide [8], and Wikstrom et al.
[9].
In the following section we describe our model and in the subsequent section we present its
governing equations. Example movements are presented in the next section and the final
section has a discussion with concluding remarks.
8.1 The Model
In the models depicted in Figures 8.1 and 8.2 the human body is represented by a series of
connected bodies (lumped parameters) simulating the human frame. For the most part these
bodies are ellipsoids, elliptical cylinders, and frustums of elliptical cones, representing the limbs
of the human body. Such models have been employed extensively since before 1975 to study
2
vehicle occupant behavior in motor vehicle accidents. (See for example Bartz [10], Huston et
al. [11, 12], King and Chou [13], Huston [14], Prasad [15], and King [16].)
Recently there have been significant advances in computer hardware and software which
make the use of these models more practical (easier to use), more accurate, and with applicability
in areas beyond accident reconstruction. Thus the study of whole body vibration and of jolting
and jarring is ideally suited for the use of these models.
For the research of this paper we used a model and system as in Figure 8.2 which has 17
bodies representing the feet, hands, upper and lower arms and legs, torso, head and neck.
Springs and dampers at the joints simulate the soft tissue connection. The objective is to
determine the model's response to vibration and impulse movement of the operator's seat.
8.2 Governing Equations
8.2.1 Configuration and Degrees of Freedom
The models of Figures 8.1 and 8.2 are multibody systems. As such they may be studied using
multibody dynamics techniques which have been developed during the past two decades (Huston
[17, 18, 19]). While there is theoretically no limit to the number of bodies in a model, for
practical purposes most human body models have fewer than 20 bodies. If the bodies are
connected by spherical joints, a system of N bodies will have 3N + 3 degrees of freedom (three
for translation of a reference body and three for the rotation of each of the N bodies).
The model of Figure 8.2 has 17 bodies and thus in general 54 degrees of freedom.
The degrees of freedom are represented by either translation or rotation variables called
"generalized coordinates" and are frequently designated as qr (r=1,...,n) where n is the number of
degrees of freedom.
8.2.2 Kinematics
For each body Bk of the model there are four kinematical quantities of interest. These are the
velocity and acceleration of the mass center Gk and the angular velocity and angular acceleration
of Bk itself all measured in an inertial (Newtonian) reference frame R. These quantities may be


v k  v km q  n m


and a k  (v km q   v km q  )n m
(8-1)
expressed in the forms:


k  km q  n m


 km q  )n m
and  k  (km q   
(8-2)

and where the n m (m=1,2,3) are mutually perpendicular unit vectors fixed in R and where the
vklm and the k  m are components of the "partial velocity" and "partial angular velocity"
vectors of Gk and Bk (see for example Kane and Levinson [20]). Efficient algorithms have been
3
written to computationally develop the vklm and k  m arrays and their derivatives (see for
example Huston [17, 21]). These arrays are the "building blocks" of the multibody dynamics
formulation.
Note in Equations (8-1) and (8-2) and in the sequel, unless otherwise noted, repeated
indices designate a sum over the range of the index.
8.2.3 Kinetics
Let the human body model be subjected to a field of force systems including both externally
applied forces, such as gravity and contact forces, and internally applied forces, such as inertia
forces and forces and moments exerted across connecting joints. Let these force systems be
represented on each body Bk of the model by an equivalent force system consisting of a simple


force Fk passing through the mass center Gk of Bk together with a couple with torque Tk .
Then the partial velocity and the partial angular velocity arrays may be used to determine the
generalized forces on the model for each generalized coordinate. Specifically, for body Bk the


generalized force Fq r due to Fk and Tk may be expressed as:




Fq  Fk  vkm n m  Tk  km n m  Fkmvkm  Tkmkm
(no sum on k )
(8-3)



where the Fkm and Tkm are the n m components of Fk and Tk .
It is usually convenient to separate the inertia forces from the other forces. For typical

body Bk the inertia forces are equivalent to a single force Fk* passing through Gk together with a



couple with torque Tk* where Fk* and Tk* may be expressed as:


Fk*  m k a k
and

 


Tk*   Ik   k   k  ( Ik   k )
(no sum on k )
(8-4)

where mk is the mass of Bk and Ik is its central inertia dyadic (see for example Kane and
Levinson [20]). Then in a form similar to Equation (8-3) the generalized inertia force on Bk for
the generalized coordinate ql is:
*
*
Fq*  Fkm
v km  Tkm
km
(no sum on k)
(8-5)



where F*km and T*km are the n m components of Fk* and Tk* .
By substituting from Equations (8-1), (8-2), and (8-4) into Equation (8-5) F*q may be
expressed in the form
4
Fq*  (m k v km v kpmq p  I kmn  km  kpmq p
 kpn q p
 m k v km v kpn q p  I kmn  kmn 
(8-6)
 e rsm I ksn  km  khr  kpn q h q p )
where ersm is the permutation array which may be expressed as:
e rsm  (1 / 2)( r  s)(s  m)(m  r )
(8-7)
Note that by relaxing the "no sum on k" restriction we have the generalized forces for
each ql for the entire system of bodies in the model.
8.2.4 Dynamics
Once the generalized forces are known, the governing dynamical equations are readily obtained
using Kane's equations (see for example Kane and Levinson [20]), which may be expressed in
the simple form:
Fq  Fq*  0
  1,..., n
(8-8)
Using the expression for Fq and F*q of Equations (8-3) and (8-6) these governing
equations may be expressed as
a p q p  f 
(8-9)
where the a  p and the f  are:
a p  m k v km v kpm  I kmn  km  kpn
(8-10)
and
 kpn q p
f   Fq  (m k v km v kpn q p  I kmn km 
 e rsm I ksn km khr kpn q h q p )
(8-11)
Observe the central role of the partial velocity and partial angular velocity arrays ( vk  m
and k  m ) and their derivatives in the governing equations.
8.3 Application with Whole Body Vibration (WBV) and with Jolting and Jarring (JAJ)
Heavy equipment operators and operators of off-highway equipment are repeatedly subjected to
5
WBV and JAJ. To simulate these phenomena we place our model in a sitting position as in
Figure 8.2. We connect the bodies with spherical joints -- although the knees and elbows could
well be represented by revolute or hinge joints.
We number (or label) the bodies as in Figure 8.3. Specifically, we select a major body
(the lower torso), call this our reference body, and number and label it as: 1 and B1. We then
number and label the remaining bodies of the system in ascending progression away from B1 as
in Figure 8.3. With this numbering system each body is connected to a unique lower numbered
body. [The connection to higher numbered bodies is not unique. Some bodies (such as B3)
are connected to more than one higher numbered body while others (such as B8) are not
connected to any higher numbered body.]
This connection configuration is efficiently described by the "lower body array" (Huston
[17]) which is simply a listing of the numbers of the adjoining lower numbered bodies for each
body. Specifically, for the numbering in Figure 3, the lower body array L(K) is
(K) : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
L(K) : 0
1 2 3 4 5 3 7 3
9 10
1 12 13
(8-12)
1 15 16
where (K) is the body number and where the lower body of B1 is the inertial frame R and
numbered as 0.
The array L(K) of Equation (8-12) not only defines the connection configuration, but it
also identifies extremity bodies (hands and feet), intermediate bodies (arms and legs), and
branching bodies (chest and lower torso). Specifically, if a body number does not appear in
L(K) (such as 6, 8, 11, 14, and 17) the corresponding body is an extremity. If a body number
appears only once (such as 2, 4, 5, 7, 9, 10, 12, 13, 15, and 16) the corresponding body is an
intermediate body. Finally, if a body number appears more than once (such as 1 and 3) the
corresponding body is a branching body.
The L(K) array also describes the branches of the system. If L(K) is regarded as an
operation on the array (K) then we can repeat the operation and obtain a lower body array of
L(K), or L(L(K)) [or L2(K)] and in the process defined L(0) as 0. We can repeat the process
again and again until we obtain an array of 0s as in Table 1.
6
Table 8.1
Lower Body Arrays for the Model of Figure 3.
(K)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
L(K)
0
1
2
3
4
5
3
7
3
9
10
1
12
13
1
15
16
L2(K)
0
0
1
2
3
4
2
3
2
3
9
0
1
12
0
1
15
L3(K)
0
0
0
1
2
3
1
2
1
2
3
0
0
1
0
0
1
L4(K)
0
0
0
0
1
2
0
1
0
1
2
0
0
0
0
0
0
L5(K)
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
L6(K)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
Observe in Table 8.1 that we select an extremity, say the right hand (B11), then the
numbers in column 11 (11, 10, 9, 3, 2, 1, 0) represent the body numbers of the bodies leading
from the right hand to the reference frame R. (These sequences are useful in developing the
system kinematics.)
These observations show that the lower body array of Equation (8-12) is equivalent to the
body arrangement in the model of Figure 8.3.
Tables 8.2 and 8.3 provide geometric and inertia data for the model of Figure 8.3.
In modeling the human joints we have also incorporated motion constraints to simulate
the movement limitation provided by the soft tissue (ligaments, tendons, discs, and cartilage).
Specifically, we have incorporated one-way dampers at the joints which create resistive moments
once the range of motion is exceeded. These moments are of the form:
8
Table 8.2
Body Mass on Mass Center/Reference Point Location for the Human Body Model1
Mass Center Location2
(m)
Reference Point Location3
(m)
Body
Numb
er
Label
Mass
(kg)
1
Lower Torso
13.36
0.0
0.0
0.0
0.0
0.0
0.0
2
Mid Torso
13.36
0.0
0.0
0.103
0.0
0.0
0.103
3
Upper Torso
10.94
0.0
0.0
0.100
0.0
0.0
0.206
4
Left Upper
Arm
2.60
0.0
0.0
-0.113
0.0
0.212
0.147
5
Left Lower
Arm
2.09
0.0
0.0
-0.178
0.0
0.0
-0.297
6
Left Hand
0.50
0.0
0.0
-0.076
0.0
0.0
-0.357
7
Neck
0.48
0.0
0.0
0.060
0.0
0.0
0.201
8
Head
5.84
0.0
0.0
0.102
0.0
0.0
0.119
9
Right Upper
Arm
2.60
0.0
0.0
-0.113
0.0
-0.212
0.147
10
Right Lower
Arm
2.09
0.0
0.0
-0.178
0.0
0.0
-0.297
11
Right Hand
0.50
0.0
0.0
-0.076
0.0
0.0
-0.357
12
Right Upper
Leg
8.10
0.0
0.0
-0.251
0.0
-0.078
0.017
13
Right Lower
Leg
5.22
0.0
0.0
-0.224
0.0
0.0
-0.472
14
Right Foot
1.021
0.089
0.0
-0.025
0.0
0.0
-0.457
9
15
Left Upper Leg
8.16
0.0
0.0
-0.25
0.0
0.078
0.017
16
Left Lower
Leg
5.22
0.0
0.0
-0.224
0.0
0.0
-0.472
17
Left Foot
1.021
0.089
0.0
-0.025
0.0
0.0
-0.457
Notes:
1
See Figure 8.3
In local body-fixed X, Y, Z coordinates (in meters) (X - forward, Y - left, Z - up)
3
In local body-fixed X, Y, Z coordinates (in meters) in adjacent lower numbered body (X
- forward, Y - left, Z - up)
2
10
Table 8.3
Inertia Matrices for the Human Body Model1
Body Number
(Name)
1
(Lower Torso)
Matrix
Elements
0.148
0.0
0.0
Body Number
(Name)
Matrix
Elements
Body Number
(Name)
Matrix
Elements
Body Number
(Name)
Matrix
Elements
Body Number
(Name)
Matrix
0.0
0.0
0.144
4
(Left Upper Arm)
0.027
0.0
0.0
Body Number
(Name)
Matrix
Elements
0.0
0.090
0.0
2
(Mid Thorax)
0.0
0.027
0.0
0.0
0.0
0.001
0.148
0.0
0.0
0.0
0.015
0.0
0.029
0.0
0.0
0.0
0.029
0.0
0.0
0.0
0.003
0.0
0.0
0.001
13
(Right Lower Leg)
0.115
0.0
0.0
0.0
0.116
0.0
0.0
0.0
0.002
0.038
0.0
0.0
0.0
0.0
0.0
0.001
0.0
0.0
0.038
0.0
0.001
0.0
0.0
0.0
0.001
0.0
0.0
0.0
0.019
0.0
0.006
0.0
0.0
0.0
0.000
0.0
0.0
0.0
0.105
6
(Left Hand)
0.001
0.0
0.0
0.0
0.001
0.0
0.0
0.0
0.000
0.027
0.0
0.0
0.0
0.027
0.0
0.0
0.0
0.001
0.096
0.0
0.0
0.0
0.096
0.0
0.0
0.0
0.003
15
(Left Upper Leg)
0.0
0.0
0.006
17
(Left Foot)
0.001
0.0
0.073
0.0
12
(Right Upper Leg)
14
(Right Foot)
0.001
0.0
0.0
0.106
0.0
0.0
9
(Right Upper Arm)
11
(Right Hand)
16
(Left Lower Leg
0.115
0.0
0.029
0.0
8
(Head)
10
(Right Lower Arm)
0.029
0.0
0.0
0.0
0.0
0.144
5
(Left Lower Arm)
7
(Neck)
0.015
0.0
0.0
0.0
0.090
0.0
3
(Upper Torso)
0.0
0.096
0.0
0.0
0.0
0.096
0.0
0.0
0.0
0.003
11
Elements
0.0
0.0
0.116
0.0
0.0
0.002
0.0
0.0
0.006
0.0
0.0
0.006
Note:
1
See Figure 8.3. The matrices are referred to local X, Y, Z body fixed coordinates
(in kilogram meters squared) (X - forward, Y - left, Z - up)
12
0


M =  K 

 - K 
0min <  < max
 < 0 and  < min
(8-13)
 > 0 and  > max
where  is a typical joint rotation angle and where min and max are the minimum and
maximum values of theta.
The simulation of whole body vibration (WBV) can be obtained by imposing an
acceleration on the operator seat and thus accelerating the lower torso (body B1). For example
for a simple oscillation we can exert acceleration of the form
a  A sin t
(8-14)
where A is the amplitude,  is the circular frequency, and t is time.
For a more random vibration we can exert acceleration in the form shown in Figure 8.4
[taken from NIOSH field measurements (Li et al. [22])]. To simulate jarring and jolting (JAJ)
we can superimpose accelerations with triangular profiles as in Figure 8.5.
8.4 Validation and Examples
While Equations (8-9) have been used extensively in crash victim simulation and have been
repeatedly validated against experimental data (Huston [23]), they have not been similarly used
and validated for WBV and JAJ. Indeed, compared with accident victim simulation there is
relatively little data available for WBV and JAJ. Nevertheless, there is some data that is useful.
First, Punjabi et al. [24] conducted a series of experiments where they measured the lumbar
vertebral response of five volunteers subjected to low frequency (between 2 and 15 Hz)
sinusoidal vertical seat acceleration of 0.1g. They found a resonance at approximately 4.3 Hz.
We used the same sinusoidal input with our model and recorded the middle torso
response. Comparisons of the results with those of the Punjabi experiments are shown in
Figures 8.6 and 8.7.
Next, we also studied and compared the vibration response of the model in a relaxed
(reclined) and in an erect position. Figures 8.8, 8.9, and 8.10 show the resulting ratio of
response to input acceleration for the head, middle torso and knee.
Third, we used data recorded by the National Institute of Occupational Safety and Health
(NIOSH) for mining equipment operator seat acceleration as input acceleration to the model's
lower torso. We computed the motion transmitted to the head. Figure 8.4 shows the input
acceleration and Figure 8.11 shows the head response (Li, et al. [22]).
Finally, to simulate JAJ we superimposed an impulse upon the NIOSH data as in Figure
8.5. Figure 8.12 shows the model's head response.
8.5 Discussion
In view of Figures 8.4 and 8.5 we see that the numerical simulation using the model produces
results consistent with experimental data. In Figures 8.8, 8.9, and 8.10, we see that there are
13
resonances of approximately 5 Hz and 10 Hz for the head, at 5 Hz for the middle torso, and at 5
Hz for the knee. It is also seen that the acceleration is generally lower at higher frequencies
when the model is in the relaxed position.
14
Figures 8.4 and 8.11 show input and head response from random vibration as
is typically encountered in heavy equipment operation. Figures 8.5 and 8.12 show
the effect of an impulse, such as a vehicle operator striking a pot hole or a similar road
irregularity. Observe that the impulse increases the response amplitude even after
the impulse is over. Observe also that for an erect operator the impulse has little
effect upon head rotation but for an inclined operator, the impulse increases the head
rotation. Finally, as expected, the impulse significantly increases the vertical head
movement.
8.6 Practical Importance and Conclusions
These results show that multibody based computer models have the potential for
effectively studying both WBV and JAJ. A principal difference in these applications,
as compared with crash victim simulation, is the key role played by the modeling of
the soft tissue at the joints as we have done in Equation (8.8-8.13). The difficulty
arises from the absence of sufficient in vivo experimental data to accurately set the
coefficients and range of motion parameters. This is similar to problems
encountered with multibody head/neck models (Tien and Huston [25]). Moreover,
since these parameters are largely individual dependent, only approximate values are
likely to become known. Nevertheless, it is believed that even with these
approximate values, the multibody modeling and simulation procedure offer a means
for studying WBV and JAJ which previously has not been generally available.
8.7 References
[1] Griffin, M. J., 1990. Handbook of Human Vibration, London, Academic Press.
[2] Bovenzi, M. and Hulshot, C. T. J., 1998.
An Updated Review of
Epidemiological Studies on the Relationship between Exposure to Whole-Body
Vibration and Low Back Pain, Journal of Sound and Vibration, Vol. 215, pp. 595-611.
[3] Donati, P., 1998. A Procedure for Developing a Vibration Test Method for
Specific Categories of Industrial Tasks, Journal of Sound and Vibration, Vol. 215, pp.
947-958.
[4] Griffin, M. J., 1998. A Comparison of Standardized Methods for Predicting the
Hazards of Whole-Body Vibration and Repeated Shocks, Journal of Sound and
Vibration, Vol. 215, pp. 883-914.
15
[5] Lewis, C. H. and Griffin, M. J., 1998. A Comparison of Evaluations and
Assessments Obtained Using Alternative Standards for Predicting the Hazards of
Whole-Body Vibration and Repeated Shocks, Journal of Sound and Vibration, Vol.
215, pp. 947-958.
[6] Maeda, S. and Morioka, M., 1998. Measurements of Whole-Body Vibration
Exposure from Garbage Trucks, Journal of Sound and Vibration, Vol. 215, pp.
959-964.
[7] Wilder, D. G. and Pope, M. H., 1996. Epidemiological and Etiological Aspects
of Low Back Pain in Vibration Environments -- An Update, Clinical Biomechanics,
Vol. 11, pp. 61-73.
[8] Seidel, H. and Heide, R., 1986.
Long-Term Effects of Whole-Body Vibration:
A Critical Survey of the Literature, International Archives of Occupational and
Environmental Health, Vol. 58, pp. 1-26.
[9] Wilkstrom, B.O., Kjellberg, A., and Landstrom, U., 1994. Health Effects of
Long-Term Occupational Exposure to Whole Body Vibration:
A Review,
International Journal of Industrial Ergonomics, Vol. 14, pp. 273-292.
[10] Bartz, J. A., 1972. Development and Validation of a Computer Simulation of a
Crash Victim in Three Dimensions, 16th Stapp Car Crash Conference, Society of
Automotive Engineers, Warrendale, PA, pp. 105-127.
[11] Huston, R. L., Hessel, R. E., and Passerello, C. E., 1974. A Three-Dimensional
Vehicle-Man Model for Collision and High Acceleration Studies. Paper Number
740275, Society of Automotive Engineers, Warrendale, PA.
[12] Huston, R. L., Hessel, R. E., and Winget, J. M., 1976. Dynamics of a Crash
Victim -- A Finite Segment Model, AIAA Journal, Vol. 14, pp. 173-178.
[13] King, A. I. and Chou, C. C., 1976. Mathematical Modeling Simulation and
Experimental Testing of Biomechanical System Crash Response, Journal of
Biomechanics, Vol. 9, pp. 301-317.
[14] Huston, R. L., 1977. A Summary of Three-Dimensional, Gross-Motion,
Crash-Victim Simulators. Structural Mechanics Software Series, Vol. I, University
16
Press of Virginia, Charlottesville, VA, pp. 611-622.
[15] Prasad, P., 1984. An Overview of Major Occupant Simulation Models, Paper
Number 840855, Society of Automotive Engineers, Warrendale, PA.
[16] King, A. I., 1984.
A review of Biomechanical Models, Journal of
Biomechanical Engineering, Vol. 106, pp. 97-104.
[17] Huston, R. L., 1990.
MA.
Multibody Dynamics, Butterworth-Heinemann, Stoneham,
[18] Huston, R. L., 1991. Multibody Dynamics -- Modeling and Analysis Methods,
Feature Article, Applied Mechanics Reviews, Vol. 44, pp. 109-117.
[19] Huston, R. L., 1996. Multibody Dynamics since 1990, Applied Mechanics
Reviews, Vol. 49, pp. 535-540.
[20] Kane, T. R. and Levinson, D. A., 1985.
New York, McGraw Hill.
Dynamics: Theory and Applications,
[21] Huston, R. L., Passerello, C. E., and Harlow, M. W., 1978.
Dynamics of
Multi-Rigid-Body Systems, Journal of Applied Mechanics, Vol. 45, pp. 889-894.
[22] Li, F., Huston, R. L., and Waters, T. R., 2001. Mulsculoskeletal Dynamics of
Heavy Equipment Operators. CAES International Conference on Computer-Aided
Ergonomics and Safety, Maui, Hawaii.
[23] Huston, R. L., 1987. Crash Victim Simulation: Use of Computer Models,
International Journal of Industrial Ergonomics, Vol. 1, pp. 285-291.
[24] Punjabi, M. M., Anderson, G. B. J., Torneus, L. L., Huet, E., and Maltson, L.,
1986. In Vivo Measurements of Spinal Column Vibrations, Journal of Bone and
Joint Surgery, Vol. 68A, pp. 695-702.
[25] Tien, C. S. and Huston, R. L., 1987. Numerical Advances in Gross-Motion
Simulations of Head/Neck Dynamics. Journal of Biomechanical Engineering, Vol.
109, pp. 163-168.
17
Fig. 8.1. A finite segment human body model
Fig. 8.2. The model in an operator/workstation configuration
18
Fig. 8.3. A numbered human body model
300
Seat acceleration (in/sec**2)
250
200
forward (ax)
left (ay)
vertical (az)
150
100
50
0
-50
0
0.1
0.2
Figure 8.4
0.3
0.4
Time(sec)
0.5
Random Acceleration Profiles
0.6
0.7
19
300
Seat acceleration (in/sec**2)
250
200
150
forward (ax)
left (ay)
vertical (az)
100
50
0
-50
0
Figure 8.5
0.1
0.2
0.3
0.4
Time(sec)
0.5
0.6
0.7
Impulse Acceleration Superimposed upon Random Acceleration
20
1.8
Experiment
Simulation
Vertical Acceleration Amplitude Ratio
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
5
10
15
Frequency(Hz)
Fig. 8.6 Comparison of model prediction with experimental data (Punjabi et al.) for
middle torso vertical acceleration ratios.
21
1
Horizontal Acceleration Amplitude Ratio
0.9
0.8
0.7
0.6
0.5
0.4
Experiment
Simulation
0.3
0.2
0
5
10
15
Frequency(Hz)
Fig. 8.7 Comparison of model prediction with experimental data (Punjabi et al.[24])
for middle torso horizontal acceleration ratios.
22
2
sitting relaxed
sitting erect
1.8
Acceleration Amplitude Ratio
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
Frequency(Hz)
14
16
18
20
Fig. 8.8 Acceleration amplitude ratio between vertical seat acceleration and head
acceleration in two postures.
23
2.5
sitting relaxed
sitting erect
Acceleration Amplitude Ratio
2
1.5
1
0.5
0
0
2
4
6
8
10
12
Frequency(Hz)
14
16
18
20
Fig. 8.9 Acceleration amplitude ratio between vertical seat acceleration and middle
torso acceleration in two postures.
24
2
sitting relaxed
sitting erect
1.8
Acceleration Amplitude Ratio
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
Frequency(Hz)
14
16
18
20
Fig. 8.10 Acceleration amplitude ratio between vertical seat acceleration and knee
acceleration in two postures.
25
The Y direction angular response (degree)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
without seat angle
with -20 degree seat angle
-1
0
0.05
0.1
0.15
0.2
0.25
0.3
Time(sec)
0.35
0.4
0.45
0.5
0.03
The Z cervical response (inch)
0.02
0.01
0
-0.01
-0.02
without seat angle
with -20 degree seat angle
-0.03
0
0.05
0.1
0.15
0.2
0.25
0.3
Time(sec)
0.35
0.4
0.45
0.5
Figure 8.11 Vertical and Rotational Movement of the Head as a Response to the Seat
Acceleration of Figure 8.4
26
The Y direction angular response (degree)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
without seat angle
with seat angle
-1
0
0.05
0.1
0.15
0.2
0.25
0.3
Time(sec)
0.35
0.4
0.45
0.5
0.015
The cervical response (inch)
0.01
0.005
0
-0.005
without seat angle
with seat angle
-0.01
0
0.05
0.1
0.15
0.2
0.25
0.3
Time(sec)
0.35
0.4
0.45
0.5
Figure 8.12 Vertical and Rotational Movement of the Head as a Response to
Jolt/Random Acceleration of Figure 8.5