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LIBRARY
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University

 
  

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This is to certify that the

 

thesis entitled

Dynamic Response of Intervertebral Joints of
a Seated Farm Machine Operator in the Range

5 — 50 Hz.
presented by

Oscar Antonio Braunbeck

has been accepted towards fulfillment
of the requirements for

Eh, 12. degree in Agricultural Engineering

fax/yam

Major professor

 

may/44 27/77;

0-7 639

 

 

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ABSTRACT
Dynamic Response of Intervertebral Joints of a Seated

Farm Machine Operator in the Range 5 - 50 Hz.

by

Oscar Antonio Braunbeck

There are a number of reports that consider vibrations
as a cause of low back pain in subjects operating tractors,
trucks, or buses over long periods of time. No objective
explanation exists which is able to describe even qualita—
tively the mechanism by which seat vibrations generate
spinal problems. An hypothesis is proposed which suggests
that if intervertebral joint deformations present distinct
levels at frequencies encountered in the seat of farm ma—
chinery, they will create a fatigue type loading of the
intervertebral joint sufficient to induce pain sensations.

A lumped parameter dynamic model of the upper torso and
head is proposed, whose main objective is to predict lumbar
intervertebral joint deformations. The governing differen-
tial equations of motion are written for a linear system
exposed to sinusoidal small amplitude displacement excita—

tion in the vertical direction through the pelvis. A

   

in ”man! :11:an

    
 

"1 :anzgqfl qu?53r1 min:

 

Oscar A. Braunbeck

particular solution is found for the system of 58 second
order differential equations that provides an equal number
of complex amplitudes of motion, corresponding to each one
of the degrees of freedom in the system. The rheological
behavior of deformable components of the structure is
modeled by means of Kelvin viscoelastic elements. The
stiffness and damping coefficients for the axial mode of
oscillation are derived from impedance data taken from
isolated vertebral units.

The model is validated by computing seat to head trans—
missibility as well as driving point impedance coefficients
over the frequency range 5-50 Hz. The transmissibility and
impedance curves corresponding to the model closely resemble
the experimental curves even though the values differ
somewhat.

The magnitude of axial and shear deformations of inter—
vertebral joints are significantly affected by the frequency
of excitation and the characteristics of the seat or cab
suspension used.

Axial deformations can be as high as 20% of the amplitu—
de of base oscillation for an operator sitting on a bare
vibrating table. The use of a spring—damper—mass suspension
results in joint deformations about 1/4 to 1/5 those corre-
sponding to a seat with no suspension,

Cab suspension results in smaller joint deformations
than seat suspension for frequencies over 10 Hz. Between

5 and 10 Hz the seat suspension gives lower deformations.

 

arr! .aJnorrole :?':.~_=-l=«=*.---"- “i”

Oscar A. Braunbeck

Joint deformations increase with suspension damping.
The best protection is offered by low damping ratios (c=0.l)
provided the large amplitude of motion taking place at fre—

quencies close to the seat natural frequency can be controlled.

' .5”
Approved:
m

Approved: ‘9 K‘M/W

Department Chairman

 

._..rl'!" 'ajnmgly '1': ."'. - .- ' a:

Oscar A. Braunbeck

Joint deformations increase with suspension damping.
The best protection is offered by low damping ratios (t=0.l)
provided the large amplitude of motion taking place at fre—

quencies close to the seat natural frequency can be controlled.

' .évv
Approved:
Major Professor

 

Approved: é). . ‘1
Department Chairman

 

DYNAMIC RESPONSE OF INTERVERTEBRAL JOINTS OF A SEATED
FARM MACHINE OPERATOR IN THE RANGE 5 — 50 HZ.

By

Oscar Antonio Braunbeck

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Agricultural Engineering

1976

 

ACKNOWLEDGMENTS

The author wishes to express a deep appreciation to
Dr. Robert H. Wilkinson (Agricultural Engineering) for his
guidance, enthusiastic support, and friendship during the
course of the program.

Appreciation is extended to Dr. L. Segerlind
(Agricultural Engineering), Dr. W. Sharpe (Metalurgy,
Mechanics and Material Science), and Dr. L. Wolterink
(Physiology) for their time and helpful suggestions as
members of the guidance committee_

Dr. L. Kazarian (Aero Medical Research Laboratory,
Dayton, Ohio) is acknowledged for his permission to use the
impedance data collected from isolated vertebral units.

Many thanks are due to Mss Sonia Gonzalez for her
assistance in preparing paper tape and cards for data
prcessing, as well as typing the draft and original copies

of this dissertation.

ii

 

LIST OF TABLES

LIST OF FIGURES

LIST OF SYMBOLS

Chapter

II.

III.

TABLE OF CONTENTS

INTRODUCTION

.1

r4 H4 )4 P‘P‘HH‘
w m m btom

Definition ofthe Problem

Evolution of Human Environment
Vibrations as a Cause of Backache.
Dynamic Model of the Spine for
Agricultural Applications
Complexity of a Model of the Human
Torso

Main Contributions Made by the
Model . . .
Objectives

REVIEW OF LITERATURE

2
2.
2
2

2.

0‘ U1 D LONH

THE

Surveys of Spinal Problems.
Hypothesis on Low Back Pain
Existing Lumped Parameter Models
of the Human Torso

Rheological Behavior of Deformable
Components of the Human Trunk
Geometrical Data of Components
Involved in the Model

Masses Involved in the Lumped
Parameter Model .

MODEL

Mayor Aspects of the Modeling
Process

Simplified Model of the Human
Torso

Kinematics of the Model Components

Page
vi
viii

xi

\l\l U‘I -$-\ ND—‘H H

12
14
16
22
24

25

25

26
3O

 

Chapter

IV.

VI.

3.4

3.3.1 Systems of coordinates .
Rheological Behavior of Deformable
Elements of the Model

AXIAL RHEOLOGICAL BEHAVIOR OF INTERVERTE—

BRAL JOINTS

4.1 Driving Point Impedance of a Ver—
tebral Unit

4.2 Mechanical Impedance

4.3 Estimation of the Parameters of
the Kelvin Model .

4.4 Frequency Dependent Stiffness and

4 Damping Coefficients

.5

Axial Dynamic Response of the .
Posterior Spinal Arch .

DYNAMIC RESPONSE OF THE LUMPED PARAMETER
MODEL

5.1

mm U'lU'I U1
O‘U‘I DUO N

Governing Differential Equations of
Motion . . . .
5.1.1 Mass matrix .

5.1.2 Global stiffness matrix

5.1.3 Damping matrix

Solution of the System of Governing
Equations

Driving Point Impedance of the Model
Shear and Axial Deformations of
Intervertebral Joints

Seat to Head Transmissibility
Computer Program

EXPERIMENTAL DATA

6.1

6.2
6.3

Geometrical Data

6.1.1 Vertebrae .

6.1 2 Head and neck

6.1.3 Pelvis

Mass Moment of Inertia of a Vertebra

Masses in the System . .

6.3.1 Vertebrae

6. 3. 2 Suspended porsion of upper
torso .

6. 3.3 Head and neck

6. 3.4 Sacrum— —pelvis

iv

Page
33
34

36
36
37
41
44
45

 

Chapter

6.4 Rheological Behavior of Deformable
Elements .
6 4.1 Intervertebral joint. Axial

4. 2 Intervertebral joint. Bending.

6.

6.4.3 Intervertebral joint. Shear
6.4.4 Costo—vertebral joints
6.4.5 Head and neck .

VII. RESULTS AND DISCUSSION

7.1 Validation of the Model
7.2 Lumbar Intervertebral Joint
Deformations as Affected by Seat
Suspension Parameters .
7.2.1 Subject sitting on bare
seat. No suspension .
7. 2. 2 Subject sitting on a bare
seat provided with seat or
cab suspension . .
7.3 Summary of Results

VIII. CONCLUSIONS AND RECOMMENDATIONS

8.1 Conclusions
8.2 Recommendations

REFERENCES

APPENDICES

99
99

103
105
107
112
114
114
116
118
123

 

Table

6.1

LIST OF TABLES

Curvature of the thoracolumbar spine in
the sagittal plane .

Geometrical data for thoracic and lumbar
vertebrae.

Mass and mass moment of inertia respect
to the center of gravity of thoracic and
lumbar vertebrae. . . .

Body weight distribution used for the
model . . .

Parameters for estimation of frequency
dependent stiffness and damping coeffi—
cients. Experimental data from Kazarian
(1972). Axial mode of oscillation .

Parameters for estimation of frequency
dependent stiffness coefficient k, and
damping coefficient c. Bending mode of
oscillation . . . .

Shear stiffness of intervertebral discs
of thoracic and lumbar spine .

Parameters for estimation of frequency
dependent stiffness coefficient k and
damping coefficient c. Shear mode of
oscillation . . . . .
Distribution of degrees of freedom
corresponding to each rigid moving
component of the model .

Stiffness data reported by Schultz et
a1. (1973) .

Experimental mechanical impedance.
Age: less than 30, Kazarian (1972)

Experimental mechanical impedance.
Age: 30 to 50, Kazarian (1972)

Experimental mechanical impedance.
Age: over 50, Kazarian (1972)

vi

Page

71

73

79

83

87

90

93

94

127

140

 

 

vii

 

Figure

3.1

LIST OF FIGURES

Components of a seated human body having
significant dynamic interaction with the
vertebral column

Simplified structure representing a
seated human body .

Axial deformation of intervertebral joint.
The displacements between vertebrae is in
a direction perpendicular to disc middle
plane a - a . . . . .

Shear deformation of intervertebral joint.
The displacements between vertebrae is in
a direction parallel to disc middle plane
a — a . . . . . .

Bending deformation of an intervertebral
joint .

Local system of coordinates for calculation
of element stiffness matrix

Spinal unit model for calculation of
driving point impedance

Intervertebral joint modeled as two
Kelvin elements in parallel

Operator seat under sinusoidal displacement
excitation .

Displacements and rotations of two mxmecuthm
vertebrae determine the axial and shear de—
formations of the enclosed joint

a) Pendulum to measure mass moment of
inertia
b) Support of vertebra

Pendulum installed in vacuum chamber to
minimize error due to air friction

Location of the point of interaction of the

head—neck lumped mass with the upper end
of the spine . . . . . .

viii

Page

27

28

32

32

35

35

46

46

67

67

77

77

80

 

Figure

6.4

A fraction of the back muscles and other
tissues are closely attached to the spine

Loading frame for impedance testing,
Kazarian (1972 ) . . . .

Driving point impedance for seated operator.

Experimental curve from Pradko et a1.

(19)

Seat to head transmissibility. Confidence
interval from Pradko et al. (1967)

Axial deformation of L3—L4 lumbar inter-
vertebral joint . .

Shear deformation of L5-S intervertebral
joint . . . . . . .

Axial deformations of L3- L4 lumbar
intervertebral joint .

Shear deformations of L5-S intervertebral
joint . . . .

Axial deformation of L3—L4 lumbar inter—
vertebral joint . . .

Shear deformation of L5-S intervertebral
joint . .

Axial deformation of L3—L4 lumbar inter-
vertebral joint . . . . .

Shear deformation of L5—S intervertebral
joint . . . . . .

Main structural components of vertebral
column . . .

Displacements affecting the equilibrium
of vertebral mass mi in x—direction

Forces acting on an intervertebral joint
Forceschvehxmd at the intervertebral joint

as a result of unit displacements of the
adjacent vertebra . . .

ix

Page

85

102

102

106

106

109

109

110

110

111

111

124

125

128

135

 

 

 

{A}

LIST OF SYMBOLS

: Vector of complex amplitudes for all degrees of freakm

in the system (cm)
Effective cross sectional area of intervertebral disc,
(cmz)

: Amplitude of base oscillation (real), (cm)
: Complex amplitude of oscillation of the jth. degree of

freedom
A; - Real part of complex amplitude

A; : Imaginary part of complex amplitude
Complex amplitude of sacrum oscillation

: Vector of damping and stiffness coefficients at the

jth iteration
Seat-base relative motion

: Damping coefficient of chair suspension (dyn.sec/cm)

: Young's modulus of elasticity of an intervertebral

joint (dyn/cmz)

: Amplitude of sinusoidal forcing function (dyn)

F: : Real part of complex force amplitude

F: : Imaginary part of complex force amplitude

Shear modulus of elasticity of an intervertebral joint
(dyn/sz)

: Age group being less than 30 years
: Age group being between 30 and 50 years
: Age group being over 50 years

: Mass moment of inertia of a vertebra about its center

of gravity (dyn. sec2/cm)

: Mass moment of inertia of ith lumbar vertebra about its

center of gravity

: Mass moment of inertia of ith thoracic vertebra about

its center of gravity
Stiffness of chair suspension spring (dyn/cm)

xi

 

Z 0?: mN m7:

Shear stiffness of intervertebral joint (dyn/cm)

Axial stiffness of intervertebral joint (dyn/cm)
Bending stiffness of intervertebral joint (dyn.cm/rad)
Mechanical mobility (cm/dyn.sec)

Mr:

Ml: Imaginary part of mechanical mobility

Real part of mechanical mobility

Mechanical mobility of jth Kelvin element in parallel
with mass m.

Mass of chair (gm)

Rotation matrix

Transformation matrix

Sum of squares function for optimization of model
parameters

Period of oscillation of pendulum (sec)

Seat to head transmissibility

Velocity (cm/sec)

Weight of oscillating body (dyn)

Weighting matrix

Sensitivity matrix

Coordinates of vertebra superior end plate, Xl=O
Coordinates of inferior end plate, X2=0

Coordinates of inferior costo—vertebral joint
Coordinates of superior costo-vertebral joint
Complex amplitude of oscillation of jth mass

along x—axis.

Coordinates of transverse costo—vertebral joint
Experimental value of mechanical impedance

Yf: Real part of experimental impedance

Y1-
Mechanical impedance (dyn.sec/cm)

Imaginary part of experimental impedance

Modeled mechanical impedance (Zr + i Zi)
Mechanical impedance of posterior arch

Mechanical impedance of anterior spine (discs plus
vertebral bodies)

Mechanical impedance of a viscous damper

 

NNN
«aw

N
LL:

Mechanical impedance of a linear spring

Mechanical impedance of a single mass

Mechanical impedance of jth Kelvin element
Mechanical impedance of jth Kelvin element in paufllel
with mass m.

Complex amplitude of head vertical motion

Complex amplitude of oscillation of jth mass along

2 - axis

Viscous damping coefficient (dyn.sec/cm)

Damping matrix

Viscous damping coefficient posterior arch

Viscous damping coefficient intervertebral disc
Parameters for estimation of frequency dependent
viscous damping coefficients

Principal diameters of rib elliptical cross section
Time dependent forcing function (dyn)

Frequency of oscillation in (Hz)

Acceleration of gravity, (980.44 cm/secz, E. Lansing)
Linear spring stiffness coefficient (dyn/cm)

Shape factor of disc cross section

Global stiffness matrix

Element stiffness matrix in global coordinates(x,z,6)
Element stiffness matrix in local coordinates (u,w,6)

: Parameters for estimation of frequency dependent

spring stiffness coefficients

Height of intervertebral disc (cm)

Mass (gm)

Lumped mass of head and neck (gm)

Mass of jth vertebra without surrounding tissues (gm)
Mass of jth vertebra with surrounding tissues (gm)
Mass of ith lumbar vertebra with surrounding tissues
Lumped mass of sacrum and pelvis

Mass of ith thoracic vertebra with surrounding tissues
Lumped mass of upper torso and upper limbs

Mass matrix

xiii

 

q(t)

qs(t)

S
t

zb(t)
A

Oi

Generalized coordinates to describe motion of
vibrating system (cm)

Coordinate describing vertical motion of sacrum
Distance from pivot point to center of gravity of
oscillating body (cm)

Independent variable in Laplace domain

: Time (sec)
(u,w,6):
(x,z,<3):

Local coordinate system
Global coordinate system

: Vertical displacement of seat base (cm)

Complex amplitude for rotational mode of motion

: Angle made by longitudinal axis of a vertebra and

vertical 2 — axis

2 Angle made by the longitudinal axis of an interver—

tebral disc and the z axis

: Angle made by vertebra end plate and disc middle

plane a-a
Phase angle of ith degree of freedom relative to

pelvis or base motion (deg)

: Level of statistical significance

Stands for either c or k in sensitivity matrix X
Rotation of a vertebra about its c.g. in sagittal

plane.

. Angular frequency (rad/sec)

 

I. INTRODUCTION

1.1 Definition of the Problem

This investigation was primarily motivated by reports of
low back pains suffered by farmers. However, the range of
applications is much wider than just for agricultural situa—
tions. Truck drivers and operators of heavy equipment also
experience the same symptoms. Excessive intervertebral joint
deformation, over long periods of time, is probably a cause
of backaches in seated operators subjected to vibrations.
The conditions under which these deformations occur are in—
vestigated so that corrective measures may be taken.

1.2 Evolution of the Human Environment

Humans have been exposed to vibrations for centuries but,
as civilization has progressed, the range of vibrational
frequencies and amplitudes has become more severe. During
the last hundred and fifty years science has changed the hmmm
working environment more than in the thousands of years since
agriculture was first developed. So it is wise to look at
the possible consequences that environmental changes may have
on the human being.

During the period of the industrial revolution, productiv—
ity was the main concern, and very little attention was paid
to the effects of the high level of physical andgmydxfloguml

stress placed on the human being. A similar situation took

 

 

place in agriculture. Farm productivity increased with mech—
anization, but the farmer was subjected to higher levels of
physical and mental stress.

Power equipment possibly would not have become a health
hazard if farmers had continued farming about the same area,
but spent less time in field operations. However, because
of economic pressures, the number of working hours remained
about the same or in many cases increased, (machines do not
need to rest as do horses) and the cultivated area increased
to raise the economic output.

1.3 Vibrations as a Cause of Backache

Due to his inherent flexibility and great ability to
"adapt” man has, for the most part, been able to adjust to
the changing situations. But the "cost" in comfort, physical
and mental stress, and general health has often been high.
Too often solutions to environmental problems are not mxmid—
ered important until a problem becomes so acute that a solu—
tion is absolutely required.

Occupational health problems associated with operation
of farm equipment by a seated operator exposed to body vibra—
tion is a kind of problem for which there exists no quick,
easy, and conclusive evidence of damage to human health.
There is some epidemiological association between vibrations
and lumbar spine disorders, but conclusive evidence is not
available yet. Large intervertebral joint deformations
appearing over prolonged periods of time may not be the only

cause of low back pain. Nonetheless, the population at risk

 

is sufficiently large, Wasserman et a1. (1974), and some of
the associated complaints are sufficiently severe that an
attempt must be made to reduce the vibration induced joint
deformations at points where they are extreme, and toconan?
rently conduct studies seeking to explain the relationship
existing between the spine disorder and the vibration.

Even though there is no information on what magnitude of
disc deformation under sinusoidal excitation could be hmmiul
the present model will indicate the frequency ranges most
likely to present tolerance problems. This means that pro—
tective systems (seat or cab suspension) can be designed
without complete knowledge of the tolerance levels, with
assurance that whatever the tolerability, the protection
system will offer maximum protection.

Improper lifting habits are frequently considered to be
the main cause of back problems. The total bending and.mdal
load applied to the human torso when lifting a heavy weight
are undoutedly higher than loads resulting from low amplitude
seat vibration. But, in a lifting situation there is addi-
tional assistance to the spine through elevation of the intra—
abdominal pressure that transforms the thorax and abdomen
into a semi—rigid-walled cylinder, Eie (1972). This parttflr
1y counteracts the compression produced by the erector qfinae
muscles and tends to elongate and straighten the lumbar qfine
anteriorly. The high intra—abdominal pressure which occurs

when lifting heavy weights explains why certain individuals

may expose their back to extremely heavy loads without damag-
ing their spine. This type of assistance is not available
to the spine in a long duration vibratory load situation.
1.4 Dynamic Model of the Spine
for Agricultural Applications

Most of the models reported in the literature have been
developed for automobile and aerospace applications, mainly
for short duration high acceleration seat ejection or front
collision phenomenon. Farm equipment operators are subjected
to vibrations of lower accelerations but for much longer
periods of time and in a frequency range where several compo-
nents of the body reach a resonant stage.

The vibration reaching the operator through the seat is
mostly sinusoidal in nature, originating at engine, tires,
transmission or some other moving component having rotary or
reciprocating motion. Some terrain-induced random vibrations
will also reach the operator with occasional transient peaks,
mainly when crossing deep furrows where the seat suspension
may bottom out.

The vertebral column of a seated tractor driver is fre—
quently overstressed as the operator turns around to look at
the machine pulled by the tractor. Yet some controls must
be adjusted during tractor operation as a function of crop
condition. This adds an extra load on the spine while it is
simultaneously twisted and receiving a vibrational input

through the pelvis.

 

1.5 Complexity of a Model of the Human Torso

The development of a mathematical dynamic model of the
human torso involves problems such as complexity of the
system; strong limitations for testing system components
under normal operating conditions (in vivo) to collect data
to validate the model; and materials as well as loads with
poorly understood behavior.

Because of the structural complexity of the vertebral
column and the difficulty of conducting experiments "in
vivo", the dynamic behavior of the spine must be investigated
through some kind of physical or mathematical model. The
model can then be successively adjusted until it predicts
the dynamic response of the human body with sufficient
accuracy.

By working with a mathematical model rather than with a
physical model it is easier to make modifications such as
changes of size, shape or rheological properties of the
connective tissue for any of the anatomical components of
the system. A physical model would require the construction
of new components, every time a dimension, shape or material
has to be changed.

Dynamic modeling of most engineering structures is nor—
mally done for well understood material and structural mem—
bers having known dimensions. The human body is very complex,
mainly because there are large variabilities of dimensions
from person to person, because connective tissues present

non—linear viscoelastic behavior and because muscles do not

 

 

behave as passive structural components.

Measurements made on cadavers are hardly sufficient to
permit production of statistically valid geometrical data of
the structural components involved in a lumped parameter
model. For some components approximations must be made
through standard geometrical figures in order to be able to
calculate the parameters required for a dynamic analysis.
For instance, the geometry of a rib can be approximated
by an elliptical cross section with variable ratios of the
diameters, dl/dZ' over the length of the rib.

Body materials tend to change with age more than engi—
neering material do. The ideal situation would be to model
the human body using data (rheological and geometrical prop-
erties) taken in vivo from young subjects of different ages,
but in practice the properties are mainly measured from
older cadaver materials. This is a limitation since cadavers
have dynamic properties which are often different from the
in vivo case. More accurate results will become available
for modeling as new transducers are developed which are
capable of taking measurements in vivo.

The loading conditions are also quite different from
other engineering cases. This difference is mainly due to
the existance of muscle forces which load the body structure
following a stimulus mechanism not sufficiently understood
to be properly modeled. But, for steady state low-amplitude
Vibratory excitation, the back muscles can be thought as

exerting a constant axial force that contributes a great deal

 

to the stability of the spine. This assumption applies to
a subject sitting erect, and not performing any tasks that
could alter the symmetry of loads with respect to the sagit—
tal plane. This is in fact the situation for most of the
time of exposure to vibrations of a tractor driver. The
thoracolumbar spine is capable of supporting very low
compressive axial loads without muscle assistance.
1.6 Main Contributions Made by the Model

The number of approximations made when developing
models of this kind will probably lead to results signifi—
cantly less accurate than those reached in dynamic engineer-
ing structures made out of better understood materials. In
spite of these uncertainties, there are positive contribu-
tions, such as:
a) A better understanding is gained both of critically lmxbd

areas of the body and of the most critical loading condi-

tions.

b

v

The need for specific geometric as well as rheological
properties becomes evident.

C

v

Interdisciplinary interaction becomes more effective as
the contributions made by the modeling work become known
in other related fields.

1.7 Objectives
The steps followed in studying the spine problem premymed
through this chapter can be summarized in seven basic objec—
tives:

1. Study existing reports on back problems of tractor

 

A
-
'1

vibration could be considered to contribute significantly
to the back pain.

Propose an hypothesis on how low amplitude vibrations
acting vertically on the pelvis of a seated subject can
adversely affect the lumbar spine.

Find the most realistic way to predict the dynamic
response of the spine to sinusoidal input through the
sacrum. This implies the simplification of a complex
system to a model that can be mathematically implemented
and solved.

Review existing data on the geometrical and rheological
properties of the system in order to reduce experimental
work to a minimum.

Verify the proposed model with existing data on overall
dynamic response of the human torso for a body in sitting
posture.

Draw conclusions and give recommendations concerning the
most critical vibrational inputs to be minimized by a
properly designed protective system.

Give recommendations on additional data required to reach

more accurate results using the proposed model.

 

II. REVIEW OF LITERATURE

2.1 Surveys on Spinal Problems

The existing reports on back problems in subjects ex—
posed to seat vibrations justifies the development of a
model able to identify the vibratory conditions most adver—
sely affecting the spine. The reports summarized in this
section lead to the conclusion that vertical seat vibration
is to some degree responsible for certain reported back
problems.

Paulson (1949) observed some of the distressing symptoms
of tractor driving during a period of several years of rural
medical practice. The complaints ranged from neck stiffness
and extremity pain, to digestive upsets, frequent stools,
heartburn, urinary frequency and dizziness; but the most
common complaint was lower backache.

Rosegger and Rosegger (1960) examined 371 tractor drivers
to assess the correlation existing between vibrations, shocks,
stomach troubles, and degenerative deformations of the tho—
racic and lumbar spine. They concluded: ”Adolescent kyphosis
can be caused not only by lifting or carrying heavy loads or
prolonged work in a bent position during puberty and adoles—
cence, but that it is also promoted by shocks and vibrations
which act as microtrauma upon the intervertebral discs while

the body is hold in a faulty posture. The degenerative spine

 

10

deformations increase proportionately with the length of
service as tractor drivers".

Baker and Wilkinson (1974) conducted an occupational
health survey on 851 farmers. The study showed that one of
every 5 Michigan farmers suffers chronic back pain. One of
every 12 farmers had to make an adjustement in his farming
activities due to back or knee problems. Improper lifting
habits and exposure to machinery vibrations are suggested
by the authors as the factors most likely to be responsible
for the backache.

There are some types of dynamic loads acting on the
spine with sufficient frequency and time of exposure to be
considered a kind of vibratory condition. Fusco et al (1963)
examined sixty workers employed in the sheet metal stamping
industry. In 60% of the cases X—rays showed signs of lumbo—
sacral arthrosis. The vibrations are transmitted to the
operator through the legs and arms. The dynamic load is not
sinusoidal but periodic with a frequency of about 1 stroke
per second.

Long time exposure to vibration of young subjects will
very likely affect bone shape and structural characteristics.
Prives (1960) has investigated the influence of work and
sports on the skeleton of 3000 growing and scenecent orga-
nisms, over a period of 10 years. Significant variations of
bone shape and structure were found for matain<xmupatflxw and
sports.

Some other effects of vibrations have been investigated

 

11

that could be due to excessive motion of nerves in the spinal
cord. Hornick (1961) studied the effect of low frequency
vertical vibration ( 1 to 7 Hz ) on male subjects at several
intensities (.15 to .35 g). The subjects were seated on a
rigid chair upon a shake table. Tracking performance was
significantly (d<.001) affected by vibration but there was

no relationship to either the intensity or frequency of vi—
bration. Reaction time was not impaired by any frequency or
intensity until after the vibration ceased.

Several types of morbidity patterns related to whole—
body vibration were investigated by Gruber and Ziperman
(1974). The information was collected from 1448 interstate
bus drivers, and included the results of periodic physical
examinations. The results of the statistical analysis
indicate that whole-body vibration must be included in the
etiology of back disorders.

The basic bus vibrations are in the range 0-15 Hz, with
a mean acceleration of 0.05 g. A rough riding bus can reach
a mean acceleration of 0.1 g., Clevenson and Leatherwood
(1973). Equivalent information is not available for farm
machinery but it is reasonable to expect figures significant—
ly higher than those reported for buses.

Most of the work done in trying to identify etiological
factors related to various disorders of the spine are survey
type research. This approach has provided sufficient evi-
dence to support the hypothesis of existence of vibration

related spinal problems. Experimental as well as modeling

 

12

work are the next stage in the process of explaining the
mechanism relating vibrations to low back pain.
2.2 Hypothesis on Low Back Pain

There is a list of possible causes for low back pain,
but the tendency among orthopedic and neurological surgeons
is to attribute low back pain to abnormalities in the lumbar
intervertebral discs. Some clinical observations suggest
that the pain may originate within the disc, but anatomical
studies have failed to demonstrate the presence of nerve
fibers within the annulus or nucleus pulposus. This seems
to indicate that pain must originate in some of the neigh-
boring elements interacting with the discs, namely vertebral
bodies, posterior arch, and ligaments. All of them contain
nerve fibers able to sense pain, Brown et a1. (1957).

The oscillatory relative motion between vertebrae creates
a fatigue type loading on the annulus fibrosus of the inter—
vertebral disc, the cartilages of the articular facets, and
the ligaments linking both vertebrae, which may be responsi—
ble for some tissue irritation that creates pain sensations.
The axial motion of vertebrae is the main vibrational mode
of the lumbar spine for a seated subject under vertical base
motion. Due to the curvature of the spine, the vertical
motion of the sacrum will also generate rotational as well
as tangential (shear) deformations of the intervertebral
joints in the sagittal plane. Relative motion between ver—
tebrae is considerably reduced above the 10th. thoracic ver—

tebra. The ribcage increases the stiffness of the thoracic

 

13

spine. Therefore the larger deformations in the lumbar
spine together with the high incidence of complaints in this
region of the spine indicates that some relationship must
exist between backache and intervertebral joint deformations.

The sinusoidal compressive force applied to an inter—
vertebral disc simultaneously subjected to a constant com—
pressive force resulting from body weight creates a pressure
gradient between the disc and the vertebral bodies enclosing
it. According to the results of Brown et a1. (1957), 1.0
to 2.5 cm3 of volume losses occur on the lumbar interverte—
bral disc under axial (quasi—static) compressive load. It
is suggested by the authors that the transference of mass
from the nucleus of the disc across the cartilagenous plates
results in a less uniform stress distribution over the
annulus fibrosus and cartilaginous plates. A reduction of
hydrostatic pressure inside the disc means that a larger
share of the axial load must be carried by the annulus fi-
brosus. A similar phenomenon may take place when the inter—
vertebral disc is subjected to sinusoidal compressive loads
superimposed to a constant axial load.

Frequency and amplitude of motion as well as time of
exposure to vibrations are probably the main parameters to
be studied concerning spine problems. The combination of a
frequency with corresponding minimum values of amplitude and
time of exposure at which low back pain develops can be
considered as a "failure” of the spine subjected to cyclic

stresses. Such fatigue failure will not occur if the exposure

 

14

time is short enough to allow repair by healing. The inter—
vertebral discs, because of lack of reparatory power, may
be particularly susceptible to this type of failure, Kraus
and Farfan (1972).

2.3 Existing Lumped Parameter Models of the Human Torso

Some of the lumped parameter models most closely related
to the one developed in this project are summarized in this
section. The limitations of these models to predict the
response of a seated subject to sinusoidal seat excitation
are pointed out in each case.

The lumped parameter model developed by Orne and King
Liu (1971) is capable of predicting the displacements of
individual vertebra subjected to transient loading condi-
tions. The behavior of the discs under axial compression
is modeled by a three—parameter linear viscoelastic solid.
The behavior of this model under compressive load closely
resembles experimental creep and relaxation curves. How—
ever, according to the analysis of the impedance data
reported by Kazarian (1972), the same model does not provide
satisfactory results for sinusoidal excitation over the
frequency range 5-50 Hz.

The model presented by Orne and King Liu, like some
others, considers the effect of the upper torso on the spine
by eccentrically, but rigidly attaching a mass (2050.0 gm)
to each vertebra. The magnitudes and moments of inertia of
these masses were measured by Liu and Wickstrom (1973). It

was felt that under sinusoidal excitation the model would

 

 

15

be more realistic if a smaller mass, corresponding to the
tissues more closely attached to the spine, is considered
as being rigidly attached to each vertebra. The remaining
mass of the upper torso must be attached to the spine
through some kind of deformable elements that would allow
for the relative motion existing between the ribcage and
the spine.

Prasad and King (1974) developed a lumped parameter
model of the spine including Kelvin viscoelastic elements
for the three modes of motion in the sagittal plane; this
is axial, shear, and bending (rotational) motions. The
stiffness and damping coefficients of the Kelvin element
are considered to be constant. The behavior of the model is
satisfactory for transient type loads, but, according to the
results of the modeling work done using the impedance data
reported by Kazarian (1972), Kelvin viscoelastic elements
with constant damping and stiffness coefficients do not give
satisfactory results for axial sinusoidal excitation. The
upper torso is modeled in a similar way to Orne and King
Liu (1971). This model considers transmission of load
through the articular facets, which is a major new feature
when compared to other models that include only the inter—
vertebral disc.

Muksian (1970) proposes a lumped parameter model in—
cluding all the joints of the vertebral column, head, pelvis,
ribs, shoulders, viscera of the upper torso, and the action

of corresponding muscles. The proposed system with between

-L flfi~ly“.o ~‘k'1

 

16

65 to 70 separate masses is finally reduced to only seven
masses due to the complexity of the original system.
Individual vertebrae are not considered as separate masses
in the final model consequently it does not provide any
information on deformation of intervertebral joints.

Roberts and Chen (1970) developed an elastostatic finite
element of the human thoracic skeleton. It includes sternum,
ribs, costal cartilage and vertebral column. The soft
tissues were neglected. This model is acceptable for static
loads; it was used only as a first approximation toward the
development of a dynamic model for the study of anterior
chest trauma, occuring for example in automobile collisions.

2.4 Rheological Behavior of Deformable
Components of the Human Trunk

Up to the present time the need of rheological data for
human biostructural analysis has been such that almost any
result able to shed some light into the field was welcomed.
It is mainly because the required experimental material is
difficult to obtain. Moreover, the apparatus used to
measure rheological properties in most cases must be special—
ly built for the particular shape of the speciment being
tested.

It was felt that enough information is available in the
literature to model the human spine and draw some important
conclusion on its response to sinusoidal input through the
pelvis and at the same time get a better understanding of

what is the additional information more urgently needed

 

17

that would lead to more accurate results. A through
literature search was done in order to keep the experimental
work to a minimum. Christ and Dupuis (1963) investigated
the motion of cervical and lumbar spine for a seated subject
under sinusoidal seat vibration. The equipment used was an
image intensifier and X—ray film equipment. The study was
not extended to the thoracic spine because of the very
indistinct pictures produced by this area. Only one fre-
quency, 2 Hz, was used for the experiment. No data is
reported on deformation of intervertebral discs. Probably
the definition of the X—ray film was not satisfactory for
such measurements. Bulk displacements of the spine and
stomach are reported.

The model under study requires stiffness and damping
coefficients for the Kelvin elements used to predict the
rheological behavior of intervertebral joints, neck, and
costo-vertebral joints. The three principal structural
connections between adjacent vertebrae, namely the disc,
plus the posterior facets, plus the interconnecting ligaments
will be refered to as the intervertebral joints.

Most of the required information is available in the
reports described in the following paragraphs. Due to the
wide ranges of variation given by some authors for some of
the parameters, more than one source will be cited whenever
possible so more average figures can be used. Most of the
rheological data available is focused on the axial behavior

of the intervertebral discs or the intervertebral joints.

 

18

The data finally adopted for the model are given in Chapter
VI.

In those limited cases where there is a choice priority
was given to data obtained under experimental conditions
similar to the conditions of operation of the model; such
as parameters measured under harmonic loading were given
priority over those measured under transient or static
loading conditions. Measurements made on complete inter-
vertebral joints had priority over those taken separately
for discs or posterior arch. Data taken from fresh cadaver
material had priority over embalmed cadaver materials.

Data measured from cadavers being between 30 and 50 years
of age were chosen when possible.

Different components of the body trunk have different
stiffness characteristics. Ribs, head, pelvis, and verte—
brae are1nuch stiffer than intercostal tissue or interverte—
bral discs, Crocker and Higgins (1966). For the purpose of
this investigation head, vertebrae, and pelvis will be
considered as rigid bodies. Andriacchi et a1. (1974) found
that although rigid bodies were used to model calcified
portions of the ribs, vertebrae and sternum, the model
predicts cage deformations in close agreement with those
measured experimentally for static loading conditions.

The stiffness and damping coefficients involved in
modeling the axial behavior of intervertebral joints are
derived from the impedance data reported by Kazarian (1972)

for isolated vertebral units. Details on these data are

  

I -- '_.-'-'-' : ‘I_-- . --_.__:.*
afiel‘fifim [vii-Insane “hm; beatnik Nib @-

.1

   
  
   

_ -. I'Il'" ='- I "a; 1;: I I; ‘-

'Z-A"-- .J-I‘r-fi t‘u‘i‘) I!" 'r-'.- 1‘. -'_. ru:~}1ib,-1.;. 9d: a,“ .1.

4

I'!

mmmt . .
'._"-'1o.“:"q "r

.‘hsol

 

19

described as needed in sections 4.1 and 6.4. The impedance
measurements were taken from preloaded vertebral units, sub—
jected to sinusoidal axial excitation; these test conditions
make the data particularly suitable for the model being
developed herein.

The rheological data for bending mode of oscillation of
intervertebral joints in the sagittal plane is adopted from
Markolf and Steidel (1970), who have measured stiffness and
damping of the intervertebral discs under harmonic loading
for the following modes of oscillation: lateral and sagittal
bending, torsion, and tension-compression. The disc was
fixed to a table by one end, and to a mass a the opposite
end. The mass was allowed to move as a single degree of
freedom system either in axial, bending, or torsional mode
of oscillation.

The stiffness was calculated from the measured natural
frequency of the single degreee of freedom system oscilla—
ting in free vibration. The damping factor was estimated
from the rate of decay of the vibration trace. Even though
there are no data to support the hypothesis that these
coefficients are frequency dependent, an exponential equa-
tion will be adopted by analogy with the axial behavior as
explained in section 6.4.2.

The measurements made by Markolf and Steidel were done
with the posterior arch sawed off for the axial test. It
was done under the assumption that the posterior facets and

ligaments play no important role for the transmission of

 

20

vertical loads through the column. It may be the case for
static loading, but under dynamic loading conditions the
amount of damping present in the posterior arch may develop
significant velocity dependent forces being transmited
between vertebrae.

The damping data for bending oscillation is adopted from
Prasad and King (1974).

Very little is known about the shear rheological behavior
of intervertebral joints. An approximation done by Orne and
King Liu (1971) using data reported by Evans and Lissner
(1959) is used in section 6.4.3 to calculate the shear
stiffness coefficients required for the model.

No data is reported on damping coefficients for shear
mode of oscillation, so a range of values will be analyzed
in an attempt to bracket the real value, and see how sensi-
tive the structure is to this parameter.

The two dimensional model (sagittal plane) developed by
Prasad and King (1974) includes Kelvin models for axial,
shear and bending behavior of the intervertebral discs.

The axial stiffness coefficients modified from Markolf and
Steidel (1970) range from 29,000.00 x 105 dyn/cm to 14,000.00
dyn/cm from top to bottom of the thoracolumbar spine. These
data are within the range swept by the frequency dependent
stiffness coefficients derived from the impedance data
reported by Kazarian (1972), but leaning toward the lower
values. This is reasonable since Prasad and King assigned

additional stiffness to the posterior arch in parallel with

 

21

the discs.

The axial damping coefficients used by Prasad and King
for intervertebral discs range between 18.0 x 105 and
44.0 x 105 dyn.sec/cm, which is close to the lower values
calculated using frequency dependent equations derived from
impedance data. No damping was considered for the posterior
arch.

The bending stiffness coefficients reported by Markolf

5 5

and Steidel vary between 3,884.0 x 10 and 23,305.0 x 10

dyn.cm/rad. Prasad and King increased these values to

67,853.0 x 105

dyn.cm/rad. for the lumbar spine, and to
l35,706.0 x 105 dyn.cm/rad for the thoracic spine. Prasad
and King justified these stiffness increases by the fact
that Markolf and Steidel's measurements were made with no
preload. Moreover, the ribcage further increases the bend-
ing stiffness in the thoracic region of the spine. Although
this reasoning is correct the stiffness increases are much
too high.

Prasad and King considered the upper torso divided in
slices cut by a horizontal planes passing through the inter-
vertebral joints. Each slice was modeled as an excentric
rigid mass attached to the corresponding vertebra. These
excentric masses attached rigidly to a vertebra create much
more severe dynamic loads on the intervertebral discs than
those that would result from a model having some kind of
deformable element linking the mass of the upper torso to

the spine.

 

22

King and Vulcan (1971) measured the gradient of force-
deformation curves under axial dynamic loading conditions.
Two segments of the spine were tested. One segment consis—
ted of half T11, T12, plus half L1. The second segment
included half L1, L2, and half L3. The rate of loading
used was approximately 9,000.0 Kg/sec. Each segment
included two joints, so the stiffness of each joint is

twice that reported for the segment.

Se ent Fresh Embalmed

5 5
T-12, aver. stiff. = 37,162.0 x 10 38,434.0 x 10 QflVCm
L—2, aver. stiff. = 23,7260 x 105 28,962.0 x 105 dyn/cm

These values are 27.0% higher than those calculated for 50
Hz using the equations for frequency dependent axial stiff-
ness coefficients. This is a result of compounding both,
deformation as well as velocity dependent forces into a
single stiffness coefficient.

In all cases there was no significant increase of stiff—
ness from fresh to embalmed specimen conditions at T—12
level.

2.5 Geometrical Data of Components Involved in the Model
The geometrical data required for assemblage of the
model presented in Chapter III involves dimensions of ver—

tebrae and location of their center of gravity, curvature
of the spine, and location of the center of gravity of the
head—neck system.

Most of the dimensions of vertebrae reqfixed.finrthenndel

could have been obtained hitle LflfiraUnE from measurements

 

23

taken by Lanier (1939) Prasad and King (1974), Schultz and
Galante (1970), and Roberts and Chen (1970). Since no data
is available on vertebra mass moment of inertia about its
center of gravity in the sagittal plane, measurements had
to be taken from an embalmed spine. All dimensions needed
were taken from a single cadaver.

The curvature of the spine adopted for the model corre—
sponds to a subject seated in erect posture on a seat
furnished with low back rest, Clark et a1. (1963). Data
reported by Kazarian (1972), Orne and King Liu (1971), and
Schultz et a1. (1973) were consulted to gather all the
information required.

The curvature of the spine considerably affects the
amount of load to be carried by the intervertebral discs.

A proper restraint system would keep the spine in an erect
position, which makes the posterior arch to share a larger
percentage of the load. Ewing et a1. (1972) found that
anterior compression of the lumbar vertebral column can be
reduced by restraining the shoulders and pelvis to the seat
back in a moderate hyperextended position.

The spine configuration used in the present study
approximately corresponds to average tractor driving condi—
tions. Moreover, the adopted relative position between
vertebra is close to that used to collect the impedance
data involved in modeling the axial rheological behavior of
the intervertebral joints. Other configurations would have

different stiffness coefficients for the intervertebral

 

24

joints if their non—linear force-deformation behavior is
taken into consideration. Therefore more data should be
available on the rheological behavior of the joints before
any studies can be conducted in relationship to dynamic
responce and body posture.

2.6 Masses Involved in the Lumped Parameter Model

The weight distribution of components over the human
torso, head and upper limbs has been reported by a number
of authors, Dempster (1955), Damon et a1. (1966), Ingalls
(1931), Lowrance and Latimer (1967), Mnksian and Nash (1974),
and Payne (1970).

Since the spinal data used for the model was measured
on a spine corresponding to a body weight of 85.0 kg, most
of the component weights will be calculated using Dempster's

data given as percentage of body weight.

 

III. THE MODEL

3.1 Mayor Aspects of the Modeling Process
Prediction of relative motion between vertebrae of a
seated human being under sinusoidal base excitation can be
done through mathematical formulation. The human torso can
be considered as an engineering structure whose dynamic
response is the integrated response of all the components

of the structure. Even though the spine is the only element

of interest for the present work, the remaining elements of

the torso (head, thorax) are included in the model only
because of their interaction with the spine.

There are five mayor aspects to the modeling of the ver—
tebral column:

1. Reduction of the real torso to a simplified version that
will be easier to formulate but having a dynamic response
close to that of the original system

2. Study of the kinematic behavior of the moving parts of
the torso

3. Study of the rheological behavior of the deformable
elements in the system

4. Derivation of the governing differential equations of
motion

5. Solution of the system of governing equations of motion

25

 

26

3.2 Simplified Model of the Human Torso
The real system shown in Figure 3.1 has been properly
reduced to the lumped parameter model of Figure 3.2, which
is suitable for mathematical formulation. The simplifica-
tions involved in the process of reducing the actual struc—
ture to a model form are based on a number of assumptions:

1. Only the torso, head and upper limbs are included in the
model. The lower limbs rest on the floor and the seat
surface without actually loading the spine.

2. Vertebrae are considered as rigid bodies. Deformations
of the spine under load take place at the intervertebral
joints. All components of the spine are deformable, but
the stiffness of vertebrae is much higher than that of
ligaments and cartilages forming the intervertebral
joints. Bell et a1. (1967) reported a median vertebral
bone stiffness of 11.0 x 109 dyn/cm compared to about

2.0 x 109

dyn/cm for the intervertebral joint.

3. The forces developed at intervertebral joints are assumed
to be linear functions of deformations. The rheological
behavior of the joints is nonlinear, Brown et a1. (1957),
but for deformations sufficiently small the behavior
closely follows the equation of a straight line tangent
to the force—deformation curve.

4. No temperature effects are considered in the rheological
equations. The range of variation of body temperature
is very small.

5. Ribcage, internal organs of the thorax, upper limbs, and

 

27

_____——————

Head

2? and

Neck

Vertebral column
Thorax

and

Pelvis Upper limbs

 

 

Figure 3.1. Components of a seated human body having
significant dynamic interaction with the
vertebral column.

 

 

28

   
  

  
 

L—l

,2 Two degrees of freedom

X

+.
+.
.+
+.
L—5
.+
+.

211x

1:} Sinusoidal excitation

—e

0

Figure 3.2 Simplified structure representing a seated
human body.

 

29

shoulders are modeled as a single rigid mass suspended
from the upper 10 thoracic vertebrae by means of 10

Kelvin units. The most significant dynamic interactions
(resonances) between the spine and the meer ammo tmqaplace
at frequencies lower than 10 Hz, Coermann et a1. (1960),
while the largest intervertebral joint deformations

take place between 30 and 40 Hz.

Vertebrae are assumed to move only in the sagittal plane.
Forces acting in a direction perpendicular to the sagit—
tal plane on both halves of a symmetric body will mutually
cancel out.

Muscles enter the model only as passive components having
dynamic interaction with the spine as a result of their
mass and viscous behavior. Transient or random excitations
that could trigger muscle activity other than the constant
stabilizing force on the spine, are not included in the
analysis. The response of the model attempts to predict
intervertebral joint deformations for subjects under
steady state sinusoidal excitation.

Head and neck are modeled as a single mass attached to the
top end of the thoracic spine by means of three Kelvin
elements representing the normal, shear and bending rheo—
logical behavior of the neck.

Since the abdominal organs are mostly resting on the pel—
vis they are considered as part of the pelvic mass. The
dynamic response of the abdomen would mainly affect the

values of driving point impedance that are used for validation

 

30

of the model. According to the results of Coermann et
a1. (1960) the motion of the abdominal wall presents a
maximum at a frequency between 3 and 4 Hz for all subjects
studied. The abdomen displacement curve levels off at
about 7 Hz, with no other resonant conditions thereafter.
It indicates that the abdominal mass is not responsible
for any of the peculiarities presented by the impedance
curve of a seated human body.
10.The impedance data used to derive the frequency dependent
stiffness and damping coefficients were obtained from
fresh cadavers. There is no doubt the properties of
living tissues will deviate from those of the inert
material. The magnitude of the deviation has yet to be,
investigated. The overall response of the model indicates
that those deviations are not significant.
11.The intervertebral joint is modeled as a set of three
massless Kelvin elements. The mass of the disc is
divided in two parts which are considered as part of the
two vertebral bodies enclosing the disc, see drawing in
Table 6.2.
3.3 Kinematics of the Model Components
There are a number of deformable elements in the human
torso that allow relative motion among components of the
structure. The ones included in the model are: interverte—
bral joints, costo—vertebral joints and the neck.
The patterns of deformation assumed for an intervertebral

joint along the disc axis (axial) as well as perpendicular

 

""-’ 3'13

31

to it (shear) are shown in Figures 3.3 and 3.4. Pure bend—
ing deformation is assumed to occur when the relative motion
between vertebrae takes place around the point of inter—
section, 0, of the longitudinal axis of the two vertebral
bodies, see Figure 3.5. These simplified patterns of joint
kinematic behavior are adopted under the assumption of small
deformations.

The kinematic behavior of the intervertebral joint
requires special techniques to handle geometric non-linear-
ities when large relative displacements are allowed between
vertebrae; for example in an hyperextended mode the articu—
lar facets on the spinal posterior arch will bottom out.

It significantly changes the kinematic behavior of the
intervertebral joint. Configuration and location of the
articular facets and posterior arch are illustred in
Appendix A.

The intervertebral joint flexibilities just introduced
allow for each thoracic and lumbar vertebra to have three
degrees of freedom in the sagittal plane that will be
identified by coordinates x,z,é as shown in Figure 3.6.

Sacrum and pelvis are included in the model as a single
mass. Only two degrees of freedom, (x,z), are assigned to
this mass. No significant pelvis rotations are expected in
light of the following assumptions:

a) For subject seated erect the point where the seat inter—
acts with the pelvis is almost on the same vertical line as

the intervertebral joint L5 — S, so that no significant

 

. wilx~

32

 

Figure 3.3. Axial deformation of intervertebral joint.
The displacements between vertebrae is in
a direction perpendicular to disc middle
plane a — a.

 

Figure 3.4. Shear defonnation of intervertebral joint. The
displacement between vertebrae is in a
direction parallel to disc middle plane
a — a.

 

33

moments are applied to the pelvis that could generate
rotational motion, Appendix F.

b) Due to the large dimensions of the pelvis, any tangential
displacement of a peripheral point of the pelvis is
translated into a small rotation about the center of
gravity of the sacrum — pelvis mass.

The ligaments holding together ribs and vertebrae at the
costo—vertebral joints will allow significant relative
displacements and rotations between a vertebra and a rib
when loaded individually, Schultz et a1. (1974). The
ribcage as a whole behaves very much like a hollow truncated
cone having mostly longitudinal and transversal displacements
relative to the thoracic spine. Any rotational relative motflxl
will be greatly inhibited by the line of costo—vertebral
joints placed on the thoracic spine. Therefore, the mass of

the upper torso, M is given only 2 degrees of freedom;

ut’
with motion in horizontal and vertical directions.

The head has three degrees of freedom in the sagittal
plane. None of them is inhibited to a point where its
elimination from the model can be justified.

Each one of the degrees of freedom just described
implies the existence of a corresponding coordinate if the
motion of the structure is to be fully described. Therefore
coordinates will be adopted according to the number of

degrees of freedom assigned to each mass.

3.3.1 Systems of coordinates

 

A system of coordinates must be properly chosen before

 

34

writing the equations governing the motion of the structure.
The resultant system of equations will have desirable
characteristics depending on the chosen coordinates.

For the model under study, the motions of each vertebra
and the head-neck lumped mass in the sagittal plane are
completely described by three x, z, 6 as shown in Figure
3.6. The z—axis is oriented vertically while the x—axis is
oriented horizontally from the spine to the anterior part
of the torso. The deflections are measured from a fixed
point in space where the center of gravity of masses would
be at rest if there were no excitation forces acting on the
system.

A clockwise rotation 6 of a mass about its center of
gravity is assumed positive. The sacrum-pelvis and the
upper torso masses are given only two degrees of freedom
described by coordinates x and z. Coordinates u, w, 6 shown
in Figure 3.6 are auxiliary coordinates used to facilitate
the derivation of the intervertebral joint stiffness matrix.

3.4 Rheological Behavior of Deformable Elements of

the Model

The rheological behavior of all deformable elements in
the system is modeled by means of Kelvin type viscoelastic
elements as shown in Figure 3.2 to model the costo—vertebral
joints. This decision was made in light of the results
obtained from the modeling work done for the axial behavior
of intervertebral joints, which is covered in chapter IV.
The numerical values assigned to each spring and dashpot in

the system are discussed in Chapter VI.

 

Figure 3.5. Bending deformation of an intervertebral
joint.

 

Figure 3.6. Local system of coordinates for calculation
of element stiffness matrix.

 

36

IV. AXIAL RHEOLOGICAL BEHAVIOR OF INTERVERTEBRAL JOINTS

4.1 Driving Point Impedance of a Vertebral Unit

The axial rheological behavior of an intervertebral joint
can be modeled using the impedance test results reported by
Kazarian (1972). The data are available for the thoracic
and lumbar spine, with and without the posterior arch, and
for three age groups of people.

Some of the conditions under which Kazarian's tests were
conducted are summarized in this section for application in
Chapter IV. The thoracolumbar spine was divided in four
units: Tl-T6; T7-T12; Ll—L3; and L4-Sacrum for impedance
testing.

The vertebral units were tested in vertical position.
The superior end of the unit was fixed to a loading head
mounted on a ball joint designed to apply pure compression
load on the unit. The lower end of the vertebral unit was
mounted onto the shaker head, Figures 4.1 and 6.5. Each
unit was tested at a preload of 21.7 Kg. Thistestkg axfldtkm
forced the intervertebral joints to a deformed configuration
with higher values of stiffness and closer to the real
situation.

Following the impedance test of a complete spinal unit,
it was removed from the loading head, its posterior arch
separated and the vertebral bodies with intact anterior and

posterior ligaments were placed back into the loading jaws

 

37

for impedance testing. The later test was conducted to find
out how the posterior arch contributes to the dynamic load
carrying capacity of the spine.
4.2 Mechanical Impedance

Electrical circuit analysis techniques can be applied
to complex mechanical systems. The mechanical impedance (or
mobility) is a relationship between force and velocity
represented by a differential equation in the time domain.
By transforming this equation to the Laplace domain the
computational work is considerably simplified since all the
differential equations become algebraic. For the present
investigation where harmonic excitation is the main interest
and the answers sought can be obtained in the frequency
domain, the Laplace variable, 3, will be repleaced by iw,
where i stands for the imaginary number /:1, and w is the
frequency of harmonic oscillation.

The impedance, Z, of the three basic elements (spring,
dashpot, and mass), equations (4.1), are used to derive the
impedance of the complete structure. The derivation of

element impedances are given in Vernon (1967).

Mass Damper Spring
z=§=ms Zd=c Zk=k/s (4.1)
m V
where:
Zm : Impedance of a mass ; F : Force
Z Impedance of a damper ; V : Velocity

 

 

38
Zk : Impedance of a spring ; m: Mass
c : Damping coefficient ; k: Stiffness coefficient

5 : Laplace variable
It is convenient to use the concept of "mobility” at some
points in the analysis of the structure. The mechanical

mobility, M, is the reciprocal of the mechanical impedance.
M = l/Z (4.2)

Whenever simple elements of the structure are in parallel,
their velocities are the same, and the total force is the
sum of the forces applied to each element. Since impedance
is defined as the ratio: force/velocity, and the velocity is
a common denominator to all parallel elements, the overall
impedance of the parallel arrangement is simply the sum of
the impedances of the individual elements.

For elements in series the same force is applied to all
elements. Since ”mobility" is defined as the ratio velocity/
force, the denominator of all elements is equal, so the
overall mobility of the arrangement is the sum of the
mobilities of the components. The previous concepts are

summarized by equations (4.3) and (4.4).

 

 

Elements in Parallel Elements in Series
_ _ 1
Z — 21 + Z2 Z — ————————-——— (4.3)
l/Zl + 1/22
M= _———1————— M= M1+ M2 (4.4)

l/M1 + 1/M2

These simple concepts are sufficient to derive an equation

 

39

for the driving impedance of a vertebral unit composed of

N masses (vertebral bodies) plus equal number of springs

and dampers located between the masses. The viscoelastic
elements representing the intervertebral joints were modeled
as a pair spring-dashpot in parallel (Kelvin model). The
damping and stiffness coefficients of the model are given as
exponential functions of frequency in the range 5-50 Hz.

Models other than Kelvin, including three to five
parameters were used in the modeling process in an attempt
to predict the mechanical impedance of the vertebral unit
using constant parameters. No satisfactory results were
obtained from these models, so the Kelvin model with fre—
quency dependent coefficients was finally adopted.

The driving point impedance of the vertebral unit shown
in Figure 4.1 is calculated from the impedances and mobilities
of the viscoelastic elements and masses in the system.

For the spring and dashpot in parallel the resultant

impedance Z3, is:
’ J

23 = ZC + Zk = c + k/s (4.5)

The impedance, 23, of the jth. Kelvin element in parallel
with mass mj is given by equation (4.6). It must be cmmmxted
to mobility, M3, so it can be added to the mobility of the
superior part of the unit, M(j—l)’ with which it is in series,

equation (4.7).

2; = c + k/s + m.s (4.6)

J

 

 

40

The masses appear to be physically in series with the
viscoelastic elements, but they must be considered as being

in parallel for the impedance calculations using electrical

analogy.
n _ 1 _ l
Mj _ ZH — c + s + mjs (4'7)

Substituting 3 by im,

 

l c i(k/w - mm.)
H = = ——.._______J_
Mj c — 'ik'7w ——+ 1me D0 + D0 (4'8)

D0 = C2 + ( mmj - k/w)2

The mobility at the bottom end plate of the jth vertebra,
Mj, is the sum of the mobility (Mj—l) at the bottom end
plate of the (jth-l) vertebra plus that corresponding to mass

mj in parallel with a Kelvin element.

r c . i

M.=M. +M'.'=M. +—+ M. +
J (J-l) J (J-l) Do 1 (J-l)

(k/w — mm.)

_____.____1_
+ D0 (4.9)

Mr, M1 : Real and imaginary parts of mechanical mobility

respectively.

Equation (4 9) is sequentially applied from the first

 

 

41

movable vertebra at the top of the unit down to the last
vertebra at the impedance head. The mobility of the first
superior vertebra is null for being attached to a fix loading
head. This known value is used in equation (4 9) to start
the sequencial calculations from top to bottom of the unit.
The impedance of the whole vertebral unit, Z, is computed
from the mobility, M , obtained for the nth vertebra (lowest)

n

of the unit using equation (4.10).

(4.10)

_ _ r - i
Z — 1/M - Mn / anIZ' 1 Mn / anl2

n

4.3 Estimation of the Parameters of the Kelvin Model

The parameters c and k in the Kelvin model of the
intervertebral joint can be calculated using the approach
described by Beck and Arnold (1975). For the implementation
of this procedure a matrix of sensitivity coefffijgnt, [X],
must be calculated. Each sensitivity coefficient is the
rate of change of the real or imaginary part of the mechani-
cal impedance, Zr and Zi respectively, with respect to the

damping coefficient c, or the stiffness coefficient k.

 

 

aZr azr
8c 8k
[X] = (4.11)
321 921
8c Bk

If 8 stands for either c or k, the two rows of the

sensitivity matrix are given by equations, (4.12) and (4.13)

 

42

which were derived from equation (4 10).

 

 

1:
3M 2 r
azr = 5§—-|M| — 2 (fi 38 rM + 38 M i) M (4.12)
BB 1M!“
1 M 2 2 ——._. Mr + _§1fil Mi) Mi
§§__ = ;l 1 " 1% as (4.13)
as IMI2

31M12=_a_[(Mr)2+ mix] = 2[Mr aMr+Mi Mi]

88 BB 88 88

Where the partial derivatives of real and imaginary parts
of the mobility can be obtained from equation (4.9), and are

given by equations (4.14) to (4.17).

aMr

 

 

 

2
_§M: = (j—l) + D° ' 2C (4.14)
ac 8c mDoz
r aMr 2c(mm. — k/m)
_3M_ = (j-l) + 3 (4.15)
8k 8k Do2
. BMi (k/w — mm.) 2c
1
__§E_ = (j-l) _ 3 (4.16)
ac ac D02
. .1

 

 

8k 8k D02

 

43

These derivatives are successively calculated for each
intervertebral joint from the loading head down to the
impedance head. The coefficients of the sensitivity matrix
are obtained from equations (4.12) and (4.13) using the
values of impedance and its derivatives calculated for the
last vertebra of the unit.

The parameters in the model are optimized by minimizing

the sum of squares function, SS, given by equation (4.18).
33 = [Y — z (3)]T wh [Y — z (sfl (4.18)
Y: Vector including real and imaginary parts of experimental

impedance.

Z: Vector including real and imaginary parts of modeled

impedance.

Yr Zr
'Y= 2(8):

Yi zi

The weighting matrix, Wh, is taken as an identity matrix for
the present work. The minimization of the sum squares is
done by the linearization method (Gauss). From this
minimization process the optimum values of the parameters
c and k are obtained, after successive iterations using
equations (4.19).

1

T -1 -1 T _
B. + (x wh X)j xj wh ( Y — Z0 (8) )j (4.19)

B<j+1>= J

 

 

B. : 2 Vector of damping and stiffness coefficients
after j iterations

j

The sensitivity matrix X and modeled impedance Z (8) are
re—evaluated after each iteration because they are function
of the parameters c and k that change after each iteraction.
The iterative procedure continues until the parameters

change a negligible amount or until the sum of squares is
sufficiently small.

4.4 Frequency Dependent Stiffness and Damping Coefficients

A damping coefficient and a stiffness coefficient are

calculated for each data point consisting of a frequency,
impedance modulus, and phase angle. The stiffness coefficients
were found to increase exponentially with frequency while

the damping coefficients decrease exponentially in the fre—
quency range 5 to 50 Hz. Equations (4.20) and (4.21) give
good approximations for the stiffness and damping coefficients.
Coefficients k1, kg and c1, C2 are listed in Table 6.5 for
three age groups of people. They were approximated by the

least square fit method.

c = C1 (fq)C2 (4.20)

kg fq

k = k1 e (4.21)

 

 

45

4.5 Axial Dynamic Response of the Posterior Spinal Arch

The posterior vertebral arch contributes to the load
carrying capacity of the spine. The portion of the load
carried by the arch is dependent on the curvature of the
spine in the sagittal plane. The load on the posterior
arch varies with sitting posture. As the degree of
hyperextension of a seated subject increases, so does the
load on the posterior arch.

The impedance measurements made by Kazarian (1972) for
vertebral units with and without posterior arch suggest the
idea of modeling the intervertebral joint as a pair of
viscoelastic elements (Kelvin) in parallel, Figure 4.2.

The element, D simulates the intervertebral disc; the

second element, A, represents the posterior arch. The
mechanical impedance of a vertebral unit is calculated
sequentially from top to bottom of the unit adding impedances
or mobilities according to convenience, as it was done in

section 4.2 for a vertebral unit with single Kelvin elements

between masses.

1
M. = M . + 4.22
J (1—1) 2m + a + 2b ( )
Zm 2 ms : Impedance of mass representing vertebral
body (4.23)
k

Z = ——— + ca : Impedance of posterior arch (4.24)

 

46

  

Loading head
1 \—————> (fixed)

f(t) driving force

Figure 4.1. Spinal unit model for calculation of

driving point impedance.

 

 

Figure 4.2. Intervertebral joint modeled as two Kelvin

elements in parallel.

 

47

k
Zb = —§— + cd Impedance of intervertebral disc (4.25)
kd, cd Stiffness and damping coefficients of inter—
vertebral disc
ka, ca : Stiffness and damping coefficients of posteriorard1

Substituting (4.23) to (4.25) in (4.22), and after introducing
s = iw for sinusoidal excitation, the real and imaginary

parts of the mobility are:

M1— = Mr + (Ca + Cd) (4.26)
J <j-1) 13"
. . M — moo]
1 _ 1
M3. ~ M(j_1) + D0 (4.27)
(k + k ) 2
D = (ca + cd)2 + [mm — Lari] (4.28)

After the mobility for the last vertebra of the unit has
been calculated the impedance of the unit is obtained using
equation (4.10).

There are two damping coefficients, (ca, Cd), and two
stiffness coefficients (ka, kd) to be estimated in equations

(4.26) and (4.27). The coefficients corresponding to the

  

-1Mui in land-sin!!! so: ant“ but I'm
salb Itfulsfl'rdjlflI

     
  
 

'23-'41?

48

intervertebral disc (c kd) were already calculated using

d’
the impedance data ”without arch” together with equations
(4.9) and (4 10) corresponding to a model with single Kelvin
elements between masses. Consequently only the stiffness
and damping coefficients for the posterior arch, ca and k3,
are left to be calculated from the impedance data ”with
arch".

In order to estimate the parameters ca and ka, the
coefficients of the sensitivity matrix (4.11) are to be
calculated from equations (4.12) and (4.13). Some of the
partial derivatives in Equation (4.12) and (4.13) must be
further expanded for implementation on digital computer.

Equations (4.29) to (4.32) follow from (4.26) and (4 27).

 

aMr BMr [n1— Eta—fidj

' _ ('—1) L02 (4.29)
37L — _5%—+2 (ca+cd)—————————————H

a a |Mj|

r r 2 _ 2
2§i_.=_3§ii;ll + [M1] 2 (Ca + Cd) (4.30)

L.
SCa 80a |Mj1

 

49

 

 

 

(k + k )
1 Mi [M.[Z-z mw—._§___9_
& = (j-l) + J m (4.31)
aka aka le14
(k + k)
. d
. 1 2(c + c) m — —a———
3141 = PWG-1) + a d m (4.32)
8C 3c [M.[“
a a j

The sensitivity matrix is assembled with the results of
equations (4 29) to (4.32). The optimization of parameters

ka and ca is done by iteration using equation (4.19).

The stiffness coefficients obtained from the data
”without arch” were in many cases greater than those corre—
spinding to the data ”with arch”, what would apparently
indicate that the posterior arch makes a negative or at least
null contribution to the load carrying capacity of the spine.
The phenomenon that actually took place is probably as
follows: by removing the posterior arch and keeping a
constant compression bias on the vertebral unit the disc
moved to a larger deformation configuration able to carry the
total load without the assistance of the posterior arch.
Given the non—linear rheological behavior of the disc, the
larger deformation imposed on the disc explains the larger
values of stiffness obtained for the vertebral unit without
arch.

The analysis in section 4.5 would probably lead to

 

 
 

.d '
‘ ‘

-1 .,.,ivé-.lrthe’ joint“; ’with' coefficients er and -k‘ "ob‘tained'rf’i‘bnf‘

 

data with arch .

 

V. DYNAMIC RESPONSE OF THE LUMPED PARAMETER MODEL

5.1 Governing Differential Equations of Motion

The system of second order differential equations
governing the motion of an N—degree of freedom system with
viscous damping is as given by equation (5.1), Meirovitch

(1967).

[m] 111' (of + [c] 146)} + [k] 1c; (t>}= 11%)} (5.1)

Where the mass matrix [m], stiffness matrix [k], and damping
matrix [c] were calculated using the procedure described in
the three following sections. The displacement function
q(t) and the forcing function f(t) are discussed in section
5.2.

Each equation (row) of the system (5 1) can be derived
by writing the equation of dynamic equilibrium, Newton's
2nd. law, for each degree of freedom of each mass in the
system. The result will be a system of equations resembling
that shown as an example in Appendix B for the motion of a
vertebra in x — direction. A less involved procedure would
probably result from the application of Lagrange's equations.
But, due to the large number of degrees of freedom in the
system, a matrix approach was used that provides the equations
of motion for the discrete system by properly choosing a

coordinate system and applying some of the well established

51

      
 
  
   

L.-
.,. l
' i, .
I _ . .-..|.,... .
“in,

  

I I ‘_
110330! To aminnum Minn-3133383 antitrust-1*
'1'"... EMF-‘71?“ 1“. "1.0-"51:: I? z,‘ -_'_“ll-- r- 'Inh':I-a 'IG 1193.2. y.- 1:“!
"I ‘- ’*'-'I
‘ ' - '- -. .-. ’4: - - _"--'-r' :51 an] ’
. 41-51.- !nmflk '-

.jtfi.“j\

52

techniques of structural analysis to derive the stiffness
and damping matrices. This approach permits handling of the
equations in a more compact and systematic form, which are
more easily programmed for digital computers.

5.1.1 Mass matrix

All masses and mass moments of inertia entering the
system of equations can be grouped in a single matrix Efl
called the ”mass matrix". For the coordinates chosen in
section 3.2.1, the mass matrix results to be diagonal, reason
for which the system is said to be ”dynamically uncoupled”.
Coupling is not an inherent property of the structure but
depends on the coordinates used to describe the motion.

Mass matrix (5.2) corresponds to the model shown in
Figure 3.2. This matrix was assembled by lining up masses
and mass moments of inertia for all masses in the system on
the diagonal of an otherwise null matrix. The order to
follow is given in Appendix C.

5.1.2 Global stiffness matrix

 

The assemblage of the stiffness matrix is not as straight
forward as it was for the mass matrix. The stiffness matrix
can be obtained from the system of equations resulting from
application of Newton's 2nd. law, or Lagrange's equations,
but these approaches lead to quite involved calculations.

A simpler and more systematic method exists to assemble
the global stiffness matrix of the structure when it can be

done in a digital computer. This is conveniently accomplished

_ ILFL"_ .- ._ _ '1 :7 "._._ .1
15213115 :9! mm!!!

:=.-3..: wines!- .

  

.21"?leth

 

1:. «1.3M: 1831",,

2 c, 11:93.?!-

““1153

 

mli-

m I
Sp

11:

ut:

53

(5.2)

SP
mut

ut

 

Lumped mass of head and neck

Mass moment of inertia of head—neck about its center
of gravity

Mass of ith thoracic vertebra

Mass moment of inertia of ith thoracic vertebra about
its center of gravity

Mass of ith lumbar vertebra

Mass moment of inertia of ith lumbar vertebra about
its center of gravity

Lumped mass of sacrum and pelvis

Lumped mass of upper torso and upper limbs

 

54

by superimposing the stiffness matrices of the individual
deformable structural elements, Martin (1966), which will be
called "element stiffness matrices”. There is a total of 28
element stiffness matrices in the structure corresponding
to: neck, 17 intervertebral joints, and 10 costo—vertebral
joints.

A generic (6x6) element stiffness matrix that applies to
all intervertebral joints is shown in matrix (5.3), where
the parameters Ka, KS, and Kb stand for axial, shear, and
bending stiffness of the intervertebral joint respectively.

The angle Oi made by the longitudinal axis of a vertebra
and the z-axis varies along the spine. The angle 5i, the
longitudinal axis of the disc makes with the z—axis, is tak-
en as the average of the angles corresponding to the two
vertebrae enclosing the disc. The angle Oi is shown in
Figure 3.3 as the angle made by the disc middle plane a-a
and the x—axis.

Appendix D shows the steps followed in deriving the inter—
vertebral joint stiffness matrix from the equations of static
equilibrium. A similar procedure was followed to calculate
the element stiffness matrices corresponding to neck, and
costo—vertebral joints.

Since matrix (5.3) was derived using a convenient system
of coordinates u, w, 6, Figure 3.6, which is not the global
system x, z, 6, adopted for the derivation of the equations
of motion (5 1), the intervertebral joint stiffness matrix

0&1] must be subjected to a coordinate transformation, a

 

Cole

V

 

m
M+NH® Nmoo

mm_ ”N + p

:0 Guam NM.

m

m m .
MH— NHN + QM!

Nfio cam mmmmu

fiwno moo mmmML

M H

.............

M...~4o~sam «HecammmHmnfllemoo azam-u

UHMHEm

T6 «Cam
mM+N Ho Nmoo

n

m a
v; N.N+ M

6

so cam swag-”

 

 

 

56

rotation, that makes it suitable for assemblage into the
global stiffness matrix [k] . The transformed matrix [ke]
is obtained from equation (5.4) which involves the rotation

matrix [R] and its transpose PEJ , Gere and Weaver (1965).
T
[Rt] [k1] [Rt] (5.4)

[Rt] = (5.5)

'——'I
z"
(D
t—J
II

[R] = sin 6 cos 0 O (5.6)

5.1.3 Damping matrix

The same formulation developed for evaluation of the
global stiffness matrix holds for the global damping matrix.
The only difference being that stiffness coefficients must
be replaced by corresponding damping coefficients.

5.2 Solution of the System of Governing Equations

One way to solve the system of equations (5.1) would be
by finding a linear transformation of coordinates able to
uncouple the system of equations. Every equation of the
uncoupled system can be solved individually as normally done
for a single degree of freedom system.

A relatively simple method was presented by Foss (1958)

 

 

to find a matrix of orthogonal eigenvectors able to uncouple
the system of equations in an auxiliar system of coordinates.
Integration of individual equations is then done for the
forcing function of interest, and the auxiliary coordinates
transformed back to the original system of generalized
coordinates that have the physical meaning of interest.

The procedure just briefly introduced has thefxmenttfl.to
provide the response of the system to different inputs.

After the mass, stiffness and damping matrices are obtained,
a dynamical matrix is assembled. The complex eigenvalues
and eigenvectors of the dynamical matrix fully characterize
the dynamic behavior of the system, so that its response can
be calculated for a given excitation using the eigenvalues
and eigenvectors as input data together with a short
computation for integration of the uncoupled differential
equations.

This approach was tried in the present work, but some
inconsistencies were found in the results. The reason for
such behavior probably being the existance of some errors in
the eigenvectorsasearemflt of the large number of degrees of
freedom of the system with some eigenvalues not very distinct
from each other.

The main objectives of this project are equally fulfilled
by using a less general solution of the system of second
order differential equations. The complementary solution of

equations (5 1) is not of major interest in the present work.

 

 

.I

58

If the vibrational input includes only sinusoidal oscfllatfixm,
a particular solution as given by equation (5.7), Thomson
(1972), Reismann and Pafljk (1975), will provide most of the
answers sought. The particular solution consists of a set
of functions qj(t) describing the steady state harmonic
oscillation of the same frequency w as that of the emfitathxr
Each mass in the structure will oscillate about its emfilihdum
position with an amplitude [Ajl and lagging the vertical
motion of the base by an angle wj which is related to the
amount of damping existing between the excitation point and
the point where the oscillation is being studied.

qj (t) = [Ajl ei(‘*’t+1’j) (5.7)

The response equations (5.7) and their derivatives can
be written in a more suitable form for implementation of the
solution of equation (5 l) in a digital computer. The phase
angle is removed from the exponential factor and incorporated

as a complex amplitude, Aj’ equations (5.8) to (5.11).

. t = A. ei‘”t 5.8
qJ() J ( )
A.=Af+'A1 .
J J 11 (59)
9j (t) = iwAj ei‘”t (5.10)

59

63. (t) = — 62 Aj eimt (5.11)
The base excitation zb(t), equation (5.12), applied in
vertical direction (12) to the pelvis of a seated operator is
a displacement type excitation, so the forcing function f(t)

entering equation (5.1) needs to be written in a different
form to be able to characterize the excitation by a displace-
ment amplitude and a frequency instead of a force amplitude
and a frequency.
iwt

(t) = A e

Zb b

(5.12)
Ab : Real amplitude of base harmonic motion

The forcing function f(t) is calculated from the equation
of dynamic equilibrium (5.13) of forces acting on the operator
seat in z-direction. The forces applied to the seat are: the
action of the body f(t), plus those generated at the seat
suspension as a result of its stiffness Kc’ damping Cc’ and

the relative motion seat—base BS, Figure 5.1.
MC qs(t) = KC BS + CC BS — f(t) (5.13)

The displacement of the seat is assumed to be that of the

sacrum-pelvis mass qs(t), equation (5.14). This assumption

 

60

is valid for an operator seated on a bare seat where the

stiffness of the tissues located between the pelvis and the
seat is sufficiently high; about 1000 Kg/cm proved to give
satisfactory results for the present model. No expenfimmtal

data are available.

_ iwt
98 (t) — As e (5.14)

As : Complex amplitude of pelvis-sacrum oscillation

BS = zb (t) — qS (t) (5.15)

From equations (5.12), (5 14), and (5.15) BS can be

written:

BS = (Ab - AS) eiwt (5.16)

Introducing equations (5.14) and (5.16) into (5.13) the
forcing function can be written:
imt

f(t) = [MC (1)2 AS + KC (Ab ~ Ag) + iCC (Ab - AS)]e (5-17)

Separating the real part, FE, and the imaginary part, F:, of

f(t),

f (t) = (F: + iFi> ei‘”t = F ei‘“t (5.18)

 

61

r = 2 _ r 1
FS KC Ab + (m MC KC) AS + mCC AS (5.19)

i: 2 _ i_ r

FS (w MC KC) AS wCC AS + wCC Ab (5.20)
Substituting equations (5.8), (5.10) (5.11) and (5.18) in
(5.1), after cancelling exponential factors the system of

differential equations is turned into a system of algebraic

equations:

_ w2[m] {A} + i0) [cHAi + [141.41 =11“; (5.21)

§A§ = {Ar} + 1 {Al}: Vector including complex amplitudes
of oscillation for all degrees of
freedon in the system.
H

{Fri + i {F1}: Vector including complex amplitudes
of all external forces acting on the
system.

Writing all amplitudes in complex form, equation (5 21)

turns into (5.22).
-w2[ ] {Ar} - 1m2[m]{AiE + 14.11.11 wpfiflhpfig}
+i[k]{Ai}-‘—{Fr§ +1119} (5.22)

By equating real and imaginary parts of equation (5.22) the
system of N equations with complex unknowns is turned into

a system with 2N equations in all real numbers.

[[k] 4.4.1.1] {A} -w [6111.11 #51:} (5.23)

 

62

0 [c] {Ar} + [[k] 462 m]] {Ai} 4171} (5.24)

The dimension of vectors Fr and F1 is 58, but only one
component is different from zero. It corresponds to the

vertical motion of the sacrum, and is obtained from equations

(5 19) and (5.20).

 

 

 

 

 

 

o ’0 o
19 91 0
KC {0 +002 Mc'Kc) [I] <0 + (”CC [I] 6.
Ab A: A:
o o: o
2 \o , .01 0 (5.25)
'01 '01 o
o 0 0
(a w m 1. 1 - m .
o .
A A1 Ar
ob OS 08
.01 Lo, 0

 

 

 

63

There are two unknowns on the right side of equation

(5.25) that must be moved to the left side of the equation

to make the system suitable for computer solution.

Equation

(5.26) was used to program the assemblage and solution of

the system of equations into subroutine ”AMPLTD" of program

”COLSOL".

 

 

8
r I
A Kc
A: = Ab § (5.26)
wéc
8
._0 -
0
on
(5.27)
'1
0
L o
oo 1
0
I (5.28)
1
0
L o

 

 

 

64

5.3 Driving Point Impedance of the Model

The driving point impedance of the seated subject was
used for validation of the model. Since no provisions were
made to represent the rheological properties of thecbfonmmle
elements existing between the sacrum and the seat, the seat
suspension spring (Kc) and damper (Cc) were used to model
the behavior of these elements in the validation process.
If the mass of the seat, Figure 5.1, is assumed to be null
and the base is thought as the seat surface, the seat
suspension left in between them would simulate the behavior
of the deformable elements separating the seat from the
sacrum.

Under these assumptions the driving point impedance can
be calculated from the velocity of the base, equation (5.29),
and the force transmitted through the suspension, equation
(5.17), which can be calculated after solving the system of

equations (5.26).

V(t) = 10 Ab el‘”t (5.29)
The driving point impedance then results:

Z = (l — AS/Ab) [ CC — 1KC/01] (5.30)

5.4 Shear and Axial Deformations of Intervertebral Joints
A general expression describing the motion of every mass

in the structure is given by equations (5.8), which are

 

65

renamed according to the direction of motion as shown by
equations (5.31) to (5.33) for horizontal, vertical, and

rotational motion respectively.

-— iwt

. t = . . 1
XJ ( ) XJ e (5 3 )

. t = 2. iwt 5.32
zJ () J 6 ( )

iwt

6. t =A. 5.

J ( ) J 9 ( 33)

Xj’ Zj, and Aj are complex amplitudes equivalent to Aj

The shear deformation, S, of an intervertebral joint is
approximated by projecting all displacements of two adjacent
vertebrae on the disc middle plane a—a, Figure 5.2. The
axial deformation, N, is calculated by projecting all

displacements in direction perpendicular to a-a.

S = [x1(t) - x2(t)] c030 — [21(t) — zz(t)] sinO +

+ [8,(t) z2 -62 21] 603012 (5.34)

2
ll

[x1(t) - x2(t)] sin 0 + [21(t) - zz(t)] cosO —

— [61(t) 22 —62 21] sin 012 (5.35)

 

66

3 =15, -2460. 6— [2,- 2,] sin e +

+[A1 Z2 — A2Z1] cos 012} eiwt (5.36)

N ={[).(1 - >712] sin5+[21- 22] C085—

_[A1 22 + 1,21] sin012}elwt (5.37)

012: Angle made by vertebra end plate and disc middle
plane a-a

5.5 Seat to Head Transmissibility

The seat to head transmissibility is defined as the ratio
between the acceleration of the head and the input acuflrmatflxl
through the pelvis. For harmonic motion the ratio of
accelerations is equivalent to the ratio of displacements.

Considering the same assumptions made for evaluation of
driving point inpedflme, that is, null seat mass and seat
suspension representing the deformable elements located
between sacrum and seat surface, the transmissibility Tr is

as given by equation (5.38).

Z
Tr = Read (5.38)
b
2 Complex amplitude of head vertical motion

head

 

 

67

Sacrum- elvis mass
\. “‘£b_z=A elwt
s

S

 

Mass of Seat Mc f(t)

 

 

 

Seat suspension

 

_%- Ab eimt

Base

Figure 5.1. Operator seat under sinusoidal
displacement excitation.

 

 

Figure 5.2. Displacements and rotations of two axwemnjve
vertebrae determine the axial and shear de—
formations of the enclosed intervertebral
joint.

 

 

68

5.6 Computer Program

The computer program "COLSOL" assembles and solves the
system of equations (5.26), which involves 116 unknowns
resulting from the 58 magnitudes and 58 phase angles corre—
sponding to the complex amplitudes of the 58 degrees of
freedom in the system.

Since the stiffness and damping coefficients are frequency
dependent, matrices [k] and [c] must be recalculated for
every frequency analyzed. The same subroutines "HEAD", "DISC"
and "THORAX" are involved in the calculation of both matrices.
Before the computation of [k] all frequency dependent stiff-
ness coefficients are calculated by calling subroutine
"CALKDZ". Similarly, before the computation of LC] ,
subroutine "CALCDZ" is called to calculate all frequency
dependent damping coefficients.

After all required matrices have been calculated, sub—
routine "AMPLTD" is called to assemble and solve the system
of equations (5.26). Jith all amplitudes and phase angles
already known, subroutine ”OUTOUT" is called to calculate
and print seat to head transmissibility, equation (5.38),
and driving point impedance, equation (5.30), which are used
for validation of the model. Subroutine ”OUTPUT" will also
calculate and print axial and shear intervertebral joint
ckeformations, equations (5.36) and (5 37), which are the
response parameters of interest after the model has been

validated.

 

 

The flow chart shown in Figure 5.3 summarizes the steps

 

69

described in the previous paragraphs.

 

[Data input and outpuf]

 

fq = 5 Hz

 

l

 

Calculation of frequency
dependent damping coefficients

Subroutine: "CALCDZ"

 

l

 

Calculation of damping matrix
Subroutines: ”HEAD", DISC",

 

and "THORAX"
l

 

fq = fq+l I

Calculation of frequency
dependent stiffness coefficients

Subroutine: "CALKDZ"

 

l

 

Calculation of stiffness matrix
Subroutines: "HEAD", "DISC", and

 

"THORAX" l

 

Assemblage and solution of system
of linear equations

Equation (4.52)
Subroutine: "AMPLTD"

 

I

 

 

Calculation and printing of impedance,
transmissibility, phase angle, shear
and normal deformations

Subroutine: "OUTPUT"

 

 

 

Figure 5.3.

Flow chart for computer program "COLSOL".

 

 

 

 

VI. EXPERIMENTAL DATA

6.1 Geometrical Data
6.1.1 Vertebrae

Most of the geometrical data required for a lumped
parameter model of the spine is available in the literature.
The curvature of the spine in the sagittal plane, Table 6.1,
was calculated from the coordinates (uo, Wo) reported by Orne
and King Liu (1971) for a seated position.

The existing data on dimensions of vertebrae, such as
that given by Lanier (1939), does not include values of mass
of individual vertebra or location of its center of gravity.
Approximation of the geometry of a vertebra by superposition
of bodies of known configuration, such as a truncated cone
or an ellipsoid was considered, but it presents some
difficulties. For instance there is enough variation of ver—
tebra configuration through the thoracic spine to justify the
use of more than one model. The cross section of the verte—
bral body at the first thoracic vertebra is approximately
trapezoidal, toward the fifth vertebra the body cross section
becomes approximately parabolic. The geometry of vertebrae
significantly changes when passing from the thoracic to the
lumbar spine so at least three different models would be
required to be able to calculate the properties such as

center of gravity and moment of inertia for the thorafic finne.

70

 

71

Table 6.1. Curvature of the thoracolumbar spine in

the sagittal plane.

 

 

 

Vertebral 0

Level Deg.
Oi = arc tan (Eilill_;_fliil) Th1 5.0(1)

w(1+1) ‘ Wu) Th2 9.8

Th3 17.4

Th4 14.9

Th5 12.5

Th6 0.0

Th7 —4.6

Th8 —8.3

Th9 -15.1

Tth -15.2

Thll —l4.0

Th12 -l8.7

Ll -16.8

W(1+1) L2 -10.6

L3 —2.2

L4 4.7

u . L5 14.2
—1}J—-—" Sacrum 45.0(2)

 

(1) Arbitrary

(2) Kazarian (1972): 45.0 deg.; Schultz et a1.

32.5 deg.

 

(1973):

 

 

72

Even if the geometry of a vertebra coul be reasonably
approximated, there still remains the problem of estimating
the density distribution over the volume of the vertebra.
The end plates have different density from the nucleus of
the spongy vertebral body or the transverse processes.

Due to the problems previously stated, the geometrical
properties of the thoracic and lumbar vertebrae were
determined experimentally. A spine (C2 to L5) was removed
from an embalmed cadaver provided by the Anatomy Laboratory
of Michigan State University. The spine was considered
normal, with larger dimensions than the average reported by
Lanier (1939).

The moisture content was maintained by wrapping the
spine in a moist cloth and sealing it in a polyethylene bag
to avoid any drying that could change the mass or density
distribution within each vertebra; such changes would affect
the values of mass moment of inertia to be measured. The
dimensions measured on each vertebra are listed in Table 6.2.
The coordinates of the costo—vertebral joints are given in
the table, but were not used for the final version of the
model.

The location of the center of gravity in the mid—sagittal
plane was determined experimentally using the pendulum built
to measure mass moment of inertia, Figure 6.1. The vertebra
was hung from the pendulum frame by means of a thin spring

wire, .5 mm in diameter. The wire was soldered to a tiny

 

Zs Xt Zt

73
Zi Xs
cm

Xi

Geometrical data for thoracic and lumbar
Z2

vertebrae.

Zl

 

888880195305

11100111110

09755212036

................

................

 

Table 6.2.

 

Vertebra

 

 

 

 

74

wood screw (5 mm long) at one end, and to a piece of razor
blade in the opposite end. The total weight of the support
wire is .42 gm.

Every vertebra was hung from two different points in the
mid-sagittal plane, and a vertical line passing through the
pivot point was drawn for each hanging position. The point
of intersection of these lines corresponds to the location
of the center of gravity. The distance "r" from the
pendulum pivot to the center of gravity as well as the
location of the costo-vertebral points of interaction were
then measured, see Table 6.3.

6.1.2 Head and neck

In the absence of experimental data, the location of the
lower end plate of the seventh cervical vertebra is assumed
to be 17 cm below and 3.8 cm behind the center of gravity of
the head-neck system, see Figure 6.3. Orne and King Liu
(1971) reported satisfactory dynamic model results using a
head neck eccentricity of 3.8 cm.

6.1.3. Pelvis

The sacrum-pelvis mass is included in the model with only
two degrees of freedom according to the assumptions made in
section 3.2. Therefore no geometrical data is required other
than the angle the axis of the sacrum makes with the z—axis,
which is given in Table 6.1.

6.2 Mass Moment of Inertia of a Vertebra

Rotation in the sagittal plane is one of the degrees of

 

 

75

freedom considered in the model. The mass moment of inertia
of each vertebra with respect to its center of gravity in the
sagittal plane is required to write the equation corresponding
to the rotational mode of oscillation.

The moments of inertia of the thoracic and lumbar verte-
brae were calculated from the period of oscillation of the
vertebra in pendular motion in the mid-sagittal plane. The
pendulum, Figure 6.1, was constructed and then tested for
bodies of regular geometry (cylinder, ring) in order to
verify the concepts described in the next paragraphs,
particularly those relating to the accuracy required to
measure the time period and the distance from the pivot
point of the pendulum to the c.g. of the oscillating body.

The moment of inertia, Ig’ can be calculated from the
natural frequency of oscillation of the pendulum, Martin
(1969). The natural frequency, equation (6.1), is obtained
from the solution of the pendulum differential equation of

motion.

lg = w r UL) - E] (6.1)
211 g

T: Period of oscillation of the pendulum

r: Distance from pivot point to center of gravity of
oscillating vertebra.

The coefficients of sensitivity of the moment of inertia

IV I!

with respect to the period "T" and the radius r can be

76
written:
31
100 g 2 g T
t g 3T g T2 - 4 rflz )
100 31 1 4 «2 6 3
Sr=I—_fig_=100—r+—’—_ (')
g 4r1Tz-gT2

A .01 sec. error in measuring T, could give an error as high
as 120% for lg. A 1.0 mm error in measuring r could give an
error as high as 34% for lg. These figures are calculated
from equations (6.2) and (6.3) together with data from Table
6.3. The period of oscillation must be measured to within
.001 sec to keep the error of Ig below 12%.

The pendulum was first run at atmospheric pressure in an
environment with apparently no air circulation. The
variability among readings of T (over 2%) was considered too
high. By enclosing the pendulum in a glass chamber under
500 mm of vacuum, Figure 6.2, the variability of T was
reduced to 0.1%. Even though it can lead to errors as high
as 12% for lg, the final results can still be within what
could be expected for a biological material.

It was found that in order to take the variability of T
to within 0.1% the time period should be averaged over at
least 500 oscillations. An electric counter activated by a
photo—relay was used to keep track of the number of (mcilkmimm

The time elapsed was measured with a stop watch.

 

 

77

10 mm

Razor blade

 

 

 

me

1%!

.__ ¢.5mm

 

Wood
Screw /4

(a)

 

 

 

 

 

 

Figure 6.1. a) Pendulum to measure mass moment of huntia

b) Support of vertebra

 

Figure 6.2. Pendulum installed in vacuum chamber to
minimize error due to air friction.

78

The radius of oscillation, r, was measured to within 0.5
mm approximately. This resulted in an error for Ig no

greater than 17%.

6.3 Masses in the System
6.3.1 Vertebrae

The masses mj, listed in Table 6.3, correspond to the
vertebra itself. It does not include any of the peripheral
tissues normally attached to the spine that contribute to its
dynamic behavior, Figure 6.4. In order to be more realistic
the amount of mass to be ideally concentrated at the center
of gravity of each vertebra must be increased so that the
material most closely attached to the vertebrae and that
actually follows its motion is taken into account.

A total mass of 7712 gm, Muksian and Nash (1974) was
adopted for the spine and most closely attached ligaments
and muscle tissues. The distribution of the back mass on
the centers of gravity of the thoracic and lumbar vertebrae
was assumed to be proportional to the mass of each vertebra

as given by equation (6.4).

m.
m: = 7712 x _—J_ (6.4)
J tag

The numerical values are shown in Table 6.3. The mass
distribution just described is satisfactory for the lumbar
spine where the mass enclosed in the abdomen can be considered

to be resting directly on the bony basin presented by the

 

 

 

 

 

Table 6.3. Mass and mass moment of inertia respect to

Age = 51

79

the center of gravity of thoracic and lumbar
vertebrae.

Sex

Male

Body weight; 85 Kg<1>

Cause of death =

Body height: 1.82 m(1)

cardiac arrest

 

 

Vertebral mj n5 r T Ig(y)

level grams cm sec gm.cm2
T1 48.5 272.3 12 66 .722 175.61
T2 45.4 254.9 12 44 .724 326.34
T3 42.1 236.4 12 59 .727 284.06
T4 46.4 260.5 12.69 .730 320.60
T5 47.0 263.9 12.94 .732 223.26
T6 51.8 290.8 12.94 .734 294.87
T7 55.0 308.8 11.79 .710 472.88
T8 62.5 350.9 11 84 .720 765.44
T9 66.6 373.9 11.84 .717 731.23
T10 74.3 417.2 11.44 .704 738.25
T11 81.0 454.8 11.39 .703 815.20
T12 93.0 522.2 11.34 .702 947.79
L1 106.5 598.0 11.66 .711 1110.82
L2 125.3 703.5 12.36 .726 1130.32
L3 140.2 787.2 12.49 .731 1367.2
L4 147.7 829.3 12.59 .733 1401.14
L5 140.2 787.2 - 1367.2

mj: mass of vertebra

Hg: mass of vertebra plus more closely attached tissues

r : radius of oscillation (pendulum)

T ; period of oscillation

Ig: mass moment of inertia

(1): Estimate ; (2) : Arbitrary value.

 

(2)

 

 

 

 

—17 cm

 

Figure 6.3. Location of the point of interaction of the
head—neck lumped mass with the upper end
of the spine.

   
 
  

Muscular
tissue

:1 ‘
Rib , “

       

   

/_./"“
Thoracic ' ’;a\_;/x
vertebra «zaigyfi

\ \i:::é/,
i sagittal plane

Figure 6.4. A fraction of the back muscles and other
tissues are closely attached to the spine.

 

 

81

pelvis without any significant dynamic interaction with the
spine. The thoracic mass requires special consideration.

6.3.2 Suspendedgportion of upper torso

 

The rib cage, enclosed internal organs, shoulders and
arms have significant dynamic interaction with the spine of
a seated operator, mainly for excitation frequencies below
15 Hz.

A single mass attached to the first ten thoracic vertebrae
by means of viscoelastic elements, Figure 3.2, simulates the
action of the upper torso and limbs on the spine well enough
to give plots of seat to head transmissibility as well as
driving point impedance close to experimental measurements.

The suspended mass of the thorax can be estimated from

the weight distribution for head and upper torso shown in

Table 6.4.
Both arms and shoulders 9981.0 gm
Thoracic organs, blood and
diaphragm 4354.0 gm
Ribcage and muscles 16838.0 gm
Suspended thoracic mass 31173.0 gm

This mass should be reduced as a result of the arms not
being supported by the spine alone, and a fraction of the
mass of thoracic organs, blood, muscles, and diaphragm being
directly attached to the spine.

Part of the weight of the arms rests on the legs according

to the posture assumed by the subject in the transmissibility

 

 

82

and impedance tests reported by Pradko et a1. (1967), which
are used for validation of the model. It is also the
situation of a machine operator with the arms resting on the
steering wheel. From these consideration it was decided to
reduce the suspended thoracic mass from 31,173 gm to 20,000
gm.
6.3.3 Head and neck

Head and neck are included in the model as a compounded
mass of 6078.0 gm, Table 6.4. The mass moment of inertia

about the center of gravity was adopted from Liu et a1. 0971).

(Mass moment of inertia of head + Cl_Tl)c.g.= 20.56 x 105

gm cmz.

Similar results were reported by Vulcan and King (1971); the
data obtained from 3 cadavers are: 21.1 x 105; 22.76 x 105
and 39.02 x 105 gm cmz.

6.3.4 Sacrum—Pelvis

The magnitude of the mass attached to the lower end of
the spine will affect the values of driving point impedance
of the model, which are compared with experimental values
for validation of the model dynamic behavior.

Assuming the total weight of the abdomen plus 45% of the
pelvis-legs weight as being directly interacting with the
operator seat, the sacrum—pelvis mass can be calculated from
Table 6.4.

Mass of sacrum—pelvis = 6623.0 + .45 X 30394.0 = 20300.0 gm

. l-I" 7.7 I f..:g."".--I1'
ad agar-ms m 11 mthuon such -M

000.0: 0! r7: Eat, ff? pat-T7: .arrv JIHJBIM‘I bshnsqafli

  

1.

   

 

 

83

Table 6.4. Body weight distribution used for the model.

 

Element Weight Mass(5)
1b. Dyn gm
(x105) (x102)
Head and neck(2) 13.40 59.60 60.78
Both arms and
shoulders<3> 22.00 97.86 99.81
Back(1) 17.00 75.62 77.12
Thoracic organs blood
and diaphragm(2) 9.60 42.70 43.54
Ribcage and muscles
in thorax(“) 37.12 165.12 168.38
Abdomen(‘) 14.60 64.94 66.23
Pelvis and legs(1) 67.0 298.04 303.94
Total 180.72 804.0 820.00
(1) Muksian and Nash (1974).
(2) Payne (1970).
(3) Modified from Payne (1970) to include weight of shouhkxs.

 

 

(4) Modified from Payne (1970).

 

(5) All values taken from literature were multiplied by a

factor:

Factor = (82000/reported body weight in grams).

 

 

   

 

84

6.4 Rheological Behavior of Deformable Elements

6.4.1 Intervertebral joints. Axial.

Three stiffness and damping coefficients are needed to
characterize the rheology of the three modes of motion of
each intervertebral joint. The coefficients entering the
Kelvin elements that model the axial behavior of the disc are
calculated from equations (4.20) and (4.21) together with the
parameters shown in Table 6.5. These coefficients have been
calculated from the impedance data collected by Kazarian
(1972) using the loading frame shown in Figure 6.5.

The vertebral unit to be tested is placed between the
superior and inferior loading heads. The upper head was
designed in a manner so that a pure compression load could be
applied. The compression bias was adjusted by slowly
rotating the loading screw until the designated preload value
was registered on the strip chart recorder.

The impedance and phase angle data reported were calculat—
ed from force and velocity recordings taken from the load and
velocity transducer located underneath the lower loading
head. The data obtained with the experimental set up just
described corresponds to the axial mode of oscillation.

The exponential functions used to model stiffness and
damping frequency dependent coefficients present short
intervals within the range 5—50 Hz where the experimental
points separate from the curve. For most frequencies the

curves fit very well the experimental data as indicated by

   

{left i iv”; . 45X,, ' ”7‘;.¥. .-; -‘fi*
03-hihimhrianihfihildllflaaa guijifliiflfli'fr"
3m nun-am in “hot: new 1:. ad' 10 130109111- 9d: 4-
:".J 'arai'” -. ' '-i"-.'" ' -"" .'.'=‘ 12,-? 1:111mw_
1'.‘ I - - .. ’ 5' ems}: ,. 7'16

1_:- .1
' e't-s lush:

   
   
 

 

 

fin-zen

4:'-' u

 

 

 

 

h

 

Loading screw

 

 

1 Upper loading head

Loading head

/"

 

 

 

 

 

 

 

 

 

fijr—- Force transducir
Lower pre-load Efiy////
system '~_7 Exciter
77 z I
Figure 6.5. Loading frame for impedance testing,
Kazarian (1972).

 

 

 

the coefficients of correlation and standard error of esti—
mate given in Table 6.5.

The lumbar spinal units, which are shorter, only three
vertebrae, present the largest deviations from the prediction
curve. The reason for this behavior most likely being the
existence of errors in the data, mainly phase angle, which is
difficult to measure at resonant points where large changes
of angle take place for small changes in frequency.

6.4.2 Intervertebral joints. Bending

 

There are no data available to model the bending and shear
stiffness coefficients as functions of frequency as it was
the case for the axial mode of deformation. It is reasonable
to expect that similar frequency dependent parameters would
be required for the shear and bending modes of deformation
when modeled by Kelvin viscoelastic elements.

The bending stiffness coefficients were adopted from
Markolf and Steidel (1970) for the thoraco—lumbar spine,
T7 - L4:

Bending stiffness = (3884.11 — 23304.68)x 10s dyn.cm/rad

These values do not show any significant variation with
disc level. One might expect lumbar intervertebral joints to
be stiffer due to the larger cross—sectional area, but the
increased lumbar disc height compensates that factor making
the bending stiffness approximately constant.

These data were obtained using free vibration tests

carried on a single intervertebral joint. A resonant mass

   

  
  
 
 
  
  
 
 

:' n.‘ .-.%|- . I”
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88

was attached to the upper vertebra.vhose oscillations were
recorded. The stiffness was calculated from the measured
natural frequency of oscillation, while the damping factor
was estimated from the rate of decay of the vibration trace.

The frequency of free oscillation corresponding to the
single specimen in bending was 36.7 Hz, so it can be expected
that bending stiffness will be lower for lower frequencies
and higher for higher frequencies. No compression bias was
used for the tests carried out by Markolf and Steidel (1970),
so that an intervertebral joint under real loading conditions
would have higher stiffness than those that resulted from the
tests.

Assuming that the stiffness of the intervertebral joint
under normal loading conditions is equal to the top value in
the interval, (23304.68 X 105 dyn.cm/rad), and adopting the
exponent factors k2 corresponding to axial stiffness from
Table 6.5, the coefficients k1 for all units of the spine can
be calculated, Table 6.6. The coefficients corresponding to
the thoracic spine were increased by 150% to take into acanmt
the higher stiffness the ribcage gives to this portion of the
spine, Prasad and King (1974). Only coefficient k,, of for-
mula (4.20) was modified (9074.4 x 2.5 = 22686.0), and the
same value was used for both halves of the thoracic spine.

In all cases the frequency used for the calculations is
36.7 Hz because it is the frequency at which the reported
stiffness were measured.

Very little data are available in the literature on

 

   
 
 

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89

damping coefficients of intervertebral joints for bending
mode of oscillation. Prasad and King (1974) reported the

following bending damping coefficients:

(Bending damping)T1 _ T10 = 226.0 x 105 dyn.cm.sec/rad

(Bending damping)T11 _ S = 113.0 x 105 dyn.cm.sec/rad

These coefficients are not experimental; they were approxi-
mated in the process of optimizing the response of a lumped
parameter model to transient vertical accelerations. The
larger damping coefficients corresponding to the thoracic
spine are in agreement with the results obtained for axial
mode of oscillation from impedance data.

The frequency dependent damping coefficients for bending
are calculated from equation (4.20). The coefficients previ-
ously introduced from Prasad and King are assigned to an
intermediate frequency, 25 Hz. The parameters c1 are then
calculated for each one of the four thoraco—lumbar units
using the exponents c2 obtained from the impedance data for
axial mode of oscillation, Table 6.5. The results are shown
in Table 6.6.

6.4.3 Intervertebral john; Shear

No direct measurements of shear stiffness are reported in
the literature. Some insight into the shear behavior of the
intervertebral joints can be obtained from Orne and King Liu
(1971) through their analysis of the data reported by Evans

and Lissner (1959). The basic data consist of load defiectkm

  

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Table 6.6. Parameters for estimation of frequency
dependent stiffness coefficient k, and
damping coefficient c. Bending mode of
oscillation.

Vertebral Age k1 k2 c1 c2
level group
(x105) (x10—2) (XIOS)
01 10865.7 2.14 584.1 —0.590
Tl-T6 G2 7470.0 3.10 478.4 —0.466
63 5610.8 3.88 308.3 -0.193
G1 ‘ ' ' ‘
T7-T12 G2 9074.4 2.57 616.0 -0.623
G3 9552.9 2.43 525.2 -0.524
G, 14784.4 1.24 717.7 -0.718
L1—L3 G2 11646.7 1.89 628.0 —0.635
G3 10982.5 1.05 608.1 -0.615
G, 12171.1 1.77 688.3 -0.692
L4—S G2 14199.4 1.35 742.4 —0.739
G3 13941.3 1.40 680.6 -0.685
k = k1 ek2 fq dyn.cm/rad

C2
C = C1 (fQ)

dyn.cm.sec/rad

 

91

curves for thoracic and lumbar spine under bending in the
sagittal plane.

The effective area, Ae' and the effective area moment of
inertia, 18, are unknown, so the values associated with the
cross-section of the vertebral body were used for the calcu—
lations. The shape factor for the disc, ks, lies somewhere
between that of a solid circular section (kS = 1.25) and

that of a thin-walled circular section (kS = 2.0).

12 E I
Shear stiffness = —————-———£———- (6.5)
13' (4c - 3)

3 E I k
e s

C = 1 + (6.6)

2
G Ae 1
G = 1516.85 X 105 dyn/cmz; E = 4550.78 X 105 dyn/cm2;ks=l.5

The shear stiffness coefficients resulting from equation
(6.5) are shown in Table (6.7).

The data reported by Schultz et a1. (1973), Appendix C,
show significantly lower values. Even though these data
correspond to intervertebral discs alone, no posterior
aspects, it still gives a word of warning for the data in
Table (6.6), which will only be considered as an upper bound
for shear stiffness.

A stiffness frequency dependence of the type given by

equation (4.21) was adopted for the shear mode of oscillation.

 

 

92

The thoracic and lumbar spine were divided in four parts as
follows: Tl—T6; T7-T12; L1-L3 and L4-S. All the interverte—
bral joints in one unit were assigned the same stiffness
coefficient, so the values in Table 6.7 are averaged for each
vertebral unit. The resulting stiffness together with the
exponents, k2, corresponding to the axial mode, Table 6.5,
were used to evaluate k1 from equation (4.21).

Since the data in Table 6.6 is on the high side, it will
be associated with the highest frequency, 50 Hz, in the
interval under consideration. An example is given below for
the evaluation of the parameter k1 corresponding to the top
half of the thoracic spine for age group Gl. A similar

procedure is applied to the remaining units of the spine,see

 

Table 6.8.

(20426.91 + .... + 22672.91) X 105: 23324_5 x 105:
5

= k, e0.0214 x 50

It follows that,

k1 = 8000.5 X105

No data are available on damping for shear mode of oscil—
lation, so the coefficients calculated for axial mode are

used for shear as well.

    
 

"a“ 11:“ see

93

Table 6.7, Shear stiffness of intervertebral discs of
thoracic and lumbar spine.

 

 

 

Disc A 1 I Shear
e e (1)
level stiffness
cm2 cm cm“ dyn/cm (X10+s)

Tl 5.68 0.20 1.00 20426
T2 6.06 0.30 1.18 22179
T3 6.58 0.30 1.37 24337
T4 7.22 0.30 1.58 26089
T5 7.74 0.30 1.91 24240
T6 8.39 0.35 2.5 22672
T7 8.52 0.38 2.7 23338
T8 8.77 0.38 3.33 25227
T9 9.48 0.38 4.03 23070
T10 9.81 0.43 4.24 27914
T11 11.87 0.43 4.78 18102
T12 12 71 0.71 4.95 13188
L1 12.52 0.96 5.62 14480
L2 14.32 1.00 7.03 15876
L3 15.74 1.00 11.73 15916
L4 l7 16 1.22 9.15 14223
L5 17 55 0.91 12.73 19502

Ae : effective area

Ie : effective area moment of inertia

1 : height of intervertebral disc

(1) Data approximated following Orne (1970)

 

94

 

 

 

 

 

 

 

Table 6.8. Parameters for estimation of frequency
dependent stiffness coefficient k and
damping coefficient c. Shear mode of
oscillation.

Vertebral Age k1 k2 c1 c2
level group
(x105) (x10’2) (x105)
G, 8000 2.14 265.78 -0.590
Tl-T6 Gz 4950 3.10 320.56 -0.465
Ga 3351 3.88 155.83 -0.193
G1 — - - -
T7-T12 G2 6032 2.57 352.48 -0.622
G3 6470 2.43 322.04 —0.523
G, 8297 1.24 117.65 —O.718
L1-L3 G2 5995 1.89 137.38 -O.634
GS 5534 2.05 163.68 —0.614
G, 6959 1.77 55.90 -0.692
L4-S G2 8585 1.35 136.84 —0.739
G, 8373 1.40 82.83 -0.685
k = k; eszq dyn/cm

c = Cl (fq)C2 dyn.sec/cm

 

 

95

6.4.4 Costo—vertebral joints

 

Most of the dynamic interaction between the upper torso
and the thoracic spine takes place at the costo-vertebral
joints. Some data is available on the rheological behavior
of the transverse, inferior and superior costo vertebral
joints. Andriacchi et a1. (1974) reported experimental val-
ues of axial, shear and bending stiffness. These data are
more applicable to a static, large deformation type of
analysis.

Since this work is mainly focused on the lower part of
the spine, the interaction spine-thorax was modeled in a
simpler way following the description in Chapter III.

Muksian and Nash (1974) developed a lumped parameter
model to study the response of seated humans to sinusoidal
displacements of the seat. The spine was modeled as a rigid
body attached to the pelvis through a linear spring and a
linear dashpot. The thoracic cage was modeled as a rigid
mass attached to the thoracic spine through a non—linear
spring and a non-linear dashpot. The non-linearity is given
by a term proportional to a cubic power of the spring
elongation or it first derivative (dashpot), which are negli-

gible for small deformations.

(Stiffness thorax — spine)Z = 525,42 x 105 dyn/cm

(Damping thorax - spine)z = 38 - 54 X 105 dyn.sec/cm

   

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Since the thorax is going to interact mainly with the
first ten thoracic vertebrae, the stiffness and damping
coefficients representing the interaction at each costover-
tebral joint can be taken as 1/10 of the values adopted from

Muksian and Nash.

(Stiffness costovertebral joint)z= 52.54 X 105 dyn/cm

(Damping costovertebral joint)z = 3.8 — 5.4 X 105 dyn.sec/cm

The critical damping corresponding to the oscillating
system representing the thorax can be obtained from the for—

mulation for single degree of freedom systems:

 

Critical damping: 2/k7i= ZJEEEfZZ x 31173.0 = 25.6 X 105
dyn.sec/cm.
So the damping range previously adopted corresponds to an
overdamped system. The value giving the best model response
was 5.0 x 105 dyn.sec/cm.
6.4.5 Head and neck

The cervical spine consists of seven vertebrae separated
by intervertebral joints and surrounded by ligaments and
muscles. The neck can be then considered as a viscoelastic
element linking the head to the upper end of the thoracic
spine.

Payne and Band (1969), reported an undamped natural fre—
quency of the head and neck, fn = 192.3 rad/sec (30 Hz),

from which a stiffness coefficient for the neck waszqmroxflmned.

     
  

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Mass(head + neck) = 6078.0 gm

Stiffness = Mass x f; = 6078.0 X (192.3)2=

(neck)

= 2248.0 x 105 dyn/cm

Critical damping<head + neck) = 2 Mass fn = 23.38 X 105

dyn.sec/cm

From the data on stiffness reported by Prasad and King
(1974) for the cervical spine the following value of stiff—

ness was calculated:

Stiffness = 708.33 X105 dyn/cm

(neck)

The model presented by Muksian and Nash (1974) reached
satisfactory results using the following parameters for the

Kelvin model:

Stiffness = 525.42 X 105 dyn/cm

(neck)

Damping<neck) = 35 - 54 X 105 dyn.sec/cm

From the previous data the following ranges of variation

for damping and stiffness of the neck were adopted.

Stiffness<neck) = 500 - 700 X 105 dYn/Cm

 

 

 

99

VII. RESULTS AND DISCUSSION

7.1 Validation of the Model

The main objective of the model proposed in Chapter III
is prediction of intervertebral joint deformations. Direct
laboratory measurements of such deformations would be the
ideal way to validate the model. Since these measurements
are possible but not feasible with the present state of
transducer design, alternative indirect validation.tedrfiques
were used. Mechanical driving point impedance and seat-to-
head transmissibility are two techniques described by
Hopkins (1970), that were adopted for validation of the
model under investigation.

Mechanical impedance and transmissibility are two well
known tools for studying the dynamic response of biological
systems. A seated human subject can be considered a "black
box” much as one would an unkown electrical circuit. The
response of the system is described by the location of
resonant points as well as the magnitude of the impedance
or transmissibility vectors over the frequency range.

The mechanical impedance gives an indication of thetmxbl
behavior at one end of the torso, the pelvis; the coeffihjent
of transmissibility indicates the response at the opposite
end, the head. If both ends of the model (head and pelvis)

have a response close to the experimental values, it is

 

100

reasonable to expect that the assuptions made to model the
structure located in between those two points must be close
to the real situation.

Impedance—frequency curves can be obtained from transient
loading comhtimu; using Fourier analysis, Weis et a1. (1966),
Sandover (1970), or a more direct method, called steady
determination, that includes measurements of force and
velocity for discrete values of frequency over the range of
interest, Coermann (1963), Pradko et a1. (1967). The results
of steady state and transient impedance determinations re—
ported by Weis et al. (1966) show some unexplained discrep-
ancies, so the data reported by Pradko et a1. (1967), from
steady state tests, will be used for validation of the model.

There are three reasons for choosing Pradko's data over
other impedance curves existing in the literature:

a. The data cover the full frequency range of interest

b. The impedance curves are the mean of different

acceleration levels

c. The measurements were made using sinusoidal

excitation (steady state), which is the situation
being studied with the model.

The driving point impedance curve of the model is shown
in Figure 7.1 together with the experimental curve reported
by Pradko et a1. (1967). These data are in good agreement
with measurements made by Suggs et al. (1969) on 11 subjects
for frequencies under 10 Hz. The impedance data reported by

Coermann (1963) show a much higher maximum between 4 and 5

 

 

 

101

Hz. However the general shape of the curve is the same.

Even though the exponential functions, (4.20) and ULZl),
giving the stiffness and damping coefficients are based on
data in the range 5 to 50 Hz, the modeled impedance curve
is extrapolated to 3 Hz to show the sharp change in slope
taking place at 5 Hz. The impedance curve presents a steep
slope from 0 to 5 Hz which is a consequence of the structure
behaving as a rigid body at low frequencies of excitation.

The modeled transmissibility curve, Figure 7.2, closely
follows the 90% confidence interval reported by Pradko et
a1. (1967). The maximun reached by the model curve at 5 Hz
exceeds the corresponding value on the upper limit of the
experimental confidence interval by approximately 5.0%.
The model transmissibility curve present a minimum at 14 Hz
that deviates from the corresponding minimum on the lower
boundary of the 90% confidence interval by approximately
18.0%. These are the two points showing the largest devia—
tions from the experimental values. The reason for this
behavior is most likely the fact that the upper torso being
a deformable continuum has been modeled as a rigid mass.
This will reduce the accuracy of the model predictions of
intervertebral joint deformations at the lower end of the
frequency interval 5—50 Hz.

From the results presented it is concluded that the res-
ponse of the model is close enough to what can be expected
from a subject seated in erect position and subjected to

vertical sinusoidal oscillations. Therefore, the model will

 

 

Impedance

Transmissibility

102

   

 

 

L(dyn.sec/cm)
40 »
3o _
20 Experimental
10 . ., —: Model
0 10 20 30 40 50 (Hz)

Frequency

Figure 7.1. Driving point impedance for seatedcmeraflm:

Experimental curve from Pradko et a1 (1967).

..... : 90% confidence interval

: Model

 

 

 

10 20 30 40 50 ‘IHz)

Frequency

Figure 7.2. Seat to head transmissibility. Confidence

interval from Pradko et a1. (1967).

   

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be used to investigate levels of lumbar intervertebral joint
deformations as affected by seat suspension characteristics.

7.2 Lumbar Intervertebral Joint Deformations as

Affected by Seat Suspension Parameters

Only passive suspensions are considered in the analysis.
The passive suspension commonly consist of a spring, damper,
and mass. It is the simplest and most widely used for farm
machinery. The mass associated with the suspension can be
as low as 60.0 Kg (including the weight of the torso), for
a suspended seat, and as high as 500.0 Kg for a suspended
cab. The spring stiffness coefficient results from assuming
a natural frequency for the suspended system between 2 and
4 Hz. The trend in the design of suspensions is toward
lower natural frequencies in an attempt to reduce as much as
possible the range of frequencies producing seat motion
magnification, (seat displacement/chassis displacement >l.O).
The limiting factor in the process of reducing natural
frequency is the increasing static deflection permitted by
the ”soft" spring associated with low natural frequencies.
The damping coefficient is set close to critical conditions
to minimize oscillations for frequencies close to the natural
frequency of the suspension.

The suspension parameters adopted for the analysis are
given in Table 7.1.

An active suspension uses a power input to help minimize
the motion of the seat under adverse terrain conditions,

Roley and Burkhardt (1975).

 

 

104

Table 7.1. Seat and cab suspension parameters.

 

 

 

KC MC CC C 6

Type of
suspension (dyn/cm) (Kg) (dyn.sec/cm)

x 105 x 105
seat 210.0 10.0 22.96 2.3 0.1
seat 210.0 10.0 22.96 15.0 0.65
seat 210.0 10.0 22.96 22.96 1.0
cab 1420.0 400.0 160.0 16.0 0.1
cab 1420.0 400.0 160.0 104.0 0.65
cab 1420.0 400.0 160.0 160.0 1.0
Natural frequency = 2.9 Hz
KC : Suspension stiffness coefficient
Mc : Seat or cab mass
C : Suspension damping coefficient
CC : Critical damping

C : Damping ratio = Actual damping/critical damping

 

 

 

 

 

105

Joint deformations are investigated for the three
following conditions:

a) Subject sitting on a bare seat without suspension.
The seat undergoes sinusoidal vertical motion

b) Subject sitting on a bare seat attached to the
vibrating chassis through a spring-damper-mass
suspension.

c) Subject sitting on a bare seat rigidly attached
to a cab installed on a machine chassis through
a spring—damper—mass suspension

7.2.1 Subject sitting on bare seat. No suspension

 

The lumbar intervertebral joints of a subject sitting on
a bare rigid seat, subjected to vertical sinusoidal excita-
tion are subjected to shear and axial deformations whose
magnitudes are strongly dependent on the frequency of
excitation, Figures 7.3 and 7.4. All deformations are given
as percentage of chassis vertical amplitude of oscillation.

The maximum axial deformation takes place at the joint
enclosed by the third and fourth lumbar vertebrae, level
L3 — L4, while the maximum shear deformation takes place at
the lumbo-sacral joint, level L5—S.

The axial deformation, Figure 7.3, sharply increases
from 1.0 to 5.0% as frequencies changes from 3 to 5 Hz. No
significant changes in axial deformations occur when vmsdng
frequency in the range 5 to 10 Hz. From 10 to 30 Hz defor—
mation increases rapidly to reach a maximum of 20.0% of base

amplitude between 35 and 45 Hz. Toward the end of the

 

 

Axial Deformation

20

._I
O

 

 

(%) No suspension .
0 10 20 30 40 50 Hz
Frequency

Figure 7.3. Axial deformation of L3 - L4 lumbar

Shear Deformation

'..I
O

intervertebral joint.
(%): Percentage of base amplitude of motion

(%)

 

Figure 7.4. Shear deformation of L5 - S intervertebral

joint.

 

 

107

frequency range studied the curve shows a decreasing trend.

The L5-S shear deformation curve shows a similar pattern,
Figure 7.4, although the magnitudes are smaller. The 5—10
Hz plateau reaches a 4.0% deformation level. The maximum
of the curve is about 9.2% of base amplitude, and takes
place on the frequency range 30 to 35 Hz, which is lower
than the range at which the axial deformations reach a
maximum value.

The remaining lumbar intervertebral joints present
significantly lower levels of deformation, but the shape of
the curves is entirely similar; consequently only the
numerical results are given in Appendices K to N.

7.2.2 Subject sitting on a bare seat provided with seat

or cab suspension

 

The magnitude of joint deformations decreases signifi—
cantly when the operator seat is attached to the vibrating
chassis through a spring—damper-mass suspension. Figures
7.5 to 7.10 show deformation curves for suspended seat or
cab, which reach much lower levels than those shown in
Figures 7.3 and 7.4 for an operator sitting on a rigid tdfle.

Three levels of suspension damping are anlyzed corre—
sponding to 10, 65, and 100% of critical damping.

The magnitude of axial deformation at level L3-L4 are
shown in Figure 7.5, for the cases of seat and cab mnmensfixm
under critical damping conditions. For most frequencies in
the range 5-50 Hz the cab suspension results in lower joint

deformations than the seat suspension. At 6 Hz the axial

     
 

   
     

 
 

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108

(bformations corresponding to the cab suspension curve
exceed the deformations of the seat suspension curve by as
much as 22%, but for all frequencies over 9 Hz the cab
suspension offers better protection.

At 30 Hz the L3—L4 axial deformations corresponding to
seat suspension exceed those of cab suspension by as much
as 75%. A very similar situation takes place for shear
deformations, as shown by Figure 7.6.

By decreasing the amount of damping the joint deforma—
tions are reduced for both seat and cab suspension as shown
in Figures 7.7 and 7.8 which correspond to a damping coef—
ficient equal to 65% of critical. The trend is larger
deformation reductions at higher frequencies. For fre-
quencies near the natural frequency of the suspension there
is an increase of joint deformation, which can be clearly
seen when the damping coefficient is further reduced.

By reducing the damping coefficient to only 10% of
critical the deformations continue to decrease for fre-
quencies over 10 Hz, but a resonant condition becomes evi-
dent at 3 Hz which is close to the natural frequency of the
suspension system, Figures 7.9 and 7 10.

From the previous analysis it can be stated that a
damper furnished with a variable damping coefficient can
contribute to significant reductions of joint deformations.
It should provide, for example, critical damping for fre—
quencies close to the suspension natural frequency, but

otherwise very light damping.

   

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Axial Deformation

Shear Deformation

109

(%) ————— : Seat suspension

Cab suspension

./"“‘x c = 1.0

 

 

 

0 10 20 30 40 50 Hz
Frequency
Figure 7.5. Axial deformations of L3 - L4 lumbar

intervertebral joint.
(%): Percentage of base amplitude of motion

 

“(7:9 — - - - -: Seat suspension
3 ’.I" m"\_ : Cab suspension
P -/. \’\'
/_/ \.\ C = 1.0
__ ..... '\

‘\

   

“\

l 1 ~

 

A
r

0 10 20 3O 40 50 Hz
Figure 7.6. Shear deformations of L5 — S intervertebral
joint.

 

Axial Deformation

Shear Deformation

110

 

 

 

0(1) ----- : Seat suspension
6 ‘ ——————: Cab suspension
5 . _ _ t = 0.65
4 _ .
3 .
2 .
l . , . , . 14% L 1 1 4; a:

0 10 20 30 4O 50 Hz

Frequency

Figure 7.7. Axial deformation of L3 — L4 lumbar
intervertebral joint.
(%): Percentage of base amplitude of motion

 

 

 

(%) ----- : Seat suspension
1‘ Cab suspension
C = 0.65

3..

2..

1..

O 1 4r 1 A A L l g 4 ; ;
0 10 2O 30 40 50 Hz

Figure 7.8. Shear deformation of L5 — S intenmntebral
joint.

 

Axial Deformation

Shear Deformation

 

 

 

 

(%) ----- :Seat suspension
51E :Cabsnwpensflx1
4. c=0.1
3.

2.

1.

o........‘.‘7_
0 10 20 30 40 50 firhz

Frequency

Figure 7.9. Axial deformation of L3 - L4 lumbar
intervertebral joint.
(%): Percentage of base amplitude of motion

 

 

 

 

 

1(%) --—~-:Seat suspension
t :Cabsmspeuflon
= .l

3 . C 0

2 I

l r-

0 . . . , . . . - T:::; ._
0 10 20 30 40 50 "Hz

Figure 7.10. Shear deformation of L5 — S intervertebral
joint.

 

 

 

112

Since the motion of the seat at frequencies close to
its natural frequency is characterized by large amplitudes,
the damper can be designed so as to give a displacement
dependent damping coefficient capable of heavily damping the
system when the seat displacement exceeds certain levels.
But, it would provide negligible amounts of damping for
low amplitude high frequency oscillations; this means
minimum joint deformation.

7.3 Summary of Results

The main findings in this study are the following:

1. A lumped parameter model of the spine in the sagittal
plane as the one shown in Figure 3.2 can closely
predict the driving point impedance of an operator
sitting in erect position while subjected to sinusoidal
vertical oscillations.

2. The coefficient of transmissibility predicted by the
model deviates as much as 18% from an experimentally
determined 90% confidence interval reported in the
literature. These deviations take place in the range
5 to 25 Hz. For higher frequencies the model predictions
fall within the confidence interval.

3. Maximum axial intervertebral joint deformations take
place at the joint located between the third and fourth
lumbar vertebrae. The maximum shear deformation takes
place at the lumbo—sacral intervertebral joint. These
statements are valid over all the frequency range 5—50

Hz.

 

 

113

Frequencies over 15 Hz will sharply increase axial and
shear joint deformations for a subject sitting on a
bare vibrating seat. Axial deformation of joint L3 —
L4 will almost triple when frequency is increased from
10 to 35 Hz. The shear deformation of joint L5 — S
more than doubles for the same frequency increase.

The use of a spring-damper—mass suspension located
between seat and chassis or between cab and chassis
results in sharp reductions of joint deformations.

The magnitude of the reduction depends on the type of
suspension, the amount of damping, and the frequency of
excitation.

Cab suspension can reduce joint deformation to almost
half the levels corresponding to a seat suspension for
frequencies over 10 Hz. Seat suspension can give joint
deformations as much as 25% lower than cab suspension
for frequencies between 5 and 10 Hz. Both types of
suspensions were given identical damping ratios and
natural frequency (2.9 Hz).

Low damping ratios (t= 0.1) give the lowest joint
deformations for most of the frequency range, but with
very high values for frequencies near the natural

frequency of the suspension system.

 

114

VIII. CONCLUSIONS AND RECOMMENDATIONS

8.1 Conclusions
The conclusions derived from this study are as follows:
The lumped parameter model developed in this investiga-
tion has shown promising results in predicting inter-
vertebral joint deformations. It puts a word of
warning on the well established criterion for design of
seat suspension based mostly on comfort considerations.
The simplified substructure used to model the upper
torso (single rigid mass) seems to be responsible for
some discrepancies between the response of the model
and the experimental data in the lower end of the fre—
quency range 5-50 Hz.
When modeling the viscoelastic rheological behavior of
intervertebral joints by means of Kelvin elements, the
corresponding stiffness and damping coefficients vary
exponentially with frequency.
The deformations of intervertebral joints are maximum
for frequencies in the range 25 to 35 Hz. Since the
rated speed of most engines used in modern farm equipment
is between 1800 rpm (30 Hz) and 2600 rpm (40 Hz), the
operator is exposed to vibrations in the most unfavorable
range of frequencies from the stand point of joint de-

formations.

114

 

115

Ride comfort has always been the criterion for the
design of farm machinery seat suspension. This approach
has led to the use of high values of damping in the
process of minimizing the amplitude of motion at
frequencies near the natural frequency of the seat. The
result is a sharp increase of joint deformations for
frequencies over 10 Hz that do not create immediate
discomfort sensations but could be the reason for low
back pain after years of exposure.

The joint deformations predicted by the model appear to
be very small, but there are no data on what levels can
be considered damaging under long time exposure condi-
tions. The alternative left is to minimize deformations
in order to offer maximum protection.

The joint deformations reach at most a 20% of the
amplitude of chassis oscillation which is already a
small quantity for the case of vibrations generated as

a result of minor unbalanced machine components having
rotary or reciprocating motion.

The use of a spring-damper—mass suspension located
between a seated operator and the vibrating chassis
results in joint deformations about l/4 to 1/5 of the
values corresponding to a subject sitting on a seat
rigidly attached to the chassis.

The use of cab suspension is desirable over seat mnmen—
sion for the minimization of intervertebral joint defor-

mations for frequencies over 10 Hz. Below 10 Hz dxzseat

 
    
 

m- .. 4...“... .4. WI...

""T '553 -jJ 1= ins"; I . IanWsn aflJ wean Is&'—

i’--_ 1'

1‘ . _.. -.-_-' ."2 r'."..-lii"- ‘.. !

 

116

suspension offers some advantage.
The use of a suspension damper capable of giving criti—
cal damping for excitation frequencies close to the
natural frequency of the seat and very light damping
for higher frequencies is desirable from the stand
point of minimization of joint deformations.

8.2 Recommendations

Some of the changes that could be incorporated to the

model to increase its range of applications and probably

improve the occuracy of the results for the lower end of the

frequency range 5—50 Hz are listed below:

1.

The assumption made about small joint deformations must
be relaxed if predictions of joint deformations are to
be made in the range of low frequencies close to the
natural frequency of the seat. It requires additional
investigation of the kinematic and rheological behavior
of the joints.

By testing two consecutive vertebrae with the corre—
sponding intervertebral joint, the patterns of relative
motions could be studied.

After motion and load histories have been recorded, the
joint could be opened and all relevant dimensions taken
for proper modeling of the kinematic behavior of the
joint.

The seat to head transmissibility as well as the driving
point impedance curves corresponding to a seated subject

are quite sensitive to changes in bending stiffness and

 

 

117

damping coefficients. Therefore, more accurate data on
bending rheological behavior of intervertebral joints

is needed. Bending impedance tests of preloaded units
similar to those carried out by Kazarian (1972), for
axial motion, would be one approach to this problem.

If more accurate joint deformations are to be predicted
for the thoracic spine, the ribcage requires a more
elaborate model than a single mass suspended from the
first 10 thoracic vertebrae. Ribs modeled as individual
masses separated by viscoelastic elements representing
the intercostal tissues, plus beam type elements
representing the costo-vertebral and the costo-sternal
joints would be an appropriate solution. The internal
organs of the upper thorax could be modeled as rigid
masses suspended from the ribcage by viscoelastic
elements.

The joint deformations as presented in this report
correspond to a point located in the center of the inter-
vertebral disc at the intersection of the axis of the
two vertebral bodies enclosing the disc. More severe
deformations most likely occur at the articular facets
on the posterior arch or at the opposite end of the
joint on the annulus fibrosus.

Some additional geometrical data plus some formulation
could be added to the existing computer program to cahnk

late those deformations.

 

 

 

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‘. '. " . " ' .161fi-‘ifiqa” "91m:- --.‘1 '24. :- infirm U“-
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APPENDICES

123

 

 

124

APPENDIX A

  
   
    

Articular .Sagittal plane
facets
Superior and plate
Posterior
arch
Vertebral body
Intervertebral disc
Articular

facets I

Figure A.1. Main structural components of vertebral

column

 

 

125

APHQEEX B
Governing Equation Describing the Motion of a Vertebra in x—direction

 

Figure B.1. Displacements affecting the equilibrium of

vertebral mass mi in x-direction.

From Newton's 2nd. law: m ii = (Forces acting on mi in
direction)

.l = _ 2 _ _ _ . - 2—
m xi (xi+l xi) cos Oi+1 Ks(i+1) 4 (xi+1 x1) Sin %&1

— . 2 _ .—
Ka(i+1) + (Xi-1 x1) COS 91 Ks(i) + (Xi-1 xi)

 

126

. 2 " _ ‘— . 5
Sin Oi Ka(i) + (zi+l 21) cos Oi+l Sin ©i+1

Ka(i+1) ‘ (21+1 ' 21) C03 E’1+1 Sin O1+1 Ks(i+1) +

+ (zi—l — 21) cos Oi Sin Oi Ka(i) — (Zi—l — zi)

Z1

C°S Oi 51“ 91 Ks(i) + 51+1 1+1 Ks(i+1) °°S e1+1 '

- 61—1 ZZi—l Ks(i) cos Oi + f(t)

f(t) = O for all d.f. except z—motion of sacrum—pelvis mass.

 

 

127

APPENDIX C

Table C.1. Distribution of degrees of freedom
corresponding to each rigid moving
component of the model.

Element Motion Degree of freedom
number

Head—neck x 1
Head neck 2 2
Head-neck 6 3
Th1 x 4
Th1 z 5
Th1 8 6
Th2 x 7
Th12 S 39
L1 x 40
L1 2 41
L1 8 42
L5 x 52
L5 2 53
L5 8 54
Sacrum—Pelvis x 55
Sacrum—Pelvis z 56
Thorax x 57
Thorax z 58

 

128

APPENDIX D

Stiffness Matrix Corresponding to an Intervertebral Joint

 

Figure D.1. Forces acting on an intervertebral joint

The joint stiffness matrix can be obtained by applying
the definition given by Vernon (1967): " kij is the load
required in the direction of coordinate i when a unit
displacement occurs in the direction of coordinate j and all
other displacements are zero". So a unit displacement will
be given to one coordinate at a time of the system in Figure
D.1, and forces in all six directions calculated from the
equations of static equilibrium.

The equation of static equilibrium of an intervertebral

 

 

 

 

129

joint are as given by equations D-l to D—3.

ZFu = fi + f5 = 0 ; fé = — f1 (D 1)

ZFW = f? + £3 = 0 ; f3 = — f? (D.2)

2M0 = M1 + M2 - f; Z2 COS 012 - f5 21 C03 012 +
+ f? 22 sin 012 — f3 21 sin Q2 = 0 (D.3)

Calculation of stiffness coefficients kjlt

A unit displacement of the superior vertebra in u-direction,
while the inferior vertebra is maintained fixed, develops a

reaction at the intervertebral joint as shown in Figure D.2

(a).

= ' — = ' = =
ZFu f1 ks 0 f1 k11 KS
ZFW = 0 no forces in w — directions f? = k2} = 0
2M0 = M1 — f; X ZZ COS 912 = O M) = K31 = K822 COS 012

 

From equations (D.1) to (D.3):

 

130

f; = " f; = "‘ K kgl = "‘ KS
£3 = - fll’ = 0 k5} = 0
M2 = - KS Zl cos ksl = - KS Z1 cos 012

 

Calculation of coefficients ka:
A unit displacement of the superior vertebra in w—direction,
while the inferior vertebra is maintained fixed, develops a

reaction at the intervertebral joint as shown in Figure D.2

(b).

W1 = 1 k12 = fi k22 = f? kaz = M1
kuz = f5 ksz = f? ksz = M2
ZFu = 0 no forces in u—direction fi = klz = 0
= " _ = I' = =
EFW f1 Ka 0 f1 kzz Ka
2M0 = M1 + f'l' 22 Sin 012 = O 1‘11 = kaz =-KaZZ Sin 612

 

From equations (D.1) to (D.3):

f5=~fi=0 k42=0

 

131

f3 = - f1 = *Ka ksz = — Ka

ll

M2 -Ka Z1 8111012 keg = "Ka Z]. 8111012

 

Calculation of coefficients kj3:

A unit rotation of the superior vertebra, while the inferior
is maintained fixed, develops the reactions shown in Figure

D.2(c) at the intervertebral joint.

51 = l kla = fi k23 = f? kaa = M5

kua = fi ksa = f? kes = M2
ZFue fi — K.S Z2 cosOlz = 0 f; = k13 = KS ZZ cos 012
ZFW= f‘1'+ K8. 22 81.11012 = 0 f1, = k23 = - Ka Z2 sin 012

ZMO=M1 - Kb + 22(f'1'sin 012 - f1 C030”), M1 = k33 = K.b + 222( Ks

coszelz + Ka singelz)

 

From equations (D.1) to (D.3):

f5 = — f{ f5 = kua = - KS Z2 cos 012

 

£12! = _ f'l' f'Z' = k53 = Ka ZZ Sin 012

 

M2 = k63 = - Kb + 21 22 (Ka sinze12 - KS cos2 0,2)

 

 

 

132

Calculation of coefficients kj4:
Similar procedure is followed when giving unit displacements

to the inferior vertebra. Only the equations are shown below.

1.12 = 1 (Figure D.2 (d))

klu = fi kzu = f? kau = M1
kuu = fi ksu = fi ks» = M2
ZF = 0 f5‘ kqn — KS
ZFW = 0 f2= k5“ = O
2M0 = 0 M2: ksm = KSZ1 COS O12

 

From equations (D.1) to (D.3):

f; = — f5 f1 = k1“ = - Ks
fT = * f? f? = kzu = 0
M1 = - M2+ (fi Z2 + f5 21) cos 012 — (f? 22 — f3 Z1) sin 012

M1 = k3m= - KS 22 cos 012

 

 

 

133

Calculation of coefficients kj5;

W2 = 1 (Figure D.2 (e))

kls = f}
kus = fi
EFu = 0 f5
ZFW = 0 f3
2M0 = 0 M2

From equations (D.

f; = - f5 ff
fy = _ f; f?
M1 =

kzs = f" kss = Ml
kss = f? kes = M2
= kus = 0
= k55 = Ka

= f3 21 sin 912

1) to (D.3):
= k15 = O
= k25 - - Ka

 

~M2 + (f1 22 + f5 21) COS 012— (f? 22 - f3 Z1) sin 912

M1 = k35 = Ka 22 Sin 012

 

Calculation of coefficients kj6:

 

 

 

134

62 = 1 (Figure D.2 (f))

 

 

k16 = fi k26 = f? kae = M1
kks = f; kse = f2 keg = M2
ZFu = 0 f5 = kus = KS 21 cos 012
ZFW = 0 f3 = kss = Ka Zl sin 012
2M0 = 0 M2 = kss = Kb+ 212(KS cos2 012 + Ka sin2 012)

 

From equations (D.1) to (D.3):

 

 

f1 = — fé f1 = kls = - KS 21 COS 012
f? = - f3 f? = kze = - Ka Zl sin 912
M1 = k36 = — Kb + 21 22 (Ka sinze12 — KS cos2 012)

 

 

thue D.2.

 

135

 

Forces developed at the intervertebral joint as
a result of unit displacements of the adjacent
vertebra.

 

 

 

136

Appendix E. Stiffness data reported by Schultz et a1.

 

 

(1973)
Stiffness x (10'5)
Vertebral Axial Shear Bending
level dyn/cm dyn/cm dyn.cm/rad.
T1 6863.1 5882.6 1960.9
T2 11765.3 10784.8 3921.7
T3 14706.6 13726.1 5882.6
T4 20589.2 18628.4 9804.4
T5 18628.3 16667.5 9804.4
T6 17647.9 15687.0 9804.4
T7 14706.6 13726.1 9804.4
T8 14706.6 12745.7 10784.8
T9 14706.6 13726.1 10784.8
T10 14706.6 13726.1 11765.3
T11 14706.6 10784.8 9804.4
T12 17647.9 9804.4 8823.9
L1 15687.0 8823.9 8823.9
L2 14706.6 7843.5 8823.9
L3 14706.6 7843.5 8823.9
L4 13726.16 6863.1 7843.5
L5 10784.8 5882.6 6863.1

 

 

 

137

AHENDDCF

 

 
 
 

   

Pelvis

.— 0— 9—0—0

vertical harmonic excitation

Figure F.1. Vertical excitation of the spine through
the pelvis.

 

138

Ammav a ”Aan\omm.nav N MANmV an

 

 

 

 

 

0.N0 H0.0H qm.H0 0.N0 00.0H «0.00 0.Nw 00.0 00.00
0.Nw 00.0H 00.50 0.Nw 00.0H 00.00 0.Nw H0.m 0H.¢¢ 0.Hw 0H.0N mH.H0
0.Nw n0.0 00.00 0.Nw HH.0H 00.00 0.Nw Hm.w «0.0m 0.H0 00.0N 00.00
0.Nw 00.0H ma.¢¢ 0.Nw 00.na n¢.wm 0.Nw 00.0H 00.00 0.Hw e0.HN AN.H0
0.Nw m0um 00.00 0.Nw 0N.0N 00.00 0.Nw 00.HH qw.mm 0.H0 0N.NN Nn.nm
0.Nw 00.0 mnnmm 0.Nw 0n.NN 00.00 0.Nm 0H.NH 00.00 0.Hw HN.¢N 00.00
0.Nw n0 0 H0 00 0.Nw Hm.¢N 00.nN 0.N0 00.0H 0w.mm 0.Hw Nm.0N nw.0m
0.N0 «0.0 00.00 0.Nw mN.n~ H0.mN 0.Nw 00.0H mm.qN 0.Hm 00.0N 00.0N
0.N0 RH.0H n0.0m 0.Nm N0.Hm 00.0w 0.Nw «H.0N 00.HN 0.Hw 00.00 m0.mm
0.Nw 0N.NH 00.0N 0.N0 00.0N 00.0H 0.Nw an.qm 0H.0N 0.H0 N0.Hm 00.0N
0HNO owHHH anNN 0.N0 H0.mm no.0H 0.N0 Hm.qm 0H.0H 0.Hw 50.00 H0.0H
0 N0 00 NH <0 cm 0.Nw 00.0w Nn.ma 0.Nw H0.0N 0H.<H 0.Hw 0n.0m m~.<a
0.Nm 00.NH HN.0H 0.N0 00.0w NH.NH 0.Nw 0N.mm HN.NH 0.Hw 0N.0m 00.0H
0.Nw 00.NH 00.5H 0.Nm Nn.nm 00.HH 0.Nw NH.00 00.0H 0.Hw 0m.wq m0.HH
0.N0 00.HH 0H.0H 0.Nw 50.00 0H.0H 0.Nw 00.00 00.0H 0.Hw 0H.q¢ 00.HH
0.N0 H0.NH 00.0H 0.Nw NH.00 00.0 0.N0 00.50 «0.0 0.Hw 0N.mq no.0H
0.Nm 00.0H no.0H 0.Nw mm.H0 00.0 0.Nw 00.00 «0.0 0.Hw 00.00 00.0
0.Nm 00.0H Na.NH 0.Nw mn.H0 00.0 0.Nm n0.Hm 00.0 0.Hm 00.n0 00.0
0.Nw 05.0N «H.0H 0.Nm 00.N0 00.0 0.Nw 00.0m 00.0 0.Hw N0.00 no.0
0.Nw 00.0w 0H.0 0.Nm H¢.¢0 00.5 0.N0 00.05 q0.~ 0.Hw 00.00 00.5
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0.Nw «0.0m 0n.n 0.N0 0H.m0 00.0 0.Nw “0.50 00.0 0.Hw mn.m0 00.0
0.Nw H0.0m N0.0 0.N0 00.00 50.0 0.N0 00.00 H0.0 0.Hw 0H.w0 00.0
0.N0 a¢.¢¢ H0.0 0.Nw N0.Hn 00.0 0.N0 00.H0a 00.0 0.Hw m0.Hm 00.0
0.N0 Hq.N0 0H.0 0.Nw 00.0n 00.0 0.Nw m0.00H 00.0 0.H0 00.0w 00.0

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A0000 000cm ommfia “a “Aaw\oom.nav mocmwmQEH “N ”Ammv kocooumuw new
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0.00 00.0H 00.00 0.00 00.00 0H.00 0.00 H0.00 00.00 0.00 00.00 00.00
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0.00 H0 H0 00.00 0.00 0H.00 q0.a0 0.00 00.00 00.0H 0.H0 00 00 00 00
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141

APPENDIX H

Table H.1. Transmissibility data, Pradko (1967)

 

 

Standard Confidence Interval (90%)
Frequency Mean Deviation Upper Bound Lower Bound
1 1.011 .032 1.032 .989
3 1.182 .105 1.253 1.111
4 1.389 .157 1.495 1.282
5 1.298 .302 1.401 1.195
7 .901 .282 1.092 .710
10 .76 .20 .836 .684
15 .74 .23 .828 .652
20 .76 .22 .843 .677
30 .63 .18 .698 .562
40 .49 .14 .570 .410
50 .35 .12 .423 .277
60 .25 .12 .302 .198

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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