A MATHEMATICAL ANALYSIS OF THE MUSCULO-SKELETAL …

A MATHEMATICAL ANALYSIS OF THE MUSCULO-SKELETAL SYSTEM OF THE HUMAN SHOULDER JOINT

by

Young-Pil Park, B. of Engr., M. S. in M. E.

A DISSERTATION

IN

MECHANICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Accepted

May, 1977

ACKNOWLEDGEMENTS

The author wishes to express grateful acknowledgement for the

devoted help of the committee members, Dr. Clarence A. Bell, Dr.

Donald J. Helmers of the Mechanical Engineering Department, Dr.

Mohamed M. Ayoub of the Industrial Engineering Department and Dr.

William G. Seliger of the School of Medicine. Their guidance, sug-

gestions and consultations proved indispensable sources of inspira-

tion. Their limitless help and keen criticism helped in overcoming

the numerous difficulties that the author faced during this study.

Thanks are extended to Dr. James H. Strickland for his helpful

advice and constructive criticism in the final examination and to

Dr. James H. Lawrence, Jr., Chairman of the Mechanical Engineering

Department, for his encouragement and interest in this study. And

thanks to Mrs. Sue Haynes for the typing of the manuscript.

n

ABSTRACT

The purpose of this study has been to formulate a mathematical

model capable of predicting muscular tension characteristics for mus-

cles in the human shoulder joint. This was done by using the data

that were collected through dissection of a cadaver and through phy-

siological information about human skeletal muscles and anatomical

characteristics of the human shoulder joint. By using this method,

the explicit characterization of the shoulder joint was described in

terms of a three dimensional coordinate system. The mathematical

equations for the relationships between the electrical signal inten-

sities that are generated from the muscles, and muscular tensions

that are exerted by muscles at various postures during abduction of

the upper extremity were investigated. General equations that can

be applied to various individual persons who have different anthro-

pometric dimensions were developed by using scale factors.

Computer programs were developed to determine the muscular ten-

sion in muscles in the shoulder joint of various persons and to pre-

dict the linear coefficients between electromyographic electrical

signal intensities and the muscular tensions of the skeletal muscles.

According to the results and the techniques of this study, it

was determined that most of the complicated human musculo-skeletal

systems can be analyzed mathematically without invasion of the

living body.

m

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS i i

ABSTRACT iii

LIST OF TABLES vi

LIST OF FIGURES ix

I. INTRODUCTION 1

1.1. Introduction 1

1.2. Review of Previous Work 5

Biomechanical Aspects of Muscle 5

Electromyogram 10

Mathematical Analysis of Human Motion 12

Mechanism of Shoulder Joint 15

1.3. Purpose of Scope 17

II. CONCEPTUAL MODEL 20

2.1. Anatomical and Functional Aspects of the

Shoulder Joint 20

Skeletal System and Joints 20

Muscular System 24

Functional Aspect 26

Assumptions 34

2.2. Conceptual Model Postulated 36

Formulated Consideration 42

III. MATHEMATICAL ANALYSIS 46

3.1. Anatomical Consideration 46 3.2. Equilibrium 48

IV

Page

3.3. Minimal Principle 53

3.4. Solution Technique 58

IV. EXPERIMENTAL PROCEDURE 61

4.1. Anthropometric Data Characteristics 62

4.2. E.M.G. Experiment 70

V. RESULTS OF THE THEORETICAL ANALYSIS 75

5.1. Functional Equations for the Muscular Tension 76

General Equation Form (Standard Coefficients) 84

A. Abduction Case 84

B. Adduction Case 85

5.2. Simplified Functional Equations for the

Muscular Tension 86

VI. EXPERIMENTAL VERIFICATION OF THE THEORETICAL ANALYSIS ... 92

VII. SUMMARY, CONCLUSION AND RECOMMENDATION 102

7.1. Summary 102

7.2. Conclusion 103

7.3. Recommendation 104

LIST OF REFERENCES 105

APPENDIX 111

Appendix (I). Anatomical Basic Data Table for Muscles .. 112

Appendix (II). Theoretical and Experimental Results-Three Parts of Deltoid Muscle 127

Appendix (III). Coefficients of Theoretical Solution of Muscular Tension 133

Appendix (IV). Documentation of Computer Program 154

LIST OF TABLES

Table Page

4.1. Anthropometric Characteristics of the Subject 64

4.2. Anthropometirc Basic Data of the Subject 69

5.1. Generalized Equation Result 91

6.1. Linear Coefficient Values 100

6.2. Statistical Results of Curve Fitting 101

1.1. Anatomical Basic Data Table for Muscles -Deltoid Anterior 112

1.2. Anatomical Basic Data Table for Muscles -Deltoid Middle 113

1.3. Anatomical Basic Data Table for Muscles -Deltoid Posterior 114

1.4. Anatomical Basic Data Table for Muscles -Supraspinatus 115

1.5. Anatomical Basic Data Table for Muscles -Infraspinatus 116

1.6. Anatomical Basic Data Table for Muscles -Teres Major 117

1.7. Anatomical Basic Data Table for Muscles -Teres Mi nor 118

1.8. Anatomical Basic Data Table for Muscles -Subscapularis 119

1.9. Anatomical Basic Data Table for Muscles -Pectoralis Major (S) 120

1.10. Anatomical Basic Data Table for Muscles -Pectoralis Major (C) 121

1.11. Anatomical Basic Data Table for Muscles -Biceps (Long) 122

1.12. Anatomical Basic Data Table for Muscles -Biceps (Short) 123

VI

vn 1 Page

1.13. Anatomical Basic Data Table for Muscles -Triceps 124

1.14. Anatomical Basic Data Table for Muscles -Coracobiâchialis 125

1.15. Anatomical Basic Data Table for Muscles -Latissimus Dorsi 126

2.1. Theoretical and Experimental Results -Three Parts of Deltoid Muscle - Subject (1) 127






3.1. Coefficients of Theoretical Solution of Muscular Tension - Subject: Model, Weight: 0 Ibs 133



3.4. Coefficients of Theoretical Solution of Muscular Tension - Subject: (1), Weight: 0 Ibs 136





v m

Page


3.10. Coefficients of Theoretical Solution of Muscular Tension - Subject: (3), Weight: Q Ibs 142












LIST OF FIGURES

Figure Rage

1.1. Process of Muscle Shortening 6

1.2. Ideal Muscular Filament Contraction 8

1.3. Mechanical Model of Muscle Performance 9

1.4. Chaffin's Gross Body Action Model 14

2.1. Bones and Joints of the Shoulder 22

2.2. The Head of Humerus 23

2.3. Axes of Shoul der Movement 28

2.4. Length-Tension Diagram of Skeletal Muscle 29

2.5. Schematic Diagram of Shoulder Muscles -Deltoid, Teres Minor, Teres Major, Supraspinatus 31

2.6. Schematic Diagram of Shoulder Muscles -Infraspinatus, Subscapularia, Latissimus Dorsi, Biceps 32

2.7. Schematic Diagram of Shoulder Muscles -

Pectoralis Major, Coracobrachialis, Triceps 33

2.8. Direction of Reaction Forces 35

2.9. Conceptual Model 38

2.10. Location of Shoulder Muscles (Anterior View) 39

2.11. Location of Shoulder Muscles (Lateral View) 40

2.12. Location of Shoulder Muscles (Posterior View) 41

2.13. Changes of Insertion Point 42

3.1. Change of Posture 46

3.2. Schematic Diagram of Sensitivity Test 47

3.3. Analyzing System 49

4.1. Anthropometric Data 63

4.2. Upper Extremity Model 65

IX

Figure

4.3. Experimental Procedure 70

4.4. Summation Area of E.M.G. Signal 72

5.1. Muscular Tension Diagram - Subject (1) 77






6.1. Muscular Tension vs. E.M.G. Intensity - Subject (1) 94






CHAPTER I

INTRODUCTION

1.1. Introduction

In recent years, the interest in the engineering approach to en-

hance the effectiveness of human activities such as exercise, to develop

clinical techniques, and to improve safety in industry, has been consi-

dered very important. Several methods of approach have been developed

to meet these problems more practically by the engineers who study biome-

chanics. Even though the complexity of the human body with its nerves,

muscles and bones, which exist and coordinate to produce complicated

human activities, has been the subject of study by many researchers

ever since antiquity, no one could ever create any device which is

able to match the superiority and versatility of human activity to per-

form innumerable and profound activities. With the development of more

sophisticated means of studying human activities modern man has been

able to apply scientific analysis methods to this study for the physi-

cal well-being of human beings.

Recently, many researchers have been applying human activity analy-

sis techniques to many fields such as (1) industry, with the emphasis on

the effectiveness of work and on safety problems, (2) medicine and medi-

cal rehabilitation including the design of prosthetic devices, (3)

sports, particularly in the analysis of techniques, and (4) space re-

search.

Previous investigations, directed toward mathematical and descrip-

tive analysis of the developed tensile forces and electromyographic

electrical intensities of human muscles, can be divided into two classes

1

according to their methods of approach. One group aimed at isolating

the muscles as much as possible, and was directed toward a physiologi-

cal approach that would fit the mechanical and electrical phenomena of

muscle fibers, and chemical components of muscle cells. This group of

research was started with an elementary consideration of the mechanical,

chemical and electrical theories of the muscular system and the nerve

system separately, and progressed towards an increasingly more complex

consideration of the integrated neuromuscular system. Most of the

work in this group involved at least some theoretical analyses of the

mechanical characteristics of muscle fibers and the cell components,

and their relation to the nerve signals. The best known physiological

descriptions resulting from this method of approach are the length-

tension relationship of human skeletal muscle, and the membrane poten-

tial theory of living cells.

The investigations of the next group were functional rather than

physiological, and sought to explain the external performance of human

beings, such as motion and effectiveness in controlling complicated

machines, and heavy and skilled work. Here, the position of the sub-

ject and the magnitude of the subject's weight were usually considered

as the system inputs, and the electrical phenomena of muscles, kinetic

responses and fatigue characteristics were generally considered as the

system outputs. The objective was to model the total task performance,

and the behavior of muscle entered only indirectly as a modifying func-

tion which indicated, in some cases, the quality of response. The ex-

periments, however, usually represented only the qualitative data invol-

ving actions by muscle groups.

Recently, in addition to the qualitative characteristics of a

given task, the quantitative responses of muscle effort have been the

subjects of research using electromoyographic and mathematical tech-

niques. The quantitative methods of approach to muscular and electri-

cal responses of a given task are extremely valuable because they can

indicate the importance of the individual muscles and the magnitude of

the applied forces to muscle groups in an intact, nonnally operating

biological system. Consequently, dissection is not needed. However,

because of structural differences and differences in shape, these stu-

dies do not allow specific and functional elements to be unequivocally

localized or associated with specific anatomical structures.

The second group is more suitable for the investigation of human

performance research because the resulting analysis can be associated

with the actual behavior of the muscle instead of the properties of

muscle. Therefore, this method of approach was adapted for this study.

In particular, the possibility that a mathematical description of

human skeletal muscles which cross the gleno-humeral joint can be devel-

oped by vector analysis and electromyographic studies, was also inves-

tigated in this study. The idea of this investigation was that both

electromyographic and vector analyses of human skeletal muscle can be

combined to solve the complicated problems that are faced so many times

in human motion research.

The only reliable results which were obtained with the same moti-

vation as this study, have been for clinical and medical purposes. Con-

sequently, they considered only the qualitative analysis rather than

the quantitative analysis. However, quantitative analysis is necessary

for the application of biomechanics to the study of the human body for

purposes such as physical training, artificial limb design and safety

problems. Most of the former experiments were to verify the existence

of the electrical signals in a certain motion, or just to compare the

changes in the magnitude of the electrical signals which arise from

the changes of the effort of muscles.

This study involved the formulation of a musculo-skeletal

model of the human shoulder joint that can be described by mathematical

vector methods, using the data collected through dissection of cadavers

and physiological informations of human skeletal muscle. Using this

model, the explicit characterization of the mathematical equation for

the postulated mechanism of the shoulder joint was formulated in order

to describe shoulder motion in terms of a three dimensional system,

without using any other methods. The theoretical procedure of this

study was based upon the mathematical analysis of the shoulder muscles

and the analysis of anatomical and physiological characteristics of the

muscles. The experimental procedure consisted of the recording of the

electromyographic signals of the shoulder muscles via surface electrodes

during application of external forces. The work of this study also in-

cluded the analysis of the recorded results in order to formulate the

relationship between the electrical signals and muscular tensions that

are generated from the muscles at a certain motion or posture. The ex-

ternal force was applied to the arm by means of weights that varied not

only in magnitude but also in the position of the weights.

1.2. Review of Previous VJork

The scientific research in the field of biomechanics, which is

the study of the structure and function of biological systems by means

of methods of mechanics, began with the growth of science.

Biomechanical Aspect of Muscles

The muscle itself has been an object of intense scientific inter-

est for some time. There is a general agreement that it is a very com-

plex biological system. Its chemical, electrical and mechanical pro-

perties are still vague, and contraction, which is the essential physio-

logical function of the muscle, remains as a perplexing phenomena.

In recent years, the research in contraction of muscle has been

almost entirely directed at the microscopic structure of muscular fila-

ments with a tendency toward finer scrutiny at the submicroscopic level.

Outstanding research for the characteristics of the contraction mechan-

ism of the skeletal muscle was conducted in the middle of the twentieth

century owing to the advanced electron microscope and its allied tech-

niques. In order to explain the mechanism of contraction, Huxley (1958,

1965, 1969) has determined, in his Sliding Theory, that during contrac-

tion two kinds of filaments in the voluntary muscle [thick elements

(myosin) and thin ones (actin)] slide past each other so as to produce

changes in the length of muscle. This effect is illustrated in Figure

1.1. After his hypothetical description of the contraction mechanism

of muscle, more detailed levels of research have been conduced by many

researchers. This research has succeeded in verifying his theory. And,

as a result, this concept was accepted and is now used for the under-

standing of the physiological phenomena of the contraction mechanism

of human skeletal muscle.

Whatever may be the molecular organization of the contractile ele-

ments in muscle, it is necessary to have a working description of the

overall mechanical behavior of muscle, in both the passive and stimu-

lated states, if one is to understand its performance in the organism.

In fact, human muscle is very different from the solid materials with

which we are familiar in engineering fields.

Perhaps the most fundamental mechanical information concerning a

muscle is given by the length-tension relationship, in which the ex-

erted tensile force is plotted against the length of muscle. Tension-

elongation experiments, performed in large number in the past (Dubisson

and Monnier 1943; Bull 1945, 1946; Guth 1947; Wilkie 1958), were in the

nature of function versus shape studies. However, by describing the

tension only in terms of change in length alone, these experiments

i

Actin

Sarcomere

A band

Myosin

Sarcoplasmic reticulum

Calcium

A. At rest

Sarcomere

A band

O

j

« 1,

B. Contraction

O

Figure 1.1. Process of Muscle Shortening

7

gave an incomplete picture of muscle behavior and contradicting re-

sults. Gutstein (1956) developed a generalized form of Hooke's Law

for muscle elasticity. He considered purely mechanical characteristics

of skeletal muscle without reference to the thermodynamic and myographic

properties of muscle. A direct determination of stress-strain relations

in skeletal muscle was studied by Nubar (1962-A). In his paper he em-

ployed the concept of theoretical filaments which are in close contact

and continue from one end of the muscle to the other, as illustrated

in Figure 1.2. He considered muscle tissue to be a nonlinear material

which has a Hooke's Law property, and established a single mathematical

expression which is applicable to the passive as well as to the stimu-

lated muscle. He related unit tension (stress) to unit change in the

length and the thickness of the human skeletal muscle (strain) as

follows,

^ = E (^) +E2 (^)^ + E3 (^)3 (1.1)

where

f is the filament tension,

w is cross section area,

E,, Ep and E^ are generalized Young's moduli,

L is initial length of the filament, and

dL is the elongatation of the filament.

8

Filamen (length

T L, thickness w)

•-—t 0 D I

a. Muscle before stress

Filament (length L+dL, thickness w')

.Plane of Symmetry

Muscle after stress

Figure 1.2. Ideal Muscular Filament Contraction

Numerous investigations of the total tension that a skeletal muscle

is capable of developing under isometric condition at various length

have been made in the past with somewhat conflicting results (Fenn

1938; Hill 1956, 1960; Bigland and Lippold 1954; Holubar 1969). Most

of the researchers have tried to determine the relationship between

the exerted tension and biological factors such as arrangement, shape

of muscle fibers, electrical activity, and lengthening and shortening

velocities. Although they could not determine the exact relationship,

they could predict that there must be some relationship between them.

Mechanical models of muscular actions have been the subject of the

investigations of many researchers for a long time (Levin 1927; Hill

1938). These researchers applied the theoretical approach to living

I

î e(t)

l

R

Resting Length

K,

•^mno pp- -^ f(t)

x(t)

Figure 1.3. Mechanical Model of Muscle Performance.

subjects without using dissection. Recently, Parnely and Sonmebloc

(1970) have developed a mechanical model of muscle that represents

most of the mechanical properties of skeletal muscle by using springs

and dashpots to simulate the elastic property of muscle. They added

an ideal force generator to simulate the actively contractile part of

muscle, as can be seen in Figure 1.3.

In the figure, K, and K^ are springs and R is a dashpot. The

"block box," labeled f is an ideal force generator of the actively

contractile part of muscle. Its output is proportional to the neural

input e(t) and the activity of the muscle's motor neuron.

Expenditure of energy in several simultaneous forms, such as mechan-

ical, chemical and electrical, is associated with all muscular activi-

ties. Based on principles of theoretical mechanics, some researchers

10

(Nubar 1962-B; Ayoub 1971; Petruno 1972; Park 1975) characterized some

described motion and discussed stresses at certain regions in the body

in order to provide fundamental understanding and to predict patterns

of significant characteristics of human motion. Nubar and Contini (1961) T

developed a minimal principle in order to solve the equations of the

theoretical mechanics that are, by themselves, incapable of determining

the unknown functions completely.

However, most of the problems of the biomechanical aspect of the

human skeletal muscle cannot be solved independently by purely mechani-

cal methods. This is because the muscle is a complicated system which

is associated with the structure of protein, the action of enzymes, and

the energy transfer in the biological system. Most of these problems

are still not well understood and are vague and contradictory.

Electromyography

Electromyography, which involves the measurement of electrical

signals that are generated while the muscle is working, has become

accepted as a useful tool for investigating muscle actions. It has

led to indirect determination of muscle participation at a particular

posture or at a certain movement without dissection, which is impossible

for a living human body. The goal of the electromyographic research

was to examine and to establish the relationship between electromyo-

graphic intensities and the magnitude of the muscular tension in the

muscle.

The basic principles and the experimental procedure of electromo-

graphy were explained by Basmajian (1967), and MaConaill and Basmajian

(1970). They also investigated electromyographical properties of some

11

muscles involving important human movements. Many investigators (Cooper

and Eccles 1930; Inman, et al 1952; Scheving and Panly 1952; Bearn 1954;

Zuniga and Simons 1969; Messier, et al 1971) have demonstrated that the

active force generated by the muscle contractile mechanism of a speci-

fic muscle depends on the level of neural activation. By means of multi-

ple channels of electromyography, one can determine not only which speci-

fic muscle is in action, but also the extent to which it is participating

with other muscles in the performance of a certain movement.

Several experiments under several conditions have been reported

(Hill 1939; Lippold 1952; Bigland and Lippold 1954) demonstrating the

relationships between electromyograms and muscular tension, contracting

velocity, energy and the level of neural activation. The major short-

coming of those results is that they always considered only the case of

maximally activated (tetanized) muscle contraction.

Recently, an experimental investigation of the relations among

force, velocity and electromography of partially activated human skele-

tal muscle was reported by Zahalak, et al (1976) for steady motion.

Also, there have been extensive studies on the electromyographic

signals from important muscles crossing several human joints such as

the knee, hip and shoulder joints (Inman 1947, 1952; Houtz and Walsh

1959; Keasy, et al 1966; Long and Brown 1964; Sutherland 1966). A de-

tailed electromographic and morphological study of the shoulder joint

muscles has been made by Inman, Saunder and Abbott (1944). However,

they did not investigate the role played by these muscles in supporting

either the shoulder girdle or the gleno-humeral joint during static and

dynamic loading of the limb. A technique was developed by Cnockaert,

12

et al (1975) to calculate the torque generated by the individual muscles

that contribute to the isometric flexion of the elbow by using integra-

ted and rectified surface electromyographic sugnal intensities. By

using a surface stimulation technique, the dynamic characteristics of

the human skeletal muscle model was investigated by Tennant (1971). He

developed a mathematical joint model of the behavior of the muscle group

comprised of the biceps and brachialis, by taking into account the

changes in muscle tension due to the inertia of the moving masses under

various surface stimulation conditions.

Since the type of measurement and the method of coUecting data in

electromyographic studies inevitably involve considerable variation and

uncertainty, the generalized results, which are necessary for analyzing

human motion, have not to date been established unequivocally.

Mathematical Analysis of Human Motion

The main ultimate objective of the study of biomechanics is to in-

crease the efficiency of human performance by minimizing the effort re-

quired to perform the motor activities.

Even in the Renaissance period of the sixteenth and seventeenth cen-

turies, such men as Leonardo da Vinci (1500) and Borelli (1685) began

to apply scientific principles to the study of human motion. Beginning

in the nineteenth century human motion analysis, in general, has been

carried out either experimentally or theoretically as the result of

the progress of scientific research techniques (Sherrington 1893;

Braune and Fischer 1889; Ducheme 1867). The kinematic and experimental

analysis techniques of obtaining motion characteristics by using the

physical records obtained from a motion, are widely used in current

13

research in kinesiology (Pearson. et al 1963; Dempster 1955; Engen and

Spencer 1968; Karas and Stapleton 1967; Bouisset and Pertuzon 1967).

Generally, the techniques consist of the recording of human motion

by high speed cinematography. With this tool the successive positions

of the joints of the body and the orientation of its segments are plot-

ted as a function of time. These films furnish velocities and accelera-

tions which are used to evaluate, with the anthropometric data of the

segments, the forces and moments at the joints, and the kinetic and

potential energies of the segments. The work which was done by the mus-

cle is obtained from the variation of these energies or, equally, from

the products of the moments by the rate of the joint rotations, inte-

grated with respect to time. Kinematic analysis is also used for the

study of cycling, cranking, walking and lifting problems under specific

conditions and speeds.

To describe the design and the application of a spatial motion

for a detailed study of the relative motions of body segments, many

researchers (Passerello and Huston 1971; Van Sickle and Harvey 1972;

Jensen and Bellow 1976; Kinzel, et al 1972; Yagoda 1974) employed the

idea of a spatial linkage which is capable of measuring biomechanical

motion. By using this linkage system analysis method, Chaffin (1969)

developed a computer model which treats the human body as a series of

seven links articulated at the ankle, knee, shoulder, hip and wrist for

a certain gross body action as can be seen in Figure 1.4. The model

was specially designed to investigate body movements that occur during

the lifting and carrying of weights. However, in most of this research,

several constraints and assumptions, that were employed in the devel-

opment of the linkage model, limit their application in practice.

14

A : ankle

K : knee

H : hip

S : shoulder

E : elbow

W : wrist

Figure 1.4. Chaffin's Gross Body Action Model

A comprehensive, statical and dynamical analysis of human body

motion requires a set of governing equations applicable to a wide

variety of stiuations. A principal source of difficulty in developing

such equations is the complex geometry due to the shape of the body

with its abundant possible motions. Because of these reasons, a rela-

tively small number of researchers (MacLeish and Charley 1964; Merchant

1965; Morecki 1966; Thomas 1968; Troup and Chapman 1969; Choa, et al

1976) made an attempt to develop a mathematical model of the musculo-

skeletal system. Most of the contributions apparently have been in

the area of electromyographic recording of muscular electrical signals

to explain the participation ratio of the various muscles, to observe

the types of movement, or to calculate the joint forces by means of

conventional mathematics. Recently, a mathematical model for the

evaluation of the forces of the musulo-skeletal system in the lower

extremities was developed by Seireg and Arvikar (1973, 1975). These

15

papers contributed greatly in advancing the study to develop a mathe-

matical model of the musculo-skeletal system of the human body which

is capable of evaluating muscle forces and joint reactions at different

static postures.

Mechanism of Shoulder Joint

The shoulder, which is the proximal joint of the upper limb, is the

most mobile joint in the human body. Although the shoulder joint is

the one most commonly used in human activities, it is surprising that

it has been the subject of only a few studies. Furthermore, most of

the previous investigations have dealt with the magnitude of the forces

in terms of electromyographic and kinematic analyses of its movement.

The reason for this is that the mechanism of shoulder movement is much

more complicated than that of any other joint in the human body. The

general mechanism of the shoulder joint movement has been studied by

Dempster (1965), who used living subjects and ligament preparations

of cadaver material. He treated the shoulder joint as a complex com-

bination of three distinct joints, the sternoclavicular, claviscapular

and glenohumeral joints. Each joint was discussed functionally both

in terms of its range of movement and in terms of the action of asso-

ciated ligaments in restraining movement. To determine the biomechani-

cal performance which are decisive for the determination of the magni-

tude of the applied actual moment of force developed by individual mus-

cles coordinating in the upper extremities, and to establish the degrees

of their participation in the given movement, Fidelus (1967) found out

the relationship between exerted tension and the length of muscles.

Using a mirror and 35-mm motion picture camera, Engen and Spencer (1968)

16

developed two techniques for manual analysis of computer processing of

shoulder motion. In their paper, accurate diagrams of upper extremity

movement were made from photographs of a normal person. The points in

the diagram were connected to identify the patterns of movement, and

the angular velocity and acceleration of the points.

As mentioned earlier in this chapter, Inman, Saunder and Abbott

(1944) investigated shoulder movement in several ways, such as compara-

tive anatomy and roentgenographic analysis of the motion. The theore-

tical force required in shoulder motion and the action current potential

were derived from the living muscles in motion. From the data so ob-

tained, they attempted to resynthesize the whole shoulder motion. After

measuring the precise relationships of the body parts among each other,

and the relative positons which they occupy during a motion, they were

able to set up the equations and to calculate the force requirement

for the maintenance of the upper extremities during flexion and abduc-

tion in terms of electrical potentials.

The functions of individual muscles associated with the shoulder

joint were studied by several researchers (Basmajian and Latif 1957;

Wright 1962; Shevlin and Lucci 1969) to determine the role of a specific

muscle for a given motion such as swimming, golf, climbing and the swing

of the arm.

An interesting method of the kinematic analysis of the motion of the

shoulder, arm and the hand complex was first investigated by Taylor and

Blaschke (1945). In their paper, in order to analyze the axes and angles

of the idealized kinematic system, several steps were involved. These

steps were the measurement of anthropometric data, fitting the subject

17

with visual landmarks taking cinematographic pictures of the subject

performing the activities under study, and using the cartesian coor-

dinates of visual landmarks.

Recently, de Duca and Forrest (1973) developed a technique for

calculating the forces generated by the individual muscles which con-

tribute to isometric abduction of the upper limb in the coronal plane

when the humerus is rotated medially. Also, mathematical relation-

ships of the forces of the individual muscle were obtained as a func-

tion of respective effect, and that of physiological cross sectional

areas.

Park (1975) analyzed the forces in every muscle crossing the

gleno-humeral joint. He analyzed the shoulder joint by using mathe-

matical methods and vector methods based upon anatomical and physio-

logical characteristics. Also, anatomical and physiological proper-

ties of the shoulder joint, and the muscles associated with the joint

movements, were analyzed functionally. The application of these methods

to other joints in the human body was also discussed.

1.3. Purpose and Scope

The principal objectives of this investigation were: (1) to seek

a mathematical and descriptive analysis of electromyographic character-

istics and muscular tensile force distribution of human skeletal mus-

cles crossing the gleno-humeral joint, and (2) to gain a better under-

standing of the actual neuro-muscular activities of human skeletal mus-

cles and their actual mechanism. The mathematical and descriptive equa-

tions for the relationships between the electrical signal intensities

that are generated from the muscles, and the muscular tension that is

exerted by muscles at various postures during abduction and adduction

of the upper extremities, were also investigated.

This study involved the formulation of a musculo-skeletal model

of the human gleno-humeral joint that can be described by mathemati-

cal vector methods. This was done by using the data that were collec-

ted through dissections of cadavers and through physiological infor-

mation about human skeletal muscles, and anatomical characteristics

of the shoulder joint. By using this model, the explicit character-

ization of the mathematical equations for the postulated mechanism

of the gleno-humeral joint was formulated. Shoulder joint motion was

described in terms of a three dimensional coordinate system. This

research consisted of theoretical and experimental parts. The theore-

tical part of this study consisted of formulation of the model. The

experimental part consisted of the recording and analyzing of the < M

electromyograms of the shoulder surface muscles (deltoid anterior, T

middle and posterior parts) via surface electrodes during the appli-

cation of external loads. The external loads were applied to the upper ;

extremities by means of weights that varied not only in magnitude but î

also in the position of the weights which, in turn, depended upon the l

angles of abduction and adduction. The methods used for determining

the unknown functions and the parameters entering the equations were

adapted from the experimental results of other researches. Also, the

validity of the minimal principle as applied to human skeletal muscles

in the static case was investigated by using the theoretical and ex-

perimental results.

General equations that can be applied to different persons, who

have different anthropometric dimensions, were also developed. Com-

19

puter programs were developed to determine the muscular tension in

the muscles crossing the gleno-humeral joint of different persons

and to predict the linear coefficient between electromyographic elec-

trical signal intensities and the muscular tension of the skeletal

muscles. These were developed from the results of the theoretical

and experimental procedures. According to the results and the tech-

niques of this study, it was concluded that most of the complicated

human musculo-skeletal joints can be analyzed mathematically without

dissecting bodies.

n

r

CHAPTER II

CONCEPTUAL MODEL

The shoulder joint provides man with a unique mechanism to inter-

act with his environment. In comparison with other joints in the human

body, it is endowed with nearly limitless positioning ability to suit

the requirements of the environment, and the ability of adjusting the

environment to suit the requirements of the body. The fact that this

mechanism is attached to the upper-lateral part of the trunk segment

illustrates its critical positioning in the body to gain the best ad-

vantage for building on the accumulated movement of the lower limb and

the trunk. In addition, its position with respect to the head allows

for a visual, sighting and aiming control which cannot be duplicated

elsewhere in the body. The shoulder complex is an integrated portion

of the upper limb, and this joint is the place where the most impor-

tant motions can occur between trunk and arm.

2.1. Anatomical and Functional Aspects of the Shoulder Joint

Over the past hundred years, the anatomical and functional aspects

of the shoulder joint have been explored in detail because of its im-

portant role in human activities. In this section, a simplified con-

cept, based on the anatomy of human body, was employed to clarify

the functions of the shoulder motions.

Skeletal System and Joints

For the purpose of studying the skeletal structure of the shoulder

joint, the joint can be divided into several segments according to

their functional purposes. As can be seen in Figure 2.1. a pair of

20

21

clavicle and scapular bones join with the sternum at its superior edges

to form the shoulder girdle. The shoulder motions are related with the

relative motions of these bones and the humerus bones of the upper arm.

Their functional and morphological characteristics are as follows:

(1) Clavicle: The clavicle is shaped like an elongated S, extended

from the sternum to the acromion. At its medial end, it artic-

ulates with the sternum and the first rib. This articular sur-

face, which is about 2.5cm in diameter, is the only bony at-

tachment between trunk and the upper extremity.

(2) Scapula: The scapula is a flat, triangular bone overlying the

upper portion of the back. It is the site of the attachment

of the superficial muscles of the back. Its glenoid fossa,

at the lateral angle of the scapula, is modified to articu-

late with the head of the humerus, and is the only joint be-

tween scapula and humerus.

(3) Humerus: The humerus, the bone of the upper arm, is articu-

lated with the scapula at the glenoid fossa, and with the

radius and ulna at the elbow joint. The most important mo-

tion of the upper extremity occurs at this bone.

(4) Sternum: The sternum consists of three segments, which are

the manubrium, body and xiphoid process. At the side of the

sternum, articular facets are present for the clavicle at

the upper end of the manubrium, and for the upper six ribs

along the length of the sternum.

22

The clavicle, scapula and humerus form a smoothly coordinated

system in which each bone has been allotted a share in shoulder mo-

tion. This reflects the synergic action of the muscles that act upon

the bones.

The union between the glenoid fossa of the scapula and the head

of humerus is an example of a ball and socket type joint. In contrast

to the hip joint, the shoulder joint sacrifices its stability for a re-

markable degree of mobility. As can be seen in Figure 2.2, the ana-

tomical axis of the shaft of the humerus forms an angle with the true

axis of the flexion and extension of about 130 degrees.

Bones

A

B

C

D

Scapula

Humerus

Clavicle

Sternum

1. Gleno-Humeral

2. Sub-Deltoid

3. Scapulo-Thoracic

4. Acromio-Clavicular

5. Sterno-Clavicular

r. X >

I r i >

íl

Figure 2.1. Bones and Joints of the Shoulder

23

'

1

/ ^ ^0130

/ 4 5 ^

Figure 2.2. The Head of Humerus

Although the shoulder girdle consists of several joints of four

bones as can be seen in Figure 2.1, according to their anatomical posi-

tions and functional roles, we can divide them into five joints as

follows:

(1) Gleno-Humeral Joint: This is a diarthrodial joint, anatomi-

cally, and the articular surfaces consist of hyaline cartilage

It is the most important joint in the shoulder mechanism;

therefore, in this study, only this joint was considered.

(2) Sub-Deltoid Joint: This is an amphiarthrodial joint. How-

ever, it is mechanically linked to the gleno-humeral joint

because any movement in the latter brings about slight move-

ment in the former.

24

(3) Scapulo-Thoracic Joint: This is an amphiarthodial joint

which does not produce any significant motion, but slight

relative motion occurs between trunk and scapula at this

joint.

(4) Acromio-Calvicular Joint: This is a diarthrodial joint

located at the acromial end of the clavicle. Even though

it is a diarthrodial joint, it does not produce any impor-

tant motion during shoulder motions. Therefore, it can be

regarded as a amphiarthrodial joint.

(5) Sterno-Clavicular Joint: This is also a diarthrodial joint,

located at the sterno end of clavicle. But, as in the case

of the acromio-clavicular joint, it does not produce any

significant motion.

The motions of the shoulder girdle occur at all of the five joints

simultaneously, each contributing its share to the accomplishment of

the movement. To maintain the rythmn of smooth and coordinated motions,

the shoulder requires that all the five intact joints and all the proper

forces in the muscles move the bones. However, rectangular abduction,

which is the subject of this study, takes place mostly at the gleno-

humeral joint; therefore, the motion of the other joints were neglected

in this study.

Muscular System

The bones and joints of the human body are not committed to any

strictly predictable pattern of motion, so they permit an infinite

variety of motions. Muscles can momentarily constrain a joint mechanism

25

to hold body segments in static positions, or to elicit a motion by

changing forces in one direction while eliminating freedom of the

system in the other direction. Innumerable postures and various

patterns of motion are possible for human beings but all of them in-

volve some degree of muscular constraint at the joint.

The muscles which act upon the mechanism of shoulder motion can

be divided into four anatomical groups as follows:

(1) Gleno-Humeral Muscles: Those passing from the scapula

to the humerus.

(2) Axio-Humeral Muscles: Those passing from the trunk to

the humerus.

(3) Axio-Scapular Muscles: Those passing from the trunk to the

scapula.

(4) Others: Those passing from the scapula to

the ulna or radius of the lower arm.

The muscles belonging to these groups can be tablulated as follows:

(i) Gleno-Humeral Group:

(1) Supraspinatus (2) Infraspinatus (3) Subscapularis

(4) Coracobrachialis (5) Teres Minor (6) Teres Major

(7) Deltoid

(a) anterior (b) middle (c) posterior

(ii) Axio-Humeral Group:

(1) Pectoralis Major - (a) sterno (b) clavicle

(2) Latissimus Dorsi

26

(iii) Axio-Scapular Group:

(1) Trapezius

(4) Levator Scapula

(iv) Other Group:

(1) Biceps -

(2) Triceps

(2) Serratus Anterior (3) Rhomboids

(5) Pectoralis Minor

(a) long head (b) short head

Functional Aspect

The movements of the shoulder joint can be divided into two main

types: the major movement and the minor movement. The major movement

and the minor movement refer to the humeral and scapular movement,

respectively. The humeral movement can be regarded as a combination

of abduction, adduction, medial rotation, lateral rotation, flexion

and extension of the forearm. The scapular movement can be regarded

as the combination of forward movement, backward movement, upward move-

ment, downward movement, and the rotation of the scapula. The scapu-

lar movement can be done with the help of the flexion of the spinal

cord.

Therefore, the gleno-humeral, axio-humeral, and other groups of

muscles, as described in the former section, influence the humeral

movement, and the axio-scapular group of muscles influences the scap-

ular movement, respectively.

In recent years, studies of the shoulder joint motions proved that

the complete elevation of the arm, in either the coronal or the frontal

plane, is the combination of the free motions of all the joints of the

shoulder complex. Although the concept may be incorrect that rectangu-

lar abduction takes place entirely at the gleno-humeral joint, and that

>

n

r i >

27

full elevation is completed by the motion of the scapula on the chest

wall, the contribution of the scapular movement to the whole move-

ment of the shoulder is small enough to neglect in the region from

0 to 90 degrees. Since the problem of abduction from 0 to only 90

degrees was considered in this study, the scapular movement was neglec-

ted.

Once the muscle origins and insertions had been identified from

pictures of a dissected cadaver, the next task was to obtain their

proper coordinates with respect to a suitable spatial set of coordinate-

axes. This knowledge was, of course, necessary for the mathematical

analysis. The shoulder joint has three degrees of freedom which al-

low the upper limb movements with respect to three planes in space.

A brief description of the axes follows: q

(1) Transverse Axis: Lying in a f ronta l plane, i t controls the lii

movement of f lex ion and extention in a s i g i t t a l plane.

(2) Anter ior-Poster ior Axis: Lying in a sag i t ta l plane, i t

n I r

controls the movement of abduction (the upper limb moves ^ >

away from the body) and adduction (the upper limb moves J

toward the body) which are performed in a frontal plane.

(3) v'ertical Axis: Lyirig through the intersection of the sagit-

tal and frontal planes, it corresponds to the third axis in

space. It controls the movements of flexion and extension

performed in a horizontal plane while the arm is abducted

to 90 degrees.

(4) Longitudinal Axis of the Humerus: This controls the move-

ments of lateral and medial rotation of the arm.

28

The position of the reference line was selected as the line of the

upper arm hanging vertically at the side of the trunk. These axes

are shown in Figure 2.3.

In this study, only abduction and adduction movements of the

upper limb were considered; therefore, the fourth axis was not

considered.

1. Transverse

2. Anterior-Posterior

3. Vertical

4. Longitudinal

R

i i n I r 5

Figure 2.3. Axes of Shoulder Movement

As mentioned in the discussion of the muscle model, if there is

no electrical signal present during a motion, the muscle does not exert

any active force and, there is only passive force in the muscle. But,

according to the length-tension diagram of Ramsey and Street (1940),

for a skeletal muscle the passive force is so small that it can be

29

neglected compared to the active force. According to their paper,

as shown in Figure 2.4, the elastic component force is only 2 percent

of the total force at 150 percent of resting length. Even at 200 per-

cent of the resting length, it is only 47 percent of total force.

The shoulder joint muscles can be divided into three groups ac-

cording to their physiological actions. Each group consists of the

abduction muscles, adduction muscles and cuff muscles. The abduction

— — — — —

to

e T

en

sio

n tiv

R

ela

Total

rassive

/ * : — - •

/

^y ^ ^ ^ ^ -I r ••

100 200 Percent of Resting Length

X

r

;D >

Figure 2.4. Length-Tension Diagram of Skeletal Muscle

muscles and adduction muscles exert forces during only abduction and

adduction, respectively, and the cuff muscles exert forces during both

abduction and adduction.

Because of the small shoulder movement assumed in this study,

only 15 muscles are considered. During the abduction from 0 to 90

degrees, there are electrical signals only in the following muscles:

30

(1) Supraspinatus (2) Deltoid Anterior

(3) Deltoid Middle (4) Deltoid Posterior

(5) Infraspinatus (6) Teres Major

(7) Teres Minor (8) Subscapularis,

and during the adduction, there are electrical signals in the following

muscles:

(1) Pectoralis Major Sternal Part

(2) Pectoralis Major Clavicular Part

(3) Latissimus Dorsi (4) Biceps Long

(5) Biceps Short (6) Triceps

(7) Coracobrachialis

(8) Infraspinatus (9) Teres Major q

(10) Teres Minor (11) Subscapularis J (0

H Consequently, following the classification system previously discussed, |!j

T

muscles were divided into three functional groups as follows: " i

(1) Abduction Muscles: Supraspinatus, Deltoid Anterior, >

Deltoid Middle, Deltoid Posterior, ^

(2) Adduction Muscles: Pectoralis Major Sternal Part,

Pectoralis Major Clavicular Part,

Latissimus Dorsi, Biceps Long,

Biceps Short, Coracobrachialis,

Triceps,

(3) Cuff Muscles: Infraspinatus, Subscapularis,

Teres Minor, Teres Major.

31

A : Anterior B : Middle C : Posterior

Deltoid Teres Minor

r X >

n I r S >

Teres Major Supraspinatus

Figure 2.5. Schematic Diagram of Shoulder Muscles

32

Infraspinatus Subscapularis

A : Long B : Short

Latissimus Dorsi

X >

n I r i >

Biceps


33

Clacular Sternal

Pectoralis Major

Triceps Coracobrachialis

i

r i IQ >


34

According to the above analysis, it can be said that during abduc-

tion all of the muscles considered, except the adduction muscles, exert

forces and during adduction all of the muscles considered, except the

abduction muscles, exert forces. However, cuff muscles exert force

during both abduction and adduction.

Assumptions

In this study, in order to develop a suitable mathematical model

for the musculo-skeletal system of the shoulder joint, several assump-

tions are needed. These are listed below:

(1) It was assumed that the stability of the skeletal structure r

in any posture is maintained by the static equilibrium of

muscular entsion and reaction force at the joint. q r

(2) Muscles were assumed to have distinctive origin and inser- ^ in

tion points and the tensile forces which are produced by ^

muscles were assumed to be directed along the lines joining 5 u.

the origin and insertion points. This assumption resulted C

in considerable difficulty in constructing the model because ^

the muscles have innumerable shapes and do not originate or ^

insert in a straight line fashion. However, the origin and

insertion points were chosen and the lines of force were

drawn judiciously as possible to represent the model. The

origin and insertion points chosen and force lines drawn for

each muscle can be seen in Figures 2.5 to 2.7.

35

(3) It was assumed that the only bone in the shoulder mechanism

to have movement was the humerus. This is because only the

relatively small amounts of abduction and adduction were

considered in this study. Consideration of scapular move-

ment would improve the model somewhat, but it was neglected

because it would result in considerable complication in the

analysis of the problem. However, in the case of abduction

over 90 degrees, the participation of the scapular movement

would become large and probably could not be neglected.

0 : Center of Rotation R : Reaction Force

=1

Ul

n I r i ;Q >

Figure 2.8. Direction of Reaction Force

36

(4) The head of the humerus was assumed to be round as illus-

trated in Figure 2.8. Also, it was assumed that there was

no friction between the glenoid fossa and the head of the

humerus. With these assumptions the resultant reaction

force would go through the center of rotation.

(5) It was assumed that the center of rotation of the upper

limb was a stationary point which could be determined by

dissection films. However, it is known that the center of

rotation does displace slightly during the motion. As a

result, the sensitivity of the result of this analysis to

slight changes in the location of the center of rotation

was considered by referring to a previous study by Park

(1975). The study was conducted by performing a number of r. X

analyses with different centers of rotation. > tn

(6) The whole set of the upper extremity, upper arm, lower arm *i ti

and hand were assumed to be a single rigid link. I r i

2.2 The Conceptual Model Postulated J >

In addition to the usual problems facing scientists and engineers ^

when they attempt to model a man-made system which is qualitatively

known (that is,'when all the parts comprising the mechanism to be

modeled are identifiable and their functions understood), in this

research. it was necessary to be content with a system whose operation

at present is not completely known. The actual mechanism by which a

shoulder can move is not fully understood and even less understood

are functional characteristics and logics that activate the systems

of nerves and muscles of the shoulder joint.

37

The observation of force and moment equilibrium of the human body

at a certain particular posture suggests that a muscle can be modeled

as a force generator producing a certain force vector which produces

a movement. This model is the classical one that was used by the

author (1975) for a former study. The force distribution used in the

model was not unique and, in fact, some assumptions were used to solve

the indeterminate problem. Also, the model required complicated ana-

tomical data for the mathematical analysis. However, this method of

approach is quite useful and general equations of motion can be devel-

oped by using the method. Therefore, this technique was adapted in the

theoretical part of this study for the mathematical analysis of the

shoulder joint and it was also used for the formulation of the rela-

tionships between individual muscles. R

The model for the mechanical and mathematical analysis of the >

shoulder joint muscles, and the kinetic behavior of the muscles, is H

briefly described below. The model consisted of four types of basic I

elements, as follow: r i ;D >

(1) A force generator (muscle) whose output depends on the exci- ^

tation from nerves, the length and the anatomical position

of the muscle itself, and the applied weight.

(2) An electrical signal (nerve) whose output produces muscular

tension. This output was recorded by using surface elec-

trodes.

(3) A working media (body segment) whose output represents the

work done by the muscles. Sometimes the weight of the body

segment was considered as an applied weight.

38

s

F

M// /x/ x"

\ /

Figure 2.9. Conceptual Model

(4) A weight (applied force) that controls the magnitude of

muscular tension and electrical signal intensity.

The elements of the postulated model are illustrated in Figure 2.9.

In this schematic representation of the shoulder mechanism, F repre-

sents the output of the force generator (muscle), S (which is recorded

by the electromyogram) represents the stimulating electrical signal

(nerve signal) from the nerve system, and W and B represent working

load (weight) and body segment (arm), respectively. It should be

noted that there is a relationship between developed muscular tensile

forces and the lengths, directions, locations of origin and insertion

points, and thickness of individual muscles.

For the gross human shoulder joint, all the muscular tensile force

vectors were drawn for three directional views as can be seen in Figures

R

tn

fi n

r i >

2.10 to 2.12.

39

1. Supraspinatus 2. Corachobrachialis

3. Pectoralis Major-Sternal

4. Pectoralis Major-Clavicular

5. Biceps-Long 6. Biceps-Short

7. Subscapularia

> t-:

r; n

tD ;Q >

Figure 2.10. Location of Shoulder Muscles (Anterior View)

40

tn

n I

tD ;Q >

8. Deltoid-Posterior

9. Deltoid-Middle

10. Deltoid-Anterior

11. Triceps

Figure 2.11. Location of Shoulder Muscles (Lateral View)

41

fr,

til

íl n I r

;Q

;Q

12. Latissimus Dorsi 13. Infraspinatus

14. Teres Minor 15. Teres Major

Figure 2.12. Location of Shoulder Muscles (Posterior View)

42

Center of Rotation Origin Insertion

Figure 2.13. Change of Insertion Point

From the anatomical data the lengths, direction cosines, and

moment arms of e\/ery muscle were calculated by vector methods at ewery

rotational position from 0 to 90 degrees at 10 degree intervals. This

situation is illustrated in Figure 2.13 for the pectoralis major clavi-

cular muscle.

tn

fi n I

01

Formulated Equations

In this section the general forms of the equations for the postu-

lated model of the shoulder joint are given briefly. The purpose of

43

this brief description is to provide insight and guidance to the more

detailed description of the theoretical study and experiments to be

discussed later.

The objective general equation characterizing the total actual

mechanism of the shoulder joint can be represented symbolically in

the following form:

M = f (T., E., W, B., A, e) (2.1)

where

M

î

w

B. 1

A

e

f.

is the total motion characteristic of the shoulder joint,

is muscular tension in a particular muscle,

is electrical signal intensity of a muscle,

is applied weight,

is an anthropometric factor of a segment,

is an antomical and physiological factor,

is an abduction or adduction angle, and

(and any subscripted f) represents a functional relation-

ship.

Equation (2.1) expresses the fact that the whole motion of the shoulder

is a function of tension in the muscle, stimulation intensity, applied

weight, anthropometric factor, anatomical and physiological factors

and abduction and adduction angles of the muscles.

In the theoretical part of this study, the following generalized

equation was investigated for the mathematical vector representation of

a single muscular tension.

tn

n I r

s > íl

T. = f^ (L., D., W, B.,A, e). (2.2)

where L^ and D . refer to the length of the muscle and the direction of

its force application, respectively. In this study D'Alembert equili-

brium equations for forces and moments were used and, in order to make

the statically indeterminate problem into a statically determinate one,

the Minimal Principle theory of Nubar and Contini (1961) was adopted

as follows for the whole shoulder joint.

where

-îc.H,') dt + A (2.3)

44

M.

dt i

' o '

s a numerical constant,

s a muscular moment of a muscle,

s a time interval,

s an initial constant of the motion and

E is the total muscular effort which is reduced to a

minimal by the imposed minimal principle.

The purpose of the experimental part of this study was to find out

the relationships between the myographically recorded electrical signal

intensities generated by the muscles and the biomechanical factors of

the subject. These relationships were represented by the following

equation:

n

tn

fí n I r 5 ;Q

;Q

E. = f^ (L., D., W, B., A, e) (2.4)

Finally, from the Equations (2.2) and (2.4), the following general

equation for the relationship between the tension and electrical signal

45

intensity of the muscle formulated.

T = 4 (E .) (2.5)

Because the experimental work was restricted to the use of surface

electrodes, electromyograms could only be obtained for surface gleno-

humeral muscles (deltoid anterior, dpltoid middle and deltoid poster-

ior).

A detailed description and explanation of the equations is provided

in the following chapters.

n I r S ;Q >

:<

CHAPTER III

MATHEMATICAL ANALYSIS

3.1. Anatomical Considerations

As discussed earlier, muscles were assumed to have distinct origin

and insertion points and the tensile forces which are generated by mus-

cles were assumed to be directed along the lines joining the origin

and insertion points. After finding the insertion and origin points

in the reference position, the length, direction consines and moment

arms (which are necessary information for calculating mathematical solu-

tions for every muscle) were calculated by vector methods at every posi-

tion from 0 to 90 degrees in 10 degree intervals, as shown in Figure

3.1.

tfl

n I r 5

XI

Figure 3.1. Change of Posture

46

47

For each of the muscles, all the information was calculated and is

tabulated in Appendix I, Tables 1.1 to 1.15.

The author (1975) calculated the effect of the change of the cen-

ter of rotation on muscle force distribution by using the various posi-

tions for the center of rotation as described below:

(1) The center of rotation moved down 0.5cm,

(2) The center of rotation moved up 0.5cm,

(3) The center of rotation moved medially 0.5cm,

(4) The center of rotation moved laterally 0.5cm,

^ ^

tn

n I r i >

XI

Figure 3.2. Schematic Diagram of Sensitivity Test

from the dissection center which was found from the films of the dis-

sected cadaver. The directions of the variations are shown in Figure

3.2. The author found that the effects of these variations were small

enough to be neglected.

48

3.2. Equilibrium

For a system to be in equilibrium, the sum of the external and

internal forces and moments must be zero. Equilibrium can be dynamic

or static. However, for the case under consideration, the system con-

dition was static equilibrium because the skeletal structure was as-

sumed to be static and the inertia forces and moments associated with

the motion of the system did not appear. This implies that the sum

of the moments and the forces about the three space axes must be equal

to zero. In order to analyze equilibrium, the free body diagrams of

each of the bones associated with the shoulder movement were considered

The equations of equilibrium were applied to the three reactinal for-

ces in X, Y and Z directions at the joint for each of the following

muscles:

(1

(2

(3

(4

(5

(6

(7

(8

(9

Spupraspinatus

Deltoid Anterior

Deltoid Middle

Deltoid Posterior

Infraspinatus

Teres Minor

Teres Major

Subscapularis

Pectoral is Major-Sternal Part

tfl

n I r S XI

XI

49

^ J X ' L'i

Figure 3.3. Analyzing System

(10) Pectoralis Major-Clavicular Part

(11) Latissimus Dorsi

(12) Biceps Long

(13) Biceps Short

(14) Triceps

(15) Corachobrachialis.

As discussed earlier, for the abduction case eight muscles (mus-

cles 1 to 8) produce muscular tension and, for the adduction case, 11

muscles (muscles 5 to 15) produce muscular tension. Therefore, all

the mathematical equations will be developed for eight muscles in

the abduction case, and for eleven muscles in the adduction case.

R

tfl

Sl n I r S X] > XI

TEXA5 TEC; ; LIBRAR^Ú

50

The basic force model is shown in Figure 3.3. Point "0" is the

center of rotation of the upper arm and point "i" is the acting point

of the muscular tension.

From force equilibrium, 3 force equations can be written.

V F X ^ + RX = 0,

2 Fy. + Ry = Fw,

V F Z ^ + RZ = 0, (3.1)

where.

Fx.

Fz.

Rx

Ry

Rz

is the x-directional component of muscular tension F.,

is the y-directional component of muscular tension F.,

s the z-directional component of muscular tension F.,

s the x-directional component of reaction force,

s the y-directional component of reaction force,

s the z-directional component of reaction force at the

joint.

Fw is the total weight of the upper arm, including the weight

of body segments and an external weight which is applied

at the hand.

tfl

n t r S XI > X}

The summations are over all of the muscles involved; eight in abduction

and eleven in adduction.

Let the direction cosines of these forces "F." be called Dx., Dy.,

and Dz., respectively. Then,

51

Fx. = F.. Dx.

Fy^ = F.. Dy.

Fz. = F.. Dz. (3.2)

By substituting Equation (3.2) into Equation (3.1) we can get,

y F .. Dx^ + Rx = 0

2_, î' Dy - + Ry = Fw

y^F.. Dz. + Rz = 0 (3.3)

Three moment equilibrium equations can also be written:

^ Mx. + Mr.. = Mw

+ Mr^ = 0 J U

V M Z ^ . + Mr^ = 0 (3.4) n

Where, Mr , Mr , and Mr are the moments due to the reaction forces Rx, ^ ^ y ^ > ^

Ry, and Rz. Mw is the moment due to the weight of the segments and the ;Q

externally applied weight at the hand, and the summations are over all ^

of the appropriate muscles.

By the same procedure as for the force analysis, let the direction

cosines of moments "M." be called Bx•, By., and Bz., respectively, then,

Mx. = M.. Bx. 1 1 1

My. = M.. By.., and

Mz. = Mi. Bz. (3.5)

imi^

52

Then the Equations (3.4) become.

+ Mr = Mw X

^ M - . B x .

^ M . . By.

^ M . . Bz. + Mr^ = 0 (3.6)

+ Mr = 0 y

Let the moment arms of the force "F." be Lx., Ly., and Lz. in X, Y, Z 1 1 "' 1 1

directions, respectively, as can be seen in Figure 3.3. Then,

Mx. = Fy .. Lz . - Fz.. Ly.

My.j = Fz.j. Lx. - Fx.. Lz.

Mz. = Fx.. Ly.. - Fy.. Lx. (3.7)

By Equations (3.2) and Equations (3.7), R

Mx. = F. (Dy.. Lz. - Dz.. Ly.) ^

My. = F. (Dz.. Lx. - Dx.. Lz.) n

r Mz. = F. (Dx.. Ly. - Dy.. Lx.) (3.8) 5

I I I 1 I I ^

Xi

Substituting Equations (3.8) into Equations (3.4), and using the assump- ^

tion that the reaction force at the joint goes through the center of

rotation, that means Mr , Mr , and Mr are all zero, we can get the X y z

following equations:

^ F . (Dy.. Lz. - Dz.. Ly.) = Mw

) F. (Dz.. Lx. - Dx.. Lz.) = 0 y f 1 1 1 I I

. 2 , ^ (DXi' Ly. - Dy.. Lx.) = 0 (3.9)

53

Consequently, six equations result from this system: three for

force equilibrium [Equations (3.3)], and three for moment equilibrium

[Equations (3.9)]. Having thus obtained the equations of equilibrium

for the system, it was evident that there were only 6 equations with

11 unknowns for the abduction case (3 reaction forces at the joint

and 8 muscular tensions) and 14 unknowns for the adduction case (3

reaction forces at the joint and 11 muscular tensions). There were

more unknowns than equations and, hence, this problem was statically

indeterminate. This implied that there were many possible solutions

for this problem. In order to make this problem determinate, it was

necessary to make some assumption concerning which muscles were called

into play in supporting the skeletal structure in nature. The problem

was solved by the hypothesis that the human structure adjusts itself ^ r'

in such a manner so as to reduce its muscular effort to the minimum. '^. tfl

3.3 Minimal Principle f; — ^

By using Nubar and Contini's minimal principle (1961) all the "

equations that are necessary to solve this problem can be determined. ^

Accordinq to their work, muscular effort is defined as the product of ;

applied moment and its duration of application. To avoid the confu-

sion of negative and positive moments, the square of moment terms was

used in the form of c M dt as a measure of muscular effort at a joint.

For the purpose of obtaining a mathematical formulation of this theory,

they used the symbol "E" to present the sum of the muscular effort at

the joint, plus some initial constant A^ as follows:

E ^(c.M.^^dt + A^ (3.10)

54

in which the common time interval "dt" has been factored out, the sub-

script "i" denotes the several joint muscles, and the "c." are numeri-

cal constants.

For the specific position of the humerus, the equilibrium condi-

tions can be considered to be constraint conditions for this system.

Therefore, Equations (3.6) can be written and used as the constraint

equations for this system:

f (M.) = Mw,

f^ (M^) = 0,

f^ (M.) = 0 (3.11)

where f,, f^ and f^ are the equilibrium equations in the X, Y, and Z

directions.

The result of differentiating E with respect to M.j in Equation

dE = 2Y(c^. M.. dM.)dt. (3.12)

dE = 2 ^ ( 0 . M. dM.)dt = 0. (3.13)

For normal individuals, operating under normal conditions, the coeffi-

cients c can be considered to be equal, and they will drop out, so

Equation (3.13) will be

V M . dM. = 0. (3.14)

tfl

(3.10) is n I r S

XI

Muscular effort is minimum according to the principle; therefore, ^

55

The differential elements dM.j are subject to the following con-

ditions, obtained by differentiating Equation (3.11).

3f, = 0 ZsT""'

1 dfr,

9f 3

E<

2,587*1 •» (315)

The method of Lagrange's undetermined multipliers, which is a

standard technique described in many references, for example, Hilde-

brand (1963), can be used by multiplying each of the three equations

in Equation (3.15) by Lagrange's multipliers VI, V2, V3, respectively, q h

and by adding to Equation (3.14) to get: ^ in

8f-| afp afo ^ M, + VI rr^ + V 2 vT^ + V 3 d M . = 0 H

j 1 9M.| gM. ^M. 1 l U.

r Since dM.'s are not zero, and the independent in general, their coef- g

^ XI ficients must be zero. Therefore, ^

Xl 3f 9f 9f

("i ^Vl 3 M 7 ^ V 2 ^ . V 3 ^ ) =0 (3.16)

The number of Equation (3.16) is, in fact, the same as the num-

ber of muscles associated with the motion; one for each muscle. With

the equations of mathematical equilibrium, there are as many equations

as unknowns. For the abduction case, there are 14 equations in 14

unknowns (8 muscular tensions, 3 reaction forces, and 3 Lagrange's

multipliers). For the adduction case, and there are 17 equations in

56

in the 17 unknowns (11 muscular tensions, 3 reaction forces and 3 La-

grange's multipliers) for adduction case. According to this proce-

dure, the problem becomes a statically determinate one.

From Equations (3.6) and (3.11), the partial derivative terms

of Equation (3.16) can be obtained as follows:

l ^î^ ZÎî* ^î - = 0

f^ (M .) M^. By. = 0

f^ (M.) M . . Bz. = 0 (3.17)

and from Equation (3.17)

9F 1 9M.

9f,

ãM"

df,

W

= Bx.

= By,

Bz.

So Equation (3.16) becomes.

( M. + VI. Bx .. + V2. By. + V3. Bz. ) = 0

(3.18)

(3.19)

l Ifl

n T 4.

r S X] > X]

And from the phythagorean theorem,

M. = ((Mx.)^ + (My.)- + (Mz.)^)^ (3.20)

According to Equation (3.8),

M^ = F. ((Dy.. Lz. - Dz.. Ly.)^ +(Dz.. Lx. - Dx.. Lz.)^ +

57

2îí (Dx.. Ly. - Dy.. Lx.)^) (3.21)

where, Dx.., Dy.., Dz.j, Lx., Ly., and Lz. are constants, so Equati

(3.21) can be written as

on

M. =F.. K. (3.22)

where

Ki^ = (Dy^. Lz. - Dz.. Ly.)^ + (Dz.. Lx. - Dx.. Lz.)^ +

(Dx.. Ly. - Dy.. Lz.)'

and

Mx. F-(Dy.. Lz. - Dz., Ly.) (Dy., Lz. - Dz.. Ly.) DA .

By^

B z .

" i

Ox^,

D x . . 1 ~

L x .

•-yi

-

^

-

K

Dx.

Dy .

F . . K.

, L z .

, Lx . (3.23)

i tn

n CD X]

X]

Changing notation, let us designate Bx.j, By ., and Bz.j by the constants

P., q., and r., respectively. Then Equation (3.19) becomes.

F., K. + VI.p. + V2.q. + V3.r. = 0 (3.24)

where aqain, K., P., q., and r, are all constants that can be calcu-' •' i 1 1 1

lated from the anatomical and physiological data as discussed pre-

viously.

3.4 Solution Technique

The solution to this problem can be finally resolved into the

problem of solving simultaneous equations. The problem can be stated

in matrix form as follows:

58

Dx-j DXp

DYI Dy^

Dz^ Dz^

Bx, Bxp

By^ By^

Bz.| Bz^

K 0

0 K,

0 0

Dx^ 1 0 0 0 0 0

Dy„ 0 1 0 0 0 0 ^n

Dz^ 0 0 1 0 0 0

Bx^ 0 0 0 0 0 0

By^ 0 0 0 0 0 0 •'n

Bz„ 0 0 0 0 0 0 n

0 0 0 0 p., ql r

0 0 0 0 P2 ^2 ^2

0

0

0

l

O..K^_2 0 0 0 0 0 P,_2qn-2V2

0, n-l 0 0 0 0 P,_iq,.ir^.i

0 \ 0 0 0 Pn % n

8 > = <

n

R X

Ry

Rz

VI

V2

V3

^ o ^

W

0

Mw

0

0

0

0

0

0

0

0

0

0

0 V. J

tfl

íl n I r 5 X] > Xl

59

where n is the number of muscles involved in the motion (n=8 for ab-

duction and n=ll for adduction). However, because of the minimal

energy principle of the muscular effort, the determinant of this

matrix becomes zero, and the matrix is singular. This problem can

be solved by using Equation (3.19) to change the moment equations

as follows:

M . = -(VI.Bx.. + V2.By^. + V3.Bz.)

Mx . = M..BX. = -(VI.Bx."^ + V2.By..Bx. + V3.Bz..Bx.)

My. = M..By. = -(Vl.Bx..By. + V2.By.^ + V3.Bz..By.)

Mq.. = M. .Bz . = -(Vl.Bx..Bz. + V2.By. .Bz. + V3.Bz.^) (3.25)

For simplicity, let the following notation be introduced for some of

the terms on the right hand sides of these equations:

^ l ' - -

=2 = -

^3 = -

=4 = -

^5 = -

^6 = -

2»<,' Z'»i »«1

2"'r"»i

2»', I!»'r"i

2-. (3.26)

í tfl

n

r S XI > X]

And instead of the moment Equations (3.6), by using Equations (3.16)

for the moment equation. From Equations (3.25) and (3.26), the dotted

square region of the matrix will be changed as can be seen following

the matrix.

60

Dx.| Dx^

Dz^ Dz^

0 0

0 0

0 0

0 0

Dx n

1 0 0 0 0 0

Dy^ Dy^ Dy^ 0 1 0 0 0 0

Dz. n

0

0

0

K 0 0

0 K 0

0 0 K^_2 0

0 K

0 0 1 0 0 0

0 0 0 s^ S2 S3

0 0 0 S2 s^ S5

0 0 0 S3 S5 s ,

0 0 0 p q^ r

0 0 0 P2 ^2 ^ 2

/ ' _ ^

< .

0

w

= <

0 0 0 P,.2qn.2^n-2

0 0 0 K^_.,0 0 0 0 p„_-,q_-,r,

n

' n - r n - V n - l

0 0 0 p q r ^n ^n n

n

R X

R.

R.

VI

V2

V3

0

Mw

0

0

0

0

0

0

0

0

0

0

^ ^

I tfl

í n I r 5 X) X]

This equation can be solved by computer methods to obtain the

muscle tens i le forces, Fp Fp* . . .» F^. The results of th is solu-

t ion at the various angular posit ions for d i f fe ren t weight for each

subject are tabulated in Appendix I I , Tables 2.1 to 2.6 together

with the experimental resu l ts .

CHAPTER IV

EXPERIMENTAL PROCEDURES

This chapter is devoted to the description of the experiments

needed to characterize the measured anthropometric data and the re-

corded electromyograms of the shoulder joint muscles.

The textbooks of anatomy describe the deltoid as a flexor, ex-

tensor, abductor, and medial and lateral rotator of the arm. The

muscle can be roughly divided into three distinct parts on the basis

of the origins and the modes of function. The anterior part pro-

duces mainly flexion and medial rotation of the humerus, the middle

part produces abduction, and the posterior part extends and laterally

rotates the humerus. The anterior and posterior parts of the del-

a toid contain parallel fibers, while the middle part is multipennate. g These anatomical features are probably responsible for the wide range tfl

H of movements and functions which make this muscle capable of produc- Jn

ing a variety of movements. Anatomically, these three parts of the ^

deltoid are distinctive and can easily be identified during the dis- ;Q

section of cadavers. 5

The purpose of the experimental procedure was to provide anatomi-

cal data for input to the theoretical analysis and to provide experi-

mental verification for theoretically predicted muscle force distri-

bution in the shoulder. However, there was a basic problem with this

verification: muscle force distribution was calculated for the cada-

ver but, of course, it was impossible to verify these results by elec-

tromyography on the cadaver. Electromyography was used on living

subjects but the theoretical analysis (which was to be verified) could

61

62

not be conducted on the living subjects without the detailed internal

anatomical data provided by dissection. This problem was overcome by

making some external anatomical measurements on both the cadaver and

the living subjects so as to establish scale factors for each subject.

These scale factors, together with the internal anatomical data of the

cadaver, provided an indirect means to estimate internal anatomical

data for the living subjects. With these internal data, theoretical

force distributions could be calculated for each subject and checked

experimentally by electromyography for the three parts of the deltoid.

4.1. Anthropometric Data Characteristics

Six male subjects were selected for this experiment. The only

limiting factor concerning the subjects was that they were to be of

two different physical builds, i.e., three of them were of good phy-

sical build, and the rest were of average build. d

As can be seen in Figure 4.1, the following anthropometric charac- n X

teristics of each subject were determined by using the method of Snyder, r Ê

et al (1971): 5 X]

(1) Weight (W): Weight of subject unclothed (Ibs)

(2) Height (H): Height of the subject while maintaining an erect

standing posture (ft)

(3) Biacromial width (BW): The horizontal distance between the

superior lateral border of the acromial process of the left

and right scapulae (ft)

(4) Chest height: The vertical distance from the center of the

umbilicus to the superior margin of the jugular notch of the

3 tfl

63

(0 +J (13

Q

U .r— s -

+J cu E o Q . O S-

+J c: <:

CD S-13

cn

i Ifl

n I r 5 X] > XI

64

^ o C\J CVJ

LO

vo n

00 o

m CSJ

lCi o o C\J

LO

co 0 0 CO r—

+J o

13 CO

0)

«+-O

to u

•r— +J </)

cu +J o fO S-rtJ sz o o ti

+-> cu E o CL. o i-

cu

co +-> CJ

cu J 3 Z3

co

LD

00 co 0 0

o

ro 0 0 C\J

LD

LT)

ro O O 0 0

I— r— O

LO cr»

OJ

oo r^ o o cr» LO 1— I— o

o

o

220

CVJ

LO

co

•~

C\J ro •"

LO

o •—

CM

•—

del

o s:

UO

L

mens

UD

174

Ibs

CSJ CM

co

+J

LO CVJ

'

+->

.2525

'

+ J «+-

.036

'

+ j

CT>

'

+J

tn + J S-c cu cu +J e cu cn E CU "3

oo s-oj

Ci-

+ J ^ cn

• p -

 c í

^ cx

-o o o cn

JCZ +J . r -2 to c o </) S-cu 0 .

cu S-cu 2

LO

A

LO

r

^" to

+J u cu

•o J2J Z3 co

r t

s-cu > fO

• o fO o

-o cu

• M

o cu to to

. r -

•o fT3

E o S-M-• o OJ +J o cu

r— 1 —

o o cu S- rtJ

-o r -

cu • o o s:

<o . r -

o to >^

^ Q .

cu cn fT3 s_ cu > fO

o

cu s-cu 5

<:t cv^

t \

OsJ

to +J u cu

''-) J 3 3

C>0

• o c fO

« *-^ to S-cu > 1 ftJ

r— Q .

r— 1 —

fO J 3 +J o o

u->)

+J . r -

00 s-rtJ >. >>

+J •r—

to S-cu >

•r—

c r3

u cu h-to fO X cu h-'—' -o r— .r—

13 J 3

• ^-^

> í +J • r -to s-

• f—

c: ZD

j r u cu

1— to ra X 

'—' • o r— •r— 13

J 2

R

n I r tn

65

manubrium of the sternum (ft),

(5) Upper arm length (UL): The distance from the right acromion

to the inferior head of the humerus (ft),

(6) Lower arm length (LL): The distance between the tip of the

elbow (olecranon) and the cénter of the hand (ft).

tn

íi n I r tn

Figure 4.2. Upper Extremity Model

Table 4.1 shows the basic anthropometric data of the model and the

six subjects. Here, the model refers to the dissected cadavers.

All the above dimensions were necessary for the comparison of the

anthropometry of each subject. From these anthropometric data, scale

factors with respect to the cadaver were established.

66

Figure 4.2 shows the upper extremity as it was used in this in-

vestigation for the calculation and analysis of the necessary anthro-

pometric data. Here, all parts of the upper extremity were considered

as a single rigid body and inertia force and moment were neglected be-

cause only the static cases were considered in this study. Before the

experiments, all the basic data such as weight, height, etc, were

gathered according to the method described earlier, and all the subjects

were checked to see whether they were in good physical condition.

By using Dempster's (1955) anthropometric data analysis method of

body action, the weight and the location of the center of gravity of

the segments were calculated as follows:

UW = W X 0.02647

LW = W X 0.02147 3

UC = UL X 0.43569 ^

LC = LL X 0.5544 (4.1) f: n

where r S

UW is the upper arm weight íi

LW is the lower arm weight

UC is the distance of the center of gravity of the upper

arm from the proximal joint

LC is the distance of the center of gravity of the lower

arm from the proximal joint.

We can calculate the applied actual weights and moments due to the

weights of segments and the abduction or adduction weight by using geo-

metrical data and the postulated model as follows:

67

TW = UW + LW -H AW, for abduction,

TW = UW -I- LW - AW, for adduction, and

TM = (UC X UW) + (UL X LC) x (LW + (UL + LL) x AW, (4.2)

where

TW is the effective weight defined as the total actual

applied weight (AW and segment weight),

TM is the effective moment defined as the total moment due

to AW and the segment weight,

AW is the abducting (or adducting) weight, and segment re-

fers the whole upper extremity, i.e., the upper arm,

the lower arm and the hand. R

In the case of abduction through the angle "ø" (See Figure 3.1), force ^ ÍH

and moment as used in Equations (3.1) and (3.4) can be determined to fj I

be the effective weight and effective moment as described by: ^ CD

Fw = TW >

Mw = TM X sin(e) (4.3) ^

As mentioned earlier, for the living subject it is impossible to

get the exact geometrical data such as length, direction, and insertion

and origin points of the muscles. In order to get the approximate

geometrical data, an anthropometric similarity method was used in this

study by employing scale factors which were defined as follows:

- _ anthropometric dimensions of subject /* . Scaie ractor anthropometric dimensions of model ^^^^

68

where "anthropometric dimension of model" refers to the anthropometric

dimension of the cadaver that was measured through dissection.

The scale factors in X, Y and Z directions were calculated accor-

ding to the above definition as follows:

C;FY = BW of subject _ BW of subject (ft) ^^ BW of model 1.25 ft

qpY = 1 /CH of subject UL of subjectx ^ ^ ' ^CH of model UL of model '

= h /CH of subject (ft) . UL of subject (ft)x ^ 1.2525 ft 1.036 ft '

SFZ = SFX (4.5)

where

3 SFX is the scale factor in X direction g l SFY is the scale factor in Y direction tu

SFZ is the scale factor in Z direction. g X

The last equation was due to the fact that for most humans the cross r S

sections of the trunk are similar to one another. > ;Q

For a specific muscle, by using the above method, necessary ana- ^

tomical data of each subject were calculated as follows:

SDMX = MDMX X SFX

SDMY = MDMY x SFY

SMDZ = MDMZ X SFZ (4.6)

where

69

SMDMX, SMDY and SMDZ are the subject's geometrical dimensions

in the X, Y and Z direction, respectively, and MDMX, MDMY and

MDMZ are the model's geometrical dimensions in the X, Y and Z

direction, respectively.

By using these scale factors and the mathematical analysis of

Chapter III, the computer program to get all the necessary data for

the mathematical analysis of the individual subjects was developed,

as can be seen in Appendix IV (Computer Programming, Part I).

Using this method, the anthropometric data was formulated and

is shown in Table 4.2. With these data, the mathematical analysis

described in Chapter III was used to calculate all muscle tensions

for each subject.

tn Table 4.2. Anthropometric Basic Data of the Subjects H

Subjects I

Model 1

uw LW

UC

LC

SFX

SFY

SFZ

TW

TM

4.62

3.75

0.45

0.66

1.00

1.00

1.00

8.37

8.45

5.29

4.29

0.46

0.66

1.04

1.03

1.04

9.58

9.78

3.78

3.07

0.40

0.58

0.80

0.84

0.80

6.85

6.04

3.39

2.75

0.36

0.53

0.80

0.80

0.80

6.14

5.00

4.13

3.35

0.44

0.60

0.99

0.93

0.88

7.48

7.16

5.56

4.51

0.51

0.78

1.10

1.09

1.10

10.07

11.77

5.82

4.7

0.47

0.69

1.12

1.04

1.12

10.54

11.11

r 5

:

70

I

y

ntegrat

Noise Shelter

Electrode

J^Weight ^ / y

n

îi ing Preamplifier

Figure 4.3. Experimental Procedure

4.2. E.M.G. Experiment

One result of this experiment was to find out the magnitude of elec-

tromyographical signal intensities of some muscles (deltoid anterior,

deltoid middle and deltoid posterior) during abduction under the differ-

ent conditions. This experiment consisted of recording electromyographic

data during isometric contraction against the applied weight at the hand.

The basic idea of the experiment is illustrated in Figure 4.3,

which shows an idealized diagram of the upper extremity with a specific

applied weight at the hand. Two abduction positions are displayed in

the figure.

Each electromyographic recording was made with three surface elec-

trodes, one for each of the parts of the deltoid muscle. These elec-

trodes were attached on the skin directly over each of the three parts

3 tn

n I r S

.

71

of the deltoid. The locations of the electrodes were as follows:

(1) Anterior Part: 2-inches below the lateral end of the clavicle,

(2) Middle Part: 2-inches below the lateral border of the acromion,

(3) Posterior Part: 3-inches below the spine of the scapula.

Miniature electrodes, llmm in diameter (Beckman No. 650437) were

chosen because they were suitable for minimizing the interference effects

from the muscles and because, due to their small size, the exact posi-

tion of each could be determined with relative ease.

Just before each electrode was positioned, the appropriate area of

the skin was rubbed with alcohol and was covered with electrode jelly

to reduce the inter-electrode resistance and skin electrical resistance.

During the recording, the action potential which is generated from ^

the skeletal muscle cells when they are in a physiologically active J tn

state, was detected by electrodes through the skin. The magnitude of ^

the action potential varied from 25 yV to 500 yV according to their ' ^

state of activation. But, due to the fact that the electrode output r S

was an alternating voltage signal it was necessary to integrate the ab- > ÎQ

solute value of the signal and to obtain the action potential as the ^

slope of the integrated curve.

For the integration of the electrical signal from the electrodes

on the muscles, an integrating preamplifier, a Sanborn Model 1035,

(operating in the area mode) was used to perform the operation electroni-

cally, as shown in Figure 4.4. The output of the recording represented

the integrated voltage during a specific time interval. The output was

in units of volt-seconds. During the integrating process, as can be

72

Â H ii''B|Í|i«<tiiiiPi imM E.M.G. Signal

Threshold Level

Area Mode Signal

t=0 n=area of E.M.G. signal since t=0

Figure 4.4. Summation Area of E.M.G. Signal

seen in Figure 4.4, the threshold triggering circuit returned the inte-

grated signal to a zero level each time it reached the full scale of

maximum integration. Consequently, the total integrated value calcu-

lated since the beginning of the area summation process could be found

by counting the number of cycles and multiplying by the value of the

maximum displacement height of the integrator.

In this study, only the static cases of abduction (0 to 90 degrees

bylO degree intervals) were investigated. That is, the arm and applied

weight were held fixed at one position while data was being taken.

Therefore, the slope of the integrated voltage curve could be inter-

preted as the intensity of the recorded electrical signal at that pos-

ture. Or, the relative intensity (E) of the electrical signal for each

muscle could be represented in the following form.

R

î tn

íí n I r S

PQ

73

E = f (4-7)

where V is the integrated action potential curve and t is time.

The resulting relative electromyographic potential intensities of

six subjects under various conditions are shown in Appendix II, Tables

2.1 to 2.6.

Because of the low voltage of the generated electrical signal of

the muscle, it was necessary to avoid all the electrical noise effects

from lights, surrounding equipment, etc. This was achieved by doing

the experiments inside of the specially designed electrical noise shel-

ter at Texas Tech University. R

Each of the subjects participating in the experiment was instruc- tn ted and trained thoroughly with regard to his duties in the experiment. Ui

H This was done in order to familiarize each subject with the equipment h

and with the motions he would be required to do. Following the train-r

ing phase, the electromyographic records for each of the subjects under ^ >

various abduction angles for different weight were taken. Each subject ^

was asked to assume an erect posture with his feet together. The sub-

ject was asked to maintain a specific posture for five or six minutes

in order to get enough data. After each experiment, the subject was

given a five minute rest period. He was then instructed to assume an-

other posture or to use another weight. A total of ten experiments were

done for six subjects under all the abduction angles considered in this

experiment. Three weights (0, 5 and 10 Ibs) were tested for every level.

' l ^

The relationship between the muscular tension as measured in the

experiments and muscular tension as determined by theoretical methods

is discussed in more detail in the next chapter.

74

R

î tn

si n

CD

>

î<

CHAPTER V

RESULTS OF THEORETICAL ANALYSIS

The steps taken in this investigation, which were presented in the

preceding chapters, can be summarized as follows:

Step (1): Biomechanical analysis of the musculo-skeletal system

of the human shoulder muscle.

Step (2): Collection of the geometrical data which are necessary

for the vector analysis by dissection of the cadaver.

Step (3): Mathematical description of the static equilibrium

equations of the forces and moments for the shoulder

joint model by using the vector method.

Step (4): Solution of the indeterminate problems by using the •

minimal principle technique. ^ ^ tn

Step (5 ) : Formulation of the experiments that would permit the ín

ver i f icat ion of the application of the minimal pr inc i - !

ple to the living human body. C CD

Step (6): Experimental procedures and collection of data to ob- ^

tain muscle force distribution and electromyographic ;<

signal intensity on the three parts of the deltoid.

In order to complete an experimental verification of the mathe-

matical model for the muscles crossing the gleno-humeral joint, the

remaining step was the characterization of the unknown functions of

the model by using the collected data, as shown in the following

equations:

75

76

î " ^2 ( •' ^•' W' B.,A,e) (5.1)

Ti = f^ (E.) (5.2)

where the symbols are the same as those used and defined in Chapter II.

The first equation represents the results of the theoretical solutions

of the model and the six subjects as described in Chapter III and Chap-

ter IV. It represents the theoretical relationship between the muscu-

lar tension and the following: the geometrical, anthropometrical and

physiological characteristics, the applied weight, and the abduction

(adduction) angle. The theoretical solutions for the subjects were

obtained by using cadaver data and scale factors as described in Chap-

ter IV. The diagram of the theoretical muscular tension vs. abduction

angle for the three deltoid parts for each subject at various condi-

tions are presented in Figures 5.1 to 5.6. The second equation repre-

sents the relationship between the relative electromyographic potential

intensity and the exerted muscular tension as determined by experimen-

tal data.

a tn

hi n I r s >

5.1. Functional Equations for the Muscular Tension

The first relationship of concern here is the relation of the mus-

cular tension to the abduction angle of each subject under various con-

ditions, as shown in Figures 5.1 to 5.6. The relationship can be repre-

sented symbolically in the following form:

T. = fg (9) (5.3)

77

to

100-

o •r-to C <u

s -<TJ

u iq50

0 ' ^ 1

10

... ,

20 r 1 1

30 40 50

Abduction Angle

1

60 1

70 1 1

80 90

(degrees)

tn

r CD

Figure 5.1. Muscular Tension Diagram - Subject (1)

78

to JO

100 .

c o to c cu I—

s -fO

3 50 to Z3

L. O

D

: Anterior

Middle

Posterior

0

5

IC

Ibs

Ibs

) Ibs -o

-o y

.O

/

/O /

/

. - 0 ' vy /

/ A

/ ^ -

/ ^ / 1 f / / ^ i

/ ^ / ^ ^ / ' /jåt^

/ / ^ , , , ^ ^ ^ ^ Ty^— _A———"•'^

, , ,_.., — ^ , -T ' • • • !

0 10 20 30 40 50 60 70 80 90

Abduction Angle (degrees)

ri

tn

si n X r 5

'Xl

Fiqure 5.2. Muscular Tension Diagram - Subject (2)

79

to X3

100 -

c: o

.r— to c cu

S -fO

u to

50 -

3 > in

íi n

r E

r<



80

J3

100-

to

cu

S-<T3

:3 u to 3

50-

: Anterior

: Middle

Posterior

0 Ibs

5 Ibs

10 Ibs

Abduction Angle

70 80 90

(degrees)

î in

n

CD

PQ


81

to J3

100-

e o •r-tO E <D

fC

u to

50 -

A

O

D

Anterior

Middle

Posterior

0 Ibs

5 Ibs

10 Ibs

Abduction Angle

— T í

80 90

(degrees)

3 tn

U m n r fi > ÎQ


82

tn

100 -

c o to c: cu

s-

u to 3

50 -

: Anterior

Middle

01

I r æ ;Q >


Fiqure 5.6. Muscular Tension Diagram - Subject (6)

î îrf-- -4*.-: -aMMai SaaiiMBSI

83

In this study, it was assumed that fourth order polynominals of

the form

T = a, {^) . a i^)' . a3 (±)' . a, (^)' (5.4)

could represent this relationship. Only the fourth order polynominal

equation coefficients a^, a , a^, and a- were considered because, ac-

cording to the results of polynominal regression methods, it was deter-

mined that the effect of fifth or higher polynominal terms could be

neglected.

The purpose of the polynominal equation was to simplify and to

generalize the calculating porcess. The coefficients are different

for every person and every muscle at various external conditions. After

the coefficients have been found for each case, the muscular tensions ÍT:

can be calculated by use of Equation (5.4) directly without the comli- iji H

cated calculation of all the anthropometrical and geometrical data h

•L that are necessary for the theoretical solution. But, this process !»

is not simple because, in order to generalize the procedure of calcu- ^ >

lation of coefficients, we must first choose coefficients for the "^

cadaver and then find the relãtionships between these coefficients and

the coefficients for each subject.

Detailed procedures of choosing the coefficients for the cadaver

and developing the relationships between the coefficients are explained

below.

The next step was the calculation of the coefficients of each

curve for each subject by using the Least Square curve fitting methods.

The resulting coefficient values of each subject at the different weight

84

and abduction angle are tabulated in Appendix III, Tables 3.1 to 3.21

For example, for the dissected cadaver, the following nineteen

general equations for the case of 0 Ibs of lifting were developed ac-

cording to the above curve fitting method. In order to find the gen-

eralized coefficients for the various cases of each subject, the co-

efficients for the various cases of each subject, the coefficients of

the model for 0 Ibs of abduction were used as standard values.

General Equation Form (Standard Coefficients)

Y: Muscular Tension (Ibs)

X: Abduction (Adduction) Angle/10

A. Abduction Case

2. Deltoid Middle

Y = 11.20X - 2.620X^ + 0.3730X^ - 0.01880X

1. Deltoid Anterior *5

Y = 3.35X - 0.658X^ + O.OB^^X"^ - 0.00262X^ il

'5 3. Deltoid Posterior ;Q

Y = -2.32X + 3.550X^ - 0.7110X^ + 0.04320X^ ^ ts

4. Supraspinatus

Y = 19.00X - 8.880X^ + 2.0200X^ - 0.12500X^

5. Infraspinatus

Y = 19.60X - 4.860X^ + 0.9250X^ - 0.05560X^

6. Teres Major

Y = 3.52X - I.IBOX^ + 0.2690X^ - 0.01590X^

7. Teres Minor

Y = 3.16X + 1.030X^ - O.IOBOX^ + 0.00334X^

85

8. Subscapularis

Y = 31.50X - 12.600X^ + 1.9800X^ - O.IOOOOX^

B. Adduction Case

1. Infrespinatus

Y = 5.75X - 0.886X^ + 0.1250X^ - 0.00410X^

2. Teres Major

Y = 1.20X - 0.163X^ + 0.0161X"^ + 0.00053X^

3. Teres Minor

Y = 1.20X + 1.090X^ - 0.3410X^ + 0.02650X^

4. Subscapularis

Y = 16.70X - 7.050X^ + l.OÔOX"^ - .^ÔOOx"^

5. Pectoralis Major-Sterno

Y = 0.65X + 0.296X^ - 0.0771X^ + 0.00664X^ '3 :<

6. Pectoralis Major-Clavicular > 'íî

Y = 0.97X + 0.085X^ - 0.0129X^ + 0.00032X^ H n

7. Biceps Long 1 Y = 4.28X - 1.320X^ + 0.2600X^ - 0.01610X^ ig

Q > tl

Y = 5.77X - 0.906X^ + 0.0777X^ + O.OOOIOX^

8. Biceps Short

9. Triceps

Y = 2.17X - 0.150X^ - 0.0195X^ + 0.00359X^

10. Coracobrachialis

Y = 4.22X + O.OOSX^ - 0.0195X^ + 0.00359X^

11. Latissimus Dorsi

Y = 1.41X - 0.389X^ + 0.0578X^ - 0.00203X^

ESStVXi-i)^'?!"

86

In the above equations, all the independent variables, such as

L, D, W. B, A of Equation (5.1) were already considered before, for

these variables had been used for the theoretical solution of Chapter

III.

5.2. Simplified Functional Equations for the Muscular Tension

The theoretical procedure described above was fairly complicated

and required a number of external anatomical measurements for each

living subject. A somewhat simpler method, based on the procedure

above, was derived by making some simplifying assumptions. This

method is described below.

From the mathematical viewpoint, the muscular tension (T) is as-

sociated with the effective moment and effective forces that were de-''HÍ

fined by Equation (4.3). However, in the final matrix of the theore- < ,i>

tical solution of Chapter III, the magnitude of the effective force '

is negligible compared to the magnitude of the effective moment. i • . í _

Furthermore, most of the effective forces are absored by the reaction ,-

forces at the joint while the effective moments are fully effective. o

According to the definition of the moment, -<

T = M/L (5.5)

where

T is muscular tension

M is moment, and

L is the length of the moment arm,

Also, for the model and subject:

87

Tmodel = Mmodel/Lmodel

Tsubject = Msubject/Lsubject

(5.6)

(5.7)

Dividing (5.6) by (5.7), we obtain

Tmodel ^ Mmodel/Lmodel ^ Lsubject/Lmodel (5.8) Tsubject Msubject/Lsubject Msubject/Mmodel

However, as can be seen in Table 4.2, the effective moment of the model

for the case of 0 Ibs abducting is

Mmodel =8.45 (5.9)

so.

Tmodel _ Lsubject/Lmodel Tsubject Msubject/8.45

(5.10)

Here, according to the definition of the scale factor in Chapter IV,

Lsubject/Lmodel was considered as the average of the scale factor, so

:rj

..<

IJl

4

Lsubject ^ (SFX + SFY + SFZ) ^ ^^^ Lmodel 3

Define the moment ratio (MR) as follows

(5.11) B

9 • : ; <

MR = Msubject/Mmodel = Msubject/8.45 (5.12)

Then, from Equations (5.8), (5.9), (5.11), and (5.12)

Tmodel Tsubject

Sav MR

Let us define multification factor (MUL) as follows:

* < « * - t*'i-:», v - - ' T ' - 3 X : -' ••••

88

MUL = MR/sav

Then,

Tsubject = Tmodel x MUL (5.13)

Therefore, in Equation (5.4) the coefficients (a. s) for the dif-

ferent subjects can be calculated as follows

a.(subject) = a.(model) x MUL(i=l,2,3,4). (5.14)

Following to the process of calculating the effective moment, the

scale factors, and the multification factor (which was defined as the

ratio of the coefficients between subject and model), the following

procedures were used for all subjects: n

1. Collection of the anthropometric data:

(a) Height (H) i

(b) Weight (W)

(c) Biacromial width (BW) D

(d) Chest height (CH) :<

(e) Upper arm length (UL)

(f) Lower arm length (LL).

2. Calculation of the effective moment data:

(a) Upper arm center of gravity (LC)

(UC) = 0.53469 x (UL)

(b) Lower arm center of gravity (UC)

(LC) = 0.55440 X (LL)

89

(c) Upper arm weight (UW)

(UW) = 0.02647 X (W)

(d) Lower arm weight (LW)

(LW) = 0.02147 X (W)

(e) Applied weight distance (AD)

(AD) = (UL) + (LL)

(f) Lower arm effective distance (LAD)

(LAD) = (UL) + (LC)

(g) Applied weight (AW).

Calculation of effective moment (M):

(M) = (UC) X (UW) + (LAD) X (LW) + (AD) x (AW)

Calculation of scale factors:

(a) Scale factor in X-direction (SFX)

(SFX) = (BW)/1.25

(b) Scale factor in Y-direction (SFY)

(SFY) = iá((CH)/1.2525 + (UL)/1.036)

(c) Scale factor in Z-direction (SFZ)

(SFZ) = (BW)/1.25

(d) Average scale factor (Sav)

(Sav) = (SFX + SFY + SFZ)/3.0.

Calculation of moment ratio (MR):

(MR) = (M)/8.45.

Final calculation of the multiplication factor (MUL):

(MUL) = (MR)/(Sav).

'n .n H

n

:<

• f ^ " v - ^ r - " j ' • ' iiÂttåJiÉÊMXãSíÊaMÊA •

90

By using this method, we could predict the coefficients of the

relationship between the muscular tension and the abduction ( or ad-

duction) angle of different subjects under the different conditions

of applied weight.

In order to examine the validity of this method, the error per-

centage between the results of the curve fitting values and the re-

sults of this method were calculated and tabulated in Table 5.1.

From the table it can be seen that this simplified method will pro-

vide results which are almost the same as those of the more detailed

and difficult procedure described in Section 5.1. The methods differ

slightly because the simplified method neglects the force effects of

the external load and uses averaged scale factors.

'•ni ••c'

ÍU

r<

é,síavK4Â^^V> •

91

o

^

O O O O ' Í - O O C M O O ' ^ C O O J ^ ' ^ ^ O O

O O O O O O O O U 3 0 0 0 C \ J O I O C Z >

m co CJ

CO

ZJ • M

u < :

O o

CM ^

CVJ

0 0 o KO CTk

m o ro t n

^ r>. r^ o o co

cr» o ro

O r-. i n CM U3

O O

cr» co f — «;*-

0 0

<M CO I— co C\J c o CM CO CVJ f O r— CVJ cvj ro

O O

CM ^ co t o

o oo o CT>

O CO

CSJ vo o

CT» CO cn C\i

co crv vo r— «o c\j «o o

KO o «:1-CM

I— c\j ro t— C V I C O r — C V J C O O C V J C O O C M C O O C V J C O CM

to

3 to

cu cm

c o

•r—

+-> fC 3 cr

LxJ

XJ <D N

•r— 1 —

fO S-(U c

C3

• '—

t n

cu

J D fCJ

h-

i r > ^

• co

^ *

21

>

t o

ivl U -l / >

> -

o o r—

L O ^

oo

.00

f —

o o

• ^_

o o

CM CO

CM

cn l O

CTk

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C O

co r«.

o co

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r—

CT> r^ CT>

.04

1—

^ o

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co o

CT> ^

CM

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, _ CM

CM 00

co

CT> CM

CM CO

CO r^

o

^ o <o

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o

o 0 0

• o

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r>. 0 0

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0 0 r^ Lf) r—

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co

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cr» uo

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oo CT>

^

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o

o 0 0

• o

o co

^ vo r—

^ CT>

CO f —

f —

r>. CM

cr» 0 0

CM CM

L O 0 0

o

«o • —

r^

.90

o

oo 0 0

• o

co CT>

oo o CM

r^ LO

r^ 1 —

r— CO

co

cr» CT>

r>. CM

CT> co r—

r-. r^ ,— r—

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'

o '—

* '~~

cr> o

r^ oo CM

0 0 LT)

' á-CM

^ ^

«^

0 0 ^

r^ CO

CM

co t~

,— • "

,— r-~

.09

'

CM

* " *

r~

o •

cr> vo CM

LO

r^ CM CM

CM

r^

co

r—

^ r—

co

y

j> -0

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:D

t o i—

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o o 00

o 00

oo 00

CM

O I O O O L O O O L O O O í O O O l O O O L O O O L O O

<u - o o

CM CO lO LO

CHAPTER VI

EXPERIMENTAL VERIFICATION OF THE THEORETICAL ANALYSIS

All of the experimental work described in Chapter IV and V which

involved external anthropometric measurements have dealt with the prob-

lem of obtaining measured data to serve as input to the theoretical

force distribution analysis. With these data and the associated assump-

tions regarding scaling, together with the theoretical model, force ver-

sus adbuction angle (e) relations were described for each part of the

deltoid for each external weight and for each subject. The electromyo-

graph experiments were for the purpose of verifying the theoretical

results for the three parts of the deltoid muscle. The details of the

verification are described below.

On the basis of the well established fact that there is a linear

relationship between the generated electromyographic potential inten-:-\

sity and the exerted muscular tension of the muscle (Basmajian 1967; ^ :c

:D

'< » > •

:n

:<

Inman, et al 1952; Bigland and Lippold 1954), Equation (5.2) can be

written in the following form: ;Q

T = cE (6.1)

The linear coefficient "c" was to be determined by experiment.

On the basis of the above fact, it was determined that a good pre-

dictor of the magnitude of the muscular tension in each muscle would be

what was recorded as the intensity of the action potential of the elec-

tromyogram. However, for most of the muscles in the human body, it was

found that such a recording was almost impossible because of the inter-

ference of the muscles with one another during the recording, and be-

92

••^« ^, >*..

93

cause of the difficulty of recording responses from the inner muscles.

The idea of choosing the three parts of deltoid came from the fact that,

for these muscles, the recording could be done more easily without signi-

ficant interference.

The necessary data for this characterization are the experimentally

recorded electromyographical signal intensities which were defined by

Equation (4.7). In order to find the slope of the integrated curve and

because of the small mesh size of the electromyogram, a magnifying glass

was used to read accurate values of the integrated voltage curves.

The display of the data collected from the static electromyographic

recording experiments and the solution of the theoretical vector solu-

tion plots, as can be seen in Figures 6.1 to 6.6, and in Appendix II,

Tables 2.1 to 2.6 were the basis of the validity of the application of

the minimal principle to the human living body. Each of the figures is

for one of the subjects. Each of the data points in each figure was

obtained as follows: The subject assumed one posture (on abduction

angle) with one external weight and the corresponding electromyographic

intensity was determined at one of the three deltoid positions. This

electromyographic intensity was the abscissa of the plotted point. The

ordinate was obtained from the theoretical model and was the muscular

tension for that particular abduction angle, weight, and deltoid part.

In all, there are 81 points plotted on each curve: nine abduction posi-

tions, three weights at each position, and at each of the three deltoid

parts. The data points showed a remarkably linear relation between theore-

tical and experimental results and straight lines were fitted to the data

using the Least Square method.

 •íi

n c

D •0 i>

•íl <

£»2S5^:-ft..:'-^- :.-^'*-- --»"

94

1/1 JD

100-

c: o (/) c (D

S-03

O to zs

50 -

1 70 80

:c

D

a D <

E.M.G. Potential Intensity (arbitrary)

Fiqure 6.1. Muscular Tension vs. E.M.G. Intensity - Subject (1)

•'^Vr-!-----F-!7^"^-"

95

r 50 0 10 20


1 r 60 70 80

:

<

1

D tl • >

<

Figure 6.2. Muscular Tension vs. E.M.G. Intensity - Subject (2)

96

JD

100-

o •r— to c cu

h-s -03

O to zs

50 -

0

s • j>

3

<



SiíMk^X- •<•-•'•*'<-••, •'«

97

<

•'j

.1

<


Figure 6.4 Muscular Tension vs. E.M.G. Intensity - Subject (4)

p í . * s B f t , ^ - ' •'•• - T~'-"

98

to JZi

100-

to c

s-fO

=3

o Z2

-

50-

0 10 20 30


1 <

1 í l


99

T 1 1 1 1 1 r 0 10 20 30 40 50 60 70 80



100

As can be seen in the figures and the statistical results of the

linear curve fitting of Table 6.2, the relationship between the theore-

tical solution, calculated according to the minimal principle, and the

experimental results, obtained from the electromyographic experiments

on living subjects, provided the basis of the validity for the appli-

cation of the minimal principle to the living human body. This, of

course, is because of the fine linear curve fitting between these values

and the negligible deviation of each subject case. Small deviations

were expected because of the assumption involving the scale factors

that were used in the theoretical solutions and because of experimental

inaccuracies in the measurements and data.

The linear coefficients of the lines correlating the theoretical

solution and the experimental results (the slopes of the lines), which

were calculated by using the Least Square curve fitting method, are

tabulated in Table 6.1

Table 6.1. Linear Coefficient Values

Subject Linear Coefficient

1 2.0133

2 2.2077

3 2.8851

4 2.444

5 1.758

6 1.9693

The difference in the coefficients for each subject was due to the dif-

ferent physical conditions of the subjects.

1 3

/ • \

101

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1

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CHAPTER VII

SUMMARY, CONCLUSION AND RECOMMENDATION

This chapter discusses several aspects of the theoretical and ex-

perimental procedures, and the significance of the results which were

found in this research. Based on the results of this study, some spec-

ulation is made about the mathematical approach to human musculo-

skeletal problems. Also, it is indicated how other similar investiga-

tions involving complicated and indeterminate problems could be solved

by this technique.

7.1 Summary

The purpose of this study has been to formulate a mathematical

model capable of predicting muscular tension characteristics for mus-

cles in the human shoulder joint. This was done by using the data

that were collected through dissection of a cadaver and through physio-

logical information about human skeletal muscles and anatomical char-

acteristics of the shoulder joint. By using this model, the explicit

characterization of the mathematical equations for the postulated

mechanism of the shoulder joint was described in terms of a three di-

mensional coordinate system. The mathematical equations for the rela-

tionships between the electrical signal intensities that are generated

from the muscles, and muscular tensions that are exerted by muscles

at various postures during abduction of the upper extremity were in-

vestigated.

General equations that can be applied to various individual per-

sons who have different anthropometric dimensions were developed by

102

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103

using scale factors. Computer programs were developed to determine the

muscular tension of muscles in the shoulder joint of various persons

and to predict the linear coefficients between electromyographic

electrical signal intensities and the muscular tensions of the

skeletal muscles. These were developed from the results of the theore-

tical and experimental procedures of this study. According to the re-

sults and the techniques of this study, it was determined that most of

the complicated human musculo-skeletal systems can be analyzed mathe-

matically without dissecting bodies.

7.2. Conclusion

The conclusions which can be drawn from this investigation with

regard to the postulated model, the theoretical and experimental pro-

cedures, and the verification experiments, are tablulated below.

These conclusions are:

(1) The human shoulder joint mechanism can be represented by

a mathematical vector model. The geometrical input data

for the model can be obtained by dissection of cadavers.

The model provides muscle force distribution in the various

muscles crossing the gleno-humeral joint at various static

abduction and adduction angles of the arm.

(2) Input data for the application of the model to living people

can be obtained by external physical measurements and scale

factors.

(3) The Minimal Principle used in the mathematical model is valid,

as verified by electromyographic experiments.

fllll t r . . . . . . . •• ^ -

104

7.3. Recommendation

There is a continuing need for generalized mathematical models to

analyze human motion characteristics. This investigation was success-

ful in contributing to fulfilling the need by exhibiting a highly ac-

curate prediction of the distribution of muscular tension in the shoulder

mechanism for abduction to a statically held position. In addition,

this investigation points the way to new efforts for the fuller devel-

opment of mathematical models for human musculo-skeletal system analysis.

The recommendations which should be considered in further researches in-

clude the following:

(1) The range of possible movement should be extended past the

range of 0-90 degrees abduction and adduction.

(2) This technique should be applied to the combination of ab-

duction, adduction, rotation, flexion, and extension of the

upper arm.

(3) The work should be extended to all of the human musculo-

skeletal system. j

(4) The work should be extended to include dynamic analysis of i

joint movements. To do this, it will be necessary to know

the dynamic characteristics of body segments.

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APPENDIX

Appendix (I)

Anatomical Basis Data Tables for Muscles

Appendix (II)

Theoretical and Experimental Results

- Deltoid Three Parts -

Appendix (III)

Coefficients of Theoretical Solution of

Muscular Tension Tables

Appendix (IV)

Documentation of Computer Program

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APPFNDIX ( I I I ) TABLE 3 - 1 . COEFFICIENTS OF THEORETICAL S n L U r i O N OF M U S C U L A R T E N S I O M

SUBJECT N O . : 0

N^ME OF MUSCLE

ABDUCTION CASE

DELTGID ANTERIOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUDRASPINATUS

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

ADDUCTI3N CASE

INFRASPINATUS

TERES MAJOR

TERES MINOR

SURSCAPULARIS

PECTORALIS M A J . ( S )

PECTORALIS M A J ^ ( C )

BICEPS(LONG)

RICEPS (SHORT )

TRICFPS

CORACOBRACHIAL IS

LATI SSIMUS ORSI

A l

3 . 3 5

1 1 .20

- 2 . 3 2

1 9 ^ 0 0

1 9 . 6 0

3 . 5 2

3. 16

31 . 5 0

ACTING i«/EIGHT

A2 A3

- 0 . 6 5 8

- 2 . 6 2 0

3 . 5 5 0

- a . B B O

- A . 860

- 1 . 1 3 0

1.C30

• 1 2 . 6 0 0

0 . 0 8 2 3

0 . 3 7 3 0

• 0 . 7 1 1 0

2 . 0 2 0 0

0 . 9 2 5 0

0 . 2 6 9 0

•0 .1C8C

1 . 9 8 0 0

0 LBS

A^

• 0 . C 0 2 6 2

• 0 . 0 1 8 8 0

0 . 0 4 3 2 0

• 0 . 12500

• 0 . C5560

• 0 . 0 1 5 9 0

0 . C 0 3 3 4

•O^ lOCOO

5 . 7 5

1 . 2 0

1 . 2 0

1 6 . 7 0

C.65

0 . 9 7

4 . 2 8

5 . 7 7

2 . 1 7

4 . 2 2

1^41

- 0 . 8 3 6

- 0 . 163

1 .090

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0 . 2 9 6

C.085

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- 0 . 9 0 6

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C .008

- 0 . 3 8 9

0 . 1 2 5 0

0 . 0 1 6 1

- 0 . 3 4 1 0

1 .02C0

- 0 . 0 7 71

- 0 . 0 1 2 9

0 . 2 6 0 0

O.C777

- 0 . 0 1 9 5

- 0 . 1 1 4 0

0 . 0 5 7 8

- 0 . 0 0 4 1 0

0 .C0C53

0 . 0 2 6 5 0

- 4 . 6 4 C 0 0

0 . 0 0 6 6 4

0 .C0C32

- 0 . 0 1 6 1 0

C.CCCIO

0 . 0 0 3 5 9

C .C1250

- 0 . 0 0 2 0 3

133

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134

APPENDIX (III) TABLE 3- 2. COEFFICIENTS OF THEORETICAL SOLUTION OF MUSCULAR TENSIO^J

SUBJECT NO. : 0

MAME OF MUSCLE

ABDUCTION CASE

DELTOID ANTERÎOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUPRASPINATUS

INFRASPINATUS

TERES MAJOR

TERES MiNiOR

SUBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

PECTORALIS MAJ. (S)

PECTORAL IS MAJ . ( C)

B I C E P S ( L O N G )

RICEPS(SHORT )

TRICEPS

CORACOBRACHIALIS

LATI SSIMLS DORSI

A l

7 . 0 8

2 5 . 9 0

- 5 . 8 3

4 4 . 1 0

4 5 . 5 0

8 . 1 6

7 . 3 4

1 3 . 3 0

2 . 7 7

2 . 7 7

3 8 , 6 0

1 . 5 1

2 . 2 4

9 . 9 3

1 3 ^ 4 0

5 . 0 4

9 . 7 9

3 . 2 7

ACTING WEIGHT

A2 A3

- 1 . 5 2 0

- 6 . 0 7 0

8 . 2 4 0

• 2 0 . 6 0 0

1 1 . 3 C 0

- 2 . 7 4 0

2 . 3 9 0

7 3 . 0 0 - 2 9 . 3 0 0

- 2 . 0 6 0

- C . 379

2 . 5 2 0

• 1 6 . 4 0 0

0 . 6 8 4

0 . 199

- 3 . 0 7 0

- 2 . 100

- C . 3 5 0

0 . 0 1 9

- 0 . 9 3 4

0 . 1910

0 . 8 6 5 0

- 1 . 6 5 C 0

4 . 6 9 C 0

2 . 1 5 0 0

0 . 6 2 3 0

- 0 . 2 5 1 0

4 . 5 9 C 0

0 . 2 9 2 0

0 . 0 3 7 4

- 0 . 7 9 1 0

2 . 3 7 0 0

- 0 . 1 7 B 0

- 0 . 0 2 S 9

0 . 6 0 3 0

0 . 1 7 9 C

- 0 . 0 4 4 9

- 0 . 2 6 5 0

0 . 1 3 4 0

5 LRS

A4

- 0 . C 0 6 0 5

- 0 . 0 4 3 6 0

0 . lOCOO

• 0 . 2 9 C 0 0

- C . 1 2 9 0 0

• 0 . 0 3 6 7 0

O.C0773

• 0 . 2 3 300

• 0 . 0 0 9 5 7

0 . C 0 1 2 3

0 . 0 6 1 6 0

• c . i o e o o

0 . 0 1 5 4 0

0 . C 0 0 7 4

• 0 . 0 3 7 3 0

0 . C 0 0 2 7

0 . 0 0 8 3 1

C . 0 2 8 9 0

• 0 . 0 0 4 7 2

tÍitflfeÍIÉKiílMI îfSî Bâaaisa

APPENDIX ( I I I ) TABLE 3 - 3 . COEFFICIENTS OF THFORETICAL SOLUTION OF MUSCULAR TENSIONI

SUBJECT N O . : 0

MAME OF MUSCLE Al

ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MIDDLE

DELTOIl) POSTERIDR

SUPRASPINATUS

INFRASPINATUS

TERES MAJOR

TERES MÎNOR

SUBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS

TERES MAJOR

TERCS MINOR

SJRSCAPULARI S

PECTGRALIS MAJ.(S)

PECTORAL ÍS MAJ. ( C)

BICEPS(LONG)

BICEf^S (SHORT )

TRICEPS

CORACDBRACHI AL IS

LAT ISS IMUS DORSI

1 1. 10

4 0 . 7 0

- 8 . 44

1 2 . 8 0

1 1 . 50

1 1 5 . 0 0

2 0 . 9 0

4 . 3 5

4 . 3 5

2 . 3 8

3 . 52

1 5 . 6 0

2 1 . 0 0

7 . 8 9

1 5 . 4 0

5 . 14

ACTING WEIGHT

A2 A3

- 2 . 3 9 0

- 9 . 5 1 0

6 9 . 1 0 - 3 2 . 3 0 0

7 1 . 3 0 - 1 7 . 7 0 0

- 4 . 3 1 0

3 . 7 8 0

- 4 6 . 0 0 0

- 3 . 2 3 0

- 0 . 5 9 4

3 . 9 6 0

6 0 . 6 0 - 2 5 . 7 0 0

1 .0 7 0

• 4 . 8 2 0

- 3 . 300

0 . 2 9 8 0

1 . 3 6 0 0

1 2 . 9 0 0 - 2 . 5 9 0 0

7 . 3 5 C 0

3 . 3 6 0 0

0 ^ 9 7 9 0

0 . 3 9 8 0

7 . 2 0 C 0

0 . 4 5 8 0

0 . 0 5 8 7

- 1 . 2 4 C 0

3 . 7 1 C 0

- O . 2 8 C 0

C . 3 1 0 - 0 . 0 4 6 8

0 . 9 4 7C

0 . 2 8 3 0

- 0 . 5 4 3 - 0 . 0 7 1 4

C .C24 - 0 . 4 1 5 0

10 LBS

A4

- 0 . 0 0 9 4 7

- C . 0 6 8 5 0

0 . 15700

• 0 . 4 5 6 0 0

• 0 . 2 0 2 0 0

• 0 . 0 5 1 7 0

0 . 0 1 2 4 0

• 0 . 36600

- 1 . 4 2 0 0 . 2 1 1 0

• 0 . 0 1 5 0 0

0 . C 0 1 9 2

0 . 0 9 6 6 0

- 0 . 169C0

0 . 0 2 4 1 0

0 . C 0 1 1 5

- 0 . 0 5 8 6 0

0 .CC036

0 . 0 1 3 1 0

C. C4 530

- 0 . 0 0 7 4 2

135

nrî i - ~

APPGNDIX (III) TABLE 3- 4. COEFFICIENTS OF THEGRETICAL SOLUTION OF MUSCULAR TENSION

136

SUBJECT NO.: 1 ACTING WCIGHT: 0 LBS

NAME OF MUSCLE Al A2 A3 A4

ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MÎDDLE

DELTOID POSTERIOR

SUPRASPINATUS

INFRASPINATUS

TERES MAJOR

TF'ÊS MINOR

SUBSCAPULARIS

3 . 4 1

1 2 . 5 0

- 2 . 5 7

2 1 . 3 0

2 1 . 9 0

3 . 9 5

3 . 5 5

3 5 . 0 0

- 0 . 7 3 4

- 2 . 9 2 0

3 . 9 6 0

- 9 . 9 9 0

- 5 . 3 9 0

- 1 . 3 3 0

1 . 1 3 0

- 1 4 . 100

0 . 0 9 1 8

0 . 4 1 8 0

- 0 . 7 9 4 0

2 . 2 7 C C

1 . 0 3 0 0

3.C2CC

- 0 ^ 1 1 5 0

^ • 2 2 C 0

- 0 . 0 0 2 9 2

- 0 . 0 2 1 1 0

0 . 0 4 8 3 0

- 0 . 14100

- 0 . 0 6 2 1 0

- C . 0 1 7 8 0

0 . 0 0 3 3 8

- 0 . 0 1 120

ADDUCTION CASE

INFRASPINATUS

TERPS MAJOR

TERES MINOR

SUBSCAPULARI S

PECTCRALIS MAJ.(S)

PECTORALIS MAJ.(C)

RICEPS(LONG)

BICFPS(SHORT)

TRÎCFPS

COR^COBRACHIALÎS

LATISSIMUS DO^SI

6 . 3 7

U 3 4

1 . 3 3

1 8 . 5 0

0 . 7 3

1 . 0 8

4 . 7 7

6 . 4 3

2 . 4 2

4 . 7 0

1 . 5 8

- 0 . 9 6 8

- 0 . 1 8 4

1 . 2 1 0

- 7 . 8 4 0

C . 3 2 8

0 . 0 9 9

- 1 . 4 7 0

- 1 . 0 1 0

- C . 166

0 . 0 1 7

- C . 4 4 0

0 . 1 3 8 C

0 . 0 1 8 4

- 0 . 3 7 9 C

1 . 1 3 0 0

- 0 . 0 8 5 6

- 0 . 0 1 5 2

0 . 2 9 C 0

0 . 0 8 7 8

- 0 . 0 2 1 7

- 0 ^ 1 1 4 0

0 . 0 6 5 5

- 0 . 0 0 ^ 5 3

0 . 0 0 0 5 7

0 . 0 2 9 6 0

- 5 . 1 6 0 0 0

0 . C 0 7 3 8

0 . 0 0 0 4 0

- 0 . 0 1 8 0 0

0 . 0 0 0 3 0

0 . 0 0 ^ 0 0

© • 0 1 4 0 0

- 0 . 0 0 2 3 2

. . , ^ - . . . , ^ , . . . • • . 0 . . ^ . . . . - > . . ^ . - . • . . . . . ^

137

APPENDIX ( I I I ) TABLE 3- 5 . COEFFICIENTS OF THEORETICAL SOLUTION OF MUSCULAR TENSICN

SURJECT N O . : 1

NAME OF MUSCLE

ABDUCTION CASE

DELTOID ANTERIOR

D E L T n i D MIDDLE

DELTOID POSTERIOR

SUPRASPINATUS

INFRASPINATUS

TEÊS MAJOR

TERFS MiNOR

SUBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

PECTORAL ÎS MAJ . ( S)

PECTORALIS M A J . ( C )

BICEPS (LONG)

RICEPS(SHORT)

TRÎCEPS

CORAC BRACHIAL IS

L A T I S S I M U S DO^SI

A l

7 . 3 2

2 6 . 9 0

- 5 . 5 2

8 . 5 0

7 . 6 1

1 3 . 7 0

2 . 8 9

2 . 8 4

1 . 5 7

2 . 3 2

1 0 . 4 0

1 3 . 8 0

5 . 2 0

1 0 . 1 0

3 . 4 0

ACTING WEIGHT

A2 A3

- 1 . 5 8 0

- 6 . 3 0 0

4 5 . 9 0 - 2 1 . 5 0 0

4 7 . 0 0 - 1 1 . 6 0 0

- 2 . 8 6 0

7 5 . 2 0 - 3 0 . 2 0 0

- 2 . C90

- 0 . 3 9 7

2 .6C0

3 9 . 8 0 - 1 6 . 8 0 0

- 3 . 170

- 2 . 1 8 0

0 . 1 9 7 0

0 . 9 C 0 0

8 . 5 1 0 - 1 . 7 1 C 0

4 . 8 9 C 0

2 . 2 1 C 0

0 . 6 4 9 0

2^430 - 0 ^ 2 4 8 0

4 . 7 3 C 0

0 . 2 9 7 0

0 . 0 3 9 8

• 0 . 8 1 5 0

2 . 4 4 0 0

C . 7 0 8 - 0 . 1 8 4 0

0 . 2 1 2 - 0 . 0 3 2 7

0 . 6 2 3 0

0 . 1 8 9 0

- 3 . 5 8 0 - 0 . 0 4 6 5

C . 0 3 4 - 0 . 2 7 6 0

5 LBS

A4

• 0 . 0 0 6 2 5

• C . 0 4 5 5 0

0 . 1 0 ^ 0 0

• 0 . 3 0 3 0 0

• 0 . 13300

• 0 . 0 3 8 3 0

0 . 0 0 7 3 5

• 0 . 2 4 1 0 0

- C . 9 4 2 0 . 1 4 C C

•C. C0974

0 . 0 0 1 2 1

0 . 0 6 3 6 0

• 0 . 1 1 100

0 . 0 1 5 9 0

0 . 0 0 0 8 7

- 0 . C3860

0 . 0 0 0 5 7

0 . C 0 8 6 0

0 . 0 3 0 0 0

- O . C 0 ^ 9 5

138

AP=>END X (III) TABLE 3- 6. COEFFICIENTS OF THEORETICAL SOLUTION GF MUSCULAR TENSION

SUPJECT NO.: 1

NAME OF MUSCLE Al

ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MIDDLE

DELTOID POSTFRIOR

SUPÂSPINATUS

INFRASPINATUS

TERFS MAJOR

TERES MINOR

SUBSCAPULARIS

ADDUCTlON CASE

INFÂSPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

PECTORALIS MAJ. ( S)

PECTGRAL IS MAJ . (C)

B ICEPS(LONG)

BICEPS (SHORT )

T R I : E P S

CORACOBRACHI AL IS


1 1 . 2 0

4 1 . 2 0

8 . 4 8

1 3 . 10

1 1 . 7 0

2 1 . 0 0

^ . 4 3

4 . 3 8

2 . 4 3

3 . 5 7

1 5 . 7 0

2 1 . 2 0

7 . 9 8

1 5 . 5 0

5 . 2 2

ACTING WEIGHT

A2 A3

- 2 . 4 2 0

- 9 . 6 6 0

7 C . 4 0 - 3 2 . 9 0 0

7 2 . 0 0 - 1 7 . 8 0 0

- 4 . 3 9 0

1 1 5 . 0 0 - 4 6 . 4 0 0

3 . 2 0 0

6 . 1 0 0

6 1 . 0 0 - 2 5 . 8 0 0

1 . 0 8 0

- 4 . 8 6 0

- 3 . 340

0 . 3 0 3 0

1 . 3 8 0 0

1 3 . 1 0 0 - 2 . 6 2 0 0

7 . 5 C 0 0

3 , 3 9C0

0 . 9 9 6 0

3 . 6 9 0 - 0 . 0 3 7 5

7 . 2 6 C 0

0 . 4 5 8 0

0 . 0 6 1 1

3 . 9 9 0 - 1 . 2 5 0 0

3 .74CC

• 0 . 2 8 2 0

C.328 - 0 . 0 5 0 5

0 . 9 5 6 0

0 . 2 8 9 0

- 0 . 5 4 7 - 0 . 0 7 1 9

C . C 5 1 - 0 . 4 2 3 0

10 LBS

A4

- 0 . C 0 9 6 4

- 0 . 0 6 9 8 0

0. i5<;co

• 0 . 4 6 600

•0 . 2 0 5 0 0

• 0 . 0 5 8 8 0

O.OI 100

• 0 . 3 6 9 0 0

- 1 . 4 5 0 0 . 2 1 5 0

• 0 . 0 1 5 0 0

0 . 0 0 1 8 6

0 . 0 9 7 5 0

•0 . 17C00

0 . 0 2 4 3 0

0 . 0 0 1 3 5

- 0 . 0 5 9 3 0

0 . 0 0 1 4 2

0 . 0 1 3 3 0

0 . 0 4 6 0 0

- 0 . 0 0 0 7 6

•MHWiiaHMiMatMaM ;i£it:^^%:

139

APPENDIX (III) TABLE 3- 7.

C 0 5 F F I C I E N T S OF THEORETICAL SOLUTION OF MUSCULAR TENSIOM

SUBJECT NO. : 2

MAME OF MUSCLE

ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUPRASPINATUS

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS

TFRFS MAJOR

TERES MINOR

SUBSCAPULARI S


PFCTORAL IS MA) . ( C )

BICEPS(LONG)

BICEPS (SHORT )

TRICEPS

CORACOBRACHIALIS

LATI SSIMUS DORSI

A l

2 . 7 0

9 . 8 0

- 2 . 1 5

1 5 . 9 0

1 7 . 6 0

2 . 9 7

2 . 6 6

5 . 3 1

1 . 0 1

l . l l

1 5 . 3 0

0 . 5 3

0 . 8 5

^ • 8 3

5^ 12

U 9 5

3 . 8 0

1 . 2 1

ACTING WEIGHT

A2 A3

- C . 5 7 8

- 2 . 2 8 0

- 7 . 3 8 0

- ^ . 4 9 0

- 0 . 9 9 4

2 8 . 7 0 - 1 1 . 5 0 0

- 0 . 9 0 4

C . 1 3 3

0 . 9 7 4

- 6 . 4 5 0

- 1 . 2 0 0

- C . 8 C 0

0 . 0 7 2 5

0 . 3 1 6 0

3 . 1 6 0 - 0 . 7 1 1 0

1 . 6 7 0 0

0 . 8 2 9 0

3 . 2 2 6 0

1 . 0 8 0 - 0 . 1 3 3 0

1 . 7 8 0 0

0 . 1 2 2 0

0 . 0 1 2 0

• 0 . 3 0 7 0

0 . 9 2 8 0

0 . 2 7 0 - 0 . 0 7 0 1

C.C54 - 0 . C 0 6 3

0 . 2 3 2 0

0 . 0 6 4 1

- 0 . 1 4 9 - 0 . 0 1 5 9

- C . C 0 3 - 0 . 0 9 7 6

0 LRS

A4

• 0 . C 0 2 3 0

• 0 . 0 1 5 7 0

0 . 03 790

• 0 . 10200

•C. C4780

• 0 . 0 1 3 2 0

0 . C0509

0 . 0 9 0 2 0

- 0 . 0 3 3 0 . 0 4 7 7

• 0 . 0 0 3 9 0

0 . C 0 0 5 9

0 . 0 2 3 8 0

-4 .21CC0

0 . 0 0 5 9 8

0 . C 0 0 2 0

• 0 . 0 1 A 2 0

O.C0040

0 . 0 0 3 1 1

0 . C 1 0 9 0

- 0 . 0 0 162

!ii.jj.tia.--i j ip.^Y»..^.^ r. - .1 » ,.,

140

APPENDIX ( I I I ) TABLE 3 - 8 . COEFFICIENTS OF TUEORFTICAL SOLUTION OF MUSCULAR TENSION

SUBJECT N O . : 2

NAME OF MUSCLE

ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MIDDLE

DELTOÎD POSTERIOR

SUPRASPINATUS

INFÂSPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

PECTORAL IS M A J . ( S )


BI CEPS(LONG)

B I C F P S Í SHORT )

TRÎCEPS

CORACOBRACHIALIS

LAT ISSIMUS DORSI

Al

7 . 0 2

2 5 . 6 0

_ R • 62

4 5 ^ 9 0

7 . 7 5

6 . 95

7 4 ^ 8 0

1 3 ^ 9 0

2 . 6 3

2 . 9 1

3 9 . 8 0

1 . 3 9

2 . 2 2

1 0 . 0 0

1 3 , 4 0

5 . 0 8

9 . 9 0

3 . 1 6

ACTING WEIGHT

A2 A3

- 1 . 5 2 0

- 5 . 9 6 0

• 1 1 . 7 0 0

- 2 . 5 9 0

2 . 8 1 0

• 3 0 . 0 0 0

- 2 . 3 7 0

- C . 3 4 1

2 . 5 4 0

• 1 6 . 8 0 0

C.709

0 . 142

- 3 . 120

- 2 . 0 9 0

- C . 3 9 1

- 0 . 0 7 4

- 0 . 8 5 6

0 . 1 8 8 0

0 . 8 2 6 0

8 . 2 4 0 - 1 . 6 3 0 0

4 1 . 5 0 - 1 9 . 3 0 0 - 4 . 3 6 C 0

^ • 1 6 0 0

© • 5 8 9 0

- © • 3 4 6 0

4 . 6 6 C 0

0 .32CC

0 ^ 0 0 3 0

- 0 . 8 0 1 C

2 . 4 2 0 0

- 0 . 1840

- 0 . 0 1 6 7

0 . 6 0 5 0

0 . 1 6 8 0

- 0 . 0 4 1 2

- 0 . 2 5 6 0

0 . 1 2 5 0

5 LBS

A4

- 0 . 0 0 5 P 4

-C .041C0

0 . 0 9 8 9 0

- 0 . 26600

• 0 . 1 2 7 0 0

• 0 . 0 3 A 5 0

0 . 0 1 3 3 0

• 0 . 2 3 5 0 0

• 0 . 0 1 C 3 0

0 . 0 0 1 5 8

0 . 0 6 2 1 0

-O.UCOO

0 . 0 1 5 6 0

- 0 . 0 0 C 0 3

• 0 . 0 3 7 0 0

©•00 114

0 . 0 0 8 1 1

0 . 0 2 8 4 0

- 0 . 0 0 4 2 6

^ . - • • . - • » ^ . . r — ^ . — . > -

APPENDIX ( I I I ) TABLE 3- 9 . COEFFICIENTS OF THEORETICAL SOLUTION OF MUSCULAR TENSION

141

SUBJECT N O . : 2 ACTING WEIGHT: 10 LBS


ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUPRASPINATUS

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

1 1 . 4 0

4 1 . 4 0

- 9 . 0 7

- 9 . 0 7

6 7 . 2 0

7 4 . 2 0

1 2 . 6 0

1 1 . 3 0

- 2 . 4 3 0

- C . 6 4 0

1 3 . 8 0 0

1 3 . 300

- 3 1 . 2 0 0

- 1 8 . 9 0 0

- 4 . 2 0 0

4 . 5 2 0

0 . 3 0 5 0

1 .34C0

- 2 . 6 4 0 0

- 2 . 64C0

7 . 0 5 0 0

3.5CCC

3 . 9 5 4 0

- 0 . 0 5 5 7

- 0 . 0 0 9 6 3

- C . 0 6 6 3 0

0 . 1 6 0 0 0

0 . 1 6 C 0 0

- 0 . 4 3 1 0 0

- 0 . 2 0 6 0 0

- 0 . 0 5 5 9 0

0 . 0 2 1 3 0

ADDUCTION CASF

INFRASPINATUS

TERES MAJOR

TERES MiMOR

SUBSCAPULARIS

PECTORAL IS MAJ . ( S )

• OECTORAL I S MAJ . ( C)

BICEPS(LONG)

BICEPS(SHORT)

TRICFPS

CORACOBRACHIALIS


1 2 1 . 0 0

2 2 . 4 0

4 . 2 7

4 . 6 9

6 4 , 4 0

2 , 2 5

3 . 6 0

1 6 . 2 0

2 1 . 7 0

8 . 2 0

5 . 1 1

- 4 8 . 500

- 3 . 8 2 0

- 0 . 5 5 5

4 . 110

- 2 7 . 200

1 . 150

C. 231

- 5 . 0 5 0

- 3 . 3 8 0

- 0 . 6 2 6

- 1 . 380

7 . 5 4 C 0

0 , 5 1 4 0

0 , C 4 9 7

- 1 . 3 0 0 0

3 . 9 2 C 0 '

- 0 . 2 9 7 0

- 0 . 0 2 7 2

3 . 9 7 8 0

0 . 2 7 2 0

- 0 . 0 6 7 8

0 . 2 0 2 0

- 0 , 3 8 1 C 0

- 0 . 0 1 6 6 0

0 , 0 0 2 5 3

O.IOCOO

- 0 , 1 7 8 0 0

0 , 0 2 5 3 0

0 . 0

- 0 . 0 5 9 8 0

0 . C 0 1 8 1

0 . 0 4 5 9 0

- 0 . C 0 6 8 7

r i rSiílii

142

APPENDIX ( I I I ) TABLE 3 - 1 0 . COEFFICIFNTS OF THEORETICAL SOLUTION OF MUSCULAR TENSIOM

SU3JECT N O . : 3

NAME OF MUSCLE

ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUPRASPINATUS

INFRASPINATUS

TERES MAJOR

TERES MirôR

SUBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS

TERFS MAJOR

TERES MINOR

SURSCAPULARIS

PECTORAL ÍS MAJ . ( S)


BICEPS(LONG)

BICEPS(SHORT)

TRICEPS

CURACOBRACHI ALIS'

L A T I S S ÍMUS DO^SI

A l

2 . 2 5

8 . 2 3

- 1 , 7 1

M,00

1 4 . 4 0

2 . 5 9

2 , 3 2

2 3 . 2 0

ACTING WEIGHT

A2 A3

- 0 . 4 3 5

- 1 , 9 3 0

2 , 6 2 0

- 6 . 5 2 0

- 3 . 5 8 0

- 0 . 8 7 3

0 . 7 6 7

- 9 , 3 2 0

0 , 0 6 C 7

0 , 2 7 4 0

- 0 , 0 5 2 4

1 , 4 8 0 0

0 , 6 8 C 0

0 , 1980

- 0 , 0 8 1 2

1 , 4 6 0 0

0 LBS

A4

- 0 , 0 0 1 9 3

- 0 . 0 1 3 8 0

0 . 0 3 1 8 0

- 0 . 0 9 190

- 0 . 0 4 C 8 0

• 0 . 0 1 170

0 . 0 0 2 5 4

• 0 . 0 7 4 0 0

4 . 2 3

0 . 8 8

0 . 8 8

2 . 2 0

0 , 4 8

0 , 7 1

3 , 1 6

4 , 2 5

1 ,60

3 , 1 1

1 ,04

- C , 6 5 6

- 0 . 1 1 9

0 . 8 0 1

- 5 . 2 0 0

0 . 2 1 9

0 , 0 6 3

- 0 . 9 7 7

- 0 . 6 6 8

- 0 . 112

0 . 0 0 3

- C . 2 8 7

0 . 0 9 2 6

0 . 0 1 1 7

- 0 . 2 5 1 0

0 . 7 5 2 0

- 0 . C 5 6 8

- 0 , 0 0 9 4

0 , 1920

0 , 0 5 7 2

- 0 , 0 1 4 1

- 0 , 0 8 4 0

0 . 0 4 2 5

- 0 . 0 0 3 0 3

0 . 0 0 0 4 0

0 . 0 1 9 6 0

- 0 . 0 3 4 2 0

0 . C 0 4 8 9

0 . 0 0 0 2 3

- 0 . 0 1 1 9 0

o.occoa

0 . C 0 2 6 3

0 . 0 0 9 1 7

- 0 . C0149

ff^ ^-^—^-——

143

APPFNDIX (III) TARLE .3-11. COFFFTCIENTS OF THEORETICAL SOLUTION OF MUSCULAR TENSIOM

SUBJECT NO.: 3

NAME OF MUSCLE

A3DUCT10N CASE

D E L T d l D ANTERIOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUPÂSPINATUS

INFRASPINATUS

TERES MAJOR

TEPES MINOR

SURSCAPULARIS

ADDUCTION CASE

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUPSCAPULARIS

PECTORALIS M A J . ( S)


B ICEPS(LONG)

BICEPSISHORT )

TRICEPS

CORACOBRACHIALIS


Al

6.29

23. 10

-4.80

39. 10

40.50

7. 24

6.49

65.CO

ACTING WEIGHT

A2 A3

- 1 , 3 6 0

- 5 , 4 C 0

7 . 3 3 0

• 1 8 . 3 0 0

•IC.OOO

- 2 . 4 3 0

2 . 1 5 0

• 2 6 . 100

0 . 1 7 C 0

0 , 7 7 0 0

1 , 4 7 0 0

• 4 , 1 6 C 0

1 ,91C0

0 , 5 5 2 0

0 , 2 2 7 0

4 , 0 8 0 0

5 LRS

A4

-0,00540

-0,03880

0,08910

-0.25700

-0.115C0

-0.03260

0.00712

-0.20700

1 1.90

2.46

2,47

3^.40

1,33

1.99

8, 83

1 1,90

4,48

8,71

2,90

-1.840

-C.334

2.240

-14.600

0.613

C. 177

-2.740

-1.870

-0.312

C.012

-0.801

0.26C0

0,0329

-0,7020

2, lOCO

-0,1590

-0.0266

0,5370

0, 16C0

-0,0398

-0,2350

0,1180

-0.00852

0.00111

0.05470

-0,09570

0,01370

0.00065

-0.0 3 320

0,00023

0.00738

C.02570

-0.00417

^,*^

144

APPENDIX (III) TABLE 3-12.

COEFFICIENTS OF THEORETICAL SOLUTION OF MUSCULAR TENSIOM

SUBJECT NO.: 3

MAME OF MUSCLE

ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUiíRASPI NATUS

INFRASPINATUS

TERES MAJOR

Al

TERES MINOR

SUBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARI S

PECTORALIS MAJ.(S)

PECTORALTS MAJ.(C)

BICEPS(LONG)

PICEPS(SHORT )

TRICFPS

CPRACOBRACHIALIS

lATÍ SSIMUS DORSI

1 9 . 5 0 .

4 . 0 4

4 , 0 7

5 6 , 5 0

2 , 1 9

3 . 2 7

1 4 . 5 0

1 9 . 5 0

7 . 3 6

1 4 . 3 0

4 . 7 7

ACTING WEIGHT: 10 LBS

A2 A3 A4

10.30

37.90

-7.86

64.20

66,40

1 1,90

10,70

07,00

-2, 220

-8,860

12,C00

-30.000

-16.400

-4.000

3.530

-42.900

0,2780

1 ,2600

-2,41CC

6,8200

3,13C0

0,9090

-0.3740

6,7000

-0,C0e82

-0,06360

0,14600

-0,42200

-0. 18800

-0,05360

C,01170

-0,34100

- 3 . 0 3 0

- 0 . 5 5 0

3 . 6 8 0

• 2 3 . 9 0 0

1 .010

C . 2 8 8

- 4 . 4 9 0

- 3 . 0 8 0

- 0 . 5 1 2

C.C22

- 1 . 3 2 0

0 , 4 2 9 0

0 , 0 5 4 0

- 1 , 1 5 C C

3 , 4 6 0 0

- 0 , 2 6 2 0

- 0 , 0 4 3 2

0 , 8 8 1 0

0 , 2 6 4 0

- 0 , 0 6 5 6

- 0 , 3 8 7 0

0 , 1 9 5 0

• 0 , 0 1 4 1 0

0 . C 0 1 8 3

0 . 0 8 9 9 0

• 0 . 1 5 7 0 0

0 , 0 2 2 5 0

0 , C 0 1 0 5

• 0 , 0 5 ^ 5 0

0 , C 0 0 3 2

0 . 0 1 2 1 0

0 . 0 4 2 2 0

• 0 . 0 0 6 8 6

ú • í '

APPENDIX (III) TABLE 3-13, COEFFICIENTS OF THEGRETICAL SOLUriON OF MUSCULAR TENSION

145

SURJECT N O , : 4 ACTING WEIGHT: 0 LBS


ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUPRASPIMATUS

INFRASPINATUS

TERFS MAJOR

TERES MINOR

SUBSCAPULARIS

2.89

10.50

- 2 , 35

1 6,70

18,90

3,14

2,80

3 1 .00

-C.618

-2.440

3.390

-7.740

-4.880

-1.C50

1.220

-12.400

0,0775

0,3350

-0.6710

1.7400

0.8930

0.2380

-0.1570

1.9300

-0.00245

-0.01650

0.04050

-0 . 106C0

-0.05210

-0.01390

0.00630

-C.09710

ADDUCTION CASE

ÎNFRASPINATUS

TERES MAJOR

TERFS MINOR

SUBSCAPULARIS



RICEPS(LONG)

BICEPSÍ SHORT)

TRICEPS

CORACORRACHIALIS

LATISSIMUS DORSI

5.77

1.07

1.21

16.60

0.55

0.91

4.12

5.31

2.09

4.09

1.29

-I.CIO

-0.138

1.C50

-6.990

C.295

C.050

-1.290

-0.859

-C, 165

-0.044

-C,346

0,134C

0,0119

-0.3320

1 .0100

-0,C763

-0.0050

0.249C

0.0673

-0.0168

-0.1040

0.C5C0

-O.0OA31

0,67800

0,02530

-0,04560

0,00648

-0,00012

-0.01520

0.00061

0,00333

0.01 160

-0,00169

IdtfMiilUMiiMaaM - — - * • * — • - * ' « • ' > »

APPENDIX ( I 11 ) TABLE 3 - 1 4 . COEFFICIENTS OF THEORETICAL SOLUTÎON OF MUSCULAR TENSIGN

SUBJECT N O . : 4

NAME OF MUSCLE

ABDUCTlON CASE

DELTOID ANTERIOR

DELTOID MIODLE

DELTOID POSTERIOR

SUPRASPINATaS

INFRASPINATUS

TERES MAJOR

TERES MINOR

SURSCAPULARIS

ADDUCTION CASE

INFRASPÎNATirS

TERES MAJOR

TERES MÎNOR

SUBSCAPULARIS

PECTORAL IS MAJ . ( S)


B ICEPS(LONG)

R ICrPS(SHORT)

TRICEPS

CORACOBRACHI ALIS'

LAT ISS IMUS UO^SI

Al


A2 A3 A4

7 . 0 9

2 5 . 7 0

- 5 . 7 5

4 1 . 0 0

4 6 . 5 0

7 . 7 0

6 . 8 9

7 6 . 2 0

- 1 . 5 2 0

- 5 . 9 8 0

8 . 3 2 0

-1<9.C00

- 1 2 . 0 0 0

- 2 . 5 7 0

2 . 9 8 0

- 3 C . 6 0 0

3 . 1 9 1 0

0 . 8 2 1 0

- 1 . 6 5 0 0

4 . 2 8 0 0

2 . 1 9 0 0

0 . 5 8 4 0

- 0 . 3 84 0

4 . 7 3 C 0

- 0 . 0 0 6 0 3

- C . C4050

0 . 0 9 9 4 0

- 0 . 2 6 1 0 0

- 0 . 1 2 8 0 0

- 0 . C 3 4 1 0

0 . 0 1 5 4 0

- 0 . 2 3 8 0 0

1 4 , 2 0

2 , 6 2

2 , 9 7

4 0 , 7 0

1 , 3 7

2 , 2 4

1 0 . 1 0

1 3 , 5 0

5 . 14

1 0 . 1 0

3 , 1 6

- 2 . 4 9 0

- 0 . 3 3 5

2 . 5 8 0

1 7 . 2 0 0

0 . 7 2 3

0 . 124

- 3 . 1 7 0

- 2 . 110

- C . 4 0 7

- 0 . 1 1 2

- 0 . 8 4 9

0 , 3 3 2 0

0 , 0 2 8 9

- 0 . 8 1 5 0

2 . 4 7 0 0

- 0 . 1870

- 0 . 0 1 2 4

0 . 6 1 1 0

0 . 1650

- 0 . 0 4 1 0

- 0 . 2 5 4 0

0 . 1230

- 0 . C 1 0 7 0

0 . 0 0 1 6 9

0 . 06290 ,

- 0 . 1 1 2 0 0

0 . 0 1 5 9 0

- 0 . 0 0 0 2 9

- 0 . 0 3 7 3 0

0 . 0 0 1 5 2

0 . C 0 8 1 6

0 . 0 2 8 5 0

- 0 . C 0 4 1 4

146

H Í iattJsláHI É

APPFNDIX ( I I ) TABLE 3 - 1 5 . COEFFICIENTS OF THEORETICAL SOLUTION GF MUSCULAR TENSIOM

SUBJECT N O . : 4

NAME OF MUSCLE

ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUPRASPINATUS

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

PECTORAL IS M A J . ( S )


BICEPS(LONG)

BICEPSISHORT)

TRICEPS

CORACOBRACHI AL I S'


A l


A2 A3 A4

1 1 , 3 0

4 1 , 0 0

- 9 . 1 6

6 5 , 50

7 4 , 0 0

1 2 , 30

1 0 , 9 0

2 1 , 0 0

- 2 . 4 1 0

- 9 . 5 3 0

1 3 , 3 0 0

- 3 C , 3 0 0

- 1 9 , 100

- 4 , C 9 0

4 , 7 7 0

- 4 8 , 7 0 0

0 , 3 0 2 0

1 , 3 1 0 0

- 2 . 6 2 C 0

6 . 8 3 C 0

3 . 4 9 0 0

0 ,93CO

- 0 , 6 1 4 0

7 . 5 4 0 0

- 0 . 0 0 9 5 4

- © • 0 6 4 6 0

0 , 15800

- 0 , 4 1 5 0 0

- 0 , 2 0 A 0 0

- 0 , 0 5 A 4 0

0 , 0 2 4 6 0

- 0 , 3 8 C 0 0

2 2 , 6 0

4 , 1 7

^ . 75

6 4 . 7 0

2 . 17

3 , 5 8

1 6 , 10

2 1 , 5 0

8 , 19

1 6 , 0 0

5 , 0 2

- 3 , 9 6 0

- 0 , 5 3 2

4 . 100

- 2 7 , 3 0 0

1 . 150

0 . 196

- 5 . 0 5 0

- 3 . 3 6 0

- C . 6 4 7

- 0 . 176

- 1 . 3 5 0

0 , 5 2 7 0

0 , 0 4 5 6

- 1 , 3 0 0 0

- 3 , 9 3 C 0

- 0 , 2 9 8 0

- 0 , 0 1 9 4

0 , 9 7 3 0

0 , 2 6 3 0

- 0 , 0 6 5 3

- 0 , 4 0 6 0

0 , 1 9 5 0

- 0 , 0 1 6 9 0

0 , 0 0 2 7 1

c'ioooo

- 0 , 1 7 8 0 0

0 , 0 2 5 3 0

- 0 , 0 0 0 4 8

- 0 . 0 0 5 9 3

0 , 0 0 2 3 5

0 , 0 1 3 0 0

0 , 0 4 5 4 0

- 0 , 0 0 6 5 7

147

ntii iBr i r - - — t w 4 ^ M ^t?mi

APPFNDIX (III) TABLE 3-16.

COEFFICIENTS OF THEORETICAL SOLUTION OF MUSCULAR TENSIOM

SUBJECT NO, : 5

NAME OF MUSCLE

A3DUCTI0N CASE

DELTOID ANTERIOR

DELTOID MIDDLE

PELTOID POSTERIOR

SUPÂSPINATUS

ÎNFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

PECTORALIS M A J , ( S )

PECTCRALIS M A J , ( C )

BICEPS(LONG)

BICEPS (SHORT )

TRICEPS

CORACOBRACHI ALIS'

L A T I S S I M U S DORSI

A l

3 . 8 3

1 4 . 10

- 2 . 8 6

2 4 , 6 0

4 . 4 9

4 . 0 2

ACTING WEIGHT

A2 A3

- 0 . 8 2 6

- 3 . 3 0 0

0 . 1030

0 , 4 7 4 0

4 . 4 6 0 - 0 . 8 9 5 0

2 4 . 3 0 - 1 1 . 4 0 0

- 6 . 0 2 0

- 1 . 5 1 0

2 . 5 9 0 0

1 , 1600

0 , ^ 4 3 0

1 ,230 - 0 , 1 2 1 0

3 9 . 2 0 - 1 5 . 7 0 0 2 , 4 7 0 0

0 LBS

A4

- 0 . 0 0 3 2 7

- 0 , 0 2 4 0 0

0 . 0 5 4 5 0

- 0 , 1 6 1 0 0

• 0 . 0 7 0 1 0

• 0 . 0 2 0 3 0

0 . 0 0 3 3 4

• 0 . 12600

7.12

1.52

1.48

20.70

0,84

1.21

5,36

7,23

2,72

5,27

1,79

-1.060

-C.210

1,360

-8.770

C.36 6

0.117

-1.650

-1. 140

-0.1B2

C.029

-C,499

0,1530

0,0213

-0,4250

1,27CC

-0,0957

-0,0185

0,3260

0,0995

-0,0249

-0.146C

0.0744

-0,00504

0,00061

0.03320

-0.05780

0.00827

0.C0054

-0.02020

-0.CCC03

0,00453

0,01580

-0,00264

148

l iiiifn n líiiiBiiii I I • Waoi !

149

APPENDIX ( I I I ) TABLE 3 - 1 7 . C O E F F I C I E N T S OF THEORETICAL SOLUTION OF MUSCULAR TENSIOM

SUBJECT NO.: 5

NAME OF MUSCLE

ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUPR^SPINATUS

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

PECTORAL IS MAJ . ( S)


BICEPS(LONG)

BICEPS(SHORT )

TRICEPS

CORACOBRACHIALIS


A l

8 . 0 7

2 9 . 7 0

- 6 . 0 4

5 1 , 0 0

5 1 . 7 0

9 . 4 5

8 . 4 6

8 2 , 5 0

ACTING WEIGHT:

A2 A3

- 1 , 7 4 0

- 6 , 9 6 0

9 . 380

- 2 3 . 9 0 0

- 1 2 . 7 0 0

- 3 . 1 3 0

2 . 5 9 0

- 3 3 . 1 0 0

0 . 2 1 7 0

1 , 0 0 0 0

- 1 , 8 8 C 0

- 5 , 4 5 0 0

2 ,A4C0

0 , 7 2 1 0

- 0 . 2 5 5 0

5 , 2 0 0 0

5 LBS

A4

- 0 , 0 0 6 9 1

- 0 , 0 5 0 7 0

0 . 1 1 5 0 0

- 0 , 3 3 9 0 0

•0 . 1 4 6 0 0

• 0 , 0 4 2 6 0

0, C0723

- 0 , 2 6 5 0 0

1 5,00

3.21

3,11

43,60

1 ,76

2,56

11,30

15,20

5,72

11,10

3.77

-2.240

-C.443

2.870

-18.500

0.774

C.247

-3.480

-2.400

-0.3B2

C.C56

-1.050

0,3230

0,0450

-3,9000

2,67CC

-0,2020

-0.039C

0,6850

0,21C0

-0,0527

-0,306C

0,1560

-0,01070

0, C0128

0,07000

-0, 12 200

0,01740

O.CO 113

-0,04260

-0,C0CO9

0,00955

0.03320

-0,00554

150

APPENDIX ( I I I ) TABLE 3 - 1 8 . COEFFICIENTS OF THEORETICAL SOLUTÎON OF MUSCULAR TENSIOVJ

SUBJECT N O . : 5

MAME OF MUSCLE A l

ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUPRASPINATUS

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS 2 2 , 9 0

TERES MAJOR 4 , 8 9

TERES MINOR 4 , 7 4

SUBSCAPULARIS 6 6 . 5 0

PECTORAUIS M A J . ( S ) 2 . 6 8

PECTORAL IS MAJ . ( C) 3 . 9 1

BICFPS(LONG) 1 7 . 2 0

BICEPS(SHORT ) 2 3 . 2 0

TRICEPS fl-73

CORACOBRACHIALIS 1 6 . 9 0

LATI SSIMUS DORSI 5 . 7 5


A2 A3 A4

1 2 . 3 0

4 5 . 2 0

- 9 . 19

7 7 . 8 0

7 8 . 7 0

1 4 . 4 0

1 2 . 9 0

1 2 6 . 0 0

- 2 . 6 5 0

- 1 0 . 6 0 0

1 4 , 3 C 0

- 3 6 , 4 0 0

- 1 9 . 3 0 0

- 4 . 8 6 0

3 , 9 5 0

- 5 0 . 5 0 0

0 . 3 3 1 0

1 , 5 2 0 0

- 2 , 8 7 C C

8 . 3 1 0 0

3 . 7 1 C 0

1 . 1 0 0 0

- 0 . 3 8 8 0

7 . 9 3 0 0

- 0 , 0 1 0 5 0

- 0 , 0 7 7 2 0

0 , 17500

- 0 , 1 7 5 0 0

- 0 . 2 2 4 0 0

- 0 . 0 6 5 1 0

0 . 0 1 0 7 0

- 0 . 0 4 0 4 0

- 3 . 4 4 0

- 0 . 6 7 8

4 . 370

• 2 8 . 2 0 0

1. 190

C , 3 7 6

- 5 , 3 1 0

- 3 . 6 5 0

- 0 . 5 8 8

0 . 0 3 9

- 1 . 6 0 0

0 , 4 9 7 0

0 . 0 6 8 9

- 1 . 3 7 0 0

4 . 0 8 0 0

- 0 . 3 0 9 0

- 0 . 0 5 9 4

1 . 0 5 0 0

0 , 3 2 0 0

- 0 , 0 7 9 4

- 0 , 4 6 7 0

0 , 2 8 3 0

• 0 . 0 1 6 5 0

C . 0 0 1 9 3

0 . 10700

• 0 . 1 8 5 0 0

0 . 0 2 6 6 0

0 . C 0 1 7 3

• 0 . 0 6 5 0 0

•O.COOll

0 . 0 1 4 5 0

C . C 5 0 6 0

- 0 , 0 0 8 4 8

rittBBSai

151

APPENDIX ( I I I ) TABLE 3 - 1 9 . COEFFICIENTS OF THEORETICAL SOLUTION OF MUSCULAR TENSIGN

SUBJECT N O . : 6

NAMF OF MUSCLE

ABDUCTION CASE

OELTOID ANTERIOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUPRASPINATUS

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

ADDUCTION CASE

TNFRASPINATUS

TERES MAJOR

TERES MINOR

SUPSCAPULARIS

PcCTORALIS M A J . ( S )

PECTORAL IS MAJ . (C)

PICEPS(LONG)

RICFPS(SHORT)

TRICEPS

CORACOBRACHIALIS

L A T I S S I M U S DORSI

A l

3 . 6 6

1 3 . 6 2

- 2 . 52

2 4 . 6 0

2 2 . 9 0

4 . 5 5

4 . 0 2

3 5 , 8 0

- 6 , 3 5

1, 54

1 . 3 0

1 8 . 70

0 . 8 8

1 . 1 7

5 . 0 1

6 . 8 4

2 . 5 4

4 . 9 1

1 , 7 8

ACTING WEIGHT

A2 A3

- C , 7 9 0

3 , 2 2 0

• 1 1 . 5 0 0

- 5 . 3 6 0

- 1 . 5 3 0

0 . 9 1 0

• 1 4 . 4 0 0

0 . 0 9 8 1

0 . 4 7 7 0

4 . 2 0 0 - 0 , 8 5 5 0

2 , 6 5 0 0

1 , 0 8 0 0

0 . 3 4 8 0

- 0 . 0 5 5 0

2 . 2 8 C 0

0 LBS

A4

• 0 . 0 0 3 1 0

• 0 . 0 2 5 0 0

0 , 0 5 2 0 0

• 0 , 1 6 7 0 0

• 0 , 0 6 8 0 0

• 0 . 0 2 0 7 0

• 0 , 0 0 0 1 9

• 0 , 1 1 720

0 . 7 8 0

C . 2 7 0

1 .280

7 . 9 8 0

0 , 3 4 0

0 . 1 5 0

1 .520

1 . 0 8 0

0 . 143

C.C98

0 . 5 1 0

0 , 1 2 9 0

0 . 0 2 5 4

- 0 . 3 9 6 0

1 , 1 6 0 0

- 0 . 0 8 8 0

- 0 . 0 2 6 9

0 . 3 0 5 0

0 . 1020

- 0 . 0 2 6 3

- 0 . 1460

0 . 0 7 8 0

- 0 . 0 0 ^ 6 0

0 , 0 0 0 3 1

0 . 0 3 1 2 0

- 0 . 0 5 3 0 0

0 . 0 0 770

0 . 0 0 1 0 8

- 0 . 0 1 9 2 0

- 0 . C 0 0 6 2

0 . 0 0 4 9 0

0 , 0 1 5 4 0

- 0 . 0 0 2 8 7

5gg:;g

152

APPFNDIX (III) TABLE 3-20. COEFFICIENTS OF THEORETICAL SOLUTTON OF MUSCULAR TENSION

SUBJECT NO.: 6

NAME OF MUSCLE

ABDUCTION CASE

DELTOID ANTERIOR

DELTOID MIDDLE

DELTOID POSTERIOR

SUPRASPINATUS

INFRASPINATUS

TERES MAJOR

TERES MINOR

SUBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS

TERES MAjnR

TERES MINOR

SUBSCAPULARIS

PECTORALIS MAJ,(S)

PECTORALIS MAJ.(C)

B I C E P S ( L O N G )

BICEPS(SHORT )

TRICEPS

CORACOBRACHIALIS

LAT ISSIMUS DORSI

A l

1 3 . 0 0

3 . 17

2 . 6 5

3 8 , 5 0

1 , 7 9

2 , 4 0

1 0 . 3 0

1 4 . 0 0

5 . 2 1

1 0 , 10

3 . 6 5


A2 A3 A4

7, 49

27,90

-5, 15

50,30

46,90

9,32

8. 25

73,40

-1,620

-6.590

8,600

-23,600

-U.CCO

-3. 140

1,860

-29.500

0.2010

0,0979

-1 ,75C0

5,4300

2,2300

0,7130

-0,1120

4,6800

-0.00638

-0.C5C70

0. 10700

-0.34300

-0.13900

-0.04250

-0.00043

-0.24CC0

- 1 . 6 2 0

- C . 4 6 4

2 . 6 3 0

• 1 6 . 3 0 0

0 . 6 8 8

C . 3 0 8

- 3 . 110

- 2 . 2 1 0

- 0 . 290

0 . 2 0 0

- 1 . 0 4 0

0 . 2 6 5 0

0 , 0 5 2 0

- 0 , 8 1 2 0

2 , 3 8 0 0

- 0 , 1810

- 0 , 0 5 5 5

- 0 , 6 2 5 0

0 , 2 0 9 0

- 0 , 0 5 4 5

- 0 , 2 9 9 0

0 . 1590

• 0 . 0 0 9 4 2

0 . 0 0 0 6 4

0 . 0 6 ^ 0 0

• 0 , 1 0 9 0 0

0 . 0 1 5 8 0

0 . 0 0 2 2 3

- 0 . 0 3 9 4 0

- 0 . 0 0 126

0 . 0 0 9 1 4

0 . 0 3 1 5 0

- 0 . 0 0 5 8 5

153

APPENDIX (III ) TABLE 3-21. COEFFICIENTS OF THEORETICAL SOLUTION OF MUSCULAR TEKSICN

SUBJECT NO.: 6

NAME OF MUSCLE A l

ABDUCTION CASE

DELTOID ANTERIOR

DELTOIO MÎODLE

DELTOID POSTERIOR

SUPRASPINATUS

INFRASPINATUS

TERES MAJOR

TERES MINOR

SJBSCAPULARIS

ADDUCTION CASE

INFRASPINATUS 1 9 . 8 0

TFRES MAJOR 4 . 7 8

TERES MINOR 4 , 0 1

SUBSCAPULARIS 5 8 . 2 0

PECTORAL IS M A J , ( S ) 2 , 7 1

PECTORALIS M A J . ( C ) 3 . 6 2

BICEPS(LONG) 1 5 . 5 0

BICEPS(SHORT) 2 1 . 2 0

TRICEPS 7 . 8 8

CORACOBRACHIALIS 1 5 . 2 0

LAT ISSIMUS DO^SÎ 5 . 5 3


A2 A3 A4

1 1 . 3 0

4 2 . 2 0

- 7 . 8 0

7 6 . 1 0

7 0 , 9 0

1 4 . 10

1 2 , 4 0

1 1 , 0 0

- 2 . 4 5 0

9 . 9 7 0

1 3 . 0 0 0

- 3 5 . 7 0 0

- 1 6 . 6 0 0

- 4 . 7 6 0

2 . 8 3 0

- 4 4 . 6 0 0

0 . 3 0 4 0

1.A8C0

- 2 . 6 5 0 0

8 . 2 1 C 0

3 , 3 6 0 0

1 ,08CC

- 0 , 1 7 3 0

7 , 0 7 C 0

- C O I C O O

- C , C 7 6 8 0

0 , 16200

- 0 . 5 1 9 0 0

- 0 . 2 1 C 0 0

- 0 . 0 6 4 3 0

- 0 . 0 0 0 4 9

- 0 , 3 6 3 0 0

- 2 . 4 9 0

- 0 . 7 0 0

3 . 9 7 0

- 2 4 . 7 0 0

1.C30

0 . 4 6 8

- 4 . 7 1 0

- 3 . 3 5 0

- 0 . 4 3 9

0 . 3 0 0

- 1 . 5 8 0

0 , 4 0 7 0

0 , 0 7 8 2

- 1 , 2 3 C C

3 , 6 0 0 0

- 0 , 2 7 3 0

- 0 , 0 8 4 5

0 , 9 4 6 0

0 , 3 1 6 0

- 0 , 0 8 2 2

- 0 , 4 5 2 0

0 . 2 4 1 C

- 0 . 0 U 5 0

0 . 0 0 1 0 0

0 , 0 9 6 7 0

- 0 , 1 6 A 0 0

0 , 0 2 3 8 0

0 , 0 0 3 4 0

- 0 . 0 5 9 6 0

- 0 , 0 0 1 9 3

0 , 0 1 3 8 0

0 , 0 4 7 7 0

- 0 , 0 0 687

Programmer:

Advisor:

Machine Used

Language:

Compiler:

APPENDIX IV

DOCUMENTATION OF COMPUTER PROGRAM

Young-Pil Park

Dr. C. A. Bell

IBM 370/145

Fortran IV

Date Completed:

Compile Time:

Computation Time

Fortran G Compiler

December 1976

Approximately 2 minutes

Part I: 2 minutes

Part II: 5 minutes

Part III: 1 minute

Lines of Output: Each 2000 lines

Purpose:

Part I (ANTHR):

Part II,(THEOR)

This program is designed to analyze the musculo^

skeletal system of the human should joint.

Calculation of the necessary anthropometric

data of a subject and geometrical data of each

muscle that are necessary for the mathematical

analysis.

Calculation of the theoretical solution of the

muscular tension in the shoulder muscles of a

subject and the relationships between muscular

tension and the abduction (adduction) angle of

the arm.

154

-.'Ví •"im

155

Part III (COEFF): Calculation of the linear coefficients re-

lating muscular tension and electromyo-

graphic signal intensity of a subject.

3. Inputs and all the necessary nomenclature used in this program

are explained and listed in the content of the program.

4. Definition of symbols used in the flowcharts:

M - Problem code

SA - Anthropometric data of a subject

SF - Scale factor of a subject

GM - Geometrical data of muscles of the cadaver

GS - Geometrical data of muscTes of the subjects

D - Direction Cosines of muscular tension and moment

CG - Center of gravity of the body segments

SW - Weight of segments

EW - Effective weight

EM - Effective moment

SF - Scale factors of a subject

N - Case code (abduction = 14, adduction = 17)

AB - Abduction coefficient matrix

AD - Adduction coefficient matrix

SUM - Moment coefficients

SUMM - Summation of moment coefficients

CE - Coefficient matrix of muscular tension

F - Solution vector of muscular tension

KK - Number of coefficients for curve fitting

m

156

KD - Number of case for curve fitting

LQ - Least square curve fitting coefficient matrix

CFM - Curve fitting for muscular tension

EMG - Electromyographic signal intensity

Program Flowcharts

157

The Main Program

0

Call

ANTHR

vattf. 'ii!

Subroutine ANTHR

0 Read

SA

158

159

Compute

GS

Write

GS,D

1 Return 3

i r r i M i i . H H W m i l l — ^ l

Subroutine THEOR

160

0 Read

SA

Compute

CG,SW

HMBKSíaKBB

^

161

0

Wri te

SA,CG,SW, EW,EM,SF

Set

AB=0

^"T™™^?

Compute

CE

AN=AN-100

1

Compute

SUM

Compute

SUMM

0

162

miw.im

163

k t / Write /

/ ' /

' !

1 Read / / KK,DK /

^ '

Compute

LQ

' f

Call

FITIT

, ^ , / Write /

/ "" 1 1 '

f Return J

Subroutine COEFF

Read

KK,KD

Write

CFM

C Return )

165

C C C C C C C C C C c c c c c c c c c c c c

c c c c c c c c c c

*

* A P P E N D I X ( I V ) - CGMPUTER PRCGRAMMING-

* T I T L E : A MATHEMATICAL ANALYSIS OF THE * MUSCULO-SKELETAL SYSTEM OF THE * 3LEN0-HUMERAL JOINT

* PROGRAMMER

*

* OATE : MAY 1977

: YGUNG P I L PARK GRADUATE STUDENT TEXAS TECH UNI VERSI TY

*

«

«:ít){t«j}c«5}c:;c«<t««««3îc:íts>::ec3{.«««:{c){t3{t««)>3{!«*:ít«>{c:ít:íc){c*«::tj{(«:ít3:tj0c

THIS PRDGRAM IS MAINLY CONSISTED OF THREE MAIN PARTS ACCORDING TO THEIR PULPOSES

READ ( 5 , 1 ) NC3DE 1 FORMAT ( 1 5 )

«*:(t:í::«c«*:0tj0ti{t;ít«;ít30c«j(c>^:íc«j{cj{tJÍ::{t««30t«j{t:{íj;ci{:!{t3}t3!t:Oíj(c){t*«:^«){c:{t«5{e

* MAIN * * îí:

* THIS PROGRAM DETERMINES WHICH * * METHODS HAS TO BE EXECUTED FOR * * THE PULPOSE OF CALCULATION « * *

*)»:4t<ts!t:{c){c:{s:{ti>:j(tj{(3!t){t:{e:ít«3!t«:í!>îej{t«*«j5c){t)0t:{tjîc«:í!«){c)ît:{c)!t:0c*«*««*«

IF (NCODE) 2 , 3 , 4

C C C C C C C c c c

2 CALL ANTHR GO TO 5

PART I SUBROUTINE ANTHR

CALCULATION OF NECESSARY ANTHRCPGMETRIC DATA OF I N D I V I D U A L SUBJECTS

DATA SET ORDER

Hl PROCESS DETERMINATION CODE ( - 1 ) H2 PERSONAL BASIC ANTHRCPOMETRIC DATA

1 ^ ^ ^ ^ — — ^ ,^^^^^^^^„1 .- ..-—.^.-^

166

C a^ MODFL GEOMETRICAL DATA FOR I N D I V I D U -C AL MUSCLES ( IND IV IDUAL MUSCLES, 0 TO C 90 DEGREES BY 10 DEGREES ÍNTERVAL ) C

C * THIS PROGRAM IS PROGRANMED FOR THE CASE C SI X SUBJECTS C C

3 CALL THEGR 3 0 TO 5

C

C PART I I SUBROUTINE THEOR C C CALCULATION DF THEORETICAL SOLUTION AND C 5TH ORDER CURVE FITTING FGR THE MUSCULAR C TENSILE FORCE VS. ABDUCTION (ADDUCTION) C ANGLES FOR A SINGLE SUBJECT C C DATA SET ORDER C C «1 PROCESS DETERMI NATIGN CODE ( 0 ) C #2 PERSONAL BASIC ANTHRGPOMETRIC DATA C ( S P E C I F I C ONE SINGLE PERSON) C H3 A N T I C I P A T I N G NUMBER OF MUSCLES C ( F I R S T ABDUCTION CASE : 14) C #4 MODEL GEOMETRICAL DATA FOR A SPECIFIC C ANGLE ( EACH ANGLE 14 MUSCLE ) C H5 AN3LE •• 1 0 0 . 0 VALUE CARD C ^6 2 1 0 . 0 C * PROCEED #4 ANC U5 UP TO 90 DEGREES C FROM 10 OEGREES BY 10 DEGREES INTERV. C C m A N T I C I P A T I N G NUMBER OF MUSCLES C (SECOND ADDUCTION CASE : 17 ) C * PROCEED «4 AND #5 UP TO 90 DEGREES AS C ABDUCTION CASE PROCESS C ^8 2 0 0 . 0 C ^9 CURVE FITTING INFORNATICN CARD C ( 5, 10 )

4 CALL COEFF C C PART ÎII SUBROUTINE CGEFF C C CALCJLATION OF THE LINEAR COEFFICIENTS C BETWEEN MUSCULAR TENSILE FORCES V S . C E.M . 3 . SIGNAL INTENSITIES C C DATA SET ORDER

m l l W B r a •raini I iJiMn_L5«Sâ^SBBWfe^

167

C C ^l PROCFSS DETERNINATION CODE ( +1 ) C #2 CURVE FITTINF INFORMATION CARD C ( 2, 81 ) C ^3 THEGRETICAL SOLUTION AND EXPERIMENTAL C RFSULT DATA ( THECPETICAL RESULTS GF C t>ART I I AND E . M . G . RESULTS OF THRE C DELTOIDS PARTS ( 81 DATA FOR THREE C PERSON, THREE WEIGHT, 9 ANGLES ) ) C C C * SUBROUTINE LINEQ C FOR THE SOLUTION OF LINEAR SIMUL-C TANEOUS EOUATIONS OF THEORETICAL PART C C C * SUBROUTINE F I F I T C FOR THE CURVE F I T T NG PROCEDURE C c

5 CALL EXI T END

168

C C C C C C C C C c c c c c c c c c c c c c c c c

SUBROUTINE ANTHR * ) 0 t * 4 e * ) { t ) { t J 0 t , » : ) ( c « j » : j ( c 4 * j { t j { C ) { c ) < t * ) { c t < t « + * + * 4 * « > » ) * ) » ) » J ^ J ^ j ) r j { . j { . * * ) > j ( c ) ^

C C c c c c C C C c c c c

*

*

*

*

*

SUBROUTINE ANTHR

THIS TS THE COMPUTER PRGGRAMMING FOR ThlE CALCULATION OF THE NECESSARY ANTHRO-TMETRIC DATA FOR ANALYSIS

*

*

*

* * J Í t * J Î : * * * * * * * * : ^ ; ( , , î t : 0 , : ^ ^ 3 j ^ : ^ i ^ 3 ^ ^ ^ 3 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 3 ^ ^ ^ ^ j ^ ^ j ^ j ^ ^

NOMENCLATURES

P W H BW CH UAL AR A l A2 A3 A4 A5 A6 A7

NUMBER OF THE SUBJECTS WEIGHT ( L B S ) HEIGHT ( F T ) BIACROMIAL WIDTH (FT) CHEST HEIGHT (FT ) UPPER ARM LENGTH (FT) LOWER ARM LENGTH (FT) A B D U : T I O N ( A D D U C T I O N ) ANGLE (DEGREE) X-COMPONENT OF MUSCLE LENGTH (MM) Y-:OMPONENT OF MUSCLE LENGTH (NM) Z-:OMPDNENT OF MUSCLE LENGTH (MM) X-COORDINATE OF INSERTION (MM) Y-COORDINATE OF INSERTION (MM) Z-C03RDINATE OF INSFRTION (MM)

DIMENSION P ( 7 ) , W ( 7 ) , H ( 7 ) , B W ( 7 ) , C H ( 7 ) , / U A L ( 7 ) ,AR( 7) , S F X ( 7 ) , S F Y ( 7 ) , S F Z ( 7 ) , Y ( 7 , 1 5 , 1 0 ) , / A l ( 1 5 , 10 ) , A 2 ( 1 5 , 1 0 ) , A 3 ( 1 5 , 1 0 ) , A 4 ( 1 5 , 1 0 ) , / A 5 ( 1 5 , 1 0 ) , A 6 ( 1 5 , 1 0 ) , A 7 ( 1 5 , 1 0 ) , X ( 7 , 1 5 , 1 0 ) , / Z ( 7 , 1 5 , 1 0 ) , X L ( 7 , 1 5 , 1 0 ) , Y L ( 7 , 1 5 , i a ) , / Z L ( 7 , 1 5 , 1 0 ) , T L ( 7 , 1 5 , 1 0 ) , D X ( 7 , 1 5 , 1 0 ) , / D Y ( 7 , 1 5 , 1 0 ) , D Z ( 7 , 1 5 , 1 0 ) , A X ( 7 , 1 5 , 1 0 ) , / A Y ( 7 , 1 5 , 1 0 ) , A Z ( 7 , 1 5 , 1 0 ) , T M ( 7 , 1 5 , 1 0 ) , / D M X ( 7 , 1 5 , 10) ,9MY( 7 , 1 5 , 1 0 ) , D M Z ( 7 , 1 5 , 1 0 )

PRINTING ORDER

I . D E L T 3 I D ANTERlOR 2 . D E L T 0 I D MIDDLE 3 . D E L T 0 I D POSTERIOR 4.SUPRASPINATUS 6 . INFRASPINATUS 7.TERES MAJOR 8.TFRES MINOR 9 . S U B S : APULARIS

lO .PECRORALIS MAJOR (STERNAL) l l . P E C T O R A L I S MAJOR(CLAVICULAR)

169

C C C C

c c c c

1 3 . B I C E P S (SHDRT) 14 ,TR ICEPS 15 .C0RAC0BRACHIAL IS

W R I T E ( 6 , 1) F O R M A T ( 3 X , « S U B J E C T ' , 3 X , • W E I G H T ' , 5 X , » H E I G H T ' ,

/ 6 X , » B W ' , 8 X , « C H « , 7 X , « U A L ' , 8X,« AR' , 7X , ' S F X ' , / 7 X , ' S F Y ' , 7 X , « S F Z ' , / / )

READ ANTHRCPOMETRIC DATA GF INDIV IDUAL ( THESE DATA ARE COLLECTED FROM MEASURING )

DO 4 1 = 1 , 7 R E A D ( 5 , 2 ) P ( I ) ,W( I ) ,H( I ) , B W ( I ) , C H ( I ) ,

/ U A L ( I ) , A R ( I ) 2 FORMAT ( 7 F 1 0 . 4 )

CALCULATION OF SCALE FACTCRS C C C C C

C C C C

SFX SFY SFZ

SCALE FACTOR IN X-DIRECTION SCALE FACTOR IN Y -D IRECTIGN SCALE FACTOR I N Z -D IRECTION

S F X ( I ) = ( B W ( I ) / 1 . 2 5 ) S F Y ( I ) = ( ( C H ( I ) / 1 . 2 5 2 5 ) + ( U A L ( I ) / 1 . 0 3 6 ) ) ' ! ' 0 , 5 SFZ( I ) = S F X ( I ) W R I T E ( 6 , 3 ) P ( I ),W ( I ) , H ( I ) , B W ( I ) ,CH( I ) ,

/ U A L d ) , A R ( I ) ,SFX( I ) ,SFY( I ) ,SFZ( I ) 3 FORMAT( 1 0 F 1 0 . 4 , / / ) 4 CONTINUE

W R I T E ( 6 , 1 2 )

READ GE3GRAPHICAL DATA OF THE MODEL ( MEASJRED FROM THE DISSECTED CADAVOR )

DO 5 1=1,15 DO 5 J = l, 10 REA0(5,6) Al (I ,J) ,A2(I ,J),A3(I,J)•A4( I,J), /A5(I,J),A6(I ,J),A7( I,J )

5 CONTINUE 6 FORMAT(7F10.4)

WRITE(6,12) DO 14 1=1,7 DO 13 J= l, 15 WRITE(6,7)

7 F0RMAT(6X,»SUBJECT» ,3X,«MUSCLE« ,5X,'ANG.' ,9X, / 'X », UX, 'YS IIX, • Z' , lOX, 'LENGTH' ,6X, 'COSXSBX, /'COSY' ,8X, 'COSZ' , //)

170

C C C

c c c c

W R I T E ( 6 , 8 ) 8 FORMAT ( 3 9 X , « L X ' , l O X , » L Y * , lOX , ' L Z ' , 1 0 X , ' T M S ^ X ,

/ • M D O S X ' , 8 X , ' M C 0 S Y « , 7 X , » MCCSZ' , / / )

CALCULATION OF GEOMETRICAL DATA FOR SUBJECTS

DO 11 K = l , 1 0 X( I , J , K ) = S F X ( I ) * A 2 ( J , K ) Y( I , J , K) = S F Y ( I ) * A 3 ( J ,K ) Z( I , J , K ) = S F Z ( I ) * A 4 ( J , K ) X L ( I , J , K ) = S F X ( I ) * A 5 ( J , K ) YL ( Î , J , < ) = SFY( I ) * A 6 ( J , K ) 71( I , J , K ) = SFZ( I ) * A 7 ( J , K ) T L ( I , J , < ) = ( X ( I , J , K ) « « 2 +Y( I , J , K ) « * 2 +

/ Z ( I , J , K ) « * 2 ) * * 0 . 5

CALCULATION OF THE DIRECTION COSINFS CF FORCES &ND MOMENTS

DX ( I , J , K ) = D Y ( I , J , K ) = DZ ( I , J , < ) = AX ( I , J , < ) =

/ Y L ( I , J , K ) AY( I , J , K ) =

/ Z L ( I , J , < ) AZ ( I , J , K ) =

/ X L ( I , J , K ) T M ( I , J , K ) =

/ A Z ( I , J , < ) * DMX( I , J , K ) D M Y d , J , K ) DMZ( I , J , K ) W R I T E ( 6 , 9 )

/ Z ( I , J , K ) ,T / D Z ( I , J , K )

9 F 0 R M A T ( 2 I 1 W R I T E ( 6 , 10

/ T M ( I , J , < ) , 10 F3RMAT(32X 11 CONTINUE

WRITE(6,12 12 FORMAT(IHl 13 CONTINIUE

W R I T E ( 6 , 12 14 CONTINUE

RETURN END

X( I , J , K ) / T L ( I , J , K ) Y ( I , J , K ) / T L ( I , J , K ) Z( I , J , K ) / T L ( T, J , K ) D Y ( I , J , K ) * Z L ( I , J , K ) - D Z ( I , J , K ) *

DZ( I , J , K ) * X L ( I , J , K ) - D X ( I , J , K ) *

D X ( I , J , K ) * Y L d , J , K ) - D Y ( I , J , K ) *

(AX( I , J , K ) * * 2 + A Y ( I , J , K ) ' í = « 2 - ^ * 2 ) « * 0 . 5 = AX( I , J , K ) / T M ( I , J , K ) = A Y ( I , J , K ) / T M ( I , J , K ) = AZ( I , J , K ) / T M ( I , J , K )

I , J , A 1 ( J , K ) , X ( I , J , K ) , Y ( I , J , K ) , L ( I , J , K ) , D X ( I , J , K ) , D Y ( I , J , K ) ,

0 , 8 F 1 2 . 5 ) ) XL ( I , J , K ) , Y L ( I , J , K ) , Z L ( I , J , K ) , D M X ( I , J , K ) , D M Y ( I , J , K ) , D M Z ( I , J , K )

, 7 F 1 2 . 5 , / / )

)

, 5X )

)

•HUB

171

C C C c c c c c c c c c c c c c c c c c c c c c c c c c c c

c c c c c c c c c c c c c c c

SUBROUTINE THEOR * * ) { t * * ) { t * j | t ) { t * ) { t * * ) f j { c * * j { t > > * ) ( c j { t * j » t J ^ j { t j > 4 : * 4 j > * j f * j O t * * « « * ) J « : * ) { t j { t

*

*

*

*

*

*

*

*

*

SUBROUTINE THE3R

THIS I S THE COMPUTER PRCGRAMMING FOR THE CALCULATION OF THE THECRETICAL CALCULAT-lON OF MUSCULAR TENSILE FORCES AND FOR THE FOUNDING OF THE RELATlONSHIPS BETW-EEN THESE VALUES VS. ABDUCTICN(ADDUCTIGN ANGLES OF THE ARM

*

*

*

*

*

* * * * ) { : * ){t)0t>!tj0e>!t*****>!t*jît)^j(cjî:*){t*j0c3îc*4jîrj}!)>«jt*<t*j!tj{c:{tj0tjj«*j0t*

NOMENCLATURES

P H LW W BW CH XUAL XLAL XUAC XLAC XUAM XLAW TW XM

NUMBE HEIGH L I F T I W EI GH BIACR CHEST UPPER LCWER UPPER LOWER UPPER LOWER EFFEC EFFEC

SUBJECTS R OF THE T (FT) NG WEIGHT T (LBS) OMIAL WIDTH

HEIGHT ( F T ) ARM LENGTH ARM LENGTH ARM C . G . (FT ) ARM C . G . ( F T ) ARM WEIGHT (LBS) ARM WEIGHT (LBS)

T IVE WEIGHT (LBS) T I V E MOMENT ( F T - L B S )

(LBS)

(FT)

(FT ) (FT)

REAL N 1 , L X , L Y , L Z , L E N G T H , K 1 , L W , J L B , P INTEGER ZZ

P R I N T I N 3 ORDER

ABDUCTION CASE

1 . REACTION FORCE IN X - D I R E C T I C N 2 . REACTION FORCE IN Y -D IRECTICN 3 . REACTION FORCE IN Z -D IRECTION 4 . LAGRANGE'S MULT IPL IER V I 5 . LAGRANGE«S MULT IPL IER V2 6 . LAGRANGE'S MULT IPL IER V3 7 . DELTOID ANTERIOR 8 . DELTOID MIDDLE 9 . DELTOID POSTERIOR

1 0 . SUPRASPINATJS

172

C 1 1 . INFRASPINATUS C 1 2 . TERES MAJ3R C 1 3 . TERES MINOR C 1 4 . SUBSCAPULARI S

DIMENSION A l ( 2 5 , 2 5 ) , D 1 ( 2 5 ) , X 9 ( 2 5 ) , C ( 1 6 ) , X D ( 1 5 0 ) , / Y D ( 1 5 0 ) , Y C d 5 0 ) , A B D ( 2 0 , 1 0 0 , 1 0 ) , A D D ( 2 0 , 1 C 0 , 1 0 )

C C

C ADDUCTION CASE C

C 1 . REACTICN FORCE IN X -D IRECTICN c 2 . R E A : T I O N F O R C E I N Y - D I R E C T Í C N

C 3 . REACTION FORCE IN Z-DIRECTION C 4 . LAGRANGE'S MULTIPL IER V I C 5 . LAGRANGE'S MULTIPL IER V2 C 6 . LAGRANIGE'S MULTIPL IER V3 C 7 . INFRASPINATJS C 8 . TERES MAJOR C 9 . TERES MINDR C 1 0 . SUBSCAPULARIS C 1 1 . PECTORALIS MAJOR ( STERNAL ) C 1 2 . PECTORALIS MAJGR ( CLAVICULAR ) C 1 3 . BICEPS ( LONG ) C 1 4 . BICFPS ( SFORT ) c 15, T R I : E P S C 16. CORACOBRACHILIS C 17, LATISSIMUS D RSI C

COMMON X(10, 150) ,A( 10) C C READING ANTHROPDMETRIC DATA FOR SINGLE C SUBJECT AND LIFTING WE IGHT C

READ(5,1) P,LW,H,W,BW,CH,XUAL,XLAL 1 F0RMAT(8F10.4)

C C CALCULATION OF C.G AND WEIGHT OF SEGMENTS C

X U A C = 0 . 4 3 5 6 9 * X U A L X L A C = 0 . 5 5 4 4 * X L A L XUAW=0,02647*W XLAW=0.02147*W

C C CALCULAT ON OF EFFECTIVE WEIGHT AND MCMENT C

XLW = XUAL-^XLAL XLAAC=XUAL-^XLAC XTW=XUAW4-XLAW

^ mÊ*m

173

C C C

C C c c

c c c

c c c

2 3 4 5

TW = LW4^XTW

XM=XUAC*XUAW4-XLAAC*XLAW + XLW«LW

DETERMINATION OF SCALE FACTGRS

SFX=(BW/ 1 . 2 5 ) S F Y = ( ( C H / 1 . 2 5 2 5 ) - » - ( X U A L / l . C 3 6 ) ) * 0 , 5 SFZ = SFX

P,LW,H,W,BW ,CH ,XUAL,XLAL XUÛC,XLAC,XUAW,XLAW,XLW,XLAAC XTW ,TW,XM SFX,SFY ,SFZ . 5 )

W R I T E ( 6 , 2 ) W R I T E ( 6 , 2 ) W R Î T E ( 6 , 2 ) WRITE( 6 , 2) F0RMAT(8F15 FORMAT ( 1 2 ) FORMAT ( 7 F 1 0 , 4 , / , 7 F 1 0 . 4 , / , 8 F 1 0 . 4 , / / / ) FORMAT ( » 1 SOLUTIGN OF ' ^ I ^ , * S IMULTANECUS,

/ L I N E A R ALGEBRIC E O U A T I O N • / / , ' 0 COEFFICIENT / MATRIX : • / / )

6 FORMAT ( « 0 SOLUTION

8 9

10

11

12

VECTOR:'//)

MAKE THE COEFFICIENT MATRIX ZERO AND DETERMINE LIFTING METHCDS

READ(5,3) M IF (M,EO.14) GO TO 8 IF (M.EO- 17) GO TO 10 NUM = 21 DO 9 1=1,14 Dl(I)=0. Al( I ,J)=0. GO TO 12 DO 11 1=1,17 Al (I ,J)=0. Dl( I ) = 0.

INITIAL DATA FGR MOMENT COEFFICIENT.

SUMXX=0. SUMXY=0. SUMXZ=0. SUMYY=0. SUMYZ=0. SUMZZ=0.

REACTION FORCE CO EF F IC I ENT S .

1=6 Al ( 1 , 1 ) = 1 , 0

áhgna TiBsaiíí møm

174

C C c c c c c c c c c c c c

c c c

Al ( 2 , 2 ) = 1.0 A l ( 3 , 3 ) = 1 . 0

C A L C U L A T I O N OF L E N G T H , D I R E C T I O N COSINES OF FORCES AND MGMENTS FROM BASIC DATA

N = ABDUCTIONI OR ADDUCTION CER3REE XX=X-COMPONENT OF LENGTH VECTOR

Y=Y-COMPONENT OF LENGTH Z=Z-COMPONENT OF LENGTH LX=X-COMPONENT 3F MOMENT LY=Y-COMPONENT OF MOMENT LZ=Z-COMPONENT OF MGÊNT

VECTOR VECTOR ARM VECTOR ARM VECTOR ARM VECTOR

READ GEOGRAPHICAL DATA (MOCEL CADAVOR)

13 READ ( 5 , 1 4 ) 14 FORMAT ( 7F

1 = 1+1 TF ( N . E O . 2 0 0 . ) I F ( N . E Q . 2 1 0 . ) I F ( N . G T . 1 0 0 . )

N , X X , Y , Z , L X , L Y , L Z 1 0 . 3 )

GO G3 G3

TO TO TO

25 7 15

CALCULATION OF ACTUAL ANTHRCPOMETRIC DATA

X X = ( S F X * X X ) / 3 0 4 . 8 Y=( SFY'î^Y) / 3 0 4 . 8 Z = ( S F Z * Z ) / 3 0 4 . 8 L X = ( S F X - ^ ^ L X ) / 3 0 4 . 8 LY = ( S F Y < ' L Y ) / 3 0 4 . 8 LZ = (SFZ'! 'LZ ) / 3 0 4 . 8 LENGTH = ( X X * * 2 - ^ Y « * 2 - ^ Z * * 2 ) * * 0 . 5 DCOSFX =XX/LENGTH DCOSFY = Y / LEN3TH DCOSFZ = Z / LENGTH AX=DCOSFY*LZ-DCOSFZ*LY AY=DCOSFZ«LX-0C0SFX*LZ AZ=DCOSFX*LY-DCOSFY*LX K l = ( A X * * 2 + A Y * * 2 - » - A Z « * 2 ) * * 0 . 5 DC0SMX=AX/K1 DC0SMY=AY/K1 DC0SMZ=AZ/K1 DCOMXX=-(DCOSMX*DCOSMX) DCOMXY=-(DCOSMX*DCOSMY) DCOMXZ=-(DC0SMX*DCOSMZ) DCOMYY=-(DCOSMY*DCOSMY) DCOMYZ=-(DC0SMY*DC0SMZ) D : 0 M 7 Z = - ( D C 0 S M Z * D C G S M Z )

175

C C C

c c c c

c c c

SUMXX=SUMXX+D:OMXX SUMXY=SUMXY+D:OMXY SUMXZ=SUMXZ+D:OMXZ SUMYY=SUMYY+D:OMYY SUMZZ=SUMZZ+D:GMZZ SUMYZ=SUMYZ+D:OMYZ DE = 0 . 0 1 7 4 5 3 2 * \ i

SETTING COEFFICIENT ÂTRIX

A l ( 1,1 ) = DCOSFX Al ( 2 , 1 ) = DCOSFY A l ( 3 , 1 ) = DCOSFZ A l ( I , 4 ) = ( ABSOCOSMX) ) / 3 0 4 . 8 A l ( 1,5 )= ( ABS(DC3SMY) ) / 3 0 4 . 8 A l ( I , 6 ) = ( A B S ( D C 0 S M Z ) ) / 3 0 4 . 8 A l ( I , 1 ) = K 1 GO TO 13

15 N = N - 1 0 0 . 0 A l ( 4 , 4 ) = ( S U M X X ) / 3 0 4 . 8 A l ( 4 , 5 ) = ( S U M X Y ) / 3 C 4 . 8 A l ( 4 , 6 ) = ( S U M X ? ) / 3 0 4 . 8 A l ( 5 , 5 ) = ( S U M Y Y ) / 3 0 4 . 8 A 1 ( 5 , 6 ) = ( S U M Y Z ) / 3 0 4 . 8 A l ( 6 , 6 ) = ( S U M Z Z ) / 3 0 4 . 8 A l ( 5 , 4 ) = A 1 ( 4 , 5 ) A l ( 6 , 4 ) = A 1 ( 4 , 6 ) A 1 ( 6 , 5 ) = A 1 ( 5 , 6 )

16 D1(2 )=TW D l ( 4 ) = X M * S I N ( D E ) WRITE ( 6 , 5 ) M DO 17 1=1 ,M

17 WRITF ( 6 , 4 ) ( A l ( I , J ) , J = 1 » M ) , D 1 ( I ) WRITE ( 6 , 1 8 )

18 FORMAT ( I H l , 5X ) WRÍTE ( 6 , 6 )

CALL FOR SGLUTION OF LINEAR SIMULTANEOUS EOUATION I N THE FORM OF MATRIX

CALL L I N E 0 ( A 1 , D 1 , X 9 , f )

WRITE THE SOLUTION VECTOR

WRITE ( 6 , 4 ) ( X 9 ( I ) , 1 = 1 , M ) WRITE ( 6 , 1 8 ) I F ( L W . E 3 . 0 . 0 ) GO TO 19 NN = N

176

C C C C C C

C C C C

GO TO 22

NW = LW GO TO 20

19 NN=N LW = LW-»-1.0 NW = LW

20 IF(M.EQ.17) DO 21 1 = 7,M

21 ABD( I,NN|,NW) = X9( I ) GO TO 24

22 00 23 1=7,M 23 ADD( I ,NN,NW) =X9( I ) 24 CONTINUE

IF(M.EQ.14) GG TO 8 IF (M.EQ. 17) GO TO 10

25 CONJTINUE DO 26 1 = 7 , 14 DO 26 NN = 1 0 , 9 0 , 1 0

26 W R I T E ( 6 , 2 7 ) I , NN , ABD ( I , NN , NW) 27 FORMAT ( 1 4 , 1 1 5 , F 1 0 . 4 )

00 28 1 = 7 , 1 7 i '\.j 1 . 1 > » — t , i f

DO 28 N N = 1 0 , 9 0 , 1 0 W R 1 T E ( 6 , 2 9 ) I , N N , A D D ( I ,NN,NW)

29 FORMAT ( 1 4 , 1 1 5 , F 1 0 . 4 ) 28 WR

CURVE F I T T I N G OF THE THEGRETICAL SCLUTIGN

KK KD

NUMBER OF ORDER ( 5TH NUMBER OF POINTS ( 10

R E A D ( 5 , 3 0 ) KK,KD 30 F O R M A T ( 2 I 4 )

KOPl = KD + 1 K K P l = <K + 1

SCALE FACTCRS FOR X AN D Y ( Y IS MUSCULAR TENSl L E ,

S C Y = 1 . 0 SCX=1. 0 PP = 1 . 0 Y = 0 . 0 0 1 X 1 = 0 . 0 YD( 1 ) = Y X D ( 1 )=X1 Y=SCY*Y X1=SCX*X1 X( 1 , 1) = 1 .0 X ( 2 , 1 ) = X 1

CATA X I S ANGLE )

^ ^

177

c c c

31 32

33 34

35

X( 3 , 1 ) = X 1 * * 2 X ( 4 , l ) = x 1 * * 3 X( 5 , 1 ) = X 1 * * 4 X ( K K P 1 , 1 ) = Y 11 = 7 X l = 1 0 . 0 DO 35 1=2 ,KD NX=X1 NW=LW I F ( P P . G T , 8 . 0 ) GO Y = A B D ( I I , N X , N W ) GO TO 34 Y=ADD( I I , N X , N ^ ) CONTINUE Y D ( I ) = Y X D ( I ) = X 1 Y=SCY*Y X1=SCX*X1 X( 1 , I ) = 1 .0 X ( 2 , I ) = X 1 X ( 3 , 1 ) = X 1 * * 2 X ( 4 » I ) = X 1 * * 3 X( 5, I ) = X 1 * * 4 X ( K K P l , 1 ) = Y X 1 = X 1 + 1 0 . 0

CALL CURVE F I T T I

TO 33

C c c

CALL F I T I T ( K D , K D P 1 , K K , K K P 1 ) PRINT 36

36 FORMATI 1 H 0 , 3 0 X , / 34HTÊ CALCULATED CCEFFICIENTS ARE AS, / 9H F O L L O W S - / / )

ACTUAL COEFFICIENTS ARE PRINTED OUT

DO 38 J = 1,KK W R I T E ( 6 , 3 7 ) J , A ( J )

37 FORMAT( 1H0 ,44X ,2HA( , 12 , 4H) 38 CONTINUE

77 = 0 JLB = 0 , 0 SS = 0 . 0 P = 0 . 0 SD = 0 . 0 SUM = 0 . 0 DO 40 J = 1 ,KD T = X ( K K - H , J)

= , 3 X , E 1 2 , 5 )

178

C

c c

39

40 41

/ / / / / /

42 43

44 45

46 / /

47

48

49

G = 0 . 0 DO 39 K = 1 , K K Ql = X ( K , J ) * A ( K ) G = G + Q 1 Y C ( J ) = G JLf i = JLB 4- ABS( ( T - G ) / T ) T = T - G

I F ( T . L T . O . 0) ZZ = ZZ - 1 S S = S S •»• T

P = P + ABS (T) SD = SD - G*G SUM = SUM 4- T * T FDRMAT( IHO,20HNUMBER OF DATA P 0 I N T , I 4 / 10X,18HSQUARED DEVIATION , E 1 2 , 5 / 10X ,10HDEVI ATION , E 1 2 , 5 / lOX,18HA3SDLUTE D E V I A T I O N , E l 2 10X,24HSUM OF THE SQ, OF CAL, 10X,30HNUM3ER OF DATA PT, GT.

5 / Y , E 1 2 . 5 / STAND. , 1 4 /

= , E 1 2 . 5 )

10X,24HSUM OF THE AVG D E V I A T I O N , E 1 2 . 5 / / ) W R I T E ( 6 , 4 1 ) K D , S U M , S S , P , S D , Z Z , J L B A(KK-e l ) = 1 . 0 DO 43 K = 1 ,KKP1 AV = 0 . 0 00 42 J = 1,KD AV = AV - X (K , J) C ( K ) = A V * A ( K ) / F L O A T ( K D ) DO 45 J = 1 ,KKP1 W R I T E ( 6 , 4 4 ) J , C ( J ) FORMAT( H O , 10X,2HC( , I 2 , 3 H ) CONTINUE PRINT 46 F O R M A T ( / / / 1 7 X , 1 6 H I N D E P E N D E N T DATA,8X , 14HDEPENDENT DATA ,8X,16HCALCULATED VALUE, 8 X , 9 H D E V I A T I 0 N , 8 X , 1 3 H P E R C E N T E R R O R / / / ) DO 49 I = 1,KD Y C ( I ) = 0 . 0 DO 47 J = 1,KK Y C ( I ) = Y C ( I ) + A ( J ) * X ( J , I )

CHAN3E Y C d ) ONLY I F DATA IS SCALED

Y C ( I ) = YC( I ) / S C Y DEV = Y D ( I ) - Y C ( I ) PCE = 1 0 0 . 0 * A B S ( D E V ) / Y D ( î ) WRITE ( 6 , 4 8 ) XD( I ) , Y D ( I ) ,YC( I ) ,DEV,PCE F n R M A T ( l H 0 , 2 1 X , F 9 . 2 , l l X , F 1 1 . 6 , l ? X , F l l , 6 ,

1 2 X , F 8 , 5 , 1 2 X , F 8 . 3) CONTINUE

^

179

50

WR I T F ( 6 , 18) P P = P P + 1 , 0 1 1 = 1 1 + 1 IF ( P P . 3 T , 1 9 , 0 ) I F ( P P , L E . 8 . 0 ) I F ( P P . E Q . 9 . 0 ) GO TO 32 CONTINUE RETURN END

3 0 TO 5 0 GD TO 32 GO TO 31

180

C C C c c c G C C C c c c

SUBROUTINE F I T IT ( N , N P 1 , M , M P l ) • « « « « : ( t < t ^ j ! t : í t : í t : > : { t « * : í c : O t * « * ) < f ) í t ) ! t : í c ) O c j O e > } t J Î i j { í ^ j O c j O t j ; t j { t j O t j O t J * ; j O t * j O t « ) O t 3 » j j c *

30 40 50

60 92

70 80

90

*

*

SUBROUT INJE F IT IT

THIS PROGRAM IS FOR THE CALCULATION OF OF INVERSE MATRIX THAT I S USED FOR THE CURVE F Î T T I N G PRGBLEM

95 100 110

) 0 t j { t * * ) 0 c j ( t * > 5 t s ( e * 4 t ) ( t * : t < : * 4 c j } e j O : j 0 t j O c O c j ^ « « « ) O t ) í t j ! t « j î í : * « : O 5 « « • * * * « « « « «

M IS NUMBER OF COEFFICIENTS N IS NUMBER OF DATA POINT A ( I ) ARE THE OESIRED COEFFICIENTS

COMMON X( 10 , 1 5 0 ) ,A ( 10) DIMENSION Z ( 1 0 , 1 5 0 ) DO 50 I = 1,M DO 40 J = 1,MP1 Z ( I , J ) = 0 . 0 DO 30 K = 1 ,N Z d , J ) = Z ( I , J ) + X ( I , K ) * X ( J , K ) CONTI NUE CONTI NUE DO 110 KM = 1 ,MP1 K = M -f 2 - KM D = 0 . 0 DO 92 I = 2 ,K Í F ( A B S ( Z( I - l , 1 ) ) . L E . D) GO TO 60 L = I - l D = A B S ( Z ( L , 1 ) ) CONTINUE CONTINJE I F ( ( L - l ) . E 0 . 0 ) GO TO 80 DO 70 J = 1 ,K D = Z ( L , J ) Z ( L , J ) = Z ( 1 , J ) Z ( 1 , J ) = D CONTINUE DO 90 I = 1 fM A d ) = Z( 1,1) DO 100 J = 2 , K D = Z( 1 , J ) / A ( 1 ) 00 95 I = 2 ,M Z ( I - 1 , J - 1 ) = Z d , J ) - A ( I ) * D Z ( M , J - 1 ) = D CONTINUE RETURN END

181

C C C C c c c c c c c c c c c c c c c

SUBROUTINE L I N EQ ( A , B ,X , N ) * * * « * * « « ' 5 c * « * * i ; ' ^ « í ! t ) < t « ) } : > ! f 4 : * j O t J O c : ^ 3 ! f « « j } t j O t 5 ! f j O c j O t « ) î t : { t « j O c j O c : { i : O t « 3 0 ! j ( t

* *

* SUBROUTINE L INEQ * * «

* T H I S IS THE SUBROUTINE FCR THE SOLUTION * * OF LINEAR SIMULTANEOUS EQUATION * * if

j O t : î : < t * « J O c 4 t « > ! t : O c « * « « * * * « « * ) { t ) { t ) ! t « ) { : ) { t J Î t ) O t J ! < J Î c « « j { t > î t > 5 i > ! e j O : : { t « : O t « « « « ' í t

FUNCTIGN RFFERENCES

THE COEFFICIEMT MATRIX ( A ) THE FORCE VECTOR (B) THE NUMBER OF EQUATICNS (N)

THE SUBÔUTINE WILL RETURN THE SOLUTIGN VECTOR (X ) TO THE CALLING PROGRAM

DIMENSION A ( 2 5 , 2 5 ) , B ( 2 5 ) , X ( 2 5 )

DO 4 I = 1 , N DO 2 K = 1 , N

F ( K . E 3 . I ) GO TO 2 CONST = - A ( K , I ) / A ( 1 , 1 ) DO 1 J = l ,N A ( K , J ) = A ( K , J ) 4 - C 0 N S T * A ( I , J ) IF ( J . E O . I ) A( K, J ) = 0 .

1 CONTINUE B ( K ) = B ( K ) + C O N S T * B ( I )

2 CONTINUE CONST=A( 1 , 1 ) DO 3 J = 1 , N

3 A( I , J ) = A ( I , J ) / C O N S T A ( I , 1 ) = 1 . B ( I ) = B ( T ) /C3NST

4 CONTINUE DO 5 I = 1 , N

5 X ( I )=B( I ) RETURN END

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