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
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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
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 < i> .n H
n
:<
• f ^ " v - ^ r - " j ' • ' ii^AttåJiÉÊMXãSíÊaMÊA •
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.
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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.
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94
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:c
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a D <
E.M.G. Potential Intensity (arbitrary)
Fiqure 6.1. Muscular Tension vs. E.M.G. Intensity - Subject (1)
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95
r 50 0 10 20
E.M.G. Potential Intensity (arbitrary)
1 r 60 70 80
:
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D tl • >
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Figure 6.2. Muscular Tension vs. E.M.G. Intensity - Subject (2)
96
JD
100-
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h-s -03
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s • j>
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Figure 6.3. Muscular Tension vs. E.M.G. Intensity - Subject (3)
SiíMk^X- •<•-•'•*'<-••, •'«
97
<
•'j
.1
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Figure 6.4 Muscular Tension vs. E.M.G. Intensity - Subject (4)
p í . * s B f t , ^ - ' •'•• - T~'-"
98
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s-fO
=3
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-
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E.M.G. Potential Intensity (arbitrary)
1 <
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Figure 6.5. Muscular Tension vs. E.M.G. Intensity - Subject (5)
99
T 1 1 1 1 1 r 0 10 20 30 40 50 60 70 80
E.M.G. Potential Intensity (arbitrary)
Figure 6.6. Muscular Tension vs. E.M.G. Intensity - Subject (6)
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.
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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
\
î
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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^S^S^flHl^^^
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tei»-jfc-^EÆW-=^-"- ^^-'
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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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o r - o j r - o o r ^ o o o x . - ( r - ( r \ j r o m m > t v r > t
i / ) rrr-i Lu m. cr o >t o o >t o CT
I r - ( , — ( r - ( r o f ^ J m m
c m r o m o o v j - O v t c ^ x
x > t L n f ^ J ^ o v i ' 0 ~ x C _( .—( 00 m m m <f o
cc OooLL ' m vo OL r> _( m o o m t-d :x. —I ,—I .—( r\.i í x r^ - J 11. 'm. ^ Ov.' i>- m. — o ^ f\i j — LC^LL • • • • • • • • • z: m c f\) < r- r-( o 00 c ;- — ( _ j r - ^ r \ j r \ i m > t
00 Lu f\i m >t vt '• r X o 00 T. •—I •—*
_J ^ - o c o j m m r - x o
0 UJ —) •£.
0 l / )
CJ. L-
0
z <
00
0 _ H
-t
0 OvJ
• 0
0 m
f—-
0 >t
' • ( ' ,
0 m.
V .-
f—t
0 0
(^J
r _ (
0 r-
n r - (
0 X '
^•v
_(
0 0
•,ir W».*-iWW»>>ig m.' *T"i
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
- 7 . C 5 0
0 . 2 9 6
C.085
- 1 . 3 2 0
- 0 . 9 0 6
- 0 . 1 5 0
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
...•-O. .-^..^.-^^.r.^..., , , ..,,>.•.,,>.-
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 B^aaaisa
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
nr^i i - ~
APPGNDIX (III) TABLE 3- 4. COEFFICIENTS OF THEGRETICAL SOLUTION OF MUSCULAR TENSION
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«S^a^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
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 )
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
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
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.
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 MG^ENT
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 ^ATRIX
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 , / 34HT^E 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 SUB^OUTINE 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