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State variable approach to the function of the muscular system supporting the hip joint during normal walking
STATE VARIABLE APPROACH TO THE FUNCTION OF THE MUSCULAR SYSTEM SUPPORTING THE HIP JOINT DURING NORMAL WALKING INTRODUCTION The motion of the hip joint is studied in the following two planes: the x-y plane, or the flexion-extension plane, along which the moment M, acts; and the y-z plane, or the abduction-adduction plane, where M, is the operating moment. The projection of the motion onto those planes is consistent with the theory of Aizerman-Andreeva’, according to which, the motion in space is a result of the activity of two or three independent moment mechanisms. The number of mechanisms is equal to the number of degrees of freedom of the jaints. The motion of the hip joint is assisted by forces produced by the subgroup of muscles supporting the joint. In this model, these forces are applied on an elliptical cross section, perpendicular to the respective muscular fibres; for flexion-extension motion, this cross section is perpendicular to the femur (F@W I) while for the abduction-adduction motion the cross section forms 45” angle with the horizontal plane X-Z (Figure 2). The active distance of each subgroup of muscles, is evaluated with the assumption that the excitation of the muscle fibres is proportional to the square of their distances
The elliptical cross section perpendicular to the Figure 1 femur used to calculate the flexion-extension forces (FJ 0 1987 Butterworth & Co (Publishers) Ltd 0141~5425/87/010088-06 $03.00 88
1. Biomed. Enn. 1987. Vol. 9, lam~5’
from the head of the femur, which is considered the centre of the axes. For a normal man these distances have already been studied in previous work3, they are: Active distance d, = 3.4 cm Active distance d, = 7.1 cm Active distance d, = 4.6 cm Active distance d4 = 6.4 cm
of the flexion muscle sub-group: of the extension
muscle sub-group:
of the abduction
muscle sub-group:
of the adduction
muscle sub-group:
The components of the flexion-reflexion with respect to the X, y, z axes are: F
kx
Fkr Fkz
as
force, Fk,
= Fk sin rh = Fk COS ‘yh =
0
where the subscript k has the value of 1 or 2, depending on whether flexion or extension is considered respectively. yh is the angle between the femur and the vertical. Similarly, for the components of the abduction-adduction force F,,,, we have: F mx
= 0
F my = F,
sin 45”
F mz
cos 45’
= F,
Figure 2 The elliptical cross section, formirig a 45’ angle with the horizontal plane x-y, used to calculate the abductionadduction forces (F,)
Figure 8 extension
Block diagram describing the role of the muscular plane and the abduction-adduction plane
where m = 3 or 4. Now, the mechanical M,, M, are given by the equations:
M,
= F,d,
+ F,d,
- F,k
M,
= F,d3
-F,d4
-F,qcoSYh
transfer function, as well as the coupling between the flexion -
moments
cos 45” - F4k cos 45’ -F,qcosyh
where k and p are the distances of the head of the femur from the main axes of the elliptical cross section on which the forces are applied, with k = 3.4 cm, q = 1.1 cm. The forces F,, F,, and F,, F,, do not appear together. A coupling existing between the two sub-systems which describes the plane flexion-extension and abduction-adduction also appears in the block diagram (Figure 3). The transfer function P(S) simulates the operation of the muscular system and is expressed as:
fw = p-f&q
simulate the muscular at: a1
=
1.4 rad s-i
a2
=
14 rad s-i
transfer function
with poles
In Figure 4 is shown the Bode diagram of this function. The values of the moments
40.
M, and M,, which are
, ,
I
I
It is known from EMG recordings that a muscle can operate at frequencies approaching 10 Hz, while the basic frequency of normal walking is 1 Hz. Therefore a second order low pass filter is used to
10 1..
Figure 4 function
: H
loo
Bode diagram of the second order muscular
J. Biomed.
transfer
Eng. 1987, Vol. 9, J~UV
89
Equations
and 2 can be written,
1
kl
Fk
M,(s) -M?(s)
f=l-pW
=
Frill(4 Mx (4 - M,*(s)
=&
or, using the inverse Laplace transform, Fk + (a+b)
&
+abFk
zm + (a+b) im
= klMz
+abF,
-KIM;
=k,MX
--KIM,*
The coupling of the two subsystems, the block diagram of Figure 3, yields M;
= Fkdk
MZ=
F,,,d,,,
+ k,k
Moments M, and M, operating in the flexionFigure 5 extension plane and abduction-adduction plane respectively
to
(8)
5 and 6 can be written - (ab+k, dk) Fk
cos 45’ F,
+ klM,
(u+b) I$,, - (ab+k 1d,)
+ k,q
(6)
(7)
-Fkqcosrh
= - (u+b) I$
Fm = -
according
- F,,, k cos 45’
so that equations Fk
(5)
(9)
F,
cos 7h Fk + klM,
(10)
Choosing as state variables of the system the forces FL and F, and their derivatives, we have the inputs of the model, are taken from the work of Paul* and they are indicated on Figure 5.
Xl
=
Fk
x2
=
Xl
x3
=
F,
xq
. = x3
. DESCRIPTION OF THE STATE VARIABLES
SYSTEM
WITH
The inputs of the model are Laplace transforms of the moments M, and M,. The error signal M, = M - M” is applied at the input of the system with transfer function P(S). The outputs of the subsystems, which are the Laplace transforms of the forces Fk or F, are given by the equations Fk
M,(s)
ts)
- Mz*@)
= P(s)
Frn(4 M,(s)
-Mx*(s)
where
k =
90
J. Biomed.
1
=
pts)
or 2 and m = 3 or 4.
Eng. 1987, Vol. 9, January
(1) (2)
and, using equations
9 and 10 we obtain
i,
=
;,
= (-ub+kldk)xl
-((a+b)x2
+k,k
+klM,
x2
cos45°x3
323
=
x3
. x4
=
k, q cos rnxl
- (a+b)x,,
- (ab+k1d,)x3
+ k 1M,
1 1)
SOLUTION
OF THE
SYSTEM
The system of equations connective form: G(t)
= Ax(t)
(11)
discrete force values; this creates a problem which may be overcome by solving the system separately for each of the intervals, as follows.
can be written in the
+ Bu(t)
(12)
where 0
1
0
0
-(ab+k,dk)
-(a+b)
k, k cos 45”
0
A= 0
0
0
k,q
OF SUBSYSTEMS
To overcome the difficulty of calculating the system of equations (12) is separated following two subsystems.
-&+b)
I
1
-(ab+k
0
c”s?‘h
SEPARATION
0
+
1dm ) -(a+b)
klk
(sl-A)-‘, into the
’
I
0
cos45’
k1 I
’
I
(134
and 0
B=
0
k,
0
0
0
0
k,
The solution equation
1
s
t$(t-A) B*U(A)d,
(14)
where
L-’
(15)
(~1 __A)-’
Such a solution exists only if matrix A is timeinvariant and not zero, but matrix A, does not vary with time, since angle yh changes during normal walking; the angle yh varies from +20” to -20” around the vertical position of the femur (yh = 0)4. The resultant variation of cos yh is of the order of 0.05, and therefore the approximation cos yh = 1 can be used for the complete period of normal walking. A second reason for the time dependence of the matrix A is the variation of the parameters dk and d,,, according to the applied forces. Thus, for some time intervals k = 1, while for some others, k = 2. (Similarly for m = 3 or m = 4). The Table I shows these forces, after reference 2. It can be seen from the table that there are five time intervals with Time intervals within the flexion-extension forces Table 1 (F,, F,) and abduction-adduction forces (F3, F4) appearing during one step cycle
0 20 70 88 95
- 20% - 70% - 88% - 95% - 100%
Force states FZ 6 FL
FZ F2
F, F, -
0
klq
SOLUTION INTERVAL
F4
Cos’Yh
Xl
.
k1 1
I I
x4
.Ki
(17)
OF THE SYSTEM FOR THE O-20% OF THE WALKING CYCLE
d2 = 0.071 d, = 0.046 Similarly,
m m
the other accepted
a = 1,4rads-‘,b
k,
= 19600
O
-15.4
0
471
+
m and q = 0.011
m.
two systems appear as follows:
0 +
k = 0.034
[ -1411
=
values are
= 14rads-‘, $
Consequently,
[::I= Fd F4
0
[
According to Table 1, over the interval O-20% of the walking cycle, forces F2 and F3 are exerted. Therefore the values of the parameters are
[ij
Time intervals
l
This separation is based on the idea that the term employing xj expresses the influence of the movement of they-z plane upon the movement on the plane x-y. Thus, parameter x3 can be considered as a secondary input for the first subsystem. Similarly, parameter xi can be considered as a secondary input for the second subsystem.
0
=
[-(ab:k&J(-(:+b)]
VW
t
4(t) = &’
=
+
of the above system is given by the
x(t) = tq+(o) +
x3 ” x4
19600
L-9023
I II x3
0
0
202
19600
M,
.
Al
(184
l
[::I
(18b)
I [I MX *Xl
J. Biomed. Eng. 1987, vd.
9, January
91
Therefore, (sZ-Al)
the results for the first subsystem =
]
IlII
s+;;,l
with det @Z-A,) = (~(-7.7
are k’
(19)
.
i
+ 37i)) (~(-7.7
- 37i) (20)
The inverse matrix is
.
F3 100.
(sZ-Ar )-’ =
s+15.4
1 (s-(-7.7+37i))(s-(-7.7-37i))
(s-(-7.7+37i))(s-(7.7-37i)) -1411 (s-(-7.7+37i))(s-(-7.7-37i)
(s-(-7.7+37i)s(s-(-7.7-37i)) I
(21) In
to find
01 0
inverse ks’
1 = cpt
+
(t-7)
3 47)
f
5
1. l-
d7
*0
Figure
and
I
The evaluated abduction-adduction
(F,,,) force
(23)
44
=
#+-to)
X0
1
&(t-7)
B u(7)
dr
to where
The matrices Gi(t--to) and @,(t-t,,), t--to = 0.05 s, are
for the step
0 *(t) has been found similarly to @i(t). 0.0006 0.66
I
(24)
t
y” #,(O.OS)
II 20@-
=
0.66
0.0003
0.55
0.65
I
The final equations
I
(25)
for the time interval 0.05% are
k*
x1(t)
= 0.67 x1 (to) + 0.108
x3(t)
=
0.66 x3(to)
+ 0.038
x3 + 4.5 M, x1 + 3.7 M,
(26)
The solution of the system for the remaining intervals is similar to the solution presented in the previous paragraph. The outputs of the model, i.e. the forces Fk and F,, are depicted in Figwes 6 and 7.
60
Figure 6
92
The evaluated flexion-extension
J. Biomed. Eng. 1987, Vol. 9, January
80
t-%
(FJ force
100
SUMMARY AND CONCLUSIONS In Figures 6 and 7, the results of the forces Fk and F, are analytically represented. The signs of these
forces show the direction of the corresponding moment which emanates from this force, since this sign is stable throughout the entire duration of the interval. The resultant moment, included in the equation 7 or 8, is expressed as Fkdk or F,,,d, respectively. The direction of this moment is positive if F, or F3 is operational, and negative if F2 or F., is operational, therefore the sign attached to this force must follow the sign of the corresponding moment. The results show that this is satisfied for all values, except for those at the 65 and 75% point of the period. This is probably because the value of the corresponding moment is very small, about 0.1, and the resultant force is primarily due to the effect of the moment M,. In this work, the flexion-extension and abductionadduction forces, Fk, F, respectively, were found to take their maximum and minimum values at times which are proximal to those found by Paul2 and other author?. On the other side there is a difference in values which fluctuates between 15 and 35% at various points; this divergence should be generally acceptable, given that it concerns models describing biological systems. These results seem to lend encouraging support to our method and suggest that the model should be reliable and effective and could be further enhanced if the time intervals were shorter but more numerous.
REFERENCES Aizermann, M.A. and Andreeva, Y.A. A study of movement control systems in living beings. In: A.S. Iberall and J.B. Reswick (eds) IFAC Congress, France, June 1972, 23-34 Paul, J.P. Forces at the human hip joint. PhD Thesis, University of Strathclyde, 1967 Bonnemay, A., Koukis, M., Couny, M., Souza, P.K. Modele analogique pour le c&u1 de forces appliquees a la tete du femur. SES/PUB/SAI/73-192, Commissariat a l’energie atomique, France, 1972, l-36 Lamoreaux, L.W. Kinematic measurements in the study of human walking. Bull prosth Res 197 1, 4, 112-l 24 Smith, A.J. A study of forces on the body in athletic activities with particular reference to jumping. PhD Thesis, University of Leeds, 1972 Morrison, J.B. The mechanics of the knee joint in relation to normal walking. J. Biomech 1970, 3, 431-451 O’Kelly, J., Unsworth, A., Dowson, D., Hall, D.A. and Wright, V.A. Study of the role of synovial fluid and its constituents in the friction and lubrication of human hip joints. Eng. in Med. 1978, 7, 73-83 Alexander, R. McN. and Vernon, A. The dimension of knee and ankle muscles and the forces they exert. J. Human Movement Studies 1975, 16, 1-19 Ellis, M.I., Seedhom, B.B. and Wright, V. Forces in the knee joint whilst rising from a seated position J. Biomed Eng 1984, 6, 113-120
Maria Grigoriadou-Koukis, University of Athens, Electronics Laboratory, Panepistimioupoli-Ktiria TYPA, 157 71 Athens. Greece
VIBRATORY PROPERTIES AND RESONANCES OF THE ISOLATED HUMAN ULNA Dear Sir, The purpose of the present letter is twofold: first to congratulate Dr E.J. Evans on his paper, Vibratory properties and resonances of the isolated human ulna, (J. Biomed, Eng 1985; 7; 144-148) and second to present some observations which may be of interest to the reader. I agree completely with the statement made by the author, ‘these results indicate that the vibratory behaviour of the ulna is strongly influenced by its complex geometry and it would be unreasonable to
0 1987 Buttenvorth & Co (Publishers) 0141-5425/87/010093-001,$03.00
assume that bones from two individuals have the same vibrational characteristics’. I would like to add that it seems reasonable to expect that for any two individuals the vibratory properties of a given bone will be influenced, perhaps in different ways, by the mechanical characteristics of the muscles attached to the bone. Certainly these muscles will possess inertial, restoring and damping characteristics which may differ greatly between individuals and therefore contribute in a significant manner to the overall vibrational characteristics of a complex dynamic system: bone and attached muscles. The differences will probably be more noticeable ‘in rho’. Yours faithfully,
Patricia A.A. Laura Director and Research Scientist, Institute of Applied Mechanics, Puerto Belgrmo Naval Base, 8 11 l-Argentina
Ltd J. Biomed. Eng. 1987, Vol. 9, January
93
The elliptical cross section perpendicular to the Figure 1 femur used to calculate the flexion-extension forces (FJ 0 1987 Butterworth & Co (Publishers) Ltd 0141~5425/87/010088-06 $03.00 88
1. Biomed. Enn. 1987. Vol. 9, lam~5’
from the head of the femur, which is considered the centre of the axes. For a normal man these distances have already been studied in previous work3, they are: Active distance d, = 3.4 cm Active distance d, = 7.1 cm Active distance d, = 4.6 cm Active distance d4 = 6.4 cm
of the flexion muscle sub-group: of the extension
muscle sub-group:
of the abduction
muscle sub-group:
of the adduction
muscle sub-group:
The components of the flexion-reflexion with respect to the X, y, z axes are: F
kx
Fkr Fkz
as
force, Fk,
= Fk sin rh = Fk COS ‘yh =
0
where the subscript k has the value of 1 or 2, depending on whether flexion or extension is considered respectively. yh is the angle between the femur and the vertical. Similarly, for the components of the abduction-adduction force F,,,, we have: F mx
= 0
F my = F,
sin 45”
F mz
cos 45’
= F,
Figure 2 The elliptical cross section, formirig a 45’ angle with the horizontal plane x-y, used to calculate the abductionadduction forces (F,)
Figure 8 extension
Block diagram describing the role of the muscular plane and the abduction-adduction plane
where m = 3 or 4. Now, the mechanical M,, M, are given by the equations:
M,
= F,d,
+ F,d,
- F,k
M,
= F,d3
-F,d4
-F,qcoSYh
transfer function, as well as the coupling between the flexion -
moments
cos 45” - F4k cos 45’ -F,qcosyh
where k and p are the distances of the head of the femur from the main axes of the elliptical cross section on which the forces are applied, with k = 3.4 cm, q = 1.1 cm. The forces F,, F,, and F,, F,, do not appear together. A coupling existing between the two sub-systems which describes the plane flexion-extension and abduction-adduction also appears in the block diagram (Figure 3). The transfer function P(S) simulates the operation of the muscular system and is expressed as:
fw = p-f&q
simulate the muscular at: a1
=
1.4 rad s-i
a2
=
14 rad s-i
transfer function
with poles
In Figure 4 is shown the Bode diagram of this function. The values of the moments
40.
M, and M,, which are
, ,
I
I
It is known from EMG recordings that a muscle can operate at frequencies approaching 10 Hz, while the basic frequency of normal walking is 1 Hz. Therefore a second order low pass filter is used to
10 1..
Figure 4 function
: H
loo
Bode diagram of the second order muscular
J. Biomed.
transfer
Eng. 1987, Vol. 9, J~UV
89
Equations
and 2 can be written,
1
kl
Fk
M,(s) -M?(s)
f=l-pW
=
Frill(4 Mx (4 - M,*(s)
=&
or, using the inverse Laplace transform, Fk + (a+b)
&
+abFk
zm + (a+b) im
= klMz
+abF,
-KIM;
=k,MX
--KIM,*
The coupling of the two subsystems, the block diagram of Figure 3, yields M;
= Fkdk
MZ=
F,,,d,,,
+ k,k
Moments M, and M, operating in the flexionFigure 5 extension plane and abduction-adduction plane respectively
to
(8)
5 and 6 can be written - (ab+k, dk) Fk
cos 45’ F,
+ klM,
(u+b) I$,, - (ab+k 1d,)
+ k,q
(6)
(7)
-Fkqcosrh
= - (u+b) I$
Fm = -
according
- F,,, k cos 45’
so that equations Fk
(5)
(9)
F,
cos 7h Fk + klM,
(10)
Choosing as state variables of the system the forces FL and F, and their derivatives, we have the inputs of the model, are taken from the work of Paul* and they are indicated on Figure 5.
Xl
=
Fk
x2
=
Xl
x3
=
F,
xq
. = x3
. DESCRIPTION OF THE STATE VARIABLES
SYSTEM
WITH
The inputs of the model are Laplace transforms of the moments M, and M,. The error signal M, = M - M” is applied at the input of the system with transfer function P(S). The outputs of the subsystems, which are the Laplace transforms of the forces Fk or F, are given by the equations Fk
M,(s)
ts)
- Mz*@)
= P(s)
Frn(4 M,(s)
-Mx*(s)
where
k =
90
J. Biomed.
1
=
pts)
or 2 and m = 3 or 4.
Eng. 1987, Vol. 9, January
(1) (2)
and, using equations
9 and 10 we obtain
i,
=
;,
= (-ub+kldk)xl
-((a+b)x2
+k,k
+klM,
x2
cos45°x3
323
=
x3
. x4
=
k, q cos rnxl
- (a+b)x,,
- (ab+k1d,)x3
+ k 1M,
1 1)
SOLUTION
OF THE
SYSTEM
The system of equations connective form: G(t)
= Ax(t)
(11)
discrete force values; this creates a problem which may be overcome by solving the system separately for each of the intervals, as follows.
can be written in the
+ Bu(t)
(12)
where 0
1
0
0
-(ab+k,dk)
-(a+b)
k, k cos 45”
0
A= 0
0
0
k,q
OF SUBSYSTEMS
To overcome the difficulty of calculating the system of equations (12) is separated following two subsystems.
-&+b)
I
1
-(ab+k
0
c”s?‘h
SEPARATION
0
+
1dm ) -(a+b)
klk
(sl-A)-‘, into the
’
I
0
cos45’
k1 I
’
I
(134
and 0
B=
0
k,
0
0
0
0
k,
The solution equation
1
s
t$(t-A) B*U(A)d,
(14)
where
L-’
(15)
(~1 __A)-’
Such a solution exists only if matrix A is timeinvariant and not zero, but matrix A, does not vary with time, since angle yh changes during normal walking; the angle yh varies from +20” to -20” around the vertical position of the femur (yh = 0)4. The resultant variation of cos yh is of the order of 0.05, and therefore the approximation cos yh = 1 can be used for the complete period of normal walking. A second reason for the time dependence of the matrix A is the variation of the parameters dk and d,,, according to the applied forces. Thus, for some time intervals k = 1, while for some others, k = 2. (Similarly for m = 3 or m = 4). The Table I shows these forces, after reference 2. It can be seen from the table that there are five time intervals with Time intervals within the flexion-extension forces Table 1 (F,, F,) and abduction-adduction forces (F3, F4) appearing during one step cycle
0 20 70 88 95
- 20% - 70% - 88% - 95% - 100%
Force states FZ 6 FL
FZ F2
F, F, -
0
klq
SOLUTION INTERVAL
F4
Cos’Yh
Xl
.
k1 1
I I
x4
.Ki
(17)
OF THE SYSTEM FOR THE O-20% OF THE WALKING CYCLE
d2 = 0.071 d, = 0.046 Similarly,
m m
the other accepted
a = 1,4rads-‘,b
k,
= 19600
O
-15.4
0
471
+
m and q = 0.011
m.
two systems appear as follows:
0 +
k = 0.034
[ -1411
=
values are
= 14rads-‘, $
Consequently,
[::I= Fd F4
0
[
According to Table 1, over the interval O-20% of the walking cycle, forces F2 and F3 are exerted. Therefore the values of the parameters are
[ij
Time intervals
l
This separation is based on the idea that the term employing xj expresses the influence of the movement of they-z plane upon the movement on the plane x-y. Thus, parameter x3 can be considered as a secondary input for the first subsystem. Similarly, parameter xi can be considered as a secondary input for the second subsystem.
0
=
[-(ab:k&J(-(:+b)]
VW
t
4(t) = &’
=
+
of the above system is given by the
x(t) = tq+(o) +
x3 ” x4
19600
L-9023
I II x3
0
0
202
19600
M,
.
Al
(184
l
[::I
(18b)
I [I MX *Xl
J. Biomed. Eng. 1987, vd.
9, January
91
Therefore, (sZ-Al)
the results for the first subsystem =
]
IlII
s+;;,l
with det @Z-A,) = (~(-7.7
are k’
(19)
.
i
+ 37i)) (~(-7.7
- 37i) (20)
The inverse matrix is
.
F3 100.
(sZ-Ar )-’ =
s+15.4
1 (s-(-7.7+37i))(s-(-7.7-37i))
(s-(-7.7+37i))(s-(7.7-37i)) -1411 (s-(-7.7+37i))(s-(-7.7-37i)
(s-(-7.7+37i)s(s-(-7.7-37i)) I
(21) In
to find
01 0
inverse ks’
1 = cpt
+
(t-7)
3 47)
f
5
1. l-
d7
*0
Figure
and
I
The evaluated abduction-adduction
(F,,,) force
(23)
44
=
#+-to)
X0
1
&(t-7)
B u(7)
dr
to where
The matrices Gi(t--to) and @,(t-t,,), t--to = 0.05 s, are
for the step
0 *(t) has been found similarly to @i(t). 0.0006 0.66
I
(24)
t
y” #,(O.OS)
II 20@-
=
0.66
0.0003
0.55
0.65
I
The final equations
I
(25)
for the time interval 0.05% are
k*
x1(t)
= 0.67 x1 (to) + 0.108
x3(t)
=
0.66 x3(to)
+ 0.038
x3 + 4.5 M, x1 + 3.7 M,
(26)
The solution of the system for the remaining intervals is similar to the solution presented in the previous paragraph. The outputs of the model, i.e. the forces Fk and F,, are depicted in Figwes 6 and 7.
60
Figure 6
92
The evaluated flexion-extension
J. Biomed. Eng. 1987, Vol. 9, January
80
t-%
(FJ force
100
SUMMARY AND CONCLUSIONS In Figures 6 and 7, the results of the forces Fk and F, are analytically represented. The signs of these
forces show the direction of the corresponding moment which emanates from this force, since this sign is stable throughout the entire duration of the interval. The resultant moment, included in the equation 7 or 8, is expressed as Fkdk or F,,,d, respectively. The direction of this moment is positive if F, or F3 is operational, and negative if F2 or F., is operational, therefore the sign attached to this force must follow the sign of the corresponding moment. The results show that this is satisfied for all values, except for those at the 65 and 75% point of the period. This is probably because the value of the corresponding moment is very small, about 0.1, and the resultant force is primarily due to the effect of the moment M,. In this work, the flexion-extension and abductionadduction forces, Fk, F, respectively, were found to take their maximum and minimum values at times which are proximal to those found by Paul2 and other author?. On the other side there is a difference in values which fluctuates between 15 and 35% at various points; this divergence should be generally acceptable, given that it concerns models describing biological systems. These results seem to lend encouraging support to our method and suggest that the model should be reliable and effective and could be further enhanced if the time intervals were shorter but more numerous.
REFERENCES Aizermann, M.A. and Andreeva, Y.A. A study of movement control systems in living beings. In: A.S. Iberall and J.B. Reswick (eds) IFAC Congress, France, June 1972, 23-34 Paul, J.P. Forces at the human hip joint. PhD Thesis, University of Strathclyde, 1967 Bonnemay, A., Koukis, M., Couny, M., Souza, P.K. Modele analogique pour le c&u1 de forces appliquees a la tete du femur. SES/PUB/SAI/73-192, Commissariat a l’energie atomique, France, 1972, l-36 Lamoreaux, L.W. Kinematic measurements in the study of human walking. Bull prosth Res 197 1, 4, 112-l 24 Smith, A.J. A study of forces on the body in athletic activities with particular reference to jumping. PhD Thesis, University of Leeds, 1972 Morrison, J.B. The mechanics of the knee joint in relation to normal walking. J. Biomech 1970, 3, 431-451 O’Kelly, J., Unsworth, A., Dowson, D., Hall, D.A. and Wright, V.A. Study of the role of synovial fluid and its constituents in the friction and lubrication of human hip joints. Eng. in Med. 1978, 7, 73-83 Alexander, R. McN. and Vernon, A. The dimension of knee and ankle muscles and the forces they exert. J. Human Movement Studies 1975, 16, 1-19 Ellis, M.I., Seedhom, B.B. and Wright, V. Forces in the knee joint whilst rising from a seated position J. Biomed Eng 1984, 6, 113-120
Maria Grigoriadou-Koukis, University of Athens, Electronics Laboratory, Panepistimioupoli-Ktiria TYPA, 157 71 Athens. Greece
VIBRATORY PROPERTIES AND RESONANCES OF THE ISOLATED HUMAN ULNA Dear Sir, The purpose of the present letter is twofold: first to congratulate Dr E.J. Evans on his paper, Vibratory properties and resonances of the isolated human ulna, (J. Biomed, Eng 1985; 7; 144-148) and second to present some observations which may be of interest to the reader. I agree completely with the statement made by the author, ‘these results indicate that the vibratory behaviour of the ulna is strongly influenced by its complex geometry and it would be unreasonable to
0 1987 Buttenvorth & Co (Publishers) 0141-5425/87/010093-001,$03.00
assume that bones from two individuals have the same vibrational characteristics’. I would like to add that it seems reasonable to expect that for any two individuals the vibratory properties of a given bone will be influenced, perhaps in different ways, by the mechanical characteristics of the muscles attached to the bone. Certainly these muscles will possess inertial, restoring and damping characteristics which may differ greatly between individuals and therefore contribute in a significant manner to the overall vibrational characteristics of a complex dynamic system: bone and attached muscles. The differences will probably be more noticeable ‘in rho’. Yours faithfully,
Patricia A.A. Laura Director and Research Scientist, Institute of Applied Mechanics, Puerto Belgrmo Naval Base, 8 11 l-Argentina
Ltd J. Biomed. Eng. 1987, Vol. 9, January
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