World Journal of Mechanics, 2012, 2, 90-97
doi:10.4236/wjm.2012.22011 Published Online April 2012 (
Modeling the Force-Velocity Relationship in Arm
Ahti Rahikainen, Janne Avela, Mikko Virmavirta
Department of Biology of Physical Activity, University of Jyväskylä, Neuromuscular Research Center, Jyväskylä, Finland
Received February 1, 2012; revised March 2, 2012; accepted March 17, 2012
Modeling the force-velocity dependence of a muscle-tendon unit has been one of the most interesting objectives in the
field of muscle mechanics. The so-called Hill’s equation [1,2] is widely used to describe the force-velocity relationship
of muscle fibers. Hill’s equation was based on the laboratory measurements of muscle fibers and its application to the
practical measurements in muscle mechanics has been problematic. Therefore, the purpose of this study was to develop
a new explicit calculation method to determine the force-velocity relationship, and test its function in experimental
measurements. The model was based on the motion analysis of arm movements. Experiments on forearm rotations and
whole arm rotations were performed downwards and upwards at maximum velocity. According to the present theory the
movement proceeds as follows: start of motion, movement proceeds at constant maximum rotational moment (Hy-
pothesis 1), movement proceeds at constant maximum power (Hypothesis 2), and stopping of motion. Theoretically
derived equation, in which the motion proceeds at constant maximum power, fitted well the experimentally measured
results. The constant maximum rotational moment hypothesis did not seem to fit the measured results and therefore a
new equation which would better fit the measured results is needed for this hypothesis.
Keywords: Muscle Mechanics; Muscle Power; Force-Velocity Relationship; Arm Movement
1. Introduction
Modeling the force-velocity relationship of muscle-tendon
unit involves many different factors. In muscle mecha-
nics force-velocity relationship of skeletal muscle is of-
ten presented by so-called Hill’s equation (F + a)(v + b)
= b(F0 + a), where F is the maximum force within mus-
cle contraction, a and b are constants, F0 the isometric
force of muscle or the constant maximum force gener-
ated by muscle with zero velocity and v is velocity,
(Figure 1) [1,2]. This equation was based on the labora-
tory measurements in which force (F) of the activated
muscle lifted different loads (F = mg) and speed of the
load (v) was then measured. In Hill’s equation F is force,
a is constants force, v is velocity, b is constant velocity
and F0 is constant force. In the equation the vectors of
forces and velocities have the same direction and there-
fore Hill’s equation can be presented in a scalar form.
The left side of Hill’s equation is the product of force and
velocity and that is power. As the right side of the equa-
tion is constant it can be seen that Hill’s equation is a
constant power model. Hill’s force-velocity relationship
is one of the most essential equations of muscle mechan-
ics and it has often been principle object in biomechani-
cal studies for about 50 years, e.g. [3-6]. Force measured
from skeletal muscle during maximum tension depends
on several internal and external factors. Internal factors
are e.g. anatomical structure of muscle (cross sectional
area, pennation etc.), fiber type distribution (fast and
slow twitch muscle fibers have different force-velocity
equations), condition of the muscle (fatigue, training) and
muscle length. External factors are e.g. contraction type
(isometric, concentric and eccentric) and contraction ve-
locity (rate of change of muscle length). Good reviews of
the above mentioned factors have been presented, e.g.
[4,7,8]. Force (F) creates a moment about the joint which
is moment arm multiplied by force (M = r × F). Length
of muscle’s moment arm depends on joint angle and it
changes as the rotation movement proceeds about the
joint axis. The combined effect of the forces of several
different muscles produces the rotation movement about
the joint axis.
Due to all the above mentioned factors it is difficult to
determine the force production [9,10], and also to deter-
mine the torque about the joint. The purpose of this study
was to develop a new explicit calculation method to de-
termine the force-velocity relationship and test its func-
tion in experimental measurements. This method is based
on the assumption that in muscle mechanics there exists a
constant maximum power which the muscle is able to
Copyright © 2012 SciRes. WJM
Figure 1. Hill’s equation (F + a)(v + b) = b(F0 + a) where F0
is so-called isometric force or force with zero velocity, v0 is
the highest possible velocity, a and b are constant force and
constant velocity. In rotational movement torque M corre-
sponds to force F and angular velocity φ corresponds to
velocity v.
generate within a certain range of velocity. The principle
of constant maximum power is the same as in Hill’s
equation except that the constant maximum power in the
present study is a characteristic of whole muscle group
instead of separate muscle fibers as in the Hill’s equation.
This study continues the development of the earlier find-
ings [11-13].
2. Methods
The experiments in the present study consisted of three
different maximum velocity arm movements: 1) forearm
rotation downwards, 2) whole arm rotation downwards
and 3) upwards. The selection of these movements was
based on the earlier findings of Rahikainen and Luhtanen
[11] where so called “constant power theory” seemed to
work at the last phase of the arm push in shot put. In or-
der to study this finding more extensively it was reason-
able to choose a simple procedure as represented by arm
rotations in the present study. The photographs of arm
movements in this study were generated by a special mo-
tion camera system [14,15] which represents the move-
ment as a series of object images. The paths of the mark
lights attached to the moving object can be seen as bro-
ken light-lines. The principle of the method is to photo-
graph the moving object through a rotating disc which
consists of one transparent opening and nine filter open-
ings serving as the shutter apertures. As the exposure
disc rotates in front of the camera lens (film camera Ca-
non T70) and the camera aperture is open, the disc serves
as the shutter. This way several overlapping exposures
are generated on the same frame. The transparent open-
ing generates images of the moving object, and the filter
openings generate the light-lines indicating the paths of
mark lights attached to the moving object (Figure 2). In
this study the speed of rotation of the exposure disc was
300 rotations per minute, exposing five (300/60) object
images per second and giving the time interval of 20 ms
Figure 2. Forearm rotation downwards with maximum
force. Angle of rotation φ and its corresponding time T (ms)
are presented on the subject image.
for nine light-lines between consecutive object images
(for more detail, see [14,15]). Figure 2 represents a fore-
arm rotation downwards. As seen in the figure the radius
of the rotation circle is not exactly the same as the radius
of forearm rotation. This is because of a slight motion of
the elbow joint. Actually the radius of forearm rotation is
slightly larger than the radius of circle on the figure and
it can be measured from the forearm image before the
start of the rotation movement. Angular velocity mea-
surements are calculated with the formula
in which the length of forearm is the radius of rotation R
and the distance measured between two successive
measured points on the path of light-lines is the distance
increment ΔS.
2.1. Measurement of Rotation Arc
For convenience the arc ΔS1 was measured as a straight
line ΔS2 (Figure 3) and the error between these two va-
lues was estimated. The arc ΔS1 can be calculated from
the straight line ΔS2 from the formula:
arccos 12
 
Formula derivation from the right-angled triangle in
Figure 3.
222 2
2sin1 cosSR R
  (2)
2sincos1 2cos
 
 (3)
sincos 1
 (5)
Copyright © 2012 SciRes. WJM
Figure 3. Measurement of the rotation arc. φ = angle of
rotation, R = length of forearm, arc ΔS1 = distance the mark
light travels during the time interval Δt and ΔS2 = the arc
ΔS1 measured as a straight line.
cos1 2
arccos 12
 
The maximum value measured from Figure 2 (corre-
sponding time 140 ms) is ΔS/R = ΔS2/R = 0.356. Substi-
tuting this value in the formula above (7) the arc of rota-
tion is obtained as ratio form ΔS1/R = 0.358. It can be
seen that ΔS2 fits with adequate accuracy to the distance
2.2. Progress of Research
The present study continues the earlier study [11] and it
is a new round in the diagram of Figure 4 presenting the
progress of research (testing the hypotheses): 1) Equation
of arm movement was derived and test predictions were
made. 2) Experiments were performed in arm rotations. 3)
Equation of arm rotation was fitted to the experimental
results and their compatibility was observed. 4) If the
present equation of motion did not fit at all the measuring
results, the hypothesis would be disproved. If the present
equation of motion fitted the measuring results in some
definite accuracy, the hypothesis would receive confir-
mation. 5) In the future, by making additional experiment
(a new round in the diagram) the hypothesis will receive
more confirmation.
2.3. Arm Rotation
Because the muscle system is able to transfer only a cer-
tain quantity of chemical energy during the time of con-
traction, it is obvious that arm rotation must have maxi-
mum power that cannot be exceeded. It can also be as-
sumed that the maximum power acts within a certain
range of velocity and it is a constant maximum power. At
the beginning of the movement angular velocity is natu-
rally zero and it takes some time to generate force. After
the start of the movement it is possible that a maximum
muscle force takes action and within rotational motion
maximum rotational moment acts as well. The constant
Figure 4. Diagram of the progress of testing the hypotheses
of arm rotations.
maximum power acts within a certain range of velocity
which cannot be at the beginning of the rotational move-
ment because power is the product of moment and angu-
lar velocity. Therefore, a constant power “theory” is pos-
sible only when the velocity is high enough. As the ve-
locity increases the motion reaches the point where the
maximum power takes action and acting rotational mo-
ment is less than the maximum moment. This way power
remains constant as the angular velocity increases and
moment decreases.
2.4. Research Hypotheses
According to the present theory and above mentioned
facts the movement proceeds as follows: 1) start of mo-
tion, 2) movement proceeds at constant maximum rota-
tional moment during the first part of the movement
[Hypothesis 1], 3) movement proceeds at constant maxi-
mum muscular power during the second part of the
movement [Hypothesis 2], 4) stopping of motion. In or-
der to test the research hypotheses, the following ex-
periments were conducted: forearm rotation downwards
at maximum velocity (1), whole arm rotation downwards
at maximum velocity (2), whole arm rotation upwards at
maximum velocity (3). The maximum power hypothesis
was tested so that the theoretical angular velocity-time
values from Equation (15) were fitted into the measured
angular velocity-time curves of arm rotations. It was as-
sumed that if the measured angular velocity-time values
matched the theoretical values within a certain velocity
range then the Hypothesis 2 would be fulfilled. The
maximum rotational moment hypothesis was tested by
Equation (8).
2.5. A Model of Arm Rotation
It was assumed (hypothesis) that in muscle mechanics
there exists the maximum power (P) which the muscle is
able to generate within a certain range of velocity. The
Copyright © 2012 SciRes. WJM
Model of arm rotation was constructed according to
Newton’s II law which was applied to rotational motion
where moment of inertia multiplied by angular accelera-
tion equals rotational moment. Rotational moment equals
moment generated by muscle force subtracted moment
generated by inner friction of muscle. The effect of gra-
vitational force is minor and it is added to the motion
mechanics afterwards in Section 2.6. The model of arm
rotation is the equation of motion:
where I is moment of inertia in arm rotation,
is an-
gular velocity, P is power generated by arm muscles, T is
time, P/
is moment generated by muscle force, C
moment generated by inner friction of muscle and C is
constant coefficient of friction.
The mass distribution of the subject’s arm sectors dif-
fered from the average values in subject mass tables.
Therefore the mass distribution of the arm sectors were
defined by sinking the arm sectors into water, and
weighing the over flowed water. The masses of the arm
sectors were calculated by means of water volume and
arm sector density (V
). The length of subject’s whole
arm was 0.64 m and the arm sectors, hand, forearm 1,
forearm 2, upper arm 1, upper arm 2 were 0.128 m each.
Arm sector densities were 1.16, 1.13, 1.07 for hand,
forearm and upper arm, respectively [6]. Moment of in-
ertia for the forearm rotation was I = 0.11 kg·m
and for
the whole arm rotation I = 0.52 kg·m
Hypothesis 1 implies that movement proceeds at a
constant maximum rotational moment. In that case the
moment generated by muscle forceP
in Equation 8 is
a constant maximum moment. Hypothesis 2 implies that
movement proceeds at a constant maximum muscular
power. In that case the power P in Equation (8) is a con-
stant maximum power. In order to determine the validity
of Hypothesis 2, Equation (8) was solved for angular
velocity-time function and this equation was employed
for validity determination:
Equation of power 2
where d
is power in arm rotation, P is power
generated by arm muscles and 2
is power consumed
by friction.
 
lnln C
 
ln PC C
2.6. Effect of Gravitational Force on the
The moment which is induced by gravity rmg
was omitted from the motion model. The power gene-
rated by this moment is
, where mg is
gravitational force of arm segments, r is distance of the
center of gravity of segments from the rotation axis and
angular velocity of arm rotation. The theoretical an-
gular velocity function, Equation (15), and the measured
angular velocity function coincide within so narrow ve-
locity range that the power induced by gravity can be
calculated as a constant factor. In this case it is included
in the power P as follows: P of rotation downwards =
power generated by muscular force + power generated by
gravitational force and P of rotation upwards: P = power
generated by muscular force-power generated by gravita-
tional force.
2.7. Finding the Matched Range of Measured
and Theoretical Angular Velocity Functions
There are two unknown variables in Equation (15), pow-
er P and kinetic friction coefficient C. In order to deter-
mine these two unknown variables, two equations were
required. These two equations were obtained from the
hypothesis according to which the movement proceeds at
constant maximum power within certain velocity range.
By substituting two angular velocity-time value pairs
from the measured angular velocity-time curve in Equa-
tion (15) the two required equations were obtained. The
zero point of time (Figure 5) is at the intersection point
of the time-axis and the broken-line curve and in order to
find that some iteration was done. From these two equa-
tions P and C could be solved. Then the constant maxi-
mum power hypothesis was tested by comparing the
calculated theoretical values from Equation (15) with the
values of measured angular velocity-time curve.
3. Results
(11) In Figure 5 the line (A-E) is connecting the experimental
data points of Figure 2. Figure 6 shows the whole arm
Copyright © 2012 SciRes. WJM
Figure 5. The measured angular velocities from forearm
rotation downwards (points on the curve A-E) and the
theoretical angular velocity values calculated from Equa-
tion 15 (broken line). The zero point of time for the theo-
retical angular velocity curve is at the intersection of the
time-axis and the broken-line curve (the same time scaling
is same for both curves).
Figure 6. Whole arm rotation downwards (a) and upwards
(b). Time of rotation is seen with the increment of 20 ms.
rotations upwards and downwards. In Figures 5 and 7
the solid line is the curve fitting to the points represent-
ing the technique to filter small digitizing errors in tradi-
tional motion analysis. This way the complicated analy-
sis of the series of the object images in the present study
0.52 kg·m
3.0 k
0.52 kg·m
3.0 kg·m
Figure 7. The measured angular velocities (points on the
curve fitting A-E) from the whole arm rotation downwards
(a) and upwards (b) and the theoretical angular velocity
values calculated from Equation 15 (broken lines). The zero
point of time for the theoretical angular velocity curve is at
the intersection of the time-axis and the broken-line curve.
could be facilitated without losing a sufficient accuracy.
Hypothesis 1 states that the rotational movement pro-
ceeds at a constant maximum rotational moment within a
certain range of velocity. This statement implies that ro-
tational moment is about constant or P
is constant.
By observing Figures 5 and 7 it can be seen that move-
ment proceeds at constant acceleration or ddT
constant approximately between the points A-B on the
velocity-time curve. The kinetic friction C
is not con-
stant. By substituting these terms in Equation (8)
it can be seen that the left side of the equation is constant
and the right side of the equation is not constant. There-
fore, we can conclude, that Hypothesis 1 is not fulfilled.
The measured values of the forearm rotation down-
wards are presented in Table 1. Angular velocities of the
forearm rotation are shown in Figure 5 as points on the
Copyright © 2012 SciRes. WJM
Table 1. Measured values of the forearm rotation down-
wards. Angular velocity is calculated with the Equation (1)
and the angular acceleration according to Figure 7. A-B
and C-D represent the estimated phases where the move-
ment proceeds at constant acceleration and constant power,
T (ms) Δφ(rad) ΣΔφ(rad)()rad s
rad s
20 0.055 0.06 2.75 114
40 A 0.110 0.17 5.50 155
60 0.170 0.35 8.52 155
80 B 0.231 0.59 11.54 155
100 0.291 0.89 14.56 128
120 0.319 1.22 15.93 72.5
140 D 0.329 1.56 16.48 12
160 0.319 1.89 15.93 -56
curve A-E. The theoretical angular velocity function with
maximum power hypothesis (Equation 15) was fitted into
the curve of the measured angular velocity-time values.
Moment of inertia of forearm rotation was calculated I =
0.11 kg·m2 (see 2.5). The values of friction coefficient C
and power and friction coefficient ratio P/C were ob-
tained within the curve fitting, C = 2.38 kg·m/s2 and P/C
=285 1/s2. In Figure 5 the movement proceeds at a con-
stant acceleration between the phases A and B (~ 40 - 80
ms) until the liquid friction begins to influence and ac-
celeration decreases between B-C. According to the Hy-
pothesis 2 the movement proceeds at a constant power
between C-D which is followed by stopping of the
movement (D-E). The theoretical angular velocity curve
(broken line) coincides with the measured angular veloc-
ity curve within section C-D. Therefore, we conclude
that Hypothesis 2 is fulfilled within this range of velo-
Figure 7 represents the curves of the measured points
of angular velocity-time values from the whole arm ro-
tations downwards and upwards (
Figure 6). The theo-
retical angular velocity functions with maximum power
hypothesis (Equation (15)) were fitted into the measured
point curves. Moment of inertia of forearm rotation was
calculated I = 0.52 kg·m2 (see 2.5). The values of friction
coefficient C and power and friction coefficient ratio P/C
were obtained within the curve fitting, whole arm rota-
tion downwards C = 3.0 kg·m/s2 , P/C =360 1/s2 and
whole arm rotation upwards C = 3.0 kg·m/s2, P/C =250
1/s2. The movement follows the hypothesized movement
pattern described in the forearm rotation above. The
theoretical angular velocity curves (broken lines) coin-
cide with the measured angular velocity curves in section
C-D (~ 150 - 190 ms and 90 - 150 ms in downward and
upward rotation, respectively, Figure 7).
Validity and Accuracy of Results
In order to confirm the accuracy of results, power P was
calculated by comparing two independent calculation
methods. Equation (9)
 
yields one power value (P1) and the other one (P2) comes
from the curve fitting used in Figures 5 and 7 (P/C).
In forearm rotation downwards the angular accelera-
tion at point T = 0.10 s,
= 14.5 rad/s was calculated
by using the tangent of the angular velocity curve (Fig-
ure 8). The tangent point can be found because the tan-
gent has only one point on the curve, otherwise there are
two intersection points. The value of angular acceleration
in Figure 8 was calculated according to
= 14.5/0.12 1/s2 = 121 1/s2.
This value of angular velocity derivative can also be
calculated using Equation (15). The time and angular
velocity of this equation corresponding to the measured
angular velocity curve time 0.10 s and velocity 14.5 rad/s
was calculated with Equation (13). Substitution of veloc-
ity 14.5 rad/s into Equation (13) gives time 0.031 s. The
derivative of Equation (15)
 
PC e
 (17)
Substituting in this equation T = 0.031 s, I = 0.11
kg·m2, C = 2.38 kg·m2/s and P = 693 W, the value of
angular acceleration of 112 1/s2 was obtained. Moment
arm of gravitational force is so short at forearm rotation
that the power generation of gravitational force has no
significance. In whole arm rotation downwards and
whole arm rotation upwards the effect of gravitational
force is within power P. The accuracy of results is pre-
sented in Table 2.
4. Conclusions
Hypothesis 1: Movement proceeds at a constant maximal
rotational moment. Measurements of the rotation move-
ments show that movement proceeds at a constant angu-
lar acceleration between A-B. Therefore, it can be con-
cluded that the torque accelerating the movement or the
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Table 2. Determination of accuracy of the results. The accuracy was obtained as a difference between the power values P1
(from Equation (16)) and P2 (from the curve fitting in Figures 5 and 7).
Forearm rotation Whole arm rotation
Down (Figure 4) Down (Figure 6) Up (Figure 6)
Time (T) 0.100 s 0.160 s 0.114 s
Angular velocity
14.5 rad/s 13.3 rad/s 11 rad/s
Angular acceleration d
14.5 / 0.12 1/s2 13 / 0.16 1/s2 11 / 0.17 1/s2
Moment of inertia (I) 0.11 kg·m2 0.52 kg·m2 0.52 kg·m2
Power into acceleration d
193 W 562 W 370 W
Coefficient of friction (C) 2.38 kg·m2/s 3.0 kg·m2/s 3.0 kg·m2/s
Power into friction
500 W 531 W 363 W
Muscle Power (P1) 693 W 1093 W 733 W
Power/Coefficient of friction (P/C) 285 1/s2 360 1/s2 250 1/s2
Muscle Power (P2) 678 W 1080 W 750 W
2.2% 1.2% 2.3%
Figure 8. Calculation of angular acceleration at point (T =
0.10 s,
= 14.5 rad/s), where the theoretical angular ve-
locity curve (broken line) coincides with the measured an-
gular velocity curve (points) between C-D.
left side of Equation (8) is constant.
Torque accelerating the movement is not the same as
muscle force which is included in the term P/
. There-
fore, we can conclude that Hypothesis 1 is not fulfilled.
However, “movement proceeds at a constant accelera-
tion” is an interesting finding which should be studied
more closely. In Equation (8) kinetic friction was as-
sumed to be directly proportional to velocity between
A-B. This is a third hypothesis included into this study,
which is not necessarily true. It is possible that kinetic
friction is constant at small velocities and at large veloci-
ties directly proportional to velocity. Then there is a con-
stant torque value accelerating the movement between
A-B. The constant acceleration of the velocity curve may
be related to the evolution of the human beings. For ex-
ample the smooth acceleration may be essential for the
accuracy of javelin throwing and targeting in fighting
and hunting. As mentioned in [10] when modeling the
control of the human limb motions, the final aim is to
estimate the force production of individual muscles in-
volved. Therefore the constant acceleration theory may
play important role in human movements.
Hypothesis 2: movement proceeds at a constant maxi-
mal muscle power. Since the matched range (C-D) of the
theoretical and measured velocity curves of arm rotation
was long enough, it can be clearly seen that the curves
did not intersect each other. Therefore it can be inferred
that the constant maximum power hypothesis is true be-
tween C-D. In addition to the present study of three dif-
ferent type of arm rotation experiments the model of
constant maximum power was also fulfilled in the previ-
ous experiments of shot put [11]. The different arm
movements used in these experiments helped to achieve
a greater certainty for the functioning ability of the pre-
sent model. This model can be considered the most in-
teresting finding of the present study.
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