World Journal of Mechanics, 2012, 2, 90-97
doi:10.4236/wjm.2012.22011 Published Online April 2012 (http://www.SciRP.org/journal/wjm)
Modeling the Force-Velocity Relationship in Arm
Movement
Ahti Rahikainen, Janne Avela, Mikko Virmavirta
Department of Biology of Physical Activity, University of Jyväskylä, Neuromuscular Research Center, Jyväskylä, Finland
Email: ahrahik.zz@kolumbus.fi
Received February 1, 2012; revised March 2, 2012; accepted March 17, 2012
ABSTRACT
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
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