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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