MUSCLE MECHANICS - Biomechanics and Motor Control of Human Movement - page 243

9.3 MUSCLE MODELING

A variety of mechanical models of muscle have evolved to describe and
predict tension, based on some input stimulation. Crowe (1970) and Gottlieb
and Agarwal (1971) proposed a contractile component in conjunction with a
linear series and a parallel elastic component plus a linear viscous damper.
Glantz (1974) proposed nonlinear elastic components plus a linear viscous
component. Winter (1976) has used a mass and a linear spring and damper
system to simulate the second-order critically damped twitch. The purpose of
this section is not to criticize or justify one model versus another but rather
to go over the principles behind the modeling and the components.
Figure 9.16 shows the force-displacement and force-velocity relationships
of linear and nonlinear springs and dampers that have been proposed. The
symbol for a viscous damper is a “piston in a cylinder,” which can be con-
sidered full of a fluid of suitable viscosity represented by the constant K .
The more common nonlinear models are exponential in form or to a power
a , which is usually greater than 1. This is especially true of viscous friction,
which often varies approximately as the velocity squared.
The total model of the passive components can take many configurations,
as shown in Figure 9.17 a . The parallel elastic component can be considered
to be in parallel with the damper or a series of combinations of the damper
and the series elastic component. For linear components, it does not make the
slightest difference which configuration is used, because they can be made
equivalent. Fung (1971) has shown that:

k 1 k 2
k 1 +

b 1
k 1 +

b 2
k 4

k 1 = k 3 + k 4 ,

k 2 =

k 3 ,

k 2 =

(9.4)

This means that if either model is known, the other equivalent model can
replace it, and it will have the same dynamic characteristics.
The total model requires the active contractile component to be repre-
sented by some form of force generator. The time course of the tension

Figure 9.16 Schematic diagram of linear and nonlinear spring and viscous damper
elements used to represent passive viscoelastic characteristics of muscle.