Page 226 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

BIODYNAMICS: A LAGRANGIAN APPROACH 203

g

B b 2

b 1

θ r
k

G 1

l

C

m

FIGURE 8.3 Single elastic body pendulum with one spring (point G represents the
1
center of gravity of the pendulum).

within the r-θ plane only. The spring pendulum system of Fig. 8.3 is another mechanically analo-
gous system that can also be used to initially model the dynamic behavior of soft tissues.
The steps needed to determine the equations of motion for this system are essentially the steps for
the previous system in Fig. 8.2. In this example, the effects of viscous damping are neglected, thereby
omitting Rayleigh’s dissipation function within the solution. This is a simple omission where removal
of the dissipation function from Lagrange’s equation will have an impact on the final solutions.
The velocity of the mass remains the same as before and is given by

v = r b + rθ b (8.27)
1 2
The kinetic energy of the system,
m
2
2
θ
T = [ r + ( r ) ] (8.28)
2
and the potential energy of the system,
k 2
V =− mgr cosθ + ( r − l) (8.29)
2