Page 228 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

BIODYNAMICS: A LAGRANGIAN APPROACH 205

j

Ω
l 3
i
l 2 b 2

F B
G 1
b 1

θ
l 1 G 0 1 C

m 1 g
a 2

A
a 1

FIGURE 8.4 Two-segment model: one rigid segment connected to a nontranslating
cylinder.

The components for the mass moment of inertia about point G in the b , b , b frame of reference are
1 1 2 3
I = 0 (8.39)
b 1
1
I = I = m l 2 (8.40)
12
b 2 b 3 13
The kinetic energy of segment BC is defined by the equation

1
T = ( I ω 2 + I ω 2 + I ω 2 ) + 1 m v • v (8.41)
1
2 b 1 b 1 b 2 b 2 b 3 b 3 2 1 G 1 G 1
where v is the velocity vector of segment taken at the center of mass. This vector is determined by
G 1
using the relative velocity relation
v = v + ω b × r G B (8.42)
/
B
G 1 1
where v is the velocity vector for point B and is
B
v =− Ω b 3 (8.43)
l
B
2
and r G 1 /B is the relative position vector for point G as defined from point B and is
1
l
r GB = 2 3 b 1 (8.44)
/
1