Page 230 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

BIODYNAMICS: A LAGRANGIAN APPROACH 207

So that
⎛
d ∂ ⎞ = 1 ml θ
L
2
1
⎜
dt ∂ ⎠ ⎟ 3 13 1 (8.55)
⎝ θ
1
The appropriate terms can be substituted into Lagrange’s equation (8.11) to give
1 2 1 2 2 1 2 1
ml θ − ml Ω sin θ cos θ − ml l Ω coosθ + mgl 3 sinθ = 0 (8.56)
1 3
1
1 2 3
1
1
1
1
13 1
3 3 2 2
since there are no externally applied torques acting on the system in the θ direction. The resulting
1
equation of motion for the one-segment system is solved as
2 θ cos θ − 3 l 2 Ω cos θ + 3 g sin θ = 0 (8.57)
2
θ − Ω sin
0
1
1
1
2 l 3 1 2 l 3 1
Next, consider an additional segment added to the two-segment system in the previous example, as
seen in Fig. 8.5. Assume that the additional segment added adjoins to the first segment at point C by
way of a revolute joint. The new segment is of length l , with point D defining the free-moving end
4
of the two-segment system and point G identifies the center of gravity for the second segment. An
2
additional moving body-fixed, coordinate system c , c , c is defined at point C and is allowed to
1 2 3
rotate about the c axis so that the unit vector c will always lie on segment CD.
3 1
j
Ω
l 3
i
l 2 b 2
F B l 4
c 2
G 1
b 1
G 2
θ
l 1 1 C c 1 D
G 0 θ
m g 2
1
a 2
m g
2
A
a 1

FIGURE 8.5 Two rigid segments connected to a nontranslating cylinder.