Page 227 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

204 BIOMECHANICS OF THE HUMAN BODY

are also the same, and therefore, the Lagrangian will not change,
m 2 2 k 2
θ
−
L = T − V = [ r + ( r ) ] + mgr cosθ − ( rl) (8.30)
2 2
With the omission of the dissipation function, Lagrange’s equation is solved for each of the chosen
generalized coordinates: q = r, Eq. (8.31), and q =θ, Eq. (8.32).
1 2
L
d ∂ ⎞ ⎛ L ∂ ⎞ Q ⇒ − 2 − ( −
⎛
mr mrθ
⎜
⎟ − ⎜
⎝
r
dt ∂ ⎠ ⎝ r ∂ ⎠ ⎟ = r mgcoosθ+ kr l ) = 0 (8.31)
L
d ∂ ⎞ ⎛ L ∂ ⎞ 2
⎛
+
⎜
⎟ − ⎜
⎝
θ
∂ ⎠
θ
dt ∂ ⎠ ⎝ θ ⎟ = Q ⇒ 2 mrr +θ mr θ + mgr sinθ = 0 (8.32)
Simplificating and rearranging Eqs. (8.31) and (8.32) yields the two equations of motion:
(
−
mgcosθ − k r l) = m r rθ 2 ) (8.33)
−
(

θ
2
(
and −mgsinθ = m rθ + r ) (8.34)
Once again, Eqs. (8.33) and (8.34) are the same equations of motion obtained by the Newtonian
approach.
8.4.3 Biodynamically Modeling the Upper or Lower Extremity
Consider the two-segment system shown in Fig. 8.4, where a single segment of length l is connected to
3
a large nontranslating cylinder at point B. For initial simplicity, it is assumed that for this system, the
connection of the segment is that of a revolute (or hinge) joint having only one axis of rotation. The free-
moving end of the segment is identified as point C, while point G identifies the center of gravity for this
1
segment. The large cylinder of height l and radius l (e.g., torso) is fixed in space and is free to rotate
2
1
about the vertical axis at some angular speed Ω. A moving coordinate system b , b , b is fixed at point
3
2
1
B and is allowed to rotate about the b axis so that the unit vector b will always lie on segment BC. This
3
1
particular system considers only one segment, which can represent the upper arm or thigh, and is
presented as an initial step for dynamically modeling the upper or lower extremity. To complete the
extremity model, additional segments are subsequently added to this initial segment. The derivations of
the models involving additional segments are presented later within this section.
The position vectors designating the locations of point B, G, and C are given as
r = l 1 + l 2 (8.35)
B
1
r = l + r
2
G 1 3 B (8.36)
and r = r + l 3 (8.37)
B
C
respectively.
The angular velocity vector of segment BC is determined to be

θ
ω =−Ω cos b + Ω sin θ b + θ b (8.38)
1 3
11
1 2
b
where θ is the angle between the segment and the vertical and θ is the time rate of change of that
1 1
angle.