Page 239 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

216 BIOMECHANICS OF THE HUMAN BODY

TABLE 8.6 The Absolute Velocity of Point C as Expressed Relative to b , b , b
1 2 3
b b b
1 2 3
b
b
b
b
b
b
ω ω =− ψ cos ψ sin ψ b ω = ψ sin ψ + b ω = ψ cos ψ cos ψ b
ψ
B B 1 1 2 3 B 2 1 2 3 B 3 1 2 3
b
b
+ ψ cos ψ b 3 + ψ sin ψ b 3
2
2
r = r 0 r 0
/
CB C C
v
B
v = r 0 0 0
/
/
CB CB
ω × r C B −r ω B 3 0 r ω B 1
C
C
B
/
1. Sum the terms to yield the velocity in the a frame.
2. Either apply a coordinate transformation between the a and b frames directly, or convert from the
a frame to the inertial frame and then from the inertial frame to the b frame.
3. Once the velocity is expressed in the b frame, it may be inserted into Table 8.6.
This process is then repeated for Tables 8.7 and 8.8 to complete the tables with the velocities of
points C and D, respectively. The derivations of the velocity equations within these tables are lengthy
and left as an exercise for the reader.
To determine the velocity at the center of gravity for each segment, additional tables can
be subsequently constructed that correspond respectively to Tables 8.5, 8.6, 8.7, and 8.8. Each new
table must take into account the position vector that defines the location of G j for each segment,
where j ranges from 0 to n − 1 and n is the number of segments considered within the system.
The kinetic energy of the entire system is determined by expanding the expression for the kinetic
energy in Eq. (8.4) and is given as
1 1
T = m v • v + m v • v
2 AG 0 G 0 2 B G 1 G 1
1 1
+ m v • v + m v G • v
2 CG 2 G 2 2 D G 3 G 3
1 1
+ {ω A } T I { }{ω A } + {ω B } T I { }{ω B }
2 G 0 2 G 1
1 T 1 T
+ {ω C } {I G }{ω C }+ {ω D } {I G }{ω D }
2 2 2 2 3 (8.87)
TABLE 8.7 The Absolute Velocity of Point D as Expressed Relative to c , c , c
1 2 3
c c c
1 2 3
c
c
c
c
c
c
ω ω =− ψ cos ψ sin ψ c ω = ψ sin ψ + ψ c ω = ψ cos ψ cos ψ c
C C 1 1 2 3 C 2 1 2 3 C 3 1 2 3
c c c c
+ ψ cos ψ 3 + ψ sin ψ 3
2
2
r = r 0 r 0
/
DC D D
v
C
v DC = r D C 0 0 0
/
/
ω × r D C −r ω C 3 0 r ω C 1
D
D
C
/