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0593_C06_fm Page 187 Monday, May 6, 2002 2:28 PM

Forces and Force Systems 187

or

 N   N 
T =  ∑ m ii × r ) − ωω ×  ∑ m ii × r ) (6.9.11)
r ×(ωω
r ×(αα
*
i
i
  i==1     i==1  
where the first term of Eq. (6.9.10) is zero because G is the mass center, and the last term
is obtained from its counterpart in the previous line by using the identity:
a ×( b c) ≡ ( a c b ) −( a b c ) (6.9.12)
⋅
×
⋅
To see this, simply expand the triple products of Eq. (6.9.10) using the identity. Specifically,
i [
ω
ωω
r × ωω ×(ωω × r )] =⋅(ωω × r ) −( r ⋅ ) × r = −( r ⋅ ) × r i (6.9.13)
ωω
ω
ω
ω
ωω
r
i
i
i
i
i
i
and
⋅(
ωω
ωω
r r
r
r
ωω ×[ r i ×( ωω × )] = ωωωω × ) −( ωω⋅ ) × r i = −( ωω⋅ ) × r i (6.9.14)
r
i
i
i
i
i
where the first terms are zero because r and ωω ωω are perpendicular to ωω ωω × r . Comparing
i
i
Eqs. (6.9.13) and (6.9.14), we see the results are the same. That is,
i [
i [
r × ωω ×(ωω × r )] = ωω × r ×(ωω × r )] (6.9.15)
i
i
Neither of the terms of Eq. (6.9.11) is in a form convenient for computation or analysis;
however, the terms have similar forms. Moreover, we can express these forms in terms of
the inertia dyadic of the body as discussed in the next chapter. This, in turn, will enable us
*
to express T in terms of the moments and products of inertia of B for its mass center G.
References
6.1. Kane, T. R., Analytical Elements of Mechanics, Vol. 1, Academic Press, New York, 1961, p. 150.
6.2. Kane, T. R., Dynamics, Holt, Rinehart & Winston, New York, 1968, pp. 92–115.
6.3. Huston, R. L., Multibody Dynamics, Butterworth-Heinemann, Stoneham, MA, 1990, pp. 153–212.
Problems
Section 6.2 Forces and Moments
P6.2.1: A force F with magnitude 12 lb acts along a line L passing through points P and
Q as in Figure P6.2.1. Let the coordinates of P and Q relative to a Cartesian reference frame
be as shown, measured in feet. Express F in terms of unit vectors n , n , and n , which are
z
y
x
parallel to the X-, Y-, and Z-axes.