Page 49 - Dynamics of Mechanical Systems

0593_C02_fm Page 30 Monday, May 6, 2002 1:46 PM

30 Dynamics of Mechanical Systems

where e is the permutation symbol defined as:
ijk

,
 1 if i j, and k are distinct and cyclic (for example: 1, 2, 3)



,
e =−1 if i j, and k are distinct and anticyclic (for example: 1, 3, 2) (2.7.7)
ijk

,

 0 if i j, and k are not distinct and cyclic (for example: 1, 1, 3)
This definition is readily seen to be equivalent to the expression:
)(
− )(
−
e = 12 i − ( j j k k i) (2.7.8)
ijk
When the unit vectors are arranged as in Figure 2.7.1, so that minus signs do not appear
in the equations with cyclic indices, the system is said to be “right-handed” or dextral.
Alternatively, when the indices are anticyclic and minus signs do not occur, the system is
said to be “left-handed” or sinistral. Figure 2.7.2 shows an example of a sinistral system.
In this book we will always use right-handed systems.
Next, consider the vector product of a vector A with a sum of vectors B + C. Let D be
the resultant of B and C, and let n be a unit vector parallel to and with the same sense
A
as A. Then, from Eq. (2.6.15), we have:

n ⋅ D = n ⋅( B C) = n ⋅ B n ⋅ C (2.7.9)
+
+
A A A A
Let D , B , and C be defined as:



D = ( n ⋅ ) B = ( n ⋅ ) C = ( n ⋅ )
A
A
|| A D n , || A B n , and || A C n A (2.7.10)
Then, from Eq. (2.7.9), we have:

D = B + C (2.7.11)
|| || ||

Z
n
n 2
3

Y n 3
n
2

n n
X 1 1
FIGURE 2.7.1 FIGURE 2.7.2
Mutually perpendicular unit vectors and coordinate axes. A sinistral unit vector system.