Page 57 - Dynamics of Mechanical Systems

0593_C02_fm Page 38 Monday, May 6, 2002 1:46 PM

38 Dynamics of Mechanical Systems

In using this convention, several rules are useful. First, in an equation or expression, a
repeated index may not be repeated more than one time. Second, any letter may be used
for a repeated index. For example, in Eq. (2.9.2) we may write:

v = v n = v n = v n (2.9.3)
i i j j k k

Finally, if an index is not repeated it is a “free” index. In an equation, there must be a
correspondence of free indices on both sides of the equation and in each term of the
equation.
In using the summation convention, we can express the scalar and vector products as
follows: If n , n , and n are mutually perpendicular unit vectors and if vectors a and b
1
2
3
are expressed as:
a = a i n and b = b i n i (2.9.4)
i
then the products a · b and a × b are:

×
⋅
ab = a b and a b = e a b n (2.9.5)
ii ijk ij k
where, as before, e is the permutation symbol (see Eq. (2.7.7)).
ijk
With a little practice, the summation convention becomes natural in analysis procedures.
We will employ it when it is convenient.

Example 2.9.1: Kronecker Delta Function Interpreted
as a Substitution Symbol
Consider the expression δ V where the δ (i, j = 1, 2, 3) are components of the Kronecker
j
ij
ij
delta function defined by Eq. (2.6.7) and where V are the components of a vector V relative
j
to a mutually perpendicular unit vector set n (i = 1, 2, 3). From the summation convention,
i
we have:
δ V = δ V + δ V + δ V ( i = 12 3 ) (2.9.6)
,
,
ij j i1 1 i22 i33
In this equation, i has one of the values 1, 2, or 3. If i is 1, the right side of the equation
reduces to V ; if i is 2, the right side becomes V ; and, if i is 3, the right side is V . Therefore,
2
3
1
the right side is simply V. That is,
i
δ V = V (2.9.7)
ij j i
The Kronecker delta function may then be interpreted as an index operator, substituting
an i for the j, thus the name substitution symbol.

2.10 Review of Matrix Procedures

In continuing our review of vector algebra, it is helpful to recall the elementary procedures
in matrix algebra. For illustration purposes, we will focus our attention primarily on square