Page 243 - Dynamics of Mechanical Systems

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224 Dynamics of Mechanical Systems

the moments and products of inertia have maximum and minimum values? To address
these questions, consider first the parallel axis theorem as expressed in Eq. (7.6.10):

I SO = I S G + I G O (7.10.1)

From this expression we see that if O is sufficiently far away from the mass center G, the
components of I G/O become arbitrarily large; thus, there are no absolute maximum
moments and products of inertia. However, if the reference point O is at the mass center
G, the moments and products of inertia may have minimum values. Thus, we have the
following question: if the mass center G is the reference point, for which directions do the
moments and products of inertia have maximum and minimum values? To answer this
question, consider the two-dimensional case and the Mohr circle analysis discussed at the
end of the previous section. From Figure 7.9.3, we see that the maximum and minimum
moments of inertia are the principal moments of inertia I and I . Also, the maximum
aa
bb
absolute value of the product of inertia I occurs at 45° to the principal moment of inertia
12
directions with value (I – I )/2. The minimum absolute value of the product of inertia
bb
aa
is zero. This occurs in directions corresponding to the directions of the principal moments
of inertia. These results can also be seen by inspection of Eqs. (7.9.56), (7.9.57), and (7.9.58).
Do similar results hold, in general, in the three-dimensional case? The answer is yes,
but the vertification is not as simple. Nevertheless, the results are readily obtained by
using the Lagrange multiplier method. For readers not familiar with this method, it is
sufficient to know that the method is used to find maximum and minimum values of
functions of several variables which in turn are related to each other by constraint equa-
tions. For example, if the maximum or minimum value of a function f(x, y, z) is to be found
subject to a constraint:
(
g xyz) = 0 (7.10.2)
,,
then a function h(x, y, z) is defined as:

h xyz,λ) = ( , , λ g xyz) (7.10.3)
(
f xyz) + (
D
,,
,
,
where λ is a parameter, called a Lagrange multiplier, to be determined along with x, y, and
z such that h has maximum or minimum values. Specifically, if h is to be a maximum or
minimum, the following equations need to be satisfied simultaneously:
λ
hy
∂∂ = 0, ∂ ∂ = 0, ∂∂ = 0, ∂ ∂ = 0 (7.10.4)
hx
hz
h
The last of these equations is identical with Eq. (7.10.2). Thus, Eq. (7.10.4) provides the
necessary conditions for f(x, y, z) to have maximum or minimum values. The solution of
these equations then determines the values of x, y, z, and λ, producing the maximum or
minimum values of f.
The advantage of the Lagrange multiplier method is that the constraint equation need
not be solved independently in the analysis. The disadvantage is that an additional
parameter λ is to be determined; however, the parameter λ may often be identified with
an important physical or geometrical parameter in a given problem.
To apply this method in obtaining maximum and minimum central (mass center refer-
ence point) moments of inertia, let I be a central inertia dyadic with components I relative
ij
to an arbitrary set of mutually perpendicular unit vectors n . Then, if n is an arbitrarily
a
i