Page 238 - Dynamics of Mechanical Systems

0593_C07_fm Page 219 Monday, May 6, 2002 2:42 PM

Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 219

where n and n are real vectors. Hence, by substituting from Eqs. (7.9.29) and (7.9.31)
α
β
into Eq. (7.9.30) (and recalling that I is real), we have:
⋅( + )( n + )
n
i
In + ) = (α i β α n i β (7.9.32)
α
β
By expanding this equation and by matching real and imaginary terms, we obtain:
In = α n − β n and In = β n + α n β (7.9.33)
⋅
⋅
β
β
α
α
β
By multiplying the first of these by n and the second by n and subtracting, we have:
β
α
n ⋅ ⋅ n − n ⋅ ⋅ n = α n n − β β α n ⋅ n = −2 β (7.9.34)
−
−
⋅
I
I
α
β
α
β
α
β
α
β
Because I is symmetric, the left side is zero. Hence, we have the conclusion:
β= 0 (7.9.35)
Therefore, because I is real and symmetric, the principal moments of inertia (the roots of
Eq. (7.7.12)) are real.
Regarding the question as to whether we always need to solve a cubic equation (Eq.
(7.7.12)), consider that if we know one of the roots (say, λ) of an equation, we can depress
the equation and obtain a quadratic equation by dividing by λ – λ . We can obtain a root
1
λ if we know a principal unit vector (say, ηη ηη ), as then from Eq. (7.7.5) λ is simply I · n .
1
1
1
1
The question then becomes: can a principal unit vector be found without solving Eq. (7.7.5)
and the associated linear system of Eqs. (7.7.10)? The answer is yes, on occasion. Specifi-
cally, if Π is a plane of symmetry of a set S of particles, then the unit vector normal to Π
is a principal unit vector for all reference points in Π.
To see this, recall that a plane of symmetry is such that for every particle on one side
of the plane there is a corresponding particle on the other side having the same mass and
at the same distance from the plane. Consider, for example, the plane Π depicted in Figure
7.9.1 with particles P and P of S equidistant from Π and on opposite sides of Π. Let the
2
1
particles each have mass m and let their distances from Π be h. Let O be any reference
point in Π, and let Q be the point of intersection of Π with the line connecting P and P .
1
2
Finally, let n be a unit vector normal to Π.
n P (m)
1
h
Q
Π
O
h
FIGURE 7.9.1
Particles on opposite sides of a plane
of symmetry. P (m)
2