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0593_C13_fm Page 457 Monday, May 6, 2002 3:21 PM

Introduction to Vibrations 457

or

ξ +( ξ ) =
˙˙
2
1 2 − )(km 1 0 (13.8.10)
or

˙˙ ξ + ω ξ =
2
1 1 1 0 (13.8.11)
˙˙
˙˙
Similarly, by considering ξ 2 and ξ 3 we have:
˙˙ ω ξ =
ξ +
2
2 2 2 0 (13.8.12)
and

˙˙ ω ξ =
ξ +
2
3 3 3 0 (13.8.13)
We can recognize Eqs. (13.8.11), (13.8.12), and (13.8.13) as being in the form of the
undamped linear oscillator equation, Eq. (13.2.1). The solutions of the equations may thus
be expressed as:

+
ξ = A cos ω t B sin ω t (13.8.14)
1 1 1 1 1
+
ξ = A cos ω t B sin ω t (13.8.15)
2 2 2 2 2
+
ξ = A cos ω t B sin ω t (13.8.16)
3 3 3 3 3
where, as before, the constants A and B (i = 1, 2, 3) are to be evaluated from auxiliary
i
i
conditions.
Equations (13.8.14), (13.8.15), and (13.8.16) show that. given suitable auxiliary conditions.
the three-particle system will have sinusoidal oscillation, with the “shape,” or mode, of
the oscillation being in the form of ξ , ξ , or ξ as defined by Eqs. (13.8.4), (13.8.5), and
3
2
1
(13.8.6).
This, however, raises a question: suppose the auxiliary conditions (say, the initial con-
ditions) are such that they do not correspond to a shape of one of the modes ξ , ξ , or ξ ;
3
2
1
what, then, is the subsequent motion of the system? The answer is that the subsequent
motion may be expressed as a linear combination of the ξ , ξ , and ξ . That is, the ξ , ξ ,
1
1
3
2
2
and ξ may be regarded as “base vectors” in a three-dimensional space (for the three
3
degrees of freedom of the system).
To see this, consider again the solutions of Eqs. (13.8 .11), (13.8.12), and (13.8.13) in the
form of Eqs. (13.8.14), (13.8.15), and (13.8.16). The derivatives of ξ , ξ , and ξ may be
2
1
3
expressed as:
˙
ξ =−A ω sinω +t B ω cosω t (13.8.17)
1 1 1 1 1 1 1
˙
ξ =−A ω sinω +t B ω cosω t (13.8.18)
2 2 2 2 2 2 2