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0593_C13_fm Page 441 Monday, May 6, 2002 3:21 PM

Introduction to Vibrations 441

As noted earlier, the constants in Eqs. (13.2.2) and (13.2.3) are to be evaluated by auxiliary
conditions. These auxiliary conditions are generally initial conditions or boundary condi-
tions. Initial conditions (where t = 0) might be expressed as:

x 0 () = x and ˙ x 0 () = ˙ x (13.2.9)
0 0

Then, by requiring the solution of Eq. (13.2.2) to meet these conditions, we have:

A = x and B x ˙ ω (13.2.10)
=
0 0
and thus,

x = x cosω x ˙ ω )sinω t (13.2.11)
0 t +( 0

Boundary conditions might be expressed as:

x 0 () = 0 and x() = 0 (13.2.12)
l

Then, by requiring the solution of Eq. (13.2.2) to meet these conditions, we see that the
constants A and B must satisfy:

A = 0 and Bsinωl = 0 (13.2.13)

The second expression is satisfied by either:

B = 0 or sinωl = 0 (13.2.14)

If B = 0, we have the trivial solution x = 0. Alternatively, if sinω = 0, ω must be an
integer multiple of π. That is,

ωl = n π or ω = ω = n π l (13.2.15)
n
Thus, there is a nontrivial solution only for selected values of ω — that is, for ω = nπ/ .
The solution for the displacement then takes the multiple forms:

π
x = x = B sin n t l n = , , … (13.2.16)
12
n n
Then, by superposition, we have:

∞
x = ∑ B sin n t l (13.2.17)
π
n
n=1
where the constants B are to be determined from other conditions of the specific system
n
being studied. (Eq. (13.2.17) is a Fourier series representation of the solution.)