Steady Periodic Vibrations

In a Nut Shell: Displacement, x(t), of an undamped mechanical system with mass, m,

and spring constant, k , (figure below) undergoes steady periodic vibrations when

subjected to a periodic forcing function, f(t).

The steady state response, (also called the steady periodic solution and is also the

particular solution) , x_p(t), is governed by the forcing function since there is

no damping to diminish the response. The differential equation of motion is:

m x’’ + k x = f(t) ----------------------------------- (1)

where m = mass of the system

x’’ = d²x/dt² = the acceleration of the mass, m

k = spring constant

x(t) = displacement of the mass

f(t) = periodic forcing function

Representation of Forcing Function, f(t)

Fourier Series provide a way to represent more complicated (more realistic) forcing

functions, f(t), with direct applications to vibration problems. Suppose f(t) is

represented as follows:

∞

f(t) = ∑ b_n sin (nπt/L) ----------------------------------- (2)

n = 1

where b_n are the Fourier coefficients (need to be determined)

Click here to continue with discussion of steady periodic vibrations.

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