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Strategy to find the steady
periodic response, x(t), for the differential
equation
governing the mechanical system
m x’’ +
k x = f(t)
---------------------------------------- (1)
where
the general, periodic forcing function is represented by
∞
f(t) =
∑ bn sin (nπt/L)
------------------------------------- (2)
n = 1
involves
four steps.
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Step 1: Find the Fourier coefficients for f(t),
of period 2L, as follows:
L
bn = (2/L) ∫ f(t) sin(nπt/L) dt and substitute into eq. (2)
0
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Step 2: Assume a steady periodic response of
the mechanical system
governed
by eq. (1), xsp(t) as follows:
∞
xsp(t) =
∑ cn sin (nπt/L)
----------------------------------- (3)
n = 1
where cn are unknown coefficients (yet to be
determined)
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Step 3: Substitute (3) into the equation of
motion, eq. (1).
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Step 4: Equate coefficients of sin (nπt/L)
and solve for cn
using
the
values obtained for bn .
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