Unformatted text preview: MA 36600 LECTURE NOTES: MONDAY, MARCH 9 Resonance. We know that the general solution to the nonhomogeneous equation
m u + k u = F (t)
is in the form u(t) = uc (t) + U (t) for a transient solution uc (t) and a steadystate solution U (t). We found
in the previous lecture that
k
uc (t) = A cos ω0 t + B sin ω0 t = R cos (ω0 t − δ )
where
ω0 =
.
m
If we try and force the mass to oscillate at a diﬀerent frequency ω = ω0 by applying an outside force,
assuming that
F (t) = F0 cos ω t
then we have the solution
F0
U (t) = A0 cos ω t + B0 sin ω t
=⇒
A0 =
and
B0 = 0.
2
m (ω0 − ω 2 )
Hence the general solution to the nonhomogeneous equation is
u(t) = A cos ω0 t + B sin ω0 t
+ transient F0
cos ω t
2
m (ω0 − ω 2 )
(ω = ω0 ) steadystate Figure 1 shows a plot of a transient solution uc (t), while Figure 2 shows a plot of a steadystate solution
U (t). Figure 3 shows the solution u(t) = uc (t) + U (t). The “wiggles” in the graph show the eﬀect of the
transient solution on the steadystate. In general, if the amplitude F0 of the forcing term F (t) = F0 cos ω t
is “small” then u(t) ≈ uc (t). But if F0 is “large” then u(t) ≈ U (t).
Figure 1. Graph of Transient Solution uc (t) = cos(9 t − 1)
10 5 2.5 0 2.5 5 7.5 5 10 1 10 12.5 15 ...
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 Spring '09
 EdrayGoins
 Steady State, Frequency, transient solution uc

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