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Unformatted text preview: MA 36600 LECTURE NOTES: FRIDAY, MARCH 6 3 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 discuss
this steadystate solution for a few choices of the forcing term F (t).
We
know that the transient solution uc (t) is a periodic function that oscillates at a the natural frequency
ω0 = k/m. However, what if we try and force the mass to oscillate at a diﬀerent frequency ω = ω0 by
applying an outside force? To this end, we assume that
F (t) = F0 cos ω t for some constant ω = ω0 . That is, we wish to consider the equation
m u + k u = F0 cos ω t =⇒ 2
u + ω0 u = F0
cos ω t.
m We wish to ﬁnd a steadystate solution U (t).
We may use the Method of Variation of Parameters to ﬁnd such a solution, but instead we will use the
Method of Undetermined Coeﬃcients. To this end, we guess a solution in the form
U (t) = A0 cos ω t + B0 sin ω t.
We have the derivatives
U
U
U =
A0 cos ω t +
B0 sin ω t =
−ω A0 sin ω t +
ω B0 cos ω t = −ω 2 A0 cos ω t + −ω 2 B0 sin ω t Since ω = ω0 , we may choose =⇒ F0
and
2
m (ω0 − ω 2 )
Hence the general solution to the nonhomogeneous equation is
u(t) = A0 = A cos ω0 t + B sin ω0 t
transient + F0 cos ω t = U + ω 2 U 0
m
2
= ω0 − ω 2 A0 cos ω t 2
+ ω0 − ω 2 B0 sin ω t. B0 = 0. F0
cos ω t
2
m (ω0 − ω 2 )
steadystate (ω = ω0 ) ...
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 Spring '09
 EdrayGoins

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