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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 steady-state 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 ) steady-state Figure 1 shows a plot of a transient solution uc (t), while Figure 2 shows a plot of a steady-state 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 steady-state. 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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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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