lecture_21 (dragged) 2

lecture_21 (dragged) 2 - MA 36600 LECTURE NOTES: FRIDAY,...

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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 steady-state solution U (t). We discuss this steady-state 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 steady-state 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 ) ￿ ￿￿ ￿ steady-state (ω ￿= ω0 ) ...
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