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driven_oscillators

# driven_oscillators - DRIVER k M kx F0 Cos t bv DAMPER...

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ω ’t + δ ’) ( F Cos 0 M k -kx -bv DAMPER ω DRIVER Driven Oscillators Consider the system pictured above. A damped oscillating system is attached to a motorized contraption that exerts an oscillating force of amplitude F 0 at an angular frequency ω 0 . The phase of the contraption at t = 0 is given by δ 0 . Σ F x = ma x - kx - bv x + F 0 cos( ω 0 t + δ 0 ) = m d 2 x dt 2 0 = d 2 x dt 2 + b m dx dt + k m x - F 0 m cos( ω 0 t + δ 0 ) The ﬁrst three terms are identical to the terms in the equation for the damped oscillator. The driver merely adds a non-homogeneous term (that is, a term unrelated to x ). Trial Solution: Let’s try a solution that has a piece that looks like the solution to a damped oscillator, since the diﬀerential equation resembles that of a damped oscilla- 1

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tor, and add a piece that oscillates with the frequency of the driver. x ( t ) = A 0 e - Bt cos( ωt + δ ) + D cos( ω 0 t + φ ) dx dt = A 0 e - Bt [ - B cos( ωt + δ ) - ω sin( ωt + δ )] - ω 0 D sin( ω 0 t + φ ) d 2 x dt 2 = A 0 e - Bt
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driven_oscillators - DRIVER k M kx F0 Cos t bv DAMPER...

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