It looks like perhaps there is a solution of the form

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It looks like perhaps there is a solution of the form xp = B cos(omega t) To see what B must be, plug this into (**) : omega_n^2) xp = B cos(omega t) xp" = - B omega^2 cos(omega t) ---------------------------------- - A cos(omega t) = xp" + omega_n^2 xp = B(omega_n^2 - omega^2) cos(omega t) This works out if we take B = A / (omega_n^2 - omega^2) . The output amplitude is a multiple of the input amplitude, and the ratio is the GAIN: H = B/A = 1/|omega_n^2 - omega^2| Imagine the natural frequency of the oscillator fixed, and we slowly increase the frequency of the input signal. The graph of the gain starts when omega = 0 at H = 1/omega_n^2 and then increases to a vertical asymptote at omega = omega_n . This is RESONANCE, and then no such sinusoidal solution exists. There are solutions, of course, and we will come back to this case later. What happens with the weight and rubber band is that the nonlinear character of the spring asserts itself for large amplitude. When omega > omega_n , the gain falls back towards zero. Also: when omega < omega_n the denominator is positive, and the
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output is a positive multiple of the input. When omega > omega_n the denominator is negative, and the output signal is a negative multiple of the input: this is PHASE REVERSAL. On Friday we'll add in damping.
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