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Archived Driven, Damped Oscillator; Plot x(t)/A

  1. Oct 24, 2013 #1
    1. The problem statement, all variables and given/known data

    The problem is long so I will post the whole thing but ask only for help on part C.

    The steady-state motion of a damped oscillator driven by an applied force F0 cos(ωt) is given by xss(t) = A cos(ωt + φ).
    Consider the oscillator which is released from rest at t = 0. Its motion must satisfy x(0) = 0 and v(0) = 0. After a very long time, we expect that x(t) = xss(t). To satisfy these conditions we can take as the solution
    x(t) = xss(t) + xtr(t),
    where xtr(t) is a solution to the equation of motion of the free damped oscillator.

    (A) Show that if xss(t) satisfies the equation of motion for the forced damped oscillator, then so does x(t) = xss(t) + xtr(t), where xtr(t) satisfies the equation of motion of the free damped oscillator.

    (B) Sketch the resulting motion for the case that the oscillator is driven at resonance (i.e., ω = ω0 where ω02=k/m) with b=mω0.

    (C) Choose the arbitrary constants in xtr(t) so that x(t) satisfies the initial conditions, assuming that the oscillator is driven at resonance with b=mω0. Hand in an accurate graph of x(t)/A.

    2. Relevant equations

    x(t)/A=sin(w0t)-[itex]\frac{2}{sqrt(3)}[/itex]sin([itex]\frac{sqrt(3)}{2}[/itex]w0t)e-w_0t/2

    A=[itex]\frac{F_0}{mw_0^2}[/itex]

    3. The attempt at a solution

    I am not quite sure what I am being asked for. How do I pick the constants in xtr(t)? Does it matter what I use or can I just plug in 1 for all of them and plot the result? Doing that yields a plot that seems to exhibit the characteristics that a driven, damped oscillator would.

    Thank you!
     
  2. jcsd
  3. Jun 6, 2016 #2

    vela

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    The response of the system when ##b=m\omega_0## is given by
    $$x(t) = e^{-\omega_0 t/2}\left[c_1 \cos \left(\frac{\sqrt{3}\omega_0 t}{2}\right) + c_2 \sin \left(\frac{\sqrt{3}\omega_0 t}{2}\right)\right] + \frac{F_0}{m[(\omega_0^2-\omega^2)^2 + \omega_0^2\omega^2]}[(\omega_0^2-\omega^2)\cos \omega t + \omega_0\omega \sin \omega t]$$ where ##c_1## and ##c_2## are arbitrary constants to be determined by the initial conditions. At resonance, we have ##\omega=\omega_0##, and the solution simplifies to
    $$x(t) = e^{-\omega_0 t/2}\left[c_1 \cos \left(\frac{\sqrt{3}\omega_0 t}{2}\right) + c_2 \sin \left(\frac{\sqrt{3}\omega_0 t}{2}\right)\right] + \frac{F_0}{m\omega_0^2}\sin \omega_0 t.$$ The condition ##x(0)=0## requires that ##c_1=0##. Taking the derivative of x(t), we get
    $$v(t) = c_2e^{-\omega_0 t/2}\left[-\frac{\omega_0}{2}\sin \left(\frac{\sqrt{3}\omega_0 t}{2}\right) + \frac{\sqrt{3}\omega_0}{2} \cos \left(\frac{\sqrt{3}\omega_0 t}{2}\right)\right] + \frac{F_0}{m\omega_0}\cos \omega_0 t.$$ The condition ##v(0)=0## requires that
    $$\frac{\sqrt{3}c_2\omega_0}{2} + \frac{F_0}{m\omega_0} = 0.$$ Solving for ##c_2## gives
    $$c_2 = -\frac{2F_0}{\sqrt{3}m\omega_0^2}.$$ The solution is therefore
    $$x(t) = \frac{F_0}{m\omega_0^2}\left[\sin \omega_0 t - \frac{2}{\sqrt{3}}\sin\left(\frac{\sqrt{3}}{2}\omega_0 t\right)\right].$$ To plot the function ##x(t)/A## where ##A = \frac{F_0}{m\omega_0^2}##, define ##T_0 = \frac{2\pi}{\omega_0}## and plot in units of ##T_0##.
     
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