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Driven un-damped harmonic motion

  1. Jan 30, 2010 #1
    1. The problem statement, all variables and given/known data

    A particle of mass 'm' is subject to a spring force, '-kx', and also a driving force, Fd*cos(wdt). But there is no damping force. Find the particular solution for x(t) by guessing x(t) = A*cos(wdt) + B*sin(wdt). If you write this in the form C*cos(wdt - [tex]\phi[/tex]), where C > 0, what are C and [tex]\phi[/tex]? Be careful about the phase (there are two cases to consider).

    2. Relevant equations

    basic inhomogeneous second-order equation solving.

    3. The attempt at a solution

    So I got the equation F = Fd*cos(wdt) - k*x = m*x'', and put it in the form x'' + w2*x = F*cos(wdt), where w2[tex]\equiv[/tex] k/m, and F [tex]\equiv[/tex] Fd/m. I took the first and second derivative of the equation they told us to use for our guess, and subbed them into x'' + w2*x = F*cos(wdt). I ended up with A(w2 - wd2)*cos(wdt) + B(w2 - wd2)*sin(wdt) = F*cos(wdt). I therefore deduced that F = A(w2 - wd2). Therefore the particular solution would be xp(t) = A(w2 - wd2)*cos(wdt). Is there anyway that this can somehow simplify for the form they want us to put it in, C*cos(wdt - [tex]\phi[/tex]), or did I do something wrong?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 31, 2010 #2

    ehild

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    Gold Member

    You know F and have to express A and B in terms of F, wd and w.

    ehild
     
  4. Feb 1, 2010 #3
    ok, so then I get x(t) = [tex]\frac{F}{w^2-wd^2}[/tex]*cos(wd2*t) for the particular solution. I still can't figure out how this can take the form that is asked for.
     
  5. Feb 1, 2010 #4

    ehild

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    Homework Helper
    Gold Member

    It is [tex]x(t) = \frac{F}{w^2-wd^2}*\cos{(w_d*t)}[/tex]

    instead and has the desired form if w>wd.
    [tex]C = {\frac{F}{|w^2-wd^2|}, \Phi =0[/tex]

    In the opposite case

    [tex]x(t) =- \frac{F}{|w^2-wd^2|}*\cos{(w_d*t)}[/tex]

    you have to include a phase constant of pi:

    [tex]C = {\frac{F}{|w^2-wd^2|}, \Phi =\pi[/tex]

    [tex]x(t) =\frac{F}{|w^2-wd^2|}*\cos{(w_d*t+\pi)}[/tex]

    ehild
     
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