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Foced Vibrations - beats

  1. Mar 17, 2014 #1
    I am trying to work though an example on this topic in my book and have reached a point that I am not sure about. I was wondering if anyone could help me clear this up.

    The equation of motion for a spring-mass system with no damping and a periodic external force is

    [tex]mu'' + ku = Fcos\omega t[/tex]

    The general solution to this is:

    [tex] u = Acos\omega_0t + B sin\omega_0 t +\frac{F}{m(\omega_0^2+\omega^2)}cos\omega t[/tex]

    If the mass is initially at rest, so that u(0)=0 and u'(0)=0, then the solution to this equation is

    [tex]u=\frac{F}{m(\omega_0^2-\omega^2)}(cos\omega t -cos\omega_0 t)[/tex]


    I have managed to follow it to here but I can not see how they have completed the next step:

    "making use of the trigonometric identities for cos(A[itex]\pm[/itex]B) with [itex]A=(\omega_0+\omega)t/2[/itex] and [itex]B=(\omega_0-\omega)t/2[/itex] we can write the equation in the form":

    [tex]u=\left[\frac{2F}{m(\omega_0^2-\omega^2)}sin\frac{(\omega_0-\omega)t}{2}\right]sin\frac{(\omega_0+\omega)t}{2}[/tex]

    I can not see how the penultimate equation becomes the final equation here; can anyone tell me how this works?
     
  2. jcsd
  3. Mar 17, 2014 #2

    olivermsun

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    Science Advisor

    Write out A+B and A-B.
    Then write out the angle addition formula for cos(A±B).
    You'll see how it works.
     
  4. Mar 17, 2014 #3
    Thanks a lot for the tip, Oliver. I've got it now.
     
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