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Thread: Vibration help

  1. #1
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    Vibration help

    a machine is subject to two vibrations.

    one vibration has the form : 2 cos wt and the other has the form 3cos(wt + pie/4)

    Determine resulting vibration and express it in the general form of n cos (wt+-a)
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  2. #2
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    Re: Vibration help

    Quote Originally Posted by jdavies535 View Post
    a machine is subject to two vibrations.

    one vibration has the form : 2 cos wt and the other has the form 3cos(wt + pie/4)

    Determine resulting vibration and express it in the general form of n cos (wt+-a)
    3cos(wt + pie/4) is that apple or cherry?
    Thanks from topsquark
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  3. #3
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    Re: Vibration help

    $y=2\cos(\omega t)+3[\cos(\omega t)\cos(\pi/4)-\sin(\omega t)\sin(\pi/4)]$

    $y=2\cos(\omega t) + \dfrac{3\sqrt{2}}{2}[\cos(\omega t)-\sin(\omega t)]$

    $y=\dfrac{4+3\sqrt{2}}{2}\cos(\omega t) - \dfrac{3\sqrt{2}}{2}\sin(\omega t)$

    note ... $a\cos(x)+b\sin(x)=R\cos(x-\alpha)$

    Let $a=\dfrac{4+3\sqrt{2}}{2}$ and $b=-\dfrac{3\sqrt{2}}{2}$

    $R=\sqrt{a^2+b^2} \implies R \approx 4.635$

    $\alpha = \arctan(b/a) \approx -0.475$

    $y=4.635\cos(\omega t + 0.475)$
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    Re: Vibration help

    Quote Originally Posted by jdavies535 View Post
    a machine is subject to two vibrations.

    one vibration has the form : 2 cos wt and the other has the form 3cos(wt + pie/4)

    Determine resulting vibration and express it in the general form of n cos (wt+-a)
    I prefer key lime pie.

    Big hint:
    2 ~ cos( \omega t) + 3 ~ cos \left ( \omega t + \frac{\pi}{4} \right )

    Use the identity cos(a + b) = cos(a)~cos(b) - sin(a)~sin(b) to rewrite this in the form a~cos(\omega t) + b~sin(\omega t + \phi)
    where c ~ \sqrt{a^2 + b^2} and \phi = atan2 \left ( \frac{b}{a} \right ). This is a standard, but little used function. It is defined in terms of the usual tan^{-1}(x) function.

    Can you fill in the details?

    -Dan

    Edit: skeeter got to it first. I'll have to take him to an empty field and do some "skeet" shooting.
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