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Math Help - Mass-spring system

  1. #1
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    Mass-spring system

    Vibration Free


    Please, are correct?

    m \frac{d^2x}{dt^2} + kx = 0

    Where frequency is

    w = \sqrt{\frac{k}{m}}

    \frac{d^2x}{dt^2} + \frac{k}{m}x = 0


    The characteristic equation is:

    r^2 + w^2 = 0
    r = +or- iw where i^2 = -1

    Then:
    x(t) = C_1e^{iwt} + C_2e^{-iwt}

    Calculating I can get
    x(t) = a_1 \cos(wt) + a_2 \sin(wt)


    Now, I need to do to get the following equation ?

    x(t) = Acos(wt - \delta) (I think this is the equation we need to get the free vibration)
    Last edited by mr fantastic; December 19th 2009 at 02:21 PM. Reason: Fixed some subscripts
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  2. #2
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    Quote Originally Posted by Apprentice123 View Post
    Vibration Free


    Please, are correct?

    m \frac{d^2x}{dt^2} + kx = 0

    Where frequency is

    w = \sqrt{\frac{k}{m}}

    \frac{d^2x}{dt^2} + \frac{k}{m}x = 0


    The characteristic equation is:

    r^2 + w^2 = 0
    r = +or- iw where i^2 = -1

    Then:
    x(t) = C_1e^{iwt} + C_2e^{-iwt}

    Calculating I can get
    x(t) = a_1 \cos(wt) + a_2 \sin(wt)


    Now, I need to do to get the following equation ?

    x(t) = Acos(wt - \delta) (I think this is the equation we need to get the free vibration)
    There are several threads in the trigonometry subforum that show how to go from a_1 \cos(wt) + a_2 \sin(wt) to Acos(wt - \delta). You will need to spend the time finding them. I'm certain that you will also be able to find the required calculation using Google.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    There are several threads in the trigonometry subforum that show how to go from a_1 \cos(wt) + a_2 \sin(wt) to Acos(wt - \delta). You will need to spend the time finding them. I'm certain that you will also be able to find the required calculation using Google.

    You have the link to a Thread about this?
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  4. #4
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    Quote Originally Posted by Apprentice123 View Post
    You have the link to a Thread about this?
    That is research for you to do, not me.
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  5. #5
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    Quote Originally Posted by Apprentice123 View Post
    Vibration Free


    Please, are correct?

    m \frac{d^2x}{dt^2} + kx = 0

    Where frequency is

    w = \sqrt{\frac{k}{m}}

    \frac{d^2x}{dt^2} + \frac{k}{m}x = 0


    The characteristic equation is:

    r^2 + w^2 = 0
    r = +or- iw where i^2 = -1

    Then:
    x(t) = C_1e^{iwt} + C_2e^{-iwt}

    Calculating I can get
    x(t) = a_1 \cos(wt) + a_2 \sin(wt)


    Now, I need to do to get the following equation ?

    x(t) = Acos(wt - \delta) (I think this is the equation we need to get the free vibration)
    There is nothing to do, just observe that these are just two different ways of writing an arbitary sinusoid of angular frequency \omega

    (and by the way the angular frequency is usually represented by \omega (omega) and the phase by \varphi (phi))

    CB
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  6. #6
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    Quote Originally Posted by CaptainBlack View Post
    There is nothing to do, just observe that these are just two different ways of writing an arbitary sinusoid of angular frequency \omega

    (and by the way the angular frequency is usually represented by \omega (omega) and the phase by \varphi (phi))

    CB

    Thank you. A is a constant? What is \delta ?
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  7. #7
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    Quote Originally Posted by Apprentice123 View Post
    Thank you. A is a constant? What is \delta ?
    It is also a constant (it is the phase) Both A and \varphi (or \delta if you prefer) are determined by the initial conditions.

    What you have is a general solution to a second order ODE, and there should be two constants, these are A and \varphi in one form of the solution and C_1 and C_2 in the other form for the solution.

    CB
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  8. #8
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    Quote Originally Posted by CaptainBlack View Post
    It is also a constant (it is the phase) Both A and \varphi (or \delta if you prefer) are determined by the initial conditions.

    What you have is a general solution to a second order ODE, and there should be two constants, these are A and \varphi in one form of the solution and C_1 and C_2 in the other form for the solution.

    CB
    The initial conditions

    x(0) = Acos(- \delta) = x_o
    x'(0) = -Awsin(- \delta) = v_o

    I find \delta = tan^{-1} \frac{v_o}{x_o w}

    How I find A ?
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  9. #9
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    Look at A^2cos^2(\delta)+ A^2sin^2(\delta)= A^2.
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  10. #10
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    Quote Originally Posted by HallsofIvy View Post
    Look at A^2cos^2(\delta)+ A^2sin^2(\delta)= A^2.
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