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Math Help - Motion of a String

  1. #1
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    Motion of a String

    In class we were learining the spring equation,
    au''+gamma u'+m u=F(t)
    Can someone explain what damping is?
    Also, the meaning of all the 3 constants and the non-homegenous function.
    ---
    What I found stupid that after this equation was alchemisted a lot of the text (physics) can go to waste. Physicists love doing that. Make up some equation that works (only by luck) and think they are so smart. So they start making up formulas for each possible case which is utterly worthless if you have the primary equation. Just saying that because it made me angry.
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    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker View Post
    In class we were learining the spring equation,
    au''+gamma u'+m u=F(t)
    Can someone explain what damping is?
    Also, the meaning of all the 3 constants and the non-homegenous function.
    ---
    What I found stupid that after this equation was alchemisted a lot of the text (physics) can go to waste. Physicists love doing that. Make up some equation that works (only by luck) and think they are so smart. So they start making up formulas for each possible case which is utterly worthless if you have the primary equation. Just saying that because it made me angry.
    Experiments show that many spring like systems give a restoring force
    proportional to their displacement from equilibrium. Such a system will
    oscillate in simple harmonic motion if released (in theory and to a level
    of approximation in practice). The ODE representing this is:

    m x'' = -k x

    where m is the effective mass of the body attached to a light spring, and
    k is the spring constant.

    However we also observe that the oscillations decay in practice, and that
    a reasonable representation of this is obtained by introducing a speed dependent force opposing the motion. Now the ODE becomes:

    m x'' = -kx - cx'

    where now c is a constant characterising the damping.

    Now if an external force F(t) is applied to the system the ODE becomes:

    m x'' = -kx -cx' + F(t)

    which is what you have.

    RonL
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    Quote Originally Posted by ThePerfectHacker View Post
    In class we were learining the spring equation,
    au''+gamma u'+m u=F(t)
    Can someone explain what damping is?
    Also, the meaning of all the 3 constants and the non-homegenous function.
    ---
    What I found stupid that after this equation was alchemisted a lot of the text (physics) can go to waste. Physicists love doing that. Make up some equation that works (only by luck) and think they are so smart. So they start making up formulas for each possible case which is utterly worthless if you have the primary equation. Just saying that because it made me angry.
    If this helps, some type of harmonic motion "springs" up (ahem, sorry!) in just about any physical system. The reason is that the equation is simply a 2nd derivative level Taylor approximation to the motion, so if we take the limit of small deviations about a point of stability in the system, the motion will always be simple harmonic (damped or otherwise).

    In practically all real cases the motion will be damped, due to friction forces or other similar resistive properties in the system. We certainly don't need to have a description of all the different constants that go into this solution in order to solve it, but it is often convenient to label the type of harmonic motion then list the properties of its solution rather than give the solution and have the reader intuit the different properties. The physical properties of the motion are, after all, the important information not the solution of the equation.

    I agree that it's a hassle to remember all of them (I typically just look up the definitions rather than remember them).

    -Dan
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    But what does damping mean?
    Last edited by CaptainBlack; October 16th 2006 at 02:36 PM.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    But what does damping mean?

    damping - some thing that absorbs the energy of the vibrational modes of a system

    RonL
    What RonL said. There are a great number of cases where the harmonic motion equation is applied to systems that aren't "particles in motion" (say electric circuits.) In all cases that I know of the coefficient of the first derivative term indicates a source of energy flow out of the system. In the case of a particle in harmonic motion the term is a linear resistance friction term. In the case of the electric circuit it represents the action of ohmic resistance on the flow of charges in the circuit. etc.

    -Dan
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    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker View Post
    But what does damping mean?

    damping - some thing that absorbs the energy of the vibrational modes of a system

    RonL

    Sorry this appears out of order - I'm suffering for the hitting-edit-rather-than-quote-key syndrome
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