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Damped Harmonic Motion Equation

  1. Aug 13, 2010 #1
    I am having trouble finding out what the equation for damped harmonic motion is. I have been researching around there there are many small variations on the exponents.

    I am conducting an experiment which has involved the use of the spring constant from Hooke's Law and have used a hypothesis which relates the two together. However, I can't seem to find a credible source for a damped harmonic motion equation when using springs. Hopefully, I was wondering whether someone could give one to me here?

  2. jcsd
  3. Aug 13, 2010 #2
    It can get quite complicated and non-linear. In some cases (eg pendulum) you have to make the assumption that the angle dispacement is below a certain "small angle" limit.

    But most simply, you use hooks law; force is proportional to dispalcement from equilibrium.
    and then you think about another "damping" force, that would be proportional to velocity.

    So adding the damping force to the normal "springing" force, you have an equation with the first two derivatives of the displacement (remember force is an acceleration):

    But (as always) it depends on the specifics of your problem...
    Last edited by a moderator: Apr 25, 2017
  4. Aug 13, 2010 #3
    what happens if I need an equation with a trigonometric function in it i.e. cos?
    Last edited by a moderator: Apr 25, 2017
  5. Aug 13, 2010 #4
    Its going to be infinitely more difficult to find the exact equation you need (and its solution) on the Internet, than just understanding the physics of your problem, formulating and solving your equation.

    What to do with a cos term depends on what you need to do... I'd be glad to point you in the right direction if you can give a better description of the problem.
  6. Aug 13, 2010 #5
    The differential equation onamor gave you is a generalized expression using Newton's 2nd Law. The "c" term is the damping coefficient. Depending on how big that coefficient is, the position function could be either an exponential decay ("overdamped" or "critically damped") or it could be the cosine function you mention with an exponential decay envelope.

    I'm sure you've looked there already, but the wikipedia page is pretty good: http://en.wikipedia.org/wiki/Damping

    This is good also http://mathworld.wolfram.com/DampedSimpleHarmonicMotion.html

    And this http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html

    And this site has an applet you can play with: http://phet.colorado.edu/en/simulation/mass-spring-lab
  7. Aug 13, 2010 #6


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    It's not clear what you mean by this. Equation for what? Are you talking about a driven harmonic oscillator where the forcing term F(t) is proportional to cos ωt, or are you referring to the solution x(t), which can be oscillatory if the system is underdamped? If it's the former, you just add another term to the differential equation:

    [tex]m\ddot{x} = F(t)-kx-b\dot{x}[/tex]

    If you know how to solve differential equations, it's probably easiest in the long run if you work through solving the differential equation yourself. It's not trivial, but it's not terribly difficult either. You'll understand what constants go where in the solution and which solution applies instead of trying to guess whether you have the right equation and are using it correctly.
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