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System mass-spring-damper and Energy

  1. Jan 12, 2014 #1

    A general system mass-damper-spring is describied by differential equation:
    [tex]m \frac{d^2x}{dt^2}+\gamma \frac{dx}{dt}+kx=F[/tex]
    For calculate the energy in this system just integrate the equation wrt x. However, I know the 3th parcel integrated is the elastic potential energy of spring, but what are it the 1st and 2nd parcels integrated?

    PS: the 1st should be the kinetic energy, I think...
  2. jcsd
  3. Jan 12, 2014 #2


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    Hello Jhenrique! :wink:
    Yes, the 1st term is mv dv/dx, = d/dx (1/2 mv2), = d/dx (KE).

    (and the 3rd term is obviously d/dx (1/2 kx2), = d/dx (PE))

    The 2nd term is v, and ∫ v dx isn't anything particular (other than the energy lost to friction) …

    nor would you expect it to be …

    this is a damping term, so non-conservative, and you only expect a potential energy from a conservative force! :smile:
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