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I Non-harmonic oscillation of pendulum

  1. Apr 17, 2017 #1
    Hi,
    I would like to ask what is the formula for non-harmonic oscillation of pendulum? I know that formula for harmonic oscillation of pendulum is: (d^2 φ)/(dt^2 )+g/r sinφ=0 where φ is angle, t is time, g is gravitational acceleration, r is length of a rope. I know that harmonic oscillation means that sinφ=φ.
     
  2. jcsd
  3. Apr 17, 2017 #2
    For an arbitrary initial angle, it can be shown that the solution to such a differential equation is given by
    $$\theta(t) = 2 \sin^{-1}\bigg\{k ~\text{sn} \bigg[\sqrt{\frac{g}{L}}(t-t_0);k\bigg]\bigg\}$$
    where ##k = \sin(\theta_0/2)## and ##t_0## is the time when the pendulum is vertical (##\theta = 0##). The function ##\text{sn}(x;k)## is a Jacobian elliptic function, which is defined as follows:

    Given the function
    $$u(y;k) = \int_0^y \frac{\mathrm{d}t}{\sqrt{(1-t^2)(1-k^2t^2)}}$$
    the Jacobian elliptic function in question is defined as the inverse of this function:
    $$y = \text{sn}(u;k)$$
    Values for such functions are often found in tables.
    The derivation of this result is non-trivial but certainly possible, if you remember the chain rule and integrate twice.
     
  4. Apr 17, 2017 #3
    Thanks for answer :smile:. Basically is it correct when I use α=-g/L*φ; T=2π*sqrt(L/g) for small amplitude where sinφ=φ and α=-g/L*sinφ; T=2π*sqrt(L/g)*(1+(1/16)*φ*φ+(11/3072*φ*φ+...) for large amplitudes? α-angular acceleration; g-gravitational acceleration; L- length of rope; φ- angle; T- period
     
  5. Apr 17, 2017 #4
    Yes, that is correct. The harmonically oscillating solution and associated initial angle-independent period (##T = 2 \pi \sqrt{\ell/g})## are always approximations. The point is, the small-angle approximation solution deviates very little from the actual solution when the initial angle is small. Figure three here illustrates this nicely; as the initial angle is increased, the full equation for the period and the approximation deviate more and more.
     
  6. Apr 17, 2017 #5
    Great! Thanks a lot for explanation :smile:.

    Edit (fresh_42): The rest of the post has been deleted, because it belongs to a separate thread.
     
    Last edited by a moderator: Apr 17, 2017
  7. Apr 17, 2017 #6

    berkeman

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    Staff: Mentor

    Multiple posting is not allowed at the PF, but maybe the other post is not a duplicate. It would be better if you would notify the Mentors before creating what looks like a duplicate post. We are dealing with post reports about this -- please give us a few hours to work this out.

    Edit (fresh_42): The new subject which this warning belongs to is now in a separate thread.
    Since the original question has been answered, this thread remains closed.
     
    Last edited by a moderator: Apr 17, 2017
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