Non-harmonic oscillation of pendulum

In summary, the formula for non-harmonic oscillation of a pendulum can be derived using the equation for harmonic oscillation and the Jacobian elliptic function. For small angles, the solution closely approximates the actual solution, but for larger angles, the deviation increases.
  • #1
nneutrino
7
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.
 
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  • #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.
 
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  • #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
 
  • #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.
 
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  • #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.
 
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  • #6
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.
 
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1. What is non-harmonic oscillation of a pendulum?

Non-harmonic oscillation of a pendulum refers to the motion of a pendulum that does not follow a perfect sinusoidal pattern. This can occur due to a variety of factors, such as air resistance, friction, or non-uniform gravity fields.

2. What causes non-harmonic oscillation in pendulums?

Non-harmonic oscillation in pendulums can be caused by external factors such as air resistance, friction, or non-uniform gravity fields. It can also be caused by imperfections in the pendulum itself, such as uneven weight distribution or irregularities in the pivot point.

3. How does non-harmonic oscillation affect the period of a pendulum?

Non-harmonic oscillation can affect the period of a pendulum by causing it to deviate from the expected oscillation pattern. This can result in a longer or shorter period, depending on the factors causing the non-harmonic motion.

4. Can non-harmonic oscillation be corrected in a pendulum?

In some cases, non-harmonic oscillation in pendulums can be corrected by adjusting the external factors that are causing it, such as minimizing air resistance or reducing friction. However, if the non-harmonic motion is caused by imperfections in the pendulum itself, it may be more difficult to correct.

5. How does non-harmonic oscillation affect the accuracy of a pendulum clock?

Non-harmonic oscillation can greatly affect the accuracy of a pendulum clock. If the pendulum is not following a regular, harmonic motion, the timekeeping of the clock will be impacted. This is why pendulum clocks often need to be adjusted or calibrated to account for any non-harmonic oscillation that may occur.

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