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I'm working in the modeling of a interactive pendulum with the following characteristics;

  • the user can apply forces at any time to move the pendulum its rest pose or to accelerate it when it is already in motion
  • the pendulum can not reach an angle biger than 90 degrees
  • the pendulum will experience a dampening force at such force must be able to bring the pendulum to zero motion in a given time.

I have been searching for articles on how to model a pendulum but so far all of them deal with a pendulum in where they assume the angle is small, which is not my case. Also when dealing with external forces the articles assume a periodic horizontal force, which again is not my case as the user can apply the force at any time no matter the current state of the pendulum.

So I wonder if someone can point me or give me clues on how to model such system.

Regards

Arthur
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Juan Camilo Acosta Arango
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  • Without an external driving force (just uniform downward gravitational force), it's straightforward to derive the equation $$ \frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + \frac{g}{\ell} \sin\theta = 0, $$ and this can be modified to include a driving force as well. The small angle approximation that you describe replaces $\sin\theta by \theta$, making the equation (much) easier to solve analytically. But you can't use this approximation if angular displacement is allowed to be large. – Sammy Black Nov 25 '23 at 06:50

1 Answers1

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A typical model for a (damped) mathematical pendulum would be $$ \ddot \phi + \gamma \dot \phi + \omega^2 \sin(\phi) = f(t)\cos(\phi) \,.$$ Here, $\phi(t)$ is the angle with respect to the rest position; $\omega$ is the resonance frequency for small excitations; $\gamma$ is the damping constant; $f(t)$ is the force that is (horizontally) applied to the bob.

Note that the restriction $|\phi| \leq \pi/2$ is not included in this formula. Physically, a pendulum can have larger angles than $90^\circ$. So to enforce this condition, you should think what happens when the pendulum reaches the horizontal position. A possibility would be that there is an elastic wall which would mean that you set $\dot \phi \mapsto - \dot \phi$ whenever $|\phi|= \pi/2$.

Fabian
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