When finding the period of a pendulum beyond the small angle approximation, we have to use integration for small interval of $\theta$ and elliptical integration.
I was trying to apply this situation for the driven pendulum.
When using small angle approximation, the equation of motion would be like below.
$$I \ddot{x}(t)+b \dot x(t)+m g l x(t)=F\cos(\omega t)$$
- $I$ : moment of inertia
- $b$ : damping coefficient
- $m$ : mass
- $g$ : gravity acceleration
- $l$ : distance between CM of the system and the origin of rotation
- $F$ : amplitude of external force
- $\omega$ : angular frequency of external force
- $x$ : angular displacement
- $t$ : time
When not using small angle approximation, above equation would be changed.
$$I \ddot x(t)+b\dot x(t)+m g l\sin(x(t))=F\cos(\omega t)\cos(x(t)).$$
Then, how do I solve this differential equation?
I tried to simulate the result with Mathematica's DSolve, but it didn't show the solution. Although I plotted the graph with NDSolve, I want to know the way to solve that differential equation.
Best Answer
As was mentioned in the comments, the differential equation you give is not solvable analytically. What one can do is one can go beyond the small angle approximation in a controlled fashion by Taylor expanding the sine and cosine function and find (e.g. expanding up to order $x^3$)
$$I \ddot x + b\dot x + mgl\left( x - \frac{x^3}{6}\right) = F \cos(\omega t) \left( 1 - \frac{x^2}{2} \right) $$
This still is a non-linear differential equation and might not be solvable either, but it opens the door for Perturbation Theory, which is the major tool to use when going beyong linear approximations.