In expt.1 we have demonstrated simple harmonic motion (SHM) using the case of a simple pendulum. Now, when the angular displacement amplitude of the pendulum is large enough that the small angle approximation, as in case of SHM, no longer holds, and then the equation of motion must remain in its nonlinear form,
This differential equation does not have a closed form solution, and must be solved numerically using a computer. The small angle approximation is valid for initial angular displacements of about 20° or less. When the initial angular displacement is significantly large that the small angle approximation is no longer valid, the error between the simple harmonic solution and the actual solution becomes apparent almost immediately, and grows as time progresses.
A more detailed discussion of the above principles of the nonlinear oscillator can be found at:
Further resources on the nonlinear pendulum: