Closed-Form Solutions to a Forced Damped Rotational Pendulum Oscillator

In this investigation, some analytical solutions to both conserved and non-conserved rotational pendulum systems are reported. The exact solution to the conserved oscillator (unforced, undamped rotational pendulum oscillator), is derived in the form of a Jacobi elliptical function. Moreover, an approximate solution for the conserved case is obtained in the form of a trigonometric function. A comparison between both exact and approximate solutions to the conserved oscillator is examined. Moreover, the analytical approximations to the non-conserved oscillators including the unforced, damped rotational pendulum oscillator and forced, damped rotational pendulum oscillator are obtained. Furthermore, all mentioned oscillators (conserved and non-conserved oscillators) are linearized, and their exact solutions are derived. In addition, all obtained approximations are compared with the four-order Runge-Kutta (RK4) numerical approximations and with the exact solutions to the linearized oscillators. The obtained results can help several authors for discussing and interpreting their results.