Novel Approximations to the Damped Parametric Driven Pendulum Oscillators

The damped parametric driven nonlinear pendulum equation/oscillator (NPE), also known as the damped disturbed NPE, is examined, along with some associated oscillators for arbitrary angles with the vertical pivot. For analyzing and solving the current pendulum equation, we reduce this equation to the damped Duffing equation (DDE) with variable coefficients. After that, the DDE with variable coefficients is divided into two cases. In the first case, two analytical approximations to the damped undisturbed NPE are obtained. The first approximation is determined using the ansatz method while the second one is derived using He’s frequency formulation. In the second case, i.e., the damped disturbed NPE, three analytical approximations in terms of the trigonometric and Jacobi elliptic functions are derived and discussed using the ansatz method. The semianalytical solutions of the two mentioned cases are graphically compared with the 4th-order Runge–Kutta (RK4) approximations. In addition, the maximum error for all the derived approximations is estimated as compared with the RK4 approximation. The proposed approaches as well as the obtained solutions may greatly help in understanding the mysteries of various nonlinear phenomena that arise in different scientific fields such as fluid mechanics, plasma physics, engineering, and electronic chips.