The kinematics behaviour of coupled pendulum using differential transformation method

Abstract

The topic of coupled oscillations is rich in physical content which is both interesting and complex. In this research work we aimed to study the kinematics behaviour of an important physical system known as coupled pendulum, in which two identical pendulums were coupled together by a spring. The Lagrangian of the system was first constructed, then the equations of motion have been derived. Analytical solutions were obtained for these equations, and in addition we apply a reliable algorithm based on an adaptation of the standard differential transformation method is presented, which is the multi-step differential transformation method to obtain numerical solutions for the considered system for some selected initial conditions. The solutions of the equations of motion were obtained by the multi-step differential transform method. Of course, this method is needed if it predicts the motion of the considered system. For this reason figurative comparisons between the multi-step differential transformation method, the standard differential transformation method and the classical fourth-order Runge–Kutta method are given. The results reveal that the proposed technique is a promising tool to predict the behaviours of the considered system in long time interval. Moreover, the motion of curves represents the initial conditions that sensitive depend and how much oscillatory motion. In conclusion, the proposed technique is a reliable method to solve the considered double pendulum problem in long time interval. We believe that the system considered in this work along with the method used is interesting for physicists, mathematicians and engineers.