Normal modes of a double pendulum at low energy levels

This study is concerned with a double pendulum and its regular behaviour associated with low energy levels and the influence of the associated initial conditions on the frequency of normal modes. The case of nonlinear oscillations described by the exact equations of motion is examined. A global qualitative insight is provided via energy diagrams and Poincaré maps. Then, the case of linear oscillations, their normal modes and associated frequencies is analysed. Further, quantitative insights via two approaches (Lindstedt–Poincaré method and harmonic balancing) are also achieved to determine analytically the influence of initial amplitudes on the existence and frequency of nonlinear normal modes. These results are compared with the one corresponding to the linear normal modes as well as with the corresponding numerical solutions of the exact equations of motion.