Internal Resonances under Oscillations of a Double Pendulum

In this paper, internal resonances appearing in mechanical systems with several degrees of freedom, when the ratio between the frequencies of their oscillations represent integers, are studied. As specific examples, such systems with two degrees of freedom like a double mathematical pendulum and a double physical pendulum are considered. The relationships for the oscillation frequencies of each of these systems are presented in dimensionless form depending on two dimensionless parameters characterizing the ratio between the weights of the end loads or weighty links, as well as the ratio between the lengths of the links. The conditions that should be imposed on these dimensionless parameters in order to provide for the appearance of internal resonances have been revealed. The obtained solutions are presented graphically in the form of curves on the plane of dimensionless parameters corresponding to internal resonances. In addition, a detailed investigation into the nature of these curves is presented, too. The results found upon analyzing the two problems are compared with each other. The obtained relationships have fundamental theoretical significance, which could be useful for the case of different technical applications.