Dynamics of a Double Pendulum with Viscous Friction at the Hinges. II. Dissipative Vibration Modes and Optimization of the Damping Parameters

This study is a continuation of the work “Dynamics of a Double Pendulum with Viscous Friction at the Hinges. I. Mathematical Model of Motion and Construction of the Regime Diagram”, in which a linear mathematical model of the motion of a double mathematical pendulum with identical parameters of its links and end loads in the presence of viscous friction at both of its joints was presented, and a diagram of dissipative modes of its motion was also constructed. The question of a particular variant of proportional damping is considered, in which the vibration modes of a dissipative system are not distorted by friction forces, and basic formulas are given that describe the dynamics of the system in this situation. For the general case of damping all key quantities characterizing the motion of the system for each of the dissipative vibration modes are identified and determined by applying a rational combination of analytical and numerical research methods. In addition, several problems of the optimal damping of system vibrations are considered, and the best dissipative parameters are selected based on the criterion of the maximum degree of stability. The results obtained are accompanied by a series of graphical illustrations, which make it possible to establish their dependence on the damping coefficients and note their main qualitative and quantitative features. The solutions found can be useful in practice when designing two-link manipulators and studying their dynamic behavior.