Dynamics of a double pendulum with viscous friction in the joints. II. Dissipative oscillation modes and optimization of damping parameters

This paper is a continuation of the article “Dynamics of a double pendulum with viscous friction in the joints. 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 links and end loads in the presence of viscous friction in both articulations was given, and a diagram of dissipative regimes of its motion was also constructed. The question of a particular variant of proportional damping is considered, in which the oscillation 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, through a rational combination of analytical and numerical research methods, all key quantities characterizing the motion of the system on each of the dissipative oscillation modes are identified and determined. In addition, several problems of optimal damping of system oscillations are considered, and the best dissipative parameters are selected based on the criterion of the maximum degree of stability. The obtained results 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.