Dynamics of a Double Pendulum with Viscous Friction at the Hinges. I. Mathematical Model of Motion and Construction of the Regime Diagram

The paper discusses the dynamic behavior of a double mathematical pendulum with identical parameters of links and end loads, which is under the influence of viscous friction at both of its hinges with generally different dissipative coefficients. A linear mathematical model of system motion for small deviations is given, and a characteristic equation containing two dimensionless dissipative parameters is derived. For the case of low damping, approximate analytical expressions are derived that make it possible to evaluate and compare with each other the damping factors during motion of the system in each of the vibration modes. A diagram of dissipative motion regimes is constructed, which arises when the plane of dimensionless parameters is divided by discriminant curves into regions with a qualitatively different character of system motion. It is noted that a dissipative internal resonance can occur in the system under consideration; the conditions for its existence are established in an analytical form, and a graphic illustration of these conditions are also displayed. This publication is the first part of the study of the dynamics of a dissipative double pendulum, the continuation of which will be presented as a separate publication “Dynamics of a Double Pendulum with Viscous Friction at the Hinges. II. Dissipative Vibration Modes and Optimization of the Damping Parameters.”