Friction Forces in the Dynamics of a Singular Pendulum

The behavior of reaction forces and Lagrange multipliers for a singular pendulum has been studied. This mechanism is a flat double mathematical pendulum whose free vertex moves along a given curve. For critical parameters of this mechanical system, the configuration space of a singular pendulum is a manifold with singularities. Near a singular point, the configuration space could be represented as two intersecting or tangent curves. For the first time, the properties of singular pendulum dynamics are considered for system parameters close to critical values. In addition, for the first time, the influence of the friction force using the Amonton–Coulomb model on the system motion near singularities has been studied. For the considered type of perturbation of the constraint equation, the configuration space with singularities splits into several smooth one-dimensional manifolds. Dynamics on smooth manifolds are described by the classical Lagrange equations. General theoretical constructions for two-dimensional systems with one holonomic constraint were considered in the author’s previous article. To apply these constructions, the configuration space must be brought to a normal form. In this article, the corresponding coordinate changes are obtained. The conditions for the properties of the motion equations were checked. It is found that for a singularity of the intersection type, the Lagrange multipliers increase without limit near the singular point when the coupling perturbation parameter tends to zero. For a tangency singularity type, several variants are possible. For first perturbation type, the Lagrange multipliers are limited near the singular point taking into account the friction force. For the second perturbation type, the Lagrange multipliers are unlimited near the singular point. The system of equations for determining reaction forces near a singularity has two solutions when the singular pendulum moves in one direction and has no solution when the pendulum moves in the opposite direction.