Motion of double pendulum colliding with an obstacle of rough surface

• Published in: Archive of applied mechanics (1991) Vol. 87; no. 5; pp. 841 - 852
• Main Authors: Sypniewska-Kamińska, Grażyna, Starosta, Roman, Awrejcewicz, Jan
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2017; Springer Nature B.V
• Subjects: Classical Mechanics; Differential equations; Engineering; Special; Surface roughness; Theoretical and Applied Mechanics; The Routh method; Collision; Double pendulum
• ISSN: 0939-1533; 1432-0681
• DOI: 10.1007/s00419-017-1230-4

The externally excited and damped vibration of the double pendulum in the vertical plane are considered. The pendulum can collide many times during the motion with a motionless obstacle having the rough surface. The double pendulum colliding with this object has been modeled as a piecewise smooth system constrained by the unilateral constraint. In the relatively long time between the collisions, the differential equations govern the motion of the pendulum. When the contact with the obstacle appears, the pendulum exhibits a discontinuous behavior. An important element of the solving algorithm is aimed on the continuous tracking of the position of the pendulum in order to detect the collision with the unilateral constraints and to determine the state vector of the pendulum at the impact time instant. A single collision is described by the Euler’s laws of motion in the integral form. The equations are supplemented by the Poisson’s hypothesis and Coulomb’s law of friction. The friction law is formulated for the instantaneous values of the reaction forces. The values of their impulses depend on the existence of a slip between the contacting bodies. Therefore, the Coulomb law cannot be generalized for the linear impulses of the forces in a simple way. We have applied the Routh method in order to solve the problem. The method has a simple geometrical interpretation in the impulse space. The angular velocities of both pendulum parts as well as the reaction forces at the joints of the system, which change in stepwise manner, have been presented in the paper.