Principle of generalized velocities in dynamics of planar separation of a rigid body

Abstract

In this paper, the dynamics of planar separation of a rigid body into two rigid bodies is investigated. Motion of the initial body is planar and its properties (position, velocity, angular velocity, and angular position) are exactly known before separation. The process of separation lasts for a short time during which it is assumed that the position and angle position of the bodies do not vary. After separation, the separated and the remainder body remain to move planar. Applying the principle of momentum and angular momentum, velocity and angular velocity of the separated and remainder body are calculated. In the paper, the analytical procedure for determining these values is developed. Introducing the virtual velocity and angular velocity, the modified D’Alembert–Lagrange equation for planar separation of a body is expressed. Virtual works of impulses caused by impact force and torque are defined and discussed. Using the generalized velocity and angular velocity and introducing the generalized impulse, modified Lagrange’s equations for planar separations are obtained. In this paper, an example of separation of a planar beam on which an impulse of impact force acts is analyzed. Applying the suggested equations, velocity and angular velocity of the remainder beam are determined and discussed. Conditions for transformation of the planar motion into pure translatory motion of the remainder beam are determined. The obtained values represent the initial conditions for long term motion of the remainder part of the beam. The analytical methods suggested in the paper have the potential to improve our knowledge in dynamics of the discontinual separation of a rigid body. The obtained results have practical significance, too.