Calculation of the Kinetic Energy of a Rigid Body in the General Case of Three-Dimensional Motion with Arbitrary Rotation

Abstract

An exact solution is obtained for the kinetic energy in the general case of spatial movement of rigid bodies with arbitrary rotation including a change in the products of inertia. The Lagrangian description of the movement and the principle of superposition, which provides the geometric summation of velocities and accelerations of cooperative Lagrangian movements for any particle at any instant of time, are used. The integrand in the equation for the kinetic energy is represented as the sum of the same velocity components of cooperative plane-parallel movements. The polar moments of inertia are not changed during movement, and they can be calculated by the current or initial state of the body. The products of inertia are changed and become zero during rotation with respect to the principal central axes only for bodies with the same principal moments of inertia, for example, for a sphere. In other cases, the difference in the principal moments of inertia leads to cyclic changes in the kinetic energy with possible manifestation of precession and nutation, the amplitudes of which are dependent on the angular rotational velocities of the body. An example using equations for a robot with one screw and two rotational kinematic pairs is presented.