Correction of the Koenig Formula for the Kinetic Energy of a Rotating Solid

Abstract

An exact solution is obtained for the kinetic energy in the general case of the spatial motion of solids with arbitrary rotation, which differs from the Koenig formula by three additional terms that take into account the change in the centrifugal moments of inertia when the body rotates. The description of motion in the Lagrange form and the superposition principle are used, which provides a geometric summation of the velocities and accelerations of the joint motions in the Lagrange form for any particle at any time. The integrand function in the equation for kinetic energy is represented as the sum of the identical velocity components of the joint plane-parallel motions. In the general case of motion with 6 degrees of freedom, the energy of rotational motion is determined by three axial moments of inertia, as in the Koenig formula, and three additional centrifugal moments, which take into account the rotation of the body. They can be calculated through 6 integral characteristics of the density distribution, determined for the initial position of the body.