Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

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

Version 1 : Received: 29 January 2021 / Approved: 2 February 2021 / Online: 2 February 2021 (13:05:52 CET)
Version 2 : Received: 29 March 2021 / Approved: 1 April 2021 / Online: 1 April 2021 (13:28:20 CEST)
Version 3 : Received: 10 May 2021 / Approved: 12 May 2021 / Online: 12 May 2021 (11:19:10 CEST)

How to cite: Alyushin, Y. Correction of the Koenig Formula for the Kinetic Energy of a Rotating Solid. Preprints 2021, 2021020086. https://doi.org/10.20944/preprints202102.0086.v2 Alyushin, Y. Correction of the Koenig Formula for the Kinetic Energy of a Rotating Solid. Preprints 2021, 2021020086. https://doi.org/10.20944/preprints202102.0086.v2

## 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.

## Keywords

kinetic energy; Lagrange variables; the principle of superposition of motions; polar; axial and centrifugal moments of inertia

## Subject

Engineering, Automotive Engineering