Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed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.v3 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.v3

## 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 with centrifugal moments of inertia. 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 by the sum of the identical velocity components of the joint plane-parallel motions. The moments of inertia in the Koenig formula do not change during movement and can be calculated from the current or initial state of the body. The centrifugal moments change and turn to 0 when rotating relative to the main central axes only for bodies with equal main moments of inertia, for example, for a ball. In other cases, the difference in the main moments of inertia leads to cyclic changes in the kinetic energy with the possible manifestation of precession and nutation, the amplitude of which depends on the angular velocities of rotation of the body. An example of using equations for a robot with one helical and two rotational kinematic pairs is given.

## Keywords

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

## Subject

Engineering, Automotive Engineering