Forced precession of a gyroscope and its application to Laithwaite's engine

The forced precession of a symmetrical gyroscope is studied for the particular case in which the axle of a flywheel is pivoted by a hinge joint and follows a horizontal circular path of a given radius. The aforementioned setup appears in the so-called Laithwaite engine, the detailed mechanics of which are still an enigma. Instead of applying Lagrangian equations, Newton's second law is applied to the rotating gyroscope with respect to its center of mass. Three novel Euler equations are developed that are much longer than those found in textbooks. In this mechanical system, which is characterized by one degree of freedom, the main nonlinear governing equation is identified and then MATLAB code is developed to obtain and visualize the numerical solution. Under particular conditions that ensure small oscillations of the gyroscope's axle (a maximum oscillation of eight degrees in the lean angle) near the horizontal plane through the pivot, a linearization is performed and is successfully compared with the aforementioned nonlinear numerical solution. The computer program facilitates the understanding and calculation of physical quantities such as the internal forces and moments, support forces and power transmission from the drive motor. In particular, it is shown that, for a hinge joint, the period of oscillation differs from that of a rotating pivot, which is crucial to the debate about whether such an engine may produce a net thrust, or not. A relevant paradox is resolved. (paper)