Dynamics of a Spinning Shaft With Non-Constant Rotating Speed, Leading to Theorems in Mechanics

Fotios Georgiades

Recent developments in nonlinear dynamic analysis of mechanical systems are discussed. The nonlinear dynamic analysis of a spinning shaft with non-constant rotating speed, as a specific type of hybrid system, in various ways is done. Due to rigid body angular rotation, this type of hybrid system admits rigid body modes associated with zero eigenvalues. Therefore the Lyapunov approximation of the nonlinear dynamics behaviour with the underlying linear system modes for low energies is not necessarily valid, and the presented two analyses are becoming more valuable. The first analysis is the well-established multiple scales nonlinear dynamic analysis. In the 2nd analysis, rigid body motion’s backbone curves have been determined and lead to additional information. The nonlinear dynamic analysis of the spinning shaft expanded further, including the new concept of perpetual points, leading to the preliminary conclusion that mechanical system’s perpetual points are associated with rigid body motions. Although the nonlinear dynamics analysis of the spinning shaft is extensive in mathematical formulation, a concrete outcome for critical situations is not established yet, and more work is needed. Moreover, based on the observation for the perpetual points, two theorems proved that the perpetual points are associated with the rigid body motions in linear natural, unforced systems, and they are forming the perpetual manifolds. With some new definitions in mechanics, a third theorem and one corollary proved with the significant outcome the conditions of wave-particle motion of flexible mechanical systems. The presented work is significant in two directions; the first is about examining the dynamics of nonlinear systems with the underlying linear system with zero eigenvalues, associated with mechanical systems with rigid body angular rotations with non-constant rotating speed. The 2nd direction is developing the perpetual mechanic’s theory, with the significant 3rd theorem in mathematics, physics/mechanics, and mechanical engineering. In mathematics, the theorem provides solutions in non-autonomous N-degrees of freedom systems. In physics/ mechanics the particle-wave motion is of high significance. Finally in mechanical engineering the rigid body motion without any oscillation is the ultimate possible type of motion, e.g., trains, cars, etc.