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.
