M.Sc Thesis

M.Sc StudentYuri Trakht
SubjectCreating Traveling Waves in Circular Structures via
Parametric Excitation
DepartmentDepartment of Mechanical Engineering
Supervisor Full Professors Bucher Izhak
Full Thesis textFull thesis text - English Version


The present research uses analytical and numerical tools to study the vibrations of an elastic ring under parametric excitation. Parametric excitation can create large amplitude vibration and traveling waves. It is shown that by enforcing periodic and uniform elongation of the neutral axis, tuned to a frequency which is the sum of bending and torsion natural frequencies, gives rise to parametric resonance and to large amplitude traveling transverse vibration waves. 

The amplitude of out of plane vibration is limited by nonlinear effects; here it is limited by the nonlinearly elevated stiffness caused by the elongation of the neutral axis.

Firstly, a formulation describing the dynamics of an elastic ring undergoing combined bending, torsion and axial deformation is developed by employing Hamilton’s principle. Applying order analysis, the small terms were neglected and the PDE’s were simplified. The parametric excitation was modeled as circumferentially uniform stretching the neutral axis.

Later, the model was verified by comparing a degenerate version without the nonlinear effects that can be neglected as small amplitudes with known results and by means of numerical finite element computations.

The PDE’s which includes the nonlinear effects and the parametric excitation were discretized using Galerkin’s method, and subsequently a modal transformation and non-dimensional coordinates were applied and state space representation of the model was created. The level of parametric excitation causing instability or parametric resonance conditions was chosen with the aid of a Floquet analysis (Ince-Strutt diagram) and the system was simulated with and without nonlinear terms. After tuning the right initial conditions to the nonlinear system it is demonstrated that indeed finite, large amplitude traveling bending waves can be formed by periodically causing uniform circumferential stretching of the neutral axis at a frequency which is the linear combination of some bending and torsion modal frequencies having the same number of nodal diameters.

 The parametric excitation of traveling waves at resonance has a potential for motoring and transporting power and displacement along their path.