Ph.D Thesis

Ph.D StudentHarel Plat
SubjectMethods for Sensing and Creating Propagating Waves and
Vibrations Using Parametric Excitation
DepartmentDepartment of Mechanical Engineering
Supervisor Full Professors Bucher Izhak
Full Thesis text - in Hebrew Full thesis text - Hebrew Version


  The parametric resonance phenomenon takes place when one of the governing parameters of the physical problem is varied periodically. Such a periodic change, small as it may be, could bring about very large response amplitudes. When parametric resonance takes place, the response amplitude increases very rapidly, the growth of the response occurs at an exponential rate and the response is theoretically unbounded. In practice the resonating amplitudes are limited by nonlinear effects.

  The current research paves the road to harnessing the parametric resonance phenomenon to create and control traveling waves and oscillatory motions.

  The first part of the thesis deals with a method to identify propagating phenomena and to separate standing from traveling waves. a non-parametric methods based on the flux of power along the structure is developed. A scalar measure, the power factor, is calculated from the identified flux of power which has a direct physical relationship with the proportion of propagating vs. standing vibrations.

  The second part of the work demonstrates the ability to harness the parametric resonance phenomenon in order to create propagating waves in a circular string-like elastic structure. When the longitudinal tension of the string has spatial and temporal variations a propagating wave can occur. In this research the necessary conditions for creating propagating waves through parametric resonance are developed. The response to the parametric excitation comprises of multiple wavelengths whose some is a traveling square-wave-like deformation. The nonlinear effect that limits the response is mostly due to the increased tension in the string that accompanies lateral vibrations. These nonlinear effects are fully accounted for in this work. The nonlinearity limits the response amplitudes thus bringing about multiple solutions. The multiple solutions are found by solving a generalized eignevalue problem where the eigenvalues represent the amplitude of the propagating wave and the eigenvectors the spatial composition.

   The question:  whether higher modes, having shorter wavelengths, can be effectively excited parametrically, is addressed in the third and final part of this thesis. A comprehensive analysis of a string to which a dynamical system is connected is carried out. The analysis shows that a dynamical system, once tuned properly, can amplify the obtained response levels for a wider range of wavelengths. Experimental results illustrate that higher modes of vibration can be effectively excited with parametric resonance in the presence of the added dynamical system.