Invitation: PhD Dissertation Defense of Bingxu Zhang

Hello ME Community,
You are invited to join the PhD Dissertation Defense of Bingxu Zhang, on Tuesday, October 20, beginning at 10:15 am.  The defense will be presented in person in the Mechanical Engineering Conference Room.  

Advisor: Dr. Weidong Zhu

Title: NONLINEAR WAVE PROPAGATION IN DISCRETE LATTICES

Abstract:

This research focuses on theoretical analysis of wave propagation in discrete nonlinear lattices. Discrete periodic lattices as a kind of metamaterials has attracted a lot of attention due to their unique properties such as propagation and attenuation zones, tunable band gaps, nonreciprocal wave propagation and so on. When the nonlinearity of the lattices is weak, perturbations can capture some basic properties of wave propagation. However, the perturbation method will be invalid, and the incremental harmonic balance(IHB) method can be used to analyze wave propagation, when increase of the amplitude or system properties lead to nonlinear enhancement. However, the IHB method may sometimes be invalid due to convergence issues. Moreover, stability and bifurcation analyses are key issues, but a systematic theoretical framework is still lacking.

A series of analytical methods are developed in this dissertation to analyze wave propagation behaviors in discrete nonlinear lattices, and the research objectives are twofold: system response and stability analyses. Due to the absence of damping and external excitation in general, the solution for wave propagation is in a hyperplane with two dimensions. Hence, the IHB method combined with the least squares method is used to address the associated rank deficiency problem. A method based on Hill’s method is used to identify the stability of solutions and determine bifurcation point positions and corresponding bifurcated directions for wave propagation in discrete strongly nonlinear structures. After determining the bifurcation points and the bifurcated directions, the modified IHB method combining the continuation method is utilized to achieve tracking of main and bifurcated branches of solutions for wave propagation in discrete strongly nonlinear structures.

The proposed methods are applied to three representative systems: a strongly nonlinear monatomic chain, a mass-in-mass chain with softening springs, and a mass-in-mass chain with hardening springs. For the monatomic chain, results demonstrate that the periodic solution is in a hyperplane with two dimensions, and the wave amplitude and the nonlinear stiffness jointly determine the effective strength of the nonlinearity. For the softening mass-in-mass chain, the nonlinear evolution of the acoustic and optical branches is characterized, and superharmonic resonances and fusion between the two branches are revealed. For the hardening mass-in-mass chain, multiple forms of periodic solutions are identified. In all the three systems, attenuation zones, stability changes, and bifurcations are determined.

       The developed methods provide an integrated framework for calculating, continuing, and evaluating periodic waves in discrete strongly nonlinear lattices, and provide predictive tools for the analysis and design of nonlinear periodic structures and adaptive mechanical metamaterials.