Nonlinear Dynamics of Thermoelastic Plates

Darshan Soni, Purdue University

Abstract

Nonlinear flexural vibrations of simply supported rectangular plates with thermal coupling are studied for the case when the plate is harmonically excited by the force acting normal to the midplane of the plate. The coupled thermo-mechanical equations are derived by applying the Galerkin procedure on the von-Karman equation and the energy equation for an element of the plate. The thermo-mechanical equations are second order in transverse displacement and first order in thermal dynamics. In our first study, we represent the transverse displacement, bending moment and membrane force due to temperature by one mode approximation, and study the response of thermoelastic plate in time and frequency domain. The analysis of forced vibration to a transverse harmonic excitation is carried out using harmonic balance as well as direct time integration coupled to a Fourier analysis for a range of excitation frequencies. The effects of thermal coupling, material nonlinearity and different amplitudes of excitation on the thermoelastic plate’s transverse displacement and thermoelastic variables are investigated. The method of averaging is applied to the one mode case to transform the nonlinear modal equations into sets of two-dimensional dynamical systems which govern the amplitudes and phases of the two modes. The averaged system is studied in detail by using pseudo arc-length continuation schemes implemented in MATCONT. The physical phenomena of interest in this study arise when a plate exhibits two distinct linear modes of vibration with nearly the same natural frequency. To analyze the dynamics of the thermoelastic plate in this scenario, we utilize a two-mode approximation. The response of the plate, as a function of excitation frequency, is determined for the two-mode model using MATCONT, and several bifurcation points are identified. Our analysis reveals two types of solutions: single-mode and coupled-mode solutions. We find that stable single-mode and coupledmode solutions can coexist over a wide range of amplitudes and excitation frequencies. Under the influence of thermal coupling, our analysis using MATCONT reveals the identification of Neimark-Sacker bifurcation points. After a detailed study of the Neimark-Sacker region using Fourier spectrum and Poincare section, we conclude that a pitchfork bifurcation occurs, resulting in stable asymmetric solutions. We further investigate the effect of in-plane forces or mechanical precompression on the thermoelastic plate, using MATCONT for a fixed value of force, damping, and excitation frequency. We find that the in-plane forces lead to buckling, which is identified as a branch point cycle (pitchfork bifurcation) in MATCONT. Consequently, the bifurcation diagram of transverse displacement as a function of in-plane forces can be divided into prebuckling and post buckling regions, with multistable solutions in each region. To validate our one mode model, we use ANSYS software to verify the transverse displacement and temperature results. We validate the frequency and time domain results for both the linear and nonlinear cases, and plot contours using ANSYS to observe the variation of displacement and temperature over the surface of the plate. Our one mode model results closely match with the ANSYS results, leading us to conclude that our one mode approximation is accurate and that the coupled thermo-mechanical equations we derived are correct.

Degree

M.Sc.

Advisors

Bajaj, Purdue University.

Subject Area

Mathematics|Thermodynamics

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