The Stability Analysis Of Elastic Beam In Large Deflection Subjected To Thermomechanical Load

Behaviors of bifurcation and chaos in nonlinear dynamic systems are key problems in the study of nonlinear dynamics. Nonlinear dynamic systems with parametrical excitations are the common but typical nonlinear dynamic systems which including very complex and abundant dynamic behaviors, such as bifurcation, fractal and chaos. In this thesis, theoretical investigation on the behaviors of dynamic stability, such as bifurcation, fractal and chaos of beams resting on nonlinear elastic foundation and subjected to both thermal shock and periodical mechanical excitation, has been presented. The main content and results are as following:1. Based on Euler-Bernoulli beam theory , by considering the effects of geometric nonlinearity and linear external damp, geometrical nonlinear dynamic governing equation for simply supported beam resting on nonlinear elastic foundation subjected to both transverse harmonic mechanical load and thermal shock were derived according to Hamilton's theory. Galerkin Discretization method is applied to truncate the nonlinear partial differential governing equation into a second-order nonlinear ordinary differential equation: Mathieu-Duffing equation. In order to decrease the primary parametric resonance of the beam with large amplitude, a nonlinear velocity feedback control is adopted.2. For the case that the uniform temperature rise less than the critical buckling load, the method of Multiple Scales is used to derive a first-order ordinary-differential equations that govern the time slowly variation of the amplitude and phase of the response of the primary resonance and primary parametric resonance of the beam to obtain the first approximate solution of the system, the steady state responses ,the stability area and the critical value of instability are analyzed. By using numerically computing , the effects of the parameters, such as the viscous damp, the elastic foundation stiffness, the amplitude and frequency of thermal shock as well as the external excitation on the response of the primary resonance and principal parametric resonance of the system. The validity of Multiple Scales and the control law based on cubic nonlinear velocity feedback on the response of the principal parametric resonance is verified by numerical simulations. Furthermore, the possibility and conditions of which the sub-harmonic response of 1/3 order and super-harmonic of 3 order exist are examined.3. At the conditions that the uniform temperature rise is great than the critical buckling load of the beam, the homoclinic and heteroclinic orbits is derived. The Melnikov technique is used to obtain the critical value of the amplitude of thermal and mechanical load at or over which the Smale horseshoes chaos will take place. Numerical simulation is used to obtain the evolution of the bifurcation process of the response. By numerical simulation and incorporating with many analytical techniques for analysis of chaotic response, such as Poincare Map, power spectrum, Lyapunov exponents, fractal dimension, time history and phase trajectory, the paths of bifurcation of the beam is plotted and the chaotic movement is studied. The conditions and regularities of the evolution of bifurcation and chaotic movement in the system are indicated.