The Study On Chaotic Behavior Of Nonlinear Columns Under Loading Excitation

The columns which are always basic element in engineering are widely applied in many fields, such as aviation, spacecraft, traffic, structural and machine engineering and so on, its dynamic problem is one of the active subjects in the filed of solid mechanics. Chaos as a new concepts and methods bring a fresh air into the problem and intrigue the researchers. In the study of chaotic behavior, the columns are always severed as first select object and the mechanics model developed from columns can reveal profound physical connotation. In this paper, the chaotic behavior of nonlinear columns under loading excitation is studied. The chaotic critical condition of the system is gived by using Melnikov method. The displacement-time history curves, phase plane and Poincarémap are acquired by Matlab, and used to describe the chaotic behavior. The conclusions are as followings:1. Chaotic behavior of viscoelastic beam with one end fixed and the other end freeed subjected to transversal and longitudinal cycle-loads were studied considering nonlinear material properties. The governing equation of viscoelastic beam was obtained and changed into second-order ordinary differential equation by the Galerkin principle. The influence of nonlinearity coefficients in constitutive relation on the chaotic behavior of beam subjected to cycle-loads was analysized. The critical condition of chaos was obtained by Melnikov method. The displacement-time history curves, phase plane and Poincarémap was gived by using Matlab and used to describe the chaotic behavior. The results show that the quadratic nonlinearity had great effect on the chaotic behavior under longitudinal cycle-loads. The stationary and chaotic motion can occur for the cantilever beam under transversal or longitudinal vibration with external excitation changes.2. The chaotic behavior of elastic beam with one end fixed and the other end sliding was studied considering finite deformation. The critical condition of chaos was gived. The influence of geometrical and physical parameters on the chaotic region was investigated.3. Accountting geometric nonlinearity, the chaotic behavior of viscoelastic beam with one end fixed and the other end sliding was studied. The critical condition of chaos was obtained. The chaotic area map was gived. The influence of geometrical parameters on the chaotic area was analysized. Further researches indicate that the beam before buckling can occur the chaos.