ABSTRACT

In fundamental research on nonlinear dynamics, it is important to understand the essential properties for nonlinear mechanics such as bifurcation or chaotic behavior, as these are physical factors in the loss of structural stability in structures. It is very useful to exactly demonstrate the nonlinear dynamics of structures by numerical analysis because such real behavior after a bifurcation point is complex. In addition, if the structural model had several degrees-of-freedom (D.O.F.) as a number of parameters, its behavior 720would be much more complex. If a real structure had some imperfection as a perturbation of the perfect design, the bifurcation point would be hidden and it might have mode-jumping phenomena to jump to another stable equilibrium path and/or a loss of stiffness, which is inappropriate behavior for the structure. There are also a number of factors that can disrupt the stability of a structure. Problems related to the elastic stability of a structure have been researched by many researchers in this field. Notably, the work of Thompson and Hunt [1] has been identified with the general theory of elastic stability using catastrophe theory in engineering, and Thompson and Stewart [2], who introduced nonlinear dynamics and chaos based upon the instability of a structural model. In particular, it is well known that shallow arches and/or trusses have unstable-symmetric bifurcation with regard to elastically geometrical nonlinearity [3–5]. Von Mises truss, which is called “a two-bar truss,” has snapthrough behavior, which is one of the structural instabilities in static nonlinear mechanics in a shallow truss without dissipation that is considered in some References 6 and 7. In addition, Cook and Simiu [8] experimentally studied the periodic and chaotic snapthrough motions of the Stoker column [9]. Ario [10] has expressed the formula of the single Duffing equation and its applications under geometrical conditions of a D.O.F. Dynamic systems with snapthrough behavior have been extensively studied because of the important phenomena of structural instability such as jumping, buckling, and snapping; see, for example, References 11–15.