Nonlinear Dynamic Response and Bifurcation Analysis of the Elastically Constrained Shallow Arch

When the ends are elastically constrained in vertical and rotation directions for the shallow arch, the natural frequencies and modes are quite different from those of the case of ideal hinged or fixed boundary condition, and the different constraint stiffness will change the nonlinear responses and the parameter fields of various bifurcations under external excitation. The dimensionless dynamic equation was established by introducing the fundamental assumptions of shallow arch, and the method that the effects by the boundary constraint stiffness were considered in the natural frequencies and modes solution was employed, then the full-basis Galerkin discretization and the multi-scale perturbation methods were used to obtain the polar- and Cartesian-type averaging equations, of which the coefficients have one-to-one correspondence with the values of constraint stiffness. With the application of numerical calculation, the dynamic behaviors of the vertical elastically constrained system in the case of one-to-two internal resonance between the lowest two modes under periodic excitation were studied. Both the comparison of calculated results with finite element results and the convergence of the coefficients in averaging equations proved the feasibility of the present method. Also, the numerical results show that there exist several bifurcation points with the variation of the amplitude and frequency of excitation, and the parameter distributions for the occurrence of bifurcations are associated with the values of constraint stiffness. Moreover, there are a series of steady-state solution, periodic solution, quasi-periodic solution and chaotic solution windows in the vicinity of the unstable areas or resonance regions which are connected by the bifurcation points, and the period-doubling bifurcation can be observed with the variation of parameters.