Numerical investigation of periodic solutions for a suspension bridge model

Numerical methods are discussed for the computation of periodic solution paths for a suspension bridge model(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\cases{&{\it u}\sb{\it tt} + \delta{\it u}\sb{\it t} + EIu\sb{xxxx} + Ku\sp+ = W + \lambda f(x,t,\mu)\cr\cr& {\it u}(0, {\it t}) = {\it u}(1, {\it t}) = {\it u}\sb{\it xx}(0, {\it t}) = {\it u}\sb{\it xx}(1,{\it t}) = 0,\cr}$$(TABLE/EQUATION ENDS)where $\delta, E, I, K,$ and $W$ are constants, $\lambda$ and $\mu$ are parameters, and $f$ is a periodic function in $t$.

The shooting method with Newton's and quasi-Newton's solver, and pseudoarclength continuation method around simple turning points are treated. The initial value problem solver that we adopted in the shooting method is implicit in the linear part, and explicit in the nonlinear part and forcing terms. We call it two-step Crank-Nicolson scheme.

Multiple periodic solutions for the suspension bridge equations are found, with their stability investigated. The unconditional stabilities of the Crank-Nicolson implicit scheme for solving suspension bridge equations in both infinite domain and finite domain are proved.