Existence and stability of large scale nonlinear oscillations in suspension bridges

Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities

“…The results of [5] show that if A> 0, p> 0, y is arbitrary, 2n/\fb < 2n/p < n/y/a + n/\fb and the ratio A/s and k > 0 are sufficiently small, then u" + ku + bu -au~ = s + A sin(pt + y)…”

“…The solution U(x,t) = (sinnx/L)u0(t) of (1.1), (1.2) represents small oscillations about the equilibrium solution [c(n/L) + d]sinnx/L of (1.1), (1.2) when the small oscillatory term f(t) = 0. Using extensions of results of Loud in [13] (Loud considered restoring forces of class C1 ) it was shown in [5] that if the form and period of / were suitably restricted, if k > 0 was sufficiently small, and if the amplitude of f(t) was sufficiently small, then there was also a large-amplitude, asymptotically stable, T-periodic solution of ( 1.3) near a translate of a nonconstant T-periodic solution of the autonomous O.D.E. u +ku +c(n/L) u + du =s.…”

“…The main motivation for this paper comes from a numerical study carried out in [5]. Consider the partial differential equation (1.1) d2U/dt2 + kdlf/dt + cd4U/dx* + dU+ = h(t,x), where U = U(x,t), 0 < x < L, r > 0 satisfies the boundary conditions (1.2) (7(0,i) = U(L,t) = (d2U/dx2)(0,t) = (d2U/dx2)(L,t) = 0, c>0,rc>0,i/>0are constants, and U+ denotes the positive part of U.…”

“…In [5], the boundary value problem (1.1), (1.2) was suggested as a model for an idealized suspension bridge. The term dU+ takes into account the fact that when the cables suspending the bridge are stretched, there is a restoring force which is assumed to be proportional to the amount of stretching (Hooke's law).…”

“…The term kdU/dt represents viscuous damping, assumed to be small. The same model with k = 0 had been considered previously in The purpose of [5] was to show how such a model could predict the existence of large amplitude, stable oscillations in suspension bridges. Assume that the external force is of the form h(x,t) = (sin nx/L)(s + f(t)), where s > 0 is a constant and f(t) is T-periodic in t, T > 0, with meanvalue zero.…”