Structural instability of nonlinear plates modelling suspension bridges: Mathematical answers to some long-standing questions

Abstract

We model the roadway of a suspension bridge as a thin rectangular plate and we study in detail its oscillating modes. The plate is assumed to be hinged on its short edges and free on its long edges. Two different kinds of oscillating modes are found: longitudinal modes and torsional modes. Then we analyze a fourth order hyperbolic-like equation describing the dynamics of the bridge. In order to emphasize the structural behavior we consider an isolated equation with no forcing and damping. Due to the nonlinear behavior of the cables and hangers, a structural instability appears. With a finite dimensional approximation we prove that the system remains stable at low energies while numerical results show that for larger energies the system becomes unstable. We analyze the energy thresholds of instability and we show that the model allows to give answers to several questions left open by the Tacoma collapse in 1940

Torsional instability in suspension bridges: The Tacoma Narrows Bridge case

“…The main tool used there are transfer maps (Poincaré maps), which highlight an instability when the characteristic multipliers exit the complex unit circle. The same phenomenon was later emphasized also for different models using results on the instability for the Hill equation and Floquet theory, see [8][9][10] . All these results were obtained by considering isolated systems, that is, by neglecting both the aerodynamic forces and the dissipation.…”

“…) is a solution of (6) -(7) - (8) . A careful look at (9) shows that its structure, its nonlinearities, and its nonlocal terms are of the same kind as those appearing in (10), (11) , and (12) .…”

Numerical Minimization of Dirichlet Laplacian Eigenvalues of Four-Dimensional Geometries

“…[11]). In Figure 3 we plot σ(λ) for λ ∈ [5,10] obtained for the 4D ball with unit volume and we can locate three eigenvalues in this interval. The method is not very sensitive to the choice of interior points M I and in all the numerical simulations we fixed M I = 50.…”

“…A naive choice could be to consider the points generated with equally spaced angles θ and φ but it is well known that this procedure does not produce a uniform distribution of points. Instead, we define an integer number M C corresponding to the number of collocation points that we would like to place and try to have the same variation locally of the angles θ and φ and that the product of these two quantities is approximately equal to the average area of each point, i.e., (10) ∆θ…”

“…A better understanding of this questions allows, for example, us to tune and improve the acoustic properties of musical instruments [8,15,25,34,36] or to find optimal designs of composite materials (e.g., [2]). A typical situation in engineering is the identification of shapes or designs of structures to prescribe their mechanical properties: being either as rigid or as soft as possible (e.g., [24]), avoiding torsional oscillations (e.g., [10]), etc. Such problems appear also in the context of electromagnetism when considering the design of optimal accelerator cavities (e.g., [1]).…”