A general mathematical framework for stochastic analysis of suspension bridges (Q5933580)

This paper deals with the stochastic model of suspension bridges which permits to study suspension bridges subject to random wind or seismic forces in addition to deterministic loads. Firstly the author considers two different deterministic models. The model A describes only the vertical motion of the coupled system of suspension cable and the bridge deck. The model B describes both the vertical and torsional motion of the deck including the positions of the suspension cables. Both these models are given by boundary value problems for the corresponding systems of partial differential equations. For example, the model A is described by the system NEWLINE\[NEWLINEm_{b}z_{tt}+\alpha D^4z-F(y-z)=f_1,\quad m_{c}y_{tt}-\beta D^2 y+F(y-z)=f_2,NEWLINE\]NEWLINE where the first equation describes the vibration of the road bed in the vertical plane and the second equation describes that of the main cable from which the road bed is suspended by tie cables. Here \(m_{b}, m_{c}\) are the mass per unit length; \(\alpha,\beta\) are the flexural rigidity of the road bed and the coefficient of elasticity of the cable, respectively; \(D^{k}\) denotes the spatial derivative of order \(k\); \(F\) represents the restraining force; \(f_{i}, i=1,2\), represent external forces. A more general abstract model covering these both cases is proposed. This model is given by the abstract first-order differential equation on the suitable Hilbert space reflective of the physical energy. Finally the stochastic model of suspension bridges is considered in the form NEWLINE\[NEWLINEd\varphi(t)=A\varphi(t)+F(t,\varphi(t)) dt+ \sigma(t)dW(t), \quad \varphi(0)=\varphi_0,NEWLINE\]NEWLINE where \(A,F,\sigma\) are some operators; \(W(t)\) is the \(H\)-valued Wiener process. The author proves the existence of a unique mild solution of last the abstract stochastic differential equation and studies the properties of this solution. The stability questions covering additive and multiplicative perturbations are also studied.