A conditional stochastic projection method applied to a parametric vibrations problem

Abstract

Parametric vibrations can be observed in cable-stayed bridges due to periodic excitations caused by a deck or a pylon. The vibrations are described by an ordinary differential equation with periodic coefficients. The paper focuses on random excitations, i.e. on the excitation amplitude and the excitation frequency which are two random variables. The excitation frequency ω_L is discretized to a finite sequence of representative points, ω_L,i Therefore, the problem is (conditionally) formulated and solved as a one-dimensional polynomial chaos expansion generated by the random excitation amplitude. The presented numerical analysis is focused on a real situation for which the problem of parametric resonance was observed (a cable of the Ben-Ahin bridge). The results obtained by the use of the conditional polynomial chaos approximations are compared with the ones based on the Monte Carlo simulation (truly two-dimensional, not conditional one). The convergence of both methods is discussed. It is found that the conditional polynomial chaos can yield a better convergence then the Monte Carlo simulation, especially if resonant vibrations are probable.

Keywords:

• stochastic dynamics
• parametric vibrations
• Hermite’s polynomial chaos
• Monte Carlo simulation

Additional information

Notes on contributors

Wlodzimierz Brzakala

Wlodzimierz BRZAKALA. DSc, PhD, Eng, Associate Professor at the Wroclaw University of Technology, Faculty of Civil Engineering. His research interests focus on structural safety and uncertainty modeling, geoengineering, numerical methods.

Aneta Herbut

Aneta HERBUT. PhD, Eng, Assistant Professor at the Wroclaw University of Technology, Faculty of Civil Engineering. Her research interests include structure dynamics; active, passive and semi-active vibration mitigation methods; cable dynamics for deterministic and stochastic cases; stochastic differential equations.