Moment stability of strongly nonlinear structural systems under random excitations

Abstract

Many structures in civil engineering are subjected to dynamic loadings. Examples include wind-induced vibrations of cables and tall buildings, wave loading on offshore structures, and seismic excitations acting on civil infrastructure. These dynamic excitations can often be described satisfactorily in probabilistic terms. Current research trends in structural dynamics are evolving from deterministic to stochastic analysis, from linear to nonlinear systems, from elastic to viscoelastic behavior, and from external excitation alone to incorporating parametric excitation. This thesis investigates the stochastic stability of strongly nonlinear structural systems subjected to random parametric excitations, with stochastic stability referring to the long-term probabilistic growth or decay of the system response. The principal stability measure employed in this study is the Moment Lyapunov Exponent (MLE), which characterizes the exponential growth or decay rate of the statistical moments of the system response. Knowledge of the MLE gives the almost-sure asymptotic stability of a stochastic dynamical system through the corresponding Lyapunov exponent. The MLE is the ideal avenue and the ultimate characteristic number for the study of the dynamic stability of stochastic dynamical systems. A unified analytical and numerical methodology is developed for the evaluation of MLEs of nonlinear oscillatory systems. The analysis is primarily based on the stochastic averaging method, which allows the original nonlinear stochastic differential equations to be reduced to averaged equations governing the slow evolution of key response variables. Particular emphasis is placed on transformations based on the system Hamiltonian or total mechanical energy, through which the response of strongly nonlinear oscillators can be represented by an energy envelope. This formulation enables the derivation of analytical expressions for the drift and diffusion coefficients governing the stochastic evolution of the system energy and provides a tractable framework for moment stability analysis. In addition to the analytical formulation, a modified Monte Carlo simulation framework is developed for the numerical estimation of MLEs. In contrast with conventional algorithms that rely on the Euclidean norm of the state vector, the proposed approach employs the square root of the system energy as the generalized amplitude used in periodic normalization procedures. This energy-based formulation offers improved numerical robustness and provides a physically meaningful measure for the response of strongly nonlinear systems, allowing reliable estimation of the MLE through long-term stochastic simulations. Using the developed framework, the stochastic stability characteristics of nonlinear systems are investigated under three representative classes of stochastic excitation: Gaussian white noise, real noise modeled by the Ornstein–Uhlenbeck process, and bounded noise with phase modulation generated by a Wiener process. These excitation models represent different spectral and amplitude characteristics and physical realism commonly encountered in engineering applications. For each case, analytical expressions or approximate eigenvalue formulations for the MLE are derived, and the corresponding stability regions are evaluated. The analytical MLE predictions are compared against independent Monte Carlo estimates obtained directly from simulations of the original stochastic equations, demonstrating strong agreement between theoretical and numerical results. Furthermore, the thesis examines the influence of viscoelastic behavior, represented by a Maxwell-type model, on the stochastic stability of nonlinear systems. Viscoelastic effects introduce additional energy dissipation mechanisms and memory-dependent dynamics that may significantly alter the stability characteristics of the system. The results demonstrate that viscoelastic parameters, including relaxation amplitude and decay rate, can modify the effective damping properties of the system and thereby shift the moment stability boundaries under stochastic excitation. Overall, this thesis provides a comprehensive investigation of stochastic stability of strongly nonlinear structural systems and extends the MLE framework to include realistic stochastic excitation models and viscoelastic material behavior. The analytical formulations and numerical methodologies developed herein contribute to a deeper understanding of stochastic parametric instability in nonlinear structures and provide practical tools for assessing the probabilistic stability and reliability of engineering systems subjected to uncertain dynamic environments.