Stochastic stability of a viscoelastic column axially loaded by a white noise force

Abstract

This paper deals with the analysis of stability of a hinged-hinged viscoelastic column subjected to a non-zero mean stochastic axial force. The randomly variable part of this is described by a stationary Gaussian white noise process. The viscosity affects the curvature of the column, for which the classic Euler–Bernoulli's model is adopted. The viscosity is described by the linear Kelvin–Voigt's model. A dynamic stability analysis is performed. Normal modes are introduced in the integro-differential equation of motion so that uncoupled modal equations are retrieved. With reference to the first mode, by using an additional state variable, three Itô’s ODE are obtained, from which the differential equations ruling the response statistical moment evolution are written by means of Itô’s differential rule. The zero solution, that is undeformed straight column, corresponds to zero moments. If the column is perturbed, it is stable when the response moments tend to zero. A necessary and sufficient condition of stability in the moments of order r is that the matrix A r of the coefficients of the ODE system ruling them has negative real eigenvalues and complex eigenvalues with negative real parts. Because of the linearity of the system the stability of the first two moments is the strongest condition of stability. If the mean axial force μ P or the white noise intensity w P are increased, there exist critical values μ Pcr , w P c r for which almost an eigenvalue is positive. The critical mean axial force is found to be inversely proportional to the parameter φ ∞ , which measures the amount of viscous deformation. The search for the critical values of w P is made numerically, and several graphs are presented for a representative column.