Transversal vibration of a parametrically excited hereditary beam

Katica Hedrih

University of Niš

Yu-18000-Niš, st. Vojvode Tankosića 3/22, Serbia

In the present paper, the stability of a hereditary viscoelastic beam subjected to parametric random bounded excitations described by stochastic processes of small intensity is investigated. The motivation for the study of these problems is the necessity to explain the influence of rotatory inertia of beam cross sections and transverse shear of beam cross section under the transverse forces on the stability of the transversal time vibrations process of the beam, and also on the stability of the deformable beam's forms.

The partial differential equation of transversal stochastic vibrations of a parametrically excited hereditary beam was derived. The beam is built by a hereditary material with known relaxation kernel, and it is subject to axial stochastic external excitation. The influence of rotatory inertia of beam cross section and transverse shear of beam cross section under the transverse force, and the corresponding members in the partial differential equation are taken into account. Bernoulli particular integral method and Lagrange method of variation of constants are used for the transformation problem. The asymptotic averaged method is used for obtaining the first approximation of Itô stochastic differential equations. The sets of Lyapunov exponents are obtained.