Abstract

Free Damped Vibrations of a Nonlinear Rectangular Thin Plate under the Conditions of Internal Combined Resonance

Y.A.Rossikhin (Department of Theoretical Mechanics, Voronezh State University of Architecture and Civil Engineering, Voronezh, Russia); M.V.Shitikova, E.I.Ovsjannikova (Department of Structural Mechanics, Voronezh State University of Architecture and Civil Engineering, Voronezh, Russia)

e-mail: shitikova@vmail.ru

Free damped vibrations of a rectangular thin plate described by a set of three nonlinear differential equations of the hyperbolic type are considered when the plate is being under the conditions of the internal combined resonance, i.e. when the doubled magnitude of the natural frequency of vertical vibrations is approximately equal to the sum of two natural frequencies of flexural in-plane vibrations. Viscous properties of the system are described by the Riemann-Liouville fractional derivative of the order less than unit. The functions of the in-plane and out-of-plane displacements are determined in terms of eigenfunctions of linear vibrations with the further utilization of the method of multiple scales. Using the constructed solutions, the influence of initial conditions on the energy exchange mechanism taking place between the vertical and flexural modes is analyzed, which is intrinsic to free vibrations of nonlinear structures being under the conditions of different internal resonances. The hydrodynamic analogy is suggested, according to which the resonance vibratory regime is interpreted by the motion of viscous fluid in the phase space. On the stream-lines playing the role of a separatrix, irreversible damping transition of energy between subsystems is observed.

Section : 2