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