Nonlinear vibrations of the orthotropic nanoplates subjected to an influence of in-plane magnetic field are considered. The model is based on the nonlocal elasticity theory. The governing equations for geometrically nonlinear vibrations use the von Kármán plate theory. Both the stress formulation and the Airy stress function are employed. The influence of the magnetic field is taken into account due to the Lorentz force yielded by Maxwell's equations. The developed approach is based on applying the Bubnov-Galerkin method and reducing partial differential equations to an ordinary differential equation. The effect of the magnetic field, elastic foundation, nonlocal parameter and plate aspect ratio on the linear frequencies and the nonlinear ratio is illustrated and discussed.
The parametric vibrations of orthotropic plates with complex forms for different types of boundary conditions are studied. The proposed novel hybrid approach is based on combination of the so called Rfunctions method and the variation method. In particular, advantages of a multimode approximation used for plate behavior analysis are addressed, among others.
The problem of nonlinear vibrations and stability analysis for the symmetric laminated plates with complex shape, loaded by static or periodic load in-plane is considered. In general case research of stability and parametric vibrations is connected with many mathematical difficulties. For this reason we propose approach based on application of R-functions theory and variational methods (RFM).The developed method takes into account pre-buckle stress state of the plate. The proposed approach is demonstrated on testing problems and applied to laminated plates with cutouts. The effects of geometrical parameters, load, boundary conditions on stability regions and nonlinear vibrations are investigated.
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