Nonlinear vibrations of FGM rectangular plates in thermal environments

Abstract

Geometrically nonlinear vibrations of FGM rectangular plates in thermal environments are investigated via multi-modal energy approach. Both nonlinear first-order shear deformation theory and von Karman theory are used to model simply supported FGM plates with movable edges. Using Lagrange equations of motion, the energy functional is reduced to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities. A pseudoarclength continuation and collocation scheme is used and it is revealed that, in order to obtain the accurate natural frequency in thermal environments, an analysis based on the full nonlinear model is unavoidable since the plate loses its original flat configuration due to thermal loads. The effect of temperature variations as well as volume fraction exponent is discussed and it is illustrated that thermally deformed FGM plates have stronger hardening behaviour; on the other hand, the effect of volume fraction exponent is not significant, but modal interactions may rise in thermally deformed FGM plates that could not be seen in their undeformed isotropic counterparts. Moreover, a bifurcation analysis is carried out using Gear's backward differentiation formula (BDF); bifurcation diagrams of Poincaré maps and maximum Lyapunov exponents are obtained in order to detect and classify bifurcations and complex nonlinear dynamics.

A Reduced Model of a Thermo-Elastic Nonlinear Circular Plate

“…The theoretical results have been validated by experimental tests. Influence of temperature on moderately thick functionally graded plates is presented in [9]. Both material properies and the temperature have been assumed as varied nonlinearly through the plate thickness.…”

Approximate Periodic Solution for the Nonlinear Helmholtz-Duffing Oscillator via Analytical Approaches

“…Surveying the literature shows that this equation has wide applications in the engineering problems. For example, due to different vibration behavior of functionally graded materials (FGMs) at positive and negative amplitudes, the governing equations of FGM beams, plates, and shells are conduced to a second-order nonlinear ordinary equation with quadratic and cubic nonlinear terms [17][18][19][20]. Moreover, Sharabiani and Yazdi [21] obtained a Helmholtz-Duffing type equation within studying of nonlinear free vibrations of functionally graded nanobeams with surface effects.…”

Editorial: Special issue on stability of non-linear dynamic structures and systems

“…In this special issue, the concept stability is also frequently encountered with respect to periodic solutions caused by periodic or pure harmonic excitation [1][2][3][4][5][6][7]. The local stability of periodic solutions is evaluated using Floquet theory, which can be extended for systems of Filippov-type by introducing saltation matrices, as reviewed in [6].…”

“…The mechanical model is coupled to thermal loads in [1], to an electrodynamic shaker in [3], to the fluid domain in [4], to fringing electrostatic fields in [9], and, for their macro-structure, to magnetic forces in [10]. We now briefly summarize the content of the papers included in this special issue.…”

“…Alijani et al [1] investigate geometrically nonlinear vibrations of rectangular plates with Functionally Graded Materials (FGM) in thermal environments via a multi-modal energy approach. Both non-linear first-order shear deformation theory and von Karman theory are used to model simply supported FGM plates with movable edges.…”