Transverse Vibration Analysis of Axially Moving Trapezoidal Plates

Transverse vibration of axially moving trapezoidal plates is investigated. The differential equation of transverse vibration for a axially moving trapezoidal plate is established by D'Alembert principle. The original trapezoid region can be replaced by regular square region by the medium parameter method for the convenience of calculation. A generalized complex eigenvalue equation is derived by a discrete method (the differential quadrature method). The complex frequency curve of trapezoidal plate is obtained by calculating the eigenvalue equation. The change of the complex frequencies of the axially moving trapezoidal plates with the dimensionless axially moving speed is analyzed. The effects of the aspect ratio and the trapezoidal angle on instability type of the trapezoidal plate are discussed under different boundary conditions. The results of numerical analysis show that there are two main instability types of axially moving trapezoidal plate: divergence and flutter. The modal orders of the two types of instability are also different, which is related to the trapezoidal angle, aspect ratio and boundary condition of the trapezoidal plate.

Effect of Trapezoidal Shapes on the Thermal Buckling Behaviour of Perforated Composite Plates

Thermal buckling study on the symmetric laminated composite trapezoidal plate with a circular cutout subjected to a uniform increase in temperature for various boundary conditions is explored in this paper. In a mathematical model, the first-order shear deformation principle is employed in accordance with the variational energy system. For acquiring the thermal buckling temperature, a nine-node heterosis plate relation has been used in the finite element formulation. By correlating the present findings with accessible literature, the effectiveness of the present formulation is verified. The impact of different parameters, such as trapezoidal shape, cutout size, ply-orientation, plate edge conditions and plate width to thickness ratio have been considered to study the effect of each parameters on the buckling characteristics of plate under various temperatures. It is observed from the study that each parametric investigation significantly affect the thermal buckling behaviour of trapezoidal plates.

Nonlinear vibration of a nanocomposite laminated piezoelectric trapezoidal actuator in subsonic airflow under combined electrical and forcing excitations

The nonlinear dynamic and vibration behaviors of a cantilevered carbon nanotube-reinforced composite trapezoidal plate with two surface-bonded piezoelectric layers as an actuator in micro air vehicles are considered in this article. The plate is reinforced by single-walled carbon nanotubes and is exposed to subsonic airflow under combined parametric and external excitations. The large deflection von Karman plate assumptions and the classical laminated plate theory are applied to derive the governing equations of the motion of the piezoelectric nanocomposite laminated trapezoidal plate by using Hamilton's principle. The geometry of the trapezoidal plate is mapped into a rectangular computational domain. The Galerkin's approach is used for transforming the nonlinear partial differential equations of motion into nonlinear two-degrees-of-freedom ordinary differential equations of cubic nonlinearities. The case of 1:3 internal resonance and primary resonance is considered, and the multiple scales method is employed. The aerodynamic pressure distribution formula is modeled by linear potential flow theory. The frequency and time history responses and phase portrait in free forced vibrations are obtained to analyze the nonlinear dynamic behavior of the plate. The effects of different parameters such as the plate geometry, volume fraction of carbon nanotubes, and different excitations on the nonlinear vibration of the thin laminated plate are also discussed. A complex softening nonlinearity with two peaks in the higher mode is observed in frequency response curves. The influence of electrical excitation with several amplitudes and frequencies on dynamic stability is investigated using time response curves.

Analysis of Nonlinear Vibrations and Dynamic Responses in a Trapezoidal Cantilever Plate Using the Rayleigh-Ritz Approach Combined with the Affine Transformation

Nonlinear vibrations of a trapezoidal cantilever plate subjected to transverse external excitation are investigated. Based on von Karman large deformation theory, the Rayleigh-Ritz approach combined with the affine transformation is developed to obtain the nonlinear ordinary differential equation of a trapezoidal plate with irregular geometries. With the variation of geometrical parameters, there exists the 1:3 internal resonance for the trapezoidal plate. The amplitude-frequency formulations of the system in three different coupled conditions are derived by using multiple scales method for 1:3 internal resonance analysis. It is found that the strong coupling of two modes can change nonlinear stiffness behaviors of modes from hardening-spring to soft-spring characteristics. The detuning parameter and excitation amplitude have significant influence on nonlinear dynamic responses of the system. The bifurcation diagrams show that there exist the periodic, quasi-periodic, and chaotic motions for the trapezoidal cantilever plate in the 1:3 internal resonance cases and the nonlinear dynamic responses are dependent on the amplitude of excitation. The possible adverse dynamic behaviors and undesired resonance can be avoided by designing appropriate excitation and system parameters.