Behavioural Study of Thin Plate Under Large Deflection Theory

For a thin plate, if the deformation is on the order of the thickness and stay elastic, linear theory might not turn out correct results because it does not predict the in plane movement of the member. Therefore, to account for the inconsistencies of geometric nonlinearity, large deflection theory is required [1]. This report pertains to the analytical study dispensed to check the behavior of thin plate under fixed and pinned edge conditions, and for diverse thicknesses, under the small and large deflection theories. The deformation is additionally studied, supported by Von-Karman equations. Non linear analysis has been performed on FE model using the ANSYS software. The consequences of geometric nonlinearities are mentioned. Outline on conclusion of the theoretical and experimental results obtained, are compared so as to review the similarity of the modeling and theory.

Hopf Bifurcation of Heated Panels Flutter in Supersonic Flow

A differential equation of panel vibration in supersonic flow is established on the basis of the thin-plate large deflection theory under the assumption of a quasi-steady temperature field. The equation is dimensionless, and the derivation of its second-order Galerkin discretization yields a four-dimensional system. The algebraic criterion of the Hopf bifurcation is applied to study the motion stability of heated panels in supersonic flow. We provide a supplementary explanation for the proof process of a theorem, and analytical expressions of flutter dynamic pressure and panel vibration frequencies are derived. The conclusion is that the algebraic criterion of Hopf bifurcation can be applied in high-dimensional problems with many parameters. Moreover, the computational intensity of the method established in this work is less than that of conventional eigenvalue computation methods using parameter variation.

Large Deflections and Buckling Loads of Cantilever Columns with Constant Volume

This paper deals with the large deflections and buckling loads of tapered cantilever columns with a constant volume. The column member has a solid regular polygonal cross-section. The depth of this cross-section is functionally varied along the column axis. Geometrical nonlinear differential equations, which govern the buckled shape of the column, are derived using the large deflection theory, considering the effect of shear deformation. The buckling load of the column is approximately equivalent to the load under which a very small tip deflection occurs. In regard to the numerical results, both the elastica and buckling loads with varying column parameters are discussed. The configurations of the strongest column are also presented.

Nonlinear vibration analysis of a membrane based on large deflection theory

This paper investigates nonlinear vibration of a simply supported rectangular membrane based on large deflection theory. Dynamic stress caused by transverse displacement of the membrane is considered in modeling the membrane. The assumed mode method and the nonlinear finite element method (FEM) are both used as discretization methods for the membrane. In the assumed mode method, an approximate analytical formula of the natural frequency is derived. In the nonlinear FEM, a three-node triangular membrane element is proposed. The difference between the membrane’s dynamical characteristics obtained by these two discretization methods is revealed. Simulation results indicate that natural frequency of the membrane will rise along with the increasing of the vibration amplitude of the membrane, and the natural frequency obtained by the nonlinear FEM is larger than that obtained by the assumed mode method. When the membrane vibration is small, the assumed mode method may achieve a reasonable result, but it may lead to a big error when the membrane vibration is large.

Analysis of Impact of Driving Amplitude on Resonance Frequency of Silicon Microgyroscope

Increasing the driving amplitude could improve the sensitivity of silicon microgyroscope, which is the effective way to enhance the performance of gyroscope. However the large driving amplitude leads to the nonlinear effect. As a result, the resonance frequency is dependent with the driving amplitude, which influences the frequency stability of gyroscope. The equivalent model of driving beam is established, and the stiffness formulas of driving beam are derivated according to the large deflection theory. The linear stiffness of driving beam is 270.1N/m and the nonlinear stiffness is 3.237×108N/m3, which are 2.7% and 6.4% different from the simulative results. Harmonic balance method can be used to establish the relationship between driving amplitude and resonance frequency. The theoretical analysis results are carried out compared with the simulative analysis results. Also the impact of driving amplitude on resonance frequency is confirmed by experiment test. This paper is significant for improving the frequency stability, and provides a theoretical basis for optimizing the structure of gyroscope.

NONLINEAR DAMPED VIBRATION OF PRE-STRESSED ORTHOTROPIC MEMBRANE STRUCTURE UNDER IMPACT LOADING

This paper is concerned with the nonlinear damped forced vibration problem of pre-stressed orthotropic membrane structure under impact loading. The governing equations of motion were derived based on the von Kármán large deflection theory and D'Alembert's principle, and solved by using the Bubnov–Galerkin method and the Krylov–Bogolubov–Mitropolsky (KBM) perturbation method. The asymptotic analytical solutions of the frequency and lateral displacement of rectangular orthotropic membrane with fixed edges were obtained. In the computational example, the frequency results were compared and analyzed. Meanwhile, the vibration mode of the membrane and the displacement and time curves of each feature point on the membrane surface were analyzed. The results obtained herein provide a simple and convenient approach to calculate the frequency and lateral displacement of the nonlinear forced vibration of rectangular orthotropic membranes with low viscous damping under impact loading. In addition, the results provide some computational basis for the vibration control and dynamic design of membrane structures.

Nonlinear Aeroelastic Panel Flutter Based on Proper Orthogonal Decomposition

It is commonly accepted that 36 in vacuo natural modes (NMS) are needed for converged, limit-cycle oscillations (LCOs) of isotropic or laminated anisotropic rectangular panels in supersonic air flow. It’s computationally costly for nonlinear aeroelastic panel response using such a large number of modes, and it also causes complexity and difficultly in designing controllers for panel flutter suppression. Based on Hamilton principle, the aeroelastic finite element motion equations of the 3-D panel are established by using the von Karman large deflection theory, first-order piston theory aerodynamics, the proper orthogonal decomposition (POD) method are adopted to construct a reduced order model of the panel, then the reduced panel flutter equations are solved in time domain using a numerical integration method. Comparing with the LCOs calculated by using 36NMS, the results obtained by using the reduced order model based on POD method (POD/ROM) show a good agreement.