Karman Large Deflection Theory Research Articles

Bifurcation and chaos analysis of a pretensioned moving printed electronic laminated membrane considering aerothermoelasticity

The bifurcation and chaotic motion characteristics are studied for a roll-to-roll flexible printed electronic laminated membrane subjected to initial tension in a steady temperature field. First, an aerodynamic model of the moving membrane is established by using an improved Newton's law for internal friction. Based on von Karman's large deflection theory, a nonlinear vibration differential equation is derived for the laminated membrane under multiple physical fields using Hamilton's principle and solved numerically using the fourth-order Runge-Kutta method. The effects of the temperature, air resistance and other parameters on the nonlinear vibration of the membrane are analyzed by using the bifurcation diagram, Lyapunov exponent diagram and corresponding phase diagram, Poincaré diagram and time history diagram. The results show that the membrane exhibits abundant chaos and bifurcation phenomena under a coupling field. The parameters of the membrane can be varied to control the nonlinear vibration characteristics. This study provides theoretical guidance for improving the transmission stability of electronic membranes and realizing high-precision overprinting.

Influence of different material parameters on nonlinear vibration of the cylindrical skeleton supported prestressed fabric composite membrane

Abstract In order to study the influence of geometric nonlinearity and material parameters on the free vibration behavior of the cylindrical skeleton supported prestressed fabric composite membrane. In this paper, based on von Karman's large deflection theory and D’Alembert's principle, the governing equations of nonlinear viscous damped prestressed free vibration of frame supported anisotropic membrane structures were established. By using Galerkin and KBM perturbation method, the analytical expressions of frequency function, displacement function and mode shape of nonlinear free vibration were obtained. In order to verify the effectiveness and effective range of the method, the fourth order Runge-Kutta method was used for numerical calculation. The calculation examples of membrane material parameters were given. The calculation and analysis of different membrane prestress, different length width ratio, different rise span ratio and different material parameters (elastic modulus ratio, viscous damping and material density) were carried out. The results were compared with the numerical results obtained by Runge-Kutta method to verify the accuracy of the method. The research results of this paper provide a theoretical reference for the selection of membrane materials, the subsequent calculation of wind-induced stability of steel skeleton membrane structure and the design of wind-driven rain resistance.

A Simple Proper Orthogonal Decomposition Method forvon Karman Plate undergoing Supersonic Flow

We apply a simple proper orthogonal decomposition (POD) method to compute the nonlinear oscillations of a degenerate two-dimensional fluttering plate undergoing supersonic flow. First, the von Karman's large deflection theory and quasi-steady aerodynamic theory are employed in constructing the governing equations of the simply supported plate. Then, the governing equations are solved by both the Galerkin method and the POD method. The Galerkin method is ac- curate but sometimes computationally expensive, since the number of degrees of freedom (dofs) required is relatively large provided that nonlinearity is strong. The POD method can be used to capture the complex dynamics of a strongly nonlinear system using very few degrees of freedom, much fewer than the Galerkin approach. The presently proposed POD method has two advantages over the conventional one. i) a simple numerical difference technique is first introduced to the POD method to avoid the complicated mode-to-mode projection between POD modes and Galerkin modes. ii) POD based reduced order models (POD/ROM) are constructed by us- ing a set of general modes which is extracted from chaotic responses. That is to say POD modes extracted from one set of parameters can be applied to various parameter variations for the same dynamic system. Moveover, results for the buck- led, LCO and chaotic responses of the plate are presented and compared with the Galerkin solutions. Numerical examples demonstrate the accuracy and efficiency of the present POD method.

Simultaneous resonances involving two mode shapes of parametrically-excited rectangular plates

It is known that, for multi-degree-of-freedom systems under time-dependent excitation, the combination of an internal resonance with an external resonance will give rise to simultaneous resonances, and these resonances are characterized by the fact that the system in question can resonate simultaneously in more than one normal mode while only one resonant mode is directly excited by the excitation. This work deals with the problem of the occurrence of simultaneous resonances in a parametrically-excited and simply-supported rectangular plate. The analysis is based on the dynamic analog of von Karman's large-deflection theory, and the governing equations are satisfied using the orthogonality properties of the assumed functions. The nonlinear temporal response of the damped system is determined by the first-order generalized asymptotic method. The solution for simply supported plates indicates the possibility of principal parametric resonances and simultaneous resonances. Simultaneous resonances involving two modes of vibration are presented in this paper, and it is shown for the first time that only one possibility out of two cases can really occur. This phenomenon is verified by experimental results and observations.

916 移動熱源が作用する直交異方性薄板の熱座屈現象に関する理論解析(GS-3 熱変形・高温疲労・熱疲労)

This paper is concerned with theoretical treatment of thermal stress and deformation of thermoformable orthotropic strip before roll forming. Especially, we treat this problem as quasi-static one. As an analytical model, we assume that orthotropic strip is subjected to partially distributed heat supply. We obtain the one-dimensional temperature solution and out-of-plane thermal deformation using von Karman's large deflection theory. The solutions of simultaneous algebraic nonlinear equations are solved by Brent Method. And we examine the phenomenon of thermal buckling caused by non-uniform temperature change near the middle portion of the strip.

On the linear and nonlinear analysis of cross-ply laminated composite plates

The cylindrical bending of a cross-ply laminated plate subjected to a uniform transverse load is evaluated via classical laminated plate theory and Von Karman's large deflection theory, respectively. It is shown that for the analysis of several cross-ply composite laminates, the common concept which expects that the linear analysis will provide a more accurate result for the plate with smaller deflection is not always correct, and the classical laminated plate theory may be inadequate even in the range of small deflection. Finally, the influence of the amplitude of deflection, temperature variation, boundary conditions, span-to-thickness ratio and stacking sequence on the accuracy of the linear and nonlinear analysis is evaluated.

Nonlinear bending of thin and thick unsymmetrically laminated composite beams using refined finite element model

Nonlinear bending behaviour of unsymmetrically laminated composite beams using von Karman's large deflection theory is investigated. For this purpose, a one-dimensional finite element having twelve degrees of freedom per node based on a simple higher shear deformation theory (HSDT) which satisfies the zero transverse shear stress at the top and bottom surface and avoids the use of shear correction factors is developed. The advantage of the present higher order theory is that first-order shear deformation theory (FSDT) and classical lamination theory (CLT) can be obtained easily by dropping the higher order terms associated with transverse shear deflection or all the terms associated with transverse shear deflection from the kinematic relations. To show the effect of transverse shear on the transverse deflection, the first-order shear deformation theory and classical lamination theory results are also presented. Unsymmetrically laminated composite beams, due to the existence of bending-extension coupling are shown to have direction dependent nonlinear bending stiffness, resulting in softening/hardening type of nonlinearity for one direction of transverse loading and hardening type nonlinearity for the other direction. It is shown that this softening type of nonlinearity at small loads changes to hardening type of nonlinearity for higher loads. Results are presented for various types of lay-ups, slenderness ratios and boundary conditions.

Hygrothermal effects on the linear and nonlinear analysis of symmetric angle-ply laminated plates

The influence of hygrothermal effects on the cylindrical bending of pinned-pinned, clamped-pinned and clamped-clamped support symmetric angle-ply laminated plates subjected to an uniform transverse load is evaluated via classical laminated plate theory and Von Karman's large deflection theory. It is shown that the classical laminated plate theory may not be adequate for the analysis of several composite laminates, even in the small deflection range. The influence of temperature variation, thickness, boundary conditions and the angle of the angle-ply laminates on the accuracy of the analysis is evaluated.