Von Karman Plate Research Articles

Mathematical modeling and numerical simulation of telephone cord buckles of elastic films

An annular sector model for the telephone cord buckles of elastic thin films on rigid substrates is established, in which the von Karman plate equations in polar coordinates are used for the elastic thin film and a discrete version of the Griffith criterion is applied to determine the shape and scale of the parameters. A numerical algorithm combining the Newmark-β scheme and the Chebyshev collocation method is designed to numerically solve the problem in a quasi-dynamic process. Numerical results are presented to show that the numerical method works well and the model agrees well with physical observations, especially successfully simulated for the first time the telephone cord buckles with two humps along the ridge of each section of a buckle.

Flutter stabilization of cantilevered plates using a bonded patch

In recent years, many researchers have studied active vibration suppression of fluttering plates using piezoelectric actuators. Lots of these researchers have focused on optimal placement of piezoelectric patches to obtain maximum controllability of the plate. Although mass and stiffness characteristics of bonded patches can alter the aeroelastic behavior of fluttering plates, few of the investigators have considered the effect of the mentioned parameters in the optimization process. This paper investigates the effect of a bonded patch on the aeroelastic behavior of cantilevered plates in supersonic flow and examines the optimal location of the patch for the best controllability performance. For mathematical simulation of the structure, linear von Karman plate theory along with first-order piston theory is employed. The results obtained through this study reveal that a bonded patch with a small mass ratio can change the system critical dynamic pressure significantly. The maximum raise of the critical dynamic pressure is acquired when the bonded patch is placed on the leading edge of the plate. A variation of the system’s aerodynamic characteristics, subsequently, influences the control performance of the bonded patch and alters the optimal patch location.

Weak and strong solutions of a nonlinear subsonic flow–structure interaction: Semigroup approach

We consider a non-rotational, subsonic flow–structure interaction describing the flow of gas above a flexible plate. A perturbed wave equation describes the flow, and a second-order nonlinear plate equation describes the plate’s displacement. It is shown that the linearization of the model generates a strongly continuous semigroup with respect to the topology generated by “finite energy” considerations. An interesting feature of the problem is that linear perturbed flow–structure interaction is not monotone with respect to the standard norm describing the finite energy space. The main tool used in overcoming this difficulty is the construction of a suitable inner product on the finite energy space which allows the application of ω -maximal monotone operator theory. The obtained result allows us to employ suitable perturbation theory in order to discuss well-posedness of weak and strong solutions corresponding to several classes of nonlinear dynamics including the full flow–structure interaction with von Karman, Berger’s and other semilinear plate equations.

MPI–CUDA parallelization of a finite-strip program for geometric nonlinear analysis: A hybrid approach

A finite-strip geometric nonlinear analysis is presented for elastic problems involving folded-plate structures. Compared with the standard finite-element method, its main advantages are in data preparation, program complexity, and execution time. The finite-strip method, which satisfies the von Karman plate equations in the nonlinear elastic range, leads to the coupling of all harmonics. However, coupling of series terms dramatically increases computation time in existing finite-strip sequential programs when a large number of series terms is used. The research reported in this paper combines various parallelization techniques and architectures (computing clusters and graphic processing units) with suitable programming models (MPI and CUDA) to speed up lengthy computations. In addition, a metric expressing the computational weight of input sets is presented. This metric allows computational complexity comparison of different inputs.

Aerothermoelastic Post-Critical and Vibration Analysis of Temperature-Dependent Functionally Graded Panels

This paper deals with the aerothermoelastic post-critical and vibration characteristics of temperature-dependent functionally graded panels in a supersonic airflow. The structural formulation is based on the von Karman plate theory and material properties are assumed to be temperature-dependent and graded in the thickness direction according to power law distribution in terms of the volume fractions of the constituents. The two-dimensional panel under study is simply supported, for which the first order piston theory is used to account for the supersonic aerodynamic loading. The Galerkin method is applied to convert the partial differential governing equation into a set of ordinary differential equations. Panel vibration responses are investigated through time history responses, state-space trajectories, frequency spectra and the bifurcation diagrams of Poincaré maps. Moreover, post-critical behaviors are detected using numerical and analytical methodologies such as bifurcation diagrams of Poincaré maps, Lyapunov exponents and Lyapunov dimension. Finally, it is shown that the Lyapunov dimensions for stable and divergence conditions are zero value, while these values for limit-cycle and chaos vibration conditions are integer and non-integer quantity, respectively.

Analytical geometrical responses in large deflection of simply supported piezoelectric layered plate under initial tension

The analytical geometrical responses in large deflection of a simply supported and layered piezoelectric circular plate under initial tension due to lateral pressure are presented. The approach follows von Karman plate theory for large deflection with a consideration of a symmetrically laminated case including a piezoelectric layer. The related nonlinear governing equations are derived in a non‐dimensional form and are simplified by neglecting the arising nonlinear terms, yielding a modified Bessel equation or a standard Bessel equation for the lateral slope. The associated analytical solutions are developed by imposing the simply supported edge conditions of the problem. For a 3‐layered nearly monolithic plate under a low pretension and a low applied voltage upon the piezoelectric layer, the results agree well with those obtained by using the classical plate theory for a single‐layered plate under pure mechanical loading, and thus the developed approach is validated. Typical 3‐layered piezoelectric plates are then implemented and the results show that, no apparent edge effect was found for the present problem. In additions, a piezoelectric effect appears to be present only up to a moderate initial tension. For a relatively high pretension, the tension effect tends to be dominant, resulting in nearly the same results for the geometrical responses, regardless of the magnitude of the applied voltage.

Nonlinear Model Reduction of von Kármán Plates Under Quasi-Steady Fluid Flow

A reduced-order model for von Karman plates, using a method of quadratic components, is presented in this paper. The method of quadratic components postulates the full kinematics of the plate as consisting of a combination of linear and quadratic components. This reduced model is then discretized in space with a Galerkin approximation and in time with an implicit integration scheme. The plate is coupled with a fluid flowing over it at a supersonic speed using a quasi-steady pressure model commonly referred to as piston theory. A static loading example is used to validate the model reduction of the plate with respect to other numerical and approximate solutions. The limit cycles of the plate coupled with piston theory are calculated via a cyclic method, and the results from parameter studies are compared to classical results. A previously unexplored regime of the limit cycle amplitudes is investigated; a second, coexisting, limit cycle is found that yields a discontinuity in the limit cycle amplitudes at a critical dynamic pressure.

Integral bounds for von Kármán's equations

AbstractIn this paper we consider the von Kármán plate equations. We show that a suitable function, defined on the solution to these equations is subharmonic. Then we derive bounds on the integral of the second gradient of the solution.

Plate models with state-dependent damping coefficient and their quasi-static limits

We study long-time dynamics of a class of plate models with a state-dependent damping coefficient and their quasi-static limits. We first present the problem in abstract form and then prove the existence of finite-dimensional global attractors and their upper semicontinuity in the quasi-static limit, i.e., in the case when the mass density of plate tends to zero. Our proofs involve a recently developed method based on “compensated” compactness and quasi-stability estimates. As an application we consider the nonlinear Kirchhoff, von Karman and Berger plate models with different types of boundary conditions and damping coefficients. Our results can be also applied to the nonlinear wave equations in an arbitrary dimension.

Adaptive stabilization of a von Karman plate equation with a boundary output feedback control

We consider a von Karman plate equation with boundary memory condition and output feedback control. We prove the existence of solutions using the Galerkin method and then investigate the stabilization of the corresponding solutions by choosing a suitable Lyapunov functional.

Solvability of Dynamic Contact Problems for Elastic von Kármán Plates

The existence of solutions is proved for unilateral dynamic contact problems of elastic von Karman plates. Boundary conditions for a free and clamped plate are considered.

Modeling of Coupled Dome-Shaped Microoscillators

Synchronized micromechanical oscillators have potential for applications in signal processing, neural computing, sensing, and other fields. This paper explores the conditions under which coupled nonlinear limit cycle microoscillators can synchronize. As an example of the modeling approach and to be able to obtain results for a system that has been experimentally realized, thermally excited dome oscillators, which are fabricated by buckling of a thin circular film of polysilicon, are modeled. Starting with the von Karman plate equations and an assumed mode shape, a Galerkin projection is performed to obtain an ordinary differential equation for the postbuckled dynamics of the mode. Because the structure is thermally excited, a thermal model is built by solving the heat equation on a disk with appropriate boundary conditions. Bifurcation analysis of the single oscillator model is performed to look for basic underlying phenomena. Conditions under which two slightly detuned coupled limit cycle dome oscillators will synchronize are explored.

Nonlinear vibrations of impacted rectangular plates. Comparison between numerical simulation and experiments

Large amplitude vibrations of free‐edge rectangular plates, subjected to an impulse excitation is addressed. In particular, this work is devoted to the analysis and simulation of a nonlinear von Karman plate equations and by associate experimental investigations. Time domain simulations are achieved using implicit finite differences (FD) schemes recently developed by Bilbao. These energy‐conserving methods guarantee the stability of the algorithm. To compare with simulations results, an experimental setup which allows reproducible impulse excitation and measurements by laser vibrometry has been developed. The time history of the force applied to a rectangular steel plate is recorded and this signal is used as excitation term in the simulations. We perform a parametric study, with both experimental and numerical approaches, by increasing gradually the amplitude of forcing. Non‐linear phenomena, such as pitch glide and chaotic behaviour are observed and discussed.

Design Tailoring for Pressure Pillowing Using Tow-Placed Steered Fibers

Manufacturing of high-quality fiber-reinforced composite structures with spatially varying fiber orientation is possible using advanced tow-placement machines. Changing the fiber-orientation angle within a layer produces variable-stiffness properties. Contrary to traditional composites with straight fibers, this method allows the designer to fully benefit from the directional material properties of the composite to improve laminate performance by determining optimal fiber paths. In this paper, design tailoring for the pressure-pillowing problem of a fuselage skin is addressed using steered fibers. The problem is modeled as a two-dimensional plate using von Karman plate equations. The analysis is performed using the Rayleigh-Ritz method, and the nonlinear response is traced using a normal flow algorithm. The design objective is to determine the optimal fiber paths over the panel for maximum failure load. Different designs are obtained for different loading cases. The results indicate that by using steered fibers, the pressure-pillowing problem can be alleviated and the load-carrying capacity of the structure can be improved, compared with designs using straight fibers.

Thermal Vibration of Functionally Graded Circular Plates

Based on the nonlinear theory of von Karman plate, axisymmetric nonlinear vibration of a functionally graded circular plate with clamped boundary condition is investigated under thermal loading. It is assumed that the mechanical and thermal properties of functionally graded materials vary continuously through the thickness of the plate and obey a simple power law related to the volume fraction of the constituents. Motion equations for the problem are derived. Existence of harmonic vibrations is assumed and then Ritz-Kantorovich method is used to convert the dynamic Von Karman equations to a set of nonlinear ordinary differential equations. Finally a shooting method is employed to numerically solve the resulting differential equations. Effects of amplitude A, thermal load parameter λ and material constant n on the vibration behavior of FGM plate are discussed in details.

Nonlinear vibration and buckling of circular sandwich plate under complex load

The nonlinear vibration fundamental equation of circular sandwich plate under uniformed load and circumjacent load and the loosely clamped boundary condition were established by von Karman plate theory, and then accordingly exact solution of static load and its numerical results were given. Based on time mode hypothesis and the variational method, the control equation of the space mode was derived, and then the amplitude frequency-load character relation of circular sandwich plate was obtained by the modified iteration method. Consequently the rule of the effect of the two kinds of load on the vibration character of the circular sandwich plate was investigated. When circumjacent load makes the lowest natural frequency zero, critical load is obtained.

A family of conservative finite difference schemes for the dynamical von Karman plate equations

AbstractNumerical methods for nonlinear plate dynamics play an important role across many disciplines. In this article, the focus is on numerical stability for numerical methods for the von Karman system, through the use of energy‐conserving methods. It is shown that one may take advantage of structure particular to the system to construct numerical methods, which are provably numerically stable, and for which computer implementation is simplified. Numerical results are presented. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Cantilevered Plate Rayleigh-Ritz Trial Function Selection for von Kármán's Plate Equations

The accuracy and convergence characteristics of the classical Rayleigh-Ritz solution of the nonlinear von Karman plate equations are studied with respect to the types of cantilever plate in-plane trial functions used in the solution. The static deflection of the cantilever plate is computed for an applied static gravity loading. Four different in-plane trial function types are studied. In each of these cases the same out-of-plane trial functions are used. It is found that in two of the four cases good convergence and accuracy are achieved when compared to the solution from a nonlinear finite element model. The degree of satisfaction of the problem natural boundary conditions is also examined, and it is shown that for the two cases that show inadequate convergence characteristics this satisfaction is poor. It is noted in particular that for these two cases at points on the problems' free boundaries, the in-plane trial functions satisfy, either exactly or approximately, the linear in-plane natural boundary conditions. At these points the addition of inplane degrees of freedom to the solution will not contribute to the satisfaction of the nonlinear natural boundary condition. Thus for these two cases the convergence is poor as more in-plane trial functions are added to the solution. The change in the statically loaded plate natural frequencies are also computed. Similar to the static deflection results, convergence to the nonlinear finite element solution is slow when in-plane trial functions are used that approximately satisfy the linear in-plane boundary conditions.

Structural Ritz-Based Simple-Polynomial Nonlinear Equivalent Plate Approach: An Assessment

Thelinearequivalentplateapproachdevelopedandusedfortheaeroelasticoptimizationofcompositewingsatthe conceptual design stage is a Ritz approach based on simple polynomial functions as generalized coordinates. Generalization to the case of geometrically nonlinear structural behavior is presented here in an effort to assess accuracy and performance of the method in the nonlinear static and linear dynamic cases. Using the von Karman plate formulation for moderately large deformations, three-dimensional assemblies of thin-plate segments are modeled using a penalty function approach to impose boundary conditions and compatibility of motion between adjacent segments. Closed-form expressions for mass and stiffness terms (linear and nonlinear) make numerical integration unnecessary. Numerical results obtained by the present method for a variety of plate structures, in both static anddynamic cases, show good correlation with published results andresults by othercomputer codes.Results by the present formulation are also compared with large-deformation results. Limits of applicability, in terms of the range of deformations still captured accurately by the equivalent plate method, are studied in all test cases.

Stacking Sequence Design of Flat Composite Panel for Flutter and Thermal Buckling

In this paper, we formulate the stacking sequence design of laminated composite flat panels for maximum supersonic flutter speed and maximum thermal buckling capacity. The design is constrained to have stable limit cycle oscillation in the vicinity of the flutter boundary. For the purpose of comparison, we consider both linear and nonlinear panel flutter analyses. Because of manufacturing constraints, the stacking sequence of the laminate is selected from a discrete set of orientation angles. The plate is modeled using von Karman plate equations and assumed to be subjected to uniform temperature differential. The aerodynamic load on the plate is computed using third-order piston theory. The plate equations of motion are derived using Hamilton's principle and discretized using the Rayleigh-Ritz method. The nonlinear panel flutter analysis is performed using an approximate perturbation approach. The Pareto fronts describing the trade-off between thermal buckling and flutter are generated for different panel aspect ratios. It is observed that optimal designs obtained for both linear and nonlinear flutter analyses are similar, with only the predicted flutter speeds different This suggests that, for the purpose of preliminary stacking sequence design, linear flutter analysis is sufficient.