Berger Approximation Research Articles

A Berger-linear Buildup Approximation for Air.

A combination of the Berger and the linear buildup approximations with the assumption of conservation of energy was fitted for air. The values predicted by this Berger-linear buildup approximation were generally more accurate than those predicted by the Berger approximation alone. Although it is not as accurate as the geometric-progression approximation, the Berger-linear approximation is analytically simpler and makes it easier to find analytical solutions for point source kernel shielding calculations.

Dynamic analysis of a Drum Charger: Large amplitude vibrations of clamped circular thin plate on a linear foundation

The Impulse Drum Charger® (IDC) represents a valid and innovative alternative in the field of the superchargers, in particular when the available space is limited, such as in motorcycles. In fact, with respect to the traditional one, which uses turbine-compressor system for engine supercharging, the IDC exploit the deflection of an elastic membrane-spring system to generate overpressure at the intake from the pressure waves generated by the exhaust gases. In this way, the aim of this work is the development of a mathematical model of the membrane-spring system, both realized in 102-RGUD600 glass fiber composite (PA matrix), of a Drum Charger® using Von Karman theory with Berger's approximation. Focusing on the central deflection of the membrane in time and frequency domain, the derived models reproduces with good accuracy the results of the complete finite-element simulations computed with Ansys™, especially in the higher frequencies. Moreover, in order the system work properly, the spring behavior must maintain in linear-elastic range. Hence, a three-point bending test of the spring was carried out, following the specifications in ASTM (D790-03), in order to verify the force-displacement linear relation. The numerical simulations shown excellent agreement with the force-displacement curve observed in the experimental tests.

Modeling strategies of electrostatically actuated initially curved bistable micro plates

Micro- and nanolectromechanical systems (MEMS/NEMS) incorporating two-dimensional structural elements such as plates and shells attracted significant interest in recent years. These structures demonstrate rich electromechanical behavior and could be advantageous in applications. In this work, we explore implementation of two models describing axisymmetric behavior of initially curved circular micro plates, subjected to a distributed nonlinear electrostatic force. While both models are based on the Kirchoff hypothesis and on the nonlinear Föppl von Kármán (FvK) strain-displacements relations, the second model employs the Berger approximation, which significantly simplifies the formulation and describes the plate by a single governing equation. In both cases, the solution is based on the Galerkin decomposition with buckling modes of an initially flat plate used as the base functions. To track the unstable branches of the equilibrium curve, continuation methods in conjunction with the Riks algorithm are implemented. The validation of the models is conducted for two loading cases, namely “mechanical” deflection-independent load, and electrostatic displacement-dependent load. Results of a finite elements (FE) analysis, as well as of a finite differences (FD) solution of the differential equations, were used as a reference. We estimate the accuracy of the RO models and provide recommendations concerning the number of degrees of freedom (DOF) required to reach a desired accuracy. We show that a simple RO model based on Berger plate theory, can be conveniently used for analysis of electrostatically actuated plates with low initial curvature and small thickness to electrostatic gap ratio.

Nonlinear modelling and analysis of thin piezoelectric plates: Buckling and post-buckling behaviour

In the present paper we discuss the stability and the post-buckling behaviour of thin piezoelastic plates. The first part of the paper is concerned with the modelling of such plates. We discuss the constitutive modelling, starting with the three-dimensional constitutive relations within Voigt\'s linearized theory of piezoelasticity. Assuming a plane state of stress and a linear distribution of the strains with respect to the thickness of the thin plate, two-dimensional constitutive relations are obtained. The specific form of the linear thickness distribution of the strain is first derived within a fully geometrically nonlinear formulation, for which a Finite Element implementation is introduced. Then, a simplified theory based on the von Karman and Tsien kinematic assumption and the Berger approximation is introduced for simply supported plates with polygonal planform. The governing equations of this theory are solved using a Galerkin procedure and cast into a non-dimensional formulation. In the second part of the paper we discuss the stability and the post-buckling behaviour for single term and multi term solutions of the non-dimensional equations. Finally, numerical results are presented using the Finite Element implementation for the fully geometrically nonlinear theory. The results from the simplified von Karman and Tsien theory are then verified by a comparison with the numerical solutions.

Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping

The dynamics of a (nonlinear) Berger plate in the absence of rotational inertia are considered with inhomogeneous boundary conditions. In our analysis, we consider boundary damping in two scenarios: (i) free plate boundary conditions, or (ii) hinged-type boundary conditions. In either situation, the nonlinearity gives rise to complicating boundary terms. In the case of free boundary conditions we show that well-posedness of finite-energy solutions can be obtained via highly nonlinear boundary dissipation. Additionally, we show the existence of a compact global attractor for the dynamics in the presence of hinged-type boundary dissipation (assuming a geometric condition on the entire boundary (Lagnese, 1989)). To obtain the existence of the attractor we explicitly construct the absorbing set for the dynamics by employing energy methods that: (i) exploit the structure of the Berger nonlinearity, and (ii) utilize sharp trace results for the Euler–Bernoulli plate in Lasiecka and Triggiani (1993).We provide a parallel commentary (from a mathematical point of view) to the discussion of modeling with Berger versus von Karman nonlinearities: to wit, we describe the derivation of each nonlinear dynamics and a discussion of the validity of the Berger approximation. We believe this discussion to be of broad value across engineering and applied mathematics communities.

Post-Buckling of Piezoelectric Thin Plates

In the present paper, the geometrically nonlinear behavior of piezoelastic thin plates is studied. First, the governing equations for the electromechanically coupled problem are derived based on the von Karman–Tsien kinematic assumption. Here, the Berger approximation is extended to the coupled piezoelastic problem. The general equations are then reduced to a single nonlinear partial differential equation for the special case of simply supported polygonal edges. The nonlinear equations are approximated by using a problem-oriented Ritz Ansatz in combination with a Galerkin procedure. Based on the resulting equations the buckling and post-buckling behavior of a polygonal simply supported plate is studied in a nondimensional form, where the special geometry of the polygonal plate enters via the eigenvalues of a Helmholtz problem with Dirichlet boundary conditions. Single term as well as multi-term solutions are discussed including the effects of piezoelectric actuation and transverse force loadings upon the solution. Novel results concerning the buckling, snap through and snap buckling behavior are presented.

Solution for large amplitude vibrations of circular plates via modified Berger's approximation

Complex mathematical formulations for the geometrically nonlinear analysis of structural members have been presented by many researchers using the von-Karman strain-displacement relations. Berger proposed a simplifying approximation to take care of the geometric nonlinearity by neglecting the second strain invariant. Although it is a reasonable approximation, it poses problems when applied to circular plates. This is mainly due to the coupling between the radial and circumferential inplane strains, by the radial displacement. In this article, a modification to the Berger's approximation, applicable to the circular plates, is proposed. The main motivation of the present study is to examine how the modified Berger's approximation works in the case of the large amplitude vibrations of circular plates. It is shown that the modified Berger's approximation gives consistently accurate results, for both the simply supported and clamped circular plates, when compared to the classical results.

A decoupled modal form Duffing equation for an orthotropic laminate in a clamped boundary condition with thermal diffusion

The non-linear thermo-elasticity response of an orthotropic laminate in a clamped boundary condition is analyzed by means of the method of weighted residuals for its behaviors subject to thermal and mechanical loading. The equation of motion for the laminate's deflection is obtained in the form of a decoupled modal form Duffing equation, without Berger's approximation. The thermal field, with both the in-plane and transverse temperature variations, is incorporated in both a steady-state and transient state. The formulation indicates that a transverse thermal field with a temperature rise produces a load onto the laminate, while the in-plane temperature variation affects the system frequency and stability. We demonstrate the numerical results of the modal response of an isotropic laminate for the thermal buckling and vibration.

Moderately large flexural vibrations of sandwich plates with thick layers

AbstractIn this paper moderately large amplitude vibrations of a polygonally shaped composite plate with thick layers are analyzed. Three homogeneous and isotropic layers with a common Poisson's ratio are perfectly bonded, and their arbitrary thickness and material properties are symmetrically disposed about the middle plane. Mindlin‐Reissner kinematic assumptions are implemented layerwise, and as such model both the global and local response. Geometric nonlinear effects arising from longitudinally constrained supports are taken into account by Berger's approximation of nonlinear strain‐displacement relations. The shear stress continuity across the interfaces is prescribed according to Hooke's law, and subsequently, a correspondence of this higher order problem to the simpler case of a homogenized shear‐deformable nonlinear plate with effective stiffness and hard hinged boundary conditions is found.

Hypar Shell on Pasternak Foundation

This paper deals with nonlinear static and dynamic analysis of hyperbolic paraboloid shells on Pasternak foundations. The governing differential equation of a hypar shell on a Pasternak foundation is reduced to that of a plate on a double elastic foundation. Moduli of double elastic foundations are presented in terms of a geometric parameter of the hypar shell and the two parameters of the Pasternak model. Nonlinear differential equations are obtained in decoupled form, using Berger's approximation. In the first part, the effects of the geometry of the shell and foundation moduli on load‐deflection characteristics for the simply supported edges are investigated. Effects of shell geometry and foundation modulus are found to be significant, but changes in shear modulus are not. In the second part, the effects of geometric and foundation parameters on time period‐amplitude relationship are studied. Application of the proposed nonlinear static and dynamic analysis method may lead to a better, more efficient d...

Multi-modal approach for large natural flexural vibrations of thermally stressed plates

By means of Berger's approximation, suitable for plates with immovable edges, the geometrically nonlinear problem is considerably simplified. Thermally loaded plates with polygonal planform under hard-hinged support conditions are considered, taking into account the effect of shear in transverse isotropy. The class of symmetric vibrations about the flat plate position is represented by a homogeneous and coupled set of Duffing-oscillators as a result of a multi-mode expansion. A unifying non-dimensional closed-form solution for the corresponding nonlinear natural vibration periods is given, which is independent of the special planform. The individual shape of the plate enters the transformation into real time through the linear natural frequencies, or, equivalently, through the linear eigen-values of an effectively prestressed membrane of the same planform.

Shallow spherical shell on Pasternak foundation subjected to elevated temperature

Nonlinear static behaviour of clamped shallow spherical shell at elevated temperature and resting on Pasternak type elastic foundation is examined, using Berger's approximation. Variation of non-dimensional central deflection with foundation parameters for a given thermal loading is investigated.

A study of nonlinear vibration of skew plates with attention to shear and rotatory inertia

The problem of nonlinear dynamic behaviour of skew plates is treated in this paper wherein consideration is given to the effects of transverse shear deformation and rotatory inertia. The formulation incorporates the well-known Berger approximation, earlier shown by the author to hold good for the analysis of skew plates. Approximate solutions to the nonlinear governing equations are obtained on the basis of an assumed single mode for w. Galerkin and numerical Runge-Kutta methods are used in the solution procedure. The single mode assumption, even though restrictive in nature, gives acceptable results for the fundamental mode of vibration. The governing equations and numerical results reported here are in agreement with those in the literature for the special cases of rectangular plates. Effects of geometric nonlinearity, transverse shear deformation and rotatory inertia are studied for isotropic and orthotropic skew plates of various aspect ratios, skew angles and thickness-to-length ratios.

Nonlinear vibration of orthotropic elliptical plates with attention to shear and rotatory inertia

A nonlinear vibration theory which includes the effects of transverse shear deformation and rotatory inertia is formulated in this paper for orthotropic elliptical plates using the well-known Berger approximation. With the aid of these equations solutions are obtained for isotropic and composite plates by the Galerkin technique and numerical Runge-Kutta procedure. Present results are as good as those obtained by using more accurate theories. It is interesting to note that the present approach to the solution of the title problem is a simplification which results in a substantial saving in the analytical and computational efforts.

Transverse shear and rotatory inertia effects on nonlinear vibration of orthotropic circular plates

In this paper a nonlinear vibration theory which includes the effects of transverse shear deformation and rotatory inertia is formulated for orthotropic circular plates using the Berger approximation. Solutions to the governing equations are obtained on the basis of a single-mode approach by use of Galerkin's method and numerical Runge-Kutta procedure. Results indicate significant influence of these effects on the nonlinear vibration behavior of orthotropic circular plates. Present results are in good agreement with those available in the literature for all special cases. It is observed that there is a substantial saving in the analytical and computational efforts in using the Berger approach and that this approach yields reasonably good results comparable with those obtained by the corresponding von Kármán type theory.

The dynamic analysis of circular plates and shallow spherical shells

In the investigation reported here an attempt has been made to study the influence of Berger's approximation on the non-linear transient response of circular plates and shallow spherical shells. The governing equations of motion obtained from Berger's approximation are solved by using the rapidly converging Chebyshev series spacewise and the Houbolt scheme for integration in the time domain. Results calculated when using Berger's approximation are compared with exact results. It is shown that Berger's method yields very accurate values for plates and shells under transient loading, in the case of immovable edge conditions.

On the berger approximation: A critical re-examination

Since the first paper by Berger some two decades ago, the simplification known as the Berger approximation has been invoked by the authors of several score papers in spite of the fact that no rational mechanical basis for the approximation could be found. Many recent papers have raised doubts on its applicability. In this paper, using certain well known results from the two dimensional theory of elasticity, a plausible explanation for the origin of the Berger method is suggested. The arguments can be developed further to show that other specious Berger-like approximations can be developed, all of them leading to uncoupled non-linear equations yielding different overall results. Further, it is shown that such methods fail to predict the non-linear behaviour with respect to important parameters and that whatever accuracy is obtained in the solution of a particular problem can at best be attributed to fortuity.

Non-linear flexural vibrations of anisotropic skew plates

The large amplitude free flexural vibrations of thin, elastic anisotropic skew plates are studied by using the von Karman field equations in which the governing non-linear dynamic equations are derived in terms of the stress function and the lateral displacement. Clamped boundary conditions are chosen and the in-plane edge conditions considered are either immovable or movable. Solutions are obtained by the Galerkin method on the basis of a one-term assumed vibration mode. The degree of non-linearity is obtained as a function of skew angle, aspect ratio and types of orthotropy. The results, on specializing for an isotropic skew plate and an orthotropic rectangular plate, agree well with those found in the literature. The use of the Berger approximation to study a skew plate with in-plane immovable edges is shown to lead to errors of both a quantitative and qualitative nature.

Effects of large amplitude, shear and rotatory inertia on vibration of rectangular plates

In this paper, the governing equations applicable for the large amplitude free, flexural vibration of orthotropic rectangular plates are formulated in terms of the displacement components u, v and w. The formulation and the solutions presented for the cases of isotropic and orthotropic simply supported rectangular plates incorporate the effects of the transverse shear deformation and the rotatory inertia on the large amplitude vibration behaviour. The influences of shear and rotatory inertia are significant in the case of moderately thick plates undergoing large amplitude vibration. The method suggested here does not require the use of the Berger approximation for plates with immovable in-plane edge conditions. A comparison with the results available in the literature indicates the inadequacy of the approach adopted earlier for the study of the large amplitude free flexural vibration of orthotropic rectangular plates.

Vibration of plates considering shear and rotatory inertia

Introduction T HE large-amplitude vibrations of plates and shells have been receiving considerable attention in the literature in recent years. In particular, rectangular plates have received the attention of many investigators. The results reported so far do not, in many cases, seem to incorporate the effects of transverse shear and rotatory inertia, Wu and Vinson investigated the effects of transverse shear and rotatory inertia on the large-amplitude vibration of isotropic, transversely isotropic, and orthotropic rectangular plates, based on assumed expressions not only for the mode shape w but also for the rotation components a. and j3. The purpose here is to suggest an improvement to the solutions given by Wu and Vinson by altogether eliminating a and 6 from the governing equations, thereby avoiding the necessity to proceed on assumed approximations for a. and /?. The equations developed here are based on Berger approximation. Based on Galerkin's method and an assumption of only mode shape w, the effects of geometric nonlinearity, transverse shear deformation, and rotatory inertia are reported for certain cases of isotropic and orthotropic rectangular plates clamped along the boundaries,