Showing papers on "Timoshenko beam theory published in 2003"

An analytical model for support loss in micromachined beam resonators with in-plane flexural vibrations

Abstract: This paper presents an analytical model for support loss in clamped–free (C–F) and clamped–clamped (C–C) micromachined beam resonators with in-plane flexural vibrations. In this model, the flexural vibration of a beam resonator is described using the beam theory. An elastic wave excited by the shear stress of the beam resonator and propagating in the support structure is described through the 2D elastic wave theory, with the assumption that the beam thickness (h) is much smaller than the transverse elastic wavelength (λT). Through the combination of these two theories and the Fourier transform, closed-form expressions for support loss in C–F and C–C beam resonators are obtained. Specifically, closed-form expression for the support loss in a C–C beam resonator is derived for the first time. The model suggests lower support quality factor (Qsupport) for higher order resonant modes compared to the fundamental mode of a beam resonator. Through comparison with experimental data, the validity of the presented analytical model is demonstrated. © 2003 Elsevier B.V. All rights reserved.

Uniform stabilization for the timoshenko beam by a locally distributed damping

Abstract: We study the uniform stabilization of a Timoshenko beam by one control force. We prove that under, one locally distributed damping, the exponential stability for this system is assured if and only if the wave speeds are the same.

A mixed finite element method for beam and frame problems

Abstract: In this work we consider solutions for the Euler- Bernoulli and Timoshenko theories of beams in which material behavior may be elastic or inelastic. The formu- lation relies on the integration of the local constitutive equation over the beam cross section to develop the relations for beam resultants. For this case we include axial, bending and shear effects. This permits consider- ation in a direct manner of elastic and inelastic behavior with or without shear deformation. A finite element solution method is presented from a three-field variational form based on an extension of the Hu-Washizu principle to permit inelastic material be- havior. The approximation for beams uses equilibrium satisfying axial force and bending moments in each ele- ment combined with discontinuous strain approximations. Shear forces are computed as derivative of bending mo- ment and, thus, also satisfy equilibrium. For quasi-static applications no interpolation is needed for the displace- ment fields, these are merely expressed in terms of nodal values. The development results in a straight forward, variationally consistent formulation which shares all the properties of so-called flexibility methods. Moreover, the approach leads to a shear deformable formulation which is free of locking effects - identical to the behavior of flexibility based elements. The advantages of the approach are illustrated with a few numerical examples.

Nonlinear generalized beam theory for cold-formed steel members

Abstract: A geometrically nonlinear Generalized Beam Theory (GBT) is formulated and its application leads to a system of equilibrium equations which are valid in the large deformation range but still retain and take advantage of the unique GBT mode decomposition feature. The proposed GBT formulation, for the elastic post-buckling analysis of isotropic thin-walled members, is able to handle various types of loading and arbitrary initial geometrical imperfections and, in particular, it can be used to perform "exact" or "approximate" (i.e., including only a few deformation modes) analyses. Concerning the solution of the system of GBT nonlinear equilibrium equations, the finite element method (FEM) constitutes the most efficient and versatile numerical technique and, thus, a beam FE is specifically developed for this purpose. The FEM implementation of the GBT post-buckling formulation is reported in some detail and then employed to obtain numerical results, which validate and illustrate the application and capabilities of the theory.

Hydroelastic response of a barge to impulsive and non-impulsive wave loads

Abstract: Global hydroelastic model for ship-type bodies is presented and validated both in frequency and time domain. Since we are interested in the global response, the structural part of the body is modelled by the so called non-uniform "Timoshenko beam model" using the finite element method, while the hydrodynamic part is modelled by the classical 3D Boundary Integral Equation (BIE) technique. The coupling is performed using the modal approach, which means that the final body response (motions/deformations) is decomposed into a series of the structural "dry" modes. An additional hydodynamic boundary value problem for each mode is created and soloved by 3D BIE method. The numerical model is validated through comparisons with data from especially dedicated model tests on an elastic barge. The barge was modelled with 12 pontoons connected with two elastic plates.

Dynamic modeling and analysis of a bimodal ultrasonic motor

Abstract: A dynamic model that includes four subsystems is developed to analyze the fundamental characteristics of a bimodal ultrasonic motor. The first subsystem is the driving circuit designed for the motor to achieve bidirectional motion. The stator is modeled as a Timoshenko beam, and the assumed mode energy method is used to obtain the dynamic equations. The normal interface force is represented by an elastic spring existing in between the tip of the stator and the moving platform. The interface forces are coupled into the dynamic formulations of the stator and the moving platform. The behavior of the force transmission between the stator and the moving platform are analyzed using the developed model. Transient and steady-state responses of the system are obtained by numerical simulation, and the results are validated by experiments. Furthermore, the existing of a nonlinear deadzone is predicted analytically, and the causes of this nonlinearity are clarified.

Electromechanical coupling correction for piezoelectric layered beams

Abstract: This paper deals with the bending of layered piezoelectric beams (multimorphs) subjected to arbitrary electrical and mechanical loading. Weinberg (1999) obtained a closed-form solution to this problem using Euler-Bernoulli beam theory and integrated equilibrium equations. In his analysis, Weinberg assumes that the electric field is constant through the thickness of the piezoelectric layers. This approximation is valid for materials with small electromechanical coupling (EMC) coefficients. In this paper, we relax this constraint and obtain a solution which accounts for the effect of strain on the electric field in the layers. We find that Weinberg's solution can be extended to arbitrary EMC with a simple correction to the moment of inertia I of the piezoelectric layers. The EMC correction amounts to replacing I with (1+/spl xi/)I, where /spl xi/ is the square of the expedient coupling coefficient. The error in beam curvature introduced by neglecting the effect of EMC is shown to be proportional to /spl xi/. This effect can be quite significant for modern piezoelectric materials which tend to have large EMC coefficients. The formulation is applied to three example cases: a cantilever unimorph, an asymmetric bimorph and a three-layer multimorph with an elastic core. The theoretical predictions for the last two examples are compared to simulations using the finite-element method (FEM) and found to be in excellent agreement.

On a geometrically exact curved/twisted beam theory under rigid cross-section assumption

Abstract: A geometrically exact curved/ twisted beam theory, that assumes that the beam cross-section remains rigid, is re-examined and extended using orthonormal frames of reference starting from a 3-D beam theory. The relevant engineering strain measures with an initial curvature correction term at any material point on the current beam cross-section, that are conjugate to the first Piola-Kirchhoff stresses, are obtained through the deformation gradient tensor of the current beam configuration relative to the initially curved beam configuration. The stress resultant and couple are defined in the classical sense and the reduced strains are obtained from the three-dimensional beam model, which are the same as obtained from the reduced differential equations of motion. The reduced differential equations of motion are also re-examined for the initially curved/twisted beams. The corresponding equations of motion include additional inertia terms as compared to previous studies. The linear and linearized nonlinear constitutive relations with couplings are considered for the engineering strain and stress conjugate pair at the three-dimensional beam level. The cross-section elasticity constants corresponding to the reduced constitutive relations are obtained with the initial curvature correction term. Along with the beam theory, some basic concepts associated with finite rotations are also summarized in a manner that is easy to understand.

Dynamic analysis of gradient elastic flexural beams

Abstract: Gradient elastic flexural beams are dynamically analysed by analytic means. The governing equation of flexural beam motion is obtained by combining the Bernoulli-Euler beam theory and the simple gradient elasticity theory due to Aifantis. All possible boundary conditions (classical and non-classical or gradient type) are obtained with the aid of a variational statement. A wave propagation analysis reveals the existence of wave dispersion in gradient elastic beams. Free vibrations of gradient elastic beams are analysed and natural frequencies and modal shapes are obtained. Forced vibrations of these beams are also analysed with the aid of the Laplace transform with respect to time and their response to loads with any time variation is obtained. Numerical examples are presented for both free and forced vibrations of a simply supported and a cantilever beam, respectively, in order to assess the gradient effect on the natural frequencies, modal shapes and beam response.

Stabilization of Timoshenko Beam by Means of Pointwise Controls

Abstract: We intend to conduct a fairly complete study on Timoshenko beams with pointwise feedback controls and seek to obtain information about the eigenvalues, eigenfunctions, Riesz-Basis-Property, spectrum-determined-growth-condition, energy decay rate and various stabilities for the beams. One major difficulty of the present problem is the non-simplicity of the eigenvalues. In fact, we shall indicate in this paper situations where the multiplicity of the eigenvalues is at least two. We build all the above-mentioned results from an effective asymptotic analysis on both the eigenvalues and the eigenfunctions, and conclude with the Riesz-Basis-Property and the spectrum-determined-growth-condition. Finally, these results are used to examine the stability effects on the system by the location of the pointwise control relative to the length of the whole beam.

Mitigation of residual film stress deformation in multilayer microelectromechanical systems cantilever devices

Abstract: An approach to compensate for the residual thin film stress deformation of multilayer microelectromechanical systems (MEMS) devices is presented based upon analytical and numerical modeling and in-process thin film characterization. Thermal and intrinsic deposition stresses can lead to the warping of released MEMS structures. This detrimental phenomenon in many cases can prevent proper device operation. Ellispsometric and laser wafer bow measurements yield thickness and film stress values that are used to update the deflection model during device fabrication, allowing for the compensation of the fabrication process variability. The derivations of linear and nonlinear residual film stress induced deflection models are presented. These models are based upon Bernoulli–Euler beam theory and are thus restricted to the associated geometric constraints. The models are initially validated by comparison with surface micro-machined sol–gel lead–zirconate–titanate cantilever structures; with initial experimental res...

Analysis of the Air-damping Effect on a Micromachined Beam Resonator

Abstract: An analysis of the air-damping effect on the frequency response of a micromachined beam resonator is presented. The motion of the beam is analyzed based on the linear elastic beam theory. The air d...

A formulation and implementation of geometrically exact curved beam elements incorporating finite strains and finite rotations

Abstract: This paper deals with the formulation and implementation of the curved beam elements based on the geometrically exact curved/twisted beam theory assuming that the beam cross-section remains rigid. The summarized beam theory is used for the slender beams or rods. Along with the beam theory, some basic concepts associated with finite rotations and their parametrizations are briefly summarized. In terms of a non-vectorial parametrization of finite rotations under spatial descriptions, a formulation is given for the virtual work equations that leads to the load residual and tangential stiffness operators. Taking the advantage of the simplicity in formulation and clear classical meanings of both rotations and moments, the non-vectorial parametrization is applied to implement a four-noded 3-D curved beam element, in which the compound rotation is represented by the unit quaternion and the incremental rotation is parametrized using the incremental rotation vector. Only static problems are considered. Conventional Lagrangian interpolation functions are adopted to approximate both the reference curve and incremental rotation of the deformed beam. Reduced integration is used to overcome locking problems. The finite element equations are developed for static structural analyses, including deformations, stress resultants/couples, and linearized/nonlinear bifurcation buckling, as well as post-buckling analyses of arches subjected to different types of loads, such as self-weight, snow, and pressure (wind) loads. Several examples are used to test the formulation and its Fortran implementation.

Diffusion-induced beam bending in hydrogen sensors

Abstract: The diffusion-induced bending of both single-layer and bilayer beam structure is analyzed by using linear elastic beam theory and the Moutier theorem. A closed form solution of the radius of curvature due to diffusion is obtained. For the single-layer beam structure, the radius of curvature is inversely proportional to the bending moment created by nonuniform concentration distribution. For the bilayer beam structure, the curvature is a linear function of the mismatch strain between the two layers and the bending moment introduced by diffusion. The mismatch strain depends on the concentration and the partial molar volume of the diffusing component in both layers. Application to microelectromechanical systems hydrogen sensors with a layer of Pd is shown.

Nonlinear Vibration Analysis of Timoshenko Beams Using the Differential Quadrature Method

Abstract: This paper addresses the large-amplitude free vibration of simplysupported Timoshenko beams with immovable ends. Various nonlineareffects are taken into account in the present formulation and thegoverning differential equations are established based on theHamilton Principle. The differential quadrature method (DQM) isemployed to solve the nonlinear differential equations. Theeffects of nonlinear terms on the frequency of the Timoshenkobeams are discussed in detail. Comparison is made with otheravailable results of the Bernoulli–Euler beams and Timoshenkobeams. It is concluded that the nonlinear term of the axial forceis the dominant factor in the nonlinear vibration of Timoshenkobeams and the nonlinear shear deformation term cannot be neglectedfor short beams, especially for large-amplitude vibrations.

Seismic Collapse Analysis of Reinforced Concrete Framed Structures Using the Finite Element Method

Abstract: A new finite element code using the Adaptively Shifted Integration (ASI) technique with a linear Timoshenko beam element is applied to the seismic collapse analysis of reinforced concrete (RC) framed structures. This technique can express member fracture as a plastic hinge located at either end of an element with simultaneous release of the resultant forces in the element. Contact between members is also considered in order to obtain results that agree more closely with actual behavior, such as intermediate-layer failure. By using the proposed code, sufficiently reliable solutions have been obtained, and the results reveal that this code can be used in the numerical estimation of the seismic design of RC framed structures. Copyright © 2003 John Wiley & Sons, Ltd.