Showing papers on "Timoshenko beam theory published in 2004"

A General Method for the Modeling of Spindle-Bearing Systems

Abstract: In this paper we outline a general method that can be used to model spindle assembly, which consists of spindle shaft, angular contact ball bearings and housing. The spindle shaft and housing are modeled as Timoshenko's beam by including the centrifugal force and gyroscopic effects. The bearing is modeled as a standard nonlinear finite element based on Jones' bearing model that includes the centrifugal force and gyroscopic effects from the rolling elements of bearings. By applying cutting forces to the spindle for a given preload, the stiffness of the bearings, contact forces on bearing balls, natural frequencies, time history response, and frequency response functions of the spindle assembly can be evaluated. In the paper we provide details of the mathematical model supported by experimental results obtained from an instrumented test spindle.

Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach

Abstract: The purpose of this paper is to show how the Timoshenko beam can be fruitfully described within the framework of distributed port Hamiltonian (dpH) systems so that rather simple and elegant considerations can be drawn regarding both the modeling and control of this mechanical system. After the dpH model of the beam is introduced, the control problem is discussed. In particular, it is shown how control approaches already presented in the literature can be unified, and a new control methodology is presented and discussed. This control methodology relies on the generalization to infinite dimensions of the concept of structural invariant (Casimir function) and on the extension to distributed systems of the so-called control by interconnection methodology. In this way, finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy, i.e., by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipative effect is introduced.

The effects of shear on delamination in layered materials

Abstract: The effect of transverse shear on delamination in layered, isotropic, linear-elastic materials has been determined. In contrast to the effects of an axial load or a bending moment on the energy-release rate for delamination, the effects of shear depend on the details of the deformation in the crack-tip region. It therefore does not appear to be possible to deduce rigorous expressions for the shear component of the energy-release rate based on steady-state energy arguments or on any type of modified beam theory. The expressions for the shear component of the energy-release rate presented in this work have been obtained using finite-element approaches. By combining these results with earlier expressions for the bending-moment and axial-force components of the energy-release rates, the framework for analyzing delamination in this type of geometry has been extended to the completely general case of any arbitrary loading. The relationship between the effects of shear and other fracture phenomena such as crack-tip rotations, elastic foundations and cohesive zones are discussed in the final sections of this paper.

Timoshenko-beam effects on transverse wave propagation in carbon nanotubes

Abstract: This paper studies effects of rotary inertia and shear deformation on transverse wave propagation in individual carbon nanotubes (CNTs) within terahertz range. Detailed results are demonstrated for transverse wave speeds of doublewall CNTs, based on Timoshenko-beam model and Euler-beam model, respectively. The present models predict some terahertz critical frequencies at which the number of wave speeds changes. The effects of rotary inertia and shear deformation are negligible and transverse wave propagation can be described satisfactorily by the existing single-Euler-beam model only when the frequency is far below the lowest critical frequency. When the frequency is below but close to the lowest critical frequency, rotary inertia and shear deformation come to significantly affect the wave speed. Furthermore, when the frequency is higher than the lowest critical frequency, more than one wave speed exists and transverse waves of given frequency could propagate at various speeds that are considerably different than the speed predicted by the single-Euler-beam model. In particular, rotary inertia and shear deformation have a significant effect on both the wave speeds and the critical frequencies especially for CNTs of larger radii. Hence, terahertz transverse wave propagation in CNTs should be better modeled by Timoshenko-beam model, instead of Euler-beam model.

Integrated Dynamic Thermo-Mechanical Modeling of High Speed Spindles, Part 1: Model Development

Abstract: This paper presents a comprehensive integrated thermo-dynamic model for various high speed spindles. The entire model consists of fully coupled three sub-models: bearing, spindle dynamic and thermal models. Using a finite element approach, a new thermal model has been generated, which can describe complex structures of high-speed motorized spindles, and can predict more accurate temperature distributions. The spindle dynamic model is constructed using finite elements based on Timoshenko beam theory and has been improved by considering shear deformation, material and bearing damping, and the spindle/tool-holder interface. Using the new thermo-dynamic model, more general and detailed bearing configurations can be modeled through a systematic coupling procedure. The thermal expansions of the shaft, housing and bearings are calculated based on predicted temperature distributions and are used to update the bearing preloads depending on the operating conditions, which are again used to update the thermal model. Therefore, the model is fully integrated and can provide solutions in terms of all the design parameters and operating conditions.

Elasticity Solutions Versus Asymptotic Sectional Analysis of Homogeneous, Isotropic, Prismatic Beams

Abstract: The original three-dimensional elasticity problem of isotropic prismatic beams has been solved analytically by the variational asymptotic method (VAM). The resulting classical model (Eiiler-Bernoulli-like) is the same as the superposition of elasticity solutions of extension, Saint-Venant torsion, and pure bending in two orthogonal directions. The resulting refined model (Timoshenko-like) is the same as the superposition of elasticity solutions of extension, Saint-Venant torsion, and both bending and transverse shear in two orthogonal directions. The fact that the VAM can reproduce results from the theory of elasticity proves that two-dimensioned finite-element-based cross-sectional analyses using the VAM, such as the variational asymptotic beam sectional analysis (VABS), have a solid mathematical foundation. One is thus able to reproduce numerically with VABS the same results for this problem as one obtains from three-dimensional elasticity, but with orders of magnitude less computational cost relative to three-dimensional finite elements.

Piezoelectric passive distributed controllers for beam flexural vibrations

Abstract: Recent technological developments have made available efficient bender transducers based on the piezoelectric effect. In this paper an electrical circuit analog to the Timoshenko beam is synthesized using a Lagrangian method and by paralleling capacitive flux linkages to rotation and transverse displacement. A Piezo-ElectroMechanical (PEM) beam is conceived by uniformly distributing piezoelectric transducers on a beam and interconnecting their electric terminals via the found analog circuit, completed with suitable resistors. The high performance features of the synthesized novel circuit include the following. (i) The circuit topology is extremely reduced, the used components are all but one two-terminal elements, and the only two-port network needed is an ideal transformer. (ii) One and the same dissipative circuit ensures a multiresonance coupling with the vibrating beam and the optimal electrical dissipation of mechanical vibrations energy. (iii) For a prototype of a PEM beam, the design of the analog ...

Estimating deflection of a simple beam model using fiber optic bragg-grating sensors

Abstract: In this paper, we develop a method to estimate the bridge deflection using fiber optic Bragg-grating (FBG) strain sensors. For most structural evaluation of bridge integrity, it is very important to measure the geometric profile, which is a major factor representing the global behavior of civil structures, especially bridges. In the past, because of the lack of appropriate methods to measure the deflection curve of bridges on site, the measurement of deflection has been restricted to just a few discrete points along the bridge, and the measuring points have been limited to the locations installed with displacement transducers. However, by applying classical beam theory, a formula is rearranged to estimate the continuously deflected profile by using strains measured directly from several points. In addition, FBG strain sensors, which are electromagnetic, noise-free and multipoint measurable, are employed to obtain flexural strains more accurately and stably. The regression analysis is performed to obtain a strain function from the measured strain data. Finally, the deflection curve can be estimated by applying the strain function to the formula. An experimental test has also been carried out to verify the developed method.

Elastic buckling of a porous beam

Abstract: The work deals with the problem a straight beam of a rectangular cross-section pivoted at both ends and loaded with a lengthwise compressive force. The beam is made of an isotropic porous material. Its properties vary through thickness of the beam. The modulus of elasticity is minimal on the beam axis and assumes maximum values at its top and bottom surfaces. The principle of stationarity of the total potential energy enables one to define a system of differential equations that govern the beam stability. The system is analytically solved, which leads to an explicit expression for the critical load of the compressed beam. Results of the solution are verified on an example beam by means of the Finite Element Method (COSMOS).

A Lyapunov-Based Piezoelectric Controller for Flexible Cartesian Robot Manipulators

Abstract: A Lyapunov-based control strategy is proposed for the regulation of a Cartesian robot manipulator, which is modeled as a flexible cantilever beam with a translational base support. The beam (arm) cross-sectional area is assumed to be uniform and Euler-Bernoulli beam theory assumptions are considered. Moreover, two types of damping mechanisms; namely viscous and structural dampings, are considered for the arm material properties. The arm base motion is controlled utilizing a linear actuator, while a piezoelectric (PZT) patch actuator is bonded on the surface of the flexible beam for suppressing residual beam vibrations. The equations of motion for the system are obtained using Hamilton's principle, which are based on the original infinite dimensional distributed system. Utilizing the Lyapunov method, the control force acting on the linear actuator and control voltage for the PZT actuator are designed such that the base is regulated to a desired set-point and the exponential stability of the system is attained. Depending on the composition of the controller, some favorable features appear such as elimination of control spillovers, controller convergence at finite time, suppression of residual oscillations and simplicity of the control implementation. The feasibility of the controller is validated through both numerical simulations and experimental testing.

Experimental investigation of a small-scale bridge model under a moving mass

Abstract: The dynamic response of a small-scale bridge model under a moving mass is investigated. The analysis is based on the continuous Euler–Bernoulli beam theory. By expanding the unknown structural response in a series of the beam eigenfunctions, the given problem is reduced to the solution of a set of second order linear differential equations with time varying coefficients. The analytical solution is validated through a series of experiments. A small-scale model is designed to satisfy both static and dynamic similitude with a selected prototype bridge structure, and a set of necessary similitude conditions for the given problem is provided. Attention is paid, in particular, to satisfaction of the mass similitude requirement, often constituting one of the main difficulties in the design of small-scale dynamic models. It is shown that experimental results are in good agreement with theoretical predictions, thus validating the analytical procedure.

Deflections of highly inflated fabric tubes

Abstract: Inflatable beams made of modern textile materials with important mechanical characteristics can be inflated at high pressure. The aim of the paper is to present experimental, analytical and numerical results on the deflections of highly inflated fabric tubes submitted to bending loads. Experiments are displayed and we show that tube behaviour looks like that of inflatable panels (Thin-Walled Struct. 40 (2002) 523–536). Equilibrium equations are once again written in the deformed state to take into account the geometrical stiffness and the following forces. The influence of the shear stress cannot be neglected and Timoshenko’s beam theory is used. A new inflatable tube theory is established and simple analytical formulas are given for a cantilever-inflated tube. Comparisons between analytical and experimental results are shown. A new inflatable finite tube element is constructed by use of algebraic operations, because the compliance matrix of the cantilever beam is not symmetric. Comparisons between experimental, analytical and numerical results prove the accuracy of this beam theory and on this new finite element for solving problems on the deflections of highly inflated tubes.

Finite element of slender beams in finite transformations: a geometrically exact approach

Abstract: This article is devoted to the modelling of thin beams undergoing finite deformations essentially due to bending and torsion and to their numerical resolution by the finite element method. The solution proposed here differs from the approaches usually implemented to treat thin beams, as it can be qualified as ‘geometrically exact’. Two numerical models are proposed. The first one is a non-linear Euler–Bernoulli model while the second one is a non-linear Rayleigh model. The finite element method is tested on several numerical examples in statics and dynamics, and validated through comparison with analytical solutions, experimental observations and the geometrically exact approach of the Reissner beam theory initiated by Simo. The numerical result shows that this approach is a good alternative to the modelling of non-linear beams, especially in statics. Copyright © 2003 John Wiley & Sons, Ltd.