Uniform Euler-Bernoulli Beam Research Articles

Study on modified differential transform method for free vibration analysis of uniform Euler-Bernoulli beam

A simulation method called modified differential transform is studied to solve the free vibration problems of uniform Euler-Bernoulli beam. First of all, the modified differential transform method is derived. Secondly, the modified differential transformation is applied to uniform Euler-Bernoulli beam free-free vibration. And then a set of differential equations are established. Through algebraic operations on these equations, we can get any natural frequency and normalized mode shape. Thirdly, the FEM is applied to obtain the numerical solutions. Finally, mode experimental method (MEM) is conducted to obtain experimental data for analysis by signal processing with LMS Test.lab Vibration testing and analysis system. Experimental data and simulation results are illustrated to be in comparison with the analytical solutions. The results show that the modified differential transform method can achieve good results in predicting the solution of such problems.

Numerical Evaluation of High-Order Modes for Stepped Beam

The numerical evaluation of high-order modes of a uniform Euler–Bernoulli beam has been studied by reformatting the classical expression of mode shapes. That method, however, is inapplicable to a stepped beam due to the nonuniform expressions of the mode shape for each beam segment. Given that concern, this study develops an alternative method for the numerical evaluation of high-order modes for stepped beams. This method effectively expands the space of high-order modal solutions by introducing local coordinate systems to replace the conventional global coordinate system. This set of local coordinate systems can significantly simplify the frequency determinant of vibration equations of a stepped beam, in turn, largely eliminating numerical round-off errors and conducive to high-order mode evaluation. The efficacy of the proposed scheme is validated using various models of Euler–Bernoulli stepped beams. The principle of the method has the potential for extension to other types of Euler–Bernoulli beams with discontinuities in material and geometry. (The Matlab code for the numerical evaluation of high-order modes for stepped beams can be provided by the corresponding author upon request.)

Exponential stability of uniform Euler–Bernoulli beams with non-collocated boundary controllers

We study the stability of a robot system composed of two Euler–Bernoulli beams with non-collocated controllers. By the detailed spectral analysis, we prove that the asymptotical spectra of the system are distributed in the complex left-half plane and there is a sequence of the generalized eigenfunctions that forms a Riesz basis in the energy space. Since there exist at most finitely many spectral points of the system in the right half-plane, to obtain the exponential stability, we show that one can choose suitable feedback gains such that all eigenvalues of the system are located in the left half-plane. Hence the Riesz basis property ensures that the system is exponentially stable. Finally we give some simulation for spectra of the system.

Exploring isospectral cantilever beams using electromagnetism inspired optimization technique

Isospectral beams have identical free vibration frequency spectrum for a specific boundary condition. The problem of finding non-uniform beams which are isospectral to a given uniform beam, with fixed-free boundary condition, leads to a multimodal optimization problem. The first Q natural frequencies of the given uniform Euler–Bernoulli beam are determined using analytical solution. The first Q natural frequencies of a non-uniform beam are obtained with the help of finite element modeling. In order to obtain the non-uniform beams isospectral to a given uniform beam, an error function is designed, which calculates the difference between the spectra of the given uniform beam and the non-uniform beam. In our study, this error function is minimized using electromagnetism inspired optimization technique, a population based iterative algorithm inspired by the attraction–repulsion physics of electromagnetism. Numerical results show the existence of the isospectral non-uniform beams for a given uniform beam, which occur as local minima. Non-uniform beams isospectral to a damaged beam, are also explored using the proposed methodology to illustrate the fact that accurate structural damage identification is difficult by just frequency measurements.

On the transient responses of floating beams and the role of radiation damping

The paper investigates the free vibration behaviour of a floating beam and its forced responses to transient wave packets. The analysis is based on the well-known simple model of a uniform Euler–Bernoulli beam, in which the coupled hydroelastic problem is solved directly. In principle the formulation is exact, within the context of linear theory and potential flow. The problem is discretised and complex eigenvalues of the system matrix are evaluated, in order to characterise the free vibration behaviour (the natural frequencies and radiation damping) over a range of dimensionless beam stiffnesses and mass ratios. The magnitudes of ‘resonant’ responses are shown to be linked to the imaginary parts of these eigenvalues. The model is then used to study the responses of the flexible beam in transient wave packets, such as the NewWave group.

Stabilization of Euler-Bernoulli beam with a nonlinear locally distributed feedback control

This paper studies the stabilization problem of uniform Euler-Bernoulli beam with a nonlinear locally distributed feedback control. By virtue of nonlinear semigroup theory, energy-perturbed approach and polynomial multiplier skill, the authors show that, corresponding to the different values of the parameters involved in the nonlinear locally distributed feedback control, the energy of the beam under the proposed feedback decays exponentially or in negative power of time t as t → ∞.

Research on the uprising dynamics of a double-carriage overhead crane system

The dynamic characteristics, in the uprising phase, of an overhead crane carrying two carriages and a cylindrical payload are studied in this article. The crane system investigated mainly consists of a twin beam, two carriages, a rigid payload, and two wire ropes. A new analytical model, in which the beam, carriages, payload, and wire ropes are, respectively, considered as uniform Euler—Bernoulli beam, lumped masses, rigid body, and springs, is presented to describe the uprising dynamics of the specific crane system. The most distinguished characteristic of the model is the dynamic coupling deriving from the presence of a flexible beam, carriages, and the cylindrical payload. The kinetic and potential energies, during the pre-tensing and lifting phases, of the components in the system are presented in particular. Then, utilizing the Rayleigh—Ritz method, one can obtain the differential equations of the dynamic system substituting the energy into Lagrange's equation. The differential equations are numerically solved by the fourth-order Runge—Kutta method, and then some useful results representing the dynamic features of the system are obtained according to the calculation. The validity of the analytical model is demonstrated by an equivalent FE model created by ANSYS and ADAMS. Comparison of the results, obtained by two distinct approaches, indicates good agreements, which can be the validity evidence of the analytical model.

The use of He’s variational iteration method for obtaining the free vibration of an Euler–Bernoulli beam

This paper presents a way of using He’s variational iteration method to solve free vibration problems for an Euler–Bernoulli beam under various supporting conditions. By employing this technique, the beam’s natural frequencies and mode shapes can be obtained and a rapidly convergent sequence is obtained during the solution. The results obtained are the same as the results obtained by the Adomian decomposition method. It is verified that the present method is accurate and it provides a simple and efficient approach for solving vibration problems for uniform Euler–Bernoulli beams. A robust and efficient algorithm is also programmed using Matlab based on the present method, which can be easily used to solve Euler Bernoulli beam problems.

On the use of negative penalty parameters in aeroelastic divergence analysis of lifting surfaces

Abstract This paper shows that a numerical modelling method in which constraints are replaced with positive and negative penalty functions, which may be regarded as artificial elastic restraints of positive and negative stiffness, may be safely used to determine the critical speed associated with aeroelastic divergence. The critical speeds of a beam with restraints of positive and negative stiffness are found to converge to that of the constrained system, from below if the stiffness is positive and from above otherwise. A uniform Euler–Bernoulli beam clamped at the rear end is analysed using an artificial restraint to enforce the constraint of zero rotation at the clamp, and the results are compared with the exact critical speed of the constrained system obtained analytically. The paper shows that, contrary to common belief that the penalty parameter must be positive, the inclusion of a negative penalty parameter enables the determination of errors due to violation of the constraints.

Corrected Formulas for Natural Frequencies of Cantilever Beams Under Uniform Axial Tension

N ATURAL frequencies of beams under axial tensile force are of interest in many applications such as rotating fan blades, turbine blades, etc. There have been extensive studies on beam dynamics considering the effects of rotation. Rosales and Filipich [1], for example, studied the dynamic stability of a spinning beam with an axial dead load.Naguleswaran [2] presented a study of lateral vibration of a centrifugally tensioned uniform Euler–Bernoulli beam. He developed the database of nondimensional parameters of speed and natural frequencies for various offset ratios of a beam. Lee [3] studied the dynamic response of a rotating Timoshenko beam subjected to axial forces and a moving load. Closed-form solutions for the natural frequencies of rotating pinned–pinned beams are available (for example, Timoshenko et al. [4]).However, other boundary conditions are not readily amenable to such a treatment. Gorman [5] presented the variation of the frequency parameter with axial tension parameter for the six modes of clamped–pinned and clamped–clamped beams. Bokaian [6] presented a comprehensive study of the effect of a constant axial tensile force on the natural frequencies andmode shapes of a uniform single-span beam, with different combinations of the end conditions. Although simple relations could capture the behavior of beams with various boundary conditions, clamped–free beams showed deviant behavior. Thus, the present study focuses on the natural frequencies of a clamped–free beam under uniform axial tension. The relations available for finding the fundamental frequency of a cantilever beam under uniform axial tension are summarized next: The Bokaian [6] characteristic equation is

3112 2自由度フレキシブルマニピュレータの手先位置制御(マニピュレータ)

The objective of this paper is to develop a novel tip position control scheme based on an exact PDE model for a two link planner flexible manipulator with all joints revolute. The techniques employed in this paper are the potential energy shaping and damping injection via the dynamic extension. We assume that each link of the flexible manipulator is uniform Euler-Bernoulli beam. The exact model that is described by ordinary and partial differential equations with non-linear boundary conditions is derived by using Hamilton's principle. Then a tip position and strain feedback strategy is developed based on the potential energy shaping plus damping injection via dynamic extension. We prove that the controller locally stabilize the closed loop system by considering an invariant level set.

Numerical evaluation of high-order modes of vibration in uniform Euler–Bernoulli beams

Many vibration text books give expressions for the mode shape functions of uniform Euler–Bernoulli beams. However, the common forms of these expressions permit the evaluation of only the first 12 modes or so due to numerical issues. This article presents alternative and approximate forms for the evaluation of beam mode shape functions that are numerically stable. Although the approximations allow numerical evaluation of the mode shapes for all modes of vibration, the penalty is that some errors occur in the calculations for low-order modes, and these errors are quantified. Beams with combinations of the classical boundary conditions of clamped, free, pinned, and sliding are covered.

Comparison of Two Types of Deflection Functions for Analysing the Responses of the Rail and the Bridge under Static or Moving Vehicles

The interaction model consisting of a train, a two-layer railway track, and a railway bridge is presented for an analysis of the responses of the rail and the bridge, and two types of deflection functions are compared. One type is the shape functions of a beam element described by the cubic Hermitian interpolation functions in the finite-element method (FEM), and the other is the deflection of a beam described by the superimposing modes in the modal analysis method (MAM). In the present model, each vehicle is modelled as a two-wheel mass-spring-damping system with four degrees of freedom (DOFs); the track is modelled as a discrete supported undamped uniform Bernoulli-Euler beam with hinged-hinged ends representing rails, discrete lumped masses representing sleepers, and discrete massless springs and dashpots representing the characteristics of pads and ballasts; and the bridge deck is modelled as a simply supported uniform Bernoulli-Euler beam. Two different equations of motion for the present model with two types of deflection functions for the rail and the bridge deck are derived from the energy principle. These equations can be easily degenerated into the equations of either a train-track or train-bridge interaction system. Numerical results show that the difference between the two types of deflection functions to analyse the responses of the bridge is very small; and FEM can, yet MAM cannot, give an acceptable result if the bending moment of the rail is taken into account.

Measuring the natural frequencies of centrifugally tensioned beam with laser doppler vibrometer

This paper describes the experimental investigation on lateral vibrations of a rotating uniform Euler–Bernoulli beam that is doubly symmetric in cross-section. Several theoretical studies have been directed to the evaluation of natural frequencies of rotating structural elements such as space frames, windmill rotors, aircraft propeller, etc. In this paperwork, a technique to simulate the presence of centrifugal force acting on the uniform beam is presented while the data about the tested material and the measurement facilities are provided. The results are in excellent agreement with the theoretical predictions about the second and third natural frequencies.

Transverse vibration of a uniform Euler-Bernoulli beam under varying axial force using differential transformation method

This paper presents the application of techniques of differential transformation method (DTM) to analyze the transverse vibration of a uniform Euler-Bernoulli beam under varying axial force. The governing differential equation of the transverse vibration of a uniform Euler-Bernoulli beam under varying axial force is derived and verified. The varying axial force was extended to the more general case which was high polynomial consisted of many terms. The concepts of DTM were briefly introduced. Numerical calculations are carried out and compared with previous published results. The accuracy and the convergence in solving the problem by DTM are discussed.

Vibration and Stability of Uniform Euler–Bernoulli Beams with Step Change in Axial Force

Beams which carry a constant axial force in one portion of the beam and a different axial force in the rest of the beam occur in engineering applications. The system parameters are the position where the axial force change occurs and the two axial force parameters. Transverse vibration of such systems (based on the Euler—Bernoulli theory of bending) is considered in this paper. Tables of the first three frequency parameters are given for various combinations of classical boundary conditions and system parameters. High-precision computing is necessary to handle large axial forces. It is possible for a natural frequency to be zero for certain combinations of the system parameters. This is Euler buckling and a table of combinations of critical system parameters is given. Transverse vibration of tie-bars (a special case of the title problem) is covered in textbooks but benchmark frequencies are not documented. In the paper, a table of frequency parameters of beams carrying tensile or compressive axial forces is given for all the combinations of classical boundary conditions.

Closed form solutions of Euler–Bernoulli beams with singularities

The problem of the integration of the static governing equations of the uniform Euler–Bernoulli beam with discontinuities is studied. In particular, two types of discontinuities have been considered: flexural stiffness and slope discontinuities. Both the above mentioned discontinuities have been modeled as singularities of the flexural stiffness by means of superimposition of suitable distributions (generalized functions) to a uniform one dimensional field. Closed form solutions of governing differential equation, requiring the knowledge of the boundary conditions only, are proposed, and no continuity conditions are enforced at intermediate cross-sections where discontinuities are shown. The continuity conditions are in fact embedded in the flexural stiffness model and are automatically accounted for by the proposed integration procedure. Finally, the proposed closed form solution for the cases of slope discontinuity is compared with the solution of a beam having an internal hinge with rotational spring reproducing the slope discontinuity.

Transverse vibration of an uniform Euler–Bernoulli beam under linearly varying axial force

This paper is concerned with the transverse vibration of uniform Euler–Bernoulli beams under linearly varying fully tensile, partly tensile or fully compressive axial force distribution. The system parameters are the constant part of the axial force and the constant of proportionality of the varying part. The mode shape differential equation is linear with variable coefficients. The general solution is derived and expressed as the super-position of four independent power series solution functions. The frequency equations of the 16 combinations of classical boundary conditions are listed. The first three frequency parameters are tabulated for example combinations of the system parameters. Increase in the values of one or both of the system parameters ‘stiffens’ the system and results in increase in the frequency parameter. If one or both system parameters are negative, combinations exist for which a frequency parameter is zero. This is the Euler buckling condition i.e., onset of dynamic instability. Example combinations of the system parameters when buckling occurs are tabulated. The results tabulated may be used to judge frequencies, buckling parameter combinations obtained by other methods.

Dynamic modelling and control of a rotating Euler–Bernoulli beam

Flexible motion of a uniform Euler–Bernoulli beam attached to a rotating rigid hub is investigated. Fully coupled non-linear integro-differential equations, describing axial, transverse and rotational motions of the beam, are derived by using the extended Hamilton's principle. The centrifugal stiffening effect is included in the derivation. A finite-dimensional model, including couplings of axial and transverse vibrations, and of elastic deformations and rigid motions, is obtained by the finite element method. By neglecting the axial motion, a simplified modelling, suitable for studying the transverse vibration and control of a beam with large angle and high-speed rotation, is presented. And suppressions of transverse vibrations of a rotating beam are simulated with the model by combining positive position feedback and momentum exchange feedback control laws. It is indicated that an improved performance for vibration control can be achieved with the method.

FREE VIBRATIONS OF SELF-STRAINED ASSEMBLIES OF BEAMS

This paper examines the natural frequencies and modes of transverse vibration of two simple redundant systems comprising straight uniform Euler–Bernoulli beams in which there are internal self-balancing axial loads (e.g., loads due to non-uniform thermal strains). The simplest system consists of two parallel beams joined at their ends and the other is a 6-beam rectangular plane frame. Symmetric mode vibration normal to the plane of the frame is studied. Transcendental frequency equations are established for the different systems. Computed frequencies and modes are presented which show the effect of (1) varying the axial loads over a wide range, up to and beyond the values which cause individual members to buckle (2) pinning or fixing the beam joints (3) varying the relative flexural stiffness of the component beams. When the internal axial loads first cause any one of the component beams to buckle, the fundamental frequency of the whole system vanishes. The critical axial loads required for this are determined. A simple criterion has been identified to predict whether a small increase from zero in the axial compressive load in any one member causes the natural frequencies of the whole system to rise or fall. It is shown that this depends on the relative flexural stiffnesses and buckling loads of the different members. Computed modes of vibration show that when the axial modes reach their critical values, the buckled beam(s) distort with large amplitudes while the unbuckled beam(s) move either as rigid bodies or with bending which decays rapidly from the ends to a near-rigid-body movement over the central part of the beam. The modes of the systems with fixed joints change very little (if at all) with changing axial load, except when the load is close to the value which maximizes or minimizes the frequency. In a narrow range around this load the mode changes rapidly. The results provide an explanation for some computed results (as yet unpublished) for the flexural modes and frequencies of flat plates with non-uniform thermal stress distributions.