The second frequency spectrum of Timoshenko beams

A second spectrum of frequencies was reported in early analytical work on the vibrations of Timoshenko beams. However, in subsequent finite element modelling this phenomenon was either ignored or not definitively classified and recorded. In fact, from a recent finite element analysis with a high precision element it was even concluded that there is no separate second spectrum of frequencies except for the special case of hinged-hinged beams and it was asserted that previous investigators had misinterpreted some frequencies thus introducing the notion of second frequencies. In this paper, a simple linear beam element with independent displacement fields and reduced integration to eliminate shear locking is used and enables one to detect the second spectrum accurately. Guidelines are provided which help to identify and classify the frequencies into two separate spectra.     

Non-linear vibration analysis of multilayer beams by incremental finite elements,Part I: Theory and numerical formulation

An incremental variational equation for non-linear motions of multilayer beams composed of n stiff layers and (n ? 1) soft cores is derived from the dynamic virtual work equation by an appropriate integration procedure. The kinematical hypotheses of Euler-Bernoulli and Timoshenko beam theories are used to describe the displacement fields of the stiff layers and cores respectively. An efficient solution procedure of incremental harmonic balance method type, with use of finite elements, is developed. To demonstrate its capability, some problems in free non-linear vibrations of multilayer beams are treated by using the procedure. Results are compared with those available in the literature. The effects of damping are also included in this investigation but are described in Part II [1] of this paper in which a number of undamped and damped forced non-linear vibration problems are studied. Results in the form of tables and plots are also presented and comparisons are made with those available in the literature.     

On the natural frequencies and mode shapes of a multispan Timoshenko beam carrying a number of various concentrated elements

The purpose of this paper is to utilize the numerical assembly method (NAM) to determine the exact natural frequencies and mode shapes of the multispan Timoshenko beam carrying a number of various concentrated elements including point masses, rotary inertias, linear springs, rotational springs and spring–mass systems. First, the coefficient matrices for an intermediate pinned support, an intermediate concentrated element, left- and right-end support of a Timoshenko beam are derived. Next, the overall coefficient matrix for the whole structural system is obtained using the numerical assembly technique of the finite element method. Finally, the exact natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the associated eigenfunctions, respectively. The effects of distribution of in-span pinned supports and various concentrated elements on the dynamic characteristics of the Timoshenko beam are also studied.     

Active control of geometrically nonlinear transient vibration of composite plates with piezoelectric actuators

In this paper, the incremental finite element equations for geometric non-linear analysis of piezoelectric smart structures are developed using a total Lagrange approach by using virtual velocity incremental variational principles. A four-node first order shear plate element model with reduced and selective integration is also developed. Geometrically non-linear transient vibration response and control of plates with piezoelectric patches subjected to pulse loads are investigated. Active damping is introduced on the plates by coupling a self-sensing and negative velocity feedback algorithm in a closed control loop. The numerical results show that piezoelectric actuators can introduce significant damping and suppress transient vibration effectively. The effects of the number and locations of the piezoelectric actuators on the control system are also discussed.     

The dynamic analysis of a cracked Timoshenko beam by the spectral element method

The aim of this paper is to introduce a new finite spectral element of a cracked Timoshenko beam for modal and elastic wave propagation analysis. The proposed approach deals with the spectral element method. This method is suitable for analyzing wave propagation problems as well as for calculating modal parameters of the structure. In the paper, the results of the change in modal parameters due to crack appearance are presented. The influence of the crack parameters, especially of the changing location of the crack, on the wave propagation is examined. Responses obtained at different points of the beam are presented. Proper analysis of these responses allows one to indicate the crack location in a very precise way. This fact is very promising for the future work in the damage detection field.     

Free vibration analysis of civil engineering structures by component-wise models

Higher-order beam models are used in this paper to carry out free vibration analysis of civil engineering structures. Refined kinematic fields are developed using the Carrera Unified Formulation (CUF), which allows for the implementation of any-order theory without the need for ad hoc formulations. The principle of virtual displacements in conjunction with the finite element method (FEM) is used to formulate stiffness and mass matrices in terms of fundamental nuclei. The nuclei depend neither on the adopted class of beam theory nor on the FEM approximation along the beam axis. This paper focuses on a particular class of CUF models that makes use of Lagrange polynomials to discretize cross-sectional displacement variables. This class of models are referred to as component-wise (CW) in recent works. According to the CW approach, each structural component (e.g. columns, walls, frame members, and floors) can be modeled by means of the same 1D formulation. A number of typical civil engineering structures (e.g. simple beams, arches, truss structures, and complete industrial and civil buildings) are analyzed and CW results are compared to classical beam theories (Euler–Bernoulli and Timoshenko), refined beam models based on Taylor-like expansions of the displacements on the cross-section, and classical solid/shell FEM solutions from the commercial code MSC Nastran. The results highlight the enhanced capabilities of the proposed formulation. It is in fact demonstrated that CW models are able to replicate 3D solid results with very low computational efforts.     

Dynamic analysis of layered composite shells using nine node degenerate shell elements

The basic objective of the work reported in this paper is to extend a nine-node degenerated shell element developed earlier for stress analysis to the free vibration analysis of thick laminated composites. The nine-noded degenerated shell element is preferable to conventional solid elements for the modeling and analysis of laminated composite shell structures since the shell element works for both thick and thin shells. An enhanced interpolation of the transverse shear strains in the natural coordinates is used in this formulation to produce a shear locking free element and an enhanced interpolation of the membrane strains in the local coordinates is used to produce a membrane locking free element. The interpolation functions used in formulating the assumed strains are based on the Lagrangian interpolation polynomials. Various numerical examples are analyzed and their results are compared with the existing exact solutions where available and the numerical solutions calculated from other shell finite element formulations, to benchmark the current formulation.     

Combined dynamic stiffness matrix and precise time integration method for transient forced vibration response analysis of beams

A method has been developed for determining the transient response of a beam. The beam is divided into several continuous Timoshenko beam elements. The overall dynamic stiffness matrix is assembled in turn. Using Leung's equation, we derive the overall mass and stiffness matrices which are more suitable for response analysis than the overall dynamic stiffness matrix. The forced vibration of the beam is computed by the precise time integration method. Three illustrative beams are discussed to evaluate the performance of the current method. Solutions calculated by the finite element method and theoretical analysis are also enumerated for comparison. In these examples, we have found that the current method can solve the forced vibration of structures with a higher precision.     

Stability analysis with lumped mass and friction effects in elastically supported pipes conveying fluid

This paper presents an accurate finite element procedure for the stability analysis of elastically supported pipes conveying fluid. With consideration of effects of lumped masses, fluid pressure and friction, the equations of motion are derived based on Hamilton's principle for the mass transport system. The kinematics of the pipe is based on Timoshenko beam theory for which the transverse shear deformation and rotary inertia of the pipe are included. The material behaviour of the pipe is described by the Kelvin viscoelastic model. The dynamic stability behaviours obtained by the present work are more conservative as compared with those evaluated by conventional Euler-Bernoulli beam theory. Also, it is found that the lumped masses, fluid pressure and friction will destabilize the system while the elastic support may have either a stabilizing or destabilizing effect depending on its stiffness and location. To demonstrate the validity and accuracy of the technique developed, several numerical examples are illustrated.     

A HYBRID FINITE SEGMENT/FINITE ELEMENT MODELLING AND EXPERIMENT ON A FLEXIBLE DEPLOYMENT SYSTEM

This paper focuses on the dynamic responses of a flexible deployment system that has a central rigid body and four articulated flexible beams and undergoes locking impact. A hybrid finite segment/finite element model and an experiment are presented for the deploy-ment system. The flexible beam components in the system are modelled with the finite segments connected by massless beam elements, wherein the finite segments describe the inertia of the large rotation flexible beam and the massless elastic elements describe the elas-ticity of the flexible beam by taking the advantage of small deformation in the relative co-ordinate system. To model the internal impacts in the articulate joints due to clearances, a continuous contact force model of locking joint is also proposed. The governing differential-algebraic equations of the system are established by the Newton-Euler method with Lagrange multipliers and are solved with the method of generalized co-ordinate partitioning. To accelerate the numerical integration, a “longitudinal constraint” is suggested to alleviate the stiff problem of the dynamic equations. In addition, a physical model of the deployment system is constructed. The deployment is released by the compressed springs in the joints. A position measuring system of linear CCD cameras is used to measure the large displacement of the system. Correlations between the mathematical model and the experiments are also presented. Reasonable results are obtained.     

Spectral element modeling for extended Timoshenko beams

Periodic lattice structures such as the large space lattice structures and carbon nanotubes may take the extension-transverse shear-bending coupled vibrations, which can be well represented by the extended Timoshenko beam theory. In this paper, the spectrally formulated finite element model (simply, spectral element model) has been developed for extended Timoshenko beams and applied to some typical periodic lattice structures such as the armchair carbon nanotube, the periodic plane truss, and the periodic space lattice beam.     

A tapered beam finite element for rotor dynamics analysis

The stiffness, mass and gyroscopic matrices of a rotating beam element are developed, a cubic function being used for the transverse displacement. Shear deflection is included by use of end nodal variables of shear strain, along with transverse displacement and cross-section rotation; rotatory inertia effects are included in the energy functional to provide a Timoshenko beam formulation. The gyroscopic effects for small perturbations are linearized as a skew symmetric damping matrix. The formulation is implemented by numerical integration for a linearly tapered circular beam. A technique of reduction of the shear nodal variable prior to global assembly is shown to provide little loss in accuracy with reduced system bandwidth. Numerical comparisons for three previously published beam models are included, with results presented for the case of forward and reverse precession to verify the gyroscopic effects. The utility of the element in a general program for rotor dynamics analysis is identified.     

A finite element method for analysis of vibration induced by maglev trains

This paper developed a finite element method to perform the maglev train–bridge–soil interaction analysis with rail irregularities. An efficient proportional integral (PI) scheme with only a simple equation is used to control the force of the maglev wheel, which is modeled as a contact node moving along a number of target nodes. The moving maglev vehicles are modeled as a combination of spring-damper elements, lumped mass and rigid links. The Newmark method with the Newton–Raphson method is then used to solve the nonlinear dynamic equation. The major advantage is that all the proposed procedures are standard in the finite element method. The analytic solution of maglev vehicles passing a Timoshenko beam was used to validate the current finite element method with good agreements. Moreover, a very large-scale finite element analysis using the proposed scheme was also tested in this paper.     

Dynamic stability of Timoshenko beams resting on an elastic foundation

A finite element model is developed for the stability analysis of a Timoshenko beam resting on an elastic foundation and subjected to periodic axial loads. The effect of an elastic foundation on the natural frequencies and static buckling loads of hinged-hinged and fixed-free Timoshenko beams is investigated. The results obtained for a Bernoulli-Euler beam which is a special case of the present analysis show excellent agreement with the available results obtained by other analytical methods. The regions of dynamic instability are determined for different values of the elastic foundation constant. As the elastic foundation constant increases the regions of dynamic instability are shifted away from the vertical axis and the width of these regions is decreased, thus making the beam less sensitive to periodic forces.     

用有限元法解龟磁波穿透非均匀有耗等离子体鞘套时的损耗

本文用有限无法求解了电磁波穿透具有任意密度剖面的非均匀等离子体鞘套时的反射和透射问題。鞘套的剖面参数选自典型再入飞行器的粘性激波层平衡流场数据^[1]。利用虚功原理和线性插值函数从Helmholtz方程导出有限元方程。鞘套按对数剖分为31个单元。然后,分别对频率f=400MHZ,4000MHZ和10GHZ时,计算了鞘套内的场值及反射,透射和吸收系数。并估算了信号的总功耗。误差分析表明,计算结果是令人满意的。与其它方法相比较,本文所给出的有限无法是简捷而精确的,所以该方程适于计算波穿透具有任意的而且变化剧烈的剖面的等离子体鞘套时的损耗     

The effects of rotation,tip mass,and hub radius on the stability of beck's column

The stability of a cantilever beam subjected to a follower force at its free end and rotating at a uniform angular velocity is investigated. The beam is assumed to be offset from the axis of rotation, carries a tip mass at its free end, and undergoes deflection in a direction perpendicular to the plane of rotation. The equations of motion are formulated within the Euler-Bernoulli and Timoshenko beam theories for the case of a Kelvin model viscoelastic beam. The associated adjoint boundary value problems are derived and appropriate adjoint variational principles are introduced. These variational principles are used for the purpose of determining approximately the values of the critical flutter load of the system as it depends upon its damping parameters, tip mass and its rotary inertia, hub radius, and speed of rotation. The variation of the critical flutter load with these parameters is revealed in a series of several graphs. The numerical results show that the critical load can be reduced significantly due to (a) the transverse and rotary inertia of the tip mass and (b) increasing values of the internal damping parameter associated with the transverse shear deformation of the rotating beam.     

A finite element for the vibration analysis of Timoshenko beams

A Timoshenko beam finite element is presented which has three nodes and two degrees of freedom per node, namely the values of the lateral deflection and the cross-sectional rotation. The element properties are based on a coupled displacement field; the lateral deflection is interpolated as a quintic polynomial function and the cross-sectional rotation is linked to the deflection by specifying satisfaction of the governing differential equation of moment equilibrium in the absence of the rotary inertia term. Numerical results confirm that this procedure does not preclude convergence to true Timoshenko theory solutions since rotary inertia is included in lumped form at element ends. The new Timoshenko beam element has good convergence characteristics and where comparison can be made in numerical studies it is shown to be generally more efficient than previous elements.     

Bending vibration of axially loaded Timoshenko beams with locally distributed Kelvin-Voigt damping

Utilizing the Timoshenko beam theory and applying Hamilton's principle, the bending vibration equations of an axially loaded beam with locally distributed internal damping of the Kelvin-Voigt type are established. The partial differential equations of motion are then discretized into linear second-order ordinary differential equations based on a finite element method. A quadratic eigenvalue problem of a damped system is formed to determine the eigenfrequencies of the damped beams. The effects of the internal damping, sizes and locations of damped segment, axial load and restraint types on the damping and oscillating parts of the damped natural frequency are investigated. It is believed that the present study is valuable for better understanding the influence of various parameters of the damped beam on its vibration characteristics.