Flexural vibration of axially loaded beams with linear or parabolic taper

Curves are presented which enable the first five natural frequencies to be found for axially loaded tapered members with an important family of cross sections. Previous curves have only included axial forces in a very limited number of cases, whereas here axial forces within the range of greatest practical importance can always be allowed for. The present curves cover 11 combinations of end conditions, three types of taper, all taper ratios between 0·2 and 1, and all axial forces between ?P_c and 0·6P_c where P_c is the critical buckling load for pure compression and can be found from curves provided (a list of principal nomenclature is given in Appendix 2). The cross sections covered include thin circular ones of constant thickness and thin-walled cross sections consisting of constant thickness flat plates which all have their breadths tapering in the same way. Truncated wedges of varying depth and constant width are also covered. The taper is linear between the ends, or linear between a maximum value at the centre and equal values at the ends, or parabolic with a maximum value at the centre. Simple examples show how the curves can be used to obtain natural frequencies or buckling loads to an accuracy which is almost always about 1%. The curves also illustrate how natural frequencies and buckling loads of thin-walled members are altered by changing the amount and type of a member's taper while keeping its wall thickness and mass constant. Finally, it is shown that, for the ranges covered by the curves, frequencies which separate the first five natural frequencies can be found a priori. The theory used to obtain the curves was simply to divide the tapered member into sufficient uniform members to ensure convergence to the tapered result to better than plotting accuracy. Exact Bernoulli-Euler dynamic stiffnesses, which allowed for axial force effects, were used for these uniform components. In Appendix 1 the effects of using exact Timoshenko stiffnesses instead of the Bernoulli-Euler ones are illustrated.     

Green's functions of the forced vibration of Timoshenko beams with damping effect

This paper is concerned with the dynamic solutions for forced vibrations of Timoshenko beams in a systematical manner. Damping effects on the vibrations of the beam are taken into consideration by introducing two characteristic parameters. Laplace transform method is applied in the present study and corresponding Green's functions are presented explicitly for beams with various boundaries. The present solutions can be readily reduced to those for others classical beam models by setting corresponding parameters to zero or infinite. Numerical calculations are performed to validate the present solutions and the effects of various important physical parameters are investigated.     

Coupled bending and torsional vibration of axially loaded Bernoulli-Euler beams including warping effects

The dynamic transfer matrix method for determining natural frequencies and mode shapes of the bending-torsion coupled vibration of axially loaded thin-walled beams with monosymmetrical cross sections is developed by using a general solution of the governing differential equations of motion based on Bernoulli-Euler beam theory. This method takes into account the effect of warping stiffness and gives allowance to the presence of axial force. The dynamic transfer matrix is derived in detail. Two illustrative examples on the application of the present theory are given for bending-torsion coupled beams with thin-walled open cross sections. The effects of axial load and warping stiffness on coupled bending-torsional frequencies are discussed. Compared with those available in the literature, numerical results demonstrate the accuracy and effectiveness of the proposed method.     

Free vibration of FGM Timoshenko beams with through-width delamination

Free vibration of functionally graded beams with a through-width delamination is investigated.It is assumed that the material property is varied in the thickness direction as power law functions and a single through-width delamination is located parallel to the beam axis.The beam is subdivided into three regions and four elements.Governing equations of the beam segments are derived based on the Timoshenko beam theory and the assumption of‘constrained mode’.By using the differential quadrature element method to solve the eigenvalue problem of ordinary differential equations governing the free vibration,numerical results for the natural frequencies of the beam are obtained.Natural frequencies of delaminated FGM beam with clamped ends are presented.Effects of parameters of the material gradients,the size and location of delamination on the natural frequency are examined in detail.     

Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams

This paper investigates dynamic stability of an axially accelerating viscoelastic beam undergoing parametric resonance. The effects of shear deformation and rotary inertia are taken into account by the Timoshenko thick beam theory. The beam material obeys the Kelvin model in which the material time derivative is used. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The governing partial-differential equations are derived from Newton's second law, Euler's angular momentum principle, and the constitutive relation. The method of multiple scales is applied to the equations to establish the solvability conditions in summation and principal parametric resonances. The sufficient and necessary condition of the stability is derived from the Routh-Hurvitz criterion. Some numerical examples are presented to demonstrate the effects of related parameters on the stability boundaries.     

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.     

Global damping of noise or vibration fields with locally synthesized controllers

A locally synthesized controller (LSC) is one that uses a local feedback signal in a noise or vibration field (VF) to synthesize the actuation signal. The global damping of a VF by available LSCs requires sensor-actuator collocation. This study presents a LSC for the global damping of a VF without requiring sensor-actuator collocation, which is important to noise control applications where a sensor may be placed away from an actuator to avoid the near field effects. It is proven that the LSC damps the entire VF instead of just a local feedback loop. This is different from other LSCs that may control local feedback loops without damping the VFs. A decentralized control law is presented here to extend the LSC to a decentralized damping system using multiple actuators.     

Analytical approach for free vibration analysis of two-layer Timoshenko beams with interlayer slip

In this paper, an analytical procedure for free vibrations of shear-deformable two-layer beams with interlayer slip is developed. The effect of transverse shear flexibility of two layers is taken into account in a general way by assuming that each layer behaves as a Timoshenko beam element. Therefore, the layers have independent shear strains that depend indeed on their own shear modulus. This is the main improvement of the proposed model compared to existing models where the transverse shear flexibility is ignored or taken into account in a simplified way in which the shear strains of both layers are assumed to be equal whatever the shear modulus of the layers. In the proposed model, the two layers are connected continuously and the partial interaction is considered by assuming a continuous relationship between the interface shear flow and the corresponding slip. Based on these key assumptions, the governing differential equation of the problem is derived using Hamilton's principle and is analytically solved. The solutions for the eigenfrequencies and eigenmodes of four single span two-layer beams with classical Euler boundary conditions, i.e. pinned-pinned, clamped-clamped, clamped-pinned and clamped-free, are presented. Next, some numerical applications dealing with these four beams are carried out in order to compare the eigenfrequencies obtained with the proposed model against two existing models which consider different kinematic assumptions. Finally, a parametric study is conducted with the aim to investigate the influence of varying material and geometric parameters on the eigenfrequencies, such as shear stiffness of the connectors, span-to-depth ratios, flexural-to-shear moduli ratios and layer shear moduli ratios.     

Optimized electric networks for vibration damping of piezoactuated beams

This paper studies the multimodal vibration damping of an elastic beam equipped with multiple piezoelectric actuators connected to an electric network. Two analytical models of the electromechanical coupled structure are considered: a homogenized one, accurate when a large number of actuators is employed, is used to derive simple design criteria for the electric network; and a discrete one, able to face real situations when few actuators are employed, is adopted to test the network performance, defined as the exponential time-decay rate of the free vibrations of the controlled structure. Some electric networks are presented and compared in simulation to networks previously proposed in the literature, in order to evaluate their performances in broadband vibration control.     

Free vibration of thin walled open section beams with constrained damping treatment

Free vibration characteristics of a thin walled, open cross-section beam, with constrained damping layers at the flanges, are investigated. Both uncoupled transverse vibration and the coupled bending-torsion oscillations, of a beam of a top-hat section, are considered. Numerical results are presented for natural frequencies and modal loss factors in the first two modes of simply supported and clamped-clamped beams. For the uncoupled mode the constrained damping treatment is more effective than an unconstrained one, but for the coupled mode the effect is just the opposite.     

A new approach for free vibration of axially functionally graded beams with non-uniform cross-section

This paper studies free vibration of axially functionally graded beams with non-uniform cross-section. A novel and simple approach is presented to solve natural frequencies of free vibration of beams with variable flexural rigidity and mass density. For various end supports including simply supported, clamped, and free ends, we transform the governing equation with varying coefficients to Fredholm integral equations. Natural frequencies can be determined by requiring that the resulting Fredholm integral equation has a non-trivial solution. Our method has fast convergence and obtained numerical results have high accuracy. The effectiveness of the method is confirmed by comparing numerical results with those available for tapered beams of linearly variable width or depth and graded beams of special polynomial non-homogeneity. Moreover, fundamental frequencies of a graded beam combined of aluminum and zirconia as two constituent phases under typical end supports are evaluated for axially varying material properties. The effects of the geometrical and gradient parameters are elucidated. The present results are of benefit to optimum design of non-homogeneous tapered beam structures.     

Free vibration analysis of corner-supported rectangular plates with symmetrically distributed edge beams

Analytical type solutions are obtained for the free vibration frequencies and mode shapes of thin corner-supported rectangular plates with symmetrically distributed reinforcing beams, or strips, attached to the plate edges. The method of superposition is employed. Equations governing reactions at plate-beam interfaces are developed in dimensionless form. The approach is comprehensive in that both lateral and rotational stiffness, and inertia, of the beam are incorporated into the analysis. For illustrative purposes computed eigenvalues and mode shapes are presented for two plate-beam systems of realistic geometries. It is shown that the method is easily extended to cover the case where the edge beams do not have a symmetrical distribution. This appears to be the first comprehensive analytical study of this problem of industrial interest.     

A description of the dynamics and thermodynamics of vibration damping in axially deformed nuclei

The Quantal Brownian Motion (QBM) model of nuclear vibration damping is adapted to describe an axially deformed nucleus where a giant oscillatory mode becomes excited. Several simplifying assumptions are imposed in order to obtain an operative version of the QBM model. Within the restrictions posed by this set of hypothesis, it is found that a system resembling the nucleide¹⁶⁶Er, as an illustration, undergoes both dynamical and thermodynamical behavior in a consistent scheme of magnitudes. The value of energies, temperatures and entropies at equilibrium fit a closed thermodynamical model concerning oscillators coupled to fermionic reservoirs. In the present approach, it is seen that the nucleonic excitations that provide the doorways to mode decay can be clearly split into two sets, or separate heat baths, for the components of the axially symmetric vibration.     

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.