Nonlinear Vibrations Of Beams Research Articles

Effect of transverse shears on complex nonlinear vibrations of elastic beams

Models of geometrically nonlinear Euler-Bernoulli, Timoshenko, and Sheremet’ev-Pelekh beams under alternating transverse loading were constructed using the variational principle and the hypothesis method. The obtained differential equation systems were analyzed based on nonlinear dynamics and the qualitative theory of differential equations with using the finite difference method with the approximation O(h2) and the Bubnov-Galerkin finite element method. It is shown that for a relative thickness λ ⩽ 50, accounting for the rotation and bending of the beam normal leads to a significant change in the beam vibration modes.

Editorial: Special issue on stability of non-linear dynamic structures and systems

The 2011 issue of Nonlinear Dynamics presents articles on stability of non-linear dynamic structures and systems. Alijani and co-researchers investigate geometrically nonlinear vibrations of rectangular plates with Functionally Graded Materials (FGM) in thermal environments via a multi-modal energy approach. Demeio and co-researchers investigate the primary non-linear resonance response of a one-dimensional continuous system, which can be regarded as a model for semiinfinite cables resting on an elastic substrate reacting in compression only, and subjected, to a constant distributed load and to a small harmonic displacement applied to the finite boundary. Stoykov and Ribeiro investigate the geometrically non-linear periodic vibrations of beams with rectangular cross section under harmonic forces using a p-version finite element method. Krylov and co-researchers investigate the feasibility of two-directional switching of an initially curved or prebuckled electrostatically actuated micro-beams using a single electrode fabricated from the same structural layer.

Stability of nonlinear periodic vibrations of 3D beams

The geometrically nonlinear periodic vibrations of beams with rectangular cross section under harmonic forces are investigated using a p-version finite element method. The beams vibrate in space; hence they experience longitudinal, torsional, and nonplanar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The theory employed is valid for moderate rotations and displacements, and physical phenomena like internal resonances and change of the stability of the solutions can be investigated. Green’s nonlinear strain tensor and Hooke’s law are considered and isotropic and elastic beams are investigated. The equation of motion is derived by the principle of virtual work. The differential equations of motion are converted into a nonlinear algebraic form employing the harmonic balance method, and then solved by the arc-length continuation method. The variation of the amplitude of vibration in space with the excitation frequency of vibration is determined and presented in the form of response curves. The stability of the solution is investigated by Floquet’s theory.

Non-linear vibration of Euler-Bernoulli beams

In this paper, variational iteration (VIM) and parametrized perturbation (PPM) methods have been used to investigate non-linear vibration of Euler-Bernoulli beams subjected to the axial loads. The proposed methods do not require small parameter in the equation which is difficult to be found for nonlinear problems. Comparison of VIM and PPM with Runge-Kutta 4th leads to highly accurate solutions.

Analytical study on the vibration frequencies of tapered beams

A vast amount of published work can be found in the field of beam vibrations dealing with analytical and numerical techniques. This paper deals with analysis of the nonlinear free vibrations of beams. The problem considered represents the governing equation of the nonlinear, large amplitude free vibrations of tapered beams. A new implementation of the ancient Chinese method called the Max-Min Approach (MMA) and Homotopy Perturbation Method (HPM) are presented to obtain natural frequency and corresponding displacement of tapered beams. The effect of vibration amplitude on the non-linear frequency is discussed. In the end to illustrate the effectiveness and convenience of the MMA and HPM, the obtained results are compared with the exact ones and shown in graphs and in tables. Those approaches are very effective and simple and with only one iteration leads to high accuracy of the solutions. It is predicted that those methods can be found wide application in engineering problems, as indicated in this paper.

Nonlinear free vibrations of beams in space due to internal resonance

The geometrically nonlinear free vibrations of beams with rectangular cross section are investigated using a p-version finite element method. The beams may vibrate in space, hence they may experience longitudinal, torsional and non-planar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The geometrical nonlinearity is taken into account by considering Green’s nonlinear strain tensor. Isotropic and elastic beams are investigated and generalised Hooke’s law is used. The equation of motion is derived by the principle of virtual work. Mostly clamped–clamped beams are investigated, although other boundary conditions are considered for validation purposes. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. One constant term, odd and even harmonics are assumed in the Fourier series and convergence with the number of harmonics is analysed. The variation of the amplitude of vibration with the frequency of vibration is determined and presented in the form of backbone curves. Coupling between modes is investigated, internal resonances are found and the ensuing multimodal oscillations are described. Some of the couplings discovered lead from planar oscillations to oscillations in the three dimensional space.

DYNAMIC RESPONSE OF AXIALLY LOADED EULER-BERNOULLI BEAMS

The current research deals with application of a new analytical technique called Energy Balance Method (EBM) for a nonlinear problem. Energy Balance Method is used to obtain the analytical solution for nonlinear vibra tion behavior of Euler-Bernoulli beams subjected to axial loads. Analytical expressions for geometrically nonlinear vibration of beams are provided. The effect of vibration amplitude on the nonlinear frequency is discussed. Com parison between Energy Balance Method results and those available in literature demonstrates the accuracy of this method. In Energy Balance Method contrary to the con ventional methods, only one iteration leads to high accu racy of the solutions which are valid for a wide range of vibration amplitudes.http://dx.doi.org/10.5755/j01.mech.17.2.335

Solution of free non-linear vibration of beams

In this paper, we have considered the nonlinear governing equation of tapered beams, attempt has been made to analyze the nonlinear behavior of tapered beams analytically. The nonlinear governing equation is solved by employing the variational approach method (VAM) and Improved AmplitudeFormulation (IAFF). Despite the increasing expenses of building structures to maintain their linear behavior, nonlinearity has been inevitable and therefore, nonlinear analysis has been of great importance to the scientists in the field. The major concern is to assess excellent approximations to the exact solutions for the whole range of the oscillation amplitude, reducing the respective error of angular frequency in comparison with the VAM and IAFF. The effect of vibration amplitude on the nonlinear frequency is discussed. It is predicted that there can be wide application of VAM and IAFF in engineering problems, as indicated in this paper.

Accurate solutions to nonlinear vibration of cantilever beam via homotopy perturbation method

In this study, analytical solutions are obtained by homotopy perturbation method (HPM) for the nonlinear vibration equation of cantilever beam excited by harmonic and EMAs forces. Novel and accurate analytical solutions for the frequency and displacement are derived. Comparison of the result obtained by the HPM with numerical solutions reveals that only the first order approximation leads to higher accurate solution.

Nonlinear vibration of beams under nonideal boundary conditions

In this paper, we investigate the influence of nonideal boundary conditions on the nonlinear vibration of damped Euler–Bernoulli beams subjected to harmonic loads. These nonidealities allow for small deflections and/or moments at the supports of the beam. Using the iteration perturbation method, analytical expressions for the case of simply supported beams with immovable end conditions are provided. The results reveal that the first order of approximation obtained by the proposed method is more accurate than the perturbation solutions. Moreover, compared with the previous studies, some interesting phenomenon is predicted. We have shown that in some special combinations of the nonidealities the nonlinear frequency of vibration as well as the frequency–response curves would be unchanged.

Large Amplitude Free Vibration of a Rotating Nonhomogeneous Beam With Nonlinear Spring and Mass System

This paper investigates the free out of plane vibration of a rotating nonhomogeneous beam with nonlinear spring and mass system. The effect of nonhomogeneity of the beam appears both in the governing equations and in the boundary conditions, but the nonlinear spring-mass effect appears in the boundary conditions only. The solution is obtained by applying the method of multiple time scales directly to the nonlinear partial differential equations and the boundary conditions. The results of the linear frequencies match well with those obtained in open literature. The effect of the nonhomogeneity of the stiffer beam (β=0.01) reduces the frequencies of vibration of the beam. A possible physical explanation of this reduced frequency of the nonhomogeneous beam is discussed. A subsequent nonlinear study of the nonhomogeneous beam indicates that the mass of the spring and its location also have a pronounced effect on the vibration of the beam. The effect of the nonhomogeneity of the beam on the relative stability of the nonlinear vibration of the beam with spring-mass system is also studied.

Nonlinear forced vibrations and static deformations of 3D beams with rectangular cross section: The influence of warping, shear deformation and longitudinal displacements

The geometrically nonlinear vibrations of beams with rectangular cross section, which may experience longitudinal, torsional and bending deformations, in any plane, are investigated by the p-version finite element method. Two main models are used: one of them is based on Bernoulli–Euler theory for beam bending and the other is based on Timoshenko theory; both models assume that the cross section rotates as a rigid body and is free to warp in longitudinal direction, as in Saint-Venants’s theory for torsion. The geometrical nonlinearity is taken into account by considering Green’s strain tensor and the importance of the longitudinal displacements of quadratic order, which are most often neglected in the strain–displacement relation, is here examined. Generalized Hooke’s law is used and the equation of motion is derived by the principle of virtual work. The importance of the warping function is analysed for different rectangular cross sections, and it is shown that its consideration can be fundamental to obtain correct results. Then, it is shown that the linearization of the trigonometric functions related with the twist angle, which is usually applied in the displacement field in models based on Saint-Venant’s hypothesis, should be done in the strain–displacement relations. Comparisons of the models for 3D beams based on Bernoulli–Euler’s and Timoshenko’s theories is presented. It is shown that if the rotations along the transverse axes are directly approximated by the respective derivatives of the transverse displacement functions, as is assumed in the model presented here that is based on Bernoulli–Euler’s theory, the additional shear stresses that appear when the bending and torsion motions are coupled, lead to wrong results. Finally, taking into account accuracy and simplicity, a model is chosen and employed to investigate the nonlinear forced vibrations of beams using direct integration of the equations of motion in the time domain. Examples of bending–torsion couplings due to the nonlinear terms are presented in dynamical problems.

Nonlinear vibration of viscoelastic sandwich beams by the harmonic balance and finite element methods

This paper deals with geometrically nonlinear vibrations of sandwich beams with viscoelastic materials. For this purpose, a new finite element formulation has been developed, in which a zig-zag model is used to describe the displacement field. The viscoelastic behaviour is handled by using hereditary integrals and their relationships with complex moduli. An efficient solution procedure based on the harmonic balance method is also developed. To demonstrate its abilities, various problems of nonlinear vibrations of sandwich beams are considered. First, the results derived from the proposed approach are compared with those of nonlinear dynamic analyses using direct time integration and to experimental data. Then, the influence of the vibration amplitude on the damping properties of sandwich beams is investigated. The effect of an initial axial strain is also examined.

Active suppression of nonlinear composite beam vibrations by selected control algorithms

This paper is focused on application of different control algorithms for a flexible, geometrically nonlinear beam-like structure with Macro Fiber Composite (MFC) actuator. Based on the mathematical model of a geometrically nonlinear beam, analytical solutions for Nonlinear Saturation Controller (NSC) are obtained using Multiple Scale Method. Effectiveness of different control strategies is evaluated by numerical simulations in Matlab–Simulink software. Then, the Digital Signal Processing (DSP) controller and selected control algorithms are implemented to the physical system to compare numerical and experimental results. Detailed analysis for the NSC system is carried out, especially for high level of amplitude and wide range of frequencies of excitation. Finally, the efficiency of the considered controllers is tested experimentally for a more complex autoparametric “ L-shape” beam system.

Low frequency and large amplitude oscillations of cantilevers in viscous fluids

We study nonlinear vibrations of cantilever beams oscillating in viscous fluids. A handleable expression for the inertial and damping loads due to the encompassing fluid is proposed. We expand on the canonical viscous diffusion theory by incorporating vortex shedding effects at large oscillation amplitudes. Comparison with experimental results on underwater low frequency and large amplitude oscillations of cantilevers is reported. The approach is applicable to the analysis of ionic polymer metal composites vibrating underwater.

A Semianalytical Method for Nonlinear Vibration of Euler-Bernoulli Beams with General Boundary Conditions

This paper presents a new semianalytical approach for geometrically nonlinear vibration analysis of Euler-Bernoulli beams with different boundary conditions. The method makes use of Linstedt-Poincaré perturbation technique to transform the nonlinear governing equations into a linear differential equation system, whose solutions are then sought through the use of differential quadrature approximation in space domain and an analytical series expansion in time domain. Validation of the present method is conducted in numerical examples through direct comparisons with existing solutions, showing that the proposed semianalytical method has excellent convergence and can give very accurate results at a long time interval.

An analytical study on the nonlinear vibration of functionally graded beams

Nonlinear vibration of beams made of functionally graded materials (FGMs) is studied in this paper based on Euler-Bernoulli beam theory and von Karman geometric nonlinearity. It is assumed that material properties follow either exponential or power law distributions through thickness direction. Galerkin procedure is used to obtain a second order nonlinear ordinary equation with quadratic and cubic nonlinear terms. The direct numerical integration method and Runge-Kutta method are employed to find the nonlinear vibration response of FGM beams with different end supports. The effects of material property distribution and end supports on the nonlinear dynamic behavior of FGM beams are discussed. It is found that unlike homogeneous beams, FGM beams show different vibration behavior at positive and negative amplitudes due to the presence of quadratic nonlinear term arising from bending-stretching coupling effect.

Non linear vibrations of stepped beam systems using artificial neural networks

In this study, the nonlinear vibrations of stepped beams having different boundary conditions were investigated. The equations of motions were obtained by using Hamilton\'s principle and made non dimensional. The stretching effect induced non-linear terms to the equations. Natural frequencies are calculated for different boundary conditions, stepped ratios and stepped locations by Newton-Raphson Method. The corresponding nonlinear correction coefficients are also calculated for the fundamental mode. At the second part, an alternative method is produced for the analysis. The calculated natural frequencies and nonlinear corrections are used for training an artificial neural network (ANN) program which has a multi-layer, feed-forward, back-propagation algorithm. The results of the algorithm produce errors less than 2.5% for linear case and 10.12% for nonlinear case. The errors are much lower for most cases except clamped-clamped end condition. By employing the ANN algorithm, the natural frequencies and nonlinear corrections are easily calculated by little errors, and the computational time is drastically reduced compared with the conventional numerical techniques.

Elasto-plastic and geometrically nonlinear vibrations of beams by the p-version finite element method

A model based on a Timoshenko beam p-version finite element is developed to analyse oscillations that are, simultaneously, elasto-plastic and geometrically nonlinear. The geometrical nonlinearity is represented by Von Kármán type strain–displacement relations and the stress–strain relation is of the bilinear type, with mixed strain hardening. The equations of motion are obtained using the principle of virtual work and are solved in the time domain by an implicit Newmark method. The Von Mises yield criterion is employed and the flow theory of plasticity applied; if plastic flow is found at a point of the domain, the total plastic strain is determined by summation. Numerical examples are carried out in order to demonstrate that the p-version element here advocated has a number of advantages and to show the influence of the plastic and geometrically nonlinear terms on the oscillations of beams.

Nonlinear vibration of edge cracked functionally graded Timoshenko beams

Nonlinear vibration of beams made of functionally graded materials (FGMs) containing an open edge crack is studied in this paper based on Timoshenko beam theory and von Kármán geometric nonlinearity. The cracked section is modeled by a massless elastic rotational spring. It is assumed that material properties follow exponential distributions through beam thickness. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear vibration frequencies of cracked FGM beams with different end supports. A detailed parametric study is conducted to study the influences of crack depth, crack location, material property gradient, slenderness ratio, and end supports on the nonlinear free vibration characteristics of cracked FGM beams. It is found that unlike isotropic homogeneous beams, both intact and cracked FGM beams show different vibration behavior at positive and negative amplitudes due to the presence of bending–extension coupling in FGM beams.