Nonlinear Free Vibration Of Beams Research Articles

Accurate analytical solution for nonlinear free vibration of beams

In this study, Hamiltonian Approach (HA) is applied to analysis the nonlinear free vibration of beams. Two well-known examples are illustrated to show the efficiency of this method. One of them deals with the Nonlinear vibration of an electrostatically actuated microbeam and the other is the nonlinear vibrations of tapered beams. This new approach prepares us to achieve the beam's natural frequencies and mode shapes easily and a rapidly convergent sequence is obtained during the solution. The effects of the small parameters on the frequency of the beams are discussed. Some comparisons are conducted between the results obtained by the Hamiltonian Approach (HA) and numerical solutions using to illustrate the effectiveness and convenience of the proposed methods.

Accurate modeling of preload discontinuity in the analytical approach of the nonlinear free vibration of beams

This article attempts to investigate the dynamical analysis of beam vibrations in the presence of preload discontinuity and proposes an innovative accurate equivalent function for this well-known nonlinearity. This approach enables us to overcome the inherent computational difficulty of the preload nonlinearity in the analytical investigations. At first, the nonlinear equation of beam vibration with preload boundary condition is considered and analytical solution is obtained using He’s parameter expanding method. The precision of the proposed equivalent function has been elucidated by comparison of our results with the obtained solutions using numerical method. Finally, the accuracy of the obtained results in the vibration analysis of suspension bridges as a realistic problem, verifies the strength of the presented modeling.

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.

Adaptive Control of Nonlinear Free Vibrations of Composite Plates Using Piezoelectric Actuators

A coupled structural-electrical finite element modal formulation is employed for the control of nonlinear free vibrations of beams and composite plates. Multiple modes of the nonlinear free vibration are considered in closed-loop simulations. Two different controllers are designed and investigated for the suppression of nonlinear free vibrations. The first is the position output feedback controller comprising a linear quadratic regulator (LQR) and an extended Kalman filter (EKF). The EKF is used to consider the nonlinear state-space matrix and has a gain sequence evaluated online. The second controller is the output feedback adaptive LQR/EKF controller. This adaptive controller includes an adaptive modal frequency identification and state estimation algorithm. Numerical simulations show that the adaptive LQR/EKF controller with system identification is effective in suppressing nonlinear free vibrations of a beam and a composite plate with unpredictable sudden frequency changes. The placement of piezoelectric self-sensing actuators is based on three approaches: one is the norm of optimal feedback control gain matrix method, the second is the H 2 norm, and the last one is the norm of Kalman filter estimation gain method.

A POWER SERIES SOLUTION FOR THE NON-LINEAR VIBRATION OF BEAMS

A power series solution is presented for the non-linear free vibration of beams with restrained ends. The analysis is based on transforming the time variable into an oscillating time which allows the motion of the beam, assumed to be periodic, to be expressed as a double power series that is convergent for all time. A recurrence relation is used to determine the series coefficients, with the initial movement satisfying the boundary conditions as its basis. Results are obtained for simply supported and clamped beams and compared with available solutions.

A symplectic Galerkin method for non-linear vibration of beams and plates

Abstract Non-linear free vibration of beams and plates is conservative in the sense that the total energy of the system remains constant. Symplectic numerical integration aims to preserve the energy, momenta and area (volume) of the phase space, and is ideal in the study of non-linear free vibration. In this paper, the Lagrangian of continuous systems is discretized by Galerkin’s method to obtain the discrete Hamiltonian and Hamilton’s equations. Symplectic numerical integration schemes are applied to the resulting ordinary differential equations to construct the phase diagram. It is shown that, due to modal coupling, bounded quasi-periodic and non-periodic free vibrations are very common for continuous systems. Comparison is made with the Runge-Kutta method. It is shown that the Runge-Kutta integration reduces the total energy of an undamped system while symplectic integration almost preserves it. The symplectic schemes are extended to ordinary differential equations by means of the Lagrangian and Hamiltonian, and the non-linear vibration of beams and plates is studied by the schemes for the first time. Non-linear modal coupling is emphasized. The extension to shell structures involves the complication of curvature effects. Research is in progress.

Non-Linear Free Vibrations of Beams By the Finite Element and Continuation Methods

A computational method is presented for determining backbone curves of arbitrary geometry. As an example, beam vibration is considered. The beam response is expanded into a truncated Fourier series with respect to time. After starting from the integral of action, the variational approach and the finite element method are used to formulate the non-linear eigenvalue problem. The continuation method is adopted to solve the resulting non-linear eigenvalue problem and to obtain the non-linear frequencies and modes of vibration. Numerical results for various beams are presented and compared with available results to demonstrate the accuracy and applicability of the method. Moreover, the bifurcation points on some beams' backbone curves are found and reported for the first time.

Application of the harmonic balance principle to the nonlinear free vibration of beams

This paper presents an analytical method for determining the vibration modes of geometrically nonlinear beams under various edge conditions. The method assumes a continuum periodic solution which allows the harmonic balance principle to be employed to derive modal components that satisfy the equation of motion exactly. Nonlinear normal modes are constructed from four such components and used to compute the natural frequencies for beams with restrained ends and for cantilevered beams. Numerical results for beams with restrained ends show good agreement with those available from other techniques.

Application of the Ritz method to the analysis of non-linear free vibrations of beams

This paper presents an analytical solution for geometrically non-linear free vibrations of beams with elastically supported ends in the horizontal direction. The equation of motion is obtained by employing Hamilton's principle and assuming that horizontal inertia forces can be neglected. The Ritz method, with a continuum solution and an iterative procedure, are used for determining the frequencies and non-linear modes of vibrations. The orthogonality conditions for these modes are also discussed. Numerical results for various beam boundary conditions are presented and compared with available results.

Nonlinear Free Vibration of Beams with Clamped Ends and with One End Clamped, Other End Simply Supported

In this paper, an approximate solution is given by the use of Galerkin's procedure for the large amplitude free vibration of a beam with immovable clamped ends, subjected to an initial axial load. The results of the vibrations of a beam with one end clamped and other end simply supported are known from those of the vibrations of the antisymmetric modes of a clamped beam. Numerical calculations are carried out for cases of the fundamental vibration of both beams. Finally, a comparison of the increasing rate of the fundamental frequency with the amplitude is made among the beams having different end conditions : both ends clamped, one end clamped and other end simply supported, and both ends simply supported.

Nonlinear Free Vibration of Beams with Clamped Ends and with One End Clamped, Other End Simply Supported

Nonlinear Free Vibration of Beams with Clamped Ends and with One End Clamped, Other End Simply Supported