Nonlinear Vibrations Of Beams Research Articles

NON-LINEAR STEADY STATE VIBRATIONS OF BEAMS EXCITED BY VORTEX SHEDDING

In this paper the non-linear vibrations of beams excited by vortex-shedding are considered. In particular, the steady state responses of beams near the synchronization region are taken into account. The main aerodynamic properties of wind are described by using the semi-empirical model proposed by Hartlen and Currie. The finite element method and the strip method are used to formulate the equation of motion of the system treated. The harmonic balance method is adopted to derive the amplitude equations. These equations are solved with the help of the continuation method which is very convenient to perform the parametric studies of the problem and to determine the response curve in the synchronization region. Moreover, the equations of motion are also integrated using the Newmark method. The results of calculations of several example problems are also shown to confirm the efficiency and accuracy of the presented method. The results obtained by the harmonic balance method and by the Newmark methods are in good agreement with each other.

HIERARCHICAL FINITE ELEMENT ANALYSES OF GEOMETRICALLY NON-LINEAR VIBRATION OF BEAMS AND PLANE FRAMES

Geometrically non-linear vibrations of beams and plane frameworks are analyzed by the hierarchical finite element method (HFEM). Two main points are of interest. The first is to compare polynomials, trigonometric functions and beam eigenfunctions as displacement shape functions for beam hierarchical finite elements. The second is to examine the suitability of the HFEM for time domain non-linear analyses.

Nonlinear vibration of buckled beams: some exact solutions

In this paper, exact solutions are obtained for the dynamics of buckled beams with different types of end conditions. The differential equation for the vibrations of buckled beams, with restrictions on axial stretch, is nonlinear. Using the modes of the corresponding linear problem, which readily satisfy the boundary conditions, the natural frequencies are obtained for the nonlinear problem by using Jacobi elliptic functions. Analytical solutions for the dynamics of buckled beams, with various rotational restraints, are presented.

Control of Nonlinear Vibration of Beams Using Active Constrained Layer Damping

Active constrained layer damping (ACLD) treatment is used to control the nonlinear vibration of beams. The beams are fully treated with ACLD treatment. A variational model of the beam /ACLD system is derived based on the von Kármán-type nonlinear deformations. The resulting governing partial differential equations of motion are reduced to nonlinear differential equations using Galerkin's method. These equations are solved by the method of harmonic balance. The effectiveness of the ACLD treatment to damp out the nonlinear vibration is demonstrated with frequency response function plots for a simply supported beam. The obtained results suggest the potential use of ACLD treatment in controlling the nonlinear vibrations of beams.

Nonlinear vibration of Timoshenko beam due to moving loads including the effects of weight and longitudinal inertia of beam

The effects of weight and axial inertia of a beam are taken into account for studying the nonlinear vibration of the Timoshenko beam due to external loads. The combination of Galerkins method and Runge-Kutta method are employed to obtain the dynamic responses of the beam. A concentrated force and a two-axle vehicle traversing on the beam are taken as two examples to investigate the response characteristics of the beam. Results show that the effect of axial inertia of the beam increases the fundamental period of the beam. Further, both the dynamic deflection and the dynamic moment of the beam obtained with including the effect of axial inertia of the beam are greater than those of the beam without including that effect of the beam.

3810 周期横励振を受ける懸垂状偏平はりのカオス振動挙動の解析

Analytical result is presented on nonlinear vibrations of suspended beams in a shape of catenary. The beam having a small deflection is simply supported at both ends and subjected to a periodic lateral force. Basic equations are reduced to the ordinary differential equation of multiple-degrees-of-freedom systems by the Galerkin procedure. To find out the effect of sag-to-span ratio on chaotic motions, first, steady-state responses are calculated by the harmonic balance method. The chaotic responses are examined by numerical integration. The following results are obtained : chaotic vibrations appears mainly in the subharmonic resonance of 1/3 order and ultra-subharmonic resonance of 3/2 order corresponding to the lowest symmetric mode of vibration. Decreasing the exciting frequency from high frequency range to low one, the chaotic vibration is generated with the two bifurcation types. Those are the periodic doubling bifurcation and the bifurcation originated for the internal resonance. As the sag-to-span ratio increases, chaotic motion appears in a wide range of frequency. Furthermore, chaotic responses are bifurcated from the subharmonic resonances both 1/2 and 1/3orders in the same frequency region.

Finite deformation beam models and triality theory in dynamical post-buckling analysis

Two new finitely deformed dynamical beam models are established for serious study on non-linear vibrations of thick beams subjected to arbitrarily given external loads. The total potentials of these beam models are non-convex with double-well structures, which can be used in post-buckling analysis and frictional contact problems. Dual extremum principles in unstable dynamic systems are developed. A pure complementary energy principle (in terms of the second Piola–Kirchhoff’s type stress only) in finite deformation mechanics is actually constructed. An interesting triality theory in post-buckling analysis is proved. This theory shows that if the gap function introduced by Gao and Strang in 1989 in positive, the generalized pure complementary energy has only one saddle point, which gives a global stable buckling state. However, if the gap function is negative, the generalized complementary energy may have two so-called super-critical points: the one which minimizes the pure complementary energy gives another relatively stable buckling state; and the other one which maximizes the complementary energy is a unstable buckling state. Application in unilateral buckling problem is illustrated, and an analytic solution for non-linear complementarity problem is obtained. Moreover, the general duality theory proposed recently is generalized into the non-linear dynamical systems. A pair of dual Duffing equations are obtained.

Infinite-Dimensional Control of Nonlinear Beam Vibrations by Piezoelectric Actuator and Sensor Layers

An infinite-dimensional approach for the active vibration control of a multilayered straight composite piezoelectric beam is presented. In order to control the excited beam vibrations, distributed piezoelectric actuator and sensor layers are spatially shaped to achieve a sensor/actuator collocation which fits the control problem. In the sense of von Karman a nonlinear formulation for the axial strain is used and a nonlinear initial boundary-value problem for the deflection is derived by means of the Hamilton formalism. Three different control strategies are proposed. The first one is an extension of the nonlinear H∞-design to the infinite-dimensional case. It will be shown that an exact solution of the corresponding Hamilton–Jacobi–Isaacs equation can be found for the beam under investigation and this leads to a control law with optimal damping properties. The second approach is a PD-controller for infinite-dimensional systems and the third strategy makes use of the disturbance compensation idea. Under certain observability assumptions of the free system, the closed loop is asymptotically stable in the sense of Lyapunov. In this way, flexural vibrations which are excited by an axial support motion or by different time varying lateral loadings, can be suppressed in an optimal manner. A numerical example serves both to illustrate the design process and to demonstrate the feasibility of the proposed methods.

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.

Nonlinear vibrations in beams and frames: The effect of the deformed equilibrium state

In the study of nonlinear vibrations of planar frames and beams with infinitesimal displacements and strains, the influence of the static displacements resulting from gravity effect and other conservative loads is usually disregarded. This paper discusses the effect of the deformed equilibrium configuration on the nonlinear vibrations through the analysis of two planar structures. Both structures present a two-to-one internal resonance and a primary response of the second mode. The equations of motion are reduced to two degrees of freedom and contain all geometrical and inertial nonlinear terms. These equations are derived by modal superposition with additional subsidiary conditions. In the two cases analyzed, the deformed equilibrium configuration virtually coincides with the undeformed configuration. Also, 2% is the maximum difference presented by the first two lower frequencies. The modes are practically coincident for the deformed and undeformed configurations. Nevertheless, the analysis of the frequency response curves clearly shows that the effect of the deformed equilibrium configuration produces a significant translation along the detuning factor axis. Such effect is even more important in the amplitude response curves. The phenomena represented by these curves may be distinct for the same excitation amplitude.

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.

Investigation of natural frequencies and mode shapes of buckled beams

The linear modes of vibration of buckled beams are investigated analytically and experimentally. Assuming a static buckled shape corresponding to the nth buckling mode, an exact solution is obtained for the linear modes and associated frequencies of initially buckled beams with fixed-fixed, fixed-hinged, and hinged-hinged boundary conditions. The analytical solution for a first-mode buckled beam is validated experimentally for the fixed-fixed case. The analytically obtained natural frequencies are in excellent agreement with those obtained experimentally. These exact linear modes provide a basis from which to study the nonlinear vibrations of buckled beams.

Symplectic Integration and Nonlinear Dynamic Symmetry Breaking of Frames

An accurate beam finite element is used to solve nonlinear vibration of arched beams and framed structures. The nonlinear governing equations of a skeletal structure are integrated numerically using symplectic integration schemes so that the Poincaré integral invariant of a Hamiltonian flow are preserved during the evolution. The element stiffness matrices are not required to be assembled into global form, because the integration is completed on an element level so that many elements can be handled in core by a small computer. Testing examples include arched beams and frames with and without damping in free and forced vibration. The dynamic symmetry breaking phenomena are noted at the dynamic buckling point.

Dynamical potential for non-linear vibrations of cantilevered beams

Recently it was shown that averaged characteristics of non-linear vibrations are potential functions of load parameters in the limit of small dissipation. The question of the range of dissipation for which potentiality takes place remained open. In this paper, we study this question for the case of non-linear vibrations of a cantilever beam excited harmonically at the unclamped end. We develop a non-linear one-degree-of-freedom beam model and show that the existence of a dynamical potential of beam vibrations can be guaranteed with acceptable accuracy even for sufficiently large dissipation.

Symplectic Integration and Nonlinear Dynamic Symmetry Breaking of Frames

An accurate beam finite element is used to solve nonlinear vibration of arched beams and framed structures. The nonlinear governing equations of a skeletal structure are integrated numerically using symplectic integration schemes so that the Poincare integral invariant of a Hamiltonian flow are preserved during the evolution. The element stiffness matrices are not required to be assembled into global form, because the integration is completed on an element level so that many elements can be handled in core by a small computer. Testing examples include arched beams and frames with and without damping in free and forced vibration. The dynamic symmetry breaking phenomena are noted at the dynamic buckling point.

Local smoothing effects for a nonlinear Timoshenko type equation

where x E IR”, t r 0. Here A denotes the usual Laplace operator, VU is the spatial gradient of u and M(s) is a nonlinear smooth function. The term M(jR” ]Vu12 du) Au is usually known as Timoshenko’s type nonlinearity and appears in a variety of models, mostly associated with nonlinear vibrations of beams [l]. Results on the existence of global solutions of (1.1) are available in the literature [2]. We are concerned here with further properties of the solutions, more specifically, we show that there is a class of solutions of (1.1) which are (locally) more regular than their initial data. We prove the result assuming a suitable decay of the data at infinity. Our main results (theorems 3.1 and 3.2) in this paper will only be given in the one-dimensional case. However, as will become clear during our proofs, several ingredients are valid in any dimension. There is a close relationship between solutions of (1 .l) and solutions of a nonlinear Schrodinger equation (at the end of this section we will give more details on this fact). Thanks to this observation our strategy was to combine classical energy estimates with the fact that the operator L = a2/at2 + aVax4 6‘ almost” commutes with the operators PM = xu + 2tD-‘u, and

An efficient algorithm for elasto-viscoplastic vibrations of multi-layered composite beams using second-order theory

An efficient time-domain algorithm for plane non-linear flexural vibrations of multi-layered composite beams, which are driven into the inelastic range by severe transverse loadings, is presented. The influence of an axial static preload is considered in the sense of the quasi-linear second-order theory of structures. The inelastic parts of strain are treated as additional sources of selfstress in the linear elastic background-structure, driving the elastic response into the inelastic one. The efficiency of this exact formulation lies in the fact that linear solution techniques can be used in their most powerful form: Rubin's useful formulation for the quasi-static second-order transfer-matrix of linear elastic structures is applied in combination with modal analysis. Having in mind multi-metal beams, the classical lamination theory is assumed to be valid. Beams with overhang composed of ideal elastic-plastic and viscoplastic layers are studied as example structures. The fictitious sources of selfstress are calculated from the different material laws of the layers in a numerical time-stepping procedure, where a generalized midpoint-rule in combination with Crisfield's secant-Newton procedure is used.

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.

Non-linear damped vibration of beams with periodic forcing

We study the forced oscillations of beams with non-linear damping. A numerical method is proposed which obtains the unique periodic solution by means of an initial-boundary value problem technique. The numerical method is motivated and justified by first studying a coupled system of ordinary differential equations.