On Mode Interactions for a Weakly Nonlinear Beam Equation

In this paper an initial-boundary value problem for a weakly nonlinear beam equation with a Rayleigh perturbation will be studied. It will be shown that the calculations to find internal resonances in this case are much more complicated than and differ substantially from the calculations for the weakly nonlinear wave equation with a Rayleigh perturbation as for instance presented in [3] or [7]. The initial-boundary value problem can be regarded as a simple model describing wind-induced oscillations of flexible structures like suspension bridges or iced overhead transmission lines. Using a two-timescales perturbation method approximations for solutions of this initial-boundary value problem will be constructed.     

The Asymptotic Perturbation Method for Nonlinear Continuous Systems

We apply the asymptotic perturbation (AP) method to the study of the vibrations of Euler--Bernoulli beam resting on a nonlinear elastic foundation. An external periodic excitation is in primary resonance or in subharmonic resonance in the order of one-half with an nth mode frequency. The AP method uses two different procedures for the solutions: introducing an asymptotic temporal rescaling and balancing the harmonic terms with a simple iteration. We obtain amplitude and phase modulation equations and determine external force-response and frequency-response curves. The validity of the method is highlighted by comparing the approximate solutions with the results of the numerical integration and multiple-scale methods.     

Nonlinear Normal Modes of a Parametrically Excited Cantilever Beam

We investigate theoretically thenonlinear normal modes of a vertical cantilever beam excited by aprincipal parametric resonance. We apply directly the method ofmultiple scales to the governing nonlinear nonautonomousintegral-partial-differential equation and associated boundary conditions.In the absence of damping, it is shown that the system has nonlinear normal modes, as defined by Rosenberg, even in the presence of the parametric excitation.We calculate the spatial correction to the linear mode shapedue to the effects of the inertia and curvature nonlinearities andthe parametric excitation. We compare the result obtained withthe direct approach with that obtained using a single-mode Galerkindiscretization.The deviation between the two predictions increases as the oscillationamplitude increases.     

Normal Modes and Boundary Layers for a Slender Tensioned Beam on a Nonlinear Foundation

In this paper, the nonlinear normal modes (NNMs) of a thin beamresting on a nonlinear spring bed subjected to an axial tension isstudied. An energy-based method is used to obtain NNMs. In conjunction with amatched asymptotic expansion, we analyze, through simple formulas, thelocal effects that a small bending stiffness has on the dynamics, alongwith the secular effects caused by a symmetric nonlinearity. Nonlinearmode shapes are computed and compared with those of the unperturbedlinear system. A double asymptotic expansion is employed to compute theboundary layers in the nonlinear mode shape due to the small bendingstiffness. Satisfactory agreement between the theoretical and numericalbackbone curves of the system in the frequency domain is observed.     

随机窄带参激下二自由度非线性系统

研究了带平方二自由度非线性系统在随机窄带参数激励下，用多尺度法分离了系统的快变项，讨论了系统的各参数对响应的影响。在一定条件下，系统具有两个均方响应值，具有跳跃现象和饱和现象，数值模拟表明提出的方法是有效的。     

On the Nonlinear Dynamics of a Buckled Beam Subjected to a Primary-Resonance Excitation

We investigate the nonlinear response of a clamped-clamped buckled beamto a primary-resonance excitation of its first vibration mode. The beamis subjected to an axial force beyond the critical load of the firstbuckling mode and a transverse harmonic excitation. We solve thenonlinear buckling problem to determine the buckled configurations as afunction of the applied axial load. A Galerkin approximation is used todiscretize the nonlinear partial-differential equation governing themotion of the beam about its buckled configuration and obtain a set ofnonlinearly coupled ordinary-differential equations governing the timeevolution of the response. Single- and multi-mode Galerkinapproximations are used. We found out that using a single-modeapproximation leads to quantitative and qualitative errors in the staticand dynamic behaviors. To investigate the global dynamics, we use ashooting method to integrate the discretized equations and obtainperiodic orbits. The stability and bifurcations of the periodic orbitsare investigated using Floquet theory. The obtained theoretical resultsare in good qualitative agreement with the experimental results obtainedby Kreider and Nayfeh (Nonlinear Dynamics 15, 1998, 155–177.     

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.     

非线性弹性地基上矩形薄板受双频参数激励作用的非线性振动

应用弹性理论和Galerkin方法建立小挠度矩形薄板在非线性弹性地基上受两对均布纵向简谐激励作用的双模态非线性动力学方程。应用多尺度法求得系统满足双频主参数共振条件的一次近似解和对应的定常解,并进行了数值计算。分析了阻尼系数、地基系数、几何参数等对系统双频主参数共振的影响。     

Nonlinear Dynamics of a Beam on Elastic Foundation

The nonlinear dynamics of a simply supported beam resting on a nonlinear spring bed with cubic stiffness is analyzed. The continuous differential operator describing the mathematical model of the system is discretized through the classical Galerkin procedure and its nonlinear dynamic behavior is investigated using the method of Normal Forms. This model can be regarded as a simple system describing the oscillations of flexural structures vibrating on nonlinear supports and then it can be considered as a simple investigation for the analysis of more complex systems of the same type. Indeed, the possibility of the model to exhibit actually interesting nonlinear phenomena (primary, superharmonic, subharmonic and internal resonances) has been shown in a range of feasibility of the physical parameters. The singular perturbation approach is used to study both the free and the forced oscillations; specifically two parameter families of stationary solutions are obtained for the forced oscillations.     

Nonlinear Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances

Nonlinear normal modes of a fixed-fixed buckled beam about its first post-buckling configuration are investigated. The cases of three-to-one and one-to-one internal resonances are analyzed. Approximate solutions for the nonlinear normal modes are computed by applying the method of multiple scales directly to the governing integral-partial-differential equation and associated boundary conditions. Curves displaying variation of the amplitude of one of the modes with the internal-resonance-detuning parameter are generated. It is shown that, for a three-to-one internal resonance between the first and third modes, the beam may possess one stable uncoupled mode (high-frequency mode) and either (a) one stable coupled mode, (b) three stable coupled modes, or (c) two stable and one unstable coupled modes. For the same resonance, the beam possesses one degenerate mode (with a multiplicity of two) and two stable and one unstable coupled modes. On the other hand, for a one-to-one internal resonance between the first and second modes, the beam possesses (a) two stable uncoupled modes and two stable and two unstable coupled modes; (b) one stable and one unstable uncoupled modes and two stable and two unstable coupled modes; and (c) two stable uncoupled and two unstable coupled modes (with a multiplicity of two). For a one-to-one internal resonance between the third and fourth modes, the beam possesses (a) two stable uncoupled modes and four stable coupled modes; (b) one stable and one unstable uncoupled modes and four stable coupled modes; (c) two unstable uncoupled modes and four stable coupled modes; and (d) two stable uncoupled modes and two stable coupled modes (each with a multiplicity of two).     

Experimental Techniques and Identification of Nonlinear and Viscoelastic Properties of Flexible Polyurethane Foam

Identification of the vibrational behavior of polyurethanefoams used in automotive seats is described. The dynamic system consistsof a rigid block mounted on a 3 cube of foam material, which serves asthe only flexible component. When constrained to undergo linearunidirectional motion, the dynamic system is modeled as a single degreeof freedom system, governed by an integro-differential equation. Inaddition to a relaxation kernel representing the linear viscoelasticbehavior of the foam, the model includes a polynomial type stiffness toaccount for the foam's strain-based nonlinearities. The relaxationkernel is assumed to be of an exponential type. Experimentalmethodologies for obtaining repeatable, accurate measurements of thesystem's response to an impulse and to single frequency harmonic baseexcitations are described. Analysis methods are then investigated forextracting the relevant linear, nonlinear, and viscoelastic parameters.Characterization of these foam properties as functions of compressionlevel is also presented.     

Nonlinear System Identification of Multi-Degree-of-Freedom Systems

A nonlinear system identification methodology based on theprinciple of harmonic balance is extended tomulti-degree-of-freedom systems. The methodology, called HarmonicBalance Nonlinearity IDentification (HBNID), is then used toidentify two theoretical two-degree-of-freedom models and anexperimental single-degree-of freedom system. The three modelsand experiments deal with self-excited motions of afluid-structure system with a subcritical Hopf bifurcation. Theperformance of HBNID in capturing the stable and unstable limitcycles in the global bifurcation behavior of these systems is alsostudied. It is found that if the model structure is well known,HBNID performs well in capturing the unknown parameters. If themodel structure is not well known, however, HBNID captures thestable limit cycle but not the unstable limit cycle.     

非线性结构动力学瞬态分析的摄动法

在用直接积分法求解非线性结构的动力响应时,常常需要做迭代运算。本文引入摄动方法后,加快了收敛速度,提高了计算效益。     

Nonlinear Responses of a Magnetoelastic Beam in a Step-Pulsed Magnetic Field

A theoretical study is carried out on the dynamics of a magnetoelastic beam being in a step-pulsed magnetic field. For this aim, the magnetic potential and elastic energies are determined for the beam and partial differential equations are established according to Hamilton's principle. It is proven that the magnetoelastic beam can give a variety of complex behavior in the case of step-pulsed field excitations. An intermediate regime of two-well chaos is observed. Theoretical findings were found to be in a good agreement with the experimental results for the specific system parameters. On leave from Institute of Physics, University of Bayreuth, 65440 Bayreuth, Germany An erratum to this article is available at .     

Parametrically Excited Vibration of a Timoshenko Beam on Random Viscoelastic Foundation jected to a Harmonic Moving Load

The vibration response of a Timoshenko beam supported by a viscoelastic foundation with randomly distributed parameters along the beam length and jected to a harmonic moving load, is studied. By means of the first-order two-dimensional regular perturbation method and employing appropriate Green's functions, the dynamic response of the beam consisting of the mean and variance of the deflection and of the bending moment are obtained analytically in integral forms. Results of a field measurement for a test track are utilized to model the uncertainty of the foundation parameters. A frequency analysis is carried out and the effect of the load speed on the response is studied. It is found that the covariance functions of the stiffness and the loss factor both have the shape of an exponential function multiplied by a cosine function. Furthermore, it is shown that in each frequency response there is a peak value for the frequency, which changes inversely with the load speed. It is also found that the peak value of the mean and also standard deviation of the deflection and bending moment can be a decreasing or increasing function of the load speed depending on its frequency. An erratum to this article is available at .     

Saddle-Node Bifurcations of Cycles in a Relief Valve

We extend the asymptotic perturbation (AP) method to the studyof a linear partial differential equation with nonlinear boundaryconditions. A relief valve under the combined effects of static anddynamic loadings is considered with the following resonances between thenth linear mode and the external periodic excitation: primaryresonance, subharmonic resonance of order one-half or one-third,superharmonic resonance of order-two or order-three and combinationresonance. The AP method uses two different procedures for thesolutions: introducing an asymptotic temporal rescaling and balancing ofthe harmonic terms with a simple iteration. We obtain amplitude andphase modulation equations and determine external force-response andfrequency-response curves. Stability of steady-state motions is alsoinvestigated. Saddle-node bifurcations of cycles are observed and underappropriate conditions the performance of the relief valve may beunsatisfactory due to the presence of jumps and hysteresis effects inthe system response. Global analysis is used in order to exclude theexistence of modulated motion. The validity of the method is highlightedby comparing approximate solutions with results of the numericalintegration.     

Nonlinear System Identification of Systems with Periodic Limit-Cycle Response

A nonlinear system identification methodology based on the principle of harmonic balance and bifurcation theory techniques like center manifold analysis and normal form reduction, is presented for multi-degree-of-freedom systems. The methodology, called Bifurcation Theory System IDentification, (BiTSID), is a general procedure for any nonlinear system that exhibits periodic limit cycle response and can be used to capture the bifurcation behavior of the nonlinear systems. The BiTSID methodology is demonstrated on an experimental system single-degree-of-freedom system that deals with self-excited motions of a fluid-structure system with a sub-critical Hopf bifurcation. It is shown that BiTSID performs excellently in capturing the stable and unstable limit cycles within the experimental regime. Its performance outside the experimental regime is also studied. The application of BiTSID to experimental multi-degree-of-freedom systems has also been very successful. However in this study only the results of the single-degree-of-freedom system are presented.     

Nonlinear Dynamics of a Cantilever Beam Carrying an Attached Mass with 1:3:9 Internal Resonances

In this paper the nonlinear response of a base-excited slender beam carrying an attached mass is investigated with 1:3:9 internal resonances for principal and combinationparametric resonances. Here the method of normal forms is used to reduce the second order nonlinear temporal differential equation of motion of the system to a set offirst order nonlinear differential equations which are used to find the fixed-point, periodic, quasi-periodic and chaotic responses of the system.Stability and bifurcation analysis of the responses are carried out and bifurcation sets are plotted. Many chaotic phenomena are reported in this paper.