Nonlinear dynamics of high-dimensional models of in-plane and out-of-plane vibration in an axially moving viscoelastic beam

Nonlinear dynamics of high-dimensional models of an axially moving viscoelastic beam with in-plane and out-of-plane vibration with combined parametric and forcing excitations are investigated by the incremental harmonic balance (IHB) method in this paper. Governing equations of transverse in-plane and out-of-plane and longitudinal vibration are obtained basing on the Hamilton's principle. The Galerkin method is used to separate time variable and spatial variable to obtain a set of multi-order differential equations. The IHB method with the fast Fourier transform (FFT) is used to solve periodic response of high-dimensional models of the beam for which convergent mode is reached. Stability of the steady-state periodic solutions is analyzed using the multivariable Floquet theory. Particular attention is paid to in-plane and out-of-plane vibration on convergent mode of the beam with combined parametric and forcing excitations. Multiple solutions are observed, and jump phenomena between in-plane and out-of-plane vibration with different transverse cross sections are discovered.     

Horizontal fast excitation in delayed van der Pol oscillator

This paper investigates the interaction effect of horizontal fast harmonic parametric excitation and time delay on self-excited vibration in van der Pol oscillator. We apply the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic of the oscillator. The method of averaging is then performed on the slow dynamic to obtain a slow flow which is analyzed for equilibria and periodic motion. This analysis provides analytical approximations of regions in parameter space where periodic self-excited vibrations can be eliminated. A numerical study is performed on the original oscillator and compared to analytical approximations. It was shown that in the delayed case, horizontal fast harmonic excitation can eliminate undesirable self-excited vibrations for moderate values of the excitation frequency. In contrast, the case without delay requires large excitation frequency to eliminate such motions. This work has application to regenerative behavior in high-speed milling.     

Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator

The discontinuous dynamics of a non-linear, friction-induced, periodically forced oscillator is studied. The analytical conditions for motion switchability at the velocity boundary in such a nonlinear oscillator are developed to understand the motion switching mechanism. Using such analytical conditions of motion switching, numerical predictions of the switching scenarios varying with excitation frequency and amplitude are carried out, and the parameter maps for specific periodic motions are presented. Chaotic and periodic motions are illustrated through phase planes and switching sections for a better understanding of motion mechanism of the nonlinear friction oscillator. The periodic motions with switching in such a nonlinear, frictional oscillator cannot be obtained from the traditional analysis (e.g., perturbation and harmonic balance method).     

Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation

A class of periodic motions of an inverted pendulum with rigid lateral barriers is analysed under the hypothesis that the system is forced by impulsed periodic excitation. Due to the piece-wise linear nature of the problem, the existence and the stability of the cycles are determined analytically. It is found that they depend on both classical (saddle-node and period-doubling) and non-classical bifurcations, the latter involving a ‘synchronization' between impulses and impacts which leads to the sudden disappearing of the orbits. Attention is paid to the physical interpretation of these bifurcations, and to the determination of analytical criteria for their occurrence. We study how the relative position (with respect to the excitation amplitude) of the local bifurcations determines the system response and the bifurcation scenario. Symmetric and unsymmetric excitations are considered and the regions of stability of the periodic solutions are analytically determined. Finally, a comparison with the case of harmonic excitation is presented showing both analogies and differences, and highlighting how the impulsed excitation allows to obtain stable periodic responses at higher values of the excitation amplitude.     

Homoclinic bifurcation and chaos control in MEMS resonators

The chaotic dynamics of a micromechanical resonator with electrostatic forces on both sides are investigated. Using the Melnikov function, an analytical criterion for homoclinic chaos in the form of an inequality is written in terms of the system parameters. Detailed numerical studies including basin of attraction, and bifurcation diagram confirm the analytical prediction and reveal the effect of parametric excitation amplitude on the system transition to chaos. The main result of this paper indicates that it is possible to reduce the electrostatically induced homoclinic and heteroclinic chaos for a range of values of the amplitude and the frequency of the parametric excitation. Different active controllers are applied to suppress the vibration of the micromechanical resonator system. Moreover, a time-varying stiffness is introduced to control the chaotic motion of the considered system. The techniques of phase portraits, time history, and Poincare maps are applied to analyze the periodic and chaotic motions.     

Multiple bifurcation trees of period-1 motions to chaos in a periodically forced,time-delayed,hardening Duffing oscillator

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.     

Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time-varying stiffness

In this paper, we investigate the Shilnikov type multi-pulse chaotic dynamics for a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time-varying in a periodic form. The dimensionless equation of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions is a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found from the numerical results that there are the phenomena of the Shilnikov type multi-pulse chaotic motions for the rotor-AMB system. A new jumping phenomenon is discovered in the rotor-AMB system with the time-varying stiffness.     

Chaos of several typical asymmetric systems

The threshold for the onset of chaos in asymmetric nonlinear dynamic systems can be determined using an extended Padé method. In this paper, a double-well asymmetric potential system with damping under external periodic excitation is investigated, as well as an asymmetric triple-well potential system under external and parametric excitation. The integrals of Melnikov functions are established to demonstrate that the motion is chaotic. Threshold values are acquired when homoclinic and heteroclinic bifurcations occur. The results of analytical and numerical integration are compared to verify the effectiveness and feasibility of the analytical method.     

Response analysis for a vibroimpact Duffing system with bilateral barriers under external and parametric Gaussian white noises

In this paper, a vibroimpact Duffing oscillator with two barriers that are symmetrical with respect to the equilibrium point of the system is considered for the cases of external and parametric Gaussian white noise random excitations. According to the levels of the system energy, the motions of the unperturbed vibroimpact system are divided into two types: oscillations without impacts and oscillations with alternate impacts on both sides. Then, under the assumption that the vibroimpact Duffing system is quasi-conservative, the stochastic averaging method for energy envelope is applied to obtain the averaged drift and diffusion coefficients for the two types of motions, respectively. The Probability Density Functions (PDFs) of stationary responses are derived by solving the corresponding Fokker-Plank-Kolmogorov (FPK) equation. Lastly, results obtained from the proposed procedure are validated by directly numerical simulation. Meanwhile, effects of the position of bilateral barriers and the random excitations on the PDFs of the stationary responses are also discussed.     

Numerical path integration method based on bubble grids for nonlinear dynamical systems

A new numerical path integration method based on bubble grids for nonlinear dynamical systems is presented in this paper. The ordinary differential equations for the first and second order moments are derived on the basis of the Gaussian closure method. Then the probability density values on the bubble nodes in the computational domain can be calculated via the obtained method. The good performance of the resulting method is finally shown in the numerical examples by using some specific nonlinear dynamical systems: Duffing oscillator subjected to harmonic and stochastic excitations, and Duffing–Rayleigh oscillator subjected to harmonic and stochastic excitations.     

Periodic bifurcation of Duffing-van der Pol oscillators having fractional derivatives and time delay

In this paper, a Duffing-van der Pol oscillator having fractional derivatives and time delays is investigated by the residue harmonic method. The angular frequencies and limit cycles of periodic motions are expanded into a power series of an order-tracking parameter and the unbalanced residues resulting from the truncated Fourier series are considered iteratively to improve the accuracy. The periodic bifurcations are examined using the fractional order, feedback gain and time delay as continuation parameters. It is shown that jumps and hysteresis phenomena can be delayed or removed. Transition from discontinuous bifurcation to continuous bifurcation is observed. The approximations are verified by numerical integration. We find that the proposed method can easily be programmed and can predict accurate periodic approximations while the system parameters being unfolded.     

Elimination of tall building vibration resulting from ground motions

In this paper, the vibration problems of tall buildings are considered. The focus is on vibration caused by earthquakes, semi–seismic phenomena and ground vibrations of other origins. The construction consists of the main system and a vibration eliminator (passive tuned mass damper – pendulum type) which is attuned to the first eigenfrequency of the main structure. The analysis focuses on elimination of structure vibration caused by horizontal components of ground motions, while the functioning of the eliminator is simultaneously influenced by the vertical component (parametric effect – the possibility of improper functioning of the device). The vertical periodic movement of the support point can cause changes of the vibration eliminator's stiffness. In such a case parametric excitation occurs in the system, which signifies that parametric resonance may appear. The numerical analysis of the problem was performed with the Newmark method in conjunction with FEM. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic chaos in a Duffing oscillator and its control

Stochastic chaos discussed here means a kind of chaotic responses in a Duffing oscillator with bounded random parameters under harmonic excitations. A system with random parameters is usually called a stochastic system. The modifier ‘stochastic’ here implies dependent on some random parameter. As the system itself is stochastic, so is the response, even under harmonic excitations alone. In this paper stochastic chaos and its control are verified by the top Lyapunov exponent of the system. A non-feedback control strategy is adopted here by adding an adjustable noisy phase to the harmonic excitation, so that the control can be realized by adjusting the noise level. It is found that by this control strategy stochastic chaos can be tamed down to the small neighborhood of a periodic trajectory or an equilibrium state. In the analysis the stochastic Duffing oscillator is first transformed into an equivalent deterministic nonlinear system by the Gegenbauer polynomial approximation, so that the problem of controlling stochastic chaos can be reduced into the problem of controlling deterministic chaos in the equivalent system. Then the top Lyapunov exponent of the equivalent system is obtained by Wolf’s method to examine the chaotic behavior of the response. Numerical simulations show that the random phase control strategy is an effective way to control stochastic chaos.     

非线性振动系统的异宿轨道分叉,次谐分叉和混沌

在参数激励与强迫激励联合作用下具有van der Pol阻尼的非线性振动系统,其动态行为是非常复杂的.本文利用Melnikov方法研究了这类系统的异宿轨道分叉、次谐分叉和混沌.对于各种不同的共振情况,系统将经过无限次奇阶次谐分叉产生Smale马蹄而进入混沌状态.最后我们利用数值计算方法研究了这类系统的混沌运动.所得结果揭示了一些新的现象.     

Bifurcating periodic solution for a class of first-order nonlinear delay differential equation after Hopf bifurcation

In this paper, we predict the accurate bifurcating periodic solution for a general class of first-order nonlinear delay differential equation with reflectional symmetry by constructing an approximate technique, named residue harmonic balance. This technique combines the features of the homotopy concept with harmonic balance which leads to easy computation and gives accurate prediction on the periodic solution to the desired accuracy. The zeroth-order solution using just one Fourier term is applied by solving a set of nonlinear algebraic equations containing the delay term. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear equations to improve the accuracy and increase the number of Fourier terms of the solutions successively. It is shown that the solutions are valid for a wide range of variation of the parameters by two examples. The second-order approximations of the periodic solutions are found to be in excellent agreement with those obtained by direct numerical integration. Moreover, the residue harmonic balance method works not only in determining the amplitude but also the frequency of the bifurcating periodic solution. The method can be easily extended to other delay differential equations.     

Nonlinear response of a beam under distributed moving contact load

In this paper, the nonlinear behavior of a one-dimensional model of the disc brake pad is examined. The contact normal force between the disc brake pad lining and rotor is represented by a second order polynomial of the relative displacement between the two elastic bodies. The frictional force due to the sliding motion of the rotor against the stationary pad is modeled as a distributed follower-type axial load with time-dependent terms. By Galerkin discretization, the equation governing the transverse motion of the beam model is reduced to a set of extended Duffing system with quasi-periodically modulated excitations. Retaining the first two vibration modes in the governing equations, frequency response curves are obtained by applying a two-dimensional spectral balance method. For the first time, it is predicted that nonlinearity resulting from the contact mechanics between the disc brake pad lining and rotor can lead to a possible irregular motion (chaotic vibration) of the pad in the neighborhood of simple and parametric resonance. This chaotic behavior is identified and quantitatively measured by examining the Poincaré maps, Fourier spectra, and Lyapunov exponents. It is also found that these chaotic motions emerge as a result of successive Hopf bifurcations characterized by the torus breakdown and torus doubling routes as the excitation frequency varies. Various aspects of the numerical difficulties in the solution of the nonlinear equations are also discussed.     

A successive integration technique for the solution of the Duffing oscillator equation with damping and excitation

In this paper, a successive integration technique is suggested for solving the Duffing oscillator equation with damping and excitation. In the technique, one performs integrations from some initial values over a wider range. Some of them may even be divergent; however, some of them will give a convergent result such that the periodic condition of motion is satisfied. In fact, the convergent result represents a stable periodic solution for the motion. If a convergent result or a stable periodic solution is obtained, the stability test is passed. A harmonic balance method in conjunction with the successive integration technique (abbreviated as HBMSIT) is also suggested. In the method, the initial values are obtained from the harmonic balance method. Therefore, the HBMSIT belongs to the successive integration techniques. Many examples with calculated results are presented.     

参数振动受迫响应的三角级数解

基于调制反馈方法,对参数周期与激励力周期不相同情况下,研究其参数系统受迫振动响应三角级数解．采用谐波的线性组合形式从数学上表达受迫振动响应解,然后通过运用谐波平衡,将参数振动方程转化成无限阶线性代数方程组,解出其谐波的系数．上述方法的特点在于:1） 用三角级数来表达振动受迫响应,十分便于参数振动的频域分析,剖析受迫响应性质;2） 从解的表达可直接推出组合谐波共振条件; 3） 采用标准的Runge Kutta算法得到的相图证实上述方法结果的精确性．研究结果表明:该方法适用于参数振动完整受迫响应解的数学表达与分析．     

Bifurcation and chaos of thin circular functionally graded plate in thermal environment

A ceramic/metal functionally graded circular plate under one-term and two-term transversal excitations in the thermal environment is investigated, respectively. The effects of geometric nonlinearity and temperature-dependent material properties are both taken into account. The material properties of the functionally graded plate are assumed to vary continuously through the thickness, according to a power law distribution of the volume fraction of the constituents. Using the principle of virtual work, the nonlinear partial differential equations of FGM plate subjected to transverse harmonic forcing excitation and thermal load are derived. For the circular plate with clamped immovable edge, the Duffing nonlinear forced vibration equation is deduced using Galerkin method. The criteria for existence of chaos under one-term and two-term periodic perturbations are given with Melnikov method. Numerical simulations are carried out to plot the bifurcation curves for the homolinic orbits. Effects of the material volume fraction index and temperature on the criterions are discussed and the existences of chaos are validated by plotting phase portraits, Poincare maps. Also, the bifurcation diagrams and corresponding maximum Lyapunov exponents are plotted. It was found that periodic, multiple periodic solutions and chaotic motions exist for the FGM plate under certain conditions.