Experimental unfolding of the nonlinear dynamics of a cable-mass suspended system around a divergence-Hopf bifurcation

Systematic experimental investigation of the finite amplitude dynamics of a multiple internally resonant suspended cable-mass, subjected to anti-phase support motion at primary resonance, is accomplished. Upon getting hints from a basic system configuration assumed as reference setup about the multiple bifurcation event possibly governing transition to complex dynamics, an improved experimental apparatus is used to make it technically accessible. Results obtained by varying three control parameters, namely the frequency and amplitude of excitation and the temperature of a thermostatic chamber embedding the experimental system, allow us to characterize in-depth various occurring classes of motion in terms of time and spatial complexity, to describe peculiar and/or persistent features of transition to nonregular dynamics, and to trace them back to a canonical scenario from bifurcation theory. Variable response paths are detected via bifurcation diagrams and spectra of singular values of measurement results, and overall behaviour charts are built in the excitation parameter space. Considering the temperature as a controllable parameter shows to be fundamental for: (i) indirectly setting cable material properties to values for which the conjectured codimension 2 bifurcation becomes apparent, (ii) qualitatively referring the experimental unfolding of regular and nonregular cable dynamics to the theoretical unfolding of the divergence-Hopf bifurcation normal form, and (iii) determining system response not only in the strict neighbourhood of the organizing divergence-Hopf bifurcation but also in the ensuing postcritical regions where the dependence of material damping on temperature affects secondary bifurcations to low-dimensional homoclinic chaos.     

Study of nonlinear system stability using eigenvalue analysis: Gyroscopic motion

General computational multibody system (MBS) algorithms allow for the linearization of the highly nonlinear equations of motion at different points in time in order to obtain the eigenvalue solution. This eigenvalue solution of the linearized equations is often used to shed light on the system stability at different configurations that correspond to different time points. Different MBS algorithms, however, employ different sets of orientation coordinates, such as Euler angles and Euler parameters, which lead to different forms of the dynamic equations of motion. As a consequence, the forms of the linearized equations and the eigenvalue solution obtained strongly depend on the set of orientation coordinates used. This paper addresses this fundamental issue by examining the effect of the use of different orientation parameters on the linearized equations of a gyroscope. The nonlinear equations of motion of the gyroscope are formulated using two different sets of orientation parameters: Euler angles and Euler parameters. In order to obtain a set of linearized equations that can be used to define the eigenvalue solution, the algebraic equations that describe the MBS constraints are systematically eliminated leading to a nonlinear form of the equations of motion expressed in terms of the system degrees of freedom. Because in MBS applications the generalized forces can be highly nonlinear and can depend on the velocities, a state space formulation is used to solve the eigenvalue problem. It is shown in this paper that the independent state equations formulated using Euler angles and Euler parameters lead to different eigenvalue solutions. This solution is also different from the solution obtained using a form of the Newton-Euler matrix equation expressed in terms of the angular accelerations and angular velocities. A time-domain solution of the linearized equations is also presented in order to compare between the solutions obtained using two different sets of orientation parameters and also to shed light on the important issue of using the eigenvalue analysis in the study of MBS stability. The validity of using the eigenvalue analysis based on the linearization of the nonlinear equations of motion in the study of the stability of railroad vehicle systems, which have known critical speeds, is examined. It is shown that such an eigenvalue analysis can lead to wrong conclusions regarding the stability of nonlinear systems.     

一类相对转动非线性动力系统的混沌运动

研究一类具有同宿轨道、异宿轨道的相对转动非线性动力系统的混沌运动. 建立具有非线性刚度、非线性阻尼和外扰激励作用的一类两质量相对转动非线性动力系统的动力学方程. 利用Melnikov方法讨论了系统的全局分岔和系统进入混沌状态的可能途径,给出了系统发生混沌的必要条件,并利用最大Lyapunov指数图,分岔图,Poincare截面图和相轨迹图进一步分析了系统的混沌行为. 关键词： 相对转动 非线性动力系统 混沌 Melnikov方法     

一类相对转动时滞非线性动力系统的稳定性分析

建立了具有时变刚度、非线性阻尼和谐波激励的一类相对转动时滞非线性动力系统的动力学方程.采用多尺度法推导出时滞动力系统的分岔响应方程,运用奇异性理论研究系统结构稳定性,得到主共振稳态响应方程的转迁集以及不同参数下分岔曲线的拓扑结构.应用Hopf分岔理论讨论了时滞动力系统动态稳定性,给出了系统产生极限环的条件,最后用数值模拟的方法研究了时滞参数对系统极限环幅值的影响。     

Experimental assessment of a 3-D plenoptic endoscopic imaging system

An endoscopic imaging system using a plenoptic technique to reconstruct 3-D information is demonstrated and analyzed in this Letter. The proposed setup integrates a clinical surgical endoscope with a plenoptic camera to achieve a depth accuracy error of about 1 mm and a precision error of about 2 mm, within a 25 mm × 25 mm field of view, operating at 11 frames per second.     

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass-spring-damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation.     

Relaxation of probability characteristics of the brownian motion of a nonlinear oscillator

We consider an oscillator with nonlinear elasticity and nonlinear damping under the action of a Gaussian delta-correlated random force. The oscillator is treated as a Brownian particle in the corresponding potential profile. We analyze the problem using the analytical-numerical method based on solving the chain of differential equations for the statistical moments, which is broken in a certain manner. For the case of nonlinear elasticity, we find the dependence of the relaxation times of the mean values and variances of both the coordinates and velocities on the system parameters and noise intensity. By analogy, the relaxation of the probability characteristics of the oscillation amplitude is studied for a system with nonlinear damping. In both cases, the evolution of the Gaussian or Rayleigh probability distributions is described on the basis of the moment relaxation. Nizhny Novgorod Architectural and Construction University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 4, pp. 468–478, September, 2000.     

Experimental study and analysis on the rising motion of grains in a vertically-vibrated pipe

Previous experimental investigations have shown that when a narrow pipe is inserted into a granular bed and is vibrated vertically but the granular bed is kept still, the grains in the bed can enter the pipe and rise against gravity along the pipe and finally stabilized at a certain height. The growth velocity and final stable height of the grain column inside the pipe can be controlled by varying the vibration conditions. In this paper, we discuss those experimental findings. We establish a mathematic relation between the grain column height(h) and time(t), and by using the relation we discuss the change of the growth velocity( dh/ dt) and acceleration( d2h/ dt2) with t and h, respectively. We also analyze the mechanism of the rising motion of the grains during vibration. Furthermore, we derive a theoretical expression for describing the final stable height(h st), which shows that the main factors influencing the height are vibration strength(Γ), bulk density of grains,inner diameter of the pipe, and vibration frequency, and that h st increases nonlinearly in the presence of air and linearly in a vacuum environment with increasing Γ.     

The analysis of a novel 3-D autonomous system and circuit implementation

This Letter presents a new three-dimensional autonomous system with four quadratic terms. The system with five equilibrium points has complex chaotic dynamics behaviors. It can generate many different single chaotic attractors and double coexisting chaotic attractors over a large range of parameters. We observe that these chaotic attractors were rarely reported in previous work. The complex dynamical behaviors of the system are further investigated by means of phase portraits, Lyapunov exponents spectrum, Lyapunov dimension, dissipativeness of system, bifurcation diagram and Poincaré map. The physical circuit experimental results of the chaotic attractors show agreement with numerical simulations. More importantly, the analysis of frequency spectrum shows that the novel system has a broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.     

Nonparametric identification of nonlinear dynamic systems using a synchronisation-based method

The present study proposes an identification method for highly nonlinear mechanical systems that does not require a priori knowledge of the underlying nonlinearities to reconstruct arbitrary restoring force surfaces between degrees of freedom. This approach is based on the master–slave synchronisation between a dynamic model of the system as the slave and the real system as the master using measurements of the latter. As the model synchronises to the measurements, it becomes an observer of the real system. The optimal observer algorithm in a least-squares sense is given by the Kalman filter. Using the well-known state augmentation technique, the Kalman filter can be turned into a dual state and parameter estimator to identify parameters of a priori characterised nonlinearities. The paper proposes an extension of this technique towards nonparametric identification. A general system model is introduced by describing the restoring forces as bilateral spring-dampers with time-variant coefficients, which are estimated as augmented states. The estimation procedure is followed by an a posteriori statistical analysis to reconstruct noise-free restoring force characteristics using the estimated states and their estimated variances. Observability is provided using only one measured mechanical quantity per degree of freedom, which makes this approach less demanding in the number of necessary measurement signals compared with truly nonparametric solutions, which typically require displacement, velocity and acceleration signals. Additionally, due to the statistical rigour of the procedure, it successfully addresses signals corrupted by significant measurement noise. In the present paper, the method is described in detail, which is followed by numerical examples of one degree of freedom (1DoF) and 2DoF mechanical systems with strong nonlinearities of vibro-impact type to demonstrate the effectiveness of the proposed technique.     

Simple real-time 3-D displacement measurement by fringe analysis using a photorefractive interferometer

A simple 3-D displacement measurement method using a photorefractive interferometer is proposed and experimentally demonstrated. A photorefractive crystal attached to the sample under test is used to record an original spherical wave that provides a shifted reconstructed wave after displacement. The shifted wave interferes with direct wave to form an interferogram. By analyzing the fringe pattern the in-place and out-of-place displacements can be determined at the same time. This method works in real-time and is easy to operate with adequate accuracy comparable with or even better than that of other techniques.     

微穿孔平板式空间吸声体的理论分析

对微穿孔平板式空间吸声体的吸声特性进行了理论预计,其中运用了半厚度模型分析和四端网络计算方法。计算结果与混响室实测数据符合较好。文中根据预测结果,分析了影响微穿孔平板式空间吸声体吸声性能的诸项主要因素,可为设计应用提供指南。     

Identification and control of chaos in nonlinear gear dynamic systems using Melnikov analysis

In this paper, the Melnikov analysis is extended to develop a practical model of gear system to control and eliminate the chaotic behavior. To this end, a nonlinear dynamic model of a spur gear pair with backlash, time-varying stiffness and static transmission error is established. Based on the Melnikov analysis the global homoclinic bifurcation and transition to chaos in this model are predicted. Then non-feedback control method is used to eliminate the chaos by applying an additional control excitation. The regions of the parameter space for the control excitation are obtained analytically. The accuracy of the theoretical predictions and also the performance of the proposed control system are verified by the comparison with the numerical simulations. The simulation results show effectiveness of the proposed control system and present some useful information to analyze and control the gear dynamical systems.     

Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber

This paper deals with the application of the concept of targeted energy transfer to the field of acoustics, providing a new approach to passive sound control in the low frequency domain, where no efficient dissipative mechanism exists. The targeted energy transfer, also called energy pumping, is a phenomenon that we observe by combining a pure nonlinear oscillator with a linear primary system. It corresponds to an almost irreversible transfer of vibration energy from the linear system to the auxiliary nonlinear one, where the energy is finally dissipated. In this study, an experimental set-up has been developed using the air inside a tube as the acoustic linear system, a thin circular visco-elastic membrane as an essentially cubic oscillator and the air inside a box as a weak coupling between those two elements. In this paper, which mainly deals with experimental results, it is shown that several regimes exist under sinusoidal forcing, corresponding to the different nonlinear normal modes of the system. One of these regimes is the quasi-periodic energy pumping regime. The targeted energy transfer phenomenon is also visible on the free oscillations of the system. Indeed, above an initial excitation threshold, the sound extinction in the tube follows a quasi-linear decrease that is much faster than the usual exponential one. During this linear decrease, the energy of the acoustic medium is irreversibly transferred to the membrane and then damped into this element called nonlinear energy sink. We present also the frequency responses of the system which shows a clipping of the original resonance peak of the acoustic medium and we finally demonstrate the ability of the nonlinear absorber to operate in a large frequency band, tuning itself to any linear system.     

具有大范围运动和非线性变形的空间柔性梁的精确动力学建模

研究具有大范围运动和非线性变形的空间柔性梁的有限元动力学建模.首先在精确描述空间柔性梁的非线性变形的基础上,采用有限元方法对梁结构进行离散,导出其动能、势能及外力对应的广义力,然后利用Lagrange方程建立了空间柔性梁的精确动力学方程.该方程在原有一次耦合模型的基础上,增加了新的表征纵向、横向、侧向弯曲变形,以及扭转变形的耦合项,同时包含了变形运动与大范围运动之间的相互耦合项.本建模方法和所得结论可为以后空间柔性梁的动力学特性分析作以参考.     

钢筋混凝土动态粘结滑移声发射特性试验

为研究不同加载速率下钢筋混凝土粘结-滑移损伤破坏过程中的损伤特征,对钢筋混凝土粘结试件进行了0.01 mm/s～10 mm/s四种不同速度的拔出试验。采用实时动态的声发射监测技术对整个试验过程进行监测,对试验过程中的声发射信号进行声发射振幅-振铃计数特性及功率谱峰值频率特性分析。结果表明,随着滑移速率的增加,声发射振幅-振铃计数拟合曲线变缓,在0～50 k Hz和100～150 k Hz频率范围内的声发射事件比重分别减小和增加。损伤过程的声发射特性对钢筋混凝土粘结滑移损伤演化规律的研究具有重要的价值。     

一类非线性相对转动周期系统的平衡稳定性分析

利用一类周期性变系数线性常微分方程解的基本矩阵的Jordan形，分析一类非线性相对转动系统扭转的运动稳定性，从而得到非线性相对转动周期系统的运动稳定准则. 运用Lyapunov函数法，对广泛存在的一类机械传动系统的相对转动运动的平衡稳定位置的稳定域进行研究，并给出数学解析表达式. 这为工程中广泛存在的这类机械传动系统稳定工作区间工作参数的选取和相似模拟提供了理论依据及方法，据此可进一步分析和评价大型复杂旋转机械主传动系统的扭振稳定性. 关键词： 相对转动 相似模拟 运动稳定 平衡稳定     

3-D FEM analysis of thermal degradation in writing and reading characteristics of a perpendicular magnetic head

In order to achieve high-density recording, the detailed behavior of thermal degradation should be investigated. In this paper, the degradation of magnetization of high-density recording medium is examined using the 3-D finite element method (FEM) combined with the modeling of Stoner–Wohlfarth (SW) particles and Neel–Arrhenius switching probability. It is shown that the anisotropy field H_k suppressed the thermal degradation and the saturation magnetization M_s enhances it. The thermal degradation is also changed by the amplitude of magnetization.     

Experimental study of nonlinear switching characteristics of conventional 2 × 2 fused tapered couplers

The nonlinear switching characteristics of fused fiber directional couplers were studied experimentally. By using femtosecond laser pulses with pulse width of 100 fs and wavelength of about 1550 nm from a system of Ti:sapphire laser and optical parametric amplifier (OPA), the nonlinear switching properties of a null coupler and a 100% coupler were measured. The experimental results were coincident with the simulations based on nonlinear propagation equations in fiber by using super-mode theory. Nonlinear loss in fiber was also measured to get the injected power at the coupler. After deducting the nonlinear loss and input efficiency, the nonlinear switching critical peak powers for a 100% and a null fused couplers were calculated to be 9410 and 9440 W, respectively. The nonlinear loss parameter PN in an expression of αNL = αP/PN was obtained to be PN = 0.23 W.