非高斯随机激励下滞迟系统响应与首次穿越失效

针对由有界噪声、泊松白噪声和高斯白噪声共同构成的非高斯随机激励,通过Monte Carlo数值模拟方法研究了此激励作用下双线性滞迟系统和Bouc-Wen滞迟系统这两类经典滞迟系统的稳态响应与首次穿越失效时间。一方面,分析了有界噪声和泊松白噪声这两种分别具有连续样本函数和非连续样本函数的非高斯随机激励,在不同激励参数条件下对双线性滞迟系统和Bouc-Wen滞迟系统的稳态响应概率密度、首次穿越失效时间概率密度及其均值的不同影响;另一方面,揭示了在这类非高斯随机激励荷载作用下,双线性滞迟系统的首次穿越失效时间概率密度将出现与Bouc-Wen滞迟系统的单峰首次穿越失效时间概率密度截然不同的双峰形式。     

Dynamic analysis of piezoelectric energy harvester under combination parametric and internal resonance: a theoretical and experimental study

In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

     

Drift response of a bilinear hysteretic system to periodic excitation under sustained load effects

This paper presents an investigation of response characteristics for hysteretic systems idealized as a bilinear hysteretic model subjected to period excitations composed of a harmonic function and a sustained load. It is shown that the displacement solution can exhibit a drift sequence persistently repeated at a frequency identical to the excitation frequency in the case of zero post-yielding stiffness. The periodic-like drift sequence is further classified into three major types according to their different hysteretic looping behaviors. An approximate solution approach based on the method of weighted residuals is proposed to analyze the drift amplitude per response cycle. The assumed response shape is composed of two concatenated harmonic functions each with a frequency slightly detuned from the excitation frequency. The method is accompanied with a subsequent first-order analysis to obtain a closed-form approximation for the drift response. Good response predictions of the proposed solution method are demonstrated through both undamped and damped drift-frequency analyses.     

Multiple parametric resonance in a non-linear two degree of freedom system

A non-linear two degrees of freedom dynamical system subjected to parametric excitation is studied. The relationship between the parametric frequency and the natural frequencies of the system are such that the first generalized coordinate is excited into parametric resonance while both generalized coordinates are excited into combination resonance at the same time. The unstable zone and response of the system under multi-resonant condition are then compared with the unstable zones and responses of the system where each type of resonance is treated on an individual basis, assuming the other type of resonance does not occur.     

Identification of nonlinear hysteretic parameters by enhanced response sensitivity approach

Hysteresis is a ubiquitous phenomenon describing the special nonlinear memory-based relation between the input and the output in many physical systems. Identifying the hysteretic parameters is the first step towards practical application of hysteretic models. In this paper, a general framework for parameter identification of nonlinear hysteretic models is developed based on the enhanced response sensitivity approach. To do so, three typical hysteretic models—Bouc–Wen model, bilinear model with kinematic hardening and bilinear model with equal yielding force are analyzed at first and the general way to model a structure with such hysteretic components is established thereafter. Then, the enhanced response sensitivity approach is presented for inverse parameter identification where the key lies in the sensitivity analysis and the trust-region constraint. Particularly, smoothing procedure is introduced to overcome the non-differentiability of bilinear hysteretic functions for sensitivity analysis of bilinear models. Numerical examples are studied to testify the feasibility and performance of the proposed approach.     

双线性滞回结构滤波问题的分段线性化法

提出了一种双线性滞回特性的分段微分表达式，根据伊藤随机微分方程和连续Markov向量过程的理论，导出了无须进行积分计算的双线性滞回结构的分段线性滤波器，简化了计算．算例表明，该法比随机等效线性化法和数字模拟的结果更为吻合．     

Drift response of bilinear hysteretic systems under two-frequency excitations

Response characteristics of bilinear hysteretic systems are investigated under two-frequency excitation. The excitation involves a predominate frequency component superimposed by a weaker high-frequency component having an integer frequency ratio. Depending on the frequency ratio, the long-term solutions show various forms of asymmetric response besides a symmetric steady-state response. In the limiting case of zero post-yielding stiffness, the displacement solution may exhibit a periodic-like response characterized by a constant drift increment occurring in each response cycle. To analyze the drift, two solution schemes, respectively, assuming uni-frequency and two-frequency response shapes are presented. Analytical results indicate that the frequency ratio and the phase difference of the two-frequency excitation highly influence not only the drift increment but also the direction of drift response. The effects associated with the addition of viscous damping are also investigated and discussed.     

Nonlinear dynamics of a viscoelastic beam traveling with pulsating speed,variable axial tension under two-frequency parametric excitations and internal resonance

This paper investigates nonlinear combined parametric transverse vibrations of a traveling viscoelastic beam. The combined parametric excitations originate from the time dependency of axial velocity as well as axial tension. Two parametric excitations are enforced into the system amid the internal resonance. Two-frequency parametric resonance is assumed to be comprised of combination parametric resonance of first two modes due to the time dependency of axial velocity, and the principal parametric resonance of first mode due to the variable tension in the axial direction in the presence of internal resonance for viscoelastic beam is considered for the first time. The higher-order integro-partial differential equation of motion is solved through direct method of multiple scales. Continuation algorithm is employed to explore the stability and various bifurcations of the nonlinear dynamic system. Focus has been made to study the effect of variations of fluctuating tension component, fluctuating velocity component independently and when combined, internal and parametric frequency detuning parameters and damping on the system response. Frequency response equilibrium curves are complex and unique in shapes which are embodied with various bifurcations. Such steady-state behavior is not seen in the existent literature. With variation in fluctuating velocity component, the number of steady-state nontrivial equilibrium curves increases to three and with variation in fluctuating axial tension, they become four. In this process, significant changes in stability, number and position of various bifurcations like supercritical and subcritical pitchfork, Hopf and saddle node are observed. Unlike the previous study, the shape, stability and bifurcations of equilibrium curves under the combined effect of axial velocity and tension closely match with the case of fluctuating axial tension component. The effect of variation in internal and parametric frequency detuning parameter is more realized for second mode compared to first mode. A comparison of the present work with a previous one where axial tension is variable reveals many qualitative and quantitative similarities and dissimilarities. But when compared with earlier work where axial velocity is constant, significant dissimilarities are surfaced. The system displays a wide ranging dynamic behavior including stable periodic, quasiperiodic and unstable chaotic behavior. The numerical computation depicts various nonlinear characteristics and oscillatory behaviors which are not found so far in the existent literature.     

Chaotic behaviors of a bilinear hysteretic oscillator

In the present research, concentration is focused on dynamic response of a bilinear hysteretic oscillator with clearance, and equivalent damping and stiffness are then determined by means of asymptotic and numerical method. A variety of dynamic responses of the nonlinear system are analyzed, and some new phenomena are discussed.     

Periodic and Non-Periodic Oscillations of a Class of Hysteretic Two Degree of Freedom Systems

The equations governing the response of hysteretic systems to sinusoidal forces, which are memory dependent in the classical phase space, can be given as a vector field over a suitable phase space with increased dimension. Hence, the stationary response can be studied with the aids of classical tools of nonlinear dynamics, as for example the Poincaré map. The particular system studied in the paper, based on hysteretic Masing rules, allows the reduction of the dimension of the phase space and the implementation of efficient algorithms. The paper summarises results on one degree of freedom systems and concentrates on a two degree of freedom system as the prototype of many degree of freedom systems. This system has been chosen to be in 1:3 internal resonance situation. Depending on the energy dissipation of the elements restoring force, the response may be more or less complex. The periodic response, described by frequency response curves for various levels of excitation intensity, is highly complex. The coupling produces a strong modification of the response around the first mode resonance, whereas it is negligible around the second mode. Quasi-periodic motion starts bifurcating for sufficiently high values of the excitation intensity; windows of periodic motions are embedded in the dominion of the quasi-periodic motion, as consequence of a locking frequency phenomenon.     

参数激励与强迫激励联合作用下非线性振动系统的分叉

本文利用多尺度法研究了参数激励与强迫激励联合作用下非线性振动系统的分叉问题,给出了分叉集和八种分叉响应曲线。     

参数激励下非线性受控系统的动力学和稳定性

研究两自由度参数激励系统的非线性动力学与控制问题．利用Lagrange方程建立含反馈控制的参激捅及其驱动机构组成的系统动力学方程，以多尺度方法获得一阶近似控制方程．然后，对系统受一阶摸态参激主共振与一、二阶模态间3：1内共振联合作用下的幅额响应及其稳定性，以及反馈参数对系统稳态行为的影响作了详细分析．结果表明，响应的稳定域位置和大小取决于位移反馈，位移立方反馈改变了系统的非线性程度，速度反馈类似于阻尼，可使系统呈现自激振动特性．     

Stability and response of a nonlinear coupled pitch-roll ship model under parametric and harmonic excitations

The present study deals with the response of a two-degree-of-freedom (2DOF) system with quadratic coupling under parametric and harmonic excitations. The method of multiple scale perturbation technique is applied to solve the nonlinear differential equations and obtain approximate solutions up to and including the second order approximations. All resonance cases are extracted and investigated. Stability of the system is studied using frequency response equations and phase-plane method. Numerical solutions are carried out and the results are presented graphically and discussed. The effects of the different parameters on both response and stability of the system are investigated. The reported results are compared to the available published work.     

一类加权全局迭代参数卡尔曼滤波算法

结合参数卡尔曼滤波算法和全局迭代推广卡尔曼滤波算法本文提出了加权全局迭代参数卡尔曼滤波算法。参数卡尔曼滤波算法可避免系统参数和状态变量之间的非线性耦合 ,同时通过带有目标函数的全局迭代算法保证能够获取到稳定、收敛的识别结果。分别针对线性结构模型和随动强化双线性结构模型进行了仿真参数识别。结果显示 ,不加权的全局迭代参数卡尔曼滤波算法对线性系统是有效的 ,而对非线性系统必须使用加权的全局迭代参数卡尔曼滤波算法。当信噪比较大 ,迭代无法得到收敛的结果时 ,目标函数保证了较好识别结果的获得     

2-dof non-linear dynamics of a rotor suspended in the magneto-hydrodynamic field in the case of soft and rigid magnetic materials

Non-linear dynamics of a high-speed rotor symmetrically supported on the magneto-hydrodynamic bearing is analyzed. In the case of soft magnetic materials the method of multiple scales [1] is applied. Non-resonant and primary resonance cases with and without internal resonance are illustrated and discussed. In the case of rigid magnetic materials the hysteretic properties are taken into consideration by means of the Bouc-Wen hysteretic model. In order to characterize the hysteretic close-loop control of the rotor motion, instability regions as well as the amplitude level contours of vertical and horizontal vibrations of the rotor are obtained in the form suitable for engineering applications, i.e. in the ‘frequency-amplitude’ parametric plane.     

Asymptotic response of a two DOF elastoplastic system under harmonic excitation. Internal resonance case

The study of a two DOF elastoplastic system is formulated in a suitable phase space, velocity and force, in which an originally multi-valued restoring force is represented by a proper function. The asymptotic response can thus be studied using the Poincaré map concept and avoiding approximate analytical techniques. On account of the peculiarity of this hysteretic system, which has a well-defined yielding point, its dynamics is studied in a reduced dimension phase space using an efficient numerical algorithm. It is shown that the asymptotic response is always periodic with the period of the driven frequency and is always stable. Thus the response of the oscillator is described by its frequency response curves at various intensities of the excitation. The results presented refer to a system with two linear frequencies in a ratio of 1 : 3. The response is highly complex with numerous peaks corresponding to higher harmonics. The effect of coupling in conditions of internal resonance is a strong modification of the frequency response curves and of the oscillation shape of the structure.     

Analytical parametric study of bi-linear hysteretic model of dry friction under harmonic,impulse and random excitations

Hysteretic behavior due to some nonlinear sources is a common phenomenon in many dynamical systems. One of the sources of this behavior in mechanical systems is dry friction. Dry friction in bolted or riveted joints of mechanical structures makes their dynamic behavior hysteretic. Bi-linear hysteresis is one of the models that can be used to study these systems which is used in this paper. A SDOF system containing a bi-linear hysteretic element called Jenkins element under harmonic, impulse and random excitations is considered. For all three types of excitations, the effects of system and excitation parameters on the defined equivalent system parameters and the response specifications are studied. Harmonic balance method is employed for harmonic excitation studies, and optimum friction threshold for minimizing response amplitude is obtained versus other system parameters and response amplitude. Energy balance method is used for impulse excitation through which the desired decaying ratio can be achieved by tuning the friction threshold, depending on stiffness ratio. System under random excitation is investigated by equivalent linearization technique in two steps. At the first step, equivalent properties are obtained versus instantaneous amplitude of response. In this step, the paper contains the parametric study of system in which the variations of equivalent parameters are described when physical parameters of system or input intensity vary. Overall variance of system response is determined in the second step, and optimum sliding threshold is obtained to have minimum overall variance of system response.     

Amplitude modulated motions in a two degree-of-freedom system with quadratic nonlinearities under parametric excitation: experimental investigation

We conducted an experimental investigation of amplitude modulated response of a two degree-of-freedom mechanical structure with quadratic nonlinearities to parametric excitation. The linear natural frequencies of the system were tuned so that they were approximately in the ratio of two-to-one, and the excitation frequency was in principal parametric resonance with the first mode. We observed periodically amplitude-modulated motions and chaotically amplitude-modulated motions.     

单自由度参数振动系统非线性响应的若干特征

用数值积分方法，分析了无阻尼单自由度参数振动系统在给定的单频刚度激励和单频外载荷激振时的非线性动力学响应特性。研究表明了参数振动问题的主要特征：1)单频激励多频响应；2)多频响应中各谐波的分布具有特殊的规律；3)系统具有多频共振特性。