On vibration mitigation and energy harvesting of a non-ideal system with autoparametric vibration absorber system

This paper demonstrates that vibration mitigation and energy harvesting can be achieved simultaneously by using of an electricity-generating from autoparametric vibration absorber system (AVAS) and non-ideal system (NIS). The NIS consists of a simple portal frame excited by a small dc motor with eccentric mass, with limited power supply and located on the top. The AVAS consists of a cantilever beam with tip mass parallel coupled to NIS. A piezoelectric material is considered for energy harvesting installed in the base of the AVAS and an electric circuit is connected to the piezoelectric material in order to produce voltage output. Several numerical simulations were carried out focusing on the passage through the resonance of NIS, when the motor rotational frequency is near the portal frame natural frequency and when the non-ideal subsystem frequency is approximately twice the absorber beam frequency (two-to-one internal resonance). The results showed the existence of Sommerfeld effect in NIS and saturation phenomenon in the NIS–AVAS.     

Imposing nodes for linear structures during harmonic excitations using SMURF method

Vibration absorbers are usually designed using the finite element (FE) model of structures. It is generally believed that the modal models are more accurate than FE models, because in modal testing the model is built by direct measurement of the test structure. In this paper, a method is proposed to design a translational vibration absorber using the measured frequency response functions of a primary structure. The designed vibration absorber imposes a node on the structure when it is excited by a harmonic force. The method is based on the structural modification using experimental frequency response functions technique and determines the required receptance of the absorber at the excitation frequency. Moreover, a procedure is developed to suppress the vibration amplitude of two arbitrary points on a linear structure subjected to harmonic excitations by attaching two sprung mass absorbers. A cantilever beam is considered for the numerical case study, and the sprung masses are designed to suppress the vibration amplitude of the beam at the selected arbitrary points. A U-shape plate was considered for the experimental validation of the method for imposing a node using one absorber. Also, a beam was tested to demonstrate the effectiveness of method for imposing two nodes on the structures. The experimental results show that the designed absorbers can considerably suppress the vibration amplitude at the selected points on the structure.     

Nonlinear Vibration Control of a System with Dry Friction and Viscous Damping Using the Saturation Phenomenon

Application of saturation to provide active nonlinear vibration control was introduced not long ago. Saturation occurs when two natural frequencies of a system with quadratic nonlinearities are in a ratio of around 2:1 and the system is excited at a frequency near its higher natural frequency. Under these conditions, there is a small upper limit for the high-frequency response and the rest of the input energy is channeled to the low-frequency mode. In this way, the vibration of one of the degrees of freedom of a coupled 2 degrees of freedom system is attenuated. In the present paper, the effect of dry friction on the response of a system that implements this vibration absorber is discussed. The system is basically a plant with a permanent magnet DC (PMDC) motor excited by a harmonic forcing term and coupled with a quadratic nonlinear controller. The absorber is built in electric circuitry and takes advantage of the saturation phenomenon. The method of multiple scales is used to find approximate solutions. Various response regimes of the closed-loop system as well as the stability of these regimes are studied and the stability boundaries are obtained. Especial attention is paid on the effect of dry friction on the stability boundaries. It is shown that while dry friction tends to shrink the stable region in some parts, it enlarges other parts of the stable region. To verify the theoretical results, they have been compared with numerical solution and good agreement between the two is observed.This work was done while the authors were associated with the Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran.     

A review of development and implementation of an active nonlinear vibration absorber

Summary We present an account of an implementation of an active nonlinear vibration absorber that we have developed. The control technique exploits the saturation phenomenon that is known to occur in quadratically-coupled multi-degree-of-freedom systems subjected to primary excitation and possessing a two-to-one internal resonance. The technique is based on introducing an absorber and coupling it with the structure through a sensor and an actuator, where the feedback and control signals are quadratic. First, we consider the case of controlling the vibrations of a single-degree-of-freedom system. We develop the equations governing the response of the closed-loop system and use the method of multiple scales to obtain an approximate solution. We investigate the performance of the control strategy by studying its steady-state and transient characteristics. Additionally, we compare the performance of the quadratic absorber with that of a linear absorber. Then, we present theoretical and experimental results that demonstrate the versatility of the technique. We design an electronic circuit to emulate the absorber and use a variety of sensors and actuators to implement the active control strategy. First, we use a motor and a potentiometer to control the vibration of a rigid beam. We develop a plant model that includes Coulomb friction and demonstrate that the closed-loop system exhibits the saturation phenomenon. Second, we extend the strategy to multi-degree-of-freedom systems. We use PZT ceramics and strain gages to suppress vibrations of flexible steel beams when subjected to single- and simultaneous two-mode excitations. Third, we employ Terfenol-D, a nonlinear actuator, and accelerometers to control the vibrations of flexible beams. In all instances, the technique is successful in reducing the response amplitude of the structures. Received 3 May 1999; accepted for publication 3 June 1999     

Dynamics of a Cubic Nonlinear Vibration Absorber

We study the dynamics of a nonlinear active vibration absorber. We consider a plant model possessing curvature and inertia nonlinearities and introduce a second-order absorber that is coupled with the plant through user-defined cubic nonlinearities. When the plant is excited at primary resonance and the absorber frequency is approximately equal to the plant natural frequency, we show the existence of a saturation phenomenon. As the forcing amplitude is increased beyond a certain threshold, the response amplitude of the directly excited mode (plant) remains constant, while the response amplitude of the indirectly excited mode (absorber) increases. We obtain an approximate solution to the governing equations using the method of multiple scales and show that the system possesses two possible saturation values. Using numerical techniques, we perform stability analyses and demonstrate that the system exhibits complicated dynamics, such as Hopf bifurcations, intermittency, and chaotic responses.     

Attractors of harmonically forced linear oscillator with attached nonlinear energy sink. II: Optimization of a nonlinear vibration absorber

This paper is the second one in the series of two papers devoted to detailed investigation of the response regimes of a linear oscillator with attached nonlinear energy sink (NES) under harmonic external forcing and assessment of possible application of the NES for vibration absorption and mitigation. In this paper, we study the performance of a strongly nonlinear, damped vibration absorber with relatively small mass attached to a periodically excited linear oscillator. We present a nonlinear absorber tuning procedure in the vicinity of (1:1) resonance which provides the best total system energy suppression, using analytical and numerical tools. A linear absorber is also tuned according to the same criterion of total system energy suppression as the nonlinear one. Both optimally tuned absorbers are compared under common parameters of damping, external forcing but different absorber stiffness characteristics; certain cases for which nonlinear absorber is preferable over the linear one are revealed and confirmed numerically.     

Amplitude reduction of primary resonance of nonlinear oscillator by a dynamic vibration absorber using nonlinear coupling

The paper presents the characteristics of a new type of nonlinear dynamic vibration absorber for a main system subjected to a nonlinear restoring force under primary resonance. The absorber is connected to the main system by a link in order to be excited with twice the frequency of the motion of the main system. The natural frequency of the absorber is tuned to be twice the natural frequency of the main system, in contrast to autoparametric vibration absorber, whose natural frequency is tuned to be one-half the natural frequency of the main system. The presented absorber is not excited through the autoparametric resonance, i.e., no trivial equilibrium state exists. Therefore, the absorber always oscillates because of the motion of the main system and cannot be trapped by Coulomb friction acting on the absorber, in contrast to the autoparametric vibration absorber. Under small excitation amplitude, this absorber does not produce an overhang in the frequency response curve, which occurs because of the use of the conventional autoparametric vibration absorber; the overhang renders the response amplitude larger than that in the case without an absorber. In addition, the absorber removes the hysteresis in the frequency response curve caused by the nonlinearity of the restoring force acting on the main system. Regarding large excitation amplitude, the response amplitude in the main system can be decreased by increasing the damping of the absorber, but that decrease is limited by the nonlinearity in the restoring force acting on the main system. This paper also describes experimental validation of the absorber under small excitation amplitude using a simple apparatus.     

Suppression of super-harmonic resonance response using a linear vibration absorber

Super-harmonic resonances may appear in the forced response of a weakly nonlinear oscillator having cubic nonlinearity, when the forcing frequency is approximately equal to one-third of the linearized natural frequency. Under super-harmonic resonance conditions, the frequency-response curve of the amplitude of the free-oscillation terms may exhibit saddle-node bifurcations, jump and hysteresis phenomena. A linear vibration absorber is used to suppress the super-harmonic resonance response of a cubically nonlinear oscillator with external excitation. The absorber can be considered as a small mass-spring-damper oscillator and thus does not adversely affect the dynamic performance of the nonlinear primary oscillator. It is shown that such a vibration absorber is effective in suppressing the super-harmonic resonance response and eliminating saddle-node bifurcations and jump phenomena of the nonlinear oscillator. Numerical examples are given to illustrate the effectiveness of the absorber in attenuating the super-harmonic resonance response.     

A Nonlinear Vibration Absorber for Flexible Structures

An approach for implementing an active nonlinear vibration absorber for flexible structures is presented. The technique exploits the saturation phenomenon exhibited by multidegree-of-freedom systems with quadratic nonlinearities possessing two-to-one autoparametric resonances. The strategy consists of introducing second-order controllers and coupling each of them with the plant through a sensor and an actuator, where both the feedback and control signals are quadratic. Once the structure is forced near its resonances, the oscillatory response is suppressed through the saturation phenomenon. We present theoretical and experimental results of the application of the proposed vibration absorber. The structure consists of a cantilever beam, the feedback signal is generated by a strain gage, and the actuation is achieved through piezoceramic patches. The equations of motion are developed and analyzed through perturbation techniques and numerical simulation. Then, the strategy is tested by assembling the controllers in electronic components and suppressing the vibrations of the first and second modes of two beams.     

Stabilization of a 1/3-order subharmonic resonance using nonlinear dynamic vibration absorber

The paper proposes a stabilization method for a 1/3-order subharmonic resonance with a new type of nonlinear vibration absorber using nonlinear coupling between a main system and the absorber. The main system with nonlinear restoring force and harmonic excitation, i.e., subjected to a sinusoidally changed magnetic force, is introduced as a model which produces a 1/3-order subharmonic resonance. A damped pendulum, whose natural frequency is tuned to be in the neighborhood of twice that of the main system, is connected through a link to the main system as a nonlinear vibration absorber. Theoretical results using the method of multiple scales show that only a stable nontrivial steady state is changed into an unstable one due to the effect of absorber. In addition, we numerically confirm the validity of the proposed absorber using Runge–Kutta method.     

利用时滞反馈控制自参数振动系统的振动

主要研究采用时滞状态反馈控制自参数动力吸振器减振系统中主系统的振动问题.系统在简谐激励作用下,采用多尺度方法得到了自参数动力吸振器减振系统中饱和控制的范围.当系统处于饱和控制时,引入时滞状态反馈控制主系统的振动.主要分析了反馈增益系数和时滞两控制参数对主系统振动的影响.结果表明,存在反馈增益系数和时滞的调节区域能够减小主系统的振动.对某一反馈增益系数,可以在某段区间内调节时滞以减小主系统的振动.在时滞的调节区间内存在一个时滞的``最大减振点',能够在该反馈增益系数下最大程度地减小主系统的振动.研究还表明,随着反馈增益系数的不断增大,时滞在``最大减振点'时系统的减振能力也不断提高.通过合理的选择反馈增益系数和时滞两参数,主系统的振动几乎可以完全消除.      

Dynamics of a mass–spring–beam with 0:1:1 internal resonance using the analytical and continuation method

The nonlinear harmonic response of an autoparametric system comprised of a linear oscillator with a vertically attached flexural beam is investigated and the capability of the beam as a vibration absorber is assessed. A weak torsional spring is used for constraining the rotation of the beam giving rise to an almost non-flexural rotational mode with a low frequency. The system parameters are also tuned to enforce the zero-to-one-to-one internal resonance condition. The Lagrange’s formulation accompanied by the assumed-mode method is used to derive the discretized equations based on the order three nonlinear Euler–Bernoulli beam theory. An analytical solution is developed based on the method of multiple scales where the generalized coordinate corresponding to the non-flexural rotational mode is approximated by higher order perturbation expansion than the other coordinates, due to much larger contribution of the non-flexural rotation to the response. Comprehensive response and bifurcation analysis are performed using analytical and direct numerical solutions. The results are obtained for vertically-aligned and also initially inclined beams and various interesting behaviors are recognized for different non-dimensional system parameters. Different types of bifurcations such as the Pitch-fork, Hopf, Period-doubling and symmetry breaking bifurcations are observed in the solution of slow-flow equations and some of them are found to be beneficial for vibration absorption in a limited range of excitation amplitudes and frequencies.     

Parametrically Excited Non-Linear Traveling Beams with and without External Forcing

The effects of parametric excitation on a traveling beam, both with and without an external harmonic excitation, have been studied including the non-linear terms. Non-linear, complex normal modes have been used for the response analysis. Detailed numerical results are presented to show the effects of non-linearity on the stability of the parametrically excited system. In the presence of both parametric and external harmonic excitations, the response characteristics are found to be similar to that of a Duffing oscillator. The results are sensitive to the relative strengths of and the phase difference between the two forms of excitations.     

有界噪声激励下单摆-谐振子系统的混沌运动

研究了具有同宿轨道和周期轨道的可积单摆-谐振子系统在弱Hamilton摄动(即弱耦合摄动)和弱非Hamilton摄动(即阻尼和有界噪声微扰)下的混沌运动．用Melnikov方程预测Hamilton系统中可能存在混沌运动的参数域，并用Poincare截面验证解析结果．用数值方法计算了有阻尼与有界噪声激励下系统的最大Lyapun0V指数和Poincare截面，结果表明有界噪声在频率上的扩散减小了引发系统产生混沌运动的效应。     

具有稳定数值解的三维谐振子

谐振子广泛应用于物理系统的描述和物理现象的数值模拟。由于二维或三维谐振子对于系统参数、初始条件和边界条件的高度敏感性,很多物理过程的动力学模拟都会出现数值解不稳定的现象。近年来发展的无网格法、物质点法和近场动力学法等数值模拟方法均绕开了对固体材料固有构形的量化描述。本文引入了定常耗散项和弹簧耗散项,考虑随机微扰效应,提出了一种三维耗散谐振子,构建了基于蛙跳法和边界松弛技术的数值积分算法。应用三维谐振子构建了耗散型弹簧摆、简化弦和简化梁三个模型,设定了13个定解问题进行动力学模拟。数值试验结果表明,三维谐振子是稳定的。基于简化弦模型,模拟了拉弦、放弦和重弦三个有界弦振动问题;其中,拉弦和放弦问题成功模拟了有界弦的三维振形;重弦问题模拟再现了悬链线在水平向的微幅振荡现象。基于简化梁模型,模拟了三维梁的拉伸、剪切和扭转行为,验证了三维谐振子对于非线性大变形问题动力学模拟的描述能力,及其对外部作用的高速响应能力。本文方法可以为弦振动问题和材料力学非线性大变形问题的动力学模拟提供一条可行的实现途径。     

Analysis of Beams with Piezoelectric Actuators

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Vibration analysis of undamped,suspended multi-beam absorber systems

The vibration analysis of an Euler-Bernoulli beam with an attached rotary unit is first carried out assuming no unbalance. For comparison purposes, two different beam end boundary conditions are considered: a simply-supported and a clamped-clamped condition. The problem is then extended to the vibration behavior of the initial beam when subjected to a harmonic load due to an unbalance in the rotary unit. To absorb the ensuing vibrations, a secondary passive beam system is suspended from the primary beam which consists of two continuous leaf springs and three discrete masses. The absorption frequency is obtained by exploring the deflection norm of the primary beam versus dimensionless frequencies of the system. To ensure the appropriateness of the procedure for similar multi-beam absorber systems, an experimental set-up is established and analytical results are verified.     

磁悬浮隔振器动态电磁力实验与建模

磁悬浮隔振器的核心部件是电磁作动器,得到其输出电磁力与线圈电流及气隙之间的模型对实现磁悬浮隔振器的主动隔振有着重要的意义.实际系统中,磁悬浮隔振器运动频率对电磁力模型参数有较大影响,为了得到比较精确的符合实际的电磁力模型,对原有模型进行了修正,给出了新的考虑频率因素的修正模型;并且设计了基于MTS809的动态电磁力实验建模方案,直接根据电磁力实时测量数据辨识得到动态电磁力修正模型,而不需要对磁路进行任何假设.实验结果表明,该方法建立的电磁力修正模型更符合实际情况.     

含机械调频式动力吸振器的准零刚度隔振系统隔振性能分析

基于动力吸振器原理,在单自由度准零刚度隔振器基础上耦合可调频动力吸振器构成两自由度隔振系统。首先,对动力吸振器工作原理进行理论分析并提出其力学模型;其次,通过静力学分析,推导出系统满足零刚度条件时,各参数间的关系并分析其对系统刚度特性的影响;然后,建立两自由度隔振系统非线性动力学方程,利用谐波平衡法进行幅频响应解析分析,得到力传递率表达式;最后,数值分析动力吸振器阻尼、刚度、质量、激励力幅值和弹簧片有效长度对力传递率的影响规律,并与单自由度准零刚度隔振系统及两自由度线性隔振系统对比分析。结果表明：通过选择适当的动力吸振器参数不仅可以减小系统的起始隔振频率,增宽隔振频带,且还能加快系统力传递率在特定频段内的衰减速率,改善系统的低频隔振性能,实现激励频率的可适应性。     

Implementation of integral fixed-time sliding mode controller for speed regulation of PMSM servo system

The paper presents a new concept of absorbing car body vibrations, which consists in a modification of the construction of the classical mono-tube hydraulic shock absorber by the introduction of an additional inner cylinder with an auxiliary piston. By making an appropriate selection of the system parameters, one may obtain the damping force characteristics dependent on the excitation amplitude and frequency. In the case of driving on a good-quality road surface, the shock absorber displays the soft characteristics which are desired as far as the driving comfort is concerned. In the case of worse-quality roads or while overcoming large obstacles, the hard characteristics ensure a higher level of safety and protect the shock absorber from getting damaged. The developed nonlinear model makes it possible to effectively analyse the system responses to harmonic, impulse and random excitations. On the basis of the analysis of the impact of harmonic excitations on the driving comfort and safety indexes, one may estimate the optimal values of the shock absorber construction parameters. Impulse and random excitations are applied in order to finally verify the effectiveness of the operation of the proposed shock absorber.