Vibration reduction of multi-parametric excited spring pendulum via a transversally tuned absorber

The use of passive control strategy is a common way to stabilize and control dangerous vibrations in a nonlinear spring pendulum which is describing the ship’s roll motion. In this paper, a tuned absorber in the transversal direction is connected to a spring pendulum with multi-parametric excitation forces to control the vibration due to some resonance cases on the system. The method of multiple scale perturbation technique (MSPT) is applied to study the periodic solution of the given system near simultaneous sub-harmonic and internal resonance case. The stability of the steady-state solution near the resonance case is investigated and studied using frequency response equations. The effects of the absorber and some system parameters on the vibrating system are studied numerically. Optimal working conditions of the system are extracted when applying passive control methods. Comparison with the available published work is reported.     

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.     

不同类型水平弹簧组合刚度非线性吸振器的性能分析及稳定性研究

动力吸振器作为一种振动控制单元被广泛运用于各种工程场合,但传统的线性吸振器只能实现窄带振动控制.文章在线性吸振器的基础上引入对称水平弹簧构建线性刚度与非线性刚度相结合的组合刚度非线性吸振器,以提升吸振器的吸振性能.考虑实际工程中可能的安装方式,分别建立水平弹簧接地安装和不接地安装的组合刚度非线性吸振器模型,利用谐波平衡法结合弧长延拓法解析求解动力学响应,并与数值结果相互验证,证明了求解结果的准确性.随后分析比较两种组合刚度非线性吸振器与线性吸振器以及非线性能量阱之间的吸振性能,发现水平弹簧接地安装类型的组合刚度非线性吸振器在保留线性吸振器优势的同时又改善其吸振频带窄的缺点,且与非线性能量阱相比在主共振频率附近的较宽频内吸振性能更优.在此基础上,讨论了水平弹簧参数以及吸振器阻尼对主结构振动幅频响应和稳定性的影响,最后观察分析主结构幅频响应曲线不稳定区内的复杂动力学行为.研究结果表明合适的设计参数能够使得主结构振动峰值较低的同时,频响曲线不稳定运动区域的范围也较小.     

接地惯容式减振器对悬臂输流管稳定性和动态响应的影响研究

输流管道广泛应用于机械、航空、核电和石油等重要工程领域.为防止管道结构因流致振动破坏造成的损失, 很有必要对其稳定性、动力学响应及其调控进行深入研究.本文提出一种由惯容器、弹簧和阻尼器并联组成的减振器模型, 研究了这种接地惯容减振器对悬臂输流管稳定性和非线性振动的影响. 首先, 基于哈密顿原理给出了带有接地惯容减振器非保守系统的非线性动力学模型; 然后, 利用高阶伽辽金方法对非线性方程进行离散化; 最后, 分别从线性和非线性角度分析了不同减振器参数下输流管道的被动控制效果, 着重讨论了惯容系数和减振器安装位置对悬臂管稳定性和动态响应的影响机制.线性理论模型的研究结果显示, 接地惯容减振器可显著影响悬臂管的失稳临界流速, 故通过调节减振器参数能有效提高输流管道的稳定性;惯容系数和弹簧刚度对系统稳定性的控制效果还与减振器的安装位置密切相关.非线性理论模型的分析结果显示, 惯容系数和减振器位置对输流管的非线性动态响应也有显著影响, 且这种影响还依赖于管道的流速取值; 在某些参数条件下, 减振器还可使输流管道由周期运动演化为复杂的混沌行为. 本文研究结果表明, 通过设计合理的惯容式减振器参数, 可提升悬臂输流管道的稳定性并有效抑制其颤振幅值.      

Competitive energy transfer between a two degree-of-freedom dynamic system and an absorber with essential nonlinearity

The background of this work is related to passive vibration control of a two degree-of-freedom master system attached to an essentially nonlinear slave absorber aimed to attenuate vibrations by irreversibly transferring energy to a localized nonlinear normal mode (nonlinear normal modes should be considered here according to the definition from Rosenberg in J. Appl. Mech., 28:275–283, 1961). Such a nonlinear absorber which has no preferential frequency is theoretically able to capture several nonlinear resonances. The main purpose is here to bring insight in what is actually going on when two linear modes are in competition for energy transfer. An original asymptotic analysis using two small parameters enables one to build a scenario that improves the understanding of resonance mechanisms and to forecast which mode will be first attenuated by means of energy transfer. Numerical benchmark simulations corroborate the reliability of obtained scenario.     

The jump phenomenon effect on the sound absorption of a nonlinear panel absorber and sound transmission loss of a nonlinear panel backed by a cavity

Theoretical analysis of the nonlinear vibration effects on the sound absorption of a panel absorber and sound transmission loss of a panel backed by a rectangular cavity is herein presented. The harmonic balance method is employed to derive a structural acoustic formulation from two-coupled partial differential equations representing the nonlinear structural forced vibration and induced acoustic pressure; one is the well-known von Karman??s plate equation and the other is the homogeneous wave equation. This method has been used in a previous study of nonlinear structural vibration, in which its results agreed well with the elliptic solution. To date, very few classical solutions for this nonlinear structural-acoustic problem have been developed, although there are many for nonlinear plate or linear structural-acoustic problems. Thus, for verification purposes, an approach based on the numerical integration method is also developed to solve the nonlinear structural-acoustic problem. The solutions obtained with the two methods agree well with each other. In the parametric study, the panel displacement amplitude converges with increases in the number of harmonic terms and acoustic and structural modes. The effects of excitation level, cavity depth, boundary condition, and damping factor are also examined. The main findings include the following: (1)?the well-known ??jump phenomenon?? in nonlinear vibration is seen in the sound absorption and transmission loss curves; (2)?the absorption peak and transmission loss dip due to the nonlinear resonance are significantly wider than those in the linear case because of the wider resonant bandwidth; and (3)?nonlinear vibration has the positive effect of widening the absorption bandwidth, but it also degrades the transmission loss at the resonant frequency.     

动力吸振器复合非线性能量阱对线性镗杆系统的振动控制

研究了线性动力吸振器复合非线性能量阱对线性镗杆在外部简谐激励下的振动控制. 忽略镗杆系统中的非线性因素, 建立了附加线性动力吸振器和非线性能量阱的镗杆系统的三自由度运动方程, 研究了附加复合式动力吸振器的镗杆系统的受迫振动. 通过平均法得到了附加复合式动力吸振器的镗杆系统的近似解析解, 并利用数值解验证了近似解析解的准确性, 两者具有很好的一致性. 利用近似解析解详细分析了线性动力吸振器和非线性能量阱的参数对镗杆振动抑制性能的影响. 对给定质量的复合式动力吸振器进行了参数优化, 其中线性动力吸振器参数采用H_∞优化方法的近似解析解进行了优化, 非线性能量阱的阻尼利用系统的近似解析解进行了优化. 分析结果表明, 线性动力吸振器与非线性能量阱组合可以有效抑制线性镗杆系统的振动, 而且采用参数优化后的复合式动力吸振器可以获得更好的减振效果. 通过附加非线性能量阱, 不但可以提高线性动力吸振器的振动抑制效果, 而且还可以提高振动控制系统的鲁棒性.      

Vibration control of nonlinear Absorber–Isolator-Combined structure with time-delayed coupling

A nonlinear combined structure consisted of isolator and absorber with time-delayed coupling active control is proposed in this study, whose vibration suppression effectiveness and control mechanism are investigated. The mathematical model of the combined structure is obtained and stability analysis for different structural parameters and time delay are firstly carried out, which provides a general guideline for the ranges of active control parameters. Then the combined effect of nonlinearity and time delay on vibration suppression and energy transfer is discussed in details based on the analysis of control mechanism by the method of multiple scales. Since the time-delayed nonlinear absorber can induce internal resonance between different modes, the vibration energy at low frequencies can be transferred to high frequency mode and the vibration of the fundamental frequency range is thus suppressed. This paper provides a novel application of internal resonance in vibration suppression of an Absorber–Isolator-Combined structure.     

Interaction of free and forced nonlinear normal modes in two-DOF dissipative systems under resonance conditions

The resonance dynamics of a dissipative spring-mass and of a dissipative spring-pendulum system is studied. Internal resonance case is considered for the first system; both external resonances and simultaneous external and internal resonance are studied for the second one. Analysis of the systems resonance behavior is made on the base of the concept of nonlinear normal vibration modes (NNMs) by Kauderer and Rosenberg, which is generalized for dissipative systems. The multiple time scales method under resonance conditions is applied. The resulting equations are reduced to a system with respect to the system energy, arctangent of the amplitudes ratio and the difference of phases of required solution in the resonance vicinity. Equilibrium positions of the reduced system correspond to nonlinear normal modes; in energy dissipation case they are quasi-equilibriums. Analysis of the equilibrium states of the reduced system permits to investigate stability of nonlinear normal modes in the resonance vicinity and to describe transfer from unstable vibration mode to stable one. New vibration regimes, which are called transient nonlinear normal modes (TNNMs) are obtained. These regimes take place only for some particular levels of the system energy. In the vicinity of values of time, corresponding to these energy levels, the TTNM attract other system motions. Then, when the energy decreases, the transient modes vanish, and the system motions tend to another nonlinear normal mode, which is stable in the resonance vicinity. The reliability of the obtained analytical results is confirmed by numerical and numerical-analytical simulations.     

Chatter mitigation using the nonlinear tuned vibration absorber

A passive vibration absorber, termed the nonlinear tuned vibration absorber (NLTVA), is designed for the suppression of chatter vibrations. Unlike most passive vibration absorbers proposed in the literature for suppressing machine tool vibrations, the NLTVA comprises both a linear and a nonlinear restoring force. Its linear characteristics are tuned in order to optimize the stability properties of the machining operation, while its nonlinear properties are chosen in order to control the bifurcation behavior of the system and guarantee robustness of stable operation. In this study, the NLTVA is applied to turning machining.     

Vibration reduction of a nonlinear spring pendulum under multi external and parametric excitations via a longitudinal absorber

The vibration of a ship pitch-roll motion described by a non-linear spring pendulum system (two degrees of freedom) subjected to multi external and parametric excitations can be reduced using a longitudinal absorber. The method of multiple scale perturbation technique (MSPT) is applied to analyze the response of this system near the simultaneous primary, sub-harmonic and internal resonance. The steady state solution near this resonance case is determined and studied applying Lyapunov’s first method. The stability of the system is investigated using frequency response equations. Numerical simulations are extensive investigations to illustrate the effects of the absorber and some system parameters at selected values on the vibrating system. The simulation results are achieved using MATLAB 7.0 programs. Results are compared to previously published work.     

On interaction of vibrating beam with essentially nonlinear absorber

The paper deals with interaction of elastic beam with essentially nonlinear vibration absorber. Forced vibrations of the beam are described by 2DOF model. We treat the motions favorable for vibration absorption as nonlinear modes in a configuration space and compute them by a modification of Rausher method. Stability of these modes is analyzed numerically with the help of the Floquet theory.     

On the Need for Control of Nonlinear Oscillations in Machine Systems

Oscillations in machines are invariably nonlinear. This is either because of inertial coupling effects between different motions of the moving components, material and constitutive phenomena giving rise to stiffness modifications, nonlinear dissipation mechanisms, large deflections, or, as is most likely, some sort of combination of all of these. The net effect of nonlinear vibrations is that at best the machine may well behave a little differently from the way the designer intended, or at worst, in a manner which renders it completely unsuitable for the job. The extent of such problems depends on the nature and the scale of the nonlinearities that are present but it is safe to say that nonlinear oscillations can rarely be completely overlooked in precision machinery analysis and design. The unifying theme in this paper is pendulum motion, firstly in the case of a mobile gantry crane for container stacking where we wish to minimise such motion and converge on a target, and then secondly in the case of a vibration absorber in which we choose to initiate pendulum motion within a special absorber, for the purposes of vibration minimisation. The third example involves the potential for pendulum motion at a very much larger scale and summarises the main control problem that is likely to be encountered in a fully deployed momentum exchange propulsion tether operating in space. The paper discusses the general mathematical issues that pertain to pendulum motion in each of the three cases. This motion is investigated initially in the context of the mobile gantry crane, in the form of a basic three dimensional dynamical model. A feedback linearised controller is shown to offer some advantages for the control of such a system and then a simulation based on data from a practical implementation of this within a real-time control system on a 1/10 laboratory scale model is discussed. It is recalled that the real-time effectiveness of the controller can be compromised by relatively slow sensing and data logging hardware but that despite this some useful performance gains can still be obtainable using this sort of control strategy. The second example comprises an autoparametric vibration absorber and here it is shown how even a simple hunting controller can exploit the mode-locking and wide detuning region effects inherent in autoparametric systems. Further experimental results are discussed in the case of a hunting controller for detuning a vertically oriented parametrically excited pendulum in order to exploit and enhance the powerful and persistent absorption available during autoparametric interaction. The paper concludes with a summary review of the third problem in which the theoretical attitude dynamics of a motorised momentum exchange space propulsion tether are summarised and it is shown that they need to be controlled for reliable and optimal payload velocity boost from both circular parking orbits and elliptical transfer orbits about the Earth.     

两自由度可调非线性减振器

研究了可调非线性减振器的优化设计.基于哈密尔顿最小势能原理建立非线性动力学模型,系统局域参数内,实现非线性系统幅值优化.利用平均法求解可调非线性减振器频响方程.分析系统解的稳定性,优化系统参数,降低系统幅值响应.     

The effects of Coulomb friction on the performance of centrifugal pendulum vibration absorbers

We investigate analytically and experimentally the effects of Coulomb friction on the performance of centrifugal pendulum vibration absorbers (CPVAs), which are used to reduce torsional vibrations in rotating machinery. The analysis is based on perturbation methods applied to the nonlinear equations of motion for a rotor subjected to an engine order applied torque and equipped with a circular path CPVA with viscous and Coulomb damping. The experimental work is based on quantifying parameters for the damping model using free vibration measurements with a viscous and Coulomb damping identification scheme that is enhanced to better handle measurement noise, and running tests for steady-state operation under a range of loading conditions. The level of Coulomb damping is varied by adjusting the friction of the absorber connection bearing. Good agreement is found between the analytical predictions and the experimental data. It is shown that the absorber sticks up to a level of excitation that allows it to release, after which the Coulomb damping acts in the expected manner, resulting in lowered response amplitudes. The results obtained are of general use in assessing absorber performance when dry friction is present in absorber suspensions.     

Vibration absorbers for a rotating flexible structure with cyclic symmetry: nonlinear path design

This paper analytically investigates the nonlinear dynamics of order-tuned vibration absorbers applied to cyclic rotating flexible structures under traveling wave (TW) engine-order excitation. The primary cyclic structure is assumed to be governed by linear vibrations and the nonlinear absorber response arises from large amplitude kinematic effects. These dynamics are captured by a lumped-parameter model that consists of N blades with one blade mode and one absorber per blade, which are arranged with cyclic symmetry on a rotating disk. The governing equations of motion are formulated for arbitrary absorber paths to allow investigation of the absorber path design for nonlinear response. This paper extends previous work by the authors, which considered the linearized blade and absorber dynamics of a similar system. Several intriguing features of the dynamics were uncovered, most notably the existence of an absorber tuning range that avoids resonance at any rotation speed. Of particular interest is the existence and stability of the steady-state TW response to TW excitation, as experienced in turbomachinery, and how these are affected by selection of the absorber paths, which fix the linear and nonlinear tuning characteristics. It is shown that the TW response, which is unique for the linearized system, also exists for the weakly nonlinear model and can be captured by an equivalent two degree of freedom model obtained using the symmetry of the excitation and system response. The forced response exhibits the usual characteristics of a weakly nonlinear system, specifically, bistability and the attendant hysteresis near resonance. More significantly, it does not experience any additional instabilities associated with the symmetry. That is, the desired TW response is robust to nonlinear effects in the absorber, which allows use of the simple equivalent model for selection of absorber tuning parameters. For good performance and robustness, the linear absorber tuning should be in the “no-resonance zone” described by the linear theory and the absorber paths should have a slightly softening nonlinear characteristic.     

Invariant Manifolds,Nonclassical Normal Modes,and Proper Orthogonal Modes in the Dynamics of the Flexible Spherical Pendulum

It is shown that the flexible spherical pendulum undergoes purely slow motions with master and slaved components. The family of slow motions is realized as a three-dimensional invariant manifold in phase space. This manifold is computed analytically by applying the method of geometric singular perturbations. This manifold is nonlinear and for all energy and angular momentum levels is characterized precisely by three PO (proper orthogonal) modes. Its submanifold of zero angular momentum is a two-dimensional invariant manifold; it is the geometric realization of a nonclassical nonlinear normal mode. This normal mode is characterized precisely by two PO modes. The slaved slow dynamics are characterized precisely by a single PO mode. The stability of the slow invariant manifold as well as interactions between fast and slow dynamics are considered.     

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.     

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.     

时滞非线性动力吸振器的减振机理

对一个带有时滞非线性动力吸振器的两自由度结构,采用多尺度法研究了时滞非线性动力吸振器对主系统的减振性能,得到了主系统的振幅-时滞响应曲线.研究结果表明,对时滞非线性动力吸振器,可以通过调节反馈增益系数和时滞控制主系统的振动. 研究还发现,对确定的反馈增益系数,可以存在时滞的一些调节区域,时滞非线性动力吸振器可以减小主系统的振动. 并且在时滞的这些可调区域里,存在一个``最大减振点'对应这一反馈增益系数下主系统振幅的最小值.对不同的反馈增益系数,``最大减振点'对应的主系统的振幅也不同.因此能够找到一组反馈增益系数和时滞量的最佳值,最大程度地减小主系统的振动.研究结果表明,当反馈增益系数和时滞量调到最佳值时,主系统的振动较无时滞非线性动力吸振器可以减少90{\%}左右, 数值模拟也证实了解析结果的正确性.