Using delayed damping to minimize transmitted vibrations

In Mokni et al. [Mokni L, Belhaq M, Lakrad F. Effect of fast parametric viscous damping excitation on vibration isolation in sdof systems. Commun Nonlinear Sci Numer Simulat 2011;16:1720-1724], it was shown that in a single degree of freedom system a fast nonlinear parametric damping enhances vibration isolation with respect to the case where the nonlinear damping is time-independent. The present work proposes additional enhancement of vibration isolation using delayed nonlinear damping. Attention is focused on assessing the contribution of a delayed nonlinear damping over a fast parametric damping in terms of minimizing transmissibility. The results show that a nonlinear damping with delay greatly improves vibration isolation.     

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.     

非线性阻尼非线性刚度隔振系统随机动力学特性研究

针对随机激励环境,同时引入刚度和阻尼非线性来提高隔振系统的隔振性能.刚度和阻尼非线性分别是由水平弹簧和水平阻尼的几何布置获得.通过求解Fokker-Planck-Kolmogorov(FPK)方程等效非线性随机振动方程来研究非线性隔振系统在随机激励下的隔振性能,并使用路径积分和Monte-Carlo数值方法进行验证.在此基础上研究刚度非线性和阻尼非线性对隔振系统在随机激励下力传递率及其概率分布的影响.研究表明随着噪声强度的增加,非线性阻尼抑制振动的能力增强,但是在较小的随机激励下线性阻尼优于非线性阻尼.     

On the characteristics of a quasi-zero-stiffness vibration isolator with viscoelastic damper

Quasi-zero-stiffness (QZS) vibration isolators have been widely studied in previous literature, in which the viscous damping was adopted. In this paper, the QZS isolator with viscoelastic damper is studied, to seek an approach for further improving the isolation performance. The governing equation of the QZS isolator with viscoelastic damper (QZS-VED isolator) is formulated. The dynamic response is analyzed analytically and numerically. The transmissibility characteristics with varying key parameters are investigated, accompanied by comparison with the conventional QZS isolator, to demonstrate the advantages of the QZS-VED isolator. It is revealed that the damping ratio can be optimized to attain the lowest peak transmissibility and meanwhile the roll-off rate of high frequency transmissibility is twice that of the conventional QZS isolator, which implies superior isolation performance both around the resonant frequency and at high frequencies.     

Effect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator

We investigate the interaction effect of fast vertical parametric excitation and time delay on self-oscillation in a van der Pol oscillator. We use the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic and then we apply the averaging technique on this slow dynamic to derive a slow flow. In particular we analyze the slow flow to analytically approximate regions where self-excited vibrations can be eliminated. Numerical integration is performed and compared to the analytical results showing a good agreement for small time delay. It was shown that vertical parametric excitation, in the presence of delay, can suppress self-excited vibrations. These vibrations, however, persist for all values of the excitation frequency in the case of a fast vertical parametric excitation without delay [Bourkha R, Belhaq M. Effect of fast harmonic excitation on a self-excited motion in van der Pol oscillator. Chaos, Solitons & Fractals, 2007;34(2):621–7.].     

Dynamic performance analysis of non-linear tuned vibration absorbers

The main purpose of this study is to evaluate the dynamic characteristics of a damping controlled semi-active tuned vibration absorber (TVA). A base-excited, single-degree-of-freedom structure coupled with a TVA model is adopted as the baseline model for our analysis. A non-model based groundhook control (displacement-based on–off control or “on–off DBG”) was used to control the damping of the TVA. After developing numerical models of a non-linear TVA along with the on–off DBG control and a passive TVA, the optimal tuning parameters of both TVA models were obtained using an optimization routine. Using the two optimally tuned models, parametric studies were preformed by varying the values of both on and off-state damping to evaluate the dynamic performances of the TVAs using the peak transmissibility criteria. The results showed that the peak transmissibility of the semi-active TVA is nearly 21% lower than that of the passive, indicating that the semi-active TVA is more effective in reducing vibration levels. The results further showed that increases in off-state damping decreases the effectiveness of semi-active TVA in reducing maximum vibration levels.     

时滞影响下受控斜拉索的参数振动稳定性

研究了轴向激励作用下受控斜拉索系统主参数共振的时滞效应,考虑了拉索垂度和几何非线性的影响,基于Hamilton变分原理建立了受控斜拉索系统轴向激励下的非线性参数振动方程,利用Galerkin方法得到时滞动力系统,运用多尺度法对受控系统的主参数共振进行了分析,得到了不同时滞值、控制增益时参数振动稳定域和受控拉索的时程曲线.研究表明,时滞影响下斜拉索振动控制系统的效果变差,参数共振的稳定域发生偏移,对受控斜拉索系统的控制效果随着时滞的增大而变差,从而对控制系统的参数设计起到指导作用.     

Sound absorption of a quadratic and cubic nonlinearly vibrating curved panel absorber

This study investigated the sound absorption of a nonlinearly vibrating curved panel backed by a cavity. Very few studies on similar nonlinear structural-acoustic problems have been conducted to date, although there have been many on nonlinear plate or linear structural-acoustic problems. A curved panel is considered because the overall absorption bandwidth can be designed by appropriately adjusting the panel curvature, which is a key factor in controlling the structural resonant frequencies and absorption peaks. The theoretical formulation is developed based on the assumptions of quadratic and cubic nonlinear structural vibrations, and the linear acoustic pressure induced within the cavity. An approach based on the numerical integration method is developed to solve the nonlinear governing equation of the structural-acoustic problem. In the parametric study, the panel displacement amplitude converges with an increasing number of modes. The effects of excitation level, cavity depth, and damping factor are also examined. The quadratic and cubic nonlinearities and their effects on the sound absorption are also investigated. An experiment was conducted. The theoretical and experimental observations correspond reasonably with each other.     

Variants of Steffensen-secant method and applications

In this paper, a parametric variant of Steffensen-secant method and three fast variants of Steffensen-secant method for solving nonlinear equations are suggested. They achieve cubic convergence or super cubic convergence for finding simple roots by only using three evaluations of the function per step. Their error equations and asymptotic convergence constants are deduced. Modified Steffensen’s method and modified parametric variant of Steffensen-secant method for finding multiple roots are also discussed. In the numerical examples, the suggested methods are supported by the solution of nonlinear equations and systems of nonlinear equations, and the application in the multiple shooting method.     

Nonlinear sliding surface based second order sliding mode controller for uncertain linear systems

A second order sliding mode (SOSM) controller using nonlinear sliding surface is proposed in this paper. The aim of the proposed controller is to guarantee stability as well as enhance the transient performance of uncertain linear systems with parametric uncertainty. The nonlinear sliding surface consists of a linear term and a nonlinear term. The linear term comprises a gain matrix which has a very low value of damping ratio and thereby facilitates fast response. The nonlinear term is introduced to accommodate a variable damping ratio to reduce overshoot and settling time of the closed loop system as the output reaches nearer the desired reference position. A major gain of the proposed SOSM controller is the elimination of chattering in the control input. The proposed nonlinear sliding surface based SOSM controller achieves fast rise, low overshoot and low settling time. Simulation results demonstrate the effectiveness of the proposed SOSM controller.     

Prediction of the dynamic response of a mini-cantilever beam partially submerged in viscous media using finite element method

In this work, the use of mini cantilever beams for characterization of rheological properties of viscous materials is demonstrated. The dynamic response of a mini cantilever beam partially submerged in air and water is measured experimentally using a duel channel PolyTec scanning vibrometer. The changes in dynamic response of the beam such as resonant frequency, and frequency amplitude are compared as functions of the rheological properties (density and viscosity) of fluid media. Next, finite element analysis (FEA) method is adopted to predict the dynamic response of the same cantilever beam. The numerical prediction is then compared with experimental results already performed to validate the FEA modeling scheme. Once the model is validated, further numerical analysis was conducted to investigate the variation in vibration response with changing fluid properties. Results obtained from this parametric study can be used to measure the rheological properties of any unknown viscous fluid.     

一类单自由度非自治系统的非线性振动

本文用奇异摄动理论多尺度法的导数展开法[1],求解了在微粘性阻尼作用下,连结在一个非线性弹簧上的一个质点的受迫振动方程.研究的是四次非线性问题,讨论了四种情况:非共振的软激发;非共振的硬激发;共振的软激发;共振的硬激发.     

非均匀变厚度梁的动力响应的一般解

在本文中提出一个新方法——阶梯折算法来研究在任意载荷下任意非均匀和任意变厚度伯努利-欧拉梁的动力响应问题.研究了自由振动和强迫振动.新方法需要将区间离散为一定数目的元素,每个元素可看作是均匀和等厚度的.因此均匀、等厚度梁的一般解可在每个元素上应用.然后用初参数表示的整个梁的一般解使之满足相邻二元素间的物理和几何连续条件,这样就可以得到解析形式的自由振动的频率方程和解析形式的强迫振动的最终解,它化为求解二元线性代数方程,与离散元素的数目无关.现在的方法可推广应用至任意非均匀及任意变厚度有粘滞性和其他种类的梁以及其他结构元件问题上去.     

非线性系统动力分析的模态综合技术

各种模态综合方法已广泛应用于线性结构的动力分析,但是,一般都不适用于非线性系统. 本文基于[20][21]提出的方法,将一种模态综合技术推广到非线性系统的动力分析.该法应用于具有连接件耦合的复杂结构系统,以往把连接件简化为线性弹簧和阻尼器.事实上,这些连接件通常具有非线性弹性和非线性阻尼特性.例如,分段线性弹簧、软特性或硬特性弹簧、库伦阻尼、弹塑性滞后阻尼等.但就各部件而言,仍属线性系统.可以通过计算或试验或兼由两者得到一组各部件的独立的自由界面主模态信息,且只保留低阶主模态.通过连接件的非线性耦合力,集合各部件运动方程而建立成总体的非线性振动方程.这样问题就成为缩减了自由度的非线性求解方程,可以达到节省计算机的存贮和运行时间的目的.对于阶次很高的非线性系统,若能缩减足够的自由度,那么问题就可在普通的计算机上得以解决. 由于一般多自由度非线性振动系统的复杂性,一般而言,这种非线性方程很难找到精确解.因此,对于任意激励下系统的瞬态响应,可以采用数值计算方法求解缩减的非线性方程.     

粘弹性Timoshenko梁的动力学普遍方程及其动态特性分析

本文用直接力法在时域内推导了粘弹性Timoshenko梁的控制微分方程,它同时计及了材料的拉伸粘性和剪切粘性.为了测定标准线性固体的复模量和三参数,对有机玻璃(PMMA)和尼龙6(PCL)试件成功地应用了强迫振动梁技术.通过大量数值计算,对粘弹性Timoshenko梁的动力特性,特别是阻尼特性进行了分析.结果表明,材料粘性对结构的动力特性,尤其是对阻尼有较大影响。对于高粘性材料,其动力学性质用标准线性固体模型来描写是合适的.     

Modelling and analysis of the nonlinear string-mass structure of the vibration absorber

ABSTRACT

In this paper a nonlinear string-mass structure of the vibration absorber is analyzed. This structure is convenient to be installed in vibration damping systems of high buildings for their protection in the case of earthquake. The considered string-mass structure contains a translator movable mass connected with two strings. Due to nonlinear geometric properties of the system the motion of the mass is described with a strong nonlinear second order differential equation. In the paper the approximate procedure for solving of the nonlinear equation of motion is developed. Based on the solution the influence of the string preloading force, slider mass and friction force on the vibration property of the string-mass system is investigated. It is concluded that variation of the preloading string force may be applied as a control parameter for vibration absorption and as the regulator of vibration decay time.     

Inertial Manifolds for von Karman Plate Equations

Inertial manifolds associated with nonlinear plate models governed by dynamical von Karman equations are considered. Three different dissipative mechanisms are discussed: viscous, structural and thermal damping. Though the systems considered are subject to some dissipation, the overall dynamics may not be dissipative. This means that the energy may not be decreasing. The main result of the paper establishes the existence of an inertial manifold subject to the spectral gap condition for linearized problems. The validity of the spectral gap condition depends on the geometry of the domain and the type of damping. It is shown that the spectral gap condition holds for plates of rectangular shape. In the case of viscous damping, which is associated with hyperbolic-like dynamics, it is also required that the damping parameter be sufficiently large. This last requirement is not needed for other types of dissipation considered in the paper.     

非线性梁结构的参数振动稳定性及其主动控制

采用压电材料研究了参数激励非线性梁结构的运动稳定性及其主动控制,通过速度反馈控制算法获得主动阻尼,利用Hamilton原理建立含阻尼的立方非线性运动方程,采用多尺度方法求解运动方程获得稳定性区域．通过数值算例,分析了控制增益、外激振力幅值等因素对稳定性区域和幅频曲线特性的影响．分析表明:控制增益增大,结构所能承受的轴向力也增大,在一定范围内结构的主动阻尼比也增加;随着控制增益的增大,响应幅值逐渐降低,但所需的控制电压存在峰值点．     

车路耦合非线性振动高阶Galerkin截断研究

将移动车辆模型化为运动的两自由度质量-弹簧-阻尼系统,道路模型化为立方非线性黏弹性地基上的弹性梁,并将路面不平度设定为简谐函数．通过受力分析,建立车路非线性耦合振动高阶偏微分方程．采用高阶Galerkin截断结合数值方法求解耦合系统的动态响应．首次研究不同截断阶数对车路耦合非线性振动动态响应的影响,确定Galerkin截断研究车路耦合振动的收敛性．研究结果表明,对于软土地基的沥青路面,耦合振动的动态响应,需要150阶以上的截断才能达到收敛效果．并通过高阶收敛的Galerkin截断研究了系统参数对车路耦合非线性振动动态响应的影响.     

Eigenwaves in a relativistic heat-conducting viscous fluid according to extended irreversible thermodynamics

We propose an approach to extended irreversible thermodynamics leading to a set of transport equations for the dissipative variables characterizing a relativistic heat-conducting viscous fluid. After this, we study the propagation eigenmodes showing the possible existence of three non-trivial modes: two longitudinal and one transverse. The fast longitudinal wave is a pressure wave associated to a slow longitudinal thermal dissipation wave and a transverse viscous dissipation wave. The general study of the longitudinal case is not possible. Natural special cases are analyzed in detail.