Analysis of sensitivity and robustness of forced response for nonlinear dynamic structures

An effective method is developed for calculation of the sensitivity and robustness of the forced response levels for strongly nonlinear structures. The sensitivity coefficients are determined with respect to parameters of friction contact interfaces, parameters of linear components of an assembled structure, frequency and level of excitation forces.Equations for determination of first- and second-order sensitivity coefficients of the forced response are derived analytically from a nonlinear multiharmonic equation of motion. The analytical derivation allows accurate and fast evaluation of the sensitivity coefficients. The sensitivity coefficients are calculated in frequency domain for each excitation frequency over the frequency range analysed simultaneously with the force response levels. The developed highly efficient method allows calculation of sensitivity characteristics without a noticeable increase of the computation time in addition to the time required for the forced response calculation.A measure of the forced response robustness is introduced. Sensitivity-based method for assessment of the forced response robustness for given ranges of uncertainty of structural and operating parameters is proposed.The methodology developed is illustrated on a set of problems including cases of forced response analysis for realistic strongly nonlinear gas-turbine structures.     

液压缸非线性约束下的轧机辊系振动行为

以四辊板带轧机为例,分析液压压下缸及弯辊缸在轧机辊系振动时表现出的分段弹性力和摩擦力两种非线性约束,建立液压缸非线性约束作用下的轧机辊系振动模型,并采用平均法求得振动系统的幅频响应。通过比较两种非线性作用下辊系振动速度和振动幅值的仿真曲线,研究辊系受分段弹性力和摩擦力影响时的行为特性。取不同分段弹性力和摩擦力,仿真分析两种非线性因素分别对轧机辊系幅频特性的影响规律。结果表明,轧机辊系振动速度受分段弹性力大小影响,系统不稳定频率区域随分段弹性力增大而变宽;摩擦力较小时,对辊系振动行为影响表现为阻尼特性,较大时,摩擦力的非线性成为影响辊系振动行为的主要特性。该结论为轧机辊系振动控制提供了理论参考。     

An approach to evaluate the effects of nonlinear traveling joints on dynamic behavior of large machine tools

In order to select the best possible design from various candidates based on structure configurations of traveling components, it is essential to evaluate the effects of traveling joints on dynamic behavior of large machine tools. Firstly, regression analysis and energy method were used to determine the nonlinear parameters of the metal–plastic joints. Then, a nonlinear receptance coupling approach was used to establish a simplified model of a large machine tool including nonlinear joints. The evaluations of the large machine tool were analyzed with different milling forces. The results show that the nonlinearity of traveling joints significantly influences the resonant frequency and the response amplitudes. Experimental verifications were performed on a prototype of a large machine tool for milling large gears.     

Parametric identification of structural nonlinearities from measured frequency response data

Structural nonlinearity is a common phenomenon encountered in engineering structures under dynamic loading. In several cases, linear theory can suffice to analyze nonlinear systems to some extent. However, there are cases where nonlinear effects and therefore nonlinear analysis become unavoidable. In most of the engineering applications it is usually very difficult if not impossible to model nonlinearity theoretically, especially for nonlinear effects stemming from structural connections. Then it becomes necessary to detect, localize and parametrically identify nonlinear elements from measured vibration data. In this study, two different methods, one being a method suggested recently by two of the authors of this paper, and the other being again a method developed in an earlier work, are implemented on a test rig containing a nonlinear element. Both methods are capable of parametrically identifying nonlinearities from measured frequency response functions. It is aimed in this paper to asses the validity of each method by applying them to a real test structure and thus parametrically identifying the nonlinear element in the system to obtain a mathematical model, and then employing the model in harmonic response analysis of the system in order to compare predicted responses with measured ones.     

Bifurcation Analysis of an Electrostatically Actuated Nano-Beam Based on Modified Couple Stress Theory

In this paper, the nonlinear size-dependent static and dynamic behavior of an electrostatically actuated nano-beam is investigated. A fully clamped nano-beam is considered for the modeling of the deformable electrode of the NEMS. The governing differential equation of the motion is derived using Hamiltonian principle based on couple stress theory; a non-classical theory for considering length scale effects. The nonlinear partial differential equation of the motion is discretized to a nonlinear Duffing type ODE’s using Galerkin method. Static and dynamic pull-in instabilities obtained by both classical theory and MCST are compared. At the second stage of analysis, shooting technique is utilized to obtain the frequency response curve, and to capture the periodic solutions of the motion; the stability of the periodic solutions are gained by Floquet theory. The nonlinear dynamic behavior of the deformable electrode due to the AC harmonic accompanied with size dependency is investigated.     

迟滞非线性振荡压实系统动态响应分析

分析了振荡压实中材料发生弹塑性变形而表现出的迟滞非线性,建立了一种新的振荡压实非线性动力学模型。在一次近似的前提下,利用等效线性化方法,将迟滞恢复力转化为等效刚度和等效阻尼表示,根据谐波平衡原则求出了单自由度振荡压实系统的频响方程。数值求解表明,随着激励幅度的增大系统出现软化趋势,共振点随激励变化而漂移。将等效线性化的结果推广到二自由度振荡压实系统,得到了频响方程组。     

NONLINER VIBRATION ANALYSIS OF THE PLATE WITH DRY FRICTION SUPPORT CONDITION

The rigid-interface friction model is usually used in the nonlinear vibration of the rectangular plate with dry friction support edges. The present study provides an extension by using a hysteretic spring friction model and taking account of the stick-slip motion of the plate. Results for a range of problem parameters have been obtained. The results show that the nonlinear frequency response behavior of the system can be quite different from the rigid-interface friction model. The effects of the stiffness at friction interfaces and the stick-slip motion on the nonlinear vibration of the plate are significant and hence cannot be neglected.     

Analysis of bilinear oscillators under harmonic loading using nonlinear output frequency response functions

In this paper, the new concept of nonlinear output frequency response functions (NOFRFs) is extended to the harmonic input case, an input-independent relationship is found between the NOFRFs and the generalized frequency response functions (GFRFs). This relationship can greatly simplify the application of the NOFRFs. Then, beginning with the demonstration that a bilinear oscillator can be approximated using a polynomial-type nonlinear oscillator, the NOFRFs are used to analyse the energy transfer phenomenon of bilinear oscillators in the frequency domain. The analysis provides insight into how new frequency generation can occur using bilinear oscillators and how the sub-resonances occur for the bilinear oscillators, and reveals that it is the resonant frequencies of the NOFRFs that dominate the occurrence of this well-known nonlinear behaviour. The results are of significance for the design and fault diagnosis of mechanical systems and structures which can be described by a bilinear oscillator model.     

The effect of speed of balanced rotor on nonlinear vibrations associated with ball bearings

The paper deals with the structural dynamic response of rotor supported by ball bearings. The mathematical model takes into account the sources of nonlinearity such as Hertzian contact force, surface waviness, varying compliance and internal radial clearance resulting transition from no contact to contact state between rolling elements and races. In terms of the feature that the nonlinear bearing forces act on the system, a new reduction method and corresponding integration technique is used to increase the numerical stability and decrease computer time for system analysis. The effects of speed of balanced rotor in which ball bearings show periodic, quasi-periodic and chaotic behavior are analyzed. The results are presented in the form of time displacement responses, frequency spectra and Poincarè maps. It is implied from the frequency spectra that peak amplitude of vibrations appear at the varying compliance frequency.     

空间机械臂锁紧机构等效线性化分析及验证

空间机械臂的锁紧机构刚度特性具有非线性特征,在开展整臂动力学分析时需要对其进行等效线性化处理,故本文提出了一种基于六维刚度等效的线性化方法并进行了分析验证。建立了锁紧机构单机在预紧状态下的非线性模型,通过非线性计算获取了锁紧机构各向刚度数值,并将其提供给用于模拟锁紧释放机构的BUSH单元。对机械臂整体模型进行了模态分析以及频率响应分析。结果表明:整臂前三阶固有频率为89 Hz、92 Hz和96 Hz,和试验结果最大相差为3%;各组件加速度响应分析结果与试验结果吻合较好,最大相差为8%,证明了提出的线性化方法的有效性。该方法考虑了各向刚度之间耦合关系,并且不必逐一处理锁紧机构接触面,有较强的工程实用价值。     

非稳态油膜力的转子-定子-轴承系统碰摩故障研究

利用拉格朗日（Lagrange）方程建立了非稳态油膜力的转子-定子-轴承系统碰摩故障的力学模型,应用数值分析对其进行研究,得出转子系统在激励频率作为唯一控制参数时系统的轴心轨迹图和分岔图;对该系统响应的非线性行为和故障机理进行分析,从而为该类转子系统故障诊断和系统的安全运行提供理论依据.研究结果表明,当激励频率作为唯一控制参数时,系统存在周期运动、拟周期运动和混沌运动等丰富的非线性现象.     

振动与纳米研究的双向发展

结合多年来与振动磨相关的研究工作,从学科交叉的视觉,论述了振动与纳米材料研究的双向发展,包括多相离散体系的碰撞振动及混沌、超微颗粒碰撞阻尼机理及应用、滚压振动磨在纳米技术发展中的应用.提出了关于振动磨的变质量振动模型,并分别利用多尺度法和四阶Runge-Kutta法求解了系统,计算结果得到了振动磨工程实践的验证.根据超微颗粒细化过程的不可逆能耗,提出了颗粒碰撞阻尼机制,建立了力学模型并分析了其混沌运动特征,证明当固有频率比为1时具有最好的减振效果.利用振动在干法室温条件下大批量制备了尺度为3～5nm的金属锌颗粒,每千克耗电仅为7～8 kw·h,并制备了晶粒度约为7.6 nm的钛-锆二元合金.     

间隙对含摩擦和时变刚度的齿轮系统动力学响应的影响

主要研究在考虑摩擦和时变刚度时,轮齿间隙对齿轮系统动力学响应的影响。建立常间隙、时变间隙和随机间隙三种不同的间隙形式。利用数值仿真的方法得到系统的幅频响应曲线和时间历程曲线。分析发现：①在低速时,随着摩擦因数的增大,系统响应的方均值和平均分量增大;并经过三次跳跃之后系统进入混沌运动状态;②时变间隙幅值增大导致系统提前进入混沌状态,而且随着时变幅值的增大跳转频率逐渐减小;时变间隙频率较小时,间隙对系统的影响较小;当时变间隙的频率较大时,系统在 =0.2,0.3时出现明显的共振响应,系统的响应以高频率分量4、5和6为主;③考虑随机间隙时,随着 的增大,齿轮系统响应的平均分量波动比较大。     

准零刚度扭转隔振器振动传递特性研究

针对一种应用连杆弹簧负刚度结构所设计的正负刚度并联扭转隔振器,通过分析其设计参数,得到了其在静平衡位置达到准零刚度的条件。针对所述系统,建立二自由度非线性动力学微分方程,应用谐波平衡法求解了准零刚度隔振器的振动响应。考虑激励幅值、系统阻尼和转动惯量等因素,从扭转振动力矩传递和扭转振动功率流传递两个角度分析了准零刚度隔振系统的非线性振动传递特性。结果表明,此隔振系统具有良好的低频隔振效果,且表现为振动传递的硬特性;从功率流角度的分析还表明,系统的瞬时传递功率流峰值与输入功率流和耗散功率流的变化趋势保持一致。在低于共振区域的低频区间内,隔振系统前后两端的转动惯量对系统振动具有较大影响。最后,对扭转隔振器的振动隔离特性进行了试验验证,试验结果表明,隔振器具有良好的低频振动隔离特性。     

A new method for the accurate measurement of higher-order frequency response functions of nonlinear structural systems

Higher-order frequency response functions (FRFs) are important to the analysis and identification of structural nonlinearities. Though much research effort has been devoted recently to their potential applications, practical issues concerning the difficulty and accuracy of higher-order FRF measurement have not been rigorously assessed to date. This paper presents a new method for the accurate measurement of higher-order FRFs. The method is developed based on sinusoidal input, which is ideal for exciting a nonlinear structure into desired regimes with flexible control, and the correlation technique, which is a novel signal processing method capable of extracting accurate frequency components present in general nonlinear responses. The correlation technique adopted is a major improvement over Fourier transform based existing methods since it eliminates leakage and aliasing errors altogether and proves to be extremely robust in the presence of measurement noise. Extensive numerical case studies have been carried out to critically assess the capability and accuracy of the proposed method and the results achieved are indeed very promising. Interesting nonlinear behavior such as frequency shift and jump have been observed in first-, second- and third-order FRFs, as well as solitary islands which have been identified over which higher-order FRFs virtually do not change as input force amplitude varies. Higher-order FRFs over such solitary islands are essentially their theoretical counterparts of Volterra transfer functions which can be measured with very low input force and can be profitably employed for the identification of physical parameters of structural nonlinearities. Subsequently, a nonlinear parameter identification method has also been developed using measured higher-order FRFs and results are presented and discussed.     

Comparisons of gear dynamic responses with rectangular mesh stiffness and its approximate form

The mesh stiffness is close to rectangular stiffness, and the first harmonic approximate term of rectangular stiffness is generally adopted in the nonlinear gear dynamic analysis. The differences between the rectangular stiffness and its approximate form are analyzed in detail. The frequency response and dynamic factor are calculated by a numerical method, to illustrate the dynamic characteristics of the gear nonlinear system with different mesh stiffness forms. The results show that: The trends of frequency response of gear dynamic system with rectangular stiffness and its approximate form are identical. The jump phenomena are detected in both cases. Without the effect of static transmission error, the dynamic factor with rectangular mesh stiffness is larger than that with approximate mesh stiffness. Under design power and speed condition, the result with approximate mesh stiffness function may deduce reasonless suggestions for a designer. The static transmission error will enlarge the vibration amplitude and dynamic factor when the approximate mesh stiffness is adopted, but the effects on the response of gear system with rectangular mesh stiffness are fractional. The mesh stiffness may excite the odd subharmonic resonance, and the static transmission error may excite the even sub-harmonic resonance respectively.     

Design of a nonlinear vibration absorber using three-to-one internal resonances

A weakly nonlinear vibration absorber is designed to attenuate the primary resonance vibrations of a single-degree-of-freedom weakly nonlinear oscillator having cubic nonlinearity. The linearised natural frequency of the nonlinear absorber is tuned to be approximately one-third the linearised natural frequency of the primary nonlinear oscillator. The low frequency mode for the absorber is favourably considered based on the fact that the nonlinear absorber can be easily realised in practice by using a light-weight mass attachment with small values of linear and nonlinear stiffness of coupling. For a given primary nonlinear oscillator and absorber mass, implementation of three-to-one internal resonances requires the smallest value of the absorber linear stiffness among three options for utilising internal resonances to design nonlinear absorber. The method of multiple scales is used to obtain the averaged equations that determine the amplitudes and phases of the first-order approximate solutions to the vibrations of the primary nonlinear oscillator and nonlinear absorber. It is found that the absorber response may admit either forced vibration having the forcing frequency or a combination of forced vibration and free-oscillation term having one third the forcing frequency. The nonlinear absorber can effectively suppress the amplitude of primary resonance response and eliminate saddle-node bifurcations occurring in the frequency-response curves of the primary nonlinear oscillator. Numerical results are given to show the effectiveness of the nonlinear absorber for suppressing nonlinear vibrations of the primary nonlinear oscillator under primary resonance conditions.     

Identification of nonlinear boundary effects using nonlinear normal modes

Local nonlinear effects due to micro-slip/slap introduced in boundaries of structures have dominant influence on their lower modal model. This paper studies these effects by experimentally observing the behavior of a clamped–free beam structure with local nonlinearities due to micro-slip at the clamped end. The structure is excited near one of its resonance frequencies and recorded responses are employed to identify the nonlinear effects at the boundary. The nonlinear response of structure is defined using an amplitude-dependent nonlinear normal mode identified from measured responses. A new method for reconstructing nonlinear normal mode is represented in this paper by relating the nonlinear normal mode to the clamped end displacement-dependent stiffness parameters using an eigensensitivity analysis. Solution of obtained equations results equivalent stiffness models at different vibration amplitudes and the corresponding nonlinear normal mode is identified. The approach results nonlinear modes with efficient capabilities in predicting dynamical behavior of the structure at different loading conditions. To evaluate the efficiency of the identified model, the structure is excited at higher excitation load levels than those employed in identification procedures and the observed responses are compared with the predictions of the model at the corresponding input force levels. The predictions are in good agreement with the observed behavior indicating success of identification procedure in capturing the physical merits involve in the boundary local nonlinearities.     

GPU-based parallel computation for structural dynamic response analysis with CUDA

Frequency response analysis is an important computational tool to simulate and understand the dynamic behavior of structures. However, for more target frequency and/or larger scale structures, the runtime is greatly increased. Furthermore, increasingly complex degree of freedom problems intended to improve the accuracy of the analysis results is creating longer. In this paper, we present efficient analysis using runtime reduction in frequency response analysis with NVIDIA GPU using the compute unified device architecture (CUDA) programming environment. The proposed method is based on the sparse conjugate gradient method and a Jacobi preconditioner. Numerical examples which implemented by three different FE model are used to verify the validity. The results show that GPU parallel implementation achieves significant speed up compared to a single CPU processor. Through these results, in the frequency response analysis, we show the possibility for efficient analysis with reduction of the solving time by using GPU parallel implementation.     

Nonlinear ultrasonic techniques for nondestructive assessment of micro damage in material: A review

The nondestructive assessment of the damage that occurs in components during service plays a key role for condition monitoring and residual life estimation of in-service components/structures. Ultrasound has been widely utilized for this; however most of these conventional methods using ultrasonic characteristics in the linear elastic region are only sensitive to gross defects but much less sensitive to micro-damage. Recently, the nonlinear ultrasonic technique, which uses nonlinear ultrasonic behavior such as higher-harmonic generation, subharmonic generation, nonlinear resonance, or mixed frequency response, has been studied as a positive method for overcoming this limitation. In this paper, overall progress in this technique is reviewed with the brief introduction of basic principle in the application of each nonlinear ultrasonic phenomenon.