Experiments and numerical results on non-linear vibrations of an impacting Hertzian contact. Part 2: random excitation

Non-linear dynamic behaviour of a normally excited preloaded Hertzian contact (including possible contact losses) is investigated using an experimental test rig. It consists of a double sphere plane contact loaded by the weight of a rigid moving mass. Contact vibrations are generated by a external Gaussian white noise and exhibit vibroimpact responses when the input level is sufficiently high. Spectral contents and statistics of the stationary transmitted normal force are analyzed. A single-degree-of-freedom non-linear oscillator including loss of contact and Hertzian non-linearities is built for modelling the experimental system. Theoretical responses are obtained by using the stationary Fokker-Planck equation and also Monte Carlo simulations. When contact loss occurrence is very occasional, numerical results show a very good agreement with experimental ones. When vibroimpacts occur, results remain in reasonable agreement with experimental ones which justify the modelling and the numerical methods described in this paper.The contact loss non-linearity appears to be rather strong compared to the Hertzian non-linearity. It actually induces a large broadening of the spectral contents of the response. This result is of great importance in noise generation for many systems such as mechanisms using contacts to transform motions and forces (gears, ball-bearings, cam systems, to name a few). It is also of great importance for tribologists preoccupied with preventing surface damage.     

具有非线性阻尼涨落的线性谐振子的随机共振

较之于线性噪声, 非线性噪声更广泛地存在于实际系统中, 但其研究远不能满足实际情况的需要. 针对作为非线性阻尼涨落噪声基本构成成分的二次阻尼涨落噪声, 本文考虑了周期信号与之共同作用下的线性谐振子, 关注这类具有基本意义的阻尼涨落噪声的非线性对系统共振行为的影响. 利用Shapiro-Loginov公式和Laplace变换推导了系统稳态响应振幅的解析表达式, 并分析了稳态响应振幅的共振行为, 且以数值仿真验证了理论分析的有效性. 研究发现: 系统稳态响应振幅关于非线性阻尼涨落噪声系数具有非单调依赖关系, 特别是非线性阻尼涨落噪声比线性阻尼涨落噪声更有助于增强系统对外部周期信号的响应程度; 而且, 非线性阻尼涨落噪声比线性阻尼涨落噪声使得稳态响应振幅关于噪声强度具有更为丰富的共振行为; 同时, 二次阻尼涨落噪声使得稳态响应振幅关于系统频率出现真正的共振现象; 而在这些现象和性质中, 非线性噪声项的非线性性质对共振行为起着关键的作用. 显然, 以二次阻尼涨落作为基本形式引入的非线性阻尼涨落噪声, 可以有助于提高微弱周期信号检测的灵敏度和实现对周期信号的频率估计.     

FREQUENCY DOMAIN ARX MODEL AND MULTI-HARMONIC FRF ESTIMATORS FOR NON-LINEAR DYNAMIC SYSTEMS

Non-linear dynamic systems respond at frequencies other than the excitation frequency; however, standard frequency response function estimators for linear systems do not accommodate this harmonic distortion. A new multi-harmonic frequency response function estimator that utilizes discrete frequency models for non-linear systems is introduced here. The multi-harmonic estimator relates the frequency response at each frequency to the input and output spectra within a given frequency band in the same way that autoregressive exogenous input models relate inputs and outputs at particular samples in the time domain. Overdetermined, least-mean-squares calculations are used to minimize model error throughout a frequency band rather than at a single frequency as in the corresponding linear estimators. The resulting multi-harmonic frequency response function models are non-parametric (e.g., vary with amplitude) when linear functions are used and parametric when non-linear functions are used. A new sensitive indicator for experimentally characterizing non-linearity is introduced.     

STOCHASTIC AVERAGING OF STRONGLY NON-LINEAR OSCILLATORS UNDER BOUNDED NOISE EXCITATION

A stochastic averaging method for strongly non-linear oscillators under external and/or parametric excitation of bounded noise is proposed by using the so-called generalized harmonics functions. The method is then applied to study the primary resonance of Duffing oscillator with hardening spring under external excitation of bounded noise. The stochastic jump and its bifurcation of the system are observed and explained by using the stationary probability density of amplitude and phase. Subsequently, the method is applied to study the dynamical instability and parametric resonance of Duffing oscillator with hardening spring under parametric excitation of bounded noise. The primary unstable region is delineated by evaluating the Lyapunov exponent of linearized system, and the response and jump of non-linear system around the unstable region are examined by using the sample functions and stationary probability density of amplitude and phase.     

柴油机轴系扭振系统的强非线性主共振

研究柴油机轴系扭振系统强非线性问题.根据拉格朗日方程建立柴油机轴系扭振系统的动力学模型,通过参数变换,应用Modified Lindstedt-Poincaré方法得到柴油机轴系扭振系统强非线性主共振的幅频响应方程,分析系统不同参数对主共振幅频响应的影响.结果表明,系统的幅频响应曲线存在跳跃,随着简谐力矩的减小和阻尼的增大,系统的非线性跳跃减弱,系统的振幅减小,系统主共振的区域也随之减小;随着调谐参数的变化,系统的主共振力幅响应曲线存在两种拓扑结构.MLP方法得出的近似解析解与龙格库塔法得出的数值解吻合.     

双稳态结构中的1/2次谐波共振及其对隔振特性的影响

以典型的双稳态系统——屈曲梁结构为例,基于等效模型,结合解析、数值和实验手段,研究了双稳态结构中的1/2次谐波共振特性、演化过程、参数调节规律及其对隔振特性的影响.研究发现,当非线性刚度系数或激励幅值增加到一定程度时,系统会在一定带宽下产生显著的1/2次谐波共振;随着激励幅值增加,阻尼系统的1/2次谐波遵循“产生-增强-衰退-消失”的过程,该过程对峰值频率和峰值传递率有重要影响;适当提高非线性强度能有效改善双稳态结构隔振特性.针对双稳态屈曲梁结构开展的实验验证了1/2次谐波特性和隔振特性变化规律.     

Simplified models of the vibration of mannequins in car seats

A simplified two-dimensional modelling approach to predict the vibration response of mannequin occupied car seats about a static settling point is demonstrated to be feasible. The goal of the research is to develop tools for car seat designers. The two-dimensional model, consisting of interconnected masses, springs and dampers is non-linear due to geometric effects but, under the excitations considered, the model behaviour is linear. In this approach to modelling, the full system is initially broken down into subsystems, and experiments are conducted with subsystems to determine approximate values for the stiffness and damping parameters. This approach is necessary because of the highly non-linear behaviour of foam where stiffness changes with compression level, and because the simplified model contains more structure than is necessary to model the relatively simple measured frequency response behaviour, thus requiring a good initial starting point from which to vary parameters. A detailed study of the effects of changing model parameters on the natural frequencies, the mode shapes and resonance locations in frequency response functions is given, highlighting the influence of particular model parameters on features in the seat-mannequin system's vibration response. Reasonable qualitative as well as good quantitative agreement between experimental and simulation frequency response estimates is obtained. In particular, the two-dimensional motions at the peaks in the frequency response, a combination of up and down and rotational behaviour is predicted well by the model. Currently research is underway to develop a similar model with non-linear springs, surface friction effects and viscoelastic elements, that predicts the static settling point, a necessary step to aid in the subsystem modelling stage in this dynamic modelling approach.     

A dynamic Lagrangian frequency-time method for the vibration of dry-friction-damped systems

A new frequency-time domain procedure, the dynamic Lagrangian mixed frequency-time method (DLFT), is proposed to calculate the non-linear steady state response to periodic excitation of structural systems subject to dry friction damping. In this formulation, the dynamic Lagrangians are defined as the non-linear contact forces obtained from the equations of motion in the frequency domain, with the adjunction of a penalization on the difference between the interface displacements calculate by the non-linear solver in the frequency domain and those calculated in the time domain from the non-linear contact forces, thus accounting for Coulomb friction and non-penetration conditions. The dynamic Lagrangians allow one to solve for the non-linear forces between two points in contact without using artifacts such as springs. The new DLFT method is thus particularly well suited to handling finite element models of structures in frictional contact, as it does not require a special model for the contact interface. Dynamic Lagrangians are also better suited to frequency-domain friction problems than the traditional time-domain method of augmented Lagrangians. Furthermore, a reduction of the non-linear system to relative interface displacements is introduced to decrease the computation time. The DLFT method is validated for a beam in contact with a flexible dry friction element connected to ground, for frictional constraints that feature two-dimensional relative motion. Results are also obtained for a large-scale structural system with a large number of one-dimensional dry-friction dampers. The DLFT method is shown to be accurate and fast, and it does not suffer from convergence problems, at least in the examples studied.     

Experimental and numerical study of a vibro-impact phenomenon in a gearshift cable

This paper presents a study of the nonlinear dynamic behavior of a gearshift cable and more specifically of the associated tizzing phenomenon. A gearshift cable is composed of an inner wire that can slide freely in an outer composite housing. When undergoing harmonic excitation, the inner wire interacts with the housing. Hammer and swept sine shaker tests are first used to estimate the characteristics of the two main components. It is shown that the behavior of the outer housing is nonlinear and depends on the amplitude of the excitation. The assembled gearshift cable is then set up on the shaker and the nonlinear vibro-impacting phenomenon is studied. Finally a finite element model, based on the Euler-Bernoulli beams and the Rayleigh damping coefficients, proves to offer good correlation with the measured data for different excitation frequencies. A period doubling bifurcation is observed both experimentally and numerically.     

基础激励下分数阶线性系统的响应特性分析

本文研究了基础激励下含分数阶阻尼的线性系统的响应特性. 当基础激励为简谐激励时, 通过待定系数方法求得系统的动力传递系数; 当基础激励为非简谐周期激励时, 首先将激励展开成傅里叶级数, 然后根据线性系统的叠加原理求得激励中各阶频率成分所引起的动力传递系数, 并根据展开的傅里叶级数解决了数值运算中的不可导问题. 用数值仿真的方法对解析结果进行了验证, 两者符合良好, 证明了解析分析的正确性. 研究表明, 基础激励引起的动力传递系数依赖于分数阶阻尼阶数的值, 通过调节阻尼阶数可以控制动力传递系数的大小. 对于基础激励为非简谐的周期激励情况, 当激励频率一定时, 激励中的高阶频率成分引起的动力传递系数可能大于激励中的低阶频率成分引起的动力传递系数. 因此, 激励中的高阶频率成分所起的作用是不可忽略的.     

Influence of non-linear forces on beam behaviour in flutter conditions

Appropriate researches on non-linear panel flutter behaviour have been already performed by many authors. In most cases the intent of them focuses on the limit cycle determination, with particular interest towards its amplitude versus the flow dynamic pressure. This paper deals first with a study of all the solutions without damping of beam flutter versus the vibration frequency in non-linear post-critical conditions. A numerical model, which takes into account the influence of the non-linear contribution of the structural forces, due to the axial stretching of the beam, has been implemented. A complete analysis of all the possible non-linear solutions without damping leads to the possibility of characterizing the most appropriate conditions for the presence of the post-critical panel flutter limit cycles. Then the complete model, which also takes into account aerodynamic damping, has been utilized, according to the “Piston Theory”, to verify the state evolution of the fluttering damped beam towards the limit cycle, which is very near to the undamped vibrating beam state with minimum amplitude. This convergence test is an interesting aspect of the numerical results.     

Three-dimensional finite element simulation of closed delaminations in composite materials

Early stage delaminations in composite materials tend to be closed at rest. Inspection with traditional linear ultrasonic techniques generally fails to diagnose and locate such imperfections. However, if undetected and left untreated, incipient defects may gradually grow within the material and eventually lead to failure of the component. Kissing bonds or clapping contacts inherently demand a non-linear diagnostic method, applying a finite excitation amplitude that is able to overcome an activation threshold to open and close the contact. In order to obtain a better understanding and analysis of the macroscopic non-linear behavior that can be observed at the component level, we developed and investigated the results of a finite element model for a composite material containing a single circular delamination. The model makes use of local node splitting and the non-linear constitutive behavior is implemented by means of spring-damper elements at the delamination interface. The results of this parametric study allow a better insight in the behavior of the excited delamination in experimental conditions, including the appearance of localized subharmonics and harmonics of the excitation frequency. Based on the developed model, two different detection and localization techniques (using either a single frequency or a sweep excitation) were demonstrated to determine position, shape, depth and orientation of one or multiple delaminations.     

Modelling the nonlinear behaviour of double walled carbon nanotube based resonator with curvature factors

This paper deals with the nonlinear vibration analysis of a double walled carbon nanotube based mass sensor with curvature factor or waviness, which is doubly clamped at a source and a drain. Nonlinear vibrational behaviour of a double-walled carbon nanotube excited harmonically near its primary resonance is considered. The double walled carbon nanotube is harmonically excited by the addition of an excitation force. The modelling involves stretching of the mid plane and damping as per phenomenon. The equation of motion involves four nonlinear terms for inner and outer tubes of DWCNT due to the curved geometry and the stretching of the central plane due to the boundary conditions. The vibrational behaviour of the double walled carbon nanotube with different surface deviations along its axis is analyzed in the context of the time response, Poincaré maps and Fast Fourier Transformation diagrams. The appearance of instability and chaos in the dynamic response is observed as the curvature factor on double walled carbon nanotube is changed. The phenomenon of Periodic doubling and intermittency are observed as the pathway to chaos. The regions of periodic, sub-harmonic and chaotic behaviour are clearly seen to be dependent on added mass and the curvature factors in the double walled carbon nanotube. Poincaré maps and frequency spectra are used to explicate and to demonstrate the miscellany of the system behaviour. With the increase in the curvature factor system excitations increases and results in an increase of the vibration amplitude with reduction in excitation frequency.     

Vibrational resonance in fractional-order overdamped multistable systems

We investigated the vibrational resonance in fractional-order overdamped multistable systems theoretically and numerically. For a given fractional order p, our results show that the response amplitude exhibits a series of peaks as the frequency or the amplitude of the high frequency input varies. However, when the low-frequency input increases, the response amplitude exhibits unimodal resonance for 1?<?p?<?2. Additionally, for different values of p, whether the response amplitude changes monotonically depends on the degree of spatial potential asymmetry. The mechanism by which p affects the resonance behavior is analyzed. Our results indicate that the value of p affects the resonance behavior by either altering the conditions under which the response amplitude attains its extreme values, or by altering the oscillating frequency of the response amplitude with varying systematic parameters.     

Vibration analysis of the tension shadow mask with wire impact damping

Non-linear vibration of the CRT shadow mask with impact damping wires is analyzed in consideration of the mask tension distribution and the effect of wire impact damping. A reduced order FEM model of the shadow mask is obtained from dynamic condensation of the mass and stiffness matrices, and damping wire is modelled using the lumped parameter method to effectively describe its contact interactions with the shadow mask. The non-linear contact-impact model is composed of spring and damper elements, of which parameters are determined from the Hertzian contact theory and the restitution coefficient, respectively. The analysis model of the shadow mask with damping wires is experimentally verified through impact tests of shadow masks performed in a vacuum chamber. Using the validated analysis model of the shadow mask with damping wires, the ‘design of experiments’ technique is applied to search for the optimal damping wire configuration so that the vibration attenuation of the shadow mask is maximized.     

Comments on nonlinear dynamics of a non-ideal Duffing-Rayleigh oscillator: Numerical and analytical approaches

An analytical and numerical investigation into the dynamic interaction between a cantilever beam with nonlinear damping and stiffness behavior, modeled by the Duffing-Rayleigh equation, and a non-ideal motor that is connected to the end of the beam, is presented. Non-stationary and steady-state responses in the resonance region as well as the passage through resonance behavior when the frequency of the excitation is varied are analyzed. The influences of nonlinear stiffness, nonlinear damping and the extent of the unbalance in the motor are examined. It is found that in this situation so-called Sommerfeld effects may be observed; the increase required by a source operating near the resonance results in a small change in the frequency, but there is a large increase in the amplitude of the resultant vibration and the jump phenomenon occurs.     

Non-linear resonances in the forced responses of plates,part II: Asymmetric responses of circular plates

The dynamic analogue of the von Karman equations is used to study the forced response, including asymmetric vibrations and traveling waves, of a clamped circular plate subjected to harmonic excitations when the frequency of excitation is near one of the natural frequencies. The method of multiple scales, a perturbation technique, is used to solve the non-linear governing equations. The approach presented provides a great deal of insight into the nature of the non-linear forced resonant response. It is shown that in the absence of internal resonance (i.e., a combination of commensurable natural frequencies) or when the frequency of excitation is near one of the lower frequencies involved in the internal resonance, the steady state response can only have the form of a standing wave. However, when the frequency of excitation is near the highest frequency involved in the internal resonance it is possible for a traveling wave component of the highest mode to appear in the steady state response.     

An adaptive harmonic balance method for predicting the nonlinear dynamic responses of mechanical systems—Application to bolted structures

Aeronautical structures are commonly assembled with bolted joints in which friction phenomena, in combination with slapping in the joint, provide damping on the dynamic behavior. Some models, mostly nonlinear, have consequently been developed and the harmonic balance method (HBM) is adapted to compute nonlinear response functions in the frequency domain. The basic idea is to develop the response as Fourier series and to solve equations linking Fourier coefficients. One specific HBM feature is that response accuracy improves as the number of harmonics increases, at the expense of larger computational time. Thus this paper presents an original adaptive HBM which adjusts the number of retained harmonics for a given precision and for each frequency value. The new proposed algorithm is based on the observation of the relative variation of an approximate strain energy for two consecutive numbers of harmonics. The developed criterion takes the advantage of being calculated from Fourier coefficients avoiding time integration and is also expressed in a condensation case. However, the convergence of the strain energy has to be smooth on tested harmonics and this constitutes a limitation of the method. Condensation and continuation methods are used to accelerate calculation. An application case is selected to illustrate the efficiency of the method and is composed of an asymmetrical two cantilever beam system linked by a bolted joint represented by a nonlinear LuGre model. The practice of adaptive HBM shows that, for a given value of the criterion, the number of harmonics increases on resonances indicating that nonlinear effects are predominant. For each frequency value, convergence of approximate strain energy is observed. Emergence of third and fifth harmonics is noticed near resonances both on vibratory responses and on approximate strain energy. Parametric studies are carried out by varying the excitation force amplitude and the threshold value of the adaptive algorithm. Maximal amplitudes of vibration and frequency response functions are plotted for three different points of the structure. Nonlinear effects become more predominant for higher force amplitudes and consequently the number of retained harmonics is increased.     

Multi-mode nonlinear resonance ultrasound spectroscopy for defect imaging: an analytical approach for the one-dimensional case

A nonlinear version of the resonance ultrasound spectroscopy (RUS) theory is presented as an extension of the RUS formalism to the treatment of microdamage characterized by nonlinear constitutive equations. General analytical equations are derived for the one-dimensional case, describing the excitation amplitude dependent shift in the resonance frequency and the generation of harmonics resulting from the interaction between bar modes due to the presence of either localized or volumetrically distributed nonlinearity. Solutions are obtained for classical cubic nonlinearity, as well as for the more interesting case of hysteresis nonlinearity. The analytical results are in excellent quantitative agreement with numerical calculations from a multiscale model. Finally, the analytical formulas are exploited to infer critical information about damage position, degree of nonlinearity, and width of the damage zone either from the shifts in resonance frequency occurring at different excitation modes, or from the shift and the harmonics predicted at a single mode. Unlike other techniques, the multi-mode-nonlinear RUS method does not require a spatial scan to locate the defect, as it lets different excitation modes, with different vibration patterns, probe the structure. Two general methods are suggested for inverting experimental data.