Experiments and numerical results on non-linear vibrations of an impacting Hertzian contact. Part 1: harmonic excitation

The purpose of this paper is to investigate experimental and numerical dynamic responses of a preloaded vibro-impacting Hertzian contact under sinusoidal excitation. Dynamic response under random excitation is analyzed in the second part of this paper. A test rig is built corresponding to a double sphere-plane contact preloaded by the weight of a moving cylinder. Typical response curves are obtained for several input levels. Time traces and spectral contents are explored. Both amplitude and phase of harmonics of the dynamic response are investigated.Linearized resonance frequency and damping ratio are identified from the almost linear behaviour under very small input amplitude. Increasing the external input amplitude, the softening behaviour induced by Hertzian non-linear stiffness is clearly demonstrated. The resonance peak is confined to a narrow frequency range. Jump discontinuities are identified for both amplitude and phase responses. The forced response spectrum exhibits several harmonics because of a non-linear Hertzian restoring force. Numerical simulations show a very good agreement with experimental results.For higher input amplitudes, the system exhibits vibro-impacts. Loss of contact non-linearity clearly dominates the dynamic behaviour of the vibro-impacting contact and leads to a wide frequency range softening resonance. The spectral content of the response is dominated by both the first and the second harmonics. Evolution of the experimental downward jump frequency vs. input amplitude allows the identification of the non-linear damping law during intermittent contact. Simulations of the vibro-impacting Hertzian contact are performed using a shooting method and show a very good agreement with experimental results.     

Time-domain prediction of impact noise from wheel flats based on measured profiles

Railway impact noise is caused by discrete rail or wheel irregularities, such as wheel flats, rail joints, switches and crossings. In order to investigate impact noise generation, a time-domain wheel/rail interaction model is needed to take account of nonlinearities in the contact zone. A nonlinear Hertzian contact spring is commonly used for wheel/rail interaction modelling but this is not sufficient to take account of actual surface defects which may include large geometry variations. A time-domain wheel/rail interaction model with a more detailed numerical non-Hertzian contact is developed here and used with surface roughness profiles from field measurements of a test wheel with a flat. The impact vibration response and noise due to the wheel flat are predicted using the numerical model and found to be in good agreement with the measurements. Moreover, compared with the Hertzian theory, a large improvement is found at high frequencies when using the detailed contact model.     

纳米粒子自组装体系I—V特性的波包演化模拟

对形成室温单电子现象的典型串联双隧道结构模型，利用含时薛定谔方程的求解，计算了其隧穿电流与偏压的关系，利用该方法对Au纳米粒子组装体系在室温下的I－V特性进行了计算机模拟，并将计算结果与实验进行了比较。     

A simplified but intuitive analytical model for intermittent-contact-mode force microscopy based on Hertzian mechanics

The forces acting on the substrate in intermittent-contact-mode (IC mode, tapping mode) atomic force microscopy are not accessible to a direct measurement. For an estimation of these forces, a simple analytical model is developed by considering only the shift of the cantilever resonance frequency caused by Hertzian (contact) forces. Based on the relationship between frequency shift and tip–sample force for large-amplitude frequency-modulation atomic force microscopy, amplitude and phase versus distance curves are calculated for the intermittent contact mode, and the forces on the substrate are calculated. The results show a qualitative agreement with numerical calculations, yielding typical maximal forces of 50–150 nN. When working above the unperturbed resonance, forces are found to be significantly larger than below the resonance.     

A HYBRID MODEL FOR THE NOISE GENERATION DUE TO RAILWAY WHEEL FLATS

A numerical model is developed to predict the wheel/rail dynamic interaction occurring due to excitation by wheel flats. A relative displacement excitation is introduced between the wheel and rail that differs from the geometric form of the wheel flat due to the finite curvature of the wheel. To allow for the non-linearity of the contact spring and the possibility of loss of contact between the wheel and the rail, a time-domain model is used to calculate the interaction force. This includes simplified dynamic models of the wheel and the track. In order to predict the consequent noise radiation, the wheel/rail interaction force is transformed into the frequency domain and then converted back to an equivalent roughness spectrum. This spectrum is used as the input to a linear, frequency-domain model of wheel/rail interaction to predict the noise. The noise level due to wheel flat excitation is found to increase with the train speed V at a rate of about 20 log₀V whereas rolling noise due to roughness excitation generally increases at about 30 log₀V. For all speeds up to at least 200 km/h the noise from typical flats exceeds that due to normal levels of roughness. When the wheel load is doubled the predicted impact noise increases by about 3 dB.     

Non-stationary electrical charging of a conductive body in the medium with the ionic component at instantaneous loss by a body an electrical link with earthed object

Theoretical and numerical research of electric charging of conducting body in the environment with ionic component at instant loss by a body of electric contact to the earthed object is carried out. Two situations are considered: the first, stationary situation which is realized when the electric current, coming on a body, through contact leaves on the earth. The second, non-stationary situation, arises when contact of a body to the earthed object disappears, and on a body the electric charge starts to grow. On the basis of electro-dynamic equations the general statement of a problem is given. Numerical modelling is executed.     

Coupled dynamics of a squeeze-film levitated mass and a vibrating piezoelectric disc: numerical analysis and experimental study

This work deals with the dynamics of a vibrating piezoelectric disc, which creates, under specific vibrating conditions, an air squeeze film that is able to levitate a freely suspended object.In such problems, the coupling effects between the various components affect the overall dynamical behaviour of the combined system. For complex systems, which combine elastic and electro-static effects together with compressible fluid effects, the coupled equations are often dealt with separately to avoid modelling and computational complexity. In this paper, the importance of handling such systems in a coupled manner is advocated by means of numerical and experimental examples. A coupled model is derived in this work making use of a concise numerical solver to allow for this investigation under several conditions. The piezoelectric part of the structure is modelled by finite elements while the squeeze film phenomenon is represented by means of finite-difference equations, to model a variant of the Reynolds equation. The numerical model was verified during each step in the development of the numerical algorithm and indeed showed good agreement with existing publications, but once the components were combined, it was found that several phenomena were misrepresented in the past due to the neglect of the coupling effects. Several physical insights are brought from the simulation and investigation of the numerical results. In the last part, the importance of coupled analysis is emphasized by introducing an experimental investigation of the dynamical behaviour while conducting a comparison with numerical simulation results. From this comparison, the limitations of state-of-the-art modelling procedures are clarified.     

Dynamics of the Langevin model subjected to colored noise: Functional-integral method

We have discussed the dynamics of Langevin model subjected to colored noise, by using the functional-integral method (FIM) combined with equations of motion for mean and variance of the state variable. Two sets of colored noise have been investigated: (a) one additive and one multiplicative colored noise, and (b) one additive and two multiplicative colored noise. The case (b) is examined with relevance to a recent controversy on the stationary subthreshold voltage distribution of an integrate-and-fire model including stochastic excitatory and inhibitory synapses and a noisy input. We have studied the stationary probability distribution and dynamical responses to time-dependent (pulse and sinusoidal) inputs of the linear Langevin model. Model calculations have shown that results of the FIM are in good agreement with those of direct simulations (DSs). A comparison is made among various approximate analytic solutions such as the universal colored noise approximation (UCNA). It has been pointed out that dynamical responses to pulse and sinusoidal inputs calculated by the UCNA are rather different from those of DS and the FIM, although they yield the same stationary distribution.     

Analysis of a non-linear structure by considering two non-linear formulations

In recent years, modal synthesis methods have been extended for solving non-linear dynamic problems subjected to harmonic excitation. These methods are based on the notion of non-linear or linearized modes and exploited in the case of structures affected by localized non-linearity. Actually, the experimental tests executed on non-linear structures are time consuming, particularly when repeated experimental tests are needed. It is often preferable to consider new non-linear methods with a view to decrease significantly the number of attempts during prototype tests and improving the accuracy of the dynamic behaviour.This article describes two fundamental non-linear formulations based on two different strategies. The first formulation exploits the eigensolutions of the associated linear system and the dynamics characteristics of each localized non-linearity. The second formulation is based on the exploitation of the linearized eigensolutions obtained using an iterative process. This article contains a numerical and an experimental study which examines the non-linear behaviour of the structure affected by localized non-linearities. The study is intended to validate the numerical algorithm and to evaluate the problems arising from the introduction of non-linearities. The complex responses are evaluated using the iterative Newton-Raphson method and for a series of discrete frequencies. The theory has been applied to a bi-dimensional structure and consists of evaluating the harmonic responses obtained using the proposed formulations by comparing measured and calculated transfer functions.     

水声波导中包络线谱强度数值预报

为了预报声呐远距离检测包络线谱的性能,本文分析在辐射噪声传播过程中包络线谱强度衰减的原因及提出在水声波导中包络线谱强度的数值预报方法。依据周期性局部平稳过程对舰船辐射噪声建模,推导出无噪声干扰及有噪声干扰下的包络线谱强度的数学表达式,获得了包络线谱强度传播衰减的规律,在水声波导中包络线谱声级传播规律与平稳谱一致,但包络线谱高出连续谱的强度的衰减取决于随信噪比变化的调幅深度修正因子。包络线谱强度数值预报方法如下:先利用传播衰减的数值方法例如用简正波方法波数积分方法PE方法等求出平稳辐射噪声的传播衰减,再求信噪比,最后依据调幅深度修正因子获得包络线谱强度降低的分贝数,预报出在水声波导中任意一点包络线谱信号高度。该理论和数值预报方法具有创新性,实用,简单,适合工程应用。      

A study on generation of noise by high-speed pulsating jets

This paper describes the noise generation in an exhaust system of a reciprocating engine and focuses on the noise generated by shock/vortex interaction. The pulsating flow through the exhaust pipe consists of the compression and expansion wave, shock wave being generated by the non-linearity of the compression wave at its head. The jet noise is produced when the pulsating flow is discharged from the pipe end into atmosphere. The numerical simulation based on a finite difference method and experiment were made, the result of both of them being compared. First, the flow field in the pipe was obtained to easily discuss the characteristic of the pulsating jet in terms of the pressure history in the pipe. The jet structure was visualized by using schlieren and shadowgraph techniques. Sound pressure measurements at various locations were made in order to survey the directivity of the noise. The comparison between the result of numerical calculation and experiment showed a good agreement. A noise source related to shock/vortex interaction was confirmed by the numerical study clearly.     

Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative

A stochastic averaging procedure for a single-degree-of-freedom (SDOF) strongly nonlinear system with light damping modeled by a fractional derivative under Gaussian white noise excitations is developed by using the so-called generalized harmonic functions. The approximate stationary probability density and the largest Lyapunov exponent of the system are obtained from the averaged Itô stochastic differential equation of the system. It is shown that the approximate stationary solutions obtained by using the stochastic averaging procedure agree well with those from the numerical simulation of original systems. The effects of system parameters on the approxiamte stationary probability density and the largest Lyapunov exponent of the system are also discussed.     

Numerical model reduction of 2D steady incompressible laminar flows: Application on the flow over a backward-facing step

The numerical solution of most fluid mechanics problems usually needs such a fine mesh that the associated computational times become non-negligible parts in any design process. In order to couple numerical modelling schemes with inversion or control algorithms, the size of such models needs to be highly reduced. The identification method is a way to build low-order models that fit with the original ones. The laminar flow over a backward-facing step is used as a test case. Presented solutions are found to be in good agreement with experimental and numerical results found in the literature.     

Colored noise in a two-dimensional nonlinear system

A two-dimensional decoupling theory is developed when colored noise is included in a nonlinear dynamical system. By a functional analysis, the colored noise is transformed to an effective noise that includes the noise correlation time, the mean dynamical variable, and the original noise strength. When the two-dimensional decoupling theory is applied to single-mode and two-mode dye laser systems, the mean, variance, and effective eigenvalue of laser intensity are calculated. Excellent agreement between theoretical analysis, numerical simulations, and experimental measurements are obtained. It is seen that the increase of noise correlation time can reduce the fluctuations in the laser system. It is also shown that there is relatively large fluctuation in the phase when the laser undergoes from thermal light to coherent light when the theory is applied to a single mode dye laser. Received 20 August 2001 and Received in final form 4 December 2001     

乘性色噪声激励下三稳态van der Pol-Duffing振子随机P-分岔

研究了乘性色噪声作用下三稳态van der Pol-Duffing振子的随机P-分岔问题. 首先应用随机平均法得到系统振动幅值稳态概率密度函数的表达式, 进而应用奇异性理论, 得到刻画随机P-分岔发生的临界参数条件的转迁集以及系统存在的典型稳态概率密度曲线, 并通过Monte-Carlo数值模拟进行了验证. 以此为基础讨论了噪声强度、相关时间、系统线性阻尼系数对随机P-分岔和系统稳态响应行为的影响.     

Frequency-domain acoustic pressure formulation for rotating source in uniform subsonic inflow with arbitrary direction

This paper presents a frequency-domain formulation for predicting noise radiated from the rotating thickness and loading sources in uniform subsonic inflow with arbitrary direction. The proposed frequency-domain formulation is an extension of the recently published frequency-domain formulation for the stationary medium. It avoids the singular integral and numerical interpolation problems encountered in the time-domain numerical method. Three test cases, i.e., noise radiation from the rotating monopole and dipole point sources and the Isom thickness noise of a transonic rotor in the subsonic uniform flow, have been carried out to validate the proposed formulation. Both the acoustic pressure spectrum and directivity pattern computed with the present frequency-domain method are in good agreement with those obtained from the time-domain method, thus validating the correctness of the present formulation. Furthermore, the numerical results indicate that the frequency-domain formulation is suitable for tonal noise prediction, while it is inefficient for broadband noise prediction.     

Prediction of airborne radiated noise from lightly loaded lubricated meshing gear teeth

This paper introduces a novel analytical method for determination of gear airborne noise under lightly loaded conditions, often promoting gear rattle of loose unengaged gear pairs. The system examined comprises a single gear pair, modelled through integrated contact tribology and inertial transient dynamics. Lubricant film thickness, structural vibration and airborne gear noise are predicted and correlated with experimental measurements undertaken in a semi-anechoic environment. Good agreement is noticed between the numerical predictions and the experimental measurements. The presented model is capable of estimating the airborne radiated gear noise levels and the dynamic behaviour of gear pairs under different operating conditions, with superimposed impulsive input speed harmonics.     

Stationary response of multi-degree-of-freedom vibro-impact systems to Poisson white noises

The stationary response of multi-degree-of-freedom (MDOF) vibro-impact (VI) systems to random pulse trains is studied. The system is formulated as a stochastically excited and dissipated Hamiltonian system. The constraints are modeled as non-linear springs according to the Hertz contact law. The random pulse trains are modeled as Poisson white noises. The approximate stationary probability density function (PDF) for the response of MDOF dissipated Hamiltonian systems to Poisson white noises is obtained by solving the fourth-order generalized Fokker-Planck-Kolmogorov (FPK) equation using perturbation approach. As examples, two-degree-of-freedom (2DOF) VI systems under external and parametric Poisson white noise excitations, respectively, are investigated. The validity of the proposed approach is confirmed by using the results obtained from Monte Carlo simulation. It is shown that the non-Gaussian behaviour depends on the product of the mean arrival rate of the impulses and the relaxation time of the oscillator.