Joël Perret-Liaudet

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 on 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 analysed. 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 shown a very good agreement with experimental ones. When vibroimpacts occur, results remain in reasonable agreement with experimental ones, that 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 a lot of 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 to prevent surface dammage.

Response of an impacting Hertzian contact to an order-2 subharmonic excitation : Theory and experiments

Abstract In this paper, the response of a normally excited preloaded Hertzian contact is investigated in order to analyse the subharmonic resonance of order 2. The nonlinearity associated with contact losses is included. The method of multiple scales is used to obtain the non-trivial steady-state solutions, their stability, and the frequency–response curves. To this end, a third-order Taylor series of the elastic Hertzian contact force is introduced over the displacement interval where the system remains in contact. A classical time integration method is also used in conjunction with a shooting method to take into account losses of contact. The theoretical results show that the subharmonic resonance constitutes a precursor of dynamic responses characterised by loss of contact, and consequently, the resonance establishes over a wide frequency range. Finally, experimental validations are also presented in this paper. To this end, a specific test rig is used. It corresponds to a double sphere–plane contact preloaded by the weight of a moving mass. Experimental results show good agreements with theoretical ones.

Friction-Induced Vibration by Stribeck's Law: Application to Wiper Blade Squeal Noise

This paper is concerned with the squeal noise of a wiper/windscreen contact. It is shown that squeal noise stems from friction-induced self-excited vibrations in the context of Stribeck’s law for friction coefficient. The study is specifically focussed on the instability range of velocities and not on the amplitude of limit cycles. The studied dynamic system consists of a single degree-of-freedom mass-spring-damper oscillator submitted to a velocity-dependent frictional force which follows the Stribeck law. The local stability is analyzed by the first Lyapunov method and results in a stability criterion. Experiments have been performed on a glass/elastomer contact lubricated with water. The tribometer ‘LUG’ provides measurements of the vibrational velocity and friction force versus sliding speed. It is found that the instability appears during the transition between boundary and elastohydrodynamic regimes where the negative gradient of the friction versus velocity curve is steep. The apparition and vanishing of instability are correctly predicted by the steady-state stability criterion.

Radiative transfer equation for multiple diffraction

This paper aims to apply the radiative transfer method to acoustical diffraction by obstacles. Some fictitious sources are introduced at diffracting wedges and a transfer equation based on energy balance determines the diffracted powers. It leads to a set of linear equations on diffracted powers which can be solved in a finite number of steps. It is then possible to calculate the diffracted field anywhere. Some applications to diffraction by obstacles of various shapes are presented. Results of this method are compared with Geometrical Theory of Diffraction and BEM reference calculations. It is shown that this method is particularly efficient in case of multiple diffraction where the ray-tracing technique involves an infinite number of rays between a source and a receiver point.

Approche globale pour l'analyse de la réponse vibratoire d'une transmission par engrenages

ABSTRACT In order to analyse dynamic response of a gearbox, we proposed a method based on the modelling of all the components (gears, shafts, bearings and housing). Forced response of the whole gearbox was computed using a spectral and iterative method. The analysis of coupling mechanisms between dynamic mesh force and mean square vibrational velocity of housing showed that a resonant excitation of modes which have a high potential energy associated with mesh stiffness and bearing stiffnesses leads to the highest vibrational responses. Bearings stiffnesses and mechanical properties of housing have an appreciable influence on these vibrational responses. Our methodology and numerical results allowed to propose new designs which lead to a significant reduction of the mean square vibrational velocity of housing of the gearbox, for a wide range of frequencies.

Direct numerical simulation of the dynamics of sliding rough surfaces

The noise generated by the friction of two rough surfaces under weak contact pressure is usually called roughness noise. The underlying vibration which produces the noise stems from numerous instantaneous shocks (in the microsecond range) between surface micro-asperities. The numerical simulation of this problem using classical mechanics requires a fine discretization in both space and time. This is why the finite element method takes much CPU time. In this study, we propose an alternative numerical approach which is based on a truncated modal decomposition of the vibration, a central difference integration scheme and two algorithms for contact: The penalty algorithm and the Lagrange multiplier algorithm. Not only does it reproduce the empirical laws of vibration level versus roughness and sliding speed found experimentally but it also provides the statistical properties of local events which are not accessible by experiment. The CPU time reduction is typically a factor of 10.

An Investigation of Steady-State Dynamic Response of a Sphere-Plane Contact Interface With Contact Loss

In this study, the dynamic behavior of an elastic sphere-plane contact interface is studied analytically and experimentally. The analytical model includes both a continuous nonlinearity associated with the Hertzian contact and a clearance-type nonlinearity due to contact loss. The dimensionless governing equation is solved analytically by using multi-term harmonic balance method in conjunction with discrete Fourier transforms. The accuracy of the dynamic model and solution methods is demonstrated through comparisons with experimental data and numerical solutions for both harmonic amplitudes of the acceleration response and the phase difference between the response and the force excitation. A single-term harmonic balance approximation is used to derive a criterion for contact loss to occur. The influence of harmonic external excitation f(r) and damping ratio (ζ on the steady state response is also demonstrated.

Vibro-Acoustic Analysis of Geared Systems—Predicting and Controlling the Whining Noise

The main source of excitation in gearboxes is generated by the meshing process. It is usually assumed that static transmission error (STE) and gear mesh stiffness fluctuations are responsible of noise radiated by the gearbox. They generate dynamic mesh forces which are transmitted to the housing through wheel bodies, shafts and bearings. Housing vibratory state is directly related to the noise radiated from the gearbox (whining noise). This work presents an efficient method to reduce the whining noise The two main strategies are to reduce the excitation source and to play on the solid-borne transfer of the generated vibration. STE results from both tooth deflection (depending of the teeth compliance) and tooth micro-geometries (voluntary profile modifications and manufacturing errors). Teeth compliance matrices are computed from a previous finite elements modeling of each toothed wheel. Then, the static equilibrium of the gear pair is computed for a set of successive positions of the driving wheel, in order to estimate static transmission error fluctuations. Finally, gear mesh stiffness fluctuations is deduced from STE obtained for different applied loads. The micro-geometry is a lever to diminish the excitation. Thus, a robust optimization of the tooth profile modifications is presented in order to reduce the STE fluctuations. The dynamic response is obtained by solving the parametric equations of motion in the frequency domain using a spectral iterative scheme, which reduces considerably the computation time. Indeed, the proposed method is efficient enough to allow a dispersion analysis or parametric studies. The inputs are the excitation sources previously computed and the modal basis of the whole gearbox, obtained by a finite element method and including gears, shafts, bearings and housing. All the different parts of this global approach have been validated with comparison to experimental data, and lead to a satisfactory correlation.

Effect of Temperature on Lubricated Steel/Steel Systems With or Without Fatty Acids Additives Using an Oscillating Dynamic Tribometer

The friction behavior of steel/steel contacts lubricated with glycerol, 150NS and PAO4 is investigated using a new technique. Then, three additives are used in PAO4 lubricant: oleic, linoleic and stearic acids. The experiments are performed at different temperatures. Moreover, nearly all the systems operate in either the mixed or boundary lubrication regime. The experimental methodology is able, with a high accuracy and without any force transducer, two contributions of friction: μ0, velocity-independent contribution and μ1, velocity-dependent one. This study shows that glycerol is the most effective in reducing friction among the three pure lubricants. In addition, only glycerol affects the viscous damping of the system at some operating conditions. Using the additives in PAO4, the friction is reduced. More precisely, the unsaturated fatty acids are more significant in reducing the friction than saturated acids. Moreover, the viscous damping of the apparatus is slightly affected as additives are used in PAO4.

Memory in random bouncing ball dynamics

The bouncing of an inelastic ball on a vibrating plate is a popular model used in various fields, from granular gases to nanometer-sized mechanical contacts. For random plate motion, so far, the model has been studied using Poincare maps in which the excitation by the plate at successive bounces is assumed to be a discrete Markovian (memoryless) process. Here, we investigate numerically the behaviour of the model for continuous random excitations with tunable correlation time. We show that the system dynamics are controlled by the ratio of the Markovian mean flight time of the ball and the mean time between successive peaks in the motion of the exciting plate. When this ratio, which depends on the bandwidth of the excitation signal, exceeds a certain value, the Markovian approach is appropriate; below, memory of preceding excitations arises, leading to a significant decrease of the jump duration; at the smallest values of the ratio, chattering occurs. Overall, our results open the way for uses of the model in the low-excitation regime, which is still poorly understood.