Emmanuel Rigaud

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.

Superharmonic resonance of order 2 for an impacting hertzian contact oscillator : Theory and experiments

The purpose of this paper is to investigate experimental responses of a preloaded vibroimpact Hertzian contact to an order 2 superharmonic excitation. A test rig is used, corresponding to a double sphere-plane contact preloaded by the weight of a moving body. Typical response curves are obtained kinder the superharmonic excitation. The Hertzian non linearity constitutes the precursor of vibroimpacts established over a wide frequency range. This behaviour can be related to the existence of a transcritical bifurcation. In conjuction with the experiments, numerical results lead to the same conclusion. In particular, the threshold level of the excitation necessary to induce vibroimpact is confirmed.Copyright

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.

Particle Swarm Optimization as an Efficient ComputationalMethod in order to Minimize Vibrations of Multimesh Gears Transmission

The aim of this work is to present the great performance of the numerical algorithm of Particle Swarm Optimization applied to find the best teeth modifications for multimesh helical gears, which are crucial for the static transmission error (STE). Indeed, STE fluctuation is the main source of vibrations and noise radiated by the geared transmission system. The microgeometrical parameters studied for each toothed wheel are the crowning, tip reliefs and start diameters for these reliefs. Minimization of added up STE amplitudes on the idler gear of a three-gear cascade is then performed using the Particle Swarm Optimization. Finally, robustness of the solutions towards manufacturing errors and applied torque is analyzed by the Particle Swarm algorithm to access to the deterioration capacity of the tested solution.

Dual characterization of boundary friction thanks to the harmonic tribometer: Identification of viscous and solid friction contributions

All the usual characterization of an interface submitted to shear is generally achieved thanks to an experiment using a pair of two loaded solids in continuous sliding. So, the friction coefficient value can be determined. But, only this friction value is quite poor information and is not able to reveal the complexity of the interfacial processes occurring in the sliding contact. A novel method has been developed at LTDS, allowing us to identify both the velocity-dependent and the solid friction contributions. It is based on a contact submitted to constant normal load, one solid being supported by a 1D mechanical oscillator, under the form of an elastic bi-blade, able to return to its equilibrium position through damped oscillations. The way the amplitude is decaying during this elastic recovery is recorded. A mechanical model has been developed, in order to fit experimental data, allowing us to identify two different contributions: the velocity-dependent contribution (typically viscous damping), and the solid-like contribution (typically solid friction). This technique has been applied here to a lubricated sphere-on-plane contact. The contacting surfaces are made of AISI 52100 polished steel. Two model products are tested: pure glycerol and 1,3-buthylene glycol. First results are presented, showing the capabilities of this technique for giving a new insight on the mixed lubrication regime description.

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.

Some Effects of Gear Eccentricities on Automotive Rattle Noise

This study deals with the problem of automotive gear rattle noise, primary caused by the engine torque fluctuations. Even if the rattling phenomenon has no consequence on reliability, it may be particularly annoying for vehicle interior sound quality and acoustic comfort. Gear rattle noise results from vibro-impacts that may occur on unloaded gears under excessive fluctuations of the engine torque and driveline dynamic responses. It is reasonable to assume very short time duration for impacts between teeth due to rattle. Then, the main parameters which govern this kind of vibrations are the restitution coefficient introduced for impact modeling, the drag torque acting on the free running wheel during its free flight motion, its inertia and the excitation source which can be modeled by an imposed displacement. This excitation source is often modeled as a harmonic or, sometimes, periodic excitation related to the acyclism behavior of the engine torque fluctuation, at twice the rotation frequency of the primary input shaft. The aim of this study is to deal on several effects of the gear eccentricities considered first as a superimposed periodic excitation source and second as a time varying backlash source. For this end, a simple non-linear oscillator is introduced, based on restitution coefficient for impact modeling. The parameters studied are the operating rotation speed of the input shaft, the inertia of the free running wheel, the drag torque, and the eccentricities compared to the acyclism source. Results are presented through dimensionless parameters.© 2007 ASME

Effect of an Original Gear Mesh Modeling on the Gearbox Dynamic Behavior

Prediction of the vibratory and acoustical behavior of gearboxes is generally based on characterization of the excitation sources, overall modeling of the gearbox, modal analysis and solving of the parametric equations of motion generated by these models. On the building process of such large degrees of freedom models, the elastic coupling induced by the gear mesh is generally described by the parametric meshing stiffness k(t) along the line of action. This kind of model is not able to take into account the load distribution along the tooth face width, even though the resulting moment can constrain rotational angles associated to wheels tilting and flexural deformation of shafts. The scope of this study is to introduce the coupling terms between wheels associated to these phenomena. Some examples show how they can influence the gear modal characteristics and dynamic response and, consequently, the vibratory and acoustical response of the gearbox.Copyright

Low noise design of a truck timing multi-stage gear: robust optimization of tooth surface modifications

This study focuses on the optimization of the micro geometry of gear tooth flank in order to minimize the Static Transmission Error level as a key of low whining noise design. The studied mechanical system is an actual truck timing multi mesh geared system. The optimization problem was efficiently addressed by the Particle Swarm Optimization. In addition, Monte Carlo simulations allowed to estimate robustness of the retained optimal solutions. On this basis, gear wheels have been manufactured within the class quality 7. Experimental tests on an actual truck engine power train reveal very significant sound power level gain.