arXiv subjects

Emmanuel Rigaud

Publications and source records attributed to Emmanuel Rigaud.

12 recordsLinked to original sources

arXiv2007

Variability of Critical Rotational Speeds of Gearbox by Misalignement and Manufacturing Errors

Noise measurement on a population of gearbox manufactured in large number often reveals high variability due to tolerances on each gear design parameter (manufacturing errors). Gearbox noise results mainly from housing vibration induced by the gear meshing process. High acoustic levels correspond to excitation in a resonant manner of some critical modes. All gearbox modes and especially these critical modes depend on tolerances related to gear design parameters through the meshing stiffness. Tolerances are considered on shaft misalignment errors, teeth profile errors and teeth longitudinal errors. We introduce these tolerances as geometric random gaussian parameters. The critical modes are extracted using an efficient procedure. The retained "statistical" method to treat tolerances is the Taguchi's method. We illustrate this methodology considering two different gearbox dynamic models fitted out with a single spur gear. Obtained dispersion results are validated by comparison with Monte-Carlo simulations.

physics.class-ph↗

arXiv2007

Gearboxes: indirect identification of dynamic forces transmitted to housing through bearings

In order to determine indirectly the generalized forces applied on the gearbox, two approaches are tested, one is a numerical method, the other is experimental. For the numerical method, the vibration response of the gearbox can be evaluated with use of a finite element analysis and a spectral and iterative method. It's confirmed that applying the generalized forces transmitted through the bearings on the empty housing allows to rebuild the housing response (reciprocity principle). The simulation on the time- and space-averaged mean square vibration velocity shows the vibration response of the housing is induced principally by the axial exciting forces, and secondarily by the moments Mx and My. transmitted on the free side (the 6 mm thick face). Realistic estimation of these excitation forces needs a modelling of the whole gearbox taking into account the mechanical properties of the housing. For the experimental approach, an inverse method based upon the FRF measurement and characterisation of gearbox vibration response is developed. Beginning from the equivalent forces obtained by this inverse method, a resulting force and a moment are derived for representing the distributed force acting on each bearing. The comparison between the results obtained by these two methods shows the effectiveness of this indirect force determination methodology.

physics.class-ph↗

arXiv2007

Response of an impacting hertzian contact to an order-2 subharmonic excitation : theory and experiments

Response of a normally excited preloaded Hertzian contact is investigated in order to analyze 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.

physics.class-ph↗

arXiv2007

Influence de l'élasticité des lignes d'arbres, des roulements et du carter sur les vitesses critiques de rotation d'une transmission par engrenages

The dynamic behaviour of a gearbox fitted out with a helical gear pair is studied. Shafts, bearings and housing are discretised using the finite element method. The elastic coupling between the toothed wheels is characterised by a 12x12 stiffness matrix. The calculation of the vibration response induced by the static transmission error shows that the highest dynamic mesh forces correspond to a resonant excitation of modes which have a high mesh stiffness energy. The numerical simulations show that a realistic prediction of the critical rotational speeds should take account of all the components of the gearbox.

physics.class-ph↗

arXiv2007

Modélisation et analyse de l'erreur statique de transmission d'un engrenage. Influence des déformations des roues et interactions entre les couples de dents en prise

We have developped a numerical tool in order to predict vibratory excitation of a gearbox induced by the meshing process. We propose a method based on a 3D Finite Element modelling of each toothed wheel. A non linear algorithm computes the fluctuations of static transmission error and mesh stiffness. Numerical simulations show the effect of wheel body deformation and the interactions between adjacent loaded teeth. The method developped allows to analyse the effect of different tooth modification types an manufacturing errors on the static transmission error.

physics.class-ph↗

arXiv2007

Résonance 2-surharmonique d'un contact unilatéral hertzien : approches théorique et expérimentale

For a preloaded impact oscillator under superharmonic excitation of order 2, the Hertzian non linearity constitutes the precursor of vibroimpacts established over a wide frequency range. The theoretical study is performed by using the shooting method with an arc length continuation. Experimental results obtained from a devoted test rig agree with theoretical ones. Particularly, the threeshold level of the excitation necessary to induce vibroimpacts is confirmed.

physics.class-ph↗

arXiv2007

Dispersion of critical rotational speeds of gearbox: effect of bearings stiffnesses

Noise measurement on a population of a gearbox manufactured in large number reveals a high variability principally due to tolerances on each design parameter of the gear. Gearbox noise results mainly from vibration of its housing excited by the transmission error. High dynamic mesh load, high vibratory response of housing and high acoustic level of the gearbox can be observed for some critical rotational speeds. These ones correspond to excitation in a resonant manner of some particular modes which store a high potential energy at the meshing stiffness. Variability of the associated critical modes depend on tolerances affected to gear design parameters. It also depends on bearings stiffnesses dispersion. The physical origins of the bearings stiffnesses variability are preloads. Bearings stiffnesses and tolerances affect the characteristics of the meshing stiffness and the dynamic transfer from dynamic meshing force to the housing vibratory response. The purpose of this paper is to study effect of the bearings stiffnesses variability by estimating the first two moments of statistical distribution of critical modes.

physics.class-ph↗

arXiv2007

Coupled computation of meshing and bearings stiffnesses : effect on radiated gearbox noise

This study focus on the acoustical behavior of gearbox induced by the static transmission error under load (STE). An original method is developed allowing to compute simultaneously STE, meshing and bearings stiffnesses. These characteristics mainly govern the vibrational and the acoustical behaviors of the geared system. The proposed procedure is based on two computational methods allowing to calculate the STE and the bearing equilibrium which are integrated in a coupled computational procedure. The technique has been implemented to treat an illustrating example and compared to standard uncoupled approach. Comparative results concern the STE, meshing stiffness, bearings stiffnesses, shafts misalignments, critical eigenmodes and associated critical running speeds, housing vibratory response and radiated noise.

physics.class-ph↗

arXiv2007

Effect of Elasticity of Shafts, Bearings, Casing and Couplings on the Critical Rotational Speeds of a Gearbox

The aim of this study is to analyse the influence of the mechanical characteristics of the set of components on the critical rotational speeds of a gearbox. The case of a gearbox fitted out with a helical gear pair was considered. The shafts and the casing were discretised using the finite element method. The elastic coupling between the toothed wheels was characterised by a 12 x12 stiffness matrix. The bearings were modelled using radial, axial and rotational stiffness elements. The calculation of the vibration response induced by the static transmission error showed that the highest dynamic mesh forces correspond to a resonant excitation of modes which have a high potential energy associated with the mesh stiffness. The numerical simulations performed showed that a realistic prediction of the critical rotational speeds should take account of all the components of the gearbox.

physics.class-ph↗

arXiv2007

Experiments and numerical results on nonlinear 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 analysed 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. Linearised 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 nonlinear stiffness is clearly demonstrated. Resonance peak is confined in a narrow frequency range. Jump discontinuities are identified for both amplitude and phase responses. Forced response spectrum exhibits several harmonics because of nonlinear Hertzian restoring force. Numerical simulations show a very good agreement with experimental results. For higher input amplitude, system exhibits vibro-impacts. Loss of contact non-linearity clearly dominates the dynamic behaviour of the vibroimpacting contact and leads to a wide frequency range softening resonance. Spectral content of the response is dominated by both the first and the second harmonics. Evolution of the experimental downward jump frequency versus input amplitude allows the identification of the nonlinear damping law during intermittent contact. Simulations of the vibroimpacting Hertzian contact are performed using a shooting method and show a very good agreement with experimental results.

physics.class-ph↗

arXiv2007

Experiments and numerical results on nonlinear 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.

physics.class-ph↗

arXiv2006

Effect of an original gear mesh modelling on the modal behaviour of gear transmission

Prediction of the vibroacoustic behaviour of geared transmissions is generally based on determining the excitation sources, the overall modelling of the transmission, the modal analysis and the solving of the parametric equations of the 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 meshing stiffness. Nevertheless, when the contact is established along the tooth width, the resulting moment can constrain the rotational angles associated to wheel tilting and flexural deformation of shafts. The scope of this study is to introduce the coupling terms between wheels associated to this phenomenum. One shows also on the basis of some examples in what manner they can influence the modal behaviour and the vibroacoustic response of the transmission.

physics.class-ph↗