Search

International audience; Weight reduction is a recurring concern in the design of modern mechanical systems. This search may lead engineers to resort to using gears with holes in order to meet their requirements. This paper presents a methodology to carry out nonlinear dynamic analyses of a gear transmission with holes in the gear blanks subjected to a multiharmonic internal excitation. This work investigates the influence of these holes on the vibration levels, occurence of contact loss and possible bifurcations. The numerical model features two flexible shafts coupled by a spur gear pair with holes. The gear model consists in two lumped masses and inertias and includes the gear backlash as well as the internal excitation sources that are the time-varying stiffness and the static transmission error (STE). The resulting mechanical system is solved in the frequency domain by the Harmonic Balance Method (HBM) coupled with an arc-length continuation algorithm and its stability is evaluated with Hill's method. Results show that adding holes not only impacts the STE but also the mesh stiffness. The interactions of these two quantities has a substantial influence on the bifurcation structure along the main solution branch and leads to a decrease of the span of vibro-impact regions. As the applied static torque is increased, it is found that using holed gear blanks can effectively prevent contact loss and lead to a linear response.

We herein propose an algorithm for tracking smooth bifurcations of nonlinear systems with interdependent parameters. The approach is based on a complex formulation of the well-known harmonic balance method (HBM). Hill’s method is used to assess the stability of the computed forced response curves, and a minimally extended system is built to allow for the parametric continuation of the detected bifurcation points. The feasibility of coupling HBM-based minimally extended systems and arc-length continuation algorithms is established and demonstrated. The method offers an efficient way of determining the stability regions of the system. The methodology is applied on a spur gear pair model including the backlash nonlinearity and subjected to transmission error and mesh stiffness fluctuation whose harmonic contents depend on several parameters that do not appear explicitly in the equations of motion.

A complete procedure for the whining noise computation of a planetary gear set induced by the multi-mesh excitations is presented. This procedure is divided into three main steps. First, the parametrical internal excitations are simultaneously characterized by considering all contacts at the multiple gear meshes. Secondly, a finite element model of the planetary gear set is built. Finally, the coupled equations of motion are projected onto the modal basis and the stationary dynamic response is computed using an iterative spectral method.

A new experimental approach is proposed to investigate the friction behaviour and highlight the relationship with the tribofilm formation. It is based on the accumulation of several free dynamic responses of a single-degree-of-freedom damped oscillating system on the same sliding surfaces. By analysing the decreasing amplitude of the damped free dynamic responses and their associated energy decays, it is not only possible to describe the evolution of the tribological behaviour, but also to accurately identify velocity-dependent and velocity-independent friction contributions as the gradually formation of tribofilm. To demonstrate the relevance of this original technique, two tribosystems were studied, each lubricated with either base oil or formulated one, and by comparing the results with those obtained by a conventional linear tribometer.

International audience; This paper presents an experimental study of gear rattle noise induced by vibroimpacts between gear teeth. A specific experimental set-up is designed to analyse the nonlinear dynamic behaviour of a spur gear submitted to input velocity fluctuation. The drag torque, the mean drive gear rotational speed, the velocity fluctuation amplitude and frequency are controlled during experiment. The dynamic transmission error is measured thanks to high resolution optical encoders. The originality of the experimental set-up consists in using a high-speed camera in order to visualize the contact zone and to identify the occurrence of successive impacts between gear teeth. The rattle threshold is identified as a function of velocity fluctuation amplitude and frequency for various operating drag torques and mean rotational speeds. Experiments show very good agreement with the theoretical master curve. Once impacts occur, stationary nonlinear gear dynamic response and rattle noise radiated by the mechanical system are investigated. Most of the time, an almost periodic response is observed with 2 impacts per period. One impact between the active flanks alternates with one impact between the reverse flanks. A contact phase between gear teeth is observed after each impact instead of an instantaneous rebound. The number of successive tooth pairs crossing the meshing zone without any contact between gear teeth varies according to the ratio of the excitation frequency to the rotation frequency. Analytical and numerical works performed using a gear rattle model show good agreement with experiments. Finally, sound pressure emitted from the gear pair is measured. The acoustic power imputable to gear rattle is found to be proportional to the total kinetic energy transferred per second to the system by the successive impacts.

Sliding motion between two rough solids under light normal loading involves myriad micro-impacts between antagonist micro-asperities. Those micro-impacts are at the origin of many emerging macroscopic phenomena, including the friction force, the slider's vibrations and the noise radiated in the surroundings. However, the individual properties of the micro-impacts (e.g. maximum force, position along the interface, duration) are essentially elusive to measurement. Here, we introduce an instrumented slider aimed at measuring the position and the normal component of the micro-impact forces during sliding against a rough track. It is based on an array of piezoelectric sensors, each placed under a single model asperity. Its dynamical characteristics are established experimentally and compared to a finite elements model. We then validate its measurement capabilities by using it against a track bearing simple, well-defined topographical features. The measurements are interpreted thanks to a simple multi-asperity contact model. Our slider is expected to be useful in future studies to provide local insights into a variety of tribological questions involving dry rough sliding interfaces., Comment: Contents are identical to that of the published paper

This study aims to implement multi-objective optimization of a gear unit in order to minimize the power loss and the vibrational excitation generated by the meshing, via a multi-scale approach that extends from gear contact to the complete transmission. All these indicators are closely linked to the macro and micro-geometry definition of the gear pair. The optimization is carried out using a genetic algorithm, namely the Non-Dominated Sorting Genetic Algorithm II (NSGA-II). The design variables chosen for the problem are the pressure angle and the helix angle, as macro-geometry characteristics of the gear, and/or the length and the amount of tooth profile modifications, as micro-geometry characteristics of the gear. Constraints are imposed in order to not exceed a maximum bending stress at the tooth root of the gear and to not fall below a minimum total contact ratio. From the results obtained, it is found that the multi-objective optimization with both micro and macro-geometry parameters simultaneously gives different results than those obtained with macro-geometry first and then micro-geometry parameters. In order to study the importance, or not, to take into account the complete gear unit, a comparison is made between the local power loss generated by gear tooth friction and the total power loss in the single stage gear unit in terms of design variables values.

This paper provides an insight into the efficiency and accuracy of a multibody approach to model gear transmission error. The multibody model is based on an augmented Lagrangian contact formulation considering a surface-to-surface contact detection. The case of spur and helical gears transmitting power between parallel shafts is considered. The static transmission error is computed without any assumptions about the contact lines positions and orientations. Tooth and wheel body flexibility and tooth profile deviations are taken into account. The main objective of this study is to evaluate the efficiency and possibilities of the proposed methodology. To this end, the static transmission error is benchmarked against the results obtained from a classical approach which is based on the computation of the equation describing the static equilibrium of the gear pair for a set of successive positions of the driving wheel.

International audience; Gear power transmissions are responsible for upsetting vibroacoustic phenomena. The gear teeth compliance, the manufacturing errors and the profile/longitudinal tooth corrections lead to a periodic transmission error fluctuation and a parametric excitation associated to gear mesh stiffness fluctuation. Under operating conditions, these excitation sources generate dynamic mesh forces transmitted to the housing through shafts and bearings. The induced housing vibration is responsible for the whining noise radiated from the gearbox. This work is focused on the multiphysics coupling between the gear mesh internal excitation and the external excitation associated with the upstream engine and downstream receiver system. These excitations mix high and low frequency phenomena: the fundamental frequency of internal excitation is the meshing frequency while the fundamental frequencies of the extrenal excitations are the shafts circular frequencies. The coupling between excitations generates an enrichment of the vibratory frequency response. The goal of this paper is to present the spectral iterative methodology used and the dynamic results induced by this coupling. The case of a root vacuum pump designed with a 1:1 ratio gearbox and submitted to a fluid narrow band excitation is highlighted.

International audience; When a rigid rough solid slides on a rigid rough surface, it experiences a random motion in the direction normal to the average contact plane. Here, through simulations of the separation at single-point contact between self-affine topographies, we characterize the statistical and spectral properties of this normal motion. In particular, its rms amplitude is much smaller than that of the equivalent roughness of the two topographies, and depends on the ratio of the slider's lateral size over a characteristic wavelength of the topography. In addition, due to the non-linearity of the sliding contact process, the normal motion's spectrum contains wavelengths smaller than the smallest wavelength present in the underlying topographies. We show that the statistical properties of the normal motion's amplitude are well captured by a simple analytic model based on the extreme value theory framework, extending its applicability to sliding-contact-related topics.

An experiment of squeal noise on a lubricated elastomer/glass contact is presented. The experimental device, stiff enough to be non-intrusive on the system response up to several kHz, provides the contact force, vibration velocity and contact image with a high sampling rate. It is shown that the self-excited oscillation responsible of squeal involves a single mode and that its apparition is induced by the velocity weakening of friction. Direct observation of the contact by microscopy highlights that the contact is highly heterogeneous and the number of contact spots decreases with the sliding velocity during squeal.

Among the possible solutions for achieving low friction, polymer brushes that are grafted onto surfaces are good candidates. The tribological characterization of such layers becomes more difficult when the friction is lower: the signal-to-noise ratio of the friction force that is measured with conventional tribometers impedes the precise quantification. Therefore, we have applied a new technique that has been developed at the Laboratory of Tribology and System Dynamics (LTDS), called the ‘oscillating relaxation tribometer’. The advantage of this original technique is that it characterizes low friction with unequalled sensitivity. The lower the friction, the better the precision, and it permits obtaining the ‘friction law’ directly from robust and rapid experimental tests. In this study, the samples that have been used are the ionic liquid-type polymer brushes (ILPBs) with different thicknesses, which have been grafted onto silicon wafers and steel coupons. The counter-face is a mirror-polished steel ball. We show that (i) a thick ILPB layer on silicon is very resistant to high contact pressure, up to 555 MPa; (ii) the friction behavior that is obtained is close to that of a Newtonian viscous one, even under maximum normal loads; (iii) poorer results are obtained for the thinner sample; and (iv) the repetition, up to 5000 oscillations on the same surface, does not affect the friction damping of the contact, which demonstrates that this film provides a favorable resistance to friction under severe contact conditions. In addition, the feasibility of grafting onto steel surfaces is demonstrated. The results are then discussed, with respect to friction and dissipation.

Friction is an important phenomenon in various dynamic systems. In order to understand the physical behavior of such systems, it is crucial to choose a friction law that can correctly model the mechanical system. This paper aims to study the dynamic responses of a new tribometer, based on a single degree-of-freedom (sdof) mass-spring system having a sliding contact. We introduce a velocity-dependence friction law in order to understand the dynamic system. The model is studied numerically and experimentally. In this paper, the friction law has been carefully investigated using a linear model (Coulomb and viscous friction) and a quadratic velocity-dependence model, introduced as a power series of the sliding velocity. Results demonstrate that when the decaying envelope is described by both concave and convex forms, a linear friction law could not properly model the system. It is shown that in some cases, a more refined friction law with higher order than quadratic expansion should be introduced to precisely model the dynamic system. In addition, some experimental results are presented using the tribometer, showing a good agreement with the numerical results. The results are then discussed in terms of different lubrication regimes.

International audience; The stationary dynamic response of a geared system with time-varying mesh stiffness subjected to stationary stochastic external excitation is examined in this paper. To this end, a spur gear pair model with periodical time-varying mesh stiffness is extended to include stochastic external force excitations. These random forces are introduced as second-order stationary and ergodic processes. In order to compute the stationary response, a very efficient method, called the iterative spectral method is used. This method is derived in the frequency domain and provides the explicit power spectral density of the response. The PSD response is expressed as a function of the bispectrum which is the bilinear Fourier transform of the bitemporal impulse response of the parametric system.

International audience; This paper presents the analysis of multiphysics coupling between periodic gear mesh excitation and upstream/downstream fluctuating loads using an iterative spectral methodology. This one is based on the resolution of the parametric equations of motion in the spectral domain. Its efficiency makes possible to treat both low and high frequency excitations for systems having a large number of degrees-of-freedom. The different excitation sources, the dynamic coupled equations of motion, the spectral methodology and the iterative resolution principle are described. The dynamic responses of a roots vacuum pump for which the spur gear high mesh frequency parametric excitation is coupled with a low fluidic drag torque frequency. The coupling between excitations generates a frequency enrichment of the dynamic response which is reflected on waterfall plots by emergence of numerous sidebands around harmonics of the mesh frequency.

International audience; This paper presents a methodology for the multi-objective optimization of both gear macro and micro-geometry parameters in order to minimize the Static Transmission Error (STE) and the mesh stiffness fluctuations generated by the meshing process. The optimization is performed using a genetic algorithm which allows testing of a high number of gears. As the repetitive calculation process requires a fast evaluation of the gear excitation sources, a semi-analytical tooth bending model is proposed in order to evaluate gear compliance from a thick Reissner-Mindlin analytical plate model for the evaluation of the tooth strain energy and from a Ritz-Galerkin approximation for the description of the tooth deflection form. For each gear design tested by the algorithm, the robustness of solution to manufacturing errors is evaluated by performing Monte Carlo simulations for a thousand samples of random manufacturing errors. The Probability Density Functions (PDF) describing the RMS values of the mesh stiffness and STE fluctuations of each gear are estimated and their average values are the two objectives minimized by the optimization procedure. Standard deviations and maximum values describing the dispersion of results are also evaluated. Finally, the methodology provides a decision help tool for the designer who intends to choose the best gear design according to the criteria retained. The method is illustrated for a mechanical system equipped with a reverse gear.

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: μ 0, velocity-independent contribution and μ 1, 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.

The friction behavior of ta-C and a-C:H as DLC coatings is investigated using a new method. Two green lubricants were used: oleic acid and glycerol. This technique is able to measure with a great accuracy, without any force transducer, two contributions of friction. The first one, μ 0 is the transient friction coefficient at zero speed (which differs from the static and dynamic friction coefficient at constant speed) and the second one, μ 1 is a velocity-dependent friction coefficient. This work aims to investigate the different friction contributions for steel/steel, steel/DLC and DLC/DLC contacts. Moreover, the results are compared with a conventional reciprocating cylinder-on-plane tribometer working at constant sliding speed. Overall, the results show the superiority of the ta-C/ta-C combination in reducing μ 0 .

10.1007/978-3-319-24055-8_5; International audience; 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.

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 Poincar{\'e} 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., Comment: Final published version, 5 pages, 4 figures

International audience; The aim of this work is to characterize the rattle noise of automotive gearboxes, resulting from impacts between toothed wheels of unselected gear ratios. These stereo-mechanical impacts are modeled by a coefficient of restitution which describes damping during the squeezing of the lubricant film for approaching surfaces, and the elastic deformation of impacting bodies. The dynamic response of the loose gear first depends on the design parameters and the engine operating conditions. The unknown parameters are the drag torque and the coefficient of restitution. They are identified experimentally through implementation of two optical encoders in an actual automotive gearbox and the operation of a specific test bench which replicates the automotive power train. Models of the different drag torque sources are validated from analysis of the free damped response of the driveline. The coefficient of restitution and its probability density function are measured from experiments under stationary operating conditions. A nonlinear model is built. The dynamic response of the loose gear depends on the dimensionless backlash, the coefficient of restitution and a dimensionless parameter proposed to describe the rattle excitation level. Experiments under controlled excitation are performed to validate the assumptions, to confirm the ability of the parameter proposed to describe the rattle noise threshold, and to characterize the dynamic response. The nonlinear model predictions are fitted with the drag torque and coefficient of restitution previously identified. They are compared with measurements to demonstrate the ability of the model to predict gear rattle for any loose gear, any gearbox and any operating condition.

A numerical study to analyze friction noise being radiated when two rough and dry surfaces are rubbed together, is presented in this paper. The explicit dynamic finite element software ABAQUS in 2D is used to simulate the roughness noise. Coulomb's friction law is used at the contact surface. This simulation provided the local contact conditions such as local contact forces and contact stresses, but also the number of impacts per second and their intensities. It is shown that roughness plays a crucial role to explain roughness noise. The shocks between antagonist asperities are responsible of the transfer between the kinetic energy of solids and the vibratory energy which produces the radiation of sound. It was also shown that roughness noise level is simultaneously an increasing function of the logarithm of the surface roughness and the sliding speed. Numerical results are in agreement with our experimental results and previous studies.

The control of vibroimpact dynamics of a single-sided Hertzian contact forced oscillator is investigated analytically and numerically in this paper. The control strategy is introduced via a fast excitation and attention is focused on the response near the primary resonance. The fast excitation is added to the basic harmonic force, either through a harmonic force applied from above, or via a harmonic base displacement added from bellow, or by considering the stiffness of the oscillator as a periodically and rapidly varying in time. The results reveal that the threshold of vibroimpact response initiated by jump phenomenon near the primary resonance can be shifted toward lower or higher frequencies of the slow dynamic system depending on the fast excitation taken into consideration. It was also shown that the most realistic and practical way for controlling the vibroimpact dynamics is the introduction of a fast harmonic base displacement.

This paper presents an experimental study of the friction noise, between two rough and dry flat surfaces. The domain of interest is the dry contact under light pressure where the roughness is the dominant cause of noise. The results show that, for sliding rough surfaces under light load, the fundamental mechanism of radiated noise is the presence of shocks occurring between antagonist asperities of sliding surfaces. The radiated roughness noise is controlled simultaneously by the detailed topography of the surfaces in contact, the sliding speed and the dynamics of the surfaces. In terms of topography and sliding speed, it was shown that the roughness noise is simultaneously an increasing linear function of the logarithm of the surface roughness and the sliding speed. In terms of dynamics, the roughness noise is generated for light dynamical coupling. Hence, the natural modes of samples are stiffer than the contact and therefore the resulting vibrations are not affected by the additional rigidity. The deformation of surfaces during contact is very light and its magnitude is negligible compared to the surface roughness.

hybrid method is proposed to study the noise radiated in the high frequency range by a gearbox inside an engine compartment. Due to its elastodynamic characteristics, the vibratory behaviour of the transmission lies in the low frequency range. However, the noise radiated is rather in the high frequency range facing the size of the acoustical cavity. Thus the proposed hybrid method allows to couple a low-frequency vibratory approach to the radiative transfer method used for the high frequency acoustical calculation. The coupling is realized through equivalent energy sources introduced on the surface of the vibrating structure.

A hybrid method is proposed to predict the noise radiated in the high-frequency range by a structure whose vibratory behavior is rather in the low-frequency range. This problem meets for instance when the structure radiates inside some closed acoustical spaces. The proposed approach is called hybrid as it couples two methods valid on different frequency bands respectively used for the structural and acoustical systems. Low-frequency structural approaches generally provide the velocity field on the vibrating surface while methods adapted to high-frequency acoustical studies are based on energy data: so the main difficulty for the coupling is a problem of data compatibility. In the proposed hybrid method, the radiated noise is predicted by using the radiative energy transfer method which is an energy boundary integral method basically equivalent to the known ray-tracing method for acoustic applications. Adequate equivalent energy sources are estimated on the surface of the vibrating structure from the surface pressure and velocity fields. These sources are then introduced in the integral equation of the radiative energy transfer method. The approach is applied to predict the radiation of a ribbed plate sandwiched between two absorbant panels: good agreement is observed with the results given by the boundary element method. Generally speaking, the hybrid method can be considered to apply the radiative energy transfer method for high-frequency acoustical predictions from the knowledge of the vibratory behavior of a structure.

The effect of electromagnetic actuation (EMA) on the dynamic of a single-sided Hertzian contact forced oscillator is studied near primary and secondary resonances. Emphasis is put on the case where two symmetric EMAs are introduced, such that one is driven by a DC actuation and the other is actuated by AC actuation with a fast frequency. An averaging technique and a perturbation analysis are performed to obtain the frequency response of the system. It is shown that for appropriate values of AC, forced Hertzian contact systems are more likely to remain operating in the linear regime without the loss of contact near certain resonances.

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.

This paper deals with the variability of the dynamic behavior induced by transmission error of a mass production gear pair. The origins of this variability are due to the manufacturing errors. The tolerances that are associated with shaft misalignment errors, gear tooth profile and lead errors, are considered as geometric independent random parameters. A procedure based on Taguchi's method is used to treat the tolerances statistically. The efficiency of this methodology is demonstrated by considering a simple dynamic model of a single spur gear pair. The predicted variations in dynamic behavior due to tolerances are verified by comparison with results obtained using Monte Carlo simulations. The analyzed parameters are firstly the static transmission error and the time-average mesh stiffness. As a consequence of the variability of the mesh stiffness, statistical variations of natural frequencies are observed for critical modes (high-energy modes). The related critical speed ranges are given too. At last, the variations of the high-energy mode shapes are also observed.

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.

This Note presents an original approach to include diffraction in the radiative transfer method when applied to acoustics. This approach leads to a better spatial description of the acoustical energy. An energetic diffraction coefficient and some diffraction sources are introduced to model the diffraction phenomena. The amplitudes of these sources are determined by solving a linear sytem of equations resulting from the power balance between all acoustical sources. The approach is applied on bidimensional examples and gives good results except at geometrical boundaries. To cite this article: E. Reboul et al., C. R. Mecanique 332 (2004).

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 depends 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.

International audience; The main source of excitation in gearboxes is generated by the meshing process, which generates vibration transmitted to the casings through shafts and bearings. Casing vibration generates leads to acoustic radiation (whining noise). It is usually assumed that the transmission error and variation of the gear mesh stiffness are the dominant excitation mechanisms. These excitations result from tooth deflection and tooth micro-geometries (voluntary profile modifications and manufacturing errors). For real cases, the prediction of noise induced by the Static Transmission Error (STE) remains a difficult problem.In this work, an original calculation procedure is implemented by using a finite element method and taking into account the parametric excitations and their coupling (Spectral Iterative Method, developed by the Ecole Centrale de Lyon). The procedure is based on a modal approach developed in the frequency domain, particularly efficient to analyze systems having many degrees of freedom.In a first step, the static transmission errors and tooth stiffness's are calculated from the knowledge of the tooth macro and micro geometry. In a second step, these data are used to calculate the dynamic transmission errors, the teeth dynamic loads, and the dynamic response of the gearbox. A finite element model of the gearbox (including housings, shafts and bearings) is used to extract the modal basis necessary for this method.Measurements carried out on a specific gearbox test bench allow comparison of simulation and experimental results. Accelerometers and optical encoders were used to measure static and dynamic errors of transmission and the vibration response of the casings.The correlation between calculations and measurements at each stage of the computation chain is used to validate the relevance of these numerical approaches.

A nonlinear dynamic model of a part of an actual truck timing multi-stage gear system which consists on a 3-gears 2-mesh configuration was built. It included gear backlash nonlinearity, parametric excitation and internal displacement excitation induced by the mesh process. A boundary value problem formulation treated with a finite difference scheme in the time domain was used in conjunction with an arc-length continuation to obtain the periodical steady state responses. Simulations demonstrate the ability of the proposed strategy to describe dynamics of the nonlinear gear system. Complex behaviours were exhibited in the area of primary and secondary softening type resonances.

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.

Resume Pour le cas fondamental d'un contact hertzien de type spheere-plan, on etudie la resonance sous-harmonique d'ordre deux. Afin de decrire les bifurcations au voisinage de cette resonance, on approche la force de contact elastique et on utilise la methode des echelles multiples. Cette approche analytique est validee par comparaison avec une methode d'integration temporelle classique. Los resultats montrent que cette resonance constitue un precurseur de regimes dynamiques de type vibro-impacts.

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., Comment: 16 pages, 16 figures, accepted version

An original method to compute the steady state forced response of linear systems with periodically varying parameters under external excitations is proposed. The procedure is based on a modal approach with developments in the frequency domain. By using an iterative scheme to construct the approximate response, the spectra and the root mean square values of the forced response at every degree of freedom can be directly obtained. The technique has been successfully tested on illustrative examples and compared to the standard Newark method. Its efficiency is a consequence of the very short computing times needed. A practical application concerning a geared transmission system which is excited by periodic meshing stiffness is presented to demonstrate that this method is very effective in analyzing systems having many degrees of freedom.

In this paper, we study a kinematic friction model of a single degree-of-freedom oscillating system. This model has the following dimensionless equation of motion: Display Formulay″+2ζy′+y=-μky′ in which the kinematic friction is described by a quadratic function of the velocity as the following: Display Formulaμky′=μ0+μ2y′2sgny′+μ1y′ The sign function preserves the asymmetry of the friction force. The model captures most of the friction behavior that has been observed experimentally using a recent apparatus known as “dynamic oscillating tribometer”. This experimental setup is based on the non-linear free dynamic response of this kind of oscillator. However, this technique is able to carefully determine, with no need for any force transducer, the velocity-independent and velocity-dependent friction coefficients, μ0 and μ1 respectively, for a linear description of the kinematic friction, μk, around y′ = 0 where: Display Formulaμky′=μ0sgny′+μ1y′ In this context, the principal aim is to investigate numerically and analytically the effect of the additional quadratic term. To analyze the free dynamic response and their corresponding envelops, different numerical methods are performed. New characteristics of the envelops are discussed in detail with respect to the form of the kinematic friction coefficient. This allows a better comprehension of the results observed experimentally.Copyright © 2012 by ASME

International audience

International audience; 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.

International audience; We consider the automotive gearbox rattle noise resulting from vibro-impacts that can occur between the idle gears under excessive velocity fluctuations of the shaft-driving gears imposed by engine torque fluctuation. Even if the rattling phenomenon has no consequence on reliability, it may be particularly annoying for vehicle interior sound quality and acoustic comfort. The main parameters governing such kind of vibrations are the excitation source associated with engine torque fluctuation which can be modeled by an imposed displacement of the driveline, the inertia of the idle gear, the drag torque acting during the free flight motion, and the impact laws. In the case of rattle, it is reasonable to assume that duration of impacts between teeth is very short compared to the excitation period. Then, these impacts are modeled by a coefficient of restitution law. The excitation source is not composed only with fundamental component but also with other harmonic components. This study presents some effects of these additional components on the dynamic response of the idle gear.

International audience; This study deals with a method which allows identification of velocity-dependent and velocity-independent friction contributions, from energy decay curve of the free response of a damped single degree-of-freedom oscillator, including tribological contact. An original device is built and experimental dynamic responses confirm that contact friction is characterized by both velocity-dependent and velocity-independent contributions. Simulations from identified values agree with experimental data. Successive peaks of the free response are very well fitted. Estimated duration of the signal before its extinction is very consistent. The methodology appears powerful, especially for low friction coefficients leading to longer signal duration before extinction.

This paper presents an experimental study of the friction noise, between two rough and dry surfaces. Two different geometries of contact have been studied: flat-flat contact and sphere-flat contact. The domain of interest is the dry contact under light pressure where the roughness is the dominant cause of noise. In agreement with some other studies, it has been observed that the sound pressure level Lp (dB) is a logarithmic function of the surface roughness and the sliding speed. However, the exponent of the logarithmic law between sound pressure level and surface roughness largely depends on the topography of the sliding surfaces. On the other hand, the exponent of the law between sound pressure level and sliding speed depends on the contact geometry.