Reduction of gear pair transmission error with tooth profile modification

The gear noise problem that widely occurs in power transmission systems is typically characterised by one or more high amplitude acoustic signals. The noise originates from the vibration of the gear pair system caused by transmission error excitation that arises from tooth profile errors, misalignment and tooth deflections. This paper aims to further research the effect of tooth profile modifications on the transmission error of gear pairs. A spur gear pair was modelled using finite elements, and the gear mesh was simulated and analysed under static conditions. The results obtained were used to study the effect of intentional tooth profile modifications on the transmission error of the gear pair. A detailed parametric study, involving development of an optimisation algorithm to design the tooth modifications, was performed to quantify the changes in the transmission error as a function of tooth profile modification parameters as compared to an unmodified gear pair baseline

Effect of planetary gearboxes on the dynamics of rotating systems

The coupled dynamic behaviour of planetary geared rotor systems is much less well understood compared to the classical geared rotor systems. For a better understanding, this research project investigates the coupled dynamic behaviour of planetary geared rotor systems and how the planetary gearbox parameters affect their global dynamics. In the numerical study, a six degrees of freedom hybrid dynamic model of a planetary geared rotor system is created in the recently developed “GEAROT” rotor dynamics software by considering gyroscopic effects. Based on the modal analysis results of the hybrid dynamic model, the vibration modes are classified as coupled torsional-axial, lateral and gearbox for the helical gear configuration, and torsional, axial, lateral and gearbox for the spur one. Modal energy analysis is used to quantify the coupling level between the shafts and planetary gearbox, which highlights the effect of a planetary gearbox on the dynamic behaviour of a rotating system. An extensive planetary gearbox parameter study including gear contact, gearbox mass and support, and planet gear parameters is conducted using the hybrid dynamic model to investigate the parameter effects on the modal behaviour of planetary geared rotors. The sensitivity of planetary geared rotor vibration modes to the gearbox parameters is determined by computing the frequency shifts and comparing the mode shapes between the two extreme cases. In the experimental study, free-free impact hammer tests are carried out on a planetary geared rotor assembly to validate the numerical modal analyses results in “GEAROT”. On the basis of both experimental and numerical modal analysis of planetary geared rotors, the lateral vibration modes are identified as “in phase” and “out of phase”. Briefly, the numerically identified lateral modal behaviour of planetary geared rotor systems is successfully validated with the experimental modal analysis results.Open Acces

Dynamic behavior of the nonlinear planetary gear model in nonstationary conditions

The nonlinear effects in gearboxes are a key concern to describe accurately their dynamic behavior. This task is difficult for complex gear systems such as planetary gearboxes. The main aim of this work is provide responses to overcome this difficulty especially in non-stationary operating regimes by investigating a back-to-back planetary gearbox in steady conditions and in run up regime. The nonlinear Hertzian contact of teeth pair is modeled in stationary and non-stationary run-up regime. Then it is incorporated to a torsional model of the planetary gearbox through the different mesh stiffness functions. In addition, motor torque and external load variation are taken into account. The nonlinear equations of motion of the back-to-back planetary gearbox are computed through the Newmark-β algorithm combined with the method of Newton-Raphson. An experimental validation of the proposed numerical model is done through a test bench for both stationary and run-up regimes. The vibration characteristics are extracted and correlated to speed and torque. Time frequency analysis is implemented to characterize the transient regime during run-up.This research work was supported by the Spanish Ministry responsible of Science and Technology through the project DPI2017-85390-P. The authors gratefully thank the University of Cantabria cooperation project which supports the doctoral trainings of students of Sfax University. The authors also acknowledge the Tunisian Project No. “19PEJC10-06”

Sources of Excitation and Models for Cylindrical Gear Dynamics: A Review

In this paper, a review of the evolution of the study of cylindrical gear dynamics is presented. After a brief historical introduction to the field, the first attempts to describe the complex interactions in those systems are analyzed introducing the dynamic factor and the first methodologies used to compute it. Next, the sources of excitation in geared systems are analyzed in detail and the models of the various contributions are discussed. Then, the paper focuses on the use of those sources in several dynamic models which are wildly different in terms of scope, applicability, complexity and methodology employed, ranging from simple analytical models, to lumped masses models up to multibody and finite element models. Finally, an outlook to the future evolution of the field is given and conclusions are drawn

Combined misalignments in spur gear transmission systems: a semi-empirical approach

Un área inexplorada en el diseño de sistemas de transmisión de engranajes es el estudio de los efectos de desalineamientos combinados en las medidas de vibración. Actualmente, las investigaciones se centran en desalineamientos individuales y no combinados los cuales reflejan mejor los escenarios de aplicaciones reales. En esta investigación se analizan los efectos de los desalineamientos combinados en las mediciones de vibración en la base de los rodamientos y en el esfuerzo de flexión de los dientes de engranajes rectos de un sistema de transmisión de una etapa. Se diseñó y construyó un banco de pruebas para generar desalineamientos radiales, axiales y angulares en un sistema de transmisión de engranajes de una etapa. Se evaluaron todas las combinaciones posibles de niveles extremos de desalineamiento para un par de engranajes rectos para identificar tendencias en la respuesta vibratoria. Se desarrolló un modelo teórico del área de contacto proyectada para estudiar la relación entre esta y la respuesta vibratoria. Al analizar el cambio en los espectros, se determinó la influencia de diferentes desalineamientos y sus interacciones en las mediciones de vibración. Finalmente, se desarrolló un modelo híbrido para estimar las aceleraciones en los rodamientos, utilizando un modelo de elementos finitos para determinar el esfuerzo de flexión en los dientes y un modelo analítico para estimar las señales de vibración en los rodamientos. El modelo demostró una alta correlación en comparación con los resultados experimentales, validando su efectividad. Finalmente, se propusieron recomendaciones de diseño considerando las zonas de esfuerzo y vibración de interés.DoctoradoDoctor en Ingeniería Mecánic

Tribo-dynamic analysis of hypoid gears in automotive differentials

Torsional vibrations in differentials of Rear Wheel Drive vehicles are of major importance for the automotive industry. Hypoid transmissions, forming the motion transfer mechanism from the driveshaft to the wheels, suffer from severe vibration issues. The latter are attributed to improper mesh between the mating gear flanks due to misalignments, variation of contact load and shifting of the effective mesh position. For certain operating conditions, the gear pair exhibits high amplitude motions accompanied with separation of the mating surfaces. Ultimately, single or even double-sided vibro-impact phenomena evolve, which have been related to noise generation. This thesis attempts to address these issues by effectively analysing the dynamic behaviour of a hypoid gear pair under torsional motion. The case study considered is focused on a commercial light truck. The major difference of the employed mathematical model to prior formulations is the usage of an alternative expression for the dynamic transmission error so that the variation of contact radii and transmission error can be accounted for. This approach combined to a correlation of the resistive torque in terms of the angular velocity of the differential enables the achievement of steady state, stable periodic solutions. The dynamic complexity of systems with gears necessitates the identification of the various response regimes. A solution continuation method (software AUTO) is employed to determine the stable/unstable branches over the operating range of the differential. The ensuing parametric studies convey the importance of the main system parameters on the dynamic behaviour of the transmission yielding crucial design guidelines. A tribo-dynamic investigation aims at expanding the dynamic model from pure dry conditions to a more integrated elastohydrodynamic (EHL) approach. Analytical and extrapolated solutions are applied for the derivation of the film thickness magnitude based on the kinematic and loading characteristics of the dynamic model. The temperature rise is governed mainly by conduction due to the thin lubricant films. The generated friction is also computed as a function of the viscous shear and asperity interactions. The effective lubricant viscosity is greatly affected by the pressure increase due to the resonant behaviour of the contact load. The final part of this work is involved with a feasibility study concerning the application of Nonlinear Energy Sinks (NES) as vibration absorbers, exploiting their ability for broadband frequency interaction. Response regimes associated with effective energy absorption are identified and encouraging results are obtained, showing the potential of the method

Transmission error in spur gears: Static and dynamic finite-element modeling and design optimization

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University, 07/10/2010.The gear noise problem that widely occurs in power transmission systems is typically characterised by one or more high amplitude acoustic signals. The noise originates from the vibration of the gear pair system caused by transmission error excitation that arises from tooth profile errors, misalignment and tooth deflections. This work aims to further research the effect of tooth profile modifications on the transmission error of gear pairs. A spur gear pair was modelled using finite elements, and the gear mesh was simulated and analysed under static conditions. The results obtained were used to study the effect of intentional tooth profile modifications on the transmission error of the gear pair. A detailed parametric study, involving development of an optimisation algorithm to design the tooth modifications, was performed to quantify the changes in the transmission error as a function of tooth profile modification parameters as compared to an unmodified gear pair baseline. The work also investigates the main differences between the static and dynamic transmission error generated during the meshing of a spur gear pair model. A combination of Finite-Element Analysis, hybrid numerical/analytical methodology and optimisation algorithms were used to scrutinise the dynamic behaviour of the gear pairs under various operating conditions

Nonlinear dynamic analysis for high speed gear-rotor-bearing system of the large scale wind turbine

In this paper, an eight-degree-of-freedom (8-DOF) lumped parameter dynamic model considering the coupled lateral-torsional vibration is proposed and the coupled multi-body dynamics of the spur gear rotor bearing system is studied containing backlash, transmission error, eccentricity, gravity and time-variant mesh stiffness. Based on the dynamical equations, the coupled dynamic response of the system is investigated using the Runge-Kutta method and the effects of error fluctuation and load fluctuation on the dynamic responses are demonstrated by 3-D frequency spectrum bifurcation diagram, etc. The results show that a diverse range of nonlinear dynamic characteristics such as periodic, chaotic behaviors and impacts exhibited in the system are strongly attributed to the interaction between internal and external excitations. For gear system, the dynamic behaviors are analyzed in light, middle and high rotational speed conditions. With the increase rotational speed, the vibration amplitude increase markedly and the region of the chaotic motion become narrow gradually. At the low rotational speed, the chaos behavior turns out more easily, and the vibration intensity relatively weak. With the increase rotational speed, the vibration amplitude obvious increase, and the characteristics of the chaos strengthen and turns backward. This study may contribute to a further understanding about the spur gear bearing system with the coupled internal and external excitation

Development of analytical-numerical methods for dynamic analysis of geared transmission systems

The main objective of the present research activity is the study of geared transmission system dynamics, which is basically represented by a system of nonlinear differential equations. First of all, the different approaches to study the nonlinear dynamics of gears are qualitatively presented. Afterwards, the realization of a lumped parameter model is discussed by analyzing two different modeling strategies linked to two different numerical resolution techniques. The first modeling strategy is based on time integration techniques and enhances the employment of a commercial software to speed-up the modeling set-up phase. The proposed method rely on a block diagram technique and it is developed in Simcenter AMESim, a commercial software widely used in industries. By starting from the single gear pair model, detailed guidelines are given to construct any type of ordinary transmission layout by connecting some pre-programmed devices between them. In order to demonstrate the reliability of the approach, an experimental validation on industrial use case is proposed with excellent outcomes. The second modeling strategy rely on a frequency domain solution technique able to capture unstable solution branches in multi-valued frequency response regions. In particular, it proposes the Asymptotic Numerical Method combined to the Harmonic Balance Method as a valuable approach to solve the nonlinear dynamics of gear pairs. Thanks to a quadratic recast of the equation of motion, the Taylor and Fourier series can be computed in a very efficient way and each step produces a continuous representation of the solution branch making the continuation very robust. Effectiveness and reliability of the method are proved by comparing the numerical outcomes with that obtained from the Runge-Kutta time integration scheme. As a result, this technique provides for excellent computational performance despite additional time is needed for the quadratic recast of the equations system. Once a detailed analysis on the modeling strategy has been conducted, rattle noise and whine noise occurrence are investigated. Regarding the rattle noise, the research activity has conducted to the introduction of a new analytical parameter as a novelty to the current state of the art. A rattle index formulation is retrieved by starting from the classical 6-DOFs equation system defining the nonlinear dynamics of a gear pair. The proposed formulation may be applied to single or multiple branch geartrain, both in idle or loaded condions. The reliability of the analytical formulation is proved by numerical experiments which demonstrate the capability of the proposed index to instantaneously describe the vibro-impacts events related to any gear pair of the driveline. In addition its magnitude may be a measure of the tooth impact severity and it is shown to be a proper indicator of the potential presence of mutual interactions between different gear pairs pertaining to the same driveline. Finally, the investigation of whine noise occurrence addresses to an analytical formulation able to forecast the main overall direction and magnitude of bearing reaction forces on idler gear. By starting from the definition of meshing forces by means of Fourier series development, idler gear bearing forces are obtained under the hypothesis of quasi-static motion. This procedure demonstrates that the alternating component of bearing forces on idler gear describes an elliptical trajectory as the prime mover rotates over a pitch angle. The formulation directly links the bearing forces elliptical trajectory with the gear spatial position, the meshing phase and the amplitude of meshing forces. By properly setting the over-mentioned parameters one may be able to control the magnitude and direction of the overall idler bearing reaction forces. Numerical experiments were performed and the obtained results confirm the author intuitionL’obiettivo principale della presente attività di ricerca riguarda lo studio della dinamica non lineare degli ingranaggi che, di fatto, è rappresentata da un sistema di equazioni differenziali. Prima di tutto, viene presentata un ‘analisi qualitativa finalizzata a valutare i diversi approcci per studiare tale fenomeno. Successivamente, viene descritto lo sviluppo di un modello a parametri concentrati analizzando due diverse strategie di modellazione basate su metodi di risoluzione numerica diversi. Il primo approccio propone l’utilizzo di un software commerciale per velocizzare la fase di set-up del modello ed è basato su tecniche di integrazione temporale. Questa strategia di modellazione è sviluppata in Simcenter AMESim, un software commerciale distribuito da Siemens. Partendo dal modello di una singola coppia di ruote, viene dettagliata una procedura per costruire qualsiasi tipo di treno di ingranaggi grazie alla tecnica dei diagrammi a blocchi. Per dimostrare l’efficacia di tale tecnica, il metodo viene applicato ad un caso industriale ottenendo un’ottima correlazione numerico-sperimentale. Il secondo approccio si basa su tecniche di risoluzione numerica nel dominio della frequenza in grado di calcolare i rami instabili della risposta dinamica. Il metodo propone la combinazione del “Asymptotic Numerical Method” con il “Harmonic Balance Method” utilizzando una formulazione quadratica del sistema di equazioni differenziali. Grazie a tale formulazione, sia la serie di Taylor che quella di Fourier possono essere sviluppate in una maniera molto efficiente rendendo la continuazione della soluzione periodica molto robusta. L’affidabilità di questa tecnica è stata dimostrata confrontando i risultati con quelli ottenuti dal metodo di Runge-Kutta, basato sull’integrazione temporale. In più, tale tecnica garantisce performance computazionali eccellenti, anche se la riformulazione quadratica del sistema iniziale non è sempre facile da ottenere. Una volta analizzate le strategie di modellazione e le tecniche numeriche risolutive, lo studio si concentra su i fenomeni di rattle e whine noise. Riguardo il rattle noise, l’attività di ricerca ha portato all’introduzione di un nuovo parametro analitico come novità rispetto allo stato dell’arte. Partendo dal sistema di equazioni che governa il moto di una coppia di ruote dentate, è stato definito un indice analitico denominato “rattle index”. Tale indice può essere applicato a qualsiasi tipo di treno di ingranaggi, a uno o più rami, sia in condizioni di trasmissione di potenza che in folle. La sua affidabilità è supportata da simulazioni numeriche che dimostrano la capacità del “rattle index” di descrivere istantaneamente la perdita di contatto tra qualsiasi coppia di ruote di una trasmissione. Infine, la sua ampiezza è un indice della severità degli urti e permette di identificare l’esistenza di mutue interazioni tra le ruote della driveline. Infine, lo studio del whine noise ha portato ad una formulazione analitica capace di prevedere la direzione e l’ampiezza delle forze sui cuscinetti delle ruote oziose. Tale formulazione viene ottenuta partendo dalla definizione delle forze di ingranamento tramite lo sviluppo in serie di Fourier e ricavando le forze sui cuscinetti sotto l’ipotesi di moto quasi-statico. Questa procedura dimostra che le componenti alterne delle forze sui cuscinetti seguono una traiettoria ellittica quando il movente ruota di un passo angolare. La formulazione mette in relazione la traiettoria delle forze sui cuscinetti con la posizione delle ruote nel piano, la fase di ingranamento e l’ampiezza delle forze di ingranamento. Agendo sui parametri descritti, è possibile pilotare e controllare la direzione delle forze sui cuscinetti delle ruote oziose. La trattazione analitica è supportata da simulazioni numeriche con un ottimo riscontro