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015 a Hybrid 3D Finite Elementlumped Parameter Model for Quasi-static and Dynamic Analyses Of

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  A hybrid 3D finite element/lumped parameter modelfor quasi-static and dynamic analysesof planetary/epicyclic gear sets V. Abousleiman, P. Velex  * Laboratoire de Me´ canique des Contacts et des Solides, UMR CNRS 5514, INSA de Lyon, Baˆ timent Jean d’Alembert, 20 Avenue Albert Einstein, 69 621 Villeurbanne Cedex, France Received 9 December 2004; received in revised form 3 August 2005; accepted 7 September 2005 Abstract A model is presented which enables the simulation of the three-dimensional dynamic behaviour of planetary/epicyclicspur and helical gears. Deformable ring-gears are introduced by using either beam elements for simple structures, or 3Dbrick elements for complex geometries. Based on a modal condensation technique, internal gear elements can be defined byconnecting the ring-gear sub-structure and a planet lumped parameter model via elastic foundations which account fortooth contacts. Discrete mesh stiffnesses and equivalent normal deviations are introduced along the contact lines, and theirvalues are re-calculated as the mating flank positions vary with time. Planetary/epicyclic gear models are completed byassembling lumped parameter sun-gear/planet elements along with shaft elements, lumped stiffnesses, masses and inertias.The corresponding equations of motion are solved by combining a time-step integration scheme and a contact algorithmfor all simultaneous meshes. Several quasi-static and dynamic results are given which illustrate the potential of the pro-posed hybrid model and the interest of taking into account ring-gear deflections.   2005 Elsevier Ltd. All rights reserved. Keywords:  Planetary gears; Flexible ring-gear; Dynamic; Finite elements; Load distribution; Sub-structure 1. Introduction In comparison with parallel shaft power transmission designs, planetary gear trains offer several advantagessuch as, shaft co-axiality, higher power density and increased efficiency which have made them very attractive,especially for aeronautical applications (turboprop engines, helicopter transmissions, etc.). Most of the papersdealing with planetary/epicyclic gears have focused on dynamic behaviour in order to optimise design, i.e.,reducing weight using lighter structural elements while ensuring system durability. Cunliffe et al. [1] usedlumped parameter models with three different mesh stiffness descriptions and obtained close correlations 0094-114X/$ - see front matter    2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2005.09.005 * Corresponding author. Tel.: +33 4 72 43 84 51; fax: +33 4 78 89 09 80. E-mail address:  Philippe.Velex@insa-lyon.fr (P. Velex).Mechanism and Machine Theory xxx (2005) xxx–xxx www.elsevier.com/locate/mechmt MechanismandMachine Theory ARTICLE IN PRESS  between theory and practice. Although their lumped parameter approach was proved to be acceptable, theauthors stressed the need to account for ring-gear flexibility which could introduce additional resonances. Bot-man [2] studied the modal content of a one-stage epicyclic train and, using a 2D constant stiffness model, hefound the axisymetrical modes to be particularly dangerous. Hidaka et al. [3–7] conducted extensive experi-mental and numerical analyses on the dynamic behaviour of planetary trains. From a 2D finite element model,ring-gear deflections were found to significantly influence tooth loads and load sharing properties. The authorsalso pointed out that ring-gear deformations can reduce the distance between adjacent teeth leading to higheractual contact ratios and possible interferences. Botman and Toda [8] studied the influence of the planet angu-lar phasing on dynamic loads induced by planet run-out errors. Significant differences were found dependingon planet indexing with certain configurations being less sensitive to manufacturing errors. Ma and Botman [9]introduced profile and run-out deviations in a 2D model for investigating planet load sharing. Hidaka et al. [6]found that the phases between the planets could modify sun-gear trajectories, and several studies [10–12] con-firmed the contribution of mesh phasing as well as the possibility of cancelling some frequency components of the global translational or torsional excitations on sun-gears and ring-gears. August and Kasuba [13] showedthat, at low speeds, sun-gear trajectories are mainly translational because of the time-varying mesh stiffnesseswhile, at high speeds, sun-gear motions become orbital. Their simulated dynamic tooth loads revealed thatfloating sun-gears may not be appropriate except in very narrow speed ranges. Saada and Velex [14] proposeda three-dimensional lumped parameter model with six degrees of freedom for each gear component and stud-ied the influence of several design parameters on the system eigenfrequencies. Following this work, Velex andFlamand [15] analysed the mesh parametric excitations in planetary trains and showed that local resonancescould appear on an individual gear mesh that cannot be detected at a global scale.However, in the majority of papers on the subject, gears are considered as rigid solids, and only a few stud-ies limited to 2D spur gear models Hidaka et al. [5], Kahraman and Vijayakar [16], Parker et al. [12] have dealt with the contributions of deformable parts. The present work attempts to introduce ring-gear elasticity in themodelling of the quasi-static and dynamic behaviours of spur and helical gear sets with tooth shape deviations(modifications and geometrical errors). It is accepted that time-varying mesh stiffnesses and manufacturing/mounting errors are prominent contributions to vibration and noise in geared transmissions. In the contextof multi-mesh systems, the concept of transmission errors as excitation sources is confusing because of themany interacting gears, so it is replaced by the simulation of the contact conditions in the various toothmeshes. Based on this approach, two srcinal hybrid finite element-lumped parameter dynamic models usingbeam and brick elements are proposed and their results, as well as those delivered by the classical lumpedparameter models, are compared for various gear geometries. 2. Models of internal gears with deformable ring-gears Two different modelling strategies are presented: (a) a ring-gear model based on straight beam finite ele-ments and (b) a more sophisticated technique using three-dimensional 20 node brick elements. The first optionis suited for simple architectures and accounts mainly for the radial deflections of the ring-gear while the sec-ond one is aimed at simulating complex mechanical arrangements like those in aeronautical applications. Forboth methods, the ring-gear model is reduced by using a modal condensation technique [17,18]. Following [19], tooth contacts are considered as line contacts on theoretical base planes which are discretised in indepen-dent cells of stiffness  k  ( M  i  ) and equivalent normal deviation  d e ( M  i  ) to simulate tooth shape deviations andmounting errors. Planets are assimilated to rigid cylinders with all six degrees of freedom characterised byscrews of infinitesimal displacements whose co-ordinates in the frame fixed to the sun-gear/planet  j   centre-lineare f s  j g ¼ u  R j !ð O  j Þ ¼  v  j S   j !þ  w  j T   j !þ  u  j  Z  ! x  j    !¼  u S   j !þ  w  j T   j !þ  h  j  Z  ! 8<: ð 1 Þ with  O  j  , centre of planet  j  ;  S  !  j ;  T  !  j ;  Z  !  j  are the unit vectors shown in Fig. 1;  v  j  ,  w  j  ,  u  j   are the translational degreesof freedom;  u  j  ,  w  j  ,  h  j   are the rotational degrees of freedom (a list of symbols is given in Appendix). 2  V. Abousleiman, P. Velex / Mechanism and Machine Theory xxx (2005) xxx–xxx ARTICLE IN PRESS   2.1. Beam element model  A ring-gear is discretised into several classical two-node beam elements having a rectangular cross-section(Fig. 2). The degrees of freedom at each node comprise three translations;  v ,  w  and  u  and three rotations;  u ,  w and  h  as described in Fig. 3. For planetary gears, i.e., a ring-gear moving with respect to the inertial frame, thegyroscopic moments and centrifugal effects caused by the rotation of the ring-gear have been neglected. Theminimum number of beam elements is set to be the number of teeth on the ring-gear in order to associate anode with every tooth (Fig. 4). According to the classical hypotheses in beam theory, all cross-sections aresupposed to remain straight after deflection and the elastic displacement field for one cross-section can bemodelled by a screw  f s  N  ck g  of the form  j j w  ψ  ,  j j u  θ  ,  j j v  φ  , S   j T   j  Z   j Planet  j O  j Sun-gearPlanet 1 T  1 Φ  j O  Z  1 O  S  1 1S Fig. 1. Degree-of-freedom and frame definition. Sc Neutralline of thering-gearBeam elements TcR nc Z Fig. 2. Ring-gear model (beam elements). V. Abousleiman, P. Velex / Mechanism and Machine Theory xxx (2005) xxx–xxx  3 ARTICLE IN PRESS  f s  N  ck g ¼ u  R c !ð  N  ck Þ ¼  v  N  ck S  c !þ  w  N  ck  T  c    !þ  u  N  ck  Z  ! x c    !¼  u  N  ck S  c !þ  w  N  ck  T  c    !þ  h  N  ck  Z  ! 8<: ð 2 Þ where  N  ck  represents the node at the centre of the cross-section as illustrated in Fig. 4.The deflection at any potential point of contact  M  i   on tooth flanks is the normal approach with respect torigid-body conditions minus the total initial separation  d e ( M  i  ). Using the shifting property of screws, oneobtains in a matrix form d ð  M  i Þ ¼f V    j ð  M  i Þgf V    N  ck ð  M  i Þg ( ) T f  X   j gf  X   N  ck g ( )   d e ð  M  i Þ ð 3 Þ Cross-section centered in N ck M i Beam elements connected in N ck S c ZT c N ck Ring-gearOne tooth Fig. 4. Detail of the ring-gear model based on beam elements and positions of potential points of contact on tooth flanks.Fig. 3. Beam element and its degrees of freedom.4  V. Abousleiman, P. Velex / Mechanism and Machine Theory xxx (2005) xxx–xxx ARTICLE IN PRESS
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