Abstract
This paper synthesizes a whole of work carried out on the spiral bevel gears from quasi-static and dynamic models viewpoint. A sophisticated quasi-static model makes it possible to calculate the tooth loads, the pressures, the instant mesh stiffness, and the deflections on the flanks of spiral bevel gears. Based on these results, two three-dimensional lumped parameter dynamic models are presented. The mechanical system under consideration comprises: a spiral bevel pinion and gear connected by a time-varying non-linear mesh stiffness function, and mounted on two shafts simulated by Timoshenko’s beams supported by bearings. Two variants are considered which rely on different contact stiffness simulations: (a) using an averaged mesh stiffness function acting at the centroid of the loaded areas on tooth flanks and (b) a more local approach based on a discrete distribution of the local mesh stiffness, elements over the contact areas. A number of results are presented and commented which illustrate the interest of these dynamic models.
Highlights
The two models incorporate actual spiral bevel gear geometries and some of the results delivered by an accurate quasi-static contact model (ASLAN, software developed by LaMCoS of INSA of Lyon)
The fundamentals of the quasi-static modeling are synthesized in the first section and it is shown how the instant mesh stiffness and kinematical error obtained using ASLAN are used as input data in the dynamic model
In a first version of the dynamic model, potential energy is calculated by assuming that mesh stiffness can be reduced to a unique stiffness element in the normal direction and located at the centroid of the contact area
Summary
Cij Cisj CiPjf , CiRjf DF eii, efi E, P, G E, 1, 2, 3, 4, S F F1 F2 Fd, Fs I1, I4 J1, J4 K ki KE, ME Me N n n1, n2 pi q ra, rb R rcPC, rcPD Ri Si TE U1i, U2i UE uE, u1, u2, u3, u4, uS V vE, wE, v1, w1, v2, w2, v3, w3, v4, w4, vS, wS yi Σ α αo, βo αPC, αPD γo φE, ψE, φ1, ψ1, φ2, ψ2, φ3, ψ3, φ4, ψ4, φS, ψS θE, θ1, θ2, θ3, θ4, θS contact area dimensions of a rectangular cell center of the contact area viscous damping matrix influence coefficient of point j on point i contact influence coefficient pinion and gear bending influence coefficient, respectively dynamic factor initial and final gap at point i pinion and gear misalignments gear pair nodes in the dynamic models global mesh force matrix additional forcing term forcing term dynamic and static load, respectively pinion and gear moments of inertia, respectively pinion and gear polar moments of inertia, respectively global mesh stiffness matrix elemental stiffness elements stiffness and mass matrices point on the mesh envelope number of rectangular cells of the discretized tangent plane normal at any point direction of the contact lines of the meshing load pressure at point i global generalized displacement vector coordinates of a point in the pinion and gear reference system, respectively tool tip radius tool reference radius of flank C and D respectively lever arm at point i surface of a rectangular cell inertial kinematics energy pinion and gear normal displacement at point i, respectively potential energy axial displacement of the gear pair nodes velocity at any point bending displacement of the nodes final gap at point i when surfaces in contact shaft angle global normal approach of the contacting surfaces parameters describing a point on the tool tool pressure angle of flank C and D respectively tool positioning parameter rotations of the nodes torsion displacements of the nodes initial model of bevel gears [9]. Gao et al [3] have proposed a model dedicated to the study of shocks in spiral bevel gear systems. Two dynamic models are presented which make it possible to obtain instantaneous displacements and tooth loads over a broad speed range. The second model provides dynamic tooth loads and contact pressures on tooth flanks. The two models incorporate actual spiral bevel gear geometries and some of the results delivered by an accurate quasi-static contact model (ASLAN, software developed by LaMCoS of INSA of Lyon). The fundamentals of the quasi-static modeling are synthesized in the first section and it is shown how the instant mesh stiffness and kinematical error obtained using ASLAN are used as input data in the dynamic model
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