An analytical study of tooth friction excitations in errorless spur and helical gears

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Abstract

Based upon a Coulomb model, an original analytical analysis of tooth friction excitations in errorless spur and helical gears is presented. Upon averaging the contributions of bending angles at pinion and gear centres and using a constant mesh stiffness per unit of contact length, it is shown that the intensity of tooth friction excitations compared to the excitations caused by time-varying contact lengths depends on three basic functions of time ΔL(τ),FF(τ) and TT(τ). ΔL(τ) represents the normalised variation of contact length between the teeth, while FF(τ) and TT(τ) account for friction excitations in translation and torsion respectively. Their exact analytical expressions are given in the form of Fourier series, valid for all spur and helical gears and, from which contour plots of excitation levels are derived. These performance diagrams enable an easy estimate of the gear set sensitivity to friction excitations as well as the consequences of modifying actual geometric parameters under load. The potentially significant contribution of tooth friction to translational vibrations of pinions and gears is pointed out, particularly in the case of high contact ratio gears.

Introduction

The prediction of the vibratory performances of geared systems has been the subject of numerous investigations in recent decades and has given rise to a vast literature containing a remarkable variety of models as shown in the reviews of Ozgüven and Houser [1], Blankenship and Singh [2], Velex and Maatar [3]. The different mathematical models to date have generally neglected the contribution of tooth friction forces to the gear set dynamics. Following the early works of Radzimosky and Miraferi [4], Iida et al. [5] on simplified lumped parameter models, sliding friction has been introduced as one of the variables in the more recent and sophisticated approaches of Vedmar and Henriksson [6], Cheng and Lim [7] but no specific conclusion has been reached. Recently, sliding friction between gear teeth has been recognised as a potentially significant excitation [8], [9] and its powerful influence on the vibro-acoustic behaviour of gears has been demonstrated by Vaishya and Houser [10]. From a modelling viewpoint, Velex and Cahouet [11] solved the motion equations of a 3D gear model with several different friction models. They combined a time-step integration scheme and a contact algorithm to predict both normal and tangential dynamic forces. Comparisons with the instantaneous bearing forces measured on a spur and a helical geared train are satisfactory and it has been proved that tooth friction can be crucial for force transmissibility through bearing mounts though its influence on transmission error is of secondary importance. Using a Coulomb model, Vaishya and Singh [12] gave analytical solutions and analysed the stability of a single degree-of-freedom torsional gear model. The authors showed that friction has only a minimal effect on torsional instabilities.

In situ measurement of instantaneous friction forces is difficult [13] and because lubrication may vary between full elasto-hydrodynamics and thin film boundary lubrication, the prediction of reliable friction coefficients in gears is still uncertain. Some relevant formulations are given in the reviews of Dowson [14], Martin [15] and a recent evaluation of a number of formulae in the context of gear meshes can be found in a paper by Vaishya and Houser [16]. However, the experimental work of Velex and Cahouet [11], Vaishya and Houser [10] led to the same conclusion; that a properly selected constant coefficient of friction should suffice for dynamic simulations.

In this context, the main objectives of the present paper are to: (1) develop an analytical approach for analysing tooth friction excitations in errorless spur and helical gears, (2) investigate the influence of gear geometry on tooth friction excitation levels and (3) give general results in terms of sensitivity to tooth friction excitations.

Section snippets

Gear model with tooth friction excitations:

Following [3], [11], the pinion (solid 1) and the gear (solid 2) of a pair are modelled as two rigid-cylinders linked by a time-varying set of stiffnesses kij which arises from the contact deformation and structural deflections of gear teeth. Elemental stiffnesses kij are distributed along the discretised instantaneous lines of contact on the base plane and, depending on the actual contact length, the total stiffness is adjusted at each time step of the meshing process. Denoting Mij (Fig. 1), a

Contact length variations ΔL(τ) [18]

Following the entry of a contact line to the base plane at τ=0, its instantaneous length can be expressed asl0(τ)=χ(τ)·bcosβbwith χ(τ) being the periodic piecewise linear function of period P shown in Fig. 2.

The instantaneous contact length between the pinion and the gear is derived asL(τ)=∑i=0P−1l0(τ−i)and its relative variation finally readsΔL(τ)=L(τ)Lm−1=∑k=1Akcos2πkτ+Bksin2πwithAk=1αεβπ2k2·[cos2πβ+cos2παcos(2πk(εαβ))−1],Bk=1αεβπ2k2·[sin2πβ+sin2παsin(2πk(εαβ))]and

Results

It is well known that contact conditions, and consequently profile and transverse contact (overlap) ratios, depend on load and tooth shape modifications. For example, tooth contact patterns can exist only in the central part of the tooth flank under light loads. Upon assuming (a) that contact conditions are nearly the same for all mating tooth pairs and (b) that actual contact areas projected onto the base plane are approximately rectangular [19], actual profile and transverse contact ratios εα

Conclusion

It is demonstrated that based on commonly used hypotheses, i.e., (i) constant mesh stiffness per unit contact length and, (ii) averaged contribution of bending angles, an analytical analysis of tooth friction excitations can be applied to errorless spur and helical gear pairs operating under Coulomb friction. As far as the authors know, the proposed analytical procedure and the results on the strength of friction excitations are original.

For a given friction coefficient, the following

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