Modelling of Hydrodynamic Journal Bearing in Spatial Multibody Systems

The model of a three dimensional journal bearing with hydrodynamic lubrication is herein presented. This model is suitable for embodiment into the equations of spatial multibody systems. Both rotational and squeeze effects together with tilting effect have been taken into account. Moreover a simplified model of friction has been also reported. The proposed methodology has been applied to an example concerning an unbalanced rotor supported by two journal bearings. INTRODUCTION Accurate analysis of rotating machinery is useful to improve performance and reliability. In many practical applications the modelling of a journal bearing as ideal revolute joint may lead to neglect some interesting dynamic effects [1]. Some authors proposed to describe such a bearing using bushing elements [2]. Often these models use linearized stiffness/damping which are not suitable for transient analysis or for heavy/variable loads. On the other hand there are many contributions about dry joints with clearances [3-9]. These concern contact/impact algorithm or penalty method approach in order to describe the interaction between bearing surfaces. Taking into account also lubrication effects leads to very complex models. One of them has been proposed in [10] and it is based on an hybrid model including both contact and squeeze effect. The model does not include effects due to rotational speed of the possibility of tilting (it can be used only in planar mechanisms with low rotational speed). The detailed and comprehensive models coming from the theory of lubrication [11-15] are too complex to be included into equations of motion of multibody systems because they require lengthy computations and very accurate numerical procedure in order to obtain correct solution. For this reason the authors developed a model which, although it contains some assumptions and simplifications, allows to simulate a three dimensional journal bearing with both effects of squeeze and sliding speed and also the possibility to exert reaction moments by tilting. The effect of friction in bearing has been also estimated using semi empirical formulas. The standard approach to describe bearing with ideal revolute joint leads to 5 scalar constraint equations (permitting 1 d.o.f. which is the relative rotation along hinge axis). The proposed model replaces these equations with two forces and three torques, thus the system increases the number of degrees of freedom. The ideal joint imposes kinematic constraints (on position), the proposed model introduces instead force constraints. This kind of force constraints often avoid structures to be kinematically overconstrained. Let us consider a shaft which rotates supported by two bearings. When modelling the bearings with revolute joints the system will be overconstrained (one joint will be overabundant) and the computation of reaction forces from Lagrange multipliers will produce wrong results. Using forces instead of position constraints one obtains correct results from computation and the system is not overconstrained. NOMENCLATURE B bearing axial length f friction coefficient {Fe} external generalized forces [J] inertia matrix w.r.t. center of mass principal axes h fluid film thickness [M] mass matrix Mx, My, Mz bearing reaction torques along x, y and z axis, respectively