【正文】 large overturning moment. This will build up an internal load on the bearings grooves (as shown in Fig. 1). At this stage of development, the intensity of the external load is not important. It is suf?cient to apply the same loading on both models: the 2D numerical model and the 3D ?nite elements model used to tune the ?rst one up. To build the numerical model, several simpli?cations were made: ? for modelling purposes, we consider only the most loaded bolt and the associated sector。 ? to characterize displacements and particularly the boundary separation of the ring from its mounting. Coupled to springs elements they are able to model the variable contact zone between the ring and the mounting according to the preload installed and the external load applied. Consequently to different contact status, the stiffness matrix will be adjusted and a nonlinear loading of the threaded element is produced. b. The socalled hybrid elements which make it possible to take into account the part pression stiffness, as well as the speci?c bending stiffness of a tube along radial direction OX. The three DOFs per node enable the structure to be loaded with a force system equivalent to the external load Fe. c. The spring elements that model the contact with the mounting. They characterize the elastic behaviour of the interface and the unilateral contact. Springs stiffness will be a parameter of the model tuning. The bolt has the formulation of an equivalent beam as described later in this paper. . Determining the axial stiffness of the bearing sector In order to calculate the axial stiffness of the bearing sector we have used the improvement made by MASSOL [13] to the formulation of Rasmussen [14] for an elementary cylindrical assembly (Fig. 3). The calculation of the equivalent section, noted Ap, makes it possible to determine the Kp stiffness of the parts. The relations used are （ 1） *** * 2 * 2 1* 2 * 2 ( 1 2 ) 1( 1 ) ( 1 ) t a n4 ( )ppp t pptllA D DDD? ? ??? ? ???? ? ? ???? Fig. 3. Dimensions of an elementary assembly. Fig. 4. Dimensions for the axial stiffness sector calculation. with the following dimensionless quantities: In the case of our bearings, the sections are not circular. Overall dimensions X and Y are considered as indicated in Fig. 4. If the diameter Dp=3*Da (3) is not inscribed in the sector section, the following expression is used: Dp=(x+y)/2 (4) The total axial stiffness of the sector is calculated with Eqs. (1) and (2) considering the length lp equal to the height of the bearing ring. Axial stiffness Kp as well as the equivalent section Ap is *p 2A PaAD? *PD paDD? * tt aDD D? * pp all D?thus obtained considering the whole angular sector. . Hybrid elements As shown in Fig. 2, the bearing ring is meshed using three hybrid elements in relation with the three main parts: one element is assigned for the part between the upper surface of the ring and the load application origin on the raceway。 ? the loading as well the formulation of the speci?c elements are considered axisymmetric。 Numerical model。 零件可由兩個或多個分區(qū)分開。對于一個軸向載荷 ,螺栓的裝配可以按照圖 8所示來代表。該矩陣轉(zhuǎn)換得到的最后形式是如圖 7 介紹的雜交元矩陣。為了準確的在軸向剛度建模，與圓柱表面單元矩陣的拉伸力的表達方式對應(yīng)的行和列已被等效剛度的梁的公式所取代。 對于我們軸承，重要的是要考慮到一個具體的圓柱表面彎曲，以及由一個徑向力（或壓力）造成的徑向位移。中間節(jié)點面對滾動體接觸點，使外部力量應(yīng)用到其中之一。考慮 LP的長度等于軸承圈的高度。所用的關(guān)系式為 （ 1） *** * 2 * 2 1* 2 * 2 ( 1 2 ) 1( 1 ) ( 1 ) t a n4 ( )ppp t pptllA D DDD? ? ??? ? ???? ? ? ???? 圖 3 一個基本部件的尺寸