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附錄
有限元素分析與設(shè)計(jì)42（2006）298 - 313
大型軸承螺栓接頭數(shù)值模式的發(fā)展
作者：奧里安 韋迪納，*，迪米特里 尼瑞巴 ，讓 癸樂特布
加拿大H3C公司3A7,魁北克，蒙特利爾，沙田，車站中心，蒙特利爾Ecole理工學(xué)院，機(jī)械工程系，P.O 6079信箱
摘要：
螺栓接頭的傳統(tǒng)理論并沒有考慮到外部負(fù)載的復(fù)雜性，既沒有其相關(guān)連接的不靈活性，也沒有接觸的非線性。本文論述了可以快速，精確的計(jì)算直徑軸承上承受很大傾覆力矩的緊固螺栓的二維數(shù)值模型。該模型的獨(dú)特性是在于一個(gè)特殊的有限元素的使用，像一環(huán)，除了在軸向方向。其軸向剛度是控制螺栓組裝方式的局部剛度。該模型調(diào)整為三維有限元的模擬，并在幾種類型的軸承中表現(xiàn)了優(yōu)異效果。
關(guān)鍵詞：螺栓接頭;數(shù)值模式;轉(zhuǎn)盤軸承;有限元分析
1、 介紹
提供快捷，準(zhǔn)確的結(jié)果，是對實(shí)際工程的挑戰(zhàn)之一，主要是在設(shè)計(jì)過程的早期階段。涉及不同的螺栓接頭的機(jī)械系統(tǒng)的制造商需要合適的計(jì)算模型，該計(jì)算模型需要考慮整體解決方案。大量的模型近似的部件和螺栓剛度使用錐體，球體，相當(dāng)于瓶裝或其他分析模型[1-4]。 根據(jù)傳統(tǒng)理論，最初是為那些居中或稍微偏離中心的負(fù)荷發(fā)展，該負(fù)荷的剛度為常數(shù)。然而，有限元模擬以及實(shí)驗(yàn)結(jié)果顯示出強(qiáng)勁的非線性由于接觸面積的變化[5-7]與外部負(fù)載。剛度非線性特性進(jìn)行了研究格洛斯[8]和吉洛[9]，他們提出了一個(gè)非線性模型，但只有板樣的配置。
另一個(gè)傳統(tǒng)理論的弱點(diǎn)在于所謂的負(fù)載系數(shù)的計(jì)算方法。負(fù)荷因素試圖測量傳送到螺栓上面的外在的力量。在外部力量的成員上應(yīng)用所在地管轄的負(fù)載因子和剛度成員的方式分配。張[10]開發(fā)了一種新的螺栓接頭分析模型，并考慮到剛性還原會(huì)與殘余力量有關(guān)，壓縮變形和尺寸變化的外力是因?yàn)槌蓡T輪換造成的。這種模式有它的局限性，并不適用于螺栓裝配時(shí)的成員有不同的幾何形狀，或在外部勢力不是在成員接口對稱的。
對于具體模型，我們提出對大型軸承可以看作為一個(gè)圓形法蘭盤，考慮到不同的非線性特性以及不同的配置或通過適當(dāng)?shù)膸缀蝿偠确植嫉耐ㄓ媚Ｐ偷幕A(chǔ)。
2、 回轉(zhuǎn)支承
本文提出的模型是對特定的大直徑螺栓軸承合適。這些大型軸承（高達(dá)13米（43英尺））也被稱為“回轉(zhuǎn)支承”，是用起重機(jī)，雷達(dá)菜，隧道掘進(jìn)機(jī)，軸承套圈等。二是夾在主框架由高強(qiáng)度的螺栓預(yù)裝。一個(gè)或兩個(gè)環(huán)是提供齒，使擺動(dòng)驅(qū)動(dòng)器進(jìn)行運(yùn)轉(zhuǎn)。連接就像大量的一個(gè)個(gè)又厚又狹小的螺栓固定一個(gè)非常嚴(yán)格的框架圓柱法蘭。
該系統(tǒng)的另一個(gè)特殊性是重要的和可變的傾覆力矩。軸承是遭受同樣重要徑向和軸向負(fù)荷。軸承所研究的三個(gè)類型，球軸承，交叉滾子軸承和三排滾子軸承，如圖 1所示。
由于結(jié)構(gòu)的復(fù)雜性和特殊性，螺栓聯(lián)合負(fù)荷，既不是傳統(tǒng)的模式也不是非線性模型是合適的。
圖 1 回轉(zhuǎn)軸承
因此，我們已經(jīng)開發(fā)出一種新的模式，同時(shí)考慮到在與框架單元的軸向方向，并與管狀分子徑向方向的彎曲剛度。此外，管內(nèi)容進(jìn)行了修改，涉及螺栓裝配的預(yù)裝行為特征。此外，在建模過程是一個(gè)原始的“混合”有限元素的定義。這個(gè)元素有一個(gè)除了在其軸向剛度有關(guān)，局部剛度，支配行為的螺栓裝配環(huán)的一般行為。嚙合同幾個(gè)元素的環(huán)狀物可以考慮到非線性剛度分布，特別是對高負(fù)荷的應(yīng)用效果。不斷演變的接觸面積是仿照通過接觸彈簧和使用迭代求解的技術(shù)。
3、 建模和假設(shè)
對這些系統(tǒng)的具體負(fù)荷是一個(gè)偏離中心的軸向載荷（正常起重機(jī)載荷型）在一個(gè)大的傾覆力矩造成的。這將建立在軸承槽（如圖1內(nèi)部負(fù)載1）。在這個(gè)發(fā)展階段，外部負(fù)荷強(qiáng)度并不重要。這足以適用于兩種型號(hào)相同的載荷：二維數(shù)值模型和三維有限元模型用于調(diào)整的第一個(gè)步驟。
要構(gòu)建數(shù)學(xué)模型，我們提出了一些簡化：
?建模的目的，我們只考慮負(fù)載最重的螺栓和相關(guān)零件;
?外圈不僅是為模型。因此，外部勢力所取代滾動(dòng)體負(fù)荷，等效負(fù)載增加螺栓的工作負(fù)荷;
?加載，以及有關(guān)的具體內(nèi)容的制定被認(rèn)為是軸對稱;
?安裝被認(rèn)為是非常嚴(yán)格的。
圖 2 基本原則提出了新的建模。在左邊的是素描和數(shù)值模型;右側(cè)的是等效有限元模型。
圖2 建模原理
正如圖 2顯示，軸承環(huán)模型包括三個(gè)要素的類型：
A：盤子模型是以由沃代安[11]提出和羅克解析式[12]發(fā)展的圓板模型為基礎(chǔ)的。他們有兩個(gè)自由度/節(jié)點(diǎn)（是翻譯和z旋轉(zhuǎn)軸對稱元素）及其作用：
?為代表的環(huán)的彎曲度是根據(jù)OY軸的方向;
?表征移位，尤其是環(huán)狀物的邊界不斷增長和它的分離。聯(lián)系彈簧元素它們能夠模擬環(huán)變量之間的接觸帶和安裝根據(jù)預(yù)安裝和外部負(fù)載的應(yīng)用。因此，以不同的接觸狀態(tài)，物體的剛度矩陣將被調(diào)整和它們對螺紋元件的非線性負(fù)荷將增大。
B：所謂的混合物原理使人們有可能考慮到物體的部分壓縮剛度，以及具體的彎曲沿徑向方向的撓度。每個(gè)節(jié)點(diǎn)的三個(gè)自由度使鐵元素的結(jié)構(gòu)與系統(tǒng)的受力相當(dāng)于外部負(fù)載。
C: 元素的彎曲和分裂是由于模型的接觸與安裝。它們的行為特征的彈性表現(xiàn)在該接口和單方面的接觸。彈簧剛度模型可作為一個(gè)調(diào)諧參數(shù)。
螺栓有一個(gè)等效梁的制定在本文件的以后會(huì)有講述。
3.1 確定軸承的軸向剛度位置
為了計(jì)算軸承的軸向剛度位置，我們已經(jīng)使用MASSOL[13]根據(jù)拉斯穆森[14]提出的一個(gè)基本圓柱集會(huì)（圖3）所作的改進(jìn)。本節(jié)計(jì)算的等效部分,記為Ap,使我們能夠確定零件剛度的Kp值。所用的關(guān)系式為
（1）
圖3 一個(gè)基本部件的尺寸
圖 4 軸向剛度部門尺寸計(jì)算
用下面的無量綱量：
對我們的知道軸承而言，有一部分不圓。外型尺寸X和Y是考慮如圖4所示。
如果直徑
Dp=3*Da (3)
不是該扇形面的部分上，下面的表達(dá)式。那么就使用
Dp=(x+y)/2 (4)
該扇形面的總軸向剛度計(jì)算是用 等式（1）和（2）?？紤]LP的長度等于軸承圈的高度。軸向剛度Kp值以及相當(dāng)于相等的面積Ap，從而得到考慮整個(gè)扇形的角度。
3.2 混合元素
如圖2所示 ，嚙合的軸承環(huán)使用的三個(gè)混合元素與三個(gè)主要部件有關(guān)：一個(gè)元素指定為環(huán)之間的上表面和軌道上負(fù)載生效的起點(diǎn)之間的區(qū)域;第二個(gè)元素是指定減少的區(qū)域是由軸承滾道和在軸承滾道及其安裝之間的下部區(qū)域的三分之一所決定。中間節(jié)點(diǎn)面對滾動(dòng)體接觸點(diǎn)，使外部力量應(yīng)用到其中之一。
v1
θ1
R - 元素的平均半徑
T - 元素的徑向厚度
L - 元素的高度
u，v，θ - 局部自由度
圖 5 管（圓柱表面）元素的參數(shù)
圖6 圓柱表面單元矩陣
3.2.1 混合單元?jiǎng)偠染仃?
由于相比于外徑和高度相對較低的徑向厚度，在外圈的運(yùn)動(dòng)方式是和其內(nèi)表面裝載的管子相似的。該管的基本自由度以及主要參數(shù)顯示如圖 5。其代表性是以圓柱表面的基本公式[15]為基礎(chǔ)。
對于我們軸承，重要的是要考慮到一個(gè)具體的圓柱表面彎曲，以及由一個(gè)徑向力（或壓力）造成的徑向位移。這種根據(jù)羅基[15]如圖6所示的元素的剛度矩陣很普遍。在圖6中，所有的表達(dá)方式kij都使用R，T，L參數(shù)（圖5），E-電子楊氏模量和-泊松數(shù)字來表達(dá)。
混合元素的剛度矩陣是以圓柱表面元素公式為基礎(chǔ)的。為了準(zhǔn)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(dá)方式對應(yīng)的行和列已被等效剛度的梁的公式所取代。因?yàn)檩p微的影響力，所以連接表達(dá)方式設(shè)置為零，正如數(shù)值試驗(yàn)表明。在圖 6（給予部分坐標(biāo)）采用的坐標(biāo)轉(zhuǎn)換程序根據(jù)整個(gè)模型的坐標(biāo)系統(tǒng)和編號(hào)矩陣圖控制的原因提出來的。
在全球CS的管狀物元素矩陣的拉伸自由度是（行和列）六方面。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。
在新的條件下截面積Ap和以前提出的橫截面計(jì)算使用改進(jìn)的RAS -穆森公式[14]是相等的。
此外，為了考慮負(fù)載點(diǎn)的應(yīng)用高度，總軸向剛度（或相反的靈活性）必須用不同的元素在非均勻模式下來分配，正如在3.3節(jié)中討論的。
圖 7 管狀物元素矩陣轉(zhuǎn)化成雜交元矩陣
上部靈活的Sp1
銻螺栓的靈活性
下部靈活的Sp2
圖 8 螺栓裝配示意圖
3.3 考慮外在負(fù)載應(yīng)用程序的起源
正如Guillot[9]和最近以來的張[10]所示的外負(fù)載應(yīng)用這個(gè)地方,對螺栓裝配行為,計(jì)算拉力的標(biāo)準(zhǔn)及帶有螺紋部件的彎曲瞬間補(bǔ)充度有極其重大影響。對于一個(gè)軸向載荷,螺栓的裝配可以按照圖8所示來代表。
圖 9 實(shí)際區(qū)域的壓縮
圖 10 自適應(yīng)的靈活性
眾所周知,和初始狀態(tài)的預(yù)加負(fù)荷Q比較,外力導(dǎo)致螺栓受力增大。螺栓所受總力Fb為
Fb=Q+Sp2*Fe/(Sp+Sb) (5)
全部零件的靈活性
Sp=Sp1+Sp2 (6)
是什么讓零件的靈活性不均勻分布的厚度的計(jì)算復(fù)雜化了，事實(shí)上,在壓縮條件下的頭螺栓的模樣,取決于裝配的水平,看起來就像一個(gè)體積接近于被切去頂端的形狀的圓錐(圖9)。
符合標(biāo)準(zhǔn)的實(shí)際情況是通過合理的算法來計(jì)算一個(gè)壓縮零件的靈活性。零件可由兩個(gè)或多個(gè)分區(qū)分開。考慮一個(gè)兩部分組裝零件隔斷案例(圖10),這個(gè)方法如下:
1、 通過改良的拉斯穆森的[14]計(jì)算橫截面面積Ap,然后全部零件的靈活性。
Ap?Sp=Lp/ApEp (7)
Finite Elements in Analysis and Design 42 (2006) 298–313
Bolted joints for very large bearings—numerical model development
Aurelian Vadean ?, Dimitri Leray , Jean Guillot
aDepartment of Mechanical Engineering, Ecole Polytechnique de Montreal, P.O. Box 6079, Station Centre-Ville,Montreal, Québec, Canada H3C 3A7
bLaboratoire de Genie Mecanique de Toulouse - COSAM, INSA Toulouse, 135 Avenue de Rangueil,Toulouse Cedex 4, 31077, France
Abstract
The conventional theory of bolted joints does not take the complexity of external loads into account, neither its related joint stiffness nor the contact non-linearities. This article deals with a 2D numerical model allowing fast and precise calculation of the fastening bolts for very large diameter bearings subjected to an overturning moment. The originality of the modelling lies in the use of a particular ?nite element that behaves like a ring, except in the axial direction. Its axial stiffness is the local stiffness that governs the behaviour of the bolted assembly. The model was tuned upon 3D ?nite elements simulations and provides excellent results for several types of bearings.
Keywords: Bolted joints; Numerical model; Slewing bearings; Finite elements analysis
1. Introduction
Providing fast and accurate results is one of the challenges of practical engineering mainly during the early stages of the design process. The manufacturers of different mechanical systems involving bolted joints need suitable calculation models that allow integrated solutions. Numerous models approximate the parts and bolt stiffness using cones,spheres, equivalent cylinders or other analytical models [1–4].
According to the conventional theory, which was originally developed for loads that are centred or slightly off-centre, the stiffness of the member is constant. However, ?nite elements simulations as well as experimental results showstrong non-linearities due to the changing contact area [5–7]with the external load. Stiffness non-linearities were studied by Grosse [8] and Guillot [9] and they propose a non-linear model but only for plate-like con?gurations.
Another weakness of the conventional theory lies in the way the so-called load factor is calculated. The load factor tries to measure the amount of the external force which is transmitted to the bolt. The location where external forces are applied on themember governs the load factor and the way themember stiffness is distributed. Zhang [10] developed a new analyticmodel of bolted joints and takes into consideration the stiffness reduction associated with the residual force on the assembly, compression deformation caused by external force and dimensions changing due to member rotation. This model has its limitation and is not applicable to bolted assemblies when the members are of different geometry or when the external forces are not symmetric about the member interface.
The speci?cmodel we are proposing for large bearings can serve as base for a genericmodel of circular ?anges which can take into account the different non-linearities as well as different con?gurations or geometries by appropriate stiffness distribution.
2. The slewing bearings
The model this article proposes is suitable for speci?c bolted joints for large diameter bearings. These large bearings (up to 13m(43 ft)) also called “slewing bearings”, are used for cranes, radar dishes, tunnel-boring machines, etc. The two bearing rings are clamped to the main frame by preloaded high strength bolts. One or both rings are provided with gear teeth to enable the swing drive to operate. The connection is like a thick and narrow cylindrical ?ange on a very rigid frame fastened with a large number of bolts.
Another particularity of the system is the important and variable overturning moment. The bearings are subjected to radial and axial loads of same importance. The three types of bearings under study, ball bearings, crossed-roller bearings and three-row roller bearings, are presented in Fig. 1.
Due to the complexity of structure and the particularity of bolted joint loading, neither traditional models nor non-linear models are appropriate.
Thus we have developed a new model that takes into account simultaneously the bending stiffness in the axial direction with shell elements and in radial direction with tube-like elements. Furthermore, the tube elements were modi?ed to consider the characteristics related to the behaviour of preloaded bolted assemblies. Therefore, the modelling process lies in the de?nition of an original “hybrid” ?nite element.This element has the general behaviour of a ring except for the axial direction where its stiffness is related to the local stiffness that governs the behaviour of the bolted assembly. Meshing the ring with several elements allows taking into account non-linear stiffness distribution in the assembly and in particular the effect of the load application height. The evolving contact area is modelled by contact springs and using
an iterative solving technique.
Fig. 1. Slewing bearings.
3. Modelling and assumptions
The speci?c loading on these systems is an off-centre axial load (a normal crane loading type) resulting in a 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;
? the outer ring only is modelled. Thus the external forces are replaced by the rolling elements load as
an equivalent load which increase the working load on bolts;
? the loading as well the formulation of the speci?c elements are considered axisymmetric;
? the mounting is considered extremely rigid.
Fig. 2 presents the principle underlying the new modelling. On the left-hand side is the sketch of the numerical model and on the right-hand side is the equivalent ?nite elements model.
Fig. 2. Modelling principle.
As Fig. 2 shows, the bearing ring model consists of three types of elements:
a. The plate elements based on the circular-plate model as described byVadean [11] and developed from Roark’s analytical formulas [12]. They are axisymmetric elements with two DOFs/node (y translation and z rotation) and their role is
? to represent the ring bending according to the axial direction OY;
? to characterize displacements and particularly the boundary separation of the ring from its mount-ing. 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 non-linear loading of the threaded element is produced.
b. The so-called hybrid elements which make it possible to take into account the part compression 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.
3.1. 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