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原文二 Contact analysis for drum brakes and disk brakes using ADINA C.Hohmann*,K.Schiffner,K.Oerter,H.Reese Institute of Engineering Mechanics and Control Engineering, University of Siegen, Paul-Bonatz Str.9±11,Fachbereich 11, Mashinentechnik,57076,Siegen, Germany Abstract Brakes in cars and trucks are safety parts.Requirements not only in performance but also in comfort,serviceability and working lifetime are high and rising.Optimal design of today's brake systems is found using additional calculations based on finite element methods.For both types of brake systems,drum brakes and disk brakes,the different parts of brakes,i.e.the brake pad with the friction material,the counter body and calliper,can be modelled.Two examples are given in this paper:a drum brake of a trailer and a typical disk brake used in passenger cars.The main problem to be solved is the calculation of the distribution of contact forces between brake pad and counter body(drum or disk).The contact problem includes friction and is solved using the ADINA 7.1 sparse solver.After the brake pressure is applied,the turning moment on the axle rises constantly until the drum or disk respectively changes from sticking to sliding condition.It is shown that the sparse solver is highly effcient for this sophisticated nonlinear problem.Results include deformation,stress distribution,contact pressure and showing which regions of the contact area are in sticking or sliding condition.#1999 Elsevier Science Ltd.All rights reserved. 1.Introduction 1.1.Brake construction Brakes in cars are expected to work properly with a minimum amount of service.The purpose of brakes is to reduce the velocity or to maintain it when the vehicle is driving downhill.Without brakes it would notbe possible to control the speed of the vehicle.Nevertheless the design of brakes is generally underestimated.In brakes high amounts of energy are transformed during short periods.This is underlined by the fact,that often the braking power is several times higher than the power of the engine. In cars and trucks different types of braking systems are used.In this paper only wheel brakes based on friction are considered.Generally two design forms are used:disk and drum brakes.The demands made by the vehicle industry are strict.The requirements will also rise with future development of lighter and more economical cars.This supports the need for efficient methods of calculating brake systems.The fnite element method is an ideal tool for this purpose.It is suited for analysis of both stress and temperature.In this paper the contact problem is described for disk brakes and drum brakes.For example a drum brake used in trucks and a disk brake used in small passenger cars are presented. In order to transfer the kinetic energy of the vehicle into heat,friction brakes are commonly used at each wheel of the vehicle.The area of contact is the origin of the heat sources.Cooling surfaces emit heat into the environment. Nevertheless the temperatures can get as high as 9008C(16508F) at the contact area. For proper operation the following criteria have to be considered: l high and stable coeffcient of friction; l good thermal capacity; l good wear resistance of the tribo system(brake linings and disk respectively drum); l mechanical resistance of material; l weight optimized construction; l and use of environmental suited materials. Despite different geometric design (see Fig.1), both types of brakes use the same principle to create the braking force: fixed brake shoes are pressed against a rotating counter body. Due to friction the brake force is acting contrary to the motion of the counter body there by reducing its velocity.The resulting friction force is proportional to the normal force N and the coeffcient of friction m.The brake parameter is defned as the relation between friction forces and the applied force and is used in the comparison of different brake designs. A typical brake used in cars consists of the operating device(pedal or hand lever),the transfer unit and the wheel brake(Fig.2).For disk and drum brakes the normal force N is applied,using mechanical,pneumatic or hydraulic transfer units.A servo unit raises the force applied by the driver: (1) The factor is the total transmission ratio of the brake.The friction force = acts on the rubbing surface.The distance between the friction force and the centre of revolution is the effective radius .The origin of the friction force for disk brakes is approximately the middle of the friction surface,depending on the shape of the friction area.In the case of drum brakes the effective radius is the inner radius of the drum. Effcient software and hardware is needed for the simulation of complete brake systems.ADINA Version 7.1 simplifes the construction of the complex three-dimensional structures together with providing reliable algorithms for the solution of the contact problem.Using the sparse solver the calculation time is reduced signifcantly. 2.Design of drum brakes 2.1.Analytical calculation The analytical calculation of drum brakes is based on the assumptions of Koessler[2,3].He describes the distribution of the normal pressure between drum and brake linings with cosine functions(see Fig.3a).The assumed distribution only if the curvatures of the brake lining and of the drum are equal.The radius of the brake lining is changed during operation due to wear.This results in irregular distributions of the contact pressure,which are shown in Fig.3b and c. At each point of the contact area the coeffcient m of sliding friction is constant.The distributed friction forces are caused by the pressure and are proportional to the coeffcient of sliding friction: = (2) The braking moment can be calculated with the distance r of the contact point to the centre of revolution. The summated friction forces make up the peripheral force R1 for the leading shoe and R2 for the trailing shoe.Both act against the rotation of the drum with the lever arm r: ， （3） The brake shoes are spread by a rotation of the S-cam clamping device.Ther- efore the forces S1 and S2 act upon the rolls(Fig.4a).They are in balance with the normal pressure and peripheral pressure acting on the lining surfaces(Fig.4b) and the resulting forces F1 and F2 acting at the bolts.The static conditions for equilibrium can be formulated for each direction(x,y and a)and both brake shoes in the form: =0 =0 =0 =0 -=0 -=0 （4） These equations have to be solved for both shoes.The solution of the integrals results in a relationship between applied forces and the braking moment.With ,the brake parameteris: ==2 （5） Type of drum brake:Simplex Duo-duplex Duo-servoBrake parameter: 。 The finite element model is used,to calculate a more realistic distribution of the contact pressure,considering the elastic deformations of both the linings and the drum. 2.2.Finite element calculation 2.2.1.Description of the model A complete three-dimensional structure of the drum brake has to be modelled.Thefnite element model consists of three independent parts:the drum,the leading shoe and the trailing shoe(see Fig.5).All parts, with the exception of the brake linings,are made of steel.Normally the brake linings are riveted to the brake shoe.In the model,however,uniform connection between lining and shoe is considered.Three dimensional elements with eight nodes are used to model the solids.The definitions of nodes and elements are created with the PATRAN-preprocessor.A FORTRAN program translates the node and element lists for ADINA-IN.Contact areas are defined for the rubbing faces of the drum and the four lining pad surfaces. The fixed bolts are modelled with truss elements. This enables the brake shoes to rotate around the fixed bolts(see Fig.6).The complete S-cam clamping device is neglected in this model.Since the S-cam provides straight spreading of the brake shoes,displacements are applied to the centre points of the rolls.The rolls are also modelled with truss elements. The correlation between S-cam rotation and the displacements of the rolls is given by the manufacturer. The connection between the rolls and the shoes is also modelled using truss elements. 2.2.2. Solution technique Although the mesh is quite coarse, the number of nodes is as high as 22,000.Although a static analysis is performed this is a large scale problem. The model is loaded in two intervals, Fig.7.In the first interval the brake shoes are spread for an initial contact between the brake linings and the drum. The value and angle of the displacements are defined by the geometry of the S-cam. (a) (b) After the spreading of the shoes, the drum is rotated in the second interval. Again displacements are used to describe the rotation. For the nodes at the anger of the drum, skew systems are defined and the displacements in the radial and the axial directions are fixed (Fig.8).The drum is still a free system with respect to the rotation. In order to avoid a singular system stiffness matrix, for the nodal points of the drum flange displacements are prescribed in the circumferential direction. 2.2.3. Results of the drum brake analysis Deformation of the drum and the brakes shoes is calculated using ADINA Version 7.0.The results are shown in respect to the rotated drum.Figs.9 and 10 show the contact forces on the brake linings. The contact surfaces are laid out from the bolt end to the end close to the rolls. It can be seen, that the contact forces have a non-uniform distribution which differs from the assumed distribution of the analytic solution by Koessler[2].High peaks are apparent at The top end of the lining, especially on the trailing shoe, close to the roll. The distribution of the contact pressure tends to that of the old lining shown in Fig.3c for the analytic solution. Furthermore the reaction forces at the fixed bolts and the rolls are compared with the results of the analytic calculation for the supporting forces F and S. Although the magnitudes of the supporting forces match, the angles of the forces do not correspond to the results of the analytical approach given by Koessler [2]. Since the calculated contact situation is different from the assumed contact situation in the analytic theory, the supporting forces of the two situations also differ. Reaction forces are also gained at the nodes of the drum, for which displacements are prescribed. These reaction forces are used to calculate the braking moment. Drum deformations and effective stresses are shown in Fig.11.The stresses are low because of the massive construction of the drum.Nevertheless, it is not over- dimensioned because most of the heat generated by friction is transferred into the drum. Maximum stresses at the shoes (Figs.12 and 13) are higher than those of the drum. The shown stress distribution can be used in optimizing the structure of the shoes. 3. Design of disk brakes 3.1. Analytical calculation Unlike the drum brakes,the friction surfaces of disk brakes are planar. The main advantage of the disk brake is that the heat can be transferred into the environment directly over the free surfaces of the disk [4].Further improvements for the heat convection are gained by ventilation channels or holes in the friction ring.However,in this example a solid disk is considered. Normal forces are generated by the hydraulic pressure in order to press the brake pads against the disk. The piston presses the pad against the disk.A reaction takes place where the caliper transfers the force to the pad on the other side of the disk(Fig.13).The normal force for each pad is: （6） The tangential stress generated on the friction surface results in the friction forces on each side of the disk. With the coeffcient m of friction, this force is: （7） As the friction forces act on both sides of the disk, the brake parameter for this type of brakes is: = （8） 3.2. FEM calculation 3.2.1. Model The finite element model consists of the disk, the caliper and the two brake pads (see also Watson and Newcomb [5]).The geometry is taken directly from the part and the model is created using the geometric elements provided by ADINA-IN. Mapped meshing is used to create the element mesh of the disk.For the complex structures of the pads and the calliper a free formed mesh with a suffcient density is constructed. The entire mesh consists out of parabolic three-dimen-sional-solid elements(21 nodes for one brick element). Linear elastic material models are used for both the lining material and the brake parts which are made of steel. The lining is glued to the supporting plates of the pads by the pressing process. The caliper as well as the brake pads are free to move in the axial direction. Together they are guided by bolts through the sprocket holes and grooves at the sides of the supporting plate. By this means they press against the supporting structure of the steering knuckle which is not modelled with finite elements but simulated by boundary conditions. The grooves and the sprocket holes are represented by nodes fixed in the directions perpendicular to the axial direction. Since the disk is free to rotate, it is fixed to a point on the axle by beam elements with high stiffness. For the contact surfaces on the linings and the friction ring of the disk a friction coeffcient of m=0.4 is defined. Although in experiments different values for static and sliding friction are found, in the calculation the value is constant due to program restrictions. 3.2.2. Solution technique Again the structure is loaded in two intervals. First the hydraulic pressure is applied on the brake pad and the caliper (Time=1.00).In the detail of Fig.13 one can see the surfaces on which the pressure is applied. In the second interval (time=2.00) the disk is rotated about the axis. The second interval corresponds to the situation of a vehicle with an automatic transmission stopping with the gear switched in driving position. The motor moment is acting against the braking moment. When the motor moment is higher than the braking moment the car begins to move. The rotation is prescribed about the centre point of the beam structure. The beam structure transfers the rotation to the disk. Since the caliper and the pads are free to move in the axial direction, truss elements with low stiffness are applied in order to avoid rigid body movement. These elements do not alter the results but make the stiffness matrix positive definite. They also stabilize the process of finding contact in the first load step. For this problem the sparse solver with automatic time stepping is used. The time steps have to be very small, especially at the start and the end of the first interval(application of brake pressure). 3.2.3. Results The sparse solver is very effcient.Contact is found easily for the plain surfaces of the linings and the disk. The effective stress and the contact pressure are compared for two states. The first state is after application of the brake pressure. Maximum stress occurs at the caliper as it is bent(Fig.14).This causes higher contact pressure at the outer radius of the linings(Fig.16).For the second interval the disk is rotated, thereby pressing the brake pads against the guidance of the steering knuckle(boundary conditions).This results in higher stresses in the supporting plate of the brake pads(Fig.15).First the disk sticks to the linings. An analysis of node displacements of lining and disk surfaces reveals that they are moved together. This condition is called static friction. The static friction forces H at each contact node are always smaller than the defined friction forces R for sliding friction: （9） After the disk is rotated through a certain angle, the disk breaks free. The peripheral forces have then reached the limit defined by the coeffcient of friction (H=R).Fig.17 shows that the contact pressure changes[6].The exact point of change from sticking to sliding condition can be found.Fig.18 marks the different regions of the contact surface as well as defining their condition. Depending upon the load step, regions of the lining surface are shown which are still in a sticking condition and those which are in a sliding condition. This is achieved by band plotting of the resultant defined in the following way: （10） In the case of sliding, this condition is equal to the coeffcient of friction input for m in ADINA-IN. This analysis is the basis for dynamic calculations of fric- Tion-induced vibrations. 4. Conclusion Only a short introduction into the fundamentals of brake constructions is presented in this paper. The basic formulas for the analytic calculation are given for different brake types. Drum and disk brakes are modelled with finite elements using ADINA-IN. It is possible to import lists of nodal coordinates and element definitions, which are created using a preprocessor such as Paranoid is more flexible to use the geometry definitions provided by ADINA-IN. The correct calculation of contact is essential for the design of friction brakes.The sparse solver implemented in ADINA Version 7.1 reduces job duration time of these large models to a few hours(DEC Alpha Server 4000).The reduction of elapsed time makes it possible to investigate the effect of altering design variables. The presented results are a small overview of what is possible. In particular the detailed analysis of the change from stick to shift is a step in understanding the friction process. Additional friction laws which distinguish between the coeffcient of static friction and of sliding friction would be a helpful tool for investigating stick/slip problems. Consideration of dynamic effects will be possible with more powerful hardware in the near future. Thermal analysis can also be performed using these models in ADINA-Tithe coupling of thermal and electromechanically calculations is a great advantage of the ADINA system. References [1]Wallentowitz H. Laèngsdynamik von Kraftfahrzeugen. Vorlesungsumdruck Kraftfahrzeugtechnik II. Institute four Kraftfahrwesen, RWTH Aachen: Schriftenreihe Automobiltechnik, 1997. [2]Koessler P.Berechnung von Innenbacken-Bremsen fuser Kraftfahrzeuge. Stuttgart: Franckh'sche, 1957. [3]Kessler. Grundlagende rFahrzeugtechnik, Originalausgabe. Munches: Heine, 1985. [4]Day AJ, Newcomb Tiptoe dissipation of frictional energy from the interface of an annular disk brake.Proc Inst Mach Engage 1984; 198D(11):201±9. [5]Watson C, Newcomb TT.A three-dimensional finite element approach to drum brake analysis.Proc Inst Mach Engng, D: J Automobile Engage 1990; 204(D2):93±101. [6]Tropic M-Day AJ.Disk brake interface pressure distribution.Proc Inst Mach Engng, D:J Automobile Engage 1991;205:137±46. 譯文二 鼓式制動器和制動盤的非線性分析 制動器是小車和卡車中的安全部分。要求不但在性能上而且在舒適上，可靠性和工作壽命要高而且要提高。今天的最佳制動器系統(tǒng)的設(shè)計是被用在以有限元方法為附加計算基礎(chǔ)上。對于各種制動系統(tǒng)，鼓式制動和盤式制動，制動器的不同部分也就是制動器接觸再摩擦材料上，主體部分和制動鉗可以被模擬。這篇論文將給出2個例子：拖車的制動鼓和典型的客車用盤式制動器。主要待解決的問題是對于摩擦片和主體部分（鼓和盤）的分布接觸力的計算。接觸問題包括摩擦和被用來解決的分析軟件ADINA7.1。在制動壓力加載后，在軸上的旋轉(zhuǎn)時刻會不斷提高直到鼓或盤各自從粘著到滑動狀態(tài)的改變。被發(fā)現(xiàn)這個求解器是很高效的對于這個復(fù)雜的非線性問題。結(jié)果包括應(yīng)力分布變化，接觸壓力和哪些接觸區(qū)域是粘著的哪些是滑行狀態(tài)的顯示。#1999Elsevier Science Ltd所有權(quán)保留。 1． 引言 1.1制動器的構(gòu)造 制動器在小車被期望在一個最小服務(wù)狀態(tài)下有效地工作。制動器的目的是減少速度或者當車輛在下坡中保持它的速度。然而通常制動器的設(shè)計還有待評估。在制動時，大量的能量在一個短時期內(nèi)被轉(zhuǎn)化。這被事實所強調(diào)，通常制動時的能量是高于引擎能量的好幾倍。 在小車和卡車中是使用不同的制動系統(tǒng)。這篇論文只是考慮摩擦中的輪式制動器?？傮w來說，2種設(shè)計形式被用到：盤式和鼓式制動。被用在汽車工業(yè)上要求是很嚴格的。這個要求也是隨著以后的輕量化的發(fā)展和更環(huán)保的小車而一同提高。這對有效地制動系統(tǒng)計算方法有很好的支持作用。對于這個目的有限元方法是一種理想的工具。它很適合分析各種應(yīng)力和溫度。在這論文中接觸問題是指盤式制動和鼓式制動。例如用于卡車的鼓式制動器和用于小型客車的盤式制動器。 為了轉(zhuǎn)化運動時速度的能量成熱量，摩擦制動器通常裝個每一個車輛的輪子上。接觸區(qū)域是熱量的起點。然后冷卻的面把熱傳送到環(huán)境中。 然而接觸面上的熱高達到9008C(