汽车磁流变制动器设计的多学科设计优化【中文8000字】
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Available online at Computers and Structures 86 (2008) 207216/locate/compstrucMultidisciplinary design optimization of an automotive magnetorheological brake designEdward J. Park, Luis Falcao da Luz, Afzal Suleman *Department of Mechanical Engineering, University of Victoria, P.O. Box 3055, STN CSC, Victoria, BC, Canada V8W 3P6Available online 3 April 2007AbstractThis paper presents the development of a new electromechanical brake system using magnetorheological (MR) uid. The proposed brake system consists of rotating disks immersed in a MR uid and enclosed in an electromagnet, where the yield stress of the uid varies as a function of the magnetic eld applied by the electromagnet. The controllable yield stress causes friction on the rotating disk surfaces, thus generating a retarding torque. The braking torque can be precisely controlled by simply changing the current applied to the elec- tromagnet. Key issues involved in the initial design of the automotive MR brake are presented such as the MR uid selection, magnetic circuit design, torque requirements, weight constraints, dimensions and temperature. A multidisciplinary nite element analysis is per- formed involving magnetostatics, uid ow, and heat transfer analysis to study the behaviour of the system, and to serve as basis for a multidisciplinary design optimization procedure. The results of the optimization procedure are presented and the nal design obtained is discussed in detail.。 2007 Elsevier Ltd. All rights reserved.Keywords: Magnetorheological uid; Automotive brake; Finite element analysis; Computational uid dynamics; Multidisciplinary design optimization; Electric brake actuator1. IntroductionThis work is concerned with the development of a new brake-by-wire system for automotive vehicles, using an electromechanical brake (EMB) that employs magnetorhe- ological (MR) uid. Brake-by-wire replaces the mechanical connection between the brake actuator on each wheel and the brake pedal with electrical components. There are many advantages of using a pure electronically controlled brake system over a conventional hydraulic brake (CHB) system. The properties and behaviour of the brake will be easy to adapt by simply changing software parameters and electrical outputs instead of adjusting mechanical com- ponents. This also allows easier integration of existing and new control features such as anti-lock braking system (ABS), vehicle stability control (VSC), electronic parking* Corresponding author. Tel.: +1 250 721 6039; fax: +1 250 721 6051.E-mail address: sulemanuvic.ca (A. Suleman).brake (EPB), etc., as well as vehicle chassis control (VCC) and adaptive cruise control (ACC). Diagnostic fea- tures and the elimination of the water polluting brake uids are additional benets 1, as well as a small number of components, simplied wiring and generalized optimized layout.In this paper, we propose a MR actuator design for the brake in each wheel. The actuator consists of a rotating disk immersed in a MR uid, enclosed in an electromagnet. In principle, the brake torque can be controlled by changing the DC current applied to the electromagnet. Magnetorhe- ological uid a compound containing ne iron particles in suspension stiens in the presence of a magnetic eld. Two important characteristics of MR uids are: (i) they exhibit approximately linear response, i.e., the increase in stiness is directly proportional to the strength of the applied mag- netic eld and (ii) they provide fast response, i.e., MR uid changes from a uid state to a near-solid state within milli- seconds of exposing a magnetic eld. CHB systems exhibit about 200300 ms of delay between the time the brake pedal0045-7949/$ - see front matter 。 2007 Elsevier Ltd. All rights reserved. doi:10.1016/pstruc.2007.01.035208E.J. Park et al. / Computers and Structures 86 (2008) 207216is pressed by the driver and the corresponding brake response is observed at the wheels due to pressure build- up within the hydraulic lines. An electric brake system has the potential to drastically reduce this time delay, conse- quently bringing a reduction in braking distance. Recently, Delphi 2 introduced an EMB with performance similar to the existing disk brakes, with the brake pads actuated by an electrical motor, instead of the hydraulic actuator.While the application of MR uid in automotive vehi- cles has been promising for years, it is only recent that MR uid-based electromechanical devices have started to displace all-mechanical or hydraulic counterparts. For instance, General Motors recently introduced the Magnetic Ride Control 3, which is a MR uid-based suspensionTotal Shear Stresst (Pa)ty h1Shear Strain Rate g (s-1)Fig. 1. Bingham plastic model.control system developed by Delphi, on the Corvette and Cadillac Seville STS and XLR. The signicance with these new systems is that the vehicle control is quickly evolving away from the limitations of traditional mechanical com- ponents, such as springs, brakes, shocks and steering gear. Instead, real-time sensors and high-speed, direct electric actuation can now adjust all these systems depending on driving conditions 4. In this regard, a MR brake (MRB) actuator is a promising technology for the automotive industry with high commercial values.The outline of this paper is as follows. In Section 2, the MR uid phenomenon is explained in detail. In Section 3, our proposed automotive MRB design is described and modelled. Sections 4 and 5 present the multidisciplinary nite element analysis and subsequent design optimization of the proposed MRB. Section 6 presents design optimiza- tion results, along with transient temperature simulations, and the resulting dimensions and parameters of the nalparticles acquire a dipole moment aligned with the applied magnetic eld to form linear chains parallel to the eld 6. This reversibly changes the free owing liquid to semi-sol- ids that have a controllable yield strength, which depends on the magnitude of the applied magnetic eld.Although MR uids have been known for decades, they had been experiencing stability and longevity issues for commercial applications. Recently, however, these prob- lems have been solved and commercial applications are starting to appear, most notably as controllable dampers in the afore-mentioned car suspensions 4 and in civil engi- neering applications for seismic response control 5.In the literature, it is found that the essential magnetic eld dependent uid characteristics of MR uids can be described by a simple Bingham plastic model 6. As illus- trated in Fig. 1, in this model, the total shear stress s is given byMRB design. Section 7 concludes the paper.s syH gc_12. MR uidswhere sy is the yield stress due to the applied magnetic eldH, g is the constant plastic viscosity, which is consideredequal to the no-eld viscosity of the uid, and c_ is theMR uids are created by adding micron-sized iron par- ticles to an appropriate carrier uid such as oil, water or sil- icon. Their rheological behaviour is nearly the same as that of the carrier uid when no external magnetic eld is pres- ent. However, when exposed to a magnetic eld, the ironshear-strain rate. Here, the plastic viscosity is dened as the slope between the shear stress and shear-strain rate, which is the traditional relationship for Newtonian uids. True behaviour of MR uids exhibits some signicant departures from the Bingham model in the absence of aFig. 2. Shear stress as a function of shear-strain rate with no magnetic eld applied: (a) MRF-132AD and (b) MRF-241ES.E.J. Park et al. / Computers and Structures 86 (2008) 207216209Table 1Key properties of Lords MRF-132AD and MRF-241ESPropertiesMRF-132ADMRF-241ESBase uidHydrocarbonWater Operating temperature40 to +130 。C10 to +70 。C Maximum yield stress, sy44.5 kPa69 kPaMRB with the exception of the MR uid, which is located in the narrow channel (part no. 7) surrounding the rotating disk (no. 3) and the stator (no. 5).Based on Eq. (1) and the given geometrical congura- tion shown in Fig. 1, the retarding torque or brakingtorque which is caused by the friction on the interfacesViscosity, g (no magneticeld applied)0.09 0.02 Pa s between500 and 800 s 12.2 0.4 Pa s 50 s 1between the MR uid and the solid surfaces within the MRB can be written as 11magnetic eld (i.e., lp lp c_ ; H ) 7. Other researchers have tried more elaborate models such as the HerschelT b 2pnZ rzrwsr2dr 2pnZ rzrwgc_ sH r2dr2Bulkely model 8,9 to accommodate the shear-strain rate dependent shear thinning and shear thickening phenomena in the uid. However, if used properly Eq. (1) provides a useful basis for the design of MR uid-based devices 10,where n is the number of surfaces of the brake disk(s) in contact with the MR uid (e.g., 2 for 1 disk with MR uid covering the both sides, 4 for 2 disks, etc.); rz and rw are the outer and inner radii of the brake disk, respectively; andand the simple Bingham model is still very suitable forthe initial design phase 5. In addition, the Lord Corpora- tions hydrocarbon-based MRF-132AD and water-basedc_ rxbhand sy kHMRF-241ES, which are analyzed and compared in this paper, have nearly linear experimental stress-shear rate curves (see Fig. 2) that are well approximated by the Bing- ham model. Table 1 summarizes some of the key properties of these two MR uids that are most suitable for the auto- motive brake application. As can be seen from the table,where x is the angular velocity of the rotating disk, h is thethickness of the MR uid gap, H is the magnetic eld inten- sity, and k and b are constant parameters that approximate the relationship between the magnetic eld intensity and the yield stress for the MR uid. Then, Eq. (2) can be rewritten asthe water-based MRF-241ES has a higher yield stress than MRF-132AD, but lower magnetic permeability.T b 2pnZ rzrwrg r xkH hb,r2dr33. Automotive MR uid brakeShown in Fig. 3 is a three-dimensional illustration of the basic conguration of the MR brake (MRB) actuator design that is proposed and analyzed in this paper. It con- sists of a disk rotating within MR uid enclosed in a static casing. In Fig. 3, a cut has been made to highlight the cross-section that was modelled and analyzed. The legend in the gure indicates the various components of theFig. 3. Basic conguration of the proposed MR brake.Eq. (3) is a more accurate form than that of the Lord Cor- porations low torque MRB used in 12, because it can take into account non-constant magnetic eld distribu- tions. This improvement is necessary in order to use a greater amount of MR uid (which causes greater varia- tions in the magnetic eld intensity) than that of 12, which was used for AC induction motor braking. Eq. (3) provides some insight into the dynamics of an MRB and shows pos- sible ways to improve the braking torque, including the use of multiple disk surfaces (increasing n) or uids with high yield stresses (increasing k and/or b). Improving the brak- ing torque by amplifying the rst term in the integral, i.e., increasing the plastic viscosity g or decreasing the gap thickness h, is not desired as this would lead to a greater residual torque (increasing the drag even without the brakes applied).Eq. (3) indicates that, while carrying a one-disk congu- ration (hence, n = 2) would be ideal in terms of the simplic- ity of the design, manufacturing and weight of the MRB, having multiple disks generates more braking torque. Hence, a total of four congurations were selected for detailed analysis, involving all possible combinations between two dierent geometry congurations, one disk or two disks, and two dierent MR uids, MRF-241ES or MRF-132AD uids. Given the number of disk surfaces, additional parameters that inuence the performance of the MRB are the physical dimensions of its components.Now, the physical dimensions of the MRB shown in Fig. 4 can be optimized for performance and weight. How- ever, its overall dimensions must be restricted so that the210E.J. Park et al. / Computers and Structures 86 (2008) 207216Fig. 4. MR brake dimensional design parameters.brake can be tted inside a wheel rim as the typical CHB does. For example, considering the fact that the general recommended minimum clearance between the wheel rim and the brake is 3 mm, the maximum acceptable MRBradius for a 1600 wheel is about 20 cm 11. In Section 6,the various dimensional parameters represented in Fig. 4 are optimized using a multidisciplinary design optimization (MDO) procedure described in Section 5.Finally, the applied magnetic eld H can be produced within the MRB when current i is supplied to the electro- magnet encircling the MR uid, i.e.,H ai4where a is a proportional gain. Then, the two contributions of the resulting braking torque, Ty due to the yield stress induced by the applied magnetic eld and Tl due to the friction and viscosity of the MR uid, can be derived by performing the integration in Eq. (3) and substituting Eq. (1), i.e.,the average length of the ux path in the steel casing. Then to maximize the braking torque, Hf has to be maximized (maximizing the magnetic eld energy in the MR uid gap), while Hs has to be minimized (minimizing the energy lost in the steel path). The proportional gain a in Eqs. (4) and (5) then can be obtained from Eq. (7).4. Finite element modellingA nite element model (FEM) of the MRB was devel- oped using ANSYS to accurately characterize the brakes behaviour. This model was a multiphysics model that accounted for magnetostatics, MR uid ow, heat transfer, structural response within the MRB. Due to the multidisci-plinary nature of the MRB, with the presence of nonlinear- ities such as magnetic saturation and non-newtonian uid behaviour and the absence of closed-form solutions, nite element modelling and analysis were an essential design step.Our nite element analysis procedure consisted of a magnetostatics study followed by a computational uid dynamics (CFD) simulation in ANSYS. The former gives the magnetic eld distribution throughout the MR brake, which allows the determination of the yield stress sy. The magnetic eld distribution is then supplied to the CFD model, which computes the wall shear stresses the friction exerted on the walls and disk surfaces and the tempera- ture distribution within the MRB.The rst step in the nite element modelling was to dene the basic brake geometry. Since our problem is axi- symmetric, meaning that the geometry, material properties and loads are all consistent along the tangential direction, only the cross-section was modelled. This way, the solution becomes that of a two-dimensional problem, allowing the use of ANSYS plane elements (i.e., the PLANE13 ele- ments for the magnetostatics modelling and the FLUID141 elements for the CFD modelling) with axisymmetric for- mulation, and thus greatly reducing the computational cost of each simulation.For the magnetostatics simulation, the BH (magnetic ux density vs. applied magnetic eld) curves for the two MR uids were obtained from the manufacturers speci-cations and the BH curve for the steel element (SAE2p33T y nkarz rwi T ii51010 steel) that makes up the casing and disk(s) was3 p44obtained from the ANSYS material library. Steel is anT l 2h nlp rz rwh_ T vh_6ideal low reluctance (or high magnetic permeability) ux conduit that can guide and focus magnetic ux into thewhere h_is the rotational speed of the disk(s) and b = 1.MR uid gap 13. Fig. 5 contains these BH curves, whichNote that the magnetomotive force which drives the mag-netic ux around the magnetic circuit within the MRB is given by 13Ishow that both the MR uid (MRF-132AD as a represen- tative) and steel have a nonlinear magnetic characteristic (i.e., saturation). In the case of the steel, the knee of the sat- uration curve starts to occur at approximately 1.6 T, whichNi H dl H f h H s Ls7should be the maximum operating point of the steel so thatHs in Eq. (7) is close to zero according to the BH curve ofwhere the subscripts ()f and ()s denote the MR uid andthe steel parts, respectively; N is the number of turns in the coil; h is the length of the MR uid gap; and Ls isthe steel in Fig. 5b, thus maximizing the braking torque.In the nite element modelling, the current in the coil was applied as an area load. Fig. 6 shows the magnetic uxE.J. Park et al. / Computers and Structures 86 (2008) 207216211Fig. 5. BH curves of the materials used in MRB design: (a) MRF-132AD (courtesy of Lord Co.) and (b) ANSI 1010 steel.Fig. 6. Magnetic ux density distribution in one-disk conguration using MRF-241ES: (a) thin casing and (b) thick casing.density distribution in the one-disk conguration using the MRF-241ES uid, with (a) thin casing and (b) thick casing, where the arrows that represent the direction of the mag- netic ux density follow the intended path around the steel casing. As Fig. 6 shows, based on the principle of continu- ity of magnetic ux, thicker casing exhibits lower magnetic ux density in the steel casing. Fig. 7a presents the distribu- tion of the magnetic eld intensity in the same congura- tion using the MRF-241ES uid. Fig. 6b illustrates the relationship between the applied magnetic eld H and the resulting yield stress sy. As the eld intensity (Hf) of the MR uid reaches about 130 kA/m, the yield stress starts to saturate. As a result, the increase in the braking torque of the MRB becomes limited.For heat transfer analysis of the CFD model, the veloc- ity of the moving disks was specied, as well as the heat generated by the current ow in the coil (so-called the Joule eect). The heat generated by the friction between the uid and solid surfaces was computed by the CFD solver. Since the brake is cooled by the ow of outside air around the casing, the convection coecient was also determined, from empirical relations based on the Nusselt number. Fig. 8 presents preliminary results obtained from the CFD analysis, again using the MRF-241ES uid. Fig. 8a presents the distribution of the wall shear stress for thetwo-disk conguration, which occurs in the MR uid gap between the rotating disks and the stationary casing and stator (the middle disk). When the hydrocarbon-based MRF-132AD uid is used instead, while the magnetic eld intensity values in the uid are higher, the wall shear stress values are actually lower due to its lower viscosity values (see Table 1). This results in a lower braking torque com- pared to that of the MRF-241ES-based MRB. Fig. 8b shows the steady-state temperature distribution associated with constant braking at a modest deceleration (0.05 g, cor- responding to braking in a long downhill road).5. Multidisciplinary design optimizationFollowing the development of the nite element models describing the behaviour of the MRB, an optimization rou- tine was written to obtain the best possible design. For suc- cessful employment of the MRB into passenger vehicles, a factor requiring the most improvement was considered to be the weight, given that the steel components of the MRB are heavy and may add excessive weight to the vehi- cle. The braking torque is also an important parameter but, at this design stage, as long as a minimum torque require- ment is met, it was deemed less important than the weight. Hence, the objective function for the optimization was212E.J. Park et al. / Computers and Structures 86 (2008) 207216Fig. 7. Magnetic eld intensity distribution in one-disk conguration using MRF-241ES: (a) magnetic eld intensity and (b) magnetic eld intensity vs.yield stress.Fig. 8. CFD analysis in two-disk conguration using MRF-241ES: (a) wall shear stress distribution and (b) steady-state temperature distribution.dened so that a much greater importance is given to the weight than to the braking torque, by assigning a greater scalar weighting factor (0.90.1). The minimum acceptable value for the braking torque and the maximum acceptable value for the brake weight were chosen as 1010 N m and 65 kg, respectively. These numbers are the constraints of the optimization problem. In this initial MRB design phase, while the minimum braking torque value corre- sponds to that of typical CHBs, the value for the maximum weight was greatly relaxed such that it would allow the optimization procedures search for a wider design space. In addition, each MRB can potentially have more weight than a comparable on-wheel CHB as it would no longer have the extra weight carried by the CHBs hydraulic com- ponents: the master cylinder, brake uid lines, and pump. The optimization problem is expressed byThe above is the objective function for the optimization, subject to the two constraints, with x containing the design variables which are the dimensional design parameters ex- pressed in Fig. 4 previously and again in Table 2 below. Tref = 1200 Nm and Wref = 30 kg were chosen as the refer- ence values for the torque and weight, respectively, and xmin and xmax represent the chosen minimum and maxi- mum values for each design variable. Table 2 shows the allowed ranges of these values.Table 2Design space for each variableDesign variables, xAllowed values, xminxmax (cm)th_disk1.05.0 (1 disk)0.52.5 (2 disks)rad_disk13.018.5rad_th_coil0.252.5Minimisef x1000:1 T b T refWW 0:9refrad_th_casing15 (1 disk)0.52.5 (2 disks)ax_th_casing0.252.5subject toT b P 1010 Nm and W 6 65 kg8with xmin 6 x 6 xmaxlength_disk3.08.0_gap0.1E.J. Park et al. / Computers and Structures 86 (2008) 207216213Table 3Best values of the objective function for each design congurationDesign congurationSubproblem approximationFirst order methodSimulated annealingOne Disk, MRF- 132ADOne Disk, MRF- 241ESTwo Disks, MRF- 132ADTwo Disks, MRF- 241ES101.7301101.7311100.8975100.9598100.7905100.7144100.5946100.4547Computation time 20 min4 h100 hFig. 9. Implemented simulated annealing procedure.Three dierent optimization methods were applied to the above problem: rst two (subproblem approximation and rst order) are built-in capabilities of ANSYS and the third is simulated annealing. The latter is a more pow- erful technique, custom-programmed for the design of the MRB, but at the expense of computation time. This is a random-search method that can nd a global minimum for the objective function f(x) in Eq. (8). The two ANSYS built-in methods, being ready to be used and requiring less computation time, gave a quick insight into the eect of each optimization variable. After evaluating with the two methods, their results were compared to the more accurate simulated annealing method, in order to obtain the best MRB design with the lowest value for the objective func- tion. An in-depth description of the theory behind the two ANSYS built-in methods is found in 14, whereas the theory behind simulated annealing is presented in 15. Fig. 9 outlines the implemented simulated annealing procedure for the MRB.6. Results of design optimizationTable 3, where a dash indicates that no solution was found that met all the prescribed constraints. The last row of Table 3 compares the computation time taken by each method in obtaining a solution.It can be seen that simulated annealing method gave the best results (lowest objective function values) for both two- disk congurations and similar results for the one-disk con- gurations compared to those of the other two methods. However, this improved result comes at a great computa- tional expense (i.e., 100 h). The rst order method seems to produce a good compromise between good results and computation time. The subproblem approximation method produces the fastest results, but no feasible solution was obtained for the one-disk congurations. Fig. 10 presents a convergence plot of the objective function using the three methods in the two-disk, MRF-241ES uid conguration. It is clear that simulated annealing method produces the best results, while subproblem approximation and rst order methods converge to similar values.It is clear from Table 3 that the two-disk conguration with the MRF-241ES uid is the best design solution which produced a simulated braking torque of 1025 N m,2disks,MRF241Subproblem Approximation First OrderSimulated Annealing102101.8101.6Objective function101.4101.2101100.8100.6100.4100.2100The implementation of the optimization methods for the optimization function in Eq. (8) led to the values listed in05101520253035IterationFig. 10. Convergence of objective function values.214E.J. Park et al. / Computers and Structures 86 (2008) 207216while weighing about 18 kg (compared to 1010 N m and 64 kg for the worst case design). However, a subsequent heat transfer analysis in Section 6.1 below showed potential heat build-up problems in this design. Considering the fact that the hydrocarbon-based MRF-132AD is higher tem- perature resistant (see Table 1) than the water-based MRF-241ES, our nal design chosen for the automotive brake application was the two-disk conguration with MRF-132AD uid. The nal design is described in detail in Section . Dynamic temperature analysisRecall from Table 1 that the operating temperature ranges of the MR uids are limited. Hence, we also carried out a temperature distribution analysis resulting from repeated use of the MRB; and for this purpose, a transient simulation was performed, as shown in Fig. 12. This simu- lates repeated cycles of pressing and releasing the brake pedal. When the pedal is pressed, the MR uid exhibits the maximum viscosity that results in full braking power; when the pedal is released, implying accelerating or cruis- ing of the vehicle, there is no applied magnetic eld and the minimum viscosity is exhibited. Since time-varying material properties cannot be explicitly dened in ANSYS, an alternate solution was found by changing the boundary conditions: the disk velocity was set to zero when the pedal is released so that the MR uid viscosity has no eect on the temperature (the brake cools o by convection). This approximate solution gave some insight into the time-vary- ing viscosity eect in the MRB. The temperature variation with time to repeated brake-release cycle in the two-disk MRB is shown in Fig. 11.Note that in Fig. 12, the duration of pressing the brake was set to 3.2 s, which is the average time that a typical automobile takes to come to a full stop from 100 km/h. The duration of the brake release was conveniently assumed to be six times the braking duration, i.e., 19.2 s. It can be seen from Fig. 12a that under these conditions, the maximum temperature within the MRF-132AD-basedFig. 11. Transient temperature analysis in two-disk MRB.MRB starts to converge to about 100 。C (or 373 K) after 12 cycles of pressing and releasing the brake, where an equilibrium in temperature is reached in which the amount of heat generated by friction during the braking cycle equals the amount dissipated by convection during the non-braking cycle. On the other hand, Fig. 12b shows thatthis equilibrium temperature is about 127 。C (or 400 K) forthe MRF-241ES-based MRB, which signicantly higher than the MR uids operating temperature range ( 10 。C to +70 。C).6.2. Final MRB designThe resulting optimal MRB design and its parameters (using simulated annealing), employing the Lord Corpora- tions hydrocarbon-based MRF-132AD uid, are given in Table 4. As shown in Fig. 6, the two-disk conguration (i.e., n = 4), with a stator between the rotating disks, was the optimal brake design that minimized the objective func- tion. This design yielded a maximum braking torque of 1013 N and a brake weight of 27.9 kg, which by itself (with- out considering the overall brake system) is twice as heavy as that of a comparable performance CHB. Table 5 lists theFig. 12. Maximum temperature variation in two-disk MRB when subjected to repeated brake-release cycle: (a) using MRF-132AD and (b) using MRF- 241ES.E.J. Park et al. / Computers and Structures 86 (2008) 207216215Table 4Optimal MRB design parametersDesign variablesOptimal valuesNumber of disks2Maximum current12 ANumber of wire turnsApprox. 80th_disk1.2 cmrad_disk16.8 cmRad_th_coil1.4 cmRad_th_casing0.5 cmax_th_casing1.6 cmlength_disk5.5 cmFl_gap (h)0.1 cmTable 5Other MRB design parametersNumber of contact surfaces, n4Outer radius of brake disk, rz0.168 mInner radius of brake disk, rw0.118 mMR uid viscosity, g0.09 Pa sMR uid thickness, h1 10 3 mElectric constant, k0.269 Pa m/AProportional gain, a12.5 103 m 1Total inertia of brake disks, Iy2.5 10 2 kg m2Overall brake mass, mb27.9 kgFig. 13. Optimal two-disk MRB conguration.remaining design parameters that was used or obtained by the simulation. Fig. 13 is an illustration of the nal MRB design, with proper physical dimensions.7. Concluding remarksA new electromechanical automotive brake has been proposed and designed. The design process included math- ematical modelling and nite element analysis of the MRB to investigate its MR uid behaviour in an applied mag- netic eld and the resulting heat transfer phenomena. The nite element analysis that included magnetostatic, uidow and heat transfer analysis formed a basis for an opti- mization procedure in which three dierent optimization methods were used and compared to obtain ideal dimen- sions for the brake, so that the sucient braking torque could be provided by a light-weight MRB. The results obtained with the three methods were compared both in terms of their nal output values and the computa
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