液压支架的最优化设计外文文献翻译、中英文翻译
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Struct Multidisc Optim 20, 7682 Springer-Verlag 2000Optimal design of hydraulic supportM. Oblak, B. Harl and B. ButinarAbstract This paper describes a procedure for optimaldetermination of two groups of parameters of a hydraulicsupport employed in the mining industry. The procedureis based on mathematical programming methods. In thefirst step, the optimal values of some parameters of theleading four-bar mechanism are found in order to ensurethe desired motion of the support with minimal transver-sal displacements. In the second step, maximal tolerancesof the optimal values of the leading four-bar mechanismare calculated, so the response of hydraulic support willbe satisfying.Key words four-bar mechanism, optimal design, math-ematical programming,approximationmethod, tolerance1IntroductionThe designer aims to find the best design for the mechan-ical system considered. Part of this effort is the optimalchoice of some selected parameters of a system. Methodsof mathematical programming can be used, if a suitablemathematical model of the system is made. Of course, itdepends on the type of the system. With this formulation,good computer support is assured to look for optimal pa-rameters of the system.The hydraulic support (Fig. 1) described by Harl(1998) is a part of the mining industry equipment inthe mine Velenje-Slovenia, used for protection of work-ing environment in the gallery. It consists of two four-barReceived April 13, 1999M. Oblak1, B. Harl2and B. Butinar31Faculty of Mechanical Engineering, Smetanova 17, 2000Maribor, Sloveniae-mail: maks.oblakuni-mb.si2M.P.P. Razvoj d.o.o., Ptujska 184, 2000 Maribor, Sloveniae-mail: bostjan.harluni-mb.si3Faculty of Chemistry and Chemical Engineering, Smetanova17, 2000 Maribor, Sloveniae-mail: branko.butinaruni-mb.simechanisms FEDG and AEDB as shown in Fig. 2. Themechanism AEDB defines the path of coupler point Cand the mechanism FEDG is used to drive the support bya hydraulic actuator.Fig. 1 Hydraulic supportIt is required that the motion of the support, moreprecisely, the motion of point C in Fig. 2, is vertical withminimal transversal displacements. If this is not the case,the hydraulic support will not work properly because it isstranded on removal of the earth machine.A prototype of the hydraulic support was tested ina laboratory (Grm 1992). The support exhibited largetransversal displacements, which would reduce its em-ployability. Therefore, a redesign was necessary. Theproject should be improved with minimal cost if pos-77Fig. 2 Two four-bar mechanismssible. It was decided to find the best values for the mostproblematic parameters a1,a2,a4of the leading four-barmechanism AEDB with methods of mathematical pro-gramming. Otherwise it would be necessary to change theproject, at least mechanism AEDB.The solution of above problem will give us the re-sponse of hydraulic support for the ideal system. Realresponse will be different because of tolerances of vari-ous parameters of the system, which is why the maximalallowed tolerances of parameters a1,a2,a4will be calcu-lated, with help of methods of mathematical program-ming.2The deterministic model of the hydraulic supportAt first it is necessary to develop an appropriate mechan-ical model of the hydraulic support. It could be based onthe following assumptions: the links are rigid bodies, the motion of individual links is relatively slow.The hydraulic support is a mechanism with one de-gree of freedom. Its kinematics can be modelled with syn-chronous motion of two four-bar mechanisms FEDG andAEDB (Oblak et al. 1998). The leading four-bar mech-anism AEDB has a decisive influence on the motion ofthe hydraulic support. Mechanism 2 is used to drive thesupport by a hydraulic actuator. The motion of the sup-port is well described by the trajectory L of the couplerpoint C. Therefore, the task is to find the optimal valuesof link lengths of mechanism 1 by requiring that the tra-jectory of the point C is as near as possible to the desiredtrajectory K.The synthesis of the four-bar mechanism 1 has beenperformed with help of kinematics equations of motiongivenby Rao and Dukkipati (1989).The generalsituationis depicted in Fig. 3.Fig. 3 Trajectory L of the point CEquations of trajectory L of the point C will be writ-ten in the coordinate frame considered. Coordinates xand y of the point C will be written with the typicalparameters of a four-bar mechanism a1,a2, ., a6. Thecoordinates of points B and D arexB= xa5cos,(1)yB= ya5sin,(2)xD= xa6cos(+),(3)yD= ya6sin(+).(4)The parameters a1,a2, ., a6are related to eachother byx2B+y2B= a22,(5)(xDa1)2+y2D= a24.(6)By substituting (1)(4) into (5)(6) the responseequations of the support are obtained as(xa5cos)2+(ya5sin)2a22= 0,(7)xa6cos(+)a12+ya6sin(+)2a24= 0.(8)This equation representsthe base of the mathematicalmodel forcalculatingthe optimalvalues ofparametersa1,a2, a4.782.1Mathematical modelThe mathematical model ofthe systemwill be formulatedin the form proposed by Haug and Arora (1979):min f(u,v),(9)subject to constraintsgi(u,v) 0,i = 1,2,. ,?,(10)and response equationshj(u,v) = 0,j = 1,2,. ,m.(11)The vector u = u1.unTis called the vector of designvariables, v = v1.vmTis the vector of response vari-ables and f in (9) is the objective function.To perform the optimal design of the leading four-barmechanism AEDB, the vector of design variables is de-fined asu = a1a2a4T,(12)and the vector of response variables asv = x yT.(13)The dimensions a3, a5, a6of the corresponding links arekept fixed.The objective function is defined as some “measure ofdifference” between the trajectory L and the desired tra-jectory K asf(u,v) = maxg0(y)f0(y)2,(14)where x = g0(y) is the equation of the curve K and x =f0(y) is the equation of the curve L.Suitable limitations for our system will be chosen. Thesystem must satisfy the well-known Grasshoffconditions(a3+a4)(a1+a2) 0,(15)(a2+a3)(a1+a4) 0.(16)Inequalities (15) and (16) express the property of a four-bar mechanism, where the links a2,a4may only oscillate.The conditionu u u(17)prescribes the lower and upper bounds of the design vari-ables.The problem (9)(11) is not directly solvable with theusual gradient-based optimization methods. This couldbe circumvented by introducing an artificial design vari-able un+1as proposed by Hsieh and Arora (1984). Thenew formulation exhibiting a more convenient form maybe written asmin un+1,(18)subject togi(u,v) 0,i = 1,2,. ,?,(19)f(u,v)un+1 0,(20)and response equationshj(u,v) = 0,j = 1,2,. ,m,(21)where u = u1.unun+1Tand v = v1.vmT.Anonlinearprogrammingproblemofthe leading four-bar mechanism AEDB can therefore be defined asmin a7,(22)subject to constraints(a3+a4)(a1+a2) 0,(23)(a2+a3)(a1+a4) 0,(24)a1 a1 a1,a2 a2 a2,a4 a4 a4,(25)g0(y)f0(y)2a7 0,(y ?y,y?),(26)and response equations(xa5cos)2+(ya5sin)2a22= 0,(27)xa6cos(+)a12+ya6sin(+)2a24= 0.(28)This formulation enables the minimization of the differ-ence between the transversal displacement of the point Cand the prescribed trajectory K. The result is the optimalvalues of the parameters a1, a2, a4.793The stochastic model of the hydraulic supportThe mathematical model (22)(28) may be used to cal-culate such values of the parameters a1, a2, a4, thatthe “difference between trajectories L and K” is mini-mal. However, the real trajectory L of the point C coulddeviate from the calculated values because of differentinfluences. The suitable mathematical model deviationcould be treated dependently on tolerances of parametersa1,a2,a4.The response equations (27)(28) allow us to calcu-late the vector of response variables v in dependence onthe vector of design variables u. This implies v =h(u).The functionh is the base of the mathematical model(22)(28), because it represents the relationship betweenthe vector of design variables u and response v of ourmechanical system. The same functionh can be used tocalculate the maximal allowed values of the tolerancesa1, a2, a4of parameters a1, a2, a4.In the stochastic model the vector u = u1.unTofdesign variables is treated as a random vector U = U1.UnT, meaning that the vector v = v1.vmTof re-sponse variablesis alsoa randomvectorV = V1.VmT,V =h(U).(29)It is supposed that the design variables U1, ., Unareindependent from the probability point of view and thatthey exhibit normal distribution, Uk N(k,k) (k =1,2,. ,n). The main parameters kand k(k = 1, 2,. , n) could be bound with technological notions suchas nominal measures, k= ukand tolerances, e.g. uk=3k, so eventskuk Uk k+uk,k = 1,2,. ,n,(30)will occur with the chosen probability.The probability distribution function of the randomvector V, that is searched for depends on the probabil-ity distribution function of the random vector U and itis practically impossible to calculate. Therefore, the ran-dom vector V will be described with help of “numberscharacteristics”, that can be estimated by Taylor approx-imation of the functionh in the point u = u1.unTorwith help of the Monte Carlo method in the papers byOblak (1982) and Harl (1998).3.1The mathematical modelThe mathematical model for calculating optimal toler-ances of the hydraulic support will be formulated asa nonlinear programmingproblemwith independent vari-ablesw = a1a2a4T,(31)and objective functionf(w) =1a1+1a2+1a4(32)with conditionsYE 0,(33)a1 a1 a1,a2 a2 a2,a4 a4 a4.(34)In (33) E is the maximal allowed standard deviation Yof coordinate x of the point C andY=16?jA?g1aj(1,2,4)?2aj,A = 1,2,4.(35)The nonlinear programming problem for calculatingthe optimal tolerances could be therefore defined asmin?1a1+1a2+1a4?,(36)subject to constraintsYE 0,(37)a1 a1 a1,a2 a2 a2,a4 a4 a4.(38)4Numerical exampleThe carrying capability of the hydraulic support is1600kN. Both four-bar mechanisms AEDB and FEDGmust fulfill the following demand: they must allow minimal transversal displacements ofthe point C, and they must provide sufficient side stability.The parameters of the hydraulic support (Fig. 2) aregiven in Table 1.The drive mechanism FEDG is specified by the vectorb1,b2,b3,b4T= 400,(1325+d),1251,1310T(mm),(39)and the mechanism AEDB bya1,a2,a3,a4T= 674,1360,382,1310T(mm).(40)In (39), the parameter d is a walk of the support withmaximal value of 925mm. Parameters for the shaft of themechanism AEDB are given in Table 2.80Table 1 Parameters of hydraulic supportSignLength (mm)M110N510O640P430Q200S1415T380Table 2 Parameters of the shaft for mechanism AEDBSigna51427.70 mma61809.68 mm179.340.520.144.1Optimal links of mechanism AEDBWith this data the mathematical model of the four-barmechanisms AEDB could be written in the form of (22)(28). A straight line is defined by x = 65 (mm) (Fig. 3) forthe desired trajectory of the point C. That is why condi-tion (26) is(x65)a7 0.(41)The angle between links AB and AE may vary be-tween 76.8and 94.8. The condition (41) will be dis-cretized by taking into accountonly the points x1, x2, .,x19in Table 3. These points correspond to the angles 21,22, ., 219ofthe interval 76.8, 94.8 at regularinter-vals of 1.The lower and upper bounds of design variables areu = 640,1330,1280,0T(mm),(42)u = 700,1390,1340,30T(mm).(43)The nonlinear programming problem is formulated inthe form of (22)(28). The problem is solved by the op-timizer described by Kegl et al. (1991) based on approx-imation method. The design derivatives are calculatednumerically by using the direct differentiation method.The starting values of design variables are?0a1,0a2,0a4,0a7?T= 674,1360,1310,30T(mm).(44)The optimal design parameters after 25 iterations areu= 676.42,1360.74,1309.88,3.65T(mm).(45)In Table 3 the coordinates x and y of the coupler pointC are listed for the starting and optimal designs, respec-tively.Table 3 Coordinates x and y of the point CAnglexstartystartxendyend2()(mm)(mm)(mm)(mm)76.866.781784.8769.471787.5077.865.911817.6768.741820.4078.864.951850.0967.931852.9279.863.921882.1567.041885.0780.862.841913.8566.121916.8781.861.751945.2065.201948.3282.860.671976.2264.291979.4483.859.652006.9163.462010.2384.858.722037.2862.722040.7085.857.922067.3562.132070.8786.857.302097.1161.732100.7487.856.912126.5961.572130.3288.856.812155.8061.722159.6389.857.062184.7462.242188.6790.857.732213.4263.212217.4691.858.912241.8764.712246.0192.860.712270.0866.852274.3393.863.212298.0969.732302.4494.866.562325.8970.502330.36Figure4illustrates the trajectoriesLofthe pointC forthe starting (hatched) and optimal (full) design as well asthe straight line K.4.2Optimal tolerances for mechanism AEDBIn the nonlinear programming problem (36)(38), thechosen lower and upper bounds of independent variablesa1, a2, a4arew = 0.001,0.001,0.001T(mm),(46)w = 3.0,3.0,3.0T(mm).(47)The starting values of the independent variables arew0= 0.1,0.1,0.1T(mm).(48)The allowed deviation of the trajectory was chosen fortwo cases as E = 0.01 and E = 0.05. In the first case, the81Fig. 4 Trajectories of the point CTable 4 Optimal tolerances for E = 0.01SignValue (mm)a10.01917a20.00868a40.00933Table 5 Optimal tolerances for E = 0.05SignValue (mm)a10.09855a20.04339a40.04667Fig. 5 Standard deviations for E = 0.01optimal tolerances for the design variables a1, a2, a4werecalculated after 9 iterations. For E = 0.05 the optimumwas obtained after 7 iterations. The results are given inTables 4 and 5.In Figs. 5 and 6 the standard deviations are calculatedby the Monte Carlo method and with Taylor approxima-tion (full line represented Taylor approximation), respec-tively.Fig. 6 Standard deviations for E = 0.055ConclusionsWith a suitable mathematical model ofthe systemand byemploying mathematical programming, the design of the82hydraulic support was improved, and better performancewas achieved. However, due to the results of optimal tol-erances, it might be reasonable to take into considerationa new construction. This is especially true for the mech-anism AEDB, since very small tolerances raise the costsof production.ReferencesGrm, V. 1992: Optimal synthesis of four-bar mechanism. MSc.Thesis. Faculty of Mechanical Engineering MariborHarl, B. 1998: Stochastic analyses of hydraulic support 2S.MSc. Thesis. Faculty of Mechanical Engineering MariborHaug, E.J.; Arora, J.S. 1979: Applied optimal design. NewYork: WileyHsieh, C.; Arora, J. 1984: Design sensitivity analysis and op-timisation of dynamic response. Comp. Meth. Appl. Mech.Engrg. 43, 195219Kegl, M.; Butinar, B.; Oblak, M. 1991: Optimization of me-chanical systems: On strategy of non-linear first-order approx-imation. Int. J. Numer. Meth. Eng. 33, 223234Oblak, M. 1982: Numerical analyses of structures part II. Fac-ulty of Mechanical Engineering MariborOblak, M.; Ciglari c, I.; Harl, B. 1998: The optimal synthesis ofhydraulic support. ZAMM 3, 10271028Rao, S.S.; Dukkipati, R.V. 1989: Mechanism and machine the-ory. New Delhi: Wiley & Sons设计（英译汉论文一） 第 12 页 共 12 页液压支架的最优化设计摘要：本论文论述了一种最优化程序，这种程序主要是针对从事采矿业液压支架两组参数的最佳测定。它是基于数学规划的方法。首先，寻求这些四连杆机构的最优参数值是为了确保支架在做所要求的运动的同时，使得横向位移最小。其次，计算出四连杆机构最佳数值的最大偏差。关键词：四连杆机构，最佳设计，数学规划法，逼近法，公差。1. 引言设计师的目的是为完美的机械系统寻求最好的设计。这种努力的一部分,是优化所选择一些特定的系统参数。如果为系统做出数学模型，就可以使用数学规划法。当然，这取决于系统的类型。在这种情况下，计算机的应用有助于能够确保找到系统的最优参数。有Harl1998年所描述的液压支架（见图1）是斯洛文尼亚矿厂采矿设备之一，它用于保护矿井巷道的工作环境。它有两组四立柱（FEDG and AEDB）所组成，如图2中所示。机构 AEDB决定藕合点的运动，机构FEDG通常石油液压缸驱动的。图1 液压支架图2 四连杆机构那就要求支架的运动更精确，如图2中C点的运动方向要垂直于最小横向位移。如果不是这种情况，液压支架将不能正常工作，因为它缺乏机械的通用性。1992年在Grm实验室对液压支架的原型机做过试验。支架表现出很大的横向位移，这就降低了它的适用性。所以，有必要对此重新设计。如果可能的话，这项工程应在提高支架性能的同时，尽量降低成本。这就决定了采用数理规划法为AEDB四连杆机构中最有争议的参数、寻求最优值。否则，有必要改变这个项目，至少改变AEDB机构。以上问题的解决将使我们重新考虑理想液压支架的系统。由于各种系统参数存在偏差，实际的考虑将是不同的，这也就是我们为什么用数理规划法计算参数、的最大允许偏差的原因。2.液压支架的选型首先，有必要开发一种合适的液压支架力学模型。它应基于以下假设：连接是刚体连接单个连杆运动速度相对缓慢液压支架是有一个自由度的机械。它的运动规律可以通过两个四杆机构FEDG 和AEDB进行模拟。(Oblak et al. 1998)AEDB四连杆机构对液压支架的运动具有决定性的影响。机构2通常通过液压缸驱动。支架的运动可以用藕合点C点的运动轨迹完全表示出来。所以，所做的工作就是通过使C点的运动轨迹尽量趋近理想的运动轨迹寻求机构1杆长的最优值.1989年Rao和Dukkipati在运动学方程的帮助下，即将四连杆机构1的综合情况完全表达了出来，大致状况如图3所示。 图3C点的运动轨迹LC点的运动轨迹L的方程已被写入坐标系，将四连杆机构具体参数，代入C点坐标x,y。B点和D点的坐标是：参数、相互关系如下：将（1）（4）式代入（5）（6）式支架方程得到如下结果：通过计算参数、最优值，这个方程表达了数学模型的基础。2.1数学模型1979年Haug 和Arora以公式化的形式提出系统的数学模型。限定条件是：响应方程是：向量 称为设计参数的向量，v = 是结果向量，在（9）中的是目标函数。为了清楚表达四连杆机构AEDB的最优设计，设计参数的向量被定义为：响应变量的矢量：尺寸、的相互关系就被确定了。目标函数被定义为介于轨迹L和理想轨迹K之间的尺寸。K被定义为：这里Xgo(y)是曲线K的方程，fo(y)是轨迹曲线L的方程。系统有特定的限制条件。系统必须满足著名的杆长条件：不等式（15）和（16）四连杆机构的特性，连杆，仅作摆动，条件是：规定了最短杆和最长杆的设计参数。（9）（10）的答案不能直接用通常基于梯度的最优化方法解出。这可以通过1984年Hsieh 和 Arora提出的引入模拟设计变量的方法来实现，新公式可以用一种更方便的形式表达，写作为：这里 ， 所以AEDB四连杆机构AEDB的非线性规划问题可以表示如下：约束条件： 响应方程：这个公式减小C点的横向位移与运动轨迹K的差别。结果使得参数，的值最优化。3. 液压支架的随机模型数学模型（22）（28）通常用于计算参数值如，运动轨迹K和L差别是最小的，然而，由于各种影响的存在C点的真实运动轨迹L会偏移与计算出来的数值。适当的数学模型偏差应当分别处理，这取决于参数，的公差。结论方程（27）（28）允许我们计算相应参数V的向量，主要取决于设计参数U。这就表示v = (u)。函数是数学模型（22）（28）的基础，因为它表示了设计参数U的向量和我门所求机械系统目标向量V之间的关系。同样的函数能够用于计算参数，所允许的最大偏差。在随机模型中，设计参数的向量作为随机变量U,这就意味着目标向量也是一个随机向量，（29）假定从概率的观点看设计参数它们服从正态分布（1，2，）。主要参数和（1，2，）一定与工艺概念上的诸如名义尺寸有关，和公差，等等。， ,（1，2，） (30) 所以这种情况只存在于选择的可能性。随机变量V概率分布函数取决于随机向量U概率分布函数，事实上它不可能计算出来。所以随机变量V将在数字特征的帮助下描述，用在点函数的泰勒逼近式计算，或者用本文所说的1982年Oblak和1988年Harl提出的Monte Carlo防
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