喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請放心下載，，有疑問咨詢QQ:414951605或者1304139763 ======================== 喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請放心下載，，有疑問咨詢QQ:414951605或者1304139763 ======================== 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