液压支架的优化设计英文翻译.pdf
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1、Struct Multidisc Optim 20, 7682 Springer-Verlag 2000 Optimal design of hydraulic support M. Oblak, B. Harl and B. Butinar Abstract This paper describes a procedure for optimal determination of two groups of parameters of a hydraulic support employed in the mining industry. The procedure is based on
2、mathematical programming methods. In the fi rst step, the optimal values of some parameters of the leading four-bar mechanism are found in order to ensure the desired motion of the support with minimal transver- sal displacements. In the second step, maximal tolerances of the optimal values of the l
3、eading four-bar mechanism are calculated, so the response of hydraulic support will be satisfying. Key words four-bar mechanism, optimal design, math- ematical programming,approximationmethod, tolerance 1 Introduction The designer aims to fi nd the best design for the mechan- ical system considered.
4、 Part of this eff ort is the optimal choice of some selected parameters of a system. Methods of mathematical programming can be used, if a suitable mathematical model of the system is made. Of course, it depends on the type of the system. With this formulation, good computer support is assured to lo
5、ok for optimal pa- rameters of the system. The hydraulic support (Fig. 1) described by Harl (1998) is a part of the mining industry equipment in the mine Velenje-Slovenia, used for protection of work- ing environment in the gallery. It consists of two four-bar Received April 13, 1999 M. Oblak1, B. H
6、arl2and B. Butinar3 1 Faculty of Mechanical Engineering, Smetanova 17, 2000 Maribor, Slovenia e-mail: maks.oblakuni-mb.si 2 M.P.P. Razvoj d.o.o., Ptujska 184, 2000 Maribor, Slovenia e-mail: bostjan.harluni-mb.si 3 Faculty of Chemistry and Chemical Engineering, Smetanova 17, 2000 Maribor, Slovenia e-
7、mail: branko.butinaruni-mb.si mechanisms FEDG and AEDB as shown in Fig. 2. The mechanism AEDB defi nes the path of coupler point C and the mechanism FEDG is used to drive the support by a hydraulic actuator. Fig. 1 Hydraulic support It is required that the motion of the support, more precisely, the
8、motion of point C in Fig. 2, is vertical with minimal transversal displacements. If this is not the case, the hydraulic support will not work properly because it is stranded on removal of the earth machine. A prototype of the hydraulic support was tested in a laboratory (Grm 1992). The support exhib
9、ited large transversal displacements, which would reduce its em- ployability. Therefore, a redesign was necessary. The project should be improved with minimal cost if pos- 77 Fig. 2 Two four-bar mechanisms sible. It was decided to fi nd the best values for the most problematic parameters a1,a2,a4of
10、the leading four-bar mechanism AEDB with methods of mathematical pro- gramming. Otherwise it would be necessary to change the project, at least mechanism AEDB. The solution of above problem will give us the re- sponse of hydraulic support for the ideal system. Real response will be diff erent becaus
11、e of tolerances of vari- ous parameters of the system, which is why the maximal allowed tolerances of parameters a1,a2,a4will be calcu- lated, with help of methods of mathematical program- ming. 2 The deterministic model of the hydraulic support At fi rst it is necessary to develop an appropriate me
12、chan- ical model of the hydraulic support. It could be based on the 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
13、 four-bar mechanisms FEDG and AEDB (Oblak et al. 1998). The leading four-bar mech- anism AEDB has a decisive infl uence on the motion of the hydraulic support. Mechanism 2 is used to drive the support by a hydraulic actuator. The motion of the sup- port is well described by the trajectory L of the c
14、oupler point C. Therefore, the task is to fi nd the optimal values of link lengths of mechanism 1 by requiring that the tra- jectory of the point C is as near as possible to the desired trajectory K. The synthesis of the four-bar mechanism 1 has been performed with help of kinematics equations of mo
15、tion givenby Rao and Dukkipati (1989).The generalsituation is depicted in Fig. 3. Fig. 3 Trajectory L of the point C Equations of trajectory L of the point C will be writ- ten in the coordinate frame considered. Coordinates x and y of the point C will be written with the typical parameters of a four
16、-bar mechanism a1,a2, ., a6. The coordinates of points B and D are xB= xa5cos,(1) yB= ya5sin,(2) xD= xa6cos(+),(3) yD= ya6sin(+).(4) The parameters a1,a2, ., a6are related to each other by x2 B+y 2 B= a 2 2, (5) (xDa1)2+y2 D= a 2 4. (6) By substituting (1)(4) into (5)(6) the response equations of th
17、e support are obtained as (xa5cos)2+(ya5sin)2a2 2= 0, (7) xa6cos(+)a12+ ya6sin(+)2a2 4= 0. (8) This equation representsthe base of the mathematical model forcalculatingthe optimalvalues ofparametersa1, a2, a4. 78 2.1 Mathematical model The mathematical model ofthe systemwill be formulated in the for
18、m proposed by Haug and Arora (1979): min f(u,v),(9) subject to constraints gi(u,v) 0,i = 1,2,. ,?,(10) and response equations hj(u,v) = 0,j = 1,2,. ,m.(11) The vector u = u1.unTis called the vector of design variables, v = v1.vmTis the vector of response vari- ables and f in (9) is the objective fun
19、ction. To perform the optimal design of the leading four-bar mechanism AEDB, the vector of design variables is de- fi ned as u = a1a2a4T,(12) and the vector of response variables as v = x yT.(13) The dimensions a3, a5, a6of the corresponding links are kept fi xed. The objective function is defi ned
20、as some “measure of diff erence” between the trajectory L and the desired tra- jectory K as f(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. The system must satisfy the well-k
21、nown Grasshoff conditions (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 condition u u u(17) prescribes the lower and upper bounds of the design vari- ables. The problem (9)(11) is not
22、 directly solvable with the usual gradient-based optimization methods. This could be circumvented by introducing an artifi cial design vari- able un+1as proposed by Hsieh and Arora (1984). The new formulation exhibiting a more convenient form may be written as min un+1,(18) subject to gi(u,v) 0,i =
23、1,2,. ,?,(19) f(u,v)un+1 0,(20) and response equations hj(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 defi ned as min a7,(22) subject to constraints (a3+a4)(a1+a2) 0,(23) (a2+a3)(a1+a4) 0,(24) a1 a
24、1 a1,a2 a2 a2, a4 a4 a4,(25) g0(y)f0(y)2a7 0,(y ? ?y,y?), (26) and response equations (xa5cos)2+(ya5sin)2a2 2= 0, (27) xa6cos(+)a12+ ya6sin(+)2a2 4= 0. (28) This formulation enables the minimization of the diff er- ence between the transversal displacement of the point C and the prescribed trajector
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