DECISION ANALYSIS OF DESIGNING GUIDELINES IN CONDITIONS OF UNCERTAINTY IN THE ANALYSIS OF DYNAMIC PROPERTIES OF MACHINE SYSTEMS

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1 DECISION ANALYSIS OF DESIGNING GUIDELINES IN CONDITIONS OF UNCERTAINTY IN THE ANALYSIS OF DYNAMIC PROPERTIES OF MACHINE SYSTEMS Adam DEPTUŁA Summary: The main prblem f dynamics is t specify the stability f the system layut and then, the real system. The analysis f the stability can be cnsidered frm the pint f vie f stability f an element r the stability f the hle system. During the prcess f changing the values f cnstructin and/r explitatin parameters it is pssible that the hle system shs instability hilst singular elements f explitatin parameters are stable. This rk presents an example f using the Quine McCluskey algrithm tgether ith cmmn minimizatin f certain and uncertain cannical prducts hile slving the cmplex decisin prcess. Keyrds: lgic functins system, Quine McCluskey algrithm, ptimizatin, verfl valve, multivalue lgic functins.. Intrductin In the cntemprary technlgy there are systems cmpsed f sets f varius physical character e.g. electric and mechanical, pneumatic, hydraulic, etc. Designing and analysing f such systems requires applicatin f apprpriate frms f descriptin and research methds in rder t apply an apprpriate ptimizatin prcedure. Overfl machines frm a large grup f systems [,, 3]. The rk f verfl machines is mst ften based n t states: the transient state (the values f the system functins change in time) and the set state (the functin values d nt underg changes in time r these changes ccur frm time t time). Changes f cnstructin parameters x, x,, x n have an influence n the behaviur f the functins f, f,, f r depending n time t. In the ptimizatin prcess, fr the same changes f cnstructin and/r explitatin parameters it is pssible t bserve different behaviur f functins depending n time [4]. Decisin tables [5], lgic functins [4, 6, 7, 8, 9,, ] and graphs [] can be applied in issues cncerning mdelling machine systems hich are described by means f (rdinary r partial). differential equatins. It results frm the fact that ccurring nt linear elements can be divided int the finished number f linear elements (parts) hat leads t receiving several linear systems in the meaning f mdelling curse frm the prime single nt linear system. A discrete ptimizatin f verfl machine systems based n lgic decisin trees is aimed at defining the imprtance f cnstructin and explitatin parameters that is guidelines cncerning the sequence f decisins taken frm the pint f vie f the realizatin f the system aim and stability functin [6,, ]. During the changing prcess f cnstructin and/r explitatin parameters values it is pssible that the hle system ill sh instability hilst single parameters ill remain stable. The ptimizatin prcedure shuld include analytic r graphic dependencies linking explitatin parameters (pressure and fl intensity) ith cnstructin parameters e.g. the 337

2 spring cnstant. If it is certain that the criteria cnditin is t be fulfilled nly by sme functins f i frm the set f functins f r f a given set, than the ptimizatin takes place in the cnditins f uncertainty. This rk presents an example f applying the Quine McCluskey algrithm tgether ith the cmmn minimizatin f certain and uncertain cannical prducts hile slving the cmplex decisin prcess.. Quine McCluskey algrithm f the minimizatin f multivalue lgic functins The Quine McCluskey algrithm makes it pssible t find all prime implicants f a given lgic functin that is there is a shrtened alternative, nrmal frm SAPN [, 3]. The terms f incmplete gluing and elementary absrptin have the main rle in the search f prime implicants and are used fr the APN f a given lgic functin. The flling transfrmatin i s called the cnsensus peratin: here: Aj ( x ) + Aj ( x ) A r + mr r =... () r =,..., n and A a partial elementary prduct, the literals f hich pssess variables belnging t the set: { x,..., x, x,..., x } r i r + i n. The flling transfrmatin is called the peratin f reductin: Aj ( x ) A A + () u r = here u mr, r n, and A a partial elementary prduct, the literals f hich pssess variables belnging t the set: { x,..., x r, x r +,..., x n } equatin takes place, then A absrbs Aj ( )). u x r.(if the abve Example Successive stages f the multivalue lgic functin minimizatin:,,,,,,,,,,,, can be presented in the flling ay: 338

3 In the end, t NAPN and MAPN f a given lgic functin are received and ritten in the frm f the mpsitin numerical system: {(), (), (), (), ()} and {(), (), (), (), ()}. 3. Decisin analysis f designing guidelines in the analysis f dynamic parameters f the verfl valve 3.. Overfl valve rk The verfl valve is used in the system in rder t let the excess f the pressed liquid g t the reservir hen it turns ut that the utput f the pump exceeds the need. The Figure presents the drive system f the engine ith the verfl valve [4]. Fig.. The driving system f the engine ith the verfl valve In the set presented in the Figure, the speed f the pistn mvement is steered by means f the thrttle D. As a result f hich, nly a part f the liquid stream pumped by the pump P cmes t the actuatr. The rest f the liquid stream ( Q = Q Q ) fls thrugh the verfl valve ZP hich must be cnstantly pen because Q Z >. During the rk f the verfl valve, it is necessary t take int cnsideratin amng thers: static frces cming frm the rk frce pressure), hydrdynamic frces, viscus frictin frces, spring frces, frces resulting frm the inertia f the liquid clumn. 339 Z P D

4 Structural mdels are built in rder t mdify dynamic prperties f fl systems and they reflect transfrmatin prperties f the system. Equatins describing the fl f liquid thrugh the valve (based n the mass cnservatin principle) are used in the cnstructin f the valve mdel. There are t types f equatins describing the valve rk: the equatin f frces acting n the valve seat, the equatin f fls. The equatin f frces having an influence n the valve seat is presented in the flling frm: Q p A dqp dx d x ρ + ρ A + ρ l = Gap + S + k x + f + m + Φ ρ cs( v) Qp p (3) dt dt dt hereas the fl equatins: here: dx V dp Q = µ K x p + A + (4) dt B dt dx = µ K x p + A (5) dt Q p K = π d m (6) ρ Equatins f the valve rk in the nnvalue frm used t make simulatin in accrdance ith rk are shn in the flling frm: Q ρ A S Φ Q p A p + S ρ cs( v) Q S Q p p p T + T Qp p dq dt p = T kx f dx T ms d + x + + S T dt T dt x + (7) T dx Q + dp A = µ x p + (8) T dt dt T Q + dx A p = µ x p (9) T dt 34

5 3.. Imprtance f cnstructin and/r explitatin parameters f the verfl valve The mdel research is made in rder t select imprtant parameters hich ill ensure stable rk t the real system. The specificatin f the imprtance f cnstructin and/r explitatin parameters and then the selectin f apprpriate ptimizatin prcedure are crucial fr the mdel verificatin. The changed cnstructin parameters f the valve are as flls: d valve diameter, m valve head mass and k spring cnstant during the bservatin f x elevatin, p pressure and Q fl intensity. Explitatin parameters ill intrduce delays hich during the applicatin f inapprpriate lp gain can cause unstable rk f the system. In rder t perfrm a discrete ptimizatin, changes in parameters are encded as flls: big reductin, small reductin, n change, 3 augmentatin, 4 big augmentatin (fr m and k ) and : small reductin, n change, augmentatin (fr d). As a result f simulatin analyses made, 75 charts shing x valve elevatin, p liquid pressure and Q fl intensity have been gained fr successive changes f parameters: m, k and d. In rder t ptimize, there is a dependency linking limits fr cnstructin and explitatin parameters [4]: I. The stabilizatin time t < 55t ; the rati f the maximum functin value t its max value after stabilizatin: <,4, stab. fr the time curse f x elevatin, p pressure and Q fl intensity. In rder t limit I, 5 charts have been chsen fr hich cde changes f cnstructin parameters m, k and d are shn in the table. Tab.. KAPN fr given changes f parameters m, k and d values m k d m k d It is necessary t ntice that the criteria cnditin I des nt all the stability f the system rk hen m= 4, s the Table presents nly t cannical prducts (as true versins f designing guidelines), in hich m parameter is encded as:,, r 3, hilst the analysis f the valve rk is fr change (the valence f hich is 5) in the cde f the m parameter. 34

6 The SAPN f functins frm the Table is shn in the Table. Tab.. SAPN f the functin frm the Table m k d 3 The NAPN f functins frm the Table is shn in the Table 3. Tab. 3. NAPN f the functin frm the Table m k d After the applicatin f the Quine McCluskey algrithm, the NAPN and MAPN f a given lgic functin frm the Table 3 is received: { 3}. 34

7 The Figure shs exemplary charts f the functins x, Q and p fr the cde changes f parameters (m, k and d) 33 and 3. a) b) Fig.. Characteristics f the valve rk fr changes in the cde f parameters m, k and d: a)33, b) Decisin analysis f designing guidelines in the cnditins f uncertainty in the analysis f dynamic prperties f the verfl valve During the changes f values f the cnstructin and/r explitatin parameters it is pssible that the hle system ill be unstable hilst single parameters ill be stable. If it is certain that the criteria cnditins can be met nly by sme functins f i frm the set f functins f r f a given system, then the ptimizatin prcess takes place in the cnditins f uncertainty. max Fr example, hen intrducing the criteria cnditin II: ( t < t ; <3,6) sme stab. time curses (x elevatin, p pressure r Q fl intensity ) are true simultaneusly fr max the criteria cnditin I ( t < 55t ; <,4) fr the same cde changes f the parameters m, k and d. stab. Figure 3 presents the time curse f the values x, p and Q (fr the cde changes m= 4, k=, d=), here the values p and x meet the criteria cnditin II and nly the value Q meets the criteria cnditin I. 343

8 Fig. 3. Characteristics f the valve rk fr changes in cde 4 (nly Q meets the criteria cnditin I) Figure 4 presents the time curse f the values x, p and Q (fr the cde changes m=, max k=, d=), here nly the values Q and x meets the criteria cnditin I ( >,4 fr p). stab. Fig. 4. Characteristics f the valve rk fr changes in cde (nly Q and x meets the criteria cnditin I) The cannical prduct f the changes in cde values f parameters fr hich nt all functins meet the criteria cnditin is called uncertain and is placed in parenthesis ( ). Uncertain cannical prducts and the certain nes take part in the gluing prcess. The Table 4 cntains certain cannical prducts frm the Table and uncertain prducts describing changes in cde fr hich nly sme time curses x, p r Q still meet the limiting criteria I ith the change t the criteria II: (), (), (), (4), (4), (), (), (4). 344

9 Tab. 4. KAPN fr certain and uncertain changes in the value f parameters m, k and d m k d m k d ( ) 3 ( ) ( ) (4 ) (4 ) ( ) ( ) 3 (4 ) The SAPN f functins frm the Table 4 is shn in the Table 5. Tab. 5. SAPN f the functin frm the Table 4 m k d 3 345

10 Tab. 6. NAPN f the functin frm the Table 5 m k d m k d ( ) ( ) ( ) (4 ) (4 ) ( ) ( ) ( ) After the applicatin f the Quine McCluskey algrithm, the NAPN and MAPN f a given lgic functin frm the Table 6 is received: { 3}. Uncertain cannical prducts (), (), (), (4), (4), (), (), (4) take part in the cnsensus prcess hen frming the SAPN lgic functin, hilst they are nt taken int cnsideratin hen lking fr prime implicants f a given functin (Table 6). As a result, the cannical prducts: and pssess blank clumns in the Table 6 (NAPN f a given functin). Elementary prducts: ; ; ; ; (earlier gluing in the SAPN f the functin frm the Table 5), pssess a mark in ne place thanks t uncertain cannical prducts that is they d nt include all pssible asterisks (*) in their clumns (Table 6)

11 When taking int cnsideratin uncertain changes in cde, e get a higher number f elementary prducts gluing in the SAPN f the lgic functin (Table 5), hereas the MAPN frm f the multivalue functin frm the Table 3 is equivalent t the MAPN fr the Table Cnclusin The mdel analysis is made in rder t select imprtant parameters hich ensure stable rk t the real system. Due t the fact that phenmena ccurring during the media fl are quite ften described nt precisely enugh, it is ften necessary t make an analysis in cnditins f uncertainty. In the analysed example f the verfl valve rk, the max changing f the criteria cnditin I (rigrus) ( t < 55t ; <,4) t the criteria cnditin II (liberal) ( t < t ; max stab. stab. <,4) causes that the number f cmbinatins f changes in cnstructin parameters m, k and d, ensuring the stable rk, increases. It means that the number f true versins increases that is the number f versins meeting ptimizatin requirements fr hich time curse f the analysed values x, p and Q simultaneusly meet the limit II. Hever, it is pssible t differentiate uncertain designing guidelines frm the hle set f true slutins that is the guidelines fr hich sme f the analysed values still meet the rigrus cnditin I. Such changes in the parameters cde tgether ith certain guidelines take part in the ptimizatin prcess. Such a prcess makes it pssible fr the analysed functins t have an influence n the variables ne by ne. In this ay, it is pssible t get mre infrmatin cncerning the imprtance f the analysed parameters. In the analysed example, the frm f the MAPN f the multivalue functin frm the Table 3 is equivalent t the MAPN frm the Table 6. Hever, it is necessary t highlight that fr certain designing guidelines (Tab. ) the valence f the m parameter is 4, hereas after taking int cnsideratin uncertain guidelines its valence is 5 (Tab. 4). Mrever, there is t small number f uncertain guidelines in the analysed example. In a general case (remving the influence f the valence and increasing the number f uncertain prducts) it is pssible t receive a smaller number f the mst imprtant designing guidelines. References. Stępnieski M.: Pmpy. WNT Warszaa, Suzuki K, Urata E.: Imprvement f Cavitatin Resistive Prperty f a Water Hydraulic Relief Valve. Prc. The Eighth SICFP,, 3, pp Francis J., Betts P. L.: Mdelling Incmpressible Fl in a Pressure Relief Valve. Prceedings f the Institutin f Mechanical Engineers, Part E: Jurnal f Prcess Mechanical Engineering, Vl., N. /997, Deptuła A.: Analiza prónacza ptymalnych zmdyfikanych drze lgicznych cenie dprnści parametró układu na zmiany arunkó pracy. XXXVIII Knf. Zast. Mat., Zakpane 9. Inst. Mat. PAN, Warszaa, Chlea W., Kaźmierczak J.: Diagnstyka techniczna maszyn. Przetarzanie cech sygnałó. Skrypt nr 94, Plitechnika Śląska, Gliice,

12 6. Deptuła A.: Rónania lgiczne z agymi spółczynnikami i kmplekse struktury rzgryające parametrycznie ptymalizacji układó maszynych. V Śrdiske Warsztaty Dktrantó Plitechniki Oplskiej, Pkrzyna. Oficyna Wydanicza Plitechniki Oplskiej, Seria: Mechanika z. 98 Nr kl. 34/, Ople,. 7. Deptuła A.: Wage rónania lgiczne ytycznych prjektania ptymalizacji dyskretnej układó maszynych. XL Knf. Zast. Mat., Zakpane. Inst. Mat. PAN, Warszaa,. 8. Giza M., Partyka M. A.: Zastsanie algrytmu Quine a Mc Cluskeya minimalizacji układu ielartściych funkcji lgicznych dla prcesó decyzyjnych zarządzania. IV Knfer. Kmput. Zintegr. Zarz., Zakpane, WNT, Warszaa,. 9. Giza M.: Rziązyanie złżneg prblemu decyzyjneg rzłączna minimalizacja układu. XXIX Knf. Zast. Matem. PAN, Zakpane. Inst. Matem. PAN, Warszaa,.. Giza M.: Rziązyanie złżneg prblemu decyzyjneg spólna minimalizacja układu. XXIX Knfer. Zast. Matem. PAN, Zakpane, Inst. Matem. PAN, Warszaa,.. Partyka M. A.: The Quine Mc Cluskey minimizatin algrithm f individual multiplevalued partial functins fr digital cntrl systems. 3rd Inter. Cnfer. Syst. Engin., Wright State University, Daytn, Deptuła A., Partyka M.A.: Applicatin f game graphs in ptimizatin f dynamic system structures. Internatinal Jurnal f Applied Mechanics and Engineering,, vl.5, N.3, pp Quine W. V..: The Prblem f Simplifying Truth Functins. American Mathematical Mnthly, 59, 95, pp Deptuła A.: Analiza rangi ażnści parametró knstrukcyjnych i eksplatacyjnych zaru przeleeg z uzględnieniem zmdyfikanych drze lgicznych. Pr. dypl. pd kier. M. A. Partyki, Wydz. Mech. Plit. Opl., Ople 9. MSc Eng. Adam DEPTUŁA Ople University f Technlgy Faculty f Prductin Engineering and Lgistic Department f Knledge Engineering 75 Ozimska Street, 4537 Ople, Pland phne/fax.: (77) a.deptula@ p.ple.pl 348