Int. J. of Applied Mechanics and Engineering, 2014, vol.19, No.3, pp DOI: /ijame

Transcription

1 Int. J. f Appled Mechancs and Engneerng, 2014, vl.19, N.3, pp DOI: /jame APPLICATION OF MULTI-VALUED WEIGHTING LOGICAL FUNCTIONS IN THE ANALYSIS OF A DEGREE OF IMPORTANCE OF CONSTRUCTION PARAMETERS ON THE EXAMPLE OF HYDRAULIC VALVES A. DEPTUŁA Ople Unversty f Technlgy Faculty f Prductn Engneerng and Lgstc 75 Ozmska Str., Ople, POLAND E-mal: a.deptula@p.ple.pl In the ptmzatn prcess, changes n the cnstructn parameters value nfluence the behavur f functns dependng n tme. Weghtng lgcal ceffcents fr the stablsatn tme are taken nt cnsderatn here,.e., a shrter (better) stablsatn tme has a mre mprtant (bgger) value f the weghtng ceffcent. An example f applyng weghtng lgcal functns n the analyss f a degree f mprtance f cnstructn parameters f a hydraulc valve s presented n the paper. Key wrds: ptmzatn prcess, weghtng lgcal ceffcents, hydraulc valves, pmps. 1. Intrductn Amng tls supprtng decsn makng prcesses, t s pssble t dfferentate decsn tables and trees, dendrtes, tree classfers as well as graphs. These tls are ncluded n the s-called decsn supprt methds based n graphs. A set f decsns (and relatns between them) s wrtten n a graphc way ut f a mathematcal mdel whch s the man bass f the decsn prcess realsatn whch a decsn-maker can use n rder t slve prblems f any knd. Mdellng f the whle prcess s necessary n the case f supprtng decsn-makng prcesses n the ptmzatn f mechancal systems. A lt f dfferent feedbacks between elements f nternal structure can be dfferentated n the bjects beng analysed (e.g., mechancal systems). It s necessary t use an apprprate graphc mdel where the cause and effect relatnshp, whch takes place nsde an analysed bject, ccurs. Flud-flw machnes frm a vast grup f sets used n ndustry (Francs and Betts, 1997; Gergel, 1990; Kurwsk, 2001). Decsn tables and lgcal functns (Deptuła, 2014; Stępnewsk, 1994) can be appled n the ssues f mdellng machne systems wth dfferental equatns (rdnary and partal nes). It results frm the fact that nn-lnear elements can be dvded nt a fnte number f lnear elements (parts) what leads t gettng several lnear systems. Dscrete ptmzatn f flud-flw machnes (Suzuk and Urata, 2003; Żak and Stefanwsk, 1994) s based n ndcatng the degree f mprtance f cnstructn and expltatn parameters. Gudelnes cncernng the sequence f makng decsns result frm mult-valued decsn trees and takng nt cnsderatn the realsatn f the assumed purpse functn (e.g., the system stablsatn). 2. Mult-valued lgcal functns wth weghtng ceffcents Graphs f utput data wth the ndcated stablsatn tme as mdellng results (e.g., n the prgrams such as: Matlab, Fluent ) depend n gven data f cnstructn parameters. Changes n such values (e.g., decreasng, ncreasng, remanng unchanged), n the prcess f desgnng the set fr dfferent wrk Unauthentcated

2 540 A.Deptuła cndtns can be wrtten n the frm f a cde n mult-valued lgc, whereas the set f desgn gudelnes can be presented as a sum f mult-valued lgcal prducts Weghtng ceffcents In the partal mult-valued lgcal functn f n varables m1,..., mn - valued, the weghtng ceffcent w befre the canncal prduct has the value wthn ths scpe w 1,...,wn, f wj wj1 wj2... w1, where j 2,..., n. Therefre, functn weghts n the graph n Fg.1 can be descrbed by the fllwng set f lgcal equatns: f2: w2( 0, 1, 2 ), whch means that the f 0 stablsatn s btaned the fastest and the f 2 stablsatn s btaned the latest, whch means w0 w1 w2 (Deptuła, 2011; Deptuła and Partyka, 2010), that s f and f j, w wj, f t t t j. Fg.1. Graphs f the functns f0, f1, f 2 dependng n tme fr the cded versn f mult-valued decsn varables x 0x, 1x, Then, an alternatve, mult-valued nrmal frm s created where a bgger lgcal weghtng ceffcent means a shrter stablsatn tme (Deptuła and Partyka, 2012). It s pssble t apply the Qune McCluskey algrthm f mult-valued functns mnmzatn n mult-valued lgcal functns wth weghtng prducts (Deptuła, 2014). 3. Sets f mult-valued lgcal equatns wth weghtng prducts In the case f flw rate calculatns (e.g., n pstve-dsplacement pumps r centrfugal pumps) many characterstcs are taken nt accunt at the same tme and they frm the set f functns f 1, f 2, f 3,..., f n. The set f equatns can be defned as R Y S R r r j... rn, R YS : R1 r r j... rn, Rm r r j... rn. (3.1) Unauthentcated

3 Applcatn f mult-valued weghtng lgcal functns 541 where: R- the set f lgcal equatns- R : R, R 1,..., Rm S : r, r,..., r, R r r,..., r., S- the set f canncal prducts as elements- 1 n, f: S R 1 n The set f mult-valued lgcal equatns s presented n the frm f a mrphlgcal table where the number f verses s equal t the number f equatns R, j=1,..., m. The set f mult-valued lgcal equatns can be slved usng cmbnatrcs n vew f mrphlgcal analyss wth mantanng pstulates f the Rsser-Turguette system (Deptuła, 2014; 2011). Example 1. Lgcally presented theretcal pssbltes f changes n the numbers f cnstructn parameters have the fllwng frm: x 1 =0, 1, x 2 =0, 1, 2, x 3 =0, 1, (where the sgn _ means t be kept unchanged and t was btaned ut f mdellng f the set f tw mult-valued lgcal equatns fr utput data (Deptuła and Partyka, 2012) j y1( t) 1001 ( ) 1011 ( ) 2021 ( ) 1000 ( ), y2 ( t) 1( 001) 2( 011) 3( 021) 1( 000), (3.2) After mnmzatn, the real slutn can be wrtten n the fllwng frm y , what results n: 1 ( 00) 1( 01) 2 ( 021). 4. Weghtng mult-valued lgcal functns n the analyss f a degree f mprtance f cnstructn parameters f the verflw valve The verflw valve s appled n the systems n rder t let the excess f flud flw t cntaner where the pump effcacy s bgger than the need. An example f a drve system f an actuatr wth an verflw valve s presented n Fg.2 (Deptuła, 2014; Żak and Stefanwsk, 1994; Smth, 2003). Unauthentcated

4 542 A.Deptuła Fg.2. A scheme f an actuatr wth an verflw valve. The equatn f frces actng n the clsng cmpnent f a valve s presented n the fllwng way (Deptuła, 2014) 2 2 Qp dqp dx d x A2 l Gap Skx f m 2cs( v) Q 2 p p, (4.1) A dt dt dt 1 whereas equatns f flws have the fllwng frm dx V dp, (4.2) Q K x p A1 dt B dt dx (4.3) Qp K x p A1 dt where K d m 2. (4.4) Equatns f the valve wrk n a dmensnless frm used t make smulatn are presented n the fllwng frm TQp dq 2 pw T f dx ms Qpw pw 1 xw 2 1 w w w Q A p kx T d x A S S T dt S T dt T dt 2 cs( vqq ) pw p pw, S (4.5) T A dx dpw Qw x pw, (4.6) T dt dt w w TA dx Qpw x pw. (4.7) T dt w Unauthentcated

5 Applcatn f mult-valued weghtng lgcal functns Weghtng ceffcents In rder t make a dscrete ptmzatn, changes n parameters have been cded n the fllwng way: 0- large decrease, 1- small decrease, 2- wthut changes, 3- ncrease, 4- large ncrease (fr m and k ) and : 0- small decrease, 1- wthut changes, 2- ncrease (fr d). Fr example, a cmbnatn f changes 122 means a small decrease n mass m, leavng the sprng cnstant wthut changes k and an ncrease f the dameter d. On the ther hand, the cmbnatn 402 means a large ncrease n mass m, a large decrease f the sprng cnstant k and an ncrease f the dameter d n relatn t the adpted arthmetc values n the early stage f desgnng. Dependng n the adpted cmbnatns f cde changes n parameters m, k and d n canncal prducts, the behavur f functns whch depend n tme s dfferent. When we lk at the behavur f the functns x, Q and p at the tme f stablzatn tw 200t(strct cndtn), f there are prducts f cde changes f parameters m, k and d the fllwng values f weghtng parameters have been adpted: w =4, t w 50t w =3, 50t tw 100t; w =2, 100t tw 150t; w =1, 150t tw 200t. ; When we ncrease the stablsatn tme t t w 800t (lberal wrk cndtns), t has been assumed that: w =4, t w 200t w =3, 200t tw 400t; w =2, 400t tw 600t; w =1, 600t tw 800t. ; The weghtng ceffcent w n the case f cde ndcatns depends n the stablsatn tme t, but l < l j f t > t j. Fgure 3 shws exemplary plts f functns x, Q and p, when apprprate changes f cde cmbnatns m, k and d ccur. Apprprate prducts f cmbnatns n cde changes have the fllwng weghtng ceffcents: - n the case f a lmtatn t w 200t : x: 2 ( 222) 1( 212) 1 ( 121) ; Q: 2 ( 222) 3 ( 212) ; p: 2 ( 222) 2 ( 212) ; - n the case f a lmtatn tw 800t: x: 4 ( 222) 4 ( 212) 4 ( 121) ; Q: 4 ( 222) 4 ( 212) 3 ( 212) ; p: 4 ( 222) 4 ( 212) 3 ( 212). Unauthentcated

6 544 A.Deptuła Fg.3. Tme functn plts x, Q, p wth the ndcated stablsatn tme and weghtng ceffcents n the case f lmtatns: t w <200 t and t w <800 t f cde changes n parameters: m, k and d: 222; 212, 121 (Deptuła, 2014). Table 1. KAPN f changes n parameter values m, k and d(t w <200 t, w w max stab. <3.6). m k d m k d m k d Unauthentcated

7 Applcatn f mult-valued weghtng lgcal functns 545 In the case f the verflw valve, a set f three mult-valued lgcal equatns f utput data x, Q, p was btaned ut f mdellng (Deptuła, 2014) respectvely fr a lmtng cndtn t 200t a) w Y tw200t x Q p t 800t b) w Ytw 800t x Q p A slutn f the set f equatns Yt 200t s 5120 (16(x) 20(Q) 16(p)) f theretcal versns f slutns. A slutn true fr the set Yt 200t btans the fllwng frm t 200t w w w f x, Q, p ( 001) 3 ( 0 2) 2 ( 011) 2 ( 021) 1( 1 2) 2 ( 122) 2 ( 112) 2 ( 102) ( 1111) 2 ( 202) 1( 212) 2 ( 222). In the case f the lmtng cndtn t w 800t, there are (30(x) 33(Q) 31(p)) real versns f slutns. Slutns can be btaned n the fllwng frm Unauthentcated

8 546 A.Deptuła t 800t w f x, Q, p ( 00) 4( 001) 4( 002) 3( 01) 4( 011) 4( 012) 3( 02) 4( 021) 4( 022) 3( 0 1) 4( 001) 4( 011) 4( 021) 4( 0 2) 41 ( 2) 1( 22) 3322 ( ) 4222 ( ) 4122 ( ) 4022 ( ) 4032 ( ) 3( 101) 4( 111) 3( 121) 1( 201) 1( 211) 4( 212) 2( 221) 3( 232) 3( 300) 3( 312) 2( 332). If we adpted a very strct cndtn n the graph f functns x, Q and p: stablsatn tme tw 100t and weghtng ceffcents values: w =4, tw 25t; w =3, 25t tw 50t; w =2, 50t tw 75t; w =1, 75t tw 100t, then a weghtng mult-valued set f equatns fr x, Q and p wuld have the fllwng frm Y tw100t x Q , p Cnclusns The real slutn wuld have the fllwng frm t 100t f ( x, Q, p) w ( 0 2). The artcle s abut a prcedure f cmbnatral slvng f weghtng mult-valued sets f lgcal equatns descrbng gudelnes f desgnng n vew f the mrphlgcal analyss wth keepng Rsser- Turguette s pstulates. Weghtng mult-valued set f lgcal equatns descrbng gudelnes f desgnng can be mnmzed separately r tgether wth keepng the lgcal equvalence. In ths way, we can als keep ndvdual prpertes f each functn. It has been prved that n a general case, mnmzatn f lgcal functns wth weghtng ceffcents can be the same as wthut weghtng ceffcents. Hwever, a better reflectn f physcal mdels f hydraulc sweep systems has been btaned n ther mathematcal mdels fr example n verflw valves r prprtnal nes. Increasng, decreasng r keepng such values unchanged n the readjustng prcess f the system t dfferent wrk cndtns can be cded n the mult-valued lgc, whereas desgnng gudelnes can be presented as a sum f mult-valued lgcal prducts. Slutns t the system f weghtng mult-valued lgcal equatns are versns f cde changes n parameters m, k and d f all functns x, Q and p dependng n tme t. In the case f such assumptns, t s als pssble t ntrduce cndtns f uncertanty fr apprprate lgcal prducts f desgnng gudelnes, whch means a partly descrbed functn n the autmata thery. Unauthentcated

9 Applcatn f mult-valued weghtng lgcal functns 547 Nmenclature d the valve dameter [m] d 1 nlet manfld dameter [m] d 2 valve seat dameter [m] F hydrdynamc reactn frce [N] K sprng cnstant, N / m m valve head mass [kg] m1,..., m n mult-valued lgcal functn f n varables m1,..., mn - valued n the number f dfferent letters n the Blean functn, P flw ntensty [m 3 /s] p tw. valve penng pressure [MPa] p p pressure n the nlet part n a partcular pstn [MPa] p z pressure ver the valve n a partcular pstn [MPa] Q 1 flw rate f the pumped lqud [m 3 /s] Q 2 flw rate f the lqud cmng ut f the valve [m 3 /s] Q m1 massve flw rate f the lqud gng nt the valve [m 3 /s] Q m2 massve flw rate f the lqud cmng ut f the valve [m 3 /s] S sprng bas frce [N] t tme [s] 3 V the valve vlume m x sprng deflectn [m] x free vbratns ampltude [m] x1, x2, x 3 decsn varables ceffcent f cmpressblty [m 3 /s] p References vscus frctn ceffcent [ Ns / m ] lqud densty 3 kg / m Deptuła A. (2014): Optmzatn f machne systems usng lgc equatns and graph-structures. Dssertatn- Department f Mechancal Engneerng, Techncal Unversty f Ople. Deptuła A. (2011): Weghted lgc equatns desgn gudelnes n dscrete ptmzatn f machne systems. XL Cnference n Applcatns f Mathematcs, Zakpane 2011, Insttute f mathematcs PAN, Warsaw. Deptuła A. and Partyka M.A. (2010): Applcatn f game graphs n ptmzatn f dynamc system structures. Internatnal Jurnal f Appled Mechancs and Engneerng, vl.15, N.3, pp Deptuła A. and Partyka M.A. (2012): Slvng weght multvalent lgc equatns descrbng the desgn gudelnes n dscrete ptmzatn f machne systems. XLI Cnference n Applcatns f Mathematcs, Zakpane 2010, Insttute f Mathematcs PAN, Warsaw Francs J. and Betts P.L. (1997): Mdellng ncmpressble flw n a pressure relef valve. prceedngs f the nsttutn f mechancal engneers. Part E: Jurnal f Prcess Mechancal Engneerng, vl.211, N.2, pp Gergel J. (1990): Identfcatn f mechancal systems. Warsaw: PWN. Kurwsk W. (2001): Mdellng techncal bjects. Manuscrpt Develpment, Płck. Unauthentcated

10 548 A.Deptuła Partyka M.A. (1984): The Qune - Mc Cluskey mnmzatn algrthm f ndvdual multple- valued partal functns fr dgtal cntrl systems. 3rd Inter. Cnfer. Syst. Engn., Wrght State Unversty, Daytn. Smth J. (2003): Parallel mechansms. Archve f Mechansms, vl.2, N.24, pp , Warsaw. Stępnewsk M. (1994): Pmps. Warsaw: WNT. Suzuk K. and Urata E. (2003): Imprvement f cavtatn resstve prperty f a water hydraulc relef valve prc. The Eghth SICFP, 1, pp Żak J. and Stefanwsk J. (1994): Determnng mantenance actvtes f mtr vehcles usng rugh sets apprach. Prc. f Eurmantenance 94 Cnference, Amsterdam. Receved: May 18, 2014 Revsed: June 25, 2014 Unauthentcated