DYNAMIC PROPERTIES OF HYDRAULIC LINES FOR FLUID POWER TRANSMISSION

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1 eek UŁANOWI Intytut Tehniny Wok otniyh ESEAH WOS OF AFIT Iue 3 pp DYNAMI OETIES OF HYDAUI INES FO FUID OWE TANSMISSION Thi paper preent the reult of a hydrauli line dynami propertie analyi taking into aount inertia of the fluid flowing in rigid and fleible line the ompreibility effet for thi fluid and the viou frition effet. The following are deribed and analyed: olution of the wave equation in the form binding four variable: preure and flow rate at the line input and preure and flow rate and the line output; two of the above-mentioned variable hould be regarded a independent (input) and the other two a dependent (output) the auray omparion of ditributed parameter model to lumped parameter model with regard to appliability range in hydrauli ytem analyi preure value at given intallation point a a repone to rapid valve loing (tranient repone) or to valve opening (fluid hammer effet). In thee onideration the hydrauli line i regarded a a two-port with two input and two output with a definite tranmittane matri. Main onideration onern the variable reitane model. A general olution i given a a funtion of the aplae operator. Adaptation of the hydrauli line model to the turbulent flow and to the flow through a line with elati boundarie i given. Introduing appropriate implifiation into the variable reitane model the ontant reitane model and lole line model i obtained. Alo general olution for three different lump parameter model are preented. Auray of individual method are ompared with eample for eleted ytem. eyword: fluid power tranmiion delivery of a pump flow rate preure flutuation hydrauli line ompreibility of hydrauli fluid viou frition of hydrauli fluid hydrauli reitane inertane apaitane fluid ontinuity equation equation of fluid motion fluid hammer.. Introdution ontemporary fluid power tranmiion mut feature high operating peed high auray and optimal energy onumption. To guarantee thee feature it i neeary to develop deign method aounting for engine dynamial propertie and alo during engine analyi and ynthei. Unauthentiated Download Date 7//8 :6 AM

2 8 eek UŁANOWI Fluid power tranmiion are deigned onerning the following: Hydrauli reeiver (atuator or hydrauli atuator) with a ontrol unit and it upply unit form a ingle ompat omponent (olution implemented in heliopter) Hydrauli reeiver i loated at an often ignifiant ditane from it ontrol and upply unit (olution implemented in airplane). In the firt ae airraft hydrauli ytem an be treated a tationary ytem with lumped parameter and phyial phenomena taking plae in thi ytem an be deribed by mean of a mathematial model nonlinear or linearied determiniti or probabiliti depending on whether input and diturbane ignal are immanently determiniti or tohati. In the eond ae there i a need to onider the hydrauli ytem a a ditributed parameter ytem. Uually during the deign routine the hydrauli ytem i regarded a a tationary ytem with lumped parameter and it dynamial propertie are not taken into aount in the omputation. Dynami propertie of hydrauli ytem (epeially airraft hydrauli intallation) generate ontinuou or tranient pulation of iruit preure and flow rate whih aue mahinery intabilitie fatigue of material and other uh harmful phenomena. Intenity of thee phenomena largely depend on working fluid flow in the hydrauli line onneting upplie with eeutive mehanim. Hydrauli line harateriti are related to their geometri length and preure or flow rate rie peed at line input or output. Maimum rie rate of thee ignal orrepond to the maimum preure and flow rate wave frequeny preent in the ytem hene to the ytem tranmiion band. nowledge of hydrauli line dynami propertie i unependable for analytial aement of preure pulation intenitie preent in the hydrauli ytem thereby allowing deign of the ytem in uh a way a not to admit harmful pulation into the ytem. In thi paper an attempt i made to analye and deribe the following problem whih are eential from a pratial point of view: ) Solution of wave equation in the form binding four variable: preure and flow rate at the line input and preure and flow rate and the line output; two of the abovementioned variable hould be regarded a independent (input) and the other two a dependent (output). ) Auray omparion of ditributed parameter model to lumped parameter model with regard to appliability range in the hydrauli ytem analyi. Thi omparion i of ubtantial importane from a purely pratial point of view ine more aurate model are more omple omputationally. nowledge of appliability range of poible implet model i thereby adviable. 3) reure and flow rate pulation damping in power hydrauli ytem. Unauthentiated Download Date 7//8 :6 AM

3 Dynami propertie of hydrauli line ) reure value at given intallation point a a repone to rapid valve loing (tranient repone) or to valve opening (fluid hammer effet).. Model for working fluid flow in hydrauli line In the ituation where a hydrauli reeiver i loated at ome often ignifiant ditane from it ontrol and upply unit (olution implemented in fluid power tranmiion) the hydrauli line hould be regarded a a line in whih the inertia of the fluid flowing in the line ompreibility effet for thi fluid and viou frition effet have eential impat on the hydrauli ytem dynami. iterature analyi [ 9 7] reveal that variou mathematial model have been adopted for hydrauli line dynami aounting for inertia of the fluid flowing in the line ompreibility effet for thi fluid and viou frition effet. A ommon feature of thee model i the aumption of one-dimenional flow in the hydrauli line. In maority of ae the hydrauli line i deribed by mean of partial differential equation hene thee model are model with ditributed parameter. The oldet and mot frequently ued model i the lole line model the o-alled Żukowki line [ ]. For thi ae in the equation of motion and in the ontinuity equation (DE ) deribing hydrauli line dynami the term ontaining viou frition are negleted. Nonviou fluid aumption implifie omputation ubtantially but in many ae it lead to erroneou reult. The eond popular model i the o-alled line with ontant reitane (hydrauli reitane) [ 7 3 6]. In thi ae in the equation of motion an additional term proportional to fluid veloity in the line i taken into aount; thi redue to aounting for viou frition effet in the form of reitane. The reitane value i determine by the Hagen-oieuille law. The other model i the line with lumped and dipered parameter preented by heng-uo Weng [5]. Thi model employ the Żukowki line where at the input and output there i a lumped reitane whih equal half of the reitane of the whole line. Therefore the viou frition effet i aounted a a lumped parameter and the inertia and ompreibility effet are aounted a dipered parameter. Yet another hydrauli line dynami model i the line model with frequenydependent reitane [3 4 8]. Here one take into onideration the veloity profile hange depending on the frequenie of preure and flow rate wave along the line and the effet of thi hange on the hange of reitane (viou frition effet) a a funtion of frequeny. In the ituation where the frequeny band tranmitted by the hydrauli ytem i mall uh line model are utilied for whih all three effet i.e. viou fri- Unauthentiated Download Date 7//8 :6 AM

4 eek UŁANOWI tion inertia and ompreibility are inorporated in the form of lumped parameter: reitane inertane and apaitane repetively [ 6 ]. Among omputational method applied by individual author to olve line wave equation three bai method an be ditinguihed: method of harateriti variable eparation and operator alulu. The operator alulu i ued in omputation mot ommonly. Hyperboli funtion preent in wave equation are a oure of omputational diffiultie thu in order to implify omputation approimating formula are ued. laial approimation onit of ubtituting hyperboli funtion for their erie epanion where only the firt epanion term are retained. However thi method i unatifatory for two reaon: approimating funtion root do not oinide with hyperboli funtion root and tability determination of the hydrauli ytem baed on uh approimation in unertain ine it frequently happen that untable model turn table and via vera. rodut approimation baed on the ubtitution of hyperboli funtion in the wave equation for infinite produt of fator whoe root oinide with root of hyperboli funtion i free of thee defet. Approimation auray depend on the number of retained epanion term. The number of term of the produt epanion to be ued depend on the frequeny band tranmitted by the ytem on the input ignal frequeny band and on the length of the line. In a typial hydrauli ytem to determine the repone to ontrol ignal (of narrow frequeny band) the approimation degree n i uffiient [4] and to determine the repone to pump-generated flow rate pulation the degree of approimation n hould be taken [4]. For tranmittane where hyperboli funtion are preent in the denominator only the approimation degree hould be higher than for tranmittane with hyperboli funtion in the denominator a well a in the numerator. A full deription of the produt approimation appliability rule for the model with dipered parameter and of auray degree of uh approimation depending on the number of epanion term retained i preented in [3 4 ]. 3. Hydrauli line with dipered parameter a a iruit model In order to derive general equation for the hydrauli line with dipered parameter the following i aumed: Ma fore ating on working fluid in the line are negligible Influene of the end of line input and output on the ytem dynami i negligible Unauthentiated Download Date 7//8 :6 AM

5 Dynami propertie of hydrauli line... The flow in the hydrauli line i laminar and aially-ymmetri The ratio of the working fluid peifi heat at ontant preure to the working fluid peifi heat at ontant volume i equal to unity Working fluid i Newtonian Working fluid temperature i ontant along the hydrauli line Ditribution of preure aro the line ro etion i ontant Outer radiu of the line i ontant. Aording to the aumption made hydrauli line propertie are fully determined by the following DE ytem in ylindrial oordinate []: Fluid ontinuity equation (onervation of ma) ρ ρrvr ρv t r r () where: ρ working fluid denity r hydrauli line running radiu hydrauli line aial oordinate V r V veloity omponent in the radial and aial diretion repetively Navier-Stoke equation of fluid motion ( ai) V V V p V µ ρ Vr V µ r t r 3r r 3 r r equation of energy onervation (fluid tate equation) V r V 4 r ( rv ) r () ρ ρ p B (3) where: B fluid ompreibility modulu. Auming that a ine preure wave propagate in the line the momentary preure value at point i given by [4] p p in t (4) where: p initial preure value at point wave frequeny veloity of the wave propagating along the line. Unauthentiated Download Date 7//8 :6 AM

6 eek UŁANOWI The equation from () to (3) are nonlinear. Taking into aount that the length of preure wave in the hydrauli line (guide) are ubtantially larger than the inner radiu of the line in the frequeny range preent in hydrauli ytem the above equation an be linearied. For lineariation the flow parameter hould be epreed in the form of um of teady value and deviation ρ ρ ρ V V V p p p (5) where: ρ ρ ρ V V V fluid denity it teady value and deviation from the teady value aial veloity omponent it teady value and deviation from the teady value p p p preure it teady value and deviation from the teady value and it mut be aumed that V «. After lineariation the equation () for ondition [4] π» r w take the form ρ V ρ t (6) where r w hydrauli line outer radiu. The veloity at whih the wave propagate along the line i a omple value of magnitude maller that the peed of ound in an ideal fluid. utting the denity differential from the tate equation (3) into (6) one obtain ρ p V ρ B t (7) Aounting for the ondition [4] π» r w the equation () redue to V p 4 V µ V ρ µ r (8) t 3 r r r Unauthentiated Download Date 7//8 :6 AM

7 Dynami propertie of hydrauli line... 3 utting a B ρ into (7) and uing the relation (4) and the equation (8) the following i obtained t p p t a p V t a p V ρ ρ o in o (9) When we take into onideration the ondition [3] ρ 3 4 µ a «(aeptable for hydrauli fluid ued in hydrauli intallation) in the equation (8) we have 3 4 V µ «p. A a reult the equation (8) take the form r V r r r p t V µ ρ () Applying the aplae tranform with repet to time with ero initial ondition p() and V ( r ) to the equation () we get V r V r r V ρ ν () Introduing a new variable to the equation () V U ρ () the following equation i obtained U r U r r U ν (3) It i a ero order modified Beel equation. It olution i a ero order Beel funtion and we onider only the Beel of the firt kind whih i finite at r [4] Unauthentiated Download Date 7//8 :6 AM

8 4 eek UŁANOWI v r F U (4) The term preent in equation (4) and ontaining the differential operator in the quare root funtion i a frational derivative. Subtituting U from the equation (4) into the equation () one obtain r F V ρ ν (5) The equation (5) hould atify the boundary ondition V at r r w [3]. It i atified if ν ρ r F w (6) Hene ν ν r r F V w (7) Multiplying both ide of the equation (7) by πr and integrating with repet to r from to r w one obtain π F r w (8) where: r w ν ;. r w Vrdr π omputing F() from the equation (6) and inerting it into equation (8) one obtain (9) Unauthentiated Download Date 7//8 :6 AM

9 Dynami propertie of hydrauli line... 5 where: ( ) () and ρ. π r w After applying the aplae tranform with repet to time with ero initial ondition p() and V ( r ) to the equation (7) then multiplying the reult by πr and integrating with repet to r from to r w one obtain ( ) Y ( ) () where: Y() () and w πrw πr a ρ B The equation (9) and () and the differential equation of the hydrauli line with variable reitane. Thee equation an be written in the matri form ( ) ( ) Y ( ) (3) Epreing the hydrauli line equation in the matri form enable it diret appliation for building a blok diagram deribing the hydrauli ytem dynami and allow employment of the automati ontrol theory for analyi and ynthei. By integrating the equation (3) with repet to from do l we obtain a olution for thi equation on the omple plane ( ) ( ) hγ h hγ Γ hγ l ( l ) (4) where: ( ) l Y Γ (5) Unauthentiated Download Date 7//8 :6 AM

10 6 eek UŁANOWI i the propagation operator and Y (6) i the wave impedane. By inerting the relation () and () into (5) and (6) the following form of the propagation operator and the wave impedane i obtained Γ T (7) (8) where: a l I T w r a π ρ In the hydrauli ytem analyi of different form of the equation (4) are ued. The equation (4) an be epreed in the form of the following two-port tranmittane matri: admittane form Γ Γ Γ l h h h l (9a) Unauthentiated Download Date 7//8 :6 AM

11 Dynami propertie of hydrauli line... 7 impedane form ( ) ( l ) h Γ ( ) ( ) h Γ ( ) h Γ ( ) ( ) l (9b) Epreing the above equation in the matri form allow u to employ the automati ontrol theory for analyi and ynthei. The epreion (7) and (8) determining the propagation operator and wave impedane are imilar to thoe obtained by F. Brown [3] but the author of [3] preent only the olution for a peial ae i.e. for a line with no refletion. In the ae of fluid flow in a line with elati boundarie the elatiity of the boundarie an be aounted for by introduing into the equation () an effetive fluid ompreibility modulu in the form B B B where: B equivalent fluid ompreibility modulu B line boundarie ompreibility modulu and in the ae of rigid (metal) line the boundary ompreibility modulu an be written in the form B r rw rw E where: r line outer radiu r w line inner radiu In the ae of fleible line analytial determination of B i etremely hard due to ophitiated truture of the line boundarie. However eperimental determination of equivalent ompreibility modulu for a peifi type of the line filled with a given type of the working fluid doe not aue diffiultie. Aiming at minimiation of hydrauli intallation weight with imultaneou preure rie (high preure upplie) may indue turbulent flow in the line. Eitene of uh a flow aue the line tati preure loe to grow. The turbulene an be taken into onideration by auming that in the equation () there i r w πµ V r dr r r 8µ q rw whih i eat for a teady flow of an inompreible fluid (Hagen oieuille law). Then the equation () take implified form Unauthentiated Download Date 7//8 :6 AM

12 8 eek UŁANOWI where: 8 4 πr w µ. p q q (3) t It i the equation of motion for the hydrauli line model with ontant reitane. The erie impedane obtained from (3) by appliation of the aplae tranform i of the form (3) Auming µ (nonviou fluid) in the equation (3) one get the mathematial model of the lole line where (3) 3 utting (3) or (3) into formula (5) and (6) in the plae of () the following i obtained a line with ontant reitane 8 Γ T T (33) 8 (34) where i nominal harateriti impedane for the line a lole line T Γ (35) (36) In the analyi of the model with ditributed parameter emphai ha been put on determination of the frequeny repone of the ytem. It i utified by the following: The already developed analyi and ynthei method in the automati ontrol theory are baed mainly on frequeny harateriti Frequeny harateriti give full although indiret information on dynamial propertie of the ytem omputation of frequeny harateriti in the ae of tranmittane with hyperboli funtion i le problemati than omputation of tranient repone Unauthentiated Download Date 7//8 :6 AM

13 Dynami propertie of hydrauli line... 9 Many input ignal (diturbane) affeting hydrauli ytem are ine in nature (in general um of trigonometri funtion); thi group of diturbane inlude flow pulation generated by hydrauli pump; in uh ae frequeny harateriti give an indiret information on ytem behaviour. Frequeny harateriti are obtained by inerting into the tranmittane or (equivalently) by ehanging the aplae tranform for the Fourier tranform. Spetral form of the propagation operator and of the wave impedane for the variable reitane model (7) and (8) are a follow β α T Γ (37) δ γ (38) where: real part of the propagation operator in the petral form equal 4 artg in b a a b T α imaginary part of the propagation operator in the petral form equal 4 artg o b a a b T β Unauthentiated Download Date 7//8 :6 AM

14 3 eek UŁANOWI real part of the wave impedane in the petral form equal γ b oartg a 4 a b imaginary part of the wave impedane in the petral form equal δ b inartg a 4 a b a b ( e ) df ( d ) ( f ) de ( d ) ber bei d ber bei e ber bei f ber bei Appliation of the above relation at high value of i inonvenient ine amplitude of ber and bei funtion and their derivative are quikly riing funtion. Thu during omputation there i a need to handle large number. Thi diffiulty an be bypaed uing the large-frequeny approimation given by F.T. Brown [3] Unauthentiated Download Date 7//8 :6 AM

15 Dynami propertie of hydrauli line... 3 Γ T (39) 3 8 (4) Fig. and Fig. preent plot of real and imaginary part of the propagation operator and the wave impedane a funtion of frequeny; the plot are given in dimenionle oordinate. Fig.. Frequeny dependene of real and imaginary part of the wave propagation operator Unauthentiated Download Date 7//8 :6 AM

16 3 eek UŁANOWI Fig.. Frequeny dependene of real and imaginary part of the wave impedane In order to implify omputation of frequeny harateriti and tranient repone produt approimation i ued. Thi approimation redue to ubtitution of the hyperboli funtion for the aplae operator polynomial. The root of thee polynomial oinide with the root of the hyperboli funtion. The approimation auray degree an be aduted by retaining the deired number of term in the produt epanion. rodut epanion of hyperboli and Beel funtion are preented in [3 4] and therefore are not given in thi paper. 4. Hydrauli line with lumped parameter a a iruit model In the deription of the hydrauli line dynamial propertie it i aumed that all ytem parameter are lumped i.e. reitane l Unauthentiated Download Date 7//8 :6 AM

17 Dynami propertie of hydrauli line inertane l apaitane l For the line with lumped parameter there are three equivalent diagram poible. Equivalent diagram for the lumped parameter line a a ymmetrial two-port i hown in Fig. 3. Equivalent diagram for the lumped parameter line a a two-port with hared reitane i depited in Fig. 4. Equivalent diagram for the lumped parameter line a a two-port with apaitane at the output i hown in Fig. 5 Fig. 3. Equivalent diagram for the lumped parameter line a a ymmetrial two-port Fig. 4. Equivalent diagram for the lumped parameter line a a ymmetrial two-port with hared reitane Fig. 5. Equivalent diagram for the lumped parameter line a a ymmetrial two-port with apaitane at the output Unauthentiated Download Date 7//8 :6 AM

18 34 eek UŁANOWI The line with lumped parameter a a ymmetrial two-port hown in Fig. 3 i deribed by the following equation in the matri form l l ; 4 4 ; 3 (4) The line with lumped parameter a a two-port with hared reitane hown in Fig. 4 i deribed by the following equation in the matri form l l ; 4 ; (4) The line with lumped parameter a a two-port with apaitane at the output hown in Fig. 5 i deribed by the following equation in the matri form l l ; ; (43) The model for the hydrauli line with lumped parameter given by the equation (4) and (43) are nonymmetri. 5. Eample of analyi of a hydrauli ytem a an impedane-loaded line The implet hydrauli ytem model whih i uitable for deription of the dynami of many real ytem i the impedane-loaded line. Impedane of the load i given by the equation l l l (44) From the equation (4) and (44) one obtain three tranmittane for the impedane-loaded line Unauthentiated Download Date 7//8 :6 AM

19 Dynami propertie of hydrauli line ( ) ( ) l ( l ) ( ) hγ hγ hγ ( l ) ( ) l hγ l l l hγ hγ hγ hγ (45) (46) (47) The tranmittane defined by the relation (45) i the output impedane of the hydrauli ytem. In the ubequent we onider everal peial ae of tranmittane given by the relation (45) (46) (47). A) Open line or line loaded with a tank with ontant preure In the ae of the open line or the line loaded with a tank with ontant preure the impedane l (). Hene the relation (44) (45) and (46) take the form ( ) ( ) thγ (48) ( l) ( ) ( l ) ( ) hγ (49) (5) B) loed line In the ae of the loed line the impedane l (). Hene the relation (44) (45) and (46) take the form ( ) ( ) thγ ( l ) ( ) hγ (5) (5) Unauthentiated Download Date 7//8 :6 AM

20 36 eek UŁANOWI ( ) l (53) ) Open line terminated with a permanent hoke In the ae of the line terminated with permanent hoke the relation between the flow rate and the preure at the end of the line i a follow ( l t ) G p( l t ) q (54) The equation (54) i nonlinear. However fouing on mall deviation from the teady tate the linearied equation an be ued whih in the operator form i given by ( l ) ( l ) (55) where: G p ( l t ) From the equation (44) and (55) one obtain l (56) 5 hene the relation (45) (46) and (47) take the form ( ) ( ) hγ hγ hγ hγ (57) ( l ) ( ) hγ hγ ( l ) hγ ( ) hγ (58) (59) Unauthentiated Download Date 7//8 :6 AM

21 Dynami propertie of hydrauli line db M.[.(Ow) (Ow ] H o o.34 o topn. 45 arg [.(w) (w)] 5 H Fig. 6. Impedane-loaded line frequeny harateriti given by the relation (57) Fig. 7. Impedane-loaded line frequeny harateriti given by the relation (58) Unauthentiated Download Date 7//8 :6 AM

22 38 eek UŁANOWI Fig. 8. Impedane-loaded line frequeny harateriti given by the relation (59) Thee onideration onern mainly the line model with frequenydependent reitane whih i mot eat from the theoretial point of view. For omputation the following line parameter are aumed: T or.34 or.5. Fig. 6 7 and 8 how the frequeny harateriti orreponding to the tranmittane (57) (58) and (59). Fig. 9. Frequeny harateriti for model of the line with dipered parameter Unauthentiated Download Date 7//8 :6 AM

23 Dynami propertie of hydrauli line Fig. 9 ompare the frequeny harateriti of variou line model with dipered parameter and the variable reitane model given by the tranmittane (58) with.34. To plot the envelope for maima and minima of the amplitude-frequeny harateriti dahed line were ued. It hould be noted that only in the ae of the variable reitane line did the amplitude of ubequent reonane peak drop. Fig.. Frequeny harateriti for model with lumped parameter Fig. how the frequeny harateriti for the dipered parameter line model. The tranmittane orreponding to thee harateriti are alulated from the equation (4) (4) (43) and (55). Thee tranmittane and the lumped parameter line tranmittane are a follow ymmetrial two-port l (6) Unauthentiated Download Date 7//8 :6 AM

24 4 eek UŁANOWI two-port with hared reitane ( l ) ( ) 4 (6) two-port with apaitane at the output ( l ) ( ) (6) From Fig. it follow that the mot aurate model i the model of the line with lumped parameter treated a a ymmetrial two-port. 6. Tranient repone In thi etion we preent the problem of tranient preure determination at given intallation point a a repone to rapid valve loing or opening (fluid hammer effet). nowing the tranmittane and the aplae tranform of the input ignal the tranient repone an be obtained by omputing the invere aplae tranform of the produt for the afore-mentioned funtion. In ae where the tranformed funtion i ompliated and ontain hyperboli funtion with the argument being a ophitiated funtion of eeuting thi operation i partiularly diffiult. The implet ae for omputing the invere aplae tranform i the lole line model. In the eample below we omputed the preure tranient at the line output a a repone to the tep preure hange at the input of the hoke-loaded line. The aplae tranform of the input ignal i of the form ( ) (63) Inerting (63) into the equation (58) with the propagation operator and the wave impedane given by the equation (35) and (36) the following aplae tranform of the preure tranient at the line output i obtained Unauthentiated Download Date 7//8 :6 AM

25 Dynami propertie of hydrauli line... 4 T T e e p ht ht p l (64) By applying the invere aplae tranform to the equation (64) one obtain the preure tranient a a funtion of time ( n n n nt t T t p t l p ) (65) Fig. how the preure tranient given by (65) for variou value in dimenionle oordinate. For < the tranient are oillatory for > the tranient are damped and in the ae of the impedane i mathed to line i.e. the wave do not reflet off the line output. Fig.. reure tranient at the output of the line terminated with a hoke a a repone to the tep preure input 7. onluion In thee onideration the hydrauli line wa regarded a a two-port with two input and two output with a definite tranmittane matri. Suh an approah allowed treatment of the hydrauli line a an element with a partiular tranmit- Unauthentiated Download Date 7//8 :6 AM

26 4 eek UŁANOWI tane matri whih an be eaily inorporated into blok diagram deribing the dynami of the ytem. Epreing the hydrauli line equation in the operator form allowed diret appliation of automati ontrol theory law in the analyi and ynthei. Main onideration were onerned with the variable reitane model. A general olution wa given in the form of the aplae operator. Adaptation of the model for the turbulent flow and for the flow through the line with elati boundarie wa given. The hydrauli line wa harateried by mean of the following hydrauli impedane (omple reitane) element: erie impedane oniting of an inertane and a reitane per unit length whih aount the inertia and viou frition effet and of a hunting admittane per unit length (harateried by apaitane) whih aount fluid ompreibility effet. After integration of the wave equation with repet to the line length the ombination of the above parameter yielded two bai obet harateriing model for the hydrauli line with dipered parameter: the propagation operator and harateriti impedane. The firt of thee obet deribe the time delay for ignal tranmiion along the line and damping and diperion of preure and flow rate wave. The eond obet i the line internal impedane viewed from the point of view of the load. Introduing appropriate implifiation into the variable reitane model the ontant reitane model and lole line model wa obtained. Two hydrauli line pulation damping method were onidered: one by mathing the load impedane to the harateriti impedane and two by hunting the line with a hydrauli aumulator pulation damper. The main tre wa put on hydrauli line frequeny harateriti determination. From among the model with dipered parameter the omputationally implet one i the lole line model. The lole line model i reommended for analyi of tranient proee (the o-alled fluid hammer effet) ine it i eay to ompute the invere aplae tranform. The error ariing due to negligene of diipative loe i not o ignifiant beaue it move the reult to the afe ide. It eem that model for the line with variable reitane depit the hydrauli ytem propertie in a mot atifatory way. To onfirm thi hypothei it i neeary to ompare theoretial formula with eperimental tet reult. Eperimental tet to be onduted hould onern line terminated with a permanent hoke or a hoke and a hydrauli aumulator and a ontant preure upply terminated with a valve with adutable or ontant ro etion or with a valve and an aumulator. The bai riterion for hydrauli ytem dynamial propertie optimiation hould be to math the load impedane to the hydrauli line harateriti impedane. Thi riterion an be formulated in a implified form: the load reitane hould equal the nominal hydrauli line harateriti impedane. Adoption of thi riterion demand to limit the maimum veloity of the line flow and the maimum Unauthentiated Download Date 7//8 :6 AM

27 Dynami propertie of hydrauli line tati reitane of the hydrauli line. In the ae when impedane mathing i not poible the preure pulation an be minimied by introduing an aumulator into the line; the apaity if the aumulator hould be a mall a poible. It mut be emphaied that appliation of pulation damper aue the reation time of eeutive element to drop and reult in lower rigidity of the ytem. Bibliography. Atarita G. Marrui G.: riniple of non-newtonian fluid mehani. ondon N.Y. MGraw-Hill Brodgett.E. ing.e.: Hydrauli ytem imulation uing time-delay element. ournal of Bai Engineering une Brown T.F.: The tranient repone of fluid line. ournal of Bai Engineering Deember Brown T.F. Nelon S.E.: Step repone of liquid line with frequeny dependent effet of vioity. ournal of Bai Engineering une hange uo Weng: Tranmiion of fluid power by pulating flow onept in hydrauli ytem. ournal of Bai Engineering une horin A.. Morden.E.: A mathematial introdution to fluid mehani. N.Y. Springer Gelman A.S. Marowa N..: Nietaonarnye kolebania gidrawliekih linii. Maynotroenie / Holmboe E.. ouleau W.T.: The effet of viou hear an tranient in liquid line. Brown T.F.: The tranient repone of fluid line. ournal of Bai Engineering Marh Motkow M.A.: rikladnaa gidromehanika. Goeniergoiydat Oldenburger. Goodon.: Dynamika gidrawlieko linii. Tehniekie Sriedtwa Awtomatiki II ongrie IFA Nauka auhang u.: Introdution to the mehani of viou fluid. MGraw-Hill Book omp Serwah A. wolak E.: Analia hydraulinego ailaa tałoiśnieniowego aworem prelewowym. IV raowa onferena Automatyki vol. VI raków Streeter V..: Waterhammer analyi with nonlinear fritional reitane. roeeding of the firt Autralian onferee Hydrauli and Fluid Mehani ergamon re Tarko.M.: ayiot dawlwnia w gidropriwodie pri nieutanowliennym reżimie. Stanki i intrument / Tarko.M.: Udarnoe dawlenie pri otanowkie gidropriwoda. Stanki i intrument / Walker M.. irkpatrik E.T.: Viou diperion in waterhammer. ournal of Bai Engineering Deember aharow.e. Baranow W.N.: O wlianii pulai dawlenia na dwiżenie pornia. niewno i giwroawtomatika Nauka 984. Unauthentiated Download Date 7//8 :6 AM