基于Matlab的齿轮泵脉动性的分析与优化

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Sep．2013机 床 与 液 压Hydromechatronics Engineering Vo1．41 No
． 18DOI：10．3969／j．issn．1001—3881．2013．18．002Analysis of the Optimization of Gear Pump PulsationBased on M atlabFENG Kailin ，ZHANG Guan，WANG Fei，LI Jin，ZHANG ZhonggongShandong Unive~i###ofScience and Technology，Qingdao 266590，China1．IntroductionsAbstract：The flow of pulsation is an important indicator of measuring gear pump performance
.

Through the theoretical analysis of the gear pump causes of pulsation
， the author uses the MAT.

LAB to simulate it and the MATLAB optimization toolbox fiminimax double objective optimizationfunction to optimize it．Through the contrast we get result that the performance of the gear pumphas been optimized in a certain extent.

Key words：gear pump，pulsation，optimizationGear pump is a kind of commonly used hydrau.

1ic component in hydraulic system． Because of thesimple structure， good fabrication procedure，self-priming ability，s~ong abiIity to resist pollution andthe reliable work．it has been widely used．Howeverdue to the big noise of the gear pump and unbalancedradial force caused by flow ripple．it has some limitsand cann’t be used widely．So the flow pulsation ofthe gear pump research and parameter optimizationfor wider application of the gear pump has importantsignificance．In this paper，the author analyzes theexternal gear pump flow pulsation and makes use ofthe Matlab simulation and parameters optimization.

2．Instantaneous delivery calculationsThe volume change method is adopted to analysethe instantaneous delivery of two involute straight gearexternal gear pump.

In the process of gear meshing，kfor the meshpoint，the discharge of oil cavity by driving wheel IReceived：2013—04—22FENG Kailin，Doctor，Professor．E—mail：fk162### sina
． co m tooth profile m，n，g，k，and the driven pulley Ⅱ sur-rounded by tooth profile k，g ，n ，m ． When geartransmission，the tooth profile mn and m n compressthe oil of discharge oil cavity，make its volume smal—ler，another part of the gear tooth profile and gt}cexpand the oil discharge volume in the rotation，andthe rest of the tooth will be completely surrounded byhigh pressure oil does not participate in the work，donot effect the discharge oil． Obviously，the formercompressed volume is greater than the latter expandedvolume，the overall result is the volume of dischargeoil cavity decrease constantly，and constantly put theoil into the high pressure pipeline.

Analysis of driving wheel I tooth profile fek andthe driven puleyⅡ tooth profile ke _厂forming thesuction chamber with similar method of，it can beseen that the suction cavity is expanded constantly ，sucks the oil from the low pressure pipe.

Consider two gears is the same，then in the Fig.

1 when the gears turning angle △ ，tooth profilemnand m n reduce the discharge oil cavity volume asfollows：A VI=b(r：一r；)△ (1)At the same time，due to the rotation of thetooth profile gk and g k，the oil discharge cavity vo1.

ume expanding as follows：△ =6( P +P； 2 r，2)A4, (2)Above you can see，the position of mesh point k8 Hydromechatronics Engineeringis changing constantly in the working process，so△don’t like AV1 changing with angle evenly，it is thefundamental reason of gear pump flow nonuniformity.

Fig．1 gear pump working principleBy the above two equation can be drawn fromthe oil volume discharged by oil discharge cavity：△ =△ ～△ =6(，2。一 ㈩The above equation on the time derivative，thetheory instantaneous delivery expression is：g =dV
= 【2r2 一(JD +pi)】 (4)According to the principle of gear meshing，tryto find out the relationship between instantaneous de—livery with changing turning angle．Amplify the trian—gle formed by the two center of the gear circle O1，O2in the figure，and mesh point ，as shown in the Fig.

2，r is standard pitch circle radius， is the radius ofbase circle，r
。
is the radius of addendum circle，linesegmentpk(its length in the figure isf)is used formeshing pointkto the node P distance．Because of thegear，pk is part of the mesh line，the trajectory ofmesh pointkare also on the efective line N1Ⅳ2 at thesame time.

Fig．2 involute gear mesh point changesAccording to the cosine theorem，in AO1pk andAO2pkP = 2+pk 一2r pkcosa (5)P；=r +pk 一2r pkcosflPut the above two equation into equationis available：q = (r 一r 一pk。)If pk expressed by f，then：q=bw(r：一r 一f )(6)(4)，(7)(8)The above equation expresses the instantaneousdelivery of any type tooth profile gear pump changingover meshing point changing．For involute gear，hasthe following features：f=pk： (9)Put equation(9)into equation(8)，the func—tion of the meshing point location pk transform as thefunction of turning angle：q=bto(r 一 2一r26 ) (10)3．Theory of flow calculationFrom Fig．2，each pair of gear teeth in meshingprocess，when the symmetry in the nodes P，alongthe efective meshing line N1N2(N1N2 印6，P6 isgear base circle，s is contact ratio of gear meshing；P6 Ti'mcoso／，pressure angle o／：20。，mis the mod-ule of gear)mesh，from 一— 6(coresponding to the厶1first mesh point N1)to+ 6(corresponding to justquit the meshing pointⅣ2)process，forming a dis-charge of oil phase．Suppose to discharge the liquidvolume in the cycle is△ ．then according to the e—quation(10)，and considering af=ogr6dt that we canget：AV=J ：J 6 (r 一r 一厂 )d =1bJ=‘c 2一f 2)2-PbThe displacement of the gear pump V is that thegear pump discharge the liquid volume by rotating aweek．In a week on a total of z tooth meshing drain-age，so available：V=△ =_bz f( — 2一f。)df(12)rt JDue to the base circle pitch P6：27rr6／z，afterabove equation integral，can be concluded that theoryexpression of displacement is：2耵6(r 一r ) (13)FENG Kailin，et al：Analysis of the Optimization of Gear Pump PulsationBased on Matlab 9Theory offlow：g = = r2
。
一 2
一 )4．The quality of flow(14)Evaluate the quality of hydraulic pump instanta·neous delivery generally using the flow of nonunifor—mity coeficient 。 gm“一qmin g￡ ，and the flow pulsa-tion ratefq：zn／60.

Ideally， q close to zero，fq also close to zero，represents the gear pump has best flow quality.

5．Simulation and parameters optimizationwere carried out on the gear pumpOne gear pump’s rated pump displacement is10 ml／r，its parameters are Fig．3．n=2 500 r／min，according to q bw(r 一 2一 2 )，get a instanta—neous delivery rate curve in the process of a pair ofgears meshing，using of Matlab simulation getting agraph：q
= g 一q i 448—395q 430 =0．12fq=zn／60：18×2 500／60=7504504404304204lO4003900 l 2 3 4 5 6 7gear around the corner／radFig．3 a instantaneous flow rate curve in the process ofa pair of gears meshingUse the Matlab optimization toolbox to optimize.

To q meet close to zero，andfq also close to zero，using multi—objective function optimization：Make[b z m]=[ 1 2 3]The objective function：mi ( )=6 ： 570x 1x370．74x1 ； 262xl 2x；+The constraint conditions：min，2( )= =2
— 50 0x
一
21)The root cuting constraints：According to thegear，it does not produce serious undercut：14一 2≤0.

2)The tooth width constraints：Increasing toothwidth can improve the bearing capacity of gear，but italso will increase the bearing load and improve thetooth face contact precision，so the general require—ments b≤9 m： 1—9 3≤O.

3)Top circle thickness constraints：The positivedisplacement gear is adopted by outside meshinggear，gear tooth top tip gradually， tooth thicknessshould satisfy：0．15x3一s≤0，here to take =2．2mm，0．15x3—2．2≤0，(s as addendum thickness，s。=sr／r一2 (inva 一inva)).

4)Wide tooth addendum circle size constraints：Generally，the selection of the ratio of tooth widthand addendum circle size range is between0．2～0．8，getting constraint conditions： 2 3一 3—5xl≤0， l一0．8x2 3—2．4x3≤0.

5)Drive requirements：Power transmission，thegear module should be greater than 2 mm，2一 3≤0.

6)The displacement constraints：Theory shouldbe greater than the nominal displacement displace—ment：10 000—6．28x1X,2X ≤0.

7)The velocity constraints：To reduce the vibra—tion and noise of gears，the speed of addendum circleshould beles1 s thanthe a1owed limit， _10≤0，0．13x2x3—10≤0.

8)The strength of gear teeth constraints：Gearteeth contact stress and bending stress is less than theallowable value，orⅣ = 2．5ZE ≤ 【orH】F ：
2
—
H
-
TY丌FaYs
a≤ 【or
F
] — — ～ L F J
d ‘1．03一 1≤0，2700一 ；≤0Using the fiminimax function of Matlab optimiza.

tion toolbox optimization， the main steps：Write aMatlab M file：Nonlinear constraints：function[c，ceq]=NonLinConstr(x)xl=x(1)；x2=x(2)；再Lf．∞．一吕J／享0 呐盘0 ll暑I暑呐ulHydromechatronics Engineeringx3=x(3)；e(1)：x2：Ic x3一x3—5 xl；c(2)=xl一0．8：l=x2；l：x3—2．4％x3；c(3)=一6．28：lxl x2 x3^2+10000；c(4)=0．13 x2 x3—10；c(5)：一x2 x3"2+2700ceq=[]；The objective function： ·function f=myfun(x)xl=x(1)；x2=x(2)；x3=x(3)；f(1)=570 x1 x3 2／(262 xl x2 x3 2+70．74 xl x3"2)；f(2)：2500 x2／60；Optimization program：x0=[10，18，3]；A=[0—1 0；1 0—9；0 0 0．15；0 0—1；一1 0 0]；b=[一14；0；2．2；一2；一1．03]；lb：[4；14；2]；ub=[9；40；10]；[x，f]=fminimax( myfun ，x0，A，b，[]，[]，Ib，ub， NonLinConstr ，[])The results of optimization： 1= 9．07， 2=14．67， 3=3．46，The objective function：6。=0．14，fq=611，b=9 mm，z=15，m=3 mm；The originalvalue：b=10 mm，z=18，m =3 mm ，6 =0．12，fq=750.

6．SummaryOptimization results by MATLAB shows that theflow quality of external gear pump is improved，ideal—ly 6
q andfq are close to zero，but 6q andfq contrary toa rule change，if the demand is smaler，then hadto alowfq increasing．The improvement of flow quali—ty t for a wider range of application of external gearpump has important practical significance． Appliea-tion Matlab optimization toolbox to design，it makesthe working strength is reduced，improves work effi—ciency，but also improves the design accuracy andreliability.

References：基于Matlab的齿轮泵脉动性的分析与优化冯开林 ，张 冠，王 飞，李 进，张中供山东科技大学，山东 青岛 266590LI Zhuangyun．The hydraulic components and systems[M]．Beijing：mechanical industry press，201 1：26— 29.

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CHENG Daxian．Handbook of mechanical design『M].

Beijing：chemical industry press，2001：19—129.

[USA]Edward B M．MATLAB principle and engineer—ing application[M]．Beijing：electronic industry press，2007：442—450.

WANG Chengfeng，ZHOU Changxun． Optimization de—sign based on MATLAB external gear pump[J]．Manu—facturing informatization，2011(7)：119—120.

ZOU Hongfeng，JIA Zhu．Flow pulsation mechanism a—nalysis and design of the gear pump[J]．Mechanical en一舀neeTs，2010(3)．47—48.

SU Xinping，GUO Zhong，WEI Xiaognang ，et a1．Basedon the research of compound gear pump flow pulsationcharacteristics of MATLAB 『J 1．Journal of mechanicalengineering，2009(3)：333—345.

TONG Xinhua，LI Ailing．Based oil MATLAB externalgear pump structure optimization[J]．Machinery，2010(37)：29—31摘要：流量的脉动性是衡量齿轮泵性能的重要指标。理论分析 了齿轮泵产生脉动的原因，利用 Mat.

1ab对齿轮泵进行了脉动曲线绘制，应用Matlab优化工具箱的fiminimax函数进行 了双 目标优化，并进行 了对比。结果表明：该方法在一定程度上优化了齿轮泵的性能。

关键词：齿轮泵；脉动性；／~4E中图分类号：TH325]J 1J 1J ]J 1 J ]J ]J ]j I二