FENG Kailin ，ZHANG Guan，WANG Fei，LI Jin，ZHANG Zhonggong，，

1.Introductions

Gear pump is a kind of commonly used hydraulic component in hydraulic system.Because of the simple structure，good fabrication procedure，selfpriming ability，strong ability to resist pollution and the reliable work，it has been widely used.However due to the big noise of the gear pump and unbalanced radial force caused by flow ripple，it has some limits and cann’t be used widely.So the flow pulsation of the gear pump research and parameter optimization for wider application of the gear pump has important significance.In this paper，the author analyzes the external gear pump flow pulsation and makes use of the Matlab simulation and parameters optimization.

2.Instantaneous delivery calculations

The volume change method is adopted to analyse the instantaneous delivery of two involute straight gear external gear pump.

In the process of gear meshing，kfor the mesh point，the discharge of oil cavity by driving wheelⅠtooth profilem，n，g，k，and the driven pulley Ⅱ surrounded by tooth profilek，g'，n'，m'.When gear transmission，the tooth profilemnandm'n'compress the oil of discharge oil cavity，make its volume smaller，another part of the gear tooth profilegkandg'kexpand the oil discharge volume in the rotation，and the rest of the tooth will be completely surrounded by high pressure oil does not participate in the work，do not effect the discharge oil.Obviously，the former compressed volume is greater than the latter expanded volume，the overall result is the volume of discharge oil cavity decrease constantly，and constantly put the oil into the high pressure pipeline.

Analysis of driving wheelⅠtooth profilefekand the driven pulleyⅡtooth profileke'f'forming the suction chamber with similar method of，it can be seen 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 Δφ，tooth profilemn andm'n'reduce the discharge oil cavity volume as follows:

At the same time，due to the rotation of the tooth profilegkandg'k，the oil discharge cavity volume expanding as follows:

Above you can see，the position of mesh pointkis changing constantly in the working process，so ΔV2don’t like ΔV1changing with angle evenly，it is the fundamental reason of gear pump flow nonuniformity.

Fig.1 gear pump working principle

By the above two equation can be drawn from the oil volume discharged by oil discharge cavity:

The above equation on the time derivative，the theory instantaneous delivery expression is:

According to the principle of gear meshing，try to find out the relationship between instantaneous delivery with changing turning angle.Amplify the triangle formed by the two center of the gear circleO1，O2in the figure，and mesh pointk，as shown in the Fig.2，rwis standard pitch circle radius，rbis the radius of base circle，rais the radius of addendum circle，line segment(its length in the figure isf)is used for meshing pointkto the node p distance.Because of the gear，is part of the mesh line，the trajectory of mesh pointkare also on the effective lineat the same time.

Fig.2 involute gear mesh point changes

According to the cosine theorem，in ΔO1pkand ΔO2pk

Put the above two equation into equation(4)，is available:

The above equation expresses the instantaneous delivery of any type tooth profile gear pump changing over meshing point changing.For involute gear，has the following features:

Put equation(9)into equation(8)，the function of the meshing point locationtransform as the function of turning angle:

3.Theory of flow calculation

The displacement of the gear pumpVis that the gear pump discharge the liquid volume by rotating a week.In a week on a total ofztooth meshing drainage，so available:

Due to the base circle pitchpb=2πrb/z，after above equation integral，can be concluded that theory expression of displacement is:

Theory of flow:

4.The quality of flow

Ideally，δqclose to zero，fqalso close to zero，represents the gear pump has best flow quality.

5.Simulation and parameters optimization were carried out on the gear pump

One gear pump’s rated pump displacement is 10 ml/r，its parameters are Fig.3.n=2 500 r/min，according toq=bω(--φ2)，get a instantaneous delivery rate curve in the process of a pair of gears meshing，using of Matlab simulation getting a graph:

Fig.3 a instantaneous flow rate curve in the process of a pair of gears meshing

Use the Matlab optimization toolbox to optimize.To δqmeet close to zero，andfqalso close to zero，using multi-objective function optimization:

Make［bzm］=［x1x2x3］

The objective function:

The constraint conditions:

1)The root cutting constraints:According to the gear ，it does not produce serious undercut:14-x2≤0.

2)The tooth width constraints:Increasing tooth width can improve the bearing capacity of gear，but it also will increase the bearing load and improve the tooth face contact precision，so the general requirementsb≤9 m:x1-9x3≤0.

3)Top circle thickness constraints:The positive displacement gear is adopted by outside meshing gear，gear tooth top tip gradually，tooth thickness should satisfy:0.15x3-s≤0，here to takes=2.2 mm，0.15x3-2.2≤0，(sas addendum thickness，sa=sra/r-2ra(invαa-invα)).

4)Wide tooth addendum circle size constraints:Generally，the selection of the ratio of tooth width and addendum circle size range is between0.2～0.8，getting constraint conditions:x2x3-x3-5x1≤0，x1-0.8x2x3-2.4x3≤0.

5)Drive requirements:Power transmission，the gear module should be greater than 2 mm，2-x3≤0.

6)The displacement constraints:Theory should be greater than the nominal displacement displacement:10 000-6.28x1x2≤0.

8)The strength of gear teeth constraints:Gear teeth contact stress and bending stress is less than the allowable value，

Using the fiminimax function of Matlab optimization toolbox optimization，the main steps:Write a MatlabMfile:

The results of optimization:x1=9.07，x2=14.67，x3=3.46，The objective function:δq=0.14，fq=611，b=9 mm，z=15，m=3 mm;The original value:b=10 mm，z=18，m=3 mm，δq=0.12，fq=750.

6.Summary

Optimization results by MATLAB shows that the flow quality of external gear pump is improved，ideally δqandfqare close to zero，but δqandfqcontrary to a rule change，if the demand δqis smaller，then had to allowfqincreasing.The improvement of flow qualitytfor a wider range of application of external gear pump has important practical significance.Application Matlab optimization toolbox to design，it makes the working strength is reduced，improves work efficiency，but also improves the design accuracy and reliability.

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