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Gerotor pumps are a kind of internal gear pump, widely used in applications of lubricating system of on-road or off-road engines. Normally their outer rotors are the circular arc profile while their inner rotor profile is the conjugate curve of the outer rotor. This research focuses on how to derive the combined trochoidal profile of a gerotor pump. The hypocycloidal and epicycloidal profile of the inner rotor is generated at first. And then the profile of the outer rotor, based on the inner rotor profile, is obtained by analytical approach and computationally graphical approach. The paper provides feasibility to develop the unconventional profile of a gerotor pump.

It is well known that internal gear pumps have lower noise than external gear pumps because gears of the former rotate along the same direction, yielding lower relative velocity, while that of the latter do along the inverse direction with higher relative velocity. Moreover, the overlap coefficient of internal gear pumps is greater than that of external gear pumps, therefore the former rotates much more smoothly than the latter. Gerotor pumps belong to a kind of internal gear pumps, which are positive displacement pumps. Compared with external or other internal gear pumps, they have advantages of less components, simple structure, compact size, low noise, and low pulsation of flow rate. Therefore they are widely used in applications of lubricating system or fuel supply system of on-road or off-road engines [

The inner rotor has one tooth less than the outer rotor, and they have a special type of conjugate profiles which always ensure contact of the inner and outer rotors at several points [

This paper proposes a combined trochoidal profile, i.e. hypocycloidal and epicycloidal curve, of the inner rotor, and the outer rotor is a conjugate profile of it. The generation of the combined trochoidal profile of the inner rotor is given in Section 2. And then the outer rotor profile, based on the inner rotor profile, is developed via analytical approach and computationally graphical approach stated in Section 3. The last section is conclusion.

The profile of the inner rotor is composed of a segment of hypocycloidal curve and a segment of epicycloidal curve. In order to connect the hypocycloid and epicycloid in radial direction, their base circles have the same radius of R_{1}. And the following condition should be satisfied so that the hypocycloid and epicycloid could connect on the circumference of the base circle: the sum of circumferences of the rolling circles to generate hypocycloid and epicycloid is one Z_{1}-th of the base circle circumference, namely the following equation.

Of course, the hypocycloidal and epicycloidal curves can connect at a point on the circumference of the base circle but the slopes of their tangential lines at this point may be not same. It can be solved by adding a fillet between them

It is not difficult to derive the parametric equations of the hypocycloid and epicycloid, expressed by Equation (2) and (3), where β is parameter, the angle between the x-axis and the line determined by the center of the base circle, O_{1}, and the center of the rolling circle of the hypocycloid or epicycloid, O_{h} or O_{e}, as shown in _{1} = 6, the whole profile is to repeat the segment of hypocycloidal or epicycloidal curve by six times along the circumference, as shown in

The meshing principle of two rotors can be stated as following. When the inner rotor and the outer rotor rotate together around their centers O_{1} and O_{2}, respectively, their pitch circles with radius of R_{1} and R_{2} actually rotate by contacting at the point P without slipping, as shown in _{1} = 6 and the distance between the centers of two rotors is eccentricity, e. The following relations are adopted in the research: R_{1} = Z_{1}e and R_{2} = Z_{2}e [

Base the combined trochoidal profile of the inner rotor, two different methods are proposed in this section. One is to obtain profile coordinates of the outer rotor according to the above meshing principle. The other is via graphical approach to determine the envelope curve of the outer rotor.

Now the geometrical relation of the inner and outer rotors is analyzed to derive the profile of the outer rotor, i.e. the conjugate profile of the inner rotor lobe. The pitch circle of the outer rotor is fixed and let that of the inner rotor rotate along the former. The center of the inner rotor, O_{1}, moves to O’_{1} as the contact point P moves to P’, as shown in _{2}O_{1} and line O_{2}O’_{1} is θ. Considering that the rotated arcs of the pitch circles of the inner and outer rotors are of the same length, the rotary angle of the inner rotor is

In fact, the contact point of the inner and outer rotors at this rotary angle is point N, which has corresponding coordinates for the inner and outer rotors, N_{in}(x_{in}, y_{in}) and N_{out}(x_{out}, y_{out}), respectively. Considering the polar angle of the point N_{in}(x_{in}, y_{in}) on the inner rotor, the following Equation (5) establishes.

By using the rotary matrix, coordinates of the conjugate point N_{out}(x_{out}, y_{out}) are expressed by Equation (6), which gives the analytical equation for the profile of the outer rotor.

Graphical approach to obtain the conjugate profile of the outer rotor is simple, intuitive, and practical, but with shortcoming of low precision. However, precision of graphical approach can be greatly improved with the aid of a computer. By this approach, the outer rotor is fixed at first, and let the pitch circle of the inner rotor rotate along the inside of the pitch circle of the outer rotor without slipping. With rotation of the inner rotor, its lobe profile will form family of curves, and the outer envelope curve is right the profile of the outer rotor, as illustrated in

Trochoidal profile is widely applied on gerotor pumps. However, the combined profile of a gerotor pump is investigated in the paper. The hypocycloidal and epicycloidal profile of the inner rotor is generated firstly. And then the profile of the outer rotor, the conjugate profile of the inner rotor, is developed. In order to find the con-jugate profile of the combined trochoid, two approaches are proposed. The analytical equation of the conjugate curve is obtained by analysis of geometrical relation. On the other hand computationally graphical approach is feasible via the numerical computation. The next work is to optimize the designed conjugate profiles and im prove performances of the gerotor pump, such as flow rate pulsation, volumetric efficiency, and so on.

This research was supported by the Ministry of Trade, Industry & Energy (MOTIE) and Korea Evaluation Institute of Industrial Technology (KEIT) through the Project of Industrial Core Technology Development, in which the research titled “Development of hydraulically shifted auto navigation tractor with self-diagnosis based on integrated load control”(Project No. 10049226).

The authors are also grateful to the support of the Center for Automotive Mechatronics Parts (CAMP) at Keimyung University granted by the MOTIE and Korea Institute for Advancement of Technology (KIAT).

Hao Liu,Jae-Cheon Lee, (2016) Development of Combined Trochoidal Profile of a Gerotor Pump. Journal of Applied Mathematics and Physics,04,28-32. doi: 10.4236/jamp.2016.41005