Research Article | Open Access

Volume 2020 |Article ID 7319871 | https://doi.org/10.1155/2020/7319871

Shanxin Guo, Dagui Chen, "Calculation of Unloading Area of Internal Gear Pump and Optimization", Mathematical Problems in Engineering, vol. 2020, Article ID 7319871, 9 pages, 2020. https://doi.org/10.1155/2020/7319871

# Calculation of Unloading Area of Internal Gear Pump and Optimization

Academic Editor: Xiao-Qiao He
Revised17 Jul 2020
Accepted24 Jul 2020
Published26 Aug 2020

#### Abstract

In order to obtain the calculation method of the unloading area of the internal gear pump during oil trapping, a pair of internal gears including an external gear and an internal gear was used as the research object to simulate the oil trapping process. The geometric relationship during the meshing process was established, and the unloading area expression was obtained by using the geometric pattern expansion method with the variable f as the independent variable. Guided by a mathematical model, two improved optimization schemes were proposed for the internal gear tooth profile, and the unloading area expressions sud, suda, and sudb were obtained. Taking the meshing gear pair with module 3 and number of teeth 13/19 as examples, the simulation results were very consistent with the existing literature. The reliability of the model and the feasibility of the optimization scheme are obtained based on the theoretical analysis and calculation results. This calculation method of unloading area can be applied to the same type of gear pump design in the future, providing a reference for the design of high pressure and low noise gear pumps.

#### 2. Oil Trapping Process Description and Calculation Basis of Internal Gear Pump

##### 2.1. Description of Oil Trapped Process

A pair of backlash-free gear tooth rotation processes are used as an analysis to explain the oil trapping and unloading of the internal gear pump. In this process, the unloading groove can be designed. Although its design is more flexible and has more geometric forms, its design principles are basically the same [11]. Figure 1 shows oil trapping and unloading process of internal gear pump, the external gear , and the internal gear from an oil trapped area. This article uses them as an example to explain that they pass through the boundary ud of the rectangular unloading groove [12]. p, n1, and n0 are gear joint nodes, meshing points, and backlash points. The meshing point n1 is moving along the meshing line during the rotation of the gear pair. Figure 1(a) shows the minimum oil trapped area, Figure 1(b) shows ud on the involute of gear and involute of gear , Figure 1(c) shows ud on the top circle of gear and gear transition curve, Figure 1(d) shows ud on the tooth top circle of gear and the tooth root circle of gear , Figure 1(e) shows ud on tooth top circle of gear and gear the transition curve, Figure 1(f) shows ud involute of gear and gear transition curve, and Figure 1(g) shows ud on the involute of gear and gear .

The length from point p to point is . The length of in Figure 1(a) is , and . When changes to the position of Figure 1(g) as moves, is , . When point p moves to ud along the meshing line of the gear pair, the length of at this time is and . The above equations , , , , and are the base pitch of the gear , the base circle radius, the pressure angle of the tooth top circle, the meshing angle, and the distance from ud to the line connecting the center of gear and gear .

##### 2.2. Mathematical Knowledge

Figure 2 is an outline of an internal gear processed by a forming method. With the center of the circle as the origin of the coordinates and the symmetry line of a cogging as the y-axis, a rectangular coordinate system is established. This coordinate system can be rotated around point with a rotation angle of θ. The line segment is an involute equation, and the line segment is a transition curve. Point t can slide on the line segment arbitrarily, and the direction is to move from point to point . Point r can slide on the line segment arbitrarily, and the direction is to move from point to point . Suppose , , , , and are the gear pressure angle at point x, the corresponding radius, the abscissa in the coordinate system, the ordinate in the coordinate system, and the rotation angle , . By the definition of the involute function, any two points on the involute line can be connected to the center of the circle to form a sector [13]:

Let the fan-shaped area enclosed by the line segment and the radius of point and point be expressed as , which can be deduced from (1), .

In rectangular coordinate system, . Suppose that the fan-shaped area enclosed by the contour of the line segment and the radius of point and point can be expressed as . According to [14], the above formula, and m are the tooth height coefficient and modulus of the rack cutter for processing the gear profile and r′ is the pitch circle radius.

The gears produced by the envelope method have regular geometric outer contours [15]. Figure 3(a) shows the outline of an external gear machined by generating method, and Figure 3(b) shows the profile of an internal gear machined by generating method. Let be the radius of the top circle and root circle of the external gear and internal gear. Let , , and the corresponding sector areas are , . Then, . Once the gear parameters are determined, the radius of the root and top circles of the gear and the angle between each gear tooth can be determined [16]; then, and are fixed values. Similarly, the sector areas corresponding to the top and root circles of the internal gear are and . Then, .

#### 3. Establishing the Calculation Model of Unloading Area

##### 3.1. Mathematical Model

Figure 4(a) is a pair of internal gear pairs and e is the center distance. The Cartesian coordinate systems and are established by using the gear circle centers and as the coordinate origins and the symmetry lines of the external gear teeth and internal gear grooves as the y-axis. Taking Figure 1(a) as the initial position, the rotation angles of and are and , respectively. During the rotation, the outline of the external gear, the profile of the internal gear, and ud intersect at points and , respectively. Among them, and are the radius of point , point , point , and the area enclosed by the contour line of the gear, respectively. For ease of calculation, let . Then, fold and along to get Figure 4(b). According to the graph, the unloading area can be expressed as

Let . are the pitch circle radius and the meshing point radius of and , respectively. are distances between points , , and n1. From the geometric relationship in the figure,

Let

From Heron’s formula,

Apply sine theorem in :

Deduce

Since , and f are linear functions, then from (2)–(7), only the changes of and with and with can be used to obtain the change of .

##### 3.2. External Gear Rotation Parameters

When and are located on ud, can be expressed as . When points are located on ud, these rotation angles can be expressed as . According to the structure of the external gear, when the point moves to , involute segment, according to the backlash-free meshing equation [17] and Figure 3(a), it can be obtained that

If , it means that the point is on the line segment. If , it means that the point is on the line segment.

When the point w is on the line segment, .

##### 3.3. Internal Gear Rotation Parameters

If and are located on ud, can be expressed as .

When the rotation process of , several special points , , c2, , , , is located on ud, the corresponding rotation angle changes to,

is represented, respectively, as the process from f to the end of when .

According to the internal gear structure, when the point moves to the and involute segments, the backlash-free meshing equation is combined with Figure 3(a):

If , it means that the point is on the segment. If , it means that the point is on the segment. When the point is at the transition curve , , as shown in the figure:

If , it means that the point is in the segment. If , it means that the point is in the segment. () is the coordinate value of the point in the coordinate system of the figure.

When the point is at the root circle , , and , where is the radius of the root circle. Therefore,

#### 4. Optimized Design

If the internal gear pair parameters and center distance are determined, then, are determined. The design goal hopes to optimize and increase . According to the aforementioned calculation model of , it can be achieved by increasing or decreasing . Considering the actual situation, is easier to implement, so two optimization schemes are proposed for machining the outer contour of the internal gear with a forming tool or mold. The first method is to increase the radius of the tooth root circle; the second method is to eliminate the cogging transition curve segment on the gear profile. The improvement method is shown in Figures 5(a) and 5(b). Let the radius of the root circle of Figure 5(a) be , and the corresponding sector area is . During the rotation, the sector enclosed by the radii and n1 and their contour is :

The unloading area of the first scheme is . can be derived by replacing in (2), then, obtaining .

Let the radius of the root circle of Figure 5(b) be , and its corresponding sector area is , . During its rotation, the radii of points and and their corresponding contour lines are enclosed in a fan shape, and their area is :

The unloading area of the second scheme is . can be derived by replacing in (2) with .

#### 5. Simulation

Taking the parameters of medium and high pressure internal gear pumps of Fuzhou University Hydraulic Parts Factory as an example, enveloping processing, the first optimization scheme, and the second optimization scheme, the parameters are shown in Table 1.

 Name of the parameter Parameter Value Number of teeth 13/19 Modulus (mm) m 3 Center distance (mm) e 9.278 Pressure angle (°) α 20 Coefficient of rack cutter tip height 1.25 External gear profile shift coefficient 0.432 Internal gear profile shift coefficient 0.55 Unloading groove boundary distance (mm) 2.035 The first solution root radius (mm) 32.0 The second solution root radius (mm) 33.80

During the gear rotation, the position variable f is left-right symmetric with the position shown in Figure 1(a). MATLAB software is used to draw the unloading area, which changes with f in a meshing cycle as shown in Figure 6.

The parameters provided in Table 1 are the original data of the gear pump during the actual design and manufacture, which are consistent with the parameters under the number of different teeth (z1 + z2 = 32) in [18]. From the results of the simulation of in Figure 6, the change law of is very consistent with [18] during a period in which the position variable f changes. This verifies the correctness of the calculation of the unloading area .

Once the gear parameters are determined, the outer contour of the gear teeth formed by the envelope method is a certain value. Therefore, the shaping method of the internal tooth contour machining method is improved. From the changes of and in Figure 6, it can be seen that increasing the root circle radius of the internal gear tooth profile can obtain a larger unloading area during the oil trapping process. The larger the radius of the root circle, the larger . Therefore, the unloading area obtained by the second optimization scheme is larger.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by Fujian Teacher Education Research Project (JAT170616) and Fujian Jiangxia College Cultivation Project (JXZ2019015).

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Copyright © 2020 Shanxin Guo and Dagui Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.