#### 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 *s*_{ud}, *s*_{uda}, and *s*_{udb} 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.

#### 1. Introduction

The internal gear pump has oil trapping phenomenon during the rotating work and has smaller oil trapping changes than the external gearing [1]. At present, gear pump oil trapping phenomenon is one of the key factors restricting higher performance [2]. In order to improve the design performance of the gear pump, Erturk et al. analyzed the oil trap mechanism of the hydraulic pump and found its variation rule and found that the backlash of the gear pair can affect the oil trap [3]. Sun et al. proposed the definition of backlash unloading of gear pumps and compared and verified the results [4]. In order to further reduce the trapped oil, Li and Liu calculated the unloading area of the gear pump and then performed a simulation analysis [5]. Zang et al. proposed a new method for reducing the trapped oil pressure of the gear pump [6]. Above those was an external gear pump. Tt is still a large difference in the structure between the internal gear and the external gear pump. There is less research on oil trapping and unloading of the internal gear pump. On internal gear pumps, Yanada et al. had studied the phenomenon of oil trapping between gear pairs and pointed out the source of the oil trap [7]. Zhou et al. had adopted a discrete method to study conjugate gears and obtained the oil trap of this pump [8, 9]. Song et al. designed the conjugated straight-line internal gear pairs for fluid power gear machines and got the unloading curve graphically by a discretization approach [10]. Above all, research methods are of reference to internal gear pumps with involute tooth profiles, but they are not continuous in calculation and not completely accurate. In these references, the unloading area was obtained by the graphic method, and the change law of the area size with processing and parameters was analyzed, but the results of such processing have limitations. Therefore, this paper will overcome the shortcomings mentioned above and, then, use calculation and derivation methods to obtain accurate results of unloading area. The conclusions obtained can provide theoretical guidance for the model parameters and design principles of the internal gear pump and can also optimize the unloading area according to the processing and forming methods.

#### 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*, *n*_{1}, and *n*_{0} are gear joint nodes, meshing points, and backlash points. The meshing point *n*_{1} 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 .

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

**(g)**

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, .

**(a)**

**(b)**

#### 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

**(a)**

**(b)**

Let . are the pitch circle radius and the meshing point radius of and , respectively. are distances between points , , and *n*_{1}. 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 , , *c*_{2}, , , , 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 *n*_{1} and their contour is :

**(a)**

**(b)**

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.

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 (*z*_{1} + *z*_{2} = 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.

#### 6. Conclusions

(1)The analytical method is used to accurately calculate the unloading area of the internal gear pump during oil trapping, and a calculation expression is given. This calculation result can be transplanted to a computer program, which can accurately simulate the trapped oil and unloading process of the internal gear pump. It has a guiding significance for the design of high-performance internal gear pumps.(2)The calculation formula of the unloading area of the internal gear profile formed by different processing methods during oil trapping is given.(3)The example compares the changes in the unloading area of the internal gear profile formed by the envelope method, forming tool method and die processing method during oil trapping and unloading. The larger the radius of the root circle of the internal tooth profile, the larger the unloading area, which provides a theoretical reference for the future design and improvement of the oil trapping phenomenon of the internal gear pump.

#### Nomenclature

: | Number of external gear teeth |

: | Number of internal gear teeth |

m: | Modulus |

e: | Distance between centers of gears and |

α: | Pressure angle |

: | Coefficient of rack cutter tip height |

: | External gear profile shift coefficient |

: | Internal gear profile shift coefficient |

: | Center of external gear |

: | Center of internal gear |

p: | Gear joint node |

: | Meshing point |

: | Backlash point |

ud: | Boundary of the rectangular unloading groove |

f: | Length from point p to point n_{1} |

: | Length from point p to point n_{1} |

: | Length from point p to point n_{1} |

: | Length from point p to point n_{1} |

: | Length from point p to point n_{1} |

: | Length from point p to point n_{1} |

: | Length from point p to point n_{1} |

: | Length from point p to point n_{1} |

: | Base pitch of external gear O_{1} |

: | Base circle radius of external gear O_{1} |

: | Pressure angle of tooth top circle |

: | Meshing angle |

: | Distance from ud to the line connecting O_{1}O_{2} |

xOy: | Coordinate system with O as origin |

x_{1}O_{1}y_{1}: | Coordinate system with as origin |

x_{2}O_{2}y_{2}: | Coordinate system with as origin |

θ: | Xoy rotation angle around O |

: | X_{1}o_{1}y_{1} rotation angle around |

: | X_{2}o_{2}y_{2} rotation angle around |

t_{1}, t_{2}: | The limit point of the involute of the gear outline |

t: | Sliding point from point t_{1} to point t_{2} |

t_{1}t_{2}: | Contour of involute |

r: | Sliding point from point r_{1} to point r_{2} |

r_{1},r_{2}: | The limit point of the transition curve of gear outline |

r_{1}r_{2}: | Contour of transition curve |

: | Gear pressure angle at point x |

: | Circle radius at point x |

, : | Abscissa, ordinate in the coordinate system |

: | Rotation angle at point x |

: | Angle between line and t line |

: | Angle between line and t line |

: | Area enclosed by t_{1}t_{2}, , and |

: | Area enclosed by r_{1}r_{2}, , and |

: | Pitch circle radius |

, : | Top circle of the external gear and internal gear |

, : | Root circle of the external gear and internal gear |

, : | The limit point of the addendum circle profile |

, : | The limit point of the addendum circle profile |

, : | The limit point of tooth root circle profile |

, : | The limit point of tooth root circle profile |

, : | The limit point of the transition curve of gear outline |

: | Angle between line and t line |

: | Angle between line and t line |

: | Angle between line and t line |

: | Angle between line and t line |

: | Angle between line and t line |

: | Area enclosed by , , and |

: | Area enclosed by , , and |

: | Area enclosed by , , and |

: | Area enclosed by , , and |

: | Intersection point of ud and external gear outline |

: | Intersection point of ud and internal gear outline |

: | Area enclosed by o_{1}, o_{1}n_{1} and the gear outline n_{1} |

: | Area enclosed by o_{2}, o_{2}n_{1} and the gear outline n_{1} |

: | Area enclosed by o_{2}, o_{2}n_{1}, and n_{1} |

: | Area enclosed by , n_{1}, and |

: | Area enclosed by o_{1}, o_{1}n_{1}, and n_{1} |

: | Unloading area |

: | Unloading area |

: | Unloading area |

: | Angle between line and t line |

: | Angle between line and t line |

: | Angle between line and t line |

: | Angle between line and t line |

: | Pitch circle radius of external gear O_{1} |

: | Pitch circle radius of internal gear O_{2} |

: | Meshing point radius of O_{1} |

: | Meshing point radius of O_{2} |

: | Distances between points and n_{1} |

: | Distances between points and n_{1} |

: | Distances between points and |

: | Circle radius of external gear O_{1} at point |

: | Circle radius of internal gear o_{2} at point |

: | Angle of when point a_{1} is located on ud |

: | Angle of when point a_{2} is located on ud |

, , : | Half of triangle perimeter |

: | Angle of when point is located on ud |

: | Angle of when point is located on ud |

: | Angle of when point a is located on ud |

: | Angle of when point is located on ud |

: | Angle of when point is located on ud |

: | Angle of when point is located on ud |

: | Angle of when point a is located on ud |

: | Angle of when point is located on ud |

: | Angle of when point is located on ud |

: | Angle of when point is located on ud |

: | Angle of when point is located on ud |

: | Angle of when point is located on ud |

: | Root circle of internal gear |

: | Area enclosed by o_{2}e_{3}, o_{2}e_{4} and the gear outline e_{3}e_{4} |

: | Unloading area |

: | Root circle of internal gear |

: | Area enclosed by o_{2}q_{3}, o_{2}q_{4} and the gear outline q_{3}q_{4} |

: | Unloading area. |

#### 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).