Calculation of Unloading Area of Internal Gear Pump and Optimization

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. .e 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. .e reliability of the model and the feasibility of the optimization scheme are obtained based on the theoretical analysis and calculation results. .is 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.


Introduction
e 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. ere 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. erefore, this paper will overcome the shortcomings mentioned above and, then, use calculation and derivation methods to obtain accurate results of unloading area. e 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.

Description of Oil Trapped Process.
A pair of backlashfree 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 O 1 , and the internal gear O 2 from an oil trapped area. is 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. e 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 Figure 2 is an outline of an internal gear processed by a forming method. With the center of the circle O 2 as the origin of the coordinates and the symmetry line of a cogging as the y-axis, a rectangular coordinate system xO 2 y is established.

Mathematical Knowledge.
is coordinate system can be rotated around point O 2 with a rotation angle of θ. e t 1 t 2 line segment is an involute equation, and the r 1 r 2 line segment is a transition curve. Point t can slide on the line segment t 1 t 2 arbitrarily, and the direction is to move from point t 1 to point t 2 . Point r can slide on the line segment r 1 r 2 arbitrarily, and the direction is to move from point r 1 to point r 2 . Suppose α x , r x , x x , y x , and θ x 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 ∠t 1 o 2 t � ∅, ∠r 1 o 2 t � ζ. 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 t 1 t 2 and the radius of point t 1 and point t 2 be expressed as Suppose that the fanshaped area enclosed by the contour of the line segment r 1 r 2 and the radius of point r 1 and point r 2 can be expressed as r1 . According to [14], 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. e 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 r a,1 , r a,2 , r f,1 , r f,2 be the radius of the top circle and root circle of the external gear and internal gear. 6 � ω, and the corresponding sector areas are S top2 , S root1 . en, S top2 � 0.5r 2 a,1 σ, S root1 � 0.5r 2 f,1 τ. 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, S top1 and S root1 are fixed values. Similarly, the sector areas corresponding to the top and root circles of the internal gear are S top2 and S root2 . en, S top2 � 0.5r 2 a,2 λ and S root1 � 0.5r 2 f,2 ω. Mathematical Problems in Engineering   Figure 3: Gear profile machined by unrolling. Figure 4: A pair of internal gears meshing.

Mathematical Problems in Engineering
r p,1 , r p,2 , r n1,1 , r n1,2 are the pitch circle radius and the meshing point radius of O 1 and O 2 , respectively. l vn1 , l wn1 , and l vw are distances between points v, w, and n 1 . From the geometric relationship in the figure, Let From Heron's formula, Apply sine theorem in Δwo 1 n 1 , Δvo 2 n 1 : Deduce Since θ 1 , θ 2 , and f are linear functions, then from (2)- (7), only the changes of r w and β 1 with θ 1 , r v and β 2 with θ 2 can be used to obtain the change of S ud .
When the point w is on the a 1 a 2 line segment, r w � r a,1 , β � sin − 1 (B ud /r a,1 ).
When the rotation process of O 2 , several special points n 0 , c 1 , c 2 , c 3 , c 4 , n 1 , is located on ud, the corresponding rotation angle θ 2 changes to, θ 2,a ⟶ θ 2,g is represented, respectively, as the process from f to the end of θ 2 when f a ⟶ f g .
According to the internal gear structure, when the v point moves to the n 0 c 1 and c 4 n 1 involute segments, the backlash-free meshing equation is combined with Figure 3(a):

Mathematical Problems in Engineering
If θ 2 ≥ β 2 , it means that the v point is on the n 1 c 4 segment. If θ 2 < β 2 , it means that the v point is on the c 1 n 0 segment. When the v point is at the transition curve c 1 c 2 , c 3 c 4 , as shown in the figure: If θ 2 ≥ β 2 , it means that the v point is in the c 3 c 4 segment. If θ 2 < β 2 , it means that the v point is in the c 1 c 2 segment. (x v , y v ) is the coordinate value of the v point in the coordinate system of the figure.
When the v point is at the root circle c 2 c 3 , β 2 � β 2,c2 � β 2,c3 , and r v � r f,2 , where r f,2 is the radius of the O 2 root circle. erefore,

Optimized Design
If the internal gear pair parameters and center distance are determined, then, S 2 , S 3 , and S 5 are determined. e design goal hopes to optimize and increase S ud . According to the aforementioned calculation model of S ud , it can be achieved by increasing S 1 or decreasing S 4 . Considering the actual situation, S 1 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. e 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. e improvement method is shown in Figures 5(a) and 5(b). Let the radius of the root circle of Figure 5(a) be r fa , and the corresponding sector area is S root2a � 0.5r 2 fa λ. During the rotation, the sector enclosed by the radii v and n 1 and their contour is S 1a : e unloading area of the first scheme is S uda . S 1a can be derived by replacing S 1 in (2), then, obtaining S uda .
Let the radius of the root circle of Figure 5(b) be r fb , and its corresponding sector area is s root2b , s root2b � 0.5r 2 fb λ. During its rotation, the radii of points v and n 1 and their corresponding contour lines are enclosed in a fan shape, and their area is s 1b : e unloading area of the second scheme is S udb . S udb can be derived by replacing S 1 in (2) with S 1b .

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 leftright 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. e 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 S ud in Figure 6, the change law of S ud is very consistent with [18] during a period in which the position variable f changes. is verifies the correctness of the calculation of the unloading area S ud .
Once the gear parameters are determined, the outer contour of the gear teeth formed by the envelope method is a certain value. erefore, the shaping method of the internal tooth contour machining method is improved.
From the changes of S uda and S udb 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 S ud during the oil trapping process. e larger the radius of the root circle, the larger S ud . erefore, the unloading area obtained by the second optimization scheme is larger.    Mathematical Problems in Engineering 7

Conclusions
(1) e analytical method is used to accurately calculate the unloading area of the internal gear pump during oil trapping, and a calculation expression is given. is 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) e calculation formula of the unloading area of the internal gear profile formed by different processing methods during oil trapping is given. (3) e 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. e 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.