SimulationResearch onTrappedOil Pressure of Involute Internal Gear Pump

,emain structure of an internal gear pump consisted of an internal gear pair, including an internal gear and an external gear.,e internal gear pump had oil trapping phenomenon like other hydraulic gear pumps. In order to solve the oil trapping phenomenon of involute gear pump with internal meshing tooth profile, in this paper, the mathematical equation of gear outer contour is established according to the principle of generation method, and the variation law of the trapped oil area in meshing process is deduced by theoretical instantaneous flow rate obtained by scanning method. ,en, the minimum trapped oil volume and unloading area are solved by the graphic method. Finally, based on fluid mechanics and dynamics, the trapped oil pressure model is obtained. ,e change of the trapped oil area and trapped oil pressure in a meshing cycle is simulated by MATLAB. ,e results show that the trapped oil area changes in a parabola, and the trapped pressure fluctuates in mountains and valleys. When the trapped area is the smallest, the trapped oil pressure reaches the peak at the corresponding corner.,e research results can provide guidance for the development of high-performance internal gear pumps.


Introduction
In order to ensure the continuous and uniform oil supply of the gear pump, two pairs of gear teeth engaging in meshing are needed in the internal gear pump for a certain period of time. It results in a closed dead volume which is not connected with the inlet and outlet cavities [1]. e size of the closed dead volume changes periodically with the rotation of the gear, resulting in a dramatic change of the working pressure [2].
is phenomenon is usually called the oil trapping phenomenon of a gear pump, which is the main source of gear pump noise and cavitation [3]. It can cause hazards such as pressure swing, pressure shock, and flow pulsation in the hydraulic system [4]. erefore, it is of great significance to avoid the so-called oil trapping phenomenon of the gear pump. However, because the trapped oil area is surrounded by complex curves such as the outer contour of gear, involute, or cycloid [5], it is very difficult to obtain the continuous and uniform oil supply. Up to date, there is much research on the oil trapping phenomenon of the external gear pump and cycloid gear [6], and great improvements have been made. In comparison with external gear pump and cycloid gear, internal gear pumps have advantages such as lighter weight, smaller volume, and higher pressure due to their geometric structure [7]. Because the internal gear has oil ports in the diameter direction and the oil trapping process is different from other gear pumps, the high-pressure internal gears are not easy to obtain by conventional fabrication methods of internal gears. e common internal meshing gears are arc tooth profile, conjugate tooth profile, involute tooth profile, and so forth. Most research is focused on arc tooth profile and conjugate tooth profile and there are a few studies on involute tooth profile [8]. In this study, in order to solve the oil trapping phenomenon of gear pump with involute tooth profile during the process of internal meshing, the contour of internal gear is obtained by means of an imaginary rack cutter generating internal gear, and the model of trapped oil volume is established. en, the design parameters and the change of trapped oil volume in the process of gear rotation are described, and then the trapped oil pressure is simulated. e research conclusion can provide an effective reference for the design and manufacture of the high-performance internal gear pump.

Oil Trapped Model of Internal Gear
Pump. Gear pairs are commonly manufactured by the generative method [9]. Figure 1(a) shows the processing of external gears by the rack tool generating method, and Figure 1(b) shows the processing of the internal gears by the virtual rack tool generating method. e outer contour of the rack cutter is composed of line segments AB, CD, DE, and arc BC. e parameters of h a , c n , h f , and r 0 are the addendum height, head clearance coefficient, total tooth height, and tool fillet, respectively. e parameters α, x, and p are pressure angle, the profile shift coefficient, and pitch, respectively. A rectangular coordinate system s t (O t x t y t ) is established at the center of the rack tool, and the mathematical equation of the rack tool is described as [10] e coordinate systems of gears O s and O s′ are s s (O s x s y s ) and s s′ (O s′ x s′ y s′ ), respectively. eir origins coincide with the gear axis and rotate with the gear axis. At the same time, the coordinate system s e (O e x e y e ) fixedly connected with the earth is established. During the forming process, the rack tool translates in the x t direction, and the displacement is s. At this moment, the gear O s rotates around O e and the rotation angle is φ s , which satisfies the following motion relationship: In the above equation, r s is the gear pitch radius [11]: where m and z are the modulus of the gear and the number of teeth, respectively. According to the generation principle, the relationship between the tooth profile of the gear and the tooth profile of the rack tool can be expressed by the following equation [12]: where the vector R p s (φ s , x t ) is the envelope of the tooth profile surface family of the rack tool and the matrix M st (φ s ) is the transformation matrix from the coordinate system s s to the coordinate system s t .
In the same way, the internal gear contour equation is obtained by the method of generating internal gears by imaginary rack tools: In the formula, the vector R p t (φ s′ , x t ) is the envelope of the rack tool tooth profile surface family, and the matrix M s′t (φ s′ ) is the transformation matrix from the coordinate system s s′ to the coordinate system s t .

Calculation of Trapped Oil Volume.
e principle of the internal gear pump trapping oil is shown in Figure 2. e theoretical instantaneous flow Q sh of the internal gear pump can be expressed as [13], which is derived [14] as In equation (6), B is the gear face width, ω is the angular velocity of the gear pump; r a is the addendum circle radius; r is the gear reference radius; r j is the gear base circle radius, and φ is the driving gear rotation angle. e scanning volume V sh changes with the change of the driving gear rotation angle φand the rate of change is [15] Figure 3 shows the relationship between the meshing point and the angle of rotation. e rotation angles corresponding to F, G, G ′ , and F ′ are φ F , φ G , φ G′ , and φ F′ , respectively.
e length between G and G ′ is h j . e relationship between the rotation angles e corresponding scan volume change rate is described as follows: e change of trapped oil volume V, V 1 , and V 2 with driving gear rotation angle φ is obtained by equations (8)- (11): Integrating equations (12)-(14), 2 Mathematical Problems in Engineering From equations (15)-(17), V, V 1 , and V 2 are the quadratic functions of the rotation angle φ. When φ � (π/z), (π/2z)(1 − (2h j /t j )), (3π/z), the minimum values V 0 , V 10 , and V 20 are obtained, respectively. 20 . According to the definition of the trapped oil [5] and Inlet

Graphical Method to Solve the Minimum Trapped Oil
en, the trapped oil volume can be described as In the equation, e and α ′ are the distance between the center of the gear and the meshing angle of the gear pair.
In Figures 4 and 5, as the driving gear angle φ changes, V o2F′F and V o1F′F are composed of two parts: the unchanged area V A and the changing area V B . en, the trapped oil volume V can also be expressed as In the equation, V A consists of four sectors of area. e radius of the sector are the addendum circle r a,1 , r a,2 and the tooth root circle r f,1 , r f,2 .
e corresponding included angles of the sector are β 11 , β 21 enclosed: e derivation of area V B is shown in Figure 6. e volume V Bθ occupied by the spread angle θ can be obtained by the following equation [16]: In the equation, the pressure angle corresponding to the r k radius on the tooth profile is θ.
Solve equation (21) definite integral: Figure 3: e relationship between the meshing point and the angle of rotation.

Mathematical Problems in Engineering
According to the principle of involute [17], erefore, when r � r a , equation (22) can be written as It can be derived from this that θ 1 , θ 11 , θ 12 , θ 2 , θ 21 , and θ 22 are the spread angles corresponding to the meshing points.
From equation (16) and (17), the positions of V 10 and V 20 are shown in Figures 7 and 8, respectively. V 10 and V 20 can be obtained from (19), (20), (24), and (25). Figure 9, points P, F are node and meshing points, respectively. Let the length of P, F be f, and the relationship between f and the gear rotation angle φ is [18]   Mathematical Problems in Engineering e unloading area is the area where the cross section of the trapped oil volume is located in the unloading groove [19]. In Figure 9, the calculation steps of S u1v1 and S u2v2 are as follows: (1) Calculate the expression of f(φ) at points F, 1, 2, 3, 4, 5, 6 on the gear tooth profile. (2) Find the curve equations of 12, 23, 34, 4F, F5, 56. (3) Solve the integral to calculate the unloading area, the method, and steps referring to [20].

Unloading Area. As shown in
In the calculation process of S u2v2 , f(φ) can be replaced with [t j − f(φ)], and the other steps can be calculated according to S u1v1 . V a , V b are the actual trapped oil volume: Figure 10 shows the trapped oil pressure model of the internal gear pump. Let the pressures of V 1 , V 2 , inlet chamber, and outlet chamber be P 1 , P 2 , P in , and P out , respectively. Let q 1 , q 2 , and q h be the unloading flow rate from the trapped oil cavity V 1 to the oil outlet cavity, the unloading flow rate from the trapped oil cavity V 2 to the oil inlet cavity, and the unloading flow rate from the trapped oil cavity V 2 to trapped oil cavity V 1 , respectively. According to fluid mechanics and dynamics, the following is derived:

Oil Trapped Pressure Model of Internal Gear Pump
In above equations, ρ, β are the density and bulk elastic modulus of the fluid, respectively. C is the flow coefficient 0.60-0.65. V h � h j × B.
According to dφ � ωdt, en,   (4) and (5), a pair of gears processed by the generative method is shown in Figure 11.

Simulation
Results. When the inlet pressure p i � 0 MPa and the outlet pressure p o � 10 MPa , the rotation process of the gear pair is simulated. During this process, the change trend of the trapped oil volumes V 1 , V 2 , V is shown in Figure 12. It can be seen that the change of V 1 , V 2 is parabolic. When the rotation angle is 0.12 rad, the minimum value is 2.34 mm 3 , and when the rotation angle is 0.36 rad, the minimum value is 2.12 mm 3 . V is the sum of V 1 and V 2 , and its changing trend is to first reach a low point and then present an increasing trend. Internal gear pumps and external gear pumps have similar changes [21,22]. During a period of gear tooth meshing, simulate the pressure changes in V 1 and V 2 . ere is no backlash between the front gear and the rear gear. When h j � 0, the changes of p 1 and p 2 are shown in Figure 13. It can be seen that the pressure change trend in the trapped oil zone is similar to that in Figure 13, but the change range is convergent. When the turning angle reaches 0.12 rad, p 1 reaches the peak value of 30.2 MPa. When the rotation angle reaches 0.36 rad, p 2 reaches the peak value of 20.24 MPa. If there is tooth side clearance h j � 0.06 m, and a rectangular unloading groove is designed in the oil trapped area, the position of the boundary line of the unloading groove is B v � B u � 0.5πm, and the pressure change trend in the oil trapped area is shown in Figure 15. e change trends of p 1 and p 2 are very similar to those in Figures 13 and 14, and the range of change is more convergent. When the rotation angle reaches 0.12 rad, p 1 reaches the peak value of 16.5 MPa, and when the rotation angle reaches 0.36 rad, p 2 reaches the peak value of 18.2 MPa. Internal gear pumps have lower trapped oil pressure than external gear pumps [23] and arc gear pumps [24]. e design of tooth side clearance and unloading groove can slow down trapped oil pressure and reduce peak pressure.

Conclusions
(1) For a pair of internal gear pairs processed by the generative method, there are oil trapped areas between the gear teeth during meshing, and the trapped oil volumes V 1 and V 2 present a parabolic change law, and each has a minimum value, which changes periodically when the gear rotates. (2) When the internal gear pump rotates, the trapped oil pressures p 1 and p 2 increase first and then decrease with the change of the rotation angle. ere is a maximum peak value. When the volume of the trapped oil cavity is the smallest, the trapped oil pressure reaches the maximum.
(3) e tooth side clearance will improve the oil trapping characteristics of the internal gear pump and reduce the pressure peak in the trapped oil cavity. (4) e design of the unloading groove will improve the fluidity of trapped oil, reduce the range of trapped oil pressure, and reduce the pressure peak in the trapped oil cavity.   Addendum angle r f,2 : e tooth root circle r f,1 : e tooth root circle β 11 : Tooth root circle included angle β 21 : Tooth root circle included angle θ: e spread angle r k : e radius on the tooth profile θ 1 : e spread angles corresponding to the meshing points θ 11 : e spread angles corresponding to the meshing points θ 12 : e spread angles corresponding to the meshing points θ 2 : e spread angles corresponding to the meshing points θ 21 : e spread angles corresponding to the meshing points θ 22 : e spread angles corresponding to the meshing points P: Node F: Meshing point f: e length ofP , F S u1v1 : e unloading area S u2v2 : e unloading area f(φ): e relationship between f and the gear rotation angle φ P 1 : e pressures of V 1 P 2 : e pressures of V 2 P in : e pressures of the inlet chamber P out : e pressures of the outlet chamber q 1 : e unloading flow rate from the trapped oil cavity V 1 to the oil outlet cavity q 2 : e unloading flow rate from the trapped oil cavity V 2 to the oil inlet cavity q h : e unloading flow rate from the trapped oil cavity V 2 to trapped oil cavity V 1 ρ: e density β: Bulk elastic modulus of the fluid C: e flow coefficient V h : Tooth side clearance volume c n1 : Radial clearance coefficient c n2 : Radial clearance coefficient.

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

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.