# Simulation Research on Trapped Oil Pressure of Involute Internal Gear Pump

Authors: Shanxin Guo; Xiangfeng Guan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The main structure of an internal gear pump consisted of an internal gear pair, including an internal gear and an external gear. The 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. Then, 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. The change of the trapped oil area and trapped oil pressure in a meshing cycle is simulated by MATLAB. The 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. The research results can provide guidance for the development of high-performance internal gear pumps.


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## Body

## 1. 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]. The size of the closed dead volume changes periodically with the rotation of the gear, resulting in a dramatic change of the working pressure [2]. This 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]. Therefore, 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. The 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. Then, 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. The research conclusion can provide an effective reference for the design and manufacture of the high-performance internal gear pump.


## 2. Mathematical Model

### 2.1. 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. The outer contour of the rack cutter is composed of line segments AB, CD, DE, and arc BC. The parameters of ha, cn, hf, and r0 are the addendum height, head clearance coefficient, total tooth height, and tool fillet, respectively. The parameters α, x, and p are pressure angle, the profile shift coefficient, and pitch, respectively. A rectangular coordinate system stOtxtyt is established at the center of the rack tool, and the mathematical equation of the rack tool is described as [10](1)Rtcxt=RtABRtBCRtCDRtDE=xtcytc1.Figure 1
(a) Process external gears by generating method. (b) Virtual rack tool, internal gear processing by the generative method.
(a)(b)The coordinate systems of gearsOs and Os′ are ssOsxsys and ss′Os′xs′ys′, respectively. Their origins coincide with the gear axis and rotate with the gear axis. At the same time, the coordinate system seOexeye fixedly connected with the earth is established. During the forming process, the rack tool translates in the xt direction, and the displacement is s. At this moment, the gear Os rotates around Oe and the rotation angle is φs, which satisfies the following motion relationship:(2)s=rsφs.In the above equation,rs is the gear pitch radius [11]:(3)rs=0.5mz,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]:(4)Rspφs,xt=MstφsRtcxt,where the vector Rspφs,xt is the envelope of the tooth profile surface family of the rack tool and the matrix Mstφs is the transformation matrix from the coordinate system ss to the coordinate system st.In the same way, the internal gear contour equation is obtained by the method of generating internal gears by imaginary rack tools:(5)Rtpφs′,xt=Ms′tφs′Rtcxt.In the formula, the vectorRtpφs′,xt is the envelope of the rack tool tooth profile surface family, and the matrix Ms′tφs′ is the transformation matrix from the coordinate system ss′ to the coordinate system st.


### 2.2. Calculation of Trapped Oil Volume

The principle of the internal gear pump trapping oil is shown in Figure2. The theoretical instantaneous flow Qsh of the internal gear pump can be expressed as [13], which is derived [14] as(6)Qsh=Bωra2−r2−rj2φ2.Figure 2
Oil trapping principle of the internal gear pump.In equation (6), B is the gear face width, ω is the angular velocity of the gear pump; ra is the addendum circle radius; r is the gear reference radius; rj is the gear base circle radius, and φ is the driving gear rotation angle.The scanning volumeVsh changes with the change of the driving gear rotation angle φand the rate of change is [15](7)dVshdφ=Qshω=Bra2−r2−rj2φ2.Figure3 shows the relationship between the meshing point and the angle of rotation. The rotation angles corresponding to F, G, G′, and F′ are φF, φG, φG′, and φF′, respectively. The length between G and G′ is hj. The relationship between the rotation angles isφG=π/z−φF, φG′=1+2hj/tjπ/z−φF, and φF′=2π/z−φF. The parameter of tj is the base pitch.Figure 3
The relationship between the meshing point and the angle of rotation.The corresponding scan volume change rate is described as follows:(8)dVFdφF=Bra2−r2−rj2φF2,(9)dVGdφG=Bra2−r2−rj2φG2,(10)dVG′dφG′=Bra2−r2−rj2φG′2,(11)dVF′dφF′=Bra2−r2−rj2φF′2.The change of trapped oil volumeV, V1, and V2 with driving gear rotation angle φ is obtained by equations (8)–(11):(12)dVdφ=4πzBrj2φ−πz,(13)dV1dφ=2πz1−2hjtjBrj2φ−π2z1−2hjtj,(14)dV2dφ=2πzBrj2φ−3π2z.Integrating equations (12)–(14),(15)V=V0+2πzBrj2φ−πz2,(16)V1=V10+πz1−2hjtjBrj2φ−π2z1−2hjtj2,(17)V2=V20+πzBrj2φ−3π2z2.From equations (15)–(17), V, V1, and V2 are the quadratic functions of the rotation angle φ. When φ=π/z, π/2z1−2hj/tj, 3π/z, the minimum values V0, V10, and V20 are obtained, respectively.


### 2.3. Graphical Method to Solve the Minimum Trapped Oil VolumesV0, V10, V20

According to the definition of the trapped oil [5] and Figure 3, let Vo1F′F^, Vo2F′F^, Vo1G′F3^, and Vo2GF3^ be the center of circle O1, O2 sector area, respectively. Let VΔo1o2F and VΔo1o2F′ be the triangle area enclosed by O1, O2, F, F′, G, and G′, respectively. Then, the trapped oil volume can be described as(18)V=Vo2F′F^−Vo1F′F^−VΔo1o2F+VΔo1o2F′=Vo2F′F^−Vo1F′F^−0.5etjcosα′,V1=VΔo1G′F3−Vo1G′F3^+VΔo2GF3−Vo2GF3^,V2=V−V1.In the equation,e and α′ are the distance between the center of the gear and the meshing angle of the gear pair.In Figures4 and 5, as the driving gear angle φ changes, Vo2F′F^ and Vo1F′F^ are composed of two parts: the unchanged area VA and the changing area VB. Then, the trapped oil volume V can also be expressed as(19)V=VA+VB−0.5etjcosα′.Figure 4
The relationship between the trapped oil and the rotation angle of the external gear.Figure 5
The relationship between the trapped oil and the rotation angle of the internal gear.In the equation,VA consists of four sectors of area. The radius of the sector are the addendum circle ra,1, ra,2 and the tooth root circle rf,1, rf,2. The corresponding included angles of the sector are β11, β21 enclosed:(20)VA=0.5ra,22β21+0.5rf,22β22−0.5ra,12β12−0.5rf,12β11.The derivation of areaVB is shown in Figure 6. The volume VBθ occupied by the spread angle θ can be obtained by the following equation [16]:(21)VBθ=0.5B∫rk2dθ+C.Figure 6
The relationship between area and expansion angle.In the equation, the pressure angle corresponding to therk radius on the tooth profile is θ.Solve equation (21) definite integral:(22)Vθ=Brj36ra3.According to the principle of involute [17],(23)r=θ+α=tanα.Therefore, whenr=ra, equation (22) can be written as(24)Vθ=Brj36tan3ra.It can be derived from this that(25)VB=Brj36tan3θ21+tan3θ22+tan3θ2−tan3θ11−tan3θ12−tan3θ1.θ1, θ11, θ12, θ2, θ21, and θ22 are the spread angles corresponding to the meshing points.According to equations (15) and (19), (20), and (25), V0 can be obtained.From equation (16) and (17), the positions of V10 and V20 are shown in Figures 7 and 8, respectively. V10 and V20 can be obtained from (19), (20), (24), and (25).Figure 7
The Minimum value ofV10.Figure 8
The Minimum value ofV20.


### 2.4. Unloading Area

As shown in Figure9, 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](26)fφ=0.5tj−rjφ−πz.Figure 9
Calculation of the unloading area.The 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 Su1v1 and Su2v2 are as follows: (1) Calculate the expression of fφ 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].In the calculation process ofSu2v2, fφ can be replaced with tj−fφ, and the other steps can be calculated according to Su1v1.Va, Vb are the actual trapped oil volume:(27)Va=V1−Su1v1,Vb=V2−Su2v2.


## 2.1. 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. The outer contour of the rack cutter is composed of line segments AB, CD, DE, and arc BC. The parameters of ha, cn, hf, and r0 are the addendum height, head clearance coefficient, total tooth height, and tool fillet, respectively. The parameters α, x, and p are pressure angle, the profile shift coefficient, and pitch, respectively. A rectangular coordinate system stOtxtyt is established at the center of the rack tool, and the mathematical equation of the rack tool is described as [10](1)Rtcxt=RtABRtBCRtCDRtDE=xtcytc1.Figure 1
(a) Process external gears by generating method. (b) Virtual rack tool, internal gear processing by the generative method.
(a)(b)The coordinate systems of gearsOs and Os′ are ssOsxsys and ss′Os′xs′ys′, respectively. Their origins coincide with the gear axis and rotate with the gear axis. At the same time, the coordinate system seOexeye fixedly connected with the earth is established. During the forming process, the rack tool translates in the xt direction, and the displacement is s. At this moment, the gear Os rotates around Oe and the rotation angle is φs, which satisfies the following motion relationship:(2)s=rsφs.In the above equation,rs is the gear pitch radius [11]:(3)rs=0.5mz,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]:(4)Rspφs,xt=MstφsRtcxt,where the vector Rspφs,xt is the envelope of the tooth profile surface family of the rack tool and the matrix Mstφs is the transformation matrix from the coordinate system ss to the coordinate system st.In the same way, the internal gear contour equation is obtained by the method of generating internal gears by imaginary rack tools:(5)Rtpφs′,xt=Ms′tφs′Rtcxt.In the formula, the vectorRtpφs′,xt is the envelope of the rack tool tooth profile surface family, and the matrix Ms′tφs′ is the transformation matrix from the coordinate system ss′ to the coordinate system st.


## 2.2. Calculation of Trapped Oil Volume

The principle of the internal gear pump trapping oil is shown in Figure2. The theoretical instantaneous flow Qsh of the internal gear pump can be expressed as [13], which is derived [14] as(6)Qsh=Bωra2−r2−rj2φ2.Figure 2
Oil trapping principle of the internal gear pump.In equation (6), B is the gear face width, ω is the angular velocity of the gear pump; ra is the addendum circle radius; r is the gear reference radius; rj is the gear base circle radius, and φ is the driving gear rotation angle.The scanning volumeVsh changes with the change of the driving gear rotation angle φand the rate of change is [15](7)dVshdφ=Qshω=Bra2−r2−rj2φ2.Figure3 shows the relationship between the meshing point and the angle of rotation. The rotation angles corresponding to F, G, G′, and F′ are φF, φG, φG′, and φF′, respectively. The length between G and G′ is hj. The relationship between the rotation angles isφG=π/z−φF, φG′=1+2hj/tjπ/z−φF, and φF′=2π/z−φF. The parameter of tj is the base pitch.Figure 3
The relationship between the meshing point and the angle of rotation.The corresponding scan volume change rate is described as follows:(8)dVFdφF=Bra2−r2−rj2φF2,(9)dVGdφG=Bra2−r2−rj2φG2,(10)dVG′dφG′=Bra2−r2−rj2φG′2,(11)dVF′dφF′=Bra2−r2−rj2φF′2.The change of trapped oil volumeV, V1, and V2 with driving gear rotation angle φ is obtained by equations (8)–(11):(12)dVdφ=4πzBrj2φ−πz,(13)dV1dφ=2πz1−2hjtjBrj2φ−π2z1−2hjtj,(14)dV2dφ=2πzBrj2φ−3π2z.Integrating equations (12)–(14),(15)V=V0+2πzBrj2φ−πz2,(16)V1=V10+πz1−2hjtjBrj2φ−π2z1−2hjtj2,(17)V2=V20+πzBrj2φ−3π2z2.From equations (15)–(17), V, V1, and V2 are the quadratic functions of the rotation angle φ. When φ=π/z, π/2z1−2hj/tj, 3π/z, the minimum values V0, V10, and V20 are obtained, respectively.


## 2.3. Graphical Method to Solve the Minimum Trapped Oil VolumesV0, V10, V20

According to the definition of the trapped oil [5] and Figure 3, let Vo1F′F^, Vo2F′F^, Vo1G′F3^, and Vo2GF3^ be the center of circle O1, O2 sector area, respectively. Let VΔo1o2F and VΔo1o2F′ be the triangle area enclosed by O1, O2, F, F′, G, and G′, respectively. Then, the trapped oil volume can be described as(18)V=Vo2F′F^−Vo1F′F^−VΔo1o2F+VΔo1o2F′=Vo2F′F^−Vo1F′F^−0.5etjcosα′,V1=VΔo1G′F3−Vo1G′F3^+VΔo2GF3−Vo2GF3^,V2=V−V1.In the equation,e and α′ are the distance between the center of the gear and the meshing angle of the gear pair.In Figures4 and 5, as the driving gear angle φ changes, Vo2F′F^ and Vo1F′F^ are composed of two parts: the unchanged area VA and the changing area VB. Then, the trapped oil volume V can also be expressed as(19)V=VA+VB−0.5etjcosα′.Figure 4
The relationship between the trapped oil and the rotation angle of the external gear.Figure 5
The relationship between the trapped oil and the rotation angle of the internal gear.In the equation,VA consists of four sectors of area. The radius of the sector are the addendum circle ra,1, ra,2 and the tooth root circle rf,1, rf,2. The corresponding included angles of the sector are β11, β21 enclosed:(20)VA=0.5ra,22β21+0.5rf,22β22−0.5ra,12β12−0.5rf,12β11.The derivation of areaVB is shown in Figure 6. The volume VBθ occupied by the spread angle θ can be obtained by the following equation [16]:(21)VBθ=0.5B∫rk2dθ+C.Figure 6
The relationship between area and expansion angle.In the equation, the pressure angle corresponding to therk radius on the tooth profile is θ.Solve equation (21) definite integral:(22)Vθ=Brj36ra3.According to the principle of involute [17],(23)r=θ+α=tanα.Therefore, whenr=ra, equation (22) can be written as(24)Vθ=Brj36tan3ra.It can be derived from this that(25)VB=Brj36tan3θ21+tan3θ22+tan3θ2−tan3θ11−tan3θ12−tan3θ1.θ1, θ11, θ12, θ2, θ21, and θ22 are the spread angles corresponding to the meshing points.According to equations (15) and (19), (20), and (25), V0 can be obtained.From equation (16) and (17), the positions of V10 and V20 are shown in Figures 7 and 8, respectively. V10 and V20 can be obtained from (19), (20), (24), and (25).Figure 7
The Minimum value ofV10.Figure 8
The Minimum value ofV20.


## 2.4. Unloading Area

As shown in Figure9, 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](26)fφ=0.5tj−rjφ−πz.Figure 9
Calculation of the unloading area.The 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 Su1v1 and Su2v2 are as follows: (1) Calculate the expression of fφ 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].In the calculation process ofSu2v2, fφ can be replaced with tj−fφ, and the other steps can be calculated according to Su1v1.Va, Vb are the actual trapped oil volume:(27)Va=V1−Su1v1,Vb=V2−Su2v2.


## 3. Oil Trapped Pressure Model of Internal Gear Pump

Figure10 shows the trapped oil pressure model of the internal gear pump. Let the pressures of V1, V2, inlet chamber, and outlet chamber be P1, P2, Pin, and Pout , respectively. Let q1, q2, and qh be the unloading flow rate from the trapped oil cavity V1 to the oil outlet cavity, the unloading flow rate from the trapped oil cavity V2 to the oil inlet cavity, and the unloading flow rate from the trapped oil cavity V2 to trapped oil cavity V1, respectively. According to fluid mechanics and dynamics, the following is derived:(28)q1=CVa2ρP1−PoutsignP1−Pout,q2=CVb2ρP2−PinsignP2−Pin,qh=CVh2ρP1−P2signP1−P2,q1+qh=−dVadt−VaβdP1dt,q2−qh=−dVbdt−VbβdP2dt,dP1dt=−βVaq1+qh+dVadt,dP2dt=−βVbq2−qh+dVbdt.Figure 10
Trapped oil pressure model of the 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. Vh=hj×B.According todφ=ωdt,(29)dP1dφ=−βωVaq1+qh+ωdVadφ,dP2dφ=−βωVbq2−qh+ωdVbdφ.Let,k1=β/ωC2/ρ, k2=2π/zBrj2, k3=2π/z1−2hj/tjBrj2, k4=π/2z1−2hj/tj, k5=3π/2z, k6=βk2, k7=βk3, k8=Vhk1.Then,(30)dP1dφ=−k1P1−Pout−k8P1−P2V10+0.5k3φ−k42−k7φ−k4V10+0.5k3φ−k42,dP2dφ=−k1P2−Pin+k8V20+0.5k2φ−k52P1−P2−k6φ−k5V20+0.5k2φ−k52.


## 4. Simulation Research

### 4.1. Gear Parameters and Simulation

Input parameters, modulusm = 3, pressure angle α=20°, tooth number z1/z2=13/19, addendum height coefficient ha1/ha2=1/1, radial clearance coefficient cn1/cn2=0.25/0.25, total tooth height hf1/hf2=6.734/6.766, tool fillet r0=0.25, the profile shift coefficient x1/x2=0.432/0.504, and base pitch tj=8.856 . According to MATLAB simulation by equations (4) and (5), a pair of gears processed by the generative method is shown in Figure 11.Figure 11
A pair of gears processed by the generative method.


### 4.2. Simulation Results

When the inlet pressurepi=0MPa and the outlet pressure po=10MPa , the rotation process of the gear pair is simulated. During this process, the change trend of the trapped oil volumes V1, V2, V is shown in Figure 12. It can be seen that the change of V1, V2 is parabolic. When the rotation angle is 0.12 rad, the minimum value is 2.34 mm3, and when the rotation angle is 0.36 rad, the minimum value is 2.12 mm3. V is the sum of V1 and V2, 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 V1 and V2. There is no backlash between the front gear and the rear gear. When hj=0, the changes of p1 and p2 are shown in Figure 13. p1 and p2 have a large variation range, and there are positive pressure and negative pressure in the interval. When the gear rotation angle reaches 0.12 rad, p1 reaches the peak value of 35.5 MPa. When the corner reaches 0.36 rad, p2 reaches the peak value of 40.2 MPa. If there is a backlash between the front gear and the rear gear, when hj=0.06m, the changes of p1 and p2 are shown in Figure 14. 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, p1 reaches the peak value of 30.2 MPa. When the rotation angle reaches 0.36 rad, p2 reaches the peak value of 20.24 MPa. If there is tooth side clearance hj=0.06m, and a rectangular unloading groove is designed in the oil trapped area, the position of the boundary line of the unloading groove is Bv=Bu=0.5πm, and the pressure change trend in the oil trapped area is shown in Figure 15. The change trends of p1 and p2 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, p1 reaches the peak value of 16.5 MPa, and when the rotation angle reaches 0.36 rad, p2 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]. The design of tooth side clearance and unloading groove can slow down trapped oil pressure and reduce peak pressure.Figure 12
Changes in trapped oil volume.Figure 13
hj = 0, changes in trapped oil pressure.Figure 14
hj = 0.06 m, changes in trapped oil pressure.Figure 15
hj = 0.06 m, Bv=Bu=0.5πm, changes in trapped oil pressure.


## 4.1. Gear Parameters and Simulation

Input parameters, modulusm = 3, pressure angle α=20°, tooth number z1/z2=13/19, addendum height coefficient ha1/ha2=1/1, radial clearance coefficient cn1/cn2=0.25/0.25, total tooth height hf1/hf2=6.734/6.766, tool fillet r0=0.25, the profile shift coefficient x1/x2=0.432/0.504, and base pitch tj=8.856 . According to MATLAB simulation by equations (4) and (5), a pair of gears processed by the generative method is shown in Figure 11.Figure 11
A pair of gears processed by the generative method.


## 4.2. Simulation Results

When the inlet pressurepi=0MPa and the outlet pressure po=10MPa , the rotation process of the gear pair is simulated. During this process, the change trend of the trapped oil volumes V1, V2, V is shown in Figure 12. It can be seen that the change of V1, V2 is parabolic. When the rotation angle is 0.12 rad, the minimum value is 2.34 mm3, and when the rotation angle is 0.36 rad, the minimum value is 2.12 mm3. V is the sum of V1 and V2, 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 V1 and V2. There is no backlash between the front gear and the rear gear. When hj=0, the changes of p1 and p2 are shown in Figure 13. p1 and p2 have a large variation range, and there are positive pressure and negative pressure in the interval. When the gear rotation angle reaches 0.12 rad, p1 reaches the peak value of 35.5 MPa. When the corner reaches 0.36 rad, p2 reaches the peak value of 40.2 MPa. If there is a backlash between the front gear and the rear gear, when hj=0.06m, the changes of p1 and p2 are shown in Figure 14. 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, p1 reaches the peak value of 30.2 MPa. When the rotation angle reaches 0.36 rad, p2 reaches the peak value of 20.24 MPa. If there is tooth side clearance hj=0.06m, and a rectangular unloading groove is designed in the oil trapped area, the position of the boundary line of the unloading groove is Bv=Bu=0.5πm, and the pressure change trend in the oil trapped area is shown in Figure 15. The change trends of p1 and p2 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, p1 reaches the peak value of 16.5 MPa, and when the rotation angle reaches 0.36 rad, p2 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]. The design of tooth side clearance and unloading groove can slow down trapped oil pressure and reduce peak pressure.Figure 12
Changes in trapped oil volume.Figure 13
hj = 0, changes in trapped oil pressure.Figure 14
hj = 0.06 m, changes in trapped oil pressure.Figure 15
hj = 0.06 m, Bv=Bu=0.5πm, changes in trapped oil pressure.


## 5. 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 volumesV1 and V2 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 pressuresp1 and p2 increase first and then decrease with the change of the rotation angle. There is a maximum peak value. When the volume of the trapped oil cavity is the smallest, the trapped oil pressure reaches the maximum.(3)
The 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)
The 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.



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