# Calculation of Unloading Area of Internal Gear Pump and Optimization

Authors: Shanxin Guo; Dagui Chen
Journal: Mathematical Problems in Engineering
 (2020)

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## 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 variablef 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.


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## Body

## 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 O1, and the internal gear O2 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 O1 and involute of gear O2, Figure 1(c) shows ud on the top circle of gear and gear O2 transition curve, Figure 1(d) shows ud on the tooth top circle of gear O1 and the tooth root circle of gear O2, Figure 1(e) shows ud on tooth top circle of gear O1 and gear O2 the transition curve, Figure 1(f) shows ud involute of gear O1 and gear O2 transition curve, and Figure 1(g) shows ud on the involute of gear O1 and gear O2.Figure 1
Oil trapping and unloading process of internal gear pump.
(a)(b)(c)(d)(e)(f)(g)The length from pointp to point n1 is f. The length of f in Figure 1(a) is f0, and f0=Pb/4. When f changes to the position of Figure 1(g) as n1 moves, f is fg, fg=rb,1tanαα,1−tanα′. When point p moves to ud along the meshing line of the gear pair, the length of f at this time is fa and fa=Bud/cosα′. The above equations pb, rb,1, αα,1, α′, and Bud are the base pitch of the gear O1, 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 O1 and gear O2.


### 2.2. Mathematical Knowledge

Figure2 is an outline of an internal gear processed by a forming method. With the center of the circle O2 as the origin of the coordinates and the symmetry line of a cogging as the y-axis, a rectangular coordinate system xO2y is established. This coordinate system can be rotated around point O2 with a rotation angle of θ. The t1t2 line segment is an involute equation, and the r1r2 line segment is a transition curve. Point t can slide on the line segment t1t2 arbitrarily, and the direction is to move from point t1 to point t2. Point r can slide on the line segment r1r2 arbitrarily, and the direction is to move from point r1 to point r2. Suppose αx, rx, xx, yx, and θx 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 ∠t1o2t=∅, ∠r1o2t=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]:(1)∅=tan∝t−∝t−tan∝t1−∝t1,Sinv=∫rt2ϕ2dϕ.Figure 2
Internal gear profile.Let the fan-shaped area enclosed by the line segmentt1t2 and the radius of point t1 and point t2 be expressed as Sinvt2t1, which can be deduced from (1), Sinvt1t2=∫rt2∅/2d∅=rb,12/2∫αt1αt2sec4αt−sec2αtd∝t=rb,12tanαt23−tanαt13/6.In rectangular coordinate system,ζ=tan−1xr/yr−tan−1xr1/yr1. Suppose that the fan-shaped area enclosed by the contour of the line segment r1r2 and the radius of point r1 and point r2 can be expressed as Sstar1r2. According to [14], Ssta=r2r1r′2/6θr23−θr13−mha0∗/2r′−mha0∗θr2−θr1 the above formula, ha0∗ 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 ra,1,ra,2,rf,1,rf,2 be the radius of the top circle and root circle of the external gear and internal gear. Let ∠a1o1a2=σ, ∠a5o1a6=τ,∠c1o2c4=μ,∠c2o1c3=λ,∠c5o1c6=ω, and the corresponding sector areas are Stop2, Sroot1. Then, Stop2=0.5ra,12σ,Sroot1=0.5rf,12τ. 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, Stop1 and Sroot1 are fixed values. Similarly, the sector areas corresponding to the top and root circles of the internal gear are Stop2 and Sroot2. Then, Stop2=0.5ra,22λandSroot1=0.5rf,22ω.Figure 3
Gear profile machined by unrolling.
(a)(b)


## 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 O1, and the internal gear O2 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 O1 and involute of gear O2, Figure 1(c) shows ud on the top circle of gear and gear O2 transition curve, Figure 1(d) shows ud on the tooth top circle of gear O1 and the tooth root circle of gear O2, Figure 1(e) shows ud on tooth top circle of gear O1 and gear O2 the transition curve, Figure 1(f) shows ud involute of gear O1 and gear O2 transition curve, and Figure 1(g) shows ud on the involute of gear O1 and gear O2.Figure 1
Oil trapping and unloading process of internal gear pump.
(a)(b)(c)(d)(e)(f)(g)The length from pointp to point n1 is f. The length of f in Figure 1(a) is f0, and f0=Pb/4. When f changes to the position of Figure 1(g) as n1 moves, f is fg, fg=rb,1tanαα,1−tanα′. When point p moves to ud along the meshing line of the gear pair, the length of f at this time is fa and fa=Bud/cosα′. The above equations pb, rb,1, αα,1, α′, and Bud are the base pitch of the gear O1, 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 O1 and gear O2.


## 2.2. Mathematical Knowledge

Figure2 is an outline of an internal gear processed by a forming method. With the center of the circle O2 as the origin of the coordinates and the symmetry line of a cogging as the y-axis, a rectangular coordinate system xO2y is established. This coordinate system can be rotated around point O2 with a rotation angle of θ. The t1t2 line segment is an involute equation, and the r1r2 line segment is a transition curve. Point t can slide on the line segment t1t2 arbitrarily, and the direction is to move from point t1 to point t2. Point r can slide on the line segment r1r2 arbitrarily, and the direction is to move from point r1 to point r2. Suppose αx, rx, xx, yx, and θx 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 ∠t1o2t=∅, ∠r1o2t=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]:(1)∅=tan∝t−∝t−tan∝t1−∝t1,Sinv=∫rt2ϕ2dϕ.Figure 2
Internal gear profile.Let the fan-shaped area enclosed by the line segmentt1t2 and the radius of point t1 and point t2 be expressed as Sinvt2t1, which can be deduced from (1), Sinvt1t2=∫rt2∅/2d∅=rb,12/2∫αt1αt2sec4αt−sec2αtd∝t=rb,12tanαt23−tanαt13/6.In rectangular coordinate system,ζ=tan−1xr/yr−tan−1xr1/yr1. Suppose that the fan-shaped area enclosed by the contour of the line segment r1r2 and the radius of point r1 and point r2 can be expressed as Sstar1r2. According to [14], Ssta=r2r1r′2/6θr23−θr13−mha0∗/2r′−mha0∗θr2−θr1 the above formula, ha0∗ 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 ra,1,ra,2,rf,1,rf,2 be the radius of the top circle and root circle of the external gear and internal gear. Let ∠a1o1a2=σ, ∠a5o1a6=τ,∠c1o2c4=μ,∠c2o1c3=λ,∠c5o1c6=ω, and the corresponding sector areas are Stop2, Sroot1. Then, Stop2=0.5ra,12σ,Sroot1=0.5rf,12τ. 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, Stop1 and Sroot1 are fixed values. Similarly, the sector areas corresponding to the top and root circles of the internal gear are Stop2 and Sroot2. Then, Stop2=0.5ra,22λandSroot1=0.5rf,22ω.Figure 3
Gear profile machined by unrolling.
(a)(b)


## 3. Establishing the Calculation Model of Unloading Area

### 3.1. Mathematical Model

Figure4(a) is a pair of internal gear pairs and e is the center distance. The Cartesian coordinate systems x1O1y1 and x2O2y2 are established by using the gear circle centers O1 and O2 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 O1 and O2 are θ1 and θ2, respectively. During the rotation, the outline of the external gear, the profile of the internal gear, and ud intersect at points w and v, respectively. Among them, S∩wo1n1 and S∩vo2n1 are the radius of point w, point n1, point v, and the area enclosed by the contour line of the gear, respectively. For ease of calculation, let S1=S∩vo2n1,S2=SΔo2vn1,S3=SΔwvn1,S4=S∩ωo1n1,andS5=SΔo1ωn1. Then, fold O2v and O2n1 along vn1 to get Figure 4(b). According to the graph, the unloading area Sud can be expressed as(2)Sud=S1−S2+S3−S4−S5.Figure 4
A pair of internal gears meshing.
(a)(b)Let∠po1w=β1,∠n1o1w=γ1,∠po2v=β2,∠n1o2v=γ2. rp,1,rp,2,rn1,1,rn1,2 are the pitch circle radius and the meshing point radius of O1 and O2, respectively. lvn1,lwn1,andlvw are distances between points v, w, and n1. From the geometric relationship in the figure,(3)rn1,12=rp,12+f2−2rp,1fcosπ2+α′,lwn12=rn1,12+rw2−2rn1,1rwcosβ1,rn1,22=rp,22+f2−2rp,2fcosπ2+α′,lvn1=rp,22+f2−2rp,2fcosπ2+α′,lvw=rwcosγ2−rvcosγ1−e.Let(4)ρ1=rw+rn1,1+lwn12,ρ2=rv+rn1,2+lvn12,ρ3=lwn1+lvn1+lwv2.From Heron’s formula,(5)S2=ρ2ρ2−rvρ2−rn1,2ρ2−lvn11/2,S3=ρ3ρ3−lwn1ρ3−lwvρ3−lvn11/2,S5=ρ1ρ1−rwρ1−rn1,1ρ1−lwn11/2.Apply sine theorem inΔwo1n1,Δvo2n1:(6)fsinβ1+γ1=rp,1sinπ−β1−γ1−π/2+α′,fsinβ2+γ2=rp,2sinπ−β2−γ2−π/2+α′.Deduce(7)γ1=tan−1fcosα′rp,1+fsinα′−β1,γ2=tan−1fcosα′rp,2+fsinα′−β2.Sinceθ1,θ2, and f are linear functions, then from (2)–(7), only the changes of rw and β1 with θ1,rv and β2 with θ2 can be used to obtain the change of Sud.


### 3.2. External Gear Rotation Parameters

Whena1 and a2 are located on ud, β1 can be expressed as β1,a1=β1,a2=sin−1Bud/ra,1. When points n0,a1,a2,andn1 are located on ud, these rotation angles θ1 can be expressed as θ1,a=fa−f0/rb,1,θ1,a1=β1,a1−σ/2,θ1,a2=β1,a2+σ/2,θ1,g=fg−f0/rb,1. According to the structure of the external gear, when the point w moves to n0a1, a2n1 involute segment, according to the backlash-free meshing equation [17] and Figure 3(a), it can be obtained that (8)θ1−β1=π+4k1tanα2z1−invαw−invα,sinβ1=Bud/rb,1cosαw.Ifθ1≥β1, it means that the w point is on the n0a1 line segment. If θ1<β1 , it means that the w point is on the a2n1 line segment.When the point w is on thea1a2 line segment, rw=ra,1,β=sin−1Bud/ra,1.


### 3.3. Internal Gear Rotation Parameters

Ifc1 and c4 are located on ud, β2can be expressed as β2,c1=β2,c4=sin−1Bud/rf,2 .When the rotation process ofO2, several special points n0, c1, c2, c3, c4, n1, is located on ud, the corresponding rotation angle θ2 changes to, (9)θ2,a=fa−f0rb,2,θ2,c4=β2,a4−μ2,θ2,c3=θ2,c4+λ,θ2,c2=θ2,c3+μ−2λ,θ2,c1=θ2,c2+λ,θ2,g=fg−f0rb,2.θ2,a⟶θ2,g is represented, respectively, as the process from f to the end of θ2 when fa⟶fg.According to the internal gear structure, when thev point moves to the n0c1 and c4n1 involute segments, the backlash-free meshing equation is combined with Figure 3(a):(10)θ2−β2=π−4k2tanα2z2+invαv−invα,sinβ2=Bud/rb,2cosαv.Ifθ2≥β2, it means that the v point is on the n1c4 segment. If θ2<β2, it means that the v point is on the c1n0 segment. When the v point is at the transition curve c1c2, c3c4, as shown in the figure:(11)θ2−β2=∠y2o2c3+∠c3o2v=μ2−λ+tan−1xvyv.Ifθ2≥β2 , it means that the v point is in the c3c4 segment. If θ2<β2, it means that the v point is in the c1c2 segment. (xv,yv) is the coordinate value of the v point in the coordinate system of the figure.When thev point is at the root circle c2c3, β2=β2,c2=β2,c3, and rv=rf,2, where rf,2 is the radius of the O2 root circle. Therefore,(12)S4=0,θ1≤θ1,a,Sinvt1=n1t2=w,θ1,a<θ1≤θ1,a2,Sinv1+θ1−θ1,a1σStop1,θ1,a2<θ1≤θ1,a1,Sinv1+Stop1+Sinvt1=n0t2=w,θ1,a1<θ1≤θ1,g,S1=0,θ2≤θ2,a,Sinvt1=n1t2=v,θ2,a<θ2≤θ2,c4,Sinv2+Sstar1=c4r2=v,θ2,c4<θ2≤θ2,c3,Sinv2+Ssta2+θ2−θ2,c30.5μ−λSroot2,θ2,c3<θ2≤θ2,c2,Sinv2+Ssta2+Sroot2+Sstar1=c2r2=v,θ2,c2<θ2≤θ2,c1,Sinv2+2Ssta2+Sroot2+Sinvt1=n0t2=v,θ2,c1<θ1≤θ2,g.


## 3.1. Mathematical Model

Figure4(a) is a pair of internal gear pairs and e is the center distance. The Cartesian coordinate systems x1O1y1 and x2O2y2 are established by using the gear circle centers O1 and O2 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 O1 and O2 are θ1 and θ2, respectively. During the rotation, the outline of the external gear, the profile of the internal gear, and ud intersect at points w and v, respectively. Among them, S∩wo1n1 and S∩vo2n1 are the radius of point w, point n1, point v, and the area enclosed by the contour line of the gear, respectively. For ease of calculation, let S1=S∩vo2n1,S2=SΔo2vn1,S3=SΔwvn1,S4=S∩ωo1n1,andS5=SΔo1ωn1. Then, fold O2v and O2n1 along vn1 to get Figure 4(b). According to the graph, the unloading area Sud can be expressed as(2)Sud=S1−S2+S3−S4−S5.Figure 4
A pair of internal gears meshing.
(a)(b)Let∠po1w=β1,∠n1o1w=γ1,∠po2v=β2,∠n1o2v=γ2. rp,1,rp,2,rn1,1,rn1,2 are the pitch circle radius and the meshing point radius of O1 and O2, respectively. lvn1,lwn1,andlvw are distances between points v, w, and n1. From the geometric relationship in the figure,(3)rn1,12=rp,12+f2−2rp,1fcosπ2+α′,lwn12=rn1,12+rw2−2rn1,1rwcosβ1,rn1,22=rp,22+f2−2rp,2fcosπ2+α′,lvn1=rp,22+f2−2rp,2fcosπ2+α′,lvw=rwcosγ2−rvcosγ1−e.Let(4)ρ1=rw+rn1,1+lwn12,ρ2=rv+rn1,2+lvn12,ρ3=lwn1+lvn1+lwv2.From Heron’s formula,(5)S2=ρ2ρ2−rvρ2−rn1,2ρ2−lvn11/2,S3=ρ3ρ3−lwn1ρ3−lwvρ3−lvn11/2,S5=ρ1ρ1−rwρ1−rn1,1ρ1−lwn11/2.Apply sine theorem inΔwo1n1,Δvo2n1:(6)fsinβ1+γ1=rp,1sinπ−β1−γ1−π/2+α′,fsinβ2+γ2=rp,2sinπ−β2−γ2−π/2+α′.Deduce(7)γ1=tan−1fcosα′rp,1+fsinα′−β1,γ2=tan−1fcosα′rp,2+fsinα′−β2.Sinceθ1,θ2, and f are linear functions, then from (2)–(7), only the changes of rw and β1 with θ1,rv and β2 with θ2 can be used to obtain the change of Sud.


## 3.2. External Gear Rotation Parameters

Whena1 and a2 are located on ud, β1 can be expressed as β1,a1=β1,a2=sin−1Bud/ra,1. When points n0,a1,a2,andn1 are located on ud, these rotation angles θ1 can be expressed as θ1,a=fa−f0/rb,1,θ1,a1=β1,a1−σ/2,θ1,a2=β1,a2+σ/2,θ1,g=fg−f0/rb,1. According to the structure of the external gear, when the point w moves to n0a1, a2n1 involute segment, according to the backlash-free meshing equation [17] and Figure 3(a), it can be obtained that (8)θ1−β1=π+4k1tanα2z1−invαw−invα,sinβ1=Bud/rb,1cosαw.Ifθ1≥β1, it means that the w point is on the n0a1 line segment. If θ1<β1 , it means that the w point is on the a2n1 line segment.When the point w is on thea1a2 line segment, rw=ra,1,β=sin−1Bud/ra,1.


## 3.3. Internal Gear Rotation Parameters

Ifc1 and c4 are located on ud, β2can be expressed as β2,c1=β2,c4=sin−1Bud/rf,2 .When the rotation process ofO2, several special points n0, c1, c2, c3, c4, n1, is located on ud, the corresponding rotation angle θ2 changes to, (9)θ2,a=fa−f0rb,2,θ2,c4=β2,a4−μ2,θ2,c3=θ2,c4+λ,θ2,c2=θ2,c3+μ−2λ,θ2,c1=θ2,c2+λ,θ2,g=fg−f0rb,2.θ2,a⟶θ2,g is represented, respectively, as the process from f to the end of θ2 when fa⟶fg.According to the internal gear structure, when thev point moves to the n0c1 and c4n1 involute segments, the backlash-free meshing equation is combined with Figure 3(a):(10)θ2−β2=π−4k2tanα2z2+invαv−invα,sinβ2=Bud/rb,2cosαv.Ifθ2≥β2, it means that the v point is on the n1c4 segment. If θ2<β2, it means that the v point is on the c1n0 segment. When the v point is at the transition curve c1c2, c3c4, as shown in the figure:(11)θ2−β2=∠y2o2c3+∠c3o2v=μ2−λ+tan−1xvyv.Ifθ2≥β2 , it means that the v point is in the c3c4 segment. If θ2<β2, it means that the v point is in the c1c2 segment. (xv,yv) is the coordinate value of the v point in the coordinate system of the figure.When thev point is at the root circle c2c3, β2=β2,c2=β2,c3, and rv=rf,2, where rf,2 is the radius of the O2 root circle. Therefore,(12)S4=0,θ1≤θ1,a,Sinvt1=n1t2=w,θ1,a<θ1≤θ1,a2,Sinv1+θ1−θ1,a1σStop1,θ1,a2<θ1≤θ1,a1,Sinv1+Stop1+Sinvt1=n0t2=w,θ1,a1<θ1≤θ1,g,S1=0,θ2≤θ2,a,Sinvt1=n1t2=v,θ2,a<θ2≤θ2,c4,Sinv2+Sstar1=c4r2=v,θ2,c4<θ2≤θ2,c3,Sinv2+Ssta2+θ2−θ2,c30.5μ−λSroot2,θ2,c3<θ2≤θ2,c2,Sinv2+Ssta2+Sroot2+Sstar1=c2r2=v,θ2,c2<θ2≤θ2,c1,Sinv2+2Ssta2+Sroot2+Sinvt1=n0t2=v,θ2,c1<θ1≤θ2,g.


## 4. Optimized Design

If the internal gear pair parameters and center distance are determined, then,S2,S3,andS5 are determined. The design goal hopes to optimize and increase Sud. According to the aforementioned calculation model of Sud, it can be achieved by increasing S1 or decreasing S4. Considering the actual situation, S1 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 rfa, and the corresponding sector area is Sroot2a=0.5rfa2λ. During the rotation, the sector enclosed by the radii v and n1 and their contour is S1a:(13)S1a=0,θ2≤θ2,a,Sinvt1=n1t2=v,θ2,a<θ2≤θ2,e4,Sinv2+Sstar1=vr2=e4,θ2,e4<θ2≤θ2,e3,Sinv2+Ssta2+θ2−θ2,e30.5μ−λSroot2a,θ2,e3<θ2≤θ2,e2,Sinv2+Ssta2+Sroot2a+Sstar1=vr2=e2,θ2,e2<θ2≤θ2,e1,Sinv2+2Ssta2+Sroot2a+Sinvt1=vt2=n0,θ2,e1<θ1≤θ2,g.Figure 5
Optimizing the internal gear profile. (a) The first optimization scheme and (b) the second optimization scheme.
(a)(b)The unloading area of the first scheme isSuda. S1a can be derived by replacing S1 in (2), then, obtaining Suda.Let the radius of the root circle of Figure5(b) be rfb, and its corresponding sector area is sroot2b, sroot2b=0.5rfb2λ. During its rotation, the radii of points v and n1 and their corresponding contour lines are enclosed in a fan shape, and their area is s1b :(14)s1b=0,θ2≤θ2,a,sinvt1=n1t2=v,θ2,a<θ2≤θ2,q4,sinv2+θ2−θ2,q1λsroot2b,θ2,q4<θ2≤θ2,q1,sinv2+sroot2b+sinvt1=vt2=n0,θ2,q1<θ2≤θ2,g.The unloading area of the second scheme isSudb. Sudb can be derived by replacing S1 in (2) with S1b.


## 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 Table1.Table 1
Internal gear pump parameters.
Name of the parameterParameterValueNumber of teethz1/z213/19Modulus (mm)m3Center distance (mm)e9.278Pressure angle (°)α20Coefficient of rack cutter tip heightha0∗1.25External gear profile shift coefficientK10.432Internal gear profile shift coefficientK20.55Unloading groove boundary distance (mm)Bud2.035The first solution root radius (mm)rfa32.0The second solution root radius (mm)rfb33.80During the gear rotation, the position variablef 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.Figure 6
Changes in unloaded area.The parameters provided in Table1 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 Sud in Figure 6, the change law of Sud 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 Sud.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 ofSuda and Sudb 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 Sud during the oil trapping process. The larger the radius of the root circle, the larger Sud. 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.



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*Source: 7319871-2020-08-26.xml*