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Abstract

A series of problems arise when a gear pump operates at high speed, including instability of the rotor, deformation of the chamber, and wear of the journal bearing. Among all failure modes, journal bearing wear is the most serious. The wear of journal bearings of a circular arc gear pump that operates at high speed is thus presented in this article. A journal bearing that offsets the unbalanced radial force is designed by analysis of the fluid and determination of eccentricity of the gear shaft. Experiments show that the wear of the new journal bearing is effectively reduced.

Keywords

Circular arc gear pump journal bearing unbalance radial force analysis of fluid eccentricity

Introduction

The development trend of the gear pump is that the pump is operated at high speed. However, a series of problems arise at high speed, resulting in research on the design of the tooth profile, research on processing methods, fluid analysis, and analysis of performance.^1–3 Some problems that arise when a gear pump operates at high speed have been solved theoretically. However, the wear of a gear pump is difficult to analyze theoretically. Moreover, the wear of a gear pump results in instability of the rotor, damage to the pump body, and an increase in the temperature of the hydraulic oil. Mucchi E et al.⁴ established a mathematical model that can be used to optimize the running-in process and to investigate the effect of wear of the pump casing. Koç E and Hooke C J⁵ studied the wear of loaded floating.

$\overset{\cdot}{I}$ mrek H and Düzcükoğlu H⁶ reduce noises and vibrations caused by wear in spur gears by tooth modification. Thiagarajan D et al.⁷ improve lubrication performance and reduces wear in external gear pump by tooth profile design. Kwon S-M et al.⁸ optimized the shape of a gear using a genetic algorithm to reduce contact stress and gear wear. Wang X et al.⁹ investigated the wear of a gear pump using omega theory and validated their results with experimental results. Frith RH and Scott W¹⁰ verified the wear model of a gear pump with experimental results. Koç E et al.¹¹ studied the lubrication of bush-type bearings operating at high pressure, revealing that lubrication between end faces of the gear and bush-type bearings depends on the surface irregularities and non-flatness. Cha M et al.¹² studied the effect of the tilt of journal bearings and showed the relationship between the thickness and pressure of the oil film. Yang H et al.¹³ discussed the effects of installation errors, hydraulic oil contamination, and manufacturing precision on the wear of journal bearings. Jolly P et al.¹⁴ studied the characteristics of a hydrostatic journal bearing for an aerospace turbo pump. The angle of the oil injection hole and the shape and size of the internal groove were determined by numerical analysis. At the same time, the flow rate of the sliding bearing was measured. Static and dynamic experiments verified the correctness of the theory. Naïmi S et al.¹⁵ focused on the load capacity of journal bearings using a circumferential central feeding groove and discussed the effect of the pressure of the oil film and eccentricity of journal bearings. Dhawan R and Verma S¹⁶ studied journal bearing lubrication employing a finite element method for hydrostatic. The bearing behavior of a hydrostatic bearing under a heavy load was analyzed from the viewpoint of tribology, and a method of improving the bearing capacity of the interaction bearing was discussed. Costa L et al.¹⁷ through set the oil groove on hydrostatic bearing to improve lubrication. The above studies revealed that the most common form of failure is wear between the inner wall of a journal bearing and the rotor.

In all wear of a gear pump, the journal bearings are obvious. This wear is caused by an unbalanced radial force. However, the research cited above rarely mentioned how to solve the effect of unbalanced radial force acting on a gear pump. The present paper thus designs journal bearings that offset part of the radial force. The design is verified experimentally.

Design of radial-force-compensating journal bearings

This section uses a circular arc gear pump as a basic model. The radial force is determined by the design of the teeth and obtained by establishing a mathematical model of high- and low-pressure zones. Journal bearings that offset the unbalanced radial force are then designed.

Basic model of a circular arc gear pump and journal bearing

The circular gear pump comprises a driving gear, driven gear, journal bearings, pump body, front end cover, rear end cover, and oil seal, as shown in Figure 1. A three-dimensional model of a journal bearing is shown in Figure 2.

Figure 1.

Structure of a circular arc gear pump.

Figure 2.

Three-dimensional model of journal bearing.

The circular arc tooth is formed by circular arc AA′, involute $A' B'$ , and circular arc BB′, as shown in Figure 3.

Figure 3.

Coordinate systems of the circular arc tooth.

The circular arc AA′ is expressed by³

\begin{array}{l} x_{0} (u) = r \cdot \cos ({\overset{⌢}{u}}_{1}) \\ y_{0} (u) = r \cdot \sin ({\overset{⌢}{u}}_{1}) + R \end{array}}

(1)

where R is the radius of the pitch circle, r is the radius of circular arc, and ${\overset{⌢}{u}}_{1}$ is the variable of circular arc AA′.

The involute A′B′ is expressed by³

\begin{array}{l} x_{0} (u) = u \cdot R_{b} \cdot \cos (u - θ_{0}) - R_{b} \cdot \sin (u - θ_{0}) \\ y_{0} (u) = u \cdot R_{b} \cdot \sin (u - θ_{0}) + R_{b} \cdot \cos (u - θ_{0}) \end{array}}

(2)

where $R_{b}$ is the radius of the base circle, angle $θ_{0}$ is formed by the y-axis, involute, and coordinate origin O, and u is a variable of involute A′B′.

The circular arc BB′ is expressed by³

\begin{array}{l} x_{0} (u) = - r \cdot \cos ({\overset{⌢}{u}}_{2}) + R \cdot \sin (φ_{0}) \\ y_{0} (u) = - r \cdot \sin ({\overset{⌢}{u}}_{2}) + R \cdot \cos (φ_{0}) \end{array}}

(3)

where $φ_{0} = π / Z$ , with Z being the number of teeth.

The helical surface is then expressed by³

\begin{matrix} x = x_{0} (u) \cos (θ) - y_{0} (u) \sin (θ) \\ y = x_{0} (u) \sin (θ) + y_{0} (u) \cos (θ) \\ z = \pm p \cdot θ + z_{0} (u) \end{matrix}}

(4)

where $θ$ is a variable parameter, p is a helical parameter, and $y_{0} (u)$ and $z_{0} (u)$ are coordinates of the transverse tooth profile. The symbol “+” denotes the driving gear and “–” denotes the driven gear. The helical gear is shown in Figure 4.

Figure 4.

Circular arc helical gear.

Determination of number of teeth of high- and low-pressure zones

The radial force is generated by the effect of high-pressure hydraulic oil on the surfaces of the gears, and the number of teeth of the high- and low-pressure zones must therefore be determined. The zones of high and low pressure depend on the outlet, inlet, and chamber. The mathematical model of the inlet and outlet is shown in Figure 5.

Figure 5.

Mathematical model of the inlet and outlet.

The length of the connect edge in the low-pressure zone is denoted $L'$ while the length of the connect edge in the high-pressure zone is denoted as L. The connect edges L and L′ are obtained by

{\begin{matrix} L = \frac{D_{out}}{2} \\ L' = \frac{D_{in}}{2} \end{matrix}

(5)

where $D_{out}$ is outlet diameter and $D_{in}$ is inlet diameter.

The outlet is mathematically expressed by

ϕ_{out} = {\begin{matrix} {ϕ'}_{out} = \arccos (\frac{a - L}{2 R_{e}}), 0 ⩽ z ⩽ h or B - h ⩽ z ⩽ B \\ \arccos (\frac{a - \sqrt{D_{out}^{2} - {(B - 2 z)}^{2}}}{2 R_{e}}), h ⩽ z ⩽ B - h \end{matrix}

(6)

where a is the central separation of the gears, $D_{out}$ is the diameter of the outlet, and B is the width of the chamber.

The inlet is mathematically expressed by

ϕ_{in} = {\begin{matrix} {ϕ'}_{in} = \arccos (\frac{a - L'}{2 R_{e}}), 0 ⩽ z ⩽ h' or B - h' ⩽ z ⩽ B \\ \arccos (\frac{a - \sqrt{D_{in}^{2} - {(B - 2 z)}^{2}}}{2 R_{e}}), h' ⩽ z ⩽ B - h' \end{matrix}

(7)

where $D_{in}$ is the diameter of the inlet. Angles $ϑ_{out}$ and $ϑ_{in}$ in Figure 5 are

ϑ_{out} = (\frac{2 π}{Z} n_{Fout} - φ) + \frac{z}{p}

(8)

ϑ_{in} = (\frac{2 π}{Z} n_{Fin} + φ) - \frac{z}{p}

(9)

where $n_{Fout}$ is the number of teeth in the high-pressure zone and $n_{Fin}$ is the number of teeth in the low-pressure zone. $φ$ is the rotational angle of the gear, $φ \in [0, 2 π / Z]$ . To ensure the seal along the face with, the rotational angle of the gear should meet the condition $ψ > ϕ$ .

When $0 ⩽ z ⩽ h or B - h ⩽ z ⩽ B$ , the inlet needs to meet ${ϑ_{out} |}_{z = 0} > ϕ'_{in}$

n_{Fout} > \frac{\arccos (\frac{a - L}{2 R_{e}}) + φ}{2 π / Z}

(10)

n_{Fin} > \frac{\arccos (\frac{a - L'}{2 R_{e}}) - φ}{2 π / Z}

(11)

When $0 ⩽ z ⩽ h' or B - h' ⩽ z ⩽ B$ , the outlet needs to meet ${ϑ_{out} |}_{z = B} > ϕ'_{out}$ . It follows that

n_{Fout} > \frac{\arccos (\frac{a - L}{2 R_{e}}) - \frac{B}{p} + φ}{2 π / Z}

(12)

n_{Fin} > \frac{\arccos (\frac{a - L'}{2 R_{e}}) + \frac{B}{p} - φ}{2 π / Z}

(13)

When $h ⩽ z ⩽ B - h$ , we set $z_{0}$ . If $z_{0} \in [h, B - h]$ , then the number of teeth in the high-pressure zone is

n_{Fout} > \frac{{ϕ_{out} |}_{z = z_{0}} - \frac{z_{0}}{p} + φ}{2 π / Z}

(14)

When $h' ⩽ z ⩽ B - h'$ , we set $z'_{0}$ . If ${z_{0}}^{'} \in [h', B - h']$ , then the number of teeth in the low-pressure zone is

n_{Fin} > \frac{{ϕ_{in} |}_{z = {z'}_{0}} + \frac{z_{0}}{p} - φ}{2 π / Z}

(15)

Calculation of the radial force and eccentricity

Because the tooth surface used in this article is helical, the infinitesimal tooth surface is studied to facilitate the calculation. The normal vector and unit normal vector for a helical surface are

{\begin{matrix} n_{x} = p ({x'}_{0} \sin (θ) + {y'}_{0} \cos (θ)) \\ n_{y} = - p ({x'}_{0} \cos (θ) - {y'}_{0} \sin (θ)) \\ n_{z} = x_{0} {x'}_{0} + y_{0} {y'}_{0} \\ e_{x} = \frac{n_{x}}{\sqrt{{n_{x}}^{2} + {n_{y}}^{2} + {n_{z}}^{2}}} \\ e_{y} = \frac{n_{y}}{\sqrt{{n_{x}}^{2} + {n_{y}}^{2} + {n_{z}}^{2}}} \\ e_{z} = \frac{n_{z}}{\sqrt{{n_{x}}^{2} + {n_{y}}^{2} + {n_{z}}^{2}}} \end{matrix}

(16)

The radial force is

d F_{r} = (\frac{x}{\sqrt{x^{2} + y^{2}}} e_{x} + \frac{y}{\sqrt{x^{2} + y^{2}}} e_{y}) \cdot \hat{p} \cdot ds

(17)

The radial forces generated by the hydraulic force in directions x and y are

F_{rx} = \int \int \frac{x}{\sqrt{x^{2} + y^{2}}} e_{x} \cdot \hat{p} ds

(18)

F_{ry} = \int \int \frac{y}{\sqrt{x^{2} + y^{2}}} e_{y} \cdot \hat{p} \cdot ds

(19)

where $\hat{p}$ is the hydraulic pressure that is related to the working area, and the pressure is reduced arithmetic from the high-pressure zone to low-pressure zone, as shown in Figure 6.

Figure 6.

Distribution of hydraulic pressure.

In Figure 6, the pressures of high-pressure zone and low-pressure zone are $P_{H}$ and $P_{L}$ , respectively. The number of teeth of high-pressure zone and low pressure zone are chosen by equations (8)–(15). The above analysis shows that the direction of the overall radial force acting on the gear is from the high-pressure zone to the low-pressure zone. This article develops a new journal bearing that reduces the effect of the unbalanced radial force. Figure 7 shows the structure of the journal bearings that are used to offset part of the radial force. The high-pressure oil enters the pressure chamber from the bearing inlet, the pressure in the pressure chamber generates a force, and the direction of the force is from the low-pressure zone to the high-pressure zone.

Figure 7.

Three-dimensional model of a hydrostatic journal bearing.

Figure 8 shows a force diagram of a hydrostatic journal bearing. The overall radial force Fr is

Fr = \sqrt{{Fr}_{x}^{2} + {Fr}_{y}^{2}}

(20)

where $F r_{x}$ is the component in the x direction and $F r_{y}$ is the component in the y direction.

Figure 8.

Force diagram of a hydrostatic journal bearing.

The high pressure in the pressure chamber acts on the gear shaft, producing an upward force $F j_{y}$ expressed as

F j_{y} = Δ pL B_{J}

(21)

where $Δ p$ is the pressure in the pressure chamber, L is the length of the pressure chamber, and $B_{J}$ is the width of the pressure chamber.

The force $F j_{x}$ is the bearing reaction in the x direction and is expressed by

F j_{x} = - F r_{x}

(22)

The new radial force is

Fr' = \sqrt{{Fr}_{x}^{2} + {(F r_{y} - F j_{y})}^{2}}

(23)

The maximum radial forces under different pressures are obtained using the above equations. Values are given in Table 1.

Table 1.

Maximum radial forces and eccentricity under different pressures.

Pressure (MPa)	1.5	2.0	2.5	3.0
Maximum radial force (N)	119.2	160.7	202.1	243.5
Eccentricity	0.1145	0.027	0.035	0.043

Eccentricity is caused by the weight of the gear shaft and hydraulic pressure, and it generates dynamic pressure when the gear pump rotates, as shown in Figure 9(a).

Figure 9.

Position of the gear shaft center and journal bearing radially unfold.

Figure 9(b) shows the unfold of the journal bearing along the radial direction. The oil film force is calculated using the Reynolds equation, expressed as

\frac{\partial}{\partial θ} (h^{3} \frac{\partial p}{\partial θ}) + R_{r}^{2} \frac{\partial}{\partial y} (h^{3} \frac{\partial p}{\partial y}) = - 6 μ R_{r}^{2} w \frac{dh}{d θ}

(24)

where $R_{r}$ is the radius of the gear shaft and $μ$ is the viscosity of the hydraulic oil. The eccentricity under different pressures is obtained using equation (24) and Figure (10); values are given in Table 1.

Figure 10.

The flow chart of calculation of eccentricity.

Figure 11 shows the center position of gear shaft under the different outlet pressures. The results show the offset distance increase with outlet pressures.

Figure 11.

The center of gear shaft under different pressures.

Simulation of fluid

This section conducts numerical simulation to study the internal pressure of journal bearings because it is difficult to obtain the real pressure in a pressure chamber experimentally. The oil film model of the hydrostatic journal bearing is shown in Figure 12. The position and eccentricity of oil film are obtained by Table 1 and Figure 10. The design parameters of the pressure chamber are given in Table 2.

Figure 12.

Oil film model of the hydrostatic journal bearing.

Table 2.

Geometrical parameters of journal bearings.

Radius of oil film in outer wall (mm)	Radius of oil film in inner wall (mm)	Face width (mm)	Rotational speed (r/min)
12.02	11.97	20	8000

The bearing inlet pressure is set to the pressure of the high-pressure zone, and the rotational speed of the oil film at the inner wall is set to the rotational speed of the gear shaft. The oil film at the outer wall maintains a constant state. The computation domain and mesh are shown in Figure 13. The mesh type is tetra/mixed. There are 2,750,355 cells in total. The outlet pressure is set to standard atmospheric pressure of 0.1 MPa. The computation model is laminar and oil is incompressible. A pressure-based solver and implicit method are employed.

Figure 13.

Mesh of the hydrostatic journal bearing.

Figure 14 shows contours of pressure at different inlet pressures. The pressure drops are caused from inlet into pressure chamber.

Figure 14.

Contours of static pressure at different inlet pressures: (a) 1.5 MPa, (b) 2 MPa, (c) 2.5 MPa, and (d) 3 MPa.

Figure 14 shows the pressure drops when the inlet pressure of the journal bearing is 1.5, 2, 2.5, and 3 MPa. The pressure chamber is designed according to the simulation results.

Comparison experiments of radial-force-compensating journal bearings

Experiments were carried out to verify the effect of the journal bearing with a pressure chamber. The experiment prototype and journal bearing with a pressure chamber are shown in Figure 15.

Figure 15.

Prototype of a circular arc gear pump.

Figure 16 shows the test platform. The rotational speed is obtained by a torque speed sensor and the different gear pump outlet pressures are produced by adjusting a throttle valve and overflow valve. The flow rates of the prototype at different rotational speeds are obtained by a flow sensor.

Figure 16.

Test platform.

Figure 17 shows the flow rates of the journal bearings with and without a pressure chamber at different rotational speeds. The outlet pressure of test pump is 2 MPa. The curves of the flow rate basically coincide when the rotational speed is lower than 8000 r/min.

Figure 17.

Flow rates with and without the pressure chamber.

Figure 18 shows the volume efficiency of gear pump with and without journal bearing pressure chamber. The curves of the volume efficiency basically coincide when the rotational speed is lower than 8000 r/min.

Figure 18.

Volume efficiency of gear pump with and without journal bearing pressure chamber under different rotational speeds.

Figure 19 shows the wear of journal bearings with and without a pressure chamber. The wear of the journal bearing without a pressure chamber is greater than that of the journal bearing with a chamber. Moreover, the gear shaft stuck when using the journal bearing without a pressure chamber at a rotational speed of 7500 r/min. The above analysis thus shows that the new journal presented in this article performed well.

Figure 19.

Journal bearings with and without a pressure chamber. (a) without a pressure chamber and (b) with a pressure chamber.

Conclusion

This article studied the radial force and radial force fluctuations of a circular arc gear pump and developed a new journal bearing that offsets the unbalanced radial force and reduces wear. The design of the journal bearing is based on the radial force. The radial force is mathematically obtained by determining the high-pressure and low-pressure zones and the number of teeth in high-pressure and low-pressure zones. In addition, the fluid analysis of journal bearings was carried out. Results show that there are pressure drops from the inlet into the pressure chamber, and the pressure drops increase with increasing inlet. Finally, an experiment was conducted to compare journal bearings with and without a pressure chamber. Results show that wear is effectively reduced for the journal bearing with a pressure chamber.

Footnotes

Handling editor: Francesca Russo

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Yang Zhou

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The design and analysis of a high-speed circular arc gear pump journal bearing

Abstract

Keywords

Introduction

Design of radial-force-compensating journal bearings

Basic model of a circular arc gear pump and journal bearing

Determination of number of teeth of high- and low-pressure zones

Calculation of the radial force and eccentricity

Simulation of fluid

Comparison experiments of radial-force-compensating journal bearings

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References