Internal gear pumps are widely used in hydraulic systems due to their compact design, high efficiency, and low noise levels. As an internal gear manufacturer, we recognize the importance of precise displacement calculation for optimizing pump performance. In this article, we present a detailed derivation of the displacement formula for internal gears based on the relationship between rotation angle and meshing point, incorporating gear meshing principles. This approach provides a foundation for analyzing flow characteristics and designing efficient internal gear pumps.
The displacement of an internal gear pump is a critical parameter that influences its flow output and overall efficiency. Traditional methods often rely on empirical formulas, which may not account for all influencing factors. By deriving an exact formula, we aim to enhance the accuracy of displacement calculations, which is essential for internal gear manufacturers seeking to improve pump design. Our method involves analyzing the kinematics of gear meshing, particularly how the meshing point changes with rotation angle, leading to a comprehensive displacement expression.
To begin, consider the simplified working principle of an internal gear pump. The pump consists of an external gear (pinion) and an internal gear (ring gear), which rotate in the same direction. The instantaneous flow rate can be expressed based on energy conservation and geometric relationships. The general form of the instantaneous flow rate is given by:
$$q_{sh} = \frac{1}{2} B \omega \left( R_{e1}^2 – x^2 \right) + \left( y^2 – R_{e2}^2 \right) \left( \frac{R_1}{R_2} \right)^2$$
where \( B \) is the gear width, \( \omega \) is the angular velocity of the external gear shaft, \( x \) is the distance from the meshing point to the external gear center, \( y \) is the distance from the meshing point to the internal gear center, \( R_1 \) and \( R_2 \) are the pitch circle radii of the external and internal gears, respectively, and \( R_{e1} \) and \( R_{e2} \) are the tip circle radii of the external and internal gears. The variables \( x \) and \( y \) vary with the rotation angle \( \beta \), and establishing their relationship is key to deriving the displacement.
The relationship between rotation angle and meshing point is derived from the involute profile equations of the gears. For the external gear, the involute equation is:
$$r_K = \frac{R_{b1}}{\cos(\alpha)}$$
$$\theta = \text{inv}(\alpha) = \tan(\alpha) – \alpha$$
where \( r_K \) is the distance from point K on the involute to the base circle center, \( R_{b1} \) is the base circle radius of the external gear, \( \alpha \) is the pressure angle at point K, and \( \theta \) is the roll angle. When the external gear rotates by an angle \( \beta_1 \), the meshing point moves from K1 to K2. The pressure angle at K2, denoted \( \alpha_1′ \), can be expressed as:
$$\alpha_1′ = \arctan(\tan(\alpha_1) + \beta_1)$$
Thus, the distance from the meshing point to the external gear center after rotation is:
$$x = \frac{R_{b1}}{\cos(\alpha_1′)} = \frac{R_{b1}}{\cos(\arctan(\tan(\alpha_1) + \beta_1))}$$
Similarly, for the internal gear, when the external gear rotates by \( \beta_1 \), the internal gear rotates by \( \beta_2 = \frac{R_1}{R_2} \beta_1 \). The pressure angle at the new meshing point K2 for the internal gear is:
$$\alpha_2′ = \arctan(\tan(\alpha_2) + \beta_2)$$
and the distance to the internal gear center is:
$$y = \frac{R_{b2}}{\cos(\alpha_2′)} = \frac{R_{b2}}{\cos(\arctan(\tan(\alpha_2) + \beta_2))}$$
Substituting these into the instantaneous flow rate formula, we obtain:
$$q_{sh} = \frac{1}{2} B \omega \left[ R_{e1}^2 – \left( \frac{R_{b1}^2}{\cos^2(\arctan(\tan(\alpha_1) + \beta_1))} \right) + \left( \frac{R_{b2}^2}{\cos^2(\arctan(\tan(\alpha_2) + \beta_2))} – R_{e2}^2 \right) \left( \frac{R_1}{R_2} \right)^2 \right]$$
This expression shows how the instantaneous flow varies with the rotation angle \( \beta_1 \), given initial pressure angles \( \alpha_1 \) and \( \alpha_2 \) at the starting meshing point. For internal gear manufacturers, understanding this relationship is crucial for predicting pump behavior under different operating conditions.
The displacement of the pump is the volume of fluid displaced per revolution of the external gear. Considering the continuity of meshing and the fact that the gear pair has a contact ratio \( \varepsilon \) greater than 1, the effective rotation interval for displacement calculation is from \( \beta_1 = 0 \) to \( \beta_1 = \frac{2\pi}{z_1} \), where \( z_1 \) is the number of teeth on the external gear. The volume displaced by one tooth pair is:
$$V_i = \int q_{sh} \, dt = \frac{1}{\omega} \int_0^{\frac{2\pi}{z_1}} q_{sh} \, d\beta_1$$
Thus, the total displacement per revolution is:
$$V = 10^{-3} z_1 V_i = 10^{-3} \pi B \left[ R_{e1}^2 – R_{e2}^2 \left( \frac{z_1}{z_2} \right) – \frac{B R_{b1}^2}{2} \left( \frac{8\pi^3}{3z_1^2} + \frac{4\pi^2}{z_1} \tan(\alpha_1) + 2\pi \tan^2(\alpha_1) + 2\pi \right) + \frac{B R_{b2}^2}{2} \left( \frac{z_1}{z_2} \right) \left( \frac{8\pi^3}{3z_2^2} + \frac{4\pi^2}{z_2} \tan(\alpha_2) + 2\pi \tan^2(\alpha_2) + 2\pi \right) \right]$$
To compute this accurately, the initial pressure angles \( \alpha_1 \) and \( \alpha_2 \) must be determined. These depend on the gear geometry and meshing conditions. For a gear pair with modification, the operating pressure angle \( \alpha” \) and contact ratio \( \varepsilon \) are calculated as:
$$\alpha” = \arccos\left( \frac{a \cos(\alpha_n)}{a’} \right)$$
$$\varepsilon = \frac{1}{2\pi} \left[ z_1 (\tan(\alpha_{a1}) – \tan(\alpha”)) – z_2 (\tan(\alpha_{a2}) – \tan(\alpha”)) \right]$$
where \( a \) is the standard center distance, \( a’ \) is the actual center distance after modification, \( \alpha_n \) is the standard pressure angle, and \( \alpha_{a1} \) and \( \alpha_{a2} \) are the tip pressure angles of the external and internal gears, respectively. The initial pressure angles are then:
$$\alpha_1 = \arctan\left( \tan(\alpha”) – \frac{\varepsilon \pi}{z_1} \right)$$
$$\alpha_2 = \arctan\left( \tan(\alpha”) – \frac{\varepsilon \pi}{z_2} \right)$$
This derivation highlights that displacement is influenced by parameters such as module, number of teeth, addendum coefficient, pressure angle, gear width, and modification factors. For internal gear manufacturers, this precise calculation method enables better parameter optimization and flow pulsation analysis.
To facilitate practical application, we developed an interface program using LabVIEW, which allows users to input gear parameters and compute displacement easily. This tool is valuable for internal gear manufacturers and engineers, as it simplifies the complex calculations involved. The program incorporates the derived formulas and provides instant results, aiding in quick design iterations and performance evaluations.
We validated the displacement formula using parameters from three different internal gear pump models. The table below summarizes the key parameters and calculated displacements, comparing them to nominal values. The results show a maximum error of around 3%, confirming the formula’s accuracy.
| Parameter | Pump I | Pump II | Pump III |
|---|---|---|---|
| Nominal Displacement (mL/rev) | 50 | 16 | 10.2 |
| External Gear Teeth (z₁) | 12 | 13 | 13 |
| Internal Gear Teeth (z₂) | 18 | 20 | 19 |
| Module (mm) | 3.5 | 2.75 | 2.25 |
| Gear Width (mm) | 48 | 31 | 25.5 |
| External Tip Radius (mm) | 26 | 20.24 | 16.95 |
| Internal Tip Radius (mm) | 29.25 | 25.41 | 19.25 |
| Standard Pressure Angle (rad) | π/9 | 5π/36 | 5π/36 |
| Calculated Displacement (mL/rev) | 49.375 | 15.505 | 10.216 |
| Error (%) | 1.25 | 3.09 | 0.157 |
For comparison, we also evaluated approximate displacement formulas commonly used in the industry. One such formula, proposed by He Cunxing, approximates displacement as the volume between the tip and base circles of the external gear:
$$q = 2\pi z_1 m^2 B \times 10^{-3}$$
Another formula by Li Hongwei estimates displacement based on the annular volume between the external tip circle and the internal tip circle relative to the external gear center:
$$q = 10^{-3} \pi B \left[ R_{e1}^2 – (R_{e2} – a’)^2 \right]$$
We applied these to the same pump parameters, and the results are summarized below:
| Calculation Method | Pump I (mL/rev) | Pump II (mL/rev) | Pump III (mL/rev) |
|---|---|---|---|
| Nominal Displacement | 50 | 16 | 10.2 |
| Exact Formula (Eq. 9) | 49.375 | 15.505 | 10.216 |
| He’s Formula | 49.392 | 20.298 | 11.177 |
| Li’s Formula | 50.829 | 15.630 | 10.499 |
As shown, the approximate formulas provide reasonable estimates but lack the precision of our derived method. He’s formula, which ignores internal gear parameters, shows significant errors in some cases, while Li’s formula performs better but still deviates. This underscores the importance of using exact calculations for internal gears in design and analysis.
In conclusion, the precise displacement calculation method based on rotation angle and meshing point relationship offers significant advantages for internal gear manufacturers. It accounts for all relevant geometric parameters and provides a foundation for optimizing pump performance. The derived formula, validated with real pump data, demonstrates high accuracy, with errors within 3%. This approach not only improves displacement calculation but also supports advanced studies on flow pulsation and dynamic meshing forces. By integrating this method into design tools, internal gear manufacturers can enhance product reliability and efficiency, meeting the demands of modern hydraulic systems.
Furthermore, the interface program we developed using LabVIEW makes this complex calculation accessible to engineers, promoting its adoption in practical applications. Future work could explore the impact of parameter variations on displacement and flow characteristics, further refining pump designs. As the industry moves toward higher efficiency and lower noise, such precise methods will become increasingly valuable for internal gear manufacturers striving to innovate and compete in the global market.