In the field of hydraulic power transmission, gear pumps are widely utilized due to their compact structure, reliability, and low cost. Among them, external meshing helical gear pumps exhibit superior performance compared to spur gear pumps, primarily due to the introduction of a helix angle that reduces flow pulsation, vibration, and noise. The instantaneous flow rate and displacement characteristics of helical gear pumps are directly linked to their pressure ripple, efficiency evaluation, and noise generation. Therefore, an accurate and efficient method for calculating these characteristics is essential for the optimal design and performance prediction of helical gears. Although many studies have focused on spur gear pumps, relatively few have addressed the precise calculation of instantaneous flow and displacement for helical gears. In this work, based on the theory of gear meshing, involute tooth profile properties, and the relationship between the instantaneous meshing point and the rotation angle, a set of analytical formulas for the instantaneous flow rate and displacement of standard involute external meshing helical gear pumps is derived. The proposed method is validated through MATLAB simulations and experimental tests on three KF-type helical gear pump models.
1. Theoretical Foundation: Instantaneous Flow Rate of Spur Gear Pumps
Before extending to helical gears, it is necessary to recall the derivation for spur gear pumps. According to the gear meshing principle and the conservation of energy, the work done by the driving and driven gears equals the product of the pressure difference across the pump and the displaced fluid volume. For a spur gear pump, the instantaneous flow rate \( Q_{sh} \) can be expressed as:
\[
Q_{sh} = \frac{1}{2} B \omega \left[ (r_{a1}^2 – x^2) + (r_{a2}^2 – y^2) \frac{r_1}{r_2} \right]
\]
where \( B \) is the gear width, \( \omega \) is the angular velocity, \( r_{a1} \) and \( r_{a2} \) are the addendum circle radii of the driving and driven gears, \( x \) and \( y \) are the distances from the meshing point to the centers of the driving and driven gears, respectively, and \( r_1 \), \( r_2 \) are the pitch circle radii. Using the involute properties, the distances \( x \) and \( y \) can be expressed as functions of the rotation angle \( \varphi \). For a spur gear with base circle radius \( r_b \), pressure angle \( \alpha’ \), and number of teeth \( z \), the instantaneous flow rate becomes:
\[
Q_{sh} = \frac{1}{2} B \omega \left\{ r_a^2 – \left( \frac{r_b}{\cos[\arctan(\varphi + \tan\alpha’ – \pi / z)]} \right)^2 \right\} + \frac{1}{2} B \omega \left\{ r_a^2 – \left( \frac{r_b}{\cos[\arctan(\tan\alpha’ – \varphi + \pi / z)]} \right)^2 \right\}
\]
This classical formula serves as the foundation for the subsequent analysis of helical gears.
2. Instantaneous Flow Rate Calculation for Helical Gears
A helical gear can be considered as an infinite number of infinitely thin spur gear slices stacked along the face width, each rotated by a small angle relative to the previous one. Based on the superposition principle, the instantaneous flow rate of a helical gear pump can be obtained by integrating the flow contribution of each thin slice over the entire face width. The key is to relate the rotation angle of each slice to the reference angle at one end face.
Consider a helical gear with helix angle \( \beta \) and pitch circle radius \( R_w \). At a distance \( m \) from the reference end face, the angular shift due to the helix is:
\[
\varphi_m = \frac{m \tan \beta}{R_w}
\]
Thus, the instantaneous flow rate of a thin slice of thickness \( dm \) at distance \( m \) is:
\[
Q_{x\theta} = \frac{1}{2} \omega \left[ (r_a^2 – x^2) + (r_a^2 – y^2) \right] dm
\]
where the instantaneous meshing point distances depend on the local rotation angle \( \theta \pm \varphi_m \). For the specific rotation direction and helical gear orientation used in this study, the expression becomes:
\[
Q_{x\theta} = \frac{1}{2} \omega \left\{ r_a^2 – \left( \frac{r_b}{\cos[\arctan(\theta + \varphi_m + \tan\alpha’ – \pi / z)]} \right)^2 \right\} dm + \frac{1}{2} \omega \left\{ r_a^2 – \left( \frac{r_b}{\cos[\arctan(\tan\alpha’ + \pi / z – \theta – \varphi_m)]} \right)^2 \right\} dm
\]
The meshing process of helical gears is divided into two regions: non-full-width meshing and full-width meshing, depending on whether the entire face width is engaged. Over one tooth pitch angle \( 2\pi / z \), the non-full-width meshing occupies an angular interval of \( \Delta \varphi = B \tan \beta / R_w \). The instantaneous flow rate for the entire helical gear pump is then obtained by integrating the contributions of all slices that are currently meshing, taking into account the phase difference between successive teeth.
For the non-full-width meshing region (\( 0 \leq \varphi_m \leq B\tan\beta / R_w \)), the reference angle at the end face is zero, and the instantaneous flow rate is:
\[
Q_{sh1}^{helical} = \frac{R_w}{\tan\beta} \int_{0}^{\varphi_m} \frac{1}{2} \omega \left\{ r_a^2 – \left( \frac{r_b}{\cos[\arctan(\varphi – \pi / z + \tan\alpha’)]} \right)^2 \right\} d\varphi + \frac{R_w}{\tan\beta} \int_{0}^{\varphi_m} \frac{1}{2} \omega \left\{ r_a^2 – \left( \frac{r_b}{\cos[\arctan(\pi / z + \tan\alpha’ – \varphi)]} \right)^2 \right\} d\varphi
\]
For the full-width meshing region (\( B\tan\beta / R_w \leq \varphi_m \leq 2\pi / z \)), the entire face width is engaged, and the integration over \( m \) from 0 to \( B \) gives:
\[
Q_{sh2}^{helical} = \int_{0}^{B} \frac{1}{2} \omega \left\{ r_a^2 – \left( \frac{r_b}{\cos[\arctan(\theta_0 – \pi / z + \tan\alpha’)]} \right)^2 \right\} dm + \int_{0}^{B} \frac{1}{2} \omega \left\{ r_a^2 – \left( \frac{r_b}{\cos[\arctan(\tan\alpha’ – \theta_0 + \pi / z)]} \right)^2 \right\} dm
\]
When the contact ratio exceeds unity, the expressions are modified accordingly by replacing \( \pi/z \) with \( \varepsilon_n \pi / z \), where \( \varepsilon_n \) is the transverse contact ratio. The complete instantaneous flow curve of a helical gear pump is obtained by overlapping the contributions of all teeth with phase shifts of \( 2\pi / z \).
3. Displacement Calculation for Helical Gear Pumps
Neglecting the trapped volume effect, a helical gear pump always has at least one tooth pair in full-width discharge condition. The theoretical displacement per revolution \( q \) can be derived by integrating the full-width instantaneous flow rate over the time required for one tooth pair to complete its discharge, which corresponds to a rotation angle of \( 2\pi / z \). Multiplying by the number of teeth \( z \) yields:
\[
q = z \int_{0}^{T} Q_{sh2}^{helical} dt = \frac{z}{\omega} \int_{0}^{2\pi / z} d\theta_0 \int_{0}^{B} \frac{1}{2} \left\{ r_a^2 – \left( \frac{r_b}{\cos[\arctan(\theta_0 – \pi / z + \tan\alpha’)]} \right)^2 \right\} dm + \frac{z}{\omega} \int_{0}^{2\pi / z} d\theta_0 \int_{0}^{B} \frac{1}{2} \left\{ r_a^2 – \left( \frac{r_b}{\cos[\arctan(\tan\alpha’ + \pi / z – \theta_0)]} \right)^2 \right\} dm
\]
Simplifying, the displacement becomes:
\[
q = \pi \int_{0}^{B} \left[ 2r_a^2 – \left( \frac{r_b}{\cos[\arctan(\theta_0 + \tan\alpha’ – \pi / z)]} \right)^2 – \left( \frac{r_b}{\cos[\arctan(\pi / z – \theta_0 + \tan\alpha’)]} \right)^2 \right] d\theta_0 \, dm
\]
This double integral can be evaluated numerically for given geometric parameters of the helical gears. The formula inherently accounts for the varying meshing point location and the helical configuration, making it more accurate than traditional approximate methods.
4. Experimental Validation and Comparison
To verify the accuracy of the proposed displacement formula, experimental tests were conducted on three models of KF-type helical gear pumps. The test setup, as illustrated in the accompanying schematic, included pressure transducers, temperature sensors, and a flow meter. The pump was operated at a constant motor speed of 1493.3 r/min with an inlet oil temperature maintained at (50 ± 2) °C. The system pressure was set to 0.2 MPa, and the average displacement was recorded after a steady state was reached.
The geometric parameters of the three tested helical gear pumps are listed in Table 1. These include modules, tooth numbers, face widths, center distances, addendum diameters, helix angles, and normal pressure angles. The pumps are designated as KF112RF1-304D15, KF125RF1-304D15, and KF150RF1-304D15.
| Parameter | KF112RF1-304D15 | KF125RF1-304D15 | KF150RF1-304D15 |
|---|---|---|---|
| Module (mm) | 4 | 4.75 | 4.75 |
| Number of teeth | 11 | 11 | 11 |
| Face width (mm) | 83 | 69 | 83 |
| Center distance (mm) | 47.5 | 57 | 57 |
| Addendum circle diameter (mm) | 56.13 | 66.8 | 66.8 |
| Helix angle (°) | 9 | 12.56 | 10.32 |
| Normal pressure angle (°) | 20 | 20 | 20 |
Table 2 summarizes the experimental data collected under steady-state conditions. The average displacement values represent the actual output per revolution after accounting for any leakage at low pressure.
| Parameter | KF112RF1-304D15 | KF125RF1-304D15 | KF150RF1-304D15 |
|---|---|---|---|
| Inlet temperature (°C) | 50.8 | 50.7 | 50.7 |
| Motor speed (r/min) | 1493.3 | 1493.3 | 1493.3 |
| Outlet pressure (MPa) | 0.205 | 0.205 | 0.205 |
| Average displacement (mL/r) | 113.5 | 129.4 | 155.6 |
The proposed displacement formula (Equation 27 in the original paper) was evaluated using the parameters from Table 1. For comparison, the traditional formula from Li Zhuangyun was also applied. The results are presented in Table 3, together with the nominal displacement values and the experimental measurements.
| Item | KF112RF1-304D15 | KF125RF1-304D15 | KF150RF1-304D15 |
|---|---|---|---|
| Nominal displacement (mL/r) | 112 | 125 | 150 |
| Li Zhuangyun formula (mL/r) | 110.54 | 124.39 | 149.63 |
| Proposed formula (mL/r) | 111.5 | 126.4 | 152.08 |
| Actual displacement (mL/r) | 113.5 | 129.4 | 155.6 |
| Error (proposed vs actual) (%) | 1.76 | 2.30 | 2.26 |
| Error (proposed vs nominal) (%) | 0.45 | 1.12 | 1.39 |
As seen from Table 3, the displacement values calculated by the proposed method are in good agreement with the experimental measurements. The maximum error relative to the actual displacement is less than 3%, and the error relative to the nominal displacement does not exceed 2%. In contrast, the traditional formula tends to underestimate the displacement slightly. These results confirm the correctness and superiority of the derived formulas for helical gear pumps.
5. Discussion on Flow Pulsation and Parametric Analysis
Using the instantaneous flow rate expressions derived in Section 2, the flow pulsation characteristics of helical gear pumps can be studied as functions of gear geometry. The MATLAB simulations reveal that the flow pulsation amplitude decreases with increasing helix angle and face width, because more tooth pairs share the discharge load simultaneously. However, an excessively large helix angle may lead to axial force problems and manufacturing difficulties. The contact ratio also plays a vital role: a higher transverse contact ratio reduces the pulsation but increases the pump size. The proposed method allows designers to quickly evaluate the trade-off between flow uniformity and compactness.
Furthermore, the instantaneous flow rate of helical gears is smoother than that of spur gears due to the gradual entry and exit of teeth along the face width. This is reflected in the computed flow curves, which exhibit a smaller ripple factor. The analytical approach presented here eliminates the need for complex numerical CFD simulations during the initial design phase, providing a fast and reliable tool for optimizing helical gear pumps.
6. Conclusion
In this work, a precise analytical method for calculating the instantaneous flow rate and displacement of involute external meshing helical gear pumps has been developed. By treating the helical gear as a superposition of infinitesimally thin spur gear slices and integrating the contributions over the face width, compact closed-form expressions have been derived. The key advantage of this method is its ability to account for the changing meshing point position along the tooth surface, which is critical for helical gears.
The experimental validation on three different KF-type helical gear pump models demonstrates that the proposed displacement formula yields results within 3% of actual measured values and within 2% of nominal displacement. This level of accuracy outperforms traditional approximate formulas and confirms the universality of the approach. The methodology can be readily extended to other types of helical gears with varying helix angles, tooth profiles, and contact ratios.
Future work may involve incorporating the effects of trapped volume and internal leakage to further refine the model. Nevertheless, the current contribution provides a solid foundation for the rational design and performance evaluation of helical gear pumps in hydraulic systems.