In the field of hydraulic systems, internal gear pumps play a critical role as compact and efficient power sources. As an engineer specializing in fluid power transmission, I have extensively studied the dynamic behavior of internal gears within these pumps. The radial micro-motion of internal gears, influenced by fluid film effects, is a key factor affecting pump performance, longevity, and noise levels. This article delves into the modeling and analysis of this phenomenon, leveraging numerical methods to predict gear behavior under various operating conditions. Internal gear manufacturers often face challenges in optimizing these components for high-pressure applications, and understanding the micro-vibrations can lead to improved designs. Throughout this discussion, I will emphasize the importance of internal gears in pump efficiency and reliability, incorporating mathematical models, tables, and simulations to provide a comprehensive overview. The insights gained here are vital for internal gear manufacturers aiming to enhance product performance and reduce wear in demanding environments.
Internal gear pumps are widely used in industries requiring precise control, low noise, and high reliability due to their compact structure and superior volumetric efficiency. The internal gears in these pumps are subject to complex forces during operation, leading to radial micro-motions that can impact sealing and lubrication. In my research, I focus on the wedge-shaped oil film formed between the internal gear and the pump housing, which arises from hydraulic pressures on the gear teeth. This film induces dynamic effects that cause the internal gear to vibrate slightly, affecting its positional stability. By employing MATLAB for numerical modeling based on the one-dimensional Reynolds equation, I analyze the pressure and gap height distributions in this oil film. This approach allows me to determine the eccentricity and eccentric angle of the internal gear, revealing its radial micro-motion patterns. Such analysis is crucial for internal gear manufacturers to predict operational characteristics and optimize designs for better performance. The following sections will explore the force analysis, pressure zoning, and coupling relationships in detail, supported by equations and simulation results.
The forces acting on internal gears in a pump include hydraulic pressures on the tooth surfaces, meshing forces from gear engagement, and support forces from the oil film and high-pressure chamber. To model these forces accurately, I begin with a parametric representation of the internal gear pair using MATLAB. This involves solving the meshing equations for involute gears and identifying key points such as meshing points and boundaries with separators like the crescent element. For instance, the meshing point defines the boundary between high and low-pressure zones, while intersections with the crescent element delineate transition regions. This zoning is essential for calculating the hydraulic forces on the internal gears, as the pressure distribution varies across these areas. Internal gear manufacturers must consider these zones to ensure balanced force distribution and minimize vibrations. The high-pressure zone exerts the pump’s outlet pressure on the tooth surfaces, while the transition zone features a linear pressure drop between adjacent tooth spaces. This pressure variation influences the radial unbalanced force, which tends to push the internal gear toward the housing, while the support forces from the oil film counteract this movement.
To quantify the hydraulic forces on a single tooth space of the internal gears, I perform an integration based on the properties of involute curves. Consider a point on the tooth surface with a pressure angle $\alpha_k$ and an unfolding angle $\theta$. The force element $dF$ at this point can be expressed as $dF = p \cdot s_k \cdot d\theta$, where $p$ is the pressure and $s_k$ is the length of the generating line. Using the relationship $s_k = R_b \theta$, where $R_b$ is the base circle radius, and considering the symmetry of the tooth surfaces, the resultant force in the y-direction for a tooth space is given by:
$$ F_y = 2 \int_{\alpha_{k0}}^{\alpha_{k1}} p R_b (\tan \alpha_k) \sin(\tan \alpha_k – \delta) \frac{1}{\cos^2 \alpha_k} d\alpha_k $$
Here, $\alpha_{k0}$ and $\alpha_{k1}$ are the pressure angles at the root and tip circles, respectively, and $\delta$ is an angular offset. This integral accounts for the pressure distribution across the tooth surface, which is critical for accurate force calculation. Internal gear manufacturers can use this approach to optimize tooth profiles for reduced stress and improved durability. Additionally, the meshing force between the internal and external gears arises from the torque transmitted through the engagement point. For an internal gear, the meshing force $F_t$ is derived from the torque $M_2$ acting on it:
$$ F_t = \frac{M_2}{r_{b2}} = \frac{p h_b (r_{d2}^2 – r_{f2}^2)}{2 r_{b2}} $$
where $h_b$ is the face width, $r_{d2}$ and $r_{f2}$ are the tip and root radii, and $r_{b2}$ is the base circle radius of the internal gear. This force contributes to the radial unbalanced force, which must be balanced by the support forces to maintain proper gear positioning.
The support forces from the wedge-shaped oil film between the internal gear and housing are modeled using the one-dimensional Reynolds equation. This equation describes the pressure generation in a thin fluid film due to relative motion and converging gaps. For a micro-element in the sealing zone, the pressure gradient is given by:
$$ \frac{dp}{dr} = 6\mu v \frac{h – h_2}{h^3} $$
where $\mu$ is the dynamic viscosity, $v$ is the relative linear velocity, $h$ is the local film thickness, and $h_2$ is the equivalent minimum film thickness at a reference point. The circumferential length element $dr$ relates to the angular element $d\alpha$ as $dr = r d\alpha$, and the velocity $v = 2\pi n r$ for a rotational speed $n$. The film thickness $h$ varies with the eccentricity $e$ and angle $\theta$:
$$ h = R – r – e \cos(\varepsilon – \theta) $$
where $R$ is the housing radius, $r$ is the internal gear radius, and $\varepsilon$ is the angular position. Discretizing the sealing zone into $m$ segments, the pressure difference between points $i$ and $j$ can be approximated as:
$$ p_i – p_j = \frac{6\pi n \mu r^2 e \alpha_2}{m} \frac{\cos(\varepsilon_2 – \theta) – \cos(\varepsilon_j – \theta)}{(R – r – e \cos(\varepsilon_j – \theta))^3} $$
This numerical iteration allows me to compute the pressure distribution and the resulting support force. Internal gear manufacturers must ensure that this force, combined with the high-pressure chamber force, counteracts the radial unbalanced force to prevent excessive wear and maintain a stable oil film. The coupling between the radial unbalanced force and the support force determines the internal gear’s eccentric position, which I solve iteratively for each operating condition.
In my simulations, I consider parameters typical for high-pressure internal gear pumps, as summarized in Table 1. These include pump speed, pressure, gear geometry, and sealing zone angles. The goal is to find the eccentricity $e$ and eccentric angle $\theta$ that minimize the resultant force on the internal gear, ensuring optimal performance. The radial unbalanced force $F_{\text{unbalanced}}$ is the sum of hydraulic and meshing forces, while the support force $F_{\text{support}}$ comes from the oil film and high-pressure chamber. The net force components in the x and y directions are:
$$ F_x(i,j) = F_{x,\text{hyd}} + F_{x,\text{mesh}} – F_{x,\text{support}}(i,j) $$
$$ F_y(i,j) = F_{y,\text{hyd}} + F_{y,\text{mesh}} – F_{y,\text{support}}(i,j) $$
$$ F(i,j) = \sqrt{F_x(i,j)^2 + F_y(i,j)^2} $$
The optimal $(i,j)$ corresponds to the minimum $F(i,j)$, indicating the stable position of the internal gear. This iterative process, implemented in MATLAB, reveals how the internal gear’s position changes with time during pump operation. For instance, at a speed of 2500 rpm and pressure of 25 MPa, the radial unbalanced force fluctuates between 65 kN and 68 kN due to variations in the high-pressure zone angle. These fluctuations cause micro-vibrations, with the eccentricity ranging from 10 μm to 30 μm and the eccentric angle shifting slightly. The oil film thickness remains above 10 μm, ensuring adequate lubrication, as the surface roughness of internal gears and housings is typically around 1.6 μm. Internal gear manufacturers can use these findings to design pumps with better sealing and reduced vibration, enhancing overall efficiency.
| Parameter | Value |
|---|---|
| Rotational Speed (rpm) | 2500 |
| Pressure (MPa) | 25 |
| Module (m) | 0.003 |
| Number of Teeth (External Gear) | 13 |
| Number of Teeth (Internal Gears) | 19 |
| Internal Gear Outer Radius (mm) | 39.5 |
| Clearance between Gear and Housing (μm) | 40 |
| Sealing Zone Angle $\alpha_1$ (°) | 36.81 |
| Sealing Zone Angle $\alpha_2$ (°) | 23.96 |
| High-Pressure Zone Angle $\beta$ (°) | 113.09 |
| Starting Angle $\varepsilon_1$ (°) | -31.58 |
| Ending Angle $\varepsilon_2$ (°) | 81.51 |
The simulation results highlight the dynamic nature of internal gears in response to force variations. For example, when the high-pressure zone angle suddenly decreases due to factors like flow port coverage or meshing point shifts, the radial unbalanced force drops, causing the internal gear to adjust its position. This adjustment is characterized by a reduction in eccentricity while the eccentric angle remains relatively stable. The oil film thickness profile shows that the minimum thickness occurs near the sealing zone edges, ensuring that the film does not collapse under load. Internal gear manufacturers should note that the pressure distribution in the wedge film contributes significantly to the support force, and improper design could lead to metal-to-metal contact and accelerated wear. By analyzing the coupling between forces and positions, I can predict the micro-motion trends and provide guidelines for optimizing the internal gear and housing interface. This is particularly important for internal gears operating at high speeds and pressures, where even small vibrations can affect pump longevity and noise levels.
To further illustrate the force distribution, I derive the pressure in the transition zone where the pressure drops linearly across tooth spaces. If the transition zone spans $n$ tooth spaces, the pressure difference between adjacent spaces is:
$$ \Delta p = \frac{p_o – p_i}{n} $$
where $p_o$ is the outlet pressure and $p_i$ is the inlet pressure. This linear gradient affects the hydraulic force on each tooth space, and internal gear manufacturers must account for this in force calculations. The total hydraulic force on the internal gear is obtained by summing the forces over all tooth spaces in the high-pressure and transition zones. Similarly, the meshing force varies with the torque, which depends on the pressure and gear geometry. The resultant radial unbalanced force can be expressed as a function of the pump’s operating parameters, and it tends to push the internal gear toward the housing. The support force from the oil film, however, generates a opposing pressure field that lifts the gear, creating a balance that determines the equilibrium position.
In terms of practical implications, internal gear manufacturers can use these models to select appropriate clearances and materials for internal gears. For instance, a smaller clearance might reduce leakage but increase the risk of contact if the oil film is too thin. My analysis shows that with the given parameters, the oil film thickness varies between 10 μm and 30 μm, which is sufficient for hydrodynamic lubrication given typical surface finishes. The eccentricity and eccentric angle changes are gradual, indicating stable operation, but sudden force drops can cause abrupt positional shifts. Therefore, designing the pump to minimize these force variations—for example, by optimizing the porting geometry—can enhance stability. Additionally, the use of high-viscosity fluids or surface treatments on internal gears can improve the oil film’s load-bearing capacity, reducing micro-vibrations and wear.
In conclusion, the radial micro-motion of internal gears in gear pumps is a complex phenomenon driven by the interplay of hydraulic, meshing, and support forces. Through numerical modeling and simulation, I have demonstrated how the wedge-shaped oil film influences the internal gear’s position and vibration characteristics. The methods described here provide a framework for internal gear manufacturers to predict and optimize the performance of their products, leading to more efficient and durable pumps. Future work could involve experimental validation and extension to three-dimensional models for even greater accuracy. As the demand for high-pressure, low-noise hydraulic systems grows, understanding and controlling the micro-dynamics of internal gears will remain a key focus for researchers and manufacturers alike.