Radial Forces in Internal and External Gear Motors

In gear motors, pressure energy is converted into mechanical energy, resulting in the output of torque and rotational speed. Both internal and external gear motors experience hydraulic torque acting on their meshing gears. For internal-external gear motors, which consist of both internal and external motor components, high-pressure oil is supplied to both motors, and hydraulic forces drive the gears to rotate, generating torque. This technical analysis focuses on calculating and examining the radial forces in both external and internal gear motors, with particular attention to how these forces affect oil pump gears.

Understanding radial forces is crucial for designing efficient and durable gear motors, as these forces directly impact bearing life, gear wear, and overall motor performance. The following sections provide a detailed breakdown of radial force calculations for both external and internal configurations, with specific applications to oil pump gears.

Figure 1: Pressure distribution in a typical gear motor, illustrating the origin of radial forces on oil pump gears

Radial Forces in External Gear Motors

The radial force in gear motors consists of two components: the radial force (Fₚ) generated by hydraulic pressure distributed around the gear circumference, and the radial force (Fₘ) resulting from gear meshing. This dual-component force system is particularly important in oil pump gears, where precise force calculation ensures optimal performance and longevity.

1. Radial Force from Circumferential Hydraulic Pressure on Gear Motors

The hydraulic pressure acting on the gear circumference is unevenly distributed, and as the meshing point changes during operation, the radial force on the gear circumference also changes continuously, making precise calculation quite complex. For calculation purposes, several simplifying assumptions are made, which are also applicable to oil pump gears:

• All hydraulic pressure acting on the gear circumference is assumed to act on the addendum circle
• The angle (φ₁) between the center line of a pair of meshing gears and the edge of the low-pressure oil port is constant
• The angle (φ₂) from the center line of the meshing gears, along the direction of gear rotation from the outlet side to the inlet side, is constant
• The angle (2π-φ₂) from the boundary of the high-pressure oil port, along the direction of gear rotation to the pitch point, is constant
• Except for the pressure in the inlet and outlet chambers on the gear circumference, the hydraulic pressure in the remaining circumferential angle (φ₁ ≤ φ < φ₂) of the meshing gears varies linearly
• The multiple gear shafts of the gear motor do not deform under the action of hydraulic pressure, and the radial clearance around the gear circumference is uniform

Figure 2-14: Approximate pressure distribution curve on the circumference of a large gear in an external gear motor, similar to patterns observed in oil pump gears

We first calculate the radial force on the gears of external gear motors. Figure 2-14 shows the approximate pressure distribution curve on the circumference of the large gear in an external gear motor. Figure 2-15 shows the developed view of the circumferential distribution curve of the large gear in an external gear motor. Let the addendum circle radius of the large gear be Rₐ₂. On the addendum circle of the external gear motor, we take an angle element dφ. The angle between the center line of the external gear of the meshing gears and the edge of the low-pressure oil port is φ₁, and the angle between the center line of the large gear and the edge of the low-pressure oil port is φ₁₂. The angle from the outlet side along the rotation direction of the external gear to the inlet side is φ₂, and the angle from the outlet side along the rotation direction of the large gear to the inlet side is φ₂₂. With a gear width of B, the differential area is dA = BRₐ₂dφ, and the hydraulic force acting on dA is:

When 0 ≤ φ ≤ φ₁, p = pₛ = constant. When φ₁ ≤ φ ≤ φ₂, p = pₛ + (pₚ - pₛ)/(φ₂ - φ₁)(φ - φ₁). When φ₂ ≤ φ ≤ 2π, p = pₚ = constant. These pressure distributions are critical considerations in the design of oil pump gears, where pressure management directly affects efficiency.

Figure 2-15: Developed view of the circumferential distribution curve of a large gear in an external gear motor, demonstrating pressure variations similar to those in oil pump gears

Using differential and integral calculus, the radial force generated by the hydraulic pressure on the circumference of the large gear in an external gear motor can be derived as:

Similarly, the radial force generated on the circumference of the meshing gear (pinion) in the external gear motor can be derived as:

These formulas are essential for engineers designing oil pump gears, as they allow for accurate prediction of radial loading under various operating conditions. Proper calculation ensures that oil pump gears can withstand operational pressures without excessive deflection or wear.

2. Radial Force from Gear Meshing in External Gear Motors

The torque generated by hydraulic pressure on the meshing gear (pinion) is:

The torque generated by hydraulic pressure on the large gear is:

The torque (M₁) generated by hydraulic pressure acting on the tooth surfaces of the meshing gear is directly transmitted to the meshing gear (as the gears rotate in mesh, the radial force caused by M₁ on the meshing gear already includes the radial force Fₘ₁ from the hydraulic pressure effect). The torque (M₂) acting on the tooth surfaces of the large gear through hydraulic pressure is first transmitted to the large gear, then through gear meshing to the meshing gear. This force transmission mechanism is critical in oil pump gears, where efficient torque transfer directly impacts pump performance.

According to mechanical principles, equal and opposite meshing forces act along the meshing line of the external gear motor. In this motor, torque and rotational speed are output through the meshing gear. The meshing forces acting on the meshing gear and the large gear are:

In actual operation of gear motors, the position of the meshing point constantly changes, which in turn changes R₂ and M₂, making calculations quite cumbersome. For calculation convenience, we generally approximate:

Substituting equation (2-41) into equation (2-40), we get:

When two external gears mesh, the meshing forces are equal in magnitude, opposite in direction, and coincide with the meshing line:

These meshing forces are critical considerations in the design of oil pump gears, as they affect not only the torque transmission but also the wear patterns and noise generation. Properly accounting for these forces ensures that oil pump gears maintain optimal meshing characteristics throughout their operational life.

Figure 2-16: Meshing force analysis in external gear motors, illustrating the radial and tangential components that also affect oil pump gears

Radial Forces in Internal Gear Motors

Internal gear motors, like their external counterparts, experience radial forces from both hydraulic pressure distribution and gear meshing. However, the internal configuration creates distinct force patterns due to the unique arrangement where one gear rotates inside another. This configuration is commonly found in certain types of oil pump gears, where space efficiency is a priority.

In internal gear motors, the radial force analysis follows similar principles to external gear motors but with important differences in pressure distribution and meshing geometry. The internal gear (annulus) and external gear (rotor) experience pressure differentials that generate both tangential forces for torque production and radial forces that must be managed through proper bearing design. This is especially true for oil pump gears, where maintaining precise clearances under varying pressure conditions is essential for efficiency.

1. Pressure Distribution in Internal Gear Motors

The pressure distribution in internal gear motors differs from external designs due to the enclosed nature of the internal gear. High-pressure oil enters the chamber between the meshing teeth, creating pressure gradients that act on both the outer diameter of the internal gear and the inner diameter of the external gear. This dual pressure application creates complex radial force patterns that engineers must account for, particularly in oil pump gears where asymmetric pressure distributions can lead to uneven wear.

For internal gear motors, the radial force calculation also involves simplifying assumptions similar to those used for external motors, but adjusted for the internal meshing geometry:

• Pressure acts uniformly across the tooth surfaces within defined pressure zones
• Pressure transitions linearly between high-pressure and low-pressure regions
• The gear shafts remain rigid under operational loads
• Clearances between mating components are uniform and maintained under load

Figure 3-1: Pressure distribution in an internal gear motor, showing the distinct zones that create radial forces in both the internal and external gears, similar to specialized oil pump gears

2. Radial Force Calculation for Internal Gear Motors

The radial force on the external gear (rotor) of an internal gear motor can be calculated using principles similar to those for external gear motors but with modified geometric parameters. The formula for the radial force from hydraulic pressure on the external gear is:

For the internal gear (annulus), the radial force acts inward rather than outward, as in the external gear design. This inward force is calculated as:

These radial forces create a net force imbalance that must be supported by the motor's bearing system. In oil pump gears designed for high-pressure applications, this force balance becomes critical to prevent excessive deflection and maintain proper gear meshing.

3. Meshing Forces in Internal Gear Motors

The meshing forces in internal gear motors also differ from external designs due to the reversed meshing geometry. The meshing force (Fₘ) can be calculated from the torque transmitted through the gear set:

This meshing force has both radial and tangential components, with the radial component contributing to the overall radial load on the gears. In internal gear configurations, this radial component acts in the opposite direction compared to external gears, creating a more complex force balance that is particularly important in oil pump gears designed for high-efficiency applications.

Figure 3-2: Meshing force components in an internal gear motor, demonstrating the unique radial force vectors that must be considered in oil pump gears design

Comparative Analysis and Practical Considerations

Both external and internal gear motors exhibit distinct radial force characteristics that influence their design and application. External gear motors typically experience higher radial loads due to their open geometry, requiring more robust bearing systems. Internal gear motors, while generally producing lower radial loads, have more complex force distributions due to their enclosed design. These differences are particularly significant when designing oil pump gears for specific applications, where load characteristics directly impact component selection and service life.

In practical applications, radial forces must be carefully managed through proper bearing selection, gear geometry optimization, and sometimes active pressure balancing techniques. This is especially true for oil pump gears operating at high pressures or in critical applications where reliability is paramount.

For oil pump gears specifically, the management of radial forces is critical to maintaining volumetric efficiency. Excessive radial deflection can increase internal clearances, leading to increased leakage and reduced efficiency. Conversely, insufficient clearance can lead to contact between moving parts, increased wear, and potential failure.

Modern computational techniques, including finite element analysis (FEA), allow engineers to accurately model radial force distributions and their effects on gear motor components. This advanced analysis capability has led to significant improvements in the design of both external and internal gear motors, as well as specialized oil pump gears, resulting in higher efficiency, longer service life, and better overall performance.

Conclusion

The analysis of radial forces in both external and internal gear motors is a complex but essential aspect of their design and performance. These forces, arising from both hydraulic pressure distribution and gear meshing, significantly impact motor efficiency, durability, and operational characteristics. The same principles apply to oil pump gears, where precise radial force calculation and management are critical to achieving optimal performance.

By understanding the origins of these radial forces and applying the appropriate calculation methods, engineers can design gear motors and oil pump gears that better withstand operational stresses while delivering efficient power transmission. The continued development of analytical techniques and computational tools will further enhance our ability to predict and manage radial forces, leading to more advanced and reliable gear motor designs.

Whether designing external or internal gear motors, a thorough understanding of radial force characteristics is essential for creating robust, efficient systems that meet the demanding requirements of modern hydraulic applications. This knowledge is particularly valuable in the development of specialized oil pump gears, where performance margins are often narrow and reliability is paramount.