Torque Analysis of Internal and External Meshing Gear Motors

For a pair of mutually meshing involute gear motors, understanding the torque characteristics is essential for optimal design and application. Similar to internal gear pumps, these hydraulic devices convert fluid pressure into mechanical rotational force, with their performance heavily dependent on geometric parameters and operational conditions. This analysis delves into the displacement calculations and torque output characteristics of both internal and external meshing gear motors, providing a thorough understanding of their operational principles.

The following technical examination covers displacement calculations, theoretical average torque output, and instantaneous torque fluctuations, with particular attention to how these factors differ between internal and external configurations. Like internal gear pumps, the efficiency and performance of these motors are intricately linked to their geometric parameters and meshing characteristics.

Displacement Characteristics of Gear Motors

For a pair of mutually meshing involute gear motors, assuming that the motor starts to discharge oil when the distance from the gear's meshing point to the pitch point is f = f₁, and continues until f = f₂ when the discharge ends, with df = ωₘdt, the volume of hydraulic oil discharged by this pair of meshing gears is:

Since the displacement of the motor is the volume of hydraulic oil discharged when the torque output gear rotates one full revolution (i.e., rotating through a tooth), we have:

The base pitch of the gear motor is tᵦ. A pair of meshing gears starts to discharge oil at f = -0.5tᵦ and ends at f = 0.5tᵦ. The displacement of the motor is:

The motor displacement calculated by equation (3-3) is the precise displacement. However, due to the complexity of the calculation process, the following method is usually used for approximate calculation of gear motor displacement in engineering applications, much like in the design of internal gear pumps where practical approximations often replace theoretical exactness.

The displacement of the motor is the sum of the working volume between the teeth on the torque output wheel and the volume of the gear teeth. This assumes that the effective volume of the gear teeth is equal to the working volume between the gear teeth. In other words, the displacement of the gear motor is equivalent to the volume of the annular cylinder enclosed by the addendum circle and the base circle of the torque output wheel, a principle that also applies to internal gear pumps.

The effective tooth height of the gear is hₑ = 2(Rₐ - Rᵦ) = 2m. Substituting into equation (3-4) gives the following expressions for gear motor displacement, which are also relevant for understanding the performance of internal gear pumps:

Figure 1: Gear motor geometric parameters influencing displacement, similar to those in internal gear pumps

A housing contains three internal motors and three external motors. Both internal and external motors can work independently or in combination. The geometric displacements of internal and external motors are as follows, with parallels to the displacement calculations used in internal gear pumps.

The geometric displacement of the external meshing gear motor is:

Where:

• z₁ - Number of teeth of the large gear
• m₁ - Module of the large gear, mm

The geometric displacement of the internal meshing gear motor is:

Where:

• z₂ - Number of teeth of the internal gear
• m₂ - Module of the internal gear, mm

Due to the special structure of the motor, internal and external motors can work in combination or independently. The motor has a total of four different working modes, which directly affect torque output capabilities similarly to how configurations affect performance in internal gear pumps. These modes are: internal motor working alone, external motor working alone, internal and external motors working in the same direction, and internal and external motors working in differential mode. The corresponding geometric displacements are Vᵢₙₜ, Vₑₓₜ, Vᵢₙₜ + Vₑₓₜ, and Vₑₓₜ - Vᵢₙₜ respectively.

Figure 2: Four operating configurations of internal and external gear motor combinations, showing displacement differences compared to standard internal gear pumps

Theoretical Average Output Torque Analysis

Since the new type of gear motor is equivalent to having three external meshing motors and three internal meshing gear motors in one housing, the formula for calculating the theoretical average output torque of ordinary gear motors applies equally to this motor, just as torque principles apply similarly to internal gear pumps:

Where:

• Δpₘ - Pressure difference between motor inlet and outlet, MPa
• Vₘ - Geometric displacement of the motor in different working states, mL/r

Here, Δpₘ = pᵢₙ - pₒᵤₜ, where pᵢₙ is the motor inlet pressure and pₒᵤₜ is the motor outlet pressure, generally taken as pₒᵤₜ = 0. This pressure differential is a critical factor in torque generation, analogous to how pressure affects performance in internal gear pumps.

Instantaneous Output Torque Analysis

The motor output torque calculated according to equation (3-10) can only reflect the average output torque of a motor and cannot reflect the fluctuation of the motor output torque. In many cases, it is necessary to understand the influencing factors of motor output torque fluctuation, so it is necessary to analyze the instantaneous output torque of the motor, much like analyzing pressure ripple in internal gear pumps.

To facilitate the analysis of the instantaneous output torque of the motor, two conditions are first set: the pressure of the pressure oil supplied to the hydraulic motor is fixed, and the supply flow is also fixed, which can be expressed as pᵢₙ = constant, Qₘ = constant. According to energy conservation, it can be expressed as:

Where:

• pᵢₙ - Motor inlet pressure, MPa
• pₒᵤₜ - Motor outlet pressure, MPa
• Qₘ - Motor supply flow rate, L/min
• ηₜₒₜ - Total motor efficiency
• Pₒᵤₜ - Actual motor output power, W
• Tᵢₙₛₜ - Instantaneous motor output torque, N·m
• ωₘ - Instantaneous motor output speed, rad/s

According to equation (3-11), when the pressure difference between the inlet and outlet of the motor and the input flow rate are constant, the instantaneous output torque of the motor is inversely proportional to the instantaneous output speed, and the variation law is similar. Therefore, we only need to study the instantaneous output torque of the motor. This relationship is crucial for understanding performance variations, much like studying flow ripple in internal gear pumps.

Since the new gear motor is essentially equivalent to having three externally meshing gear motors and three internally meshing gear motors in one housing, and the external and internal motors can work independently or in combination, we will analyze the instantaneous output torque of the internal meshing gear motor, external meshing gear motor, and the new gear motor separately, drawing comparisons to torque characteristics in internal gear pumps.

External Meshing Gear Motor Instantaneous Torque Analysis

Figure 3-3 shows a simplified diagram of the working principle of an external meshing gear motor. When high-pressure oil with pressure pᵢₙ is introduced into the left side of the motor, the torque output gear 1 of the motor rotates counterclockwise, and the driven gear 2 rotates clockwise. Low-pressure oil pₒᵤₜ is discharged from the right side. At a certain instant during the motor's operation, the meshing point of the driving gear and the driven gear is point C.

Figure 4: Working principle diagram of external meshing gear motor, illustrating pressure zones and torque generation points, with operational similarities to certain internal gear pumps configurations

Analyzing the forces on the driving gear 1 of the motor, assuming the pressure at the motor's oil return port pₒᵤₜ = 0, the force on the teeth of the motor's driving gear can be determined. This force analysis is critical for understanding torque fluctuations, as the number and position of teeth in contact with high-pressure fluid change continuously during rotation, creating periodic torque variations similar to those observed in some internal gear pumps.

The instantaneous torque produced by an external meshing gear motor results from the vector sum of forces acting on all teeth in contact with high-pressure fluid. As the gears rotate, the number of teeth exposed to high pressure changes, creating a cyclical variation in torque output. The frequency of this torque ripple is directly related to the number of teeth and rotational speed, a characteristic that designers must consider when selecting between motor types, just as they evaluate pressure ripple in internal gear pumps.

Internal Meshing Gear Motor Instantaneous Torque Analysis

Internal meshing gear motors, much like their external counterparts and internal gear pumps, exhibit unique torque characteristics due to their geometry. In internal meshing configurations, the smaller gear (pinion) rotates inside the larger internal gear, creating a more compact design with different meshing characteristics. This configuration results in a different torque ripple pattern compared to external meshing motors.

One key advantage of internal meshing designs, which they share with high-performance internal gear pumps, is the potential for reduced torque ripple. This is due to the more uniform distribution of pressure forces around the gear circumference and the possibility of having more teeth in mesh simultaneously. The overlapping pressure zones create a smoothing effect on the torque output, resulting in more consistent performance.

Figure 5: Torque ripple comparison between internal and external meshing gear motors under identical operating conditions, showing smoother characteristics in internal designs similar to high-quality internal gear pumps

The instantaneous torque calculation for internal meshing motors follows similar principles to external designs but incorporates the unique geometric relationship between the internal and external gears. The torque output is determined by the pressure acting on the surfaces of the meshing teeth, with the internal configuration allowing for more balanced force distribution.

When analyzing internal meshing motor torque, engineers must consider the different lever arms created by the internal gear geometry. The distance from the center of rotation to the point of pressure application varies differently than in external designs, affecting both average torque output and torque fluctuation characteristics. This is analogous to how internal gear pumps exhibit different pressure-flow characteristics compared to external gear pumps.

Combined Motor Configuration Torque Characteristics

The unique multi-motor configuration, combining three internal and three external meshing units, offers significant advantages in torque control and smoothness. By strategically combining the outputs of internal and external motors, the system can minimize overall torque ripple through phase cancellation, a technique that enhances performance beyond what is possible with single internal gear pumps or motors.

In the combined operating mode, the torque ripple from individual motors can be designed to occur at different phases, creating a more uniform total output. This effect is particularly pronounced in the differential mode, where the interaction between internal and external motor torques can be optimized for specific performance characteristics.

The ability to select between different operating modes provides significant flexibility in matching torque characteristics to application requirements. For precision applications requiring minimal vibration, the combined mode offers superior smoothness, while independent operation of internal or external motors provides efficiency advantages for specific torque and speed ranges. This versatility is one of the key benefits of the design, complementing the advantages of using specialized internal gear pumps in hydraulic systems.

Conclusion

The torque characteristics of both internal and external meshing gear motors are determined by their geometric parameters and operating conditions, with direct relationships between pressure differential, displacement, and output torque. The unique multi-motor configuration analyzed offers exceptional versatility through its four operating modes, providing solutions for various torque requirements.

Understanding both average and instantaneous torque characteristics is essential for proper motor selection and system design. The smoother torque output of internal meshing configurations, which they share with high-performance internal gear pumps, makes them suitable for applications requiring precision and minimal vibration, while external meshing designs offer advantages in certain torque and speed ranges. By leveraging the combined capabilities of both motor types, the system achieves a performance envelope that exceeds what is possible with single-mode designs or standard internal gear pumps.