Transient Torque Analysis in Internal and External Meshing Gear Motors

In equation (2-4), the theoretical output torque cannot reflect the transient characteristics of internal and external meshing gear motors. To further understand the theoretical performance of internal and external meshing gear motors, it is necessary to conduct a thorough analysis of the motor's transient characteristics. The instantaneous torque of an internal and external meshing gear motor refers to the periodic fluctuation of the motor's output torque around the theoretical output torque as the gear meshing point moves periodically. This phenomenon is crucial to understand not only for motor design but also for optimizing the performance of related components like the hydraulic gear pump.

Generally, it is assumed that the pressure difference Δp between the inlet and outlet of the motor is constant, and the input flow Q of the motor is constant. Under these conditions, the following relationship holds:

Where:

• η_v — volumetric efficiency
• η_m — mechanical efficiency
• T_w — instantaneous output torque, N·m
• ω_w — instantaneous output angular velocity, rad/s

From equation (2-5), it can be seen that T_w is inversely proportional to ω_w. Therefore, when analyzing the transient characteristics of the motor, the study of the motor's instantaneous torque is equivalent to the study of the motor's instantaneous rotational speed. This fundamental relationship is also critical in understanding the performance of the hydraulic gear pump, where similar torque-speed relationships govern operational efficiency. The following sections present a detailed analysis of the instantaneous output torque of internal and external meshing gear motors, with occasional references to parallel principles in the hydraulic gear pump for comparative understanding.

1. Instantaneous Torque Analysis of Internal and External Meshing Gear Motors

An internal and external meshing gear motor consists of both an internal meshing gear motor and an external meshing gear motor. Both components contribute to the overall output torque of the motor. The torque output gear of the external meshing gear motor is a common external gear, while the torque output gear of the internal meshing gear motor is a common internal gear. This common gear combines the output torques of the external and internal motors onto the output shaft, achieving the final torque output. Therefore, it is necessary to analyze the instantaneous torques of both the internal and external motors separately, much like how we would analyze the components of a hydraulic gear pump to understand its overall performance.

Figure 2-3: Principle of internal meshing gear motor operation, illustrating torque generation similar to certain hydraulic gear pump designs

As shown in Figure 2-3, which illustrates the principle of an internal meshing gear motor, high-pressure oil p_in is supplied to the upper inlet of the motor. The hydraulic pressure generates an output torque, causing both the internal and external gears to rotate clockwise, while low-pressure oil p_out is discharged from the lower outlet. During this process, a pair of internal and external gears mesh and rotate together. This operational principle shares similarities with the basic functionality of a hydraulic gear pump, though with reversed fluid flow direction.

When the internal gear rotates by dθ_i, the external gear rotates by dθ_e. The oil volumes supplied to the internal and external gears are dV_i and dV_e respectively, resulting in a total supplied volume of dV = dV_i + dV_e. The supplied pressure energy can be calculated as dE = (p_in - p_out)(dV_i + dV_e). The output mechanical energy is given by dW = T_idθ_i + T_edθ_e.

According to the principle of energy conservation, assuming no volumetric or mechanical losses, the supplied hydraulic pressure energy equals the mechanical energy output by the motor. This fundamental energy conservation principle applies equally to both hydraulic motors and the hydraulic gear pump, forming the basis of their operational efficiency calculations. Thus, we can derive:

Where:

• T_i — Torque on the internal meshing gear, N·m
• T_e — Torque on the external meshing gear, N·m

This energy balance equation is crucial for understanding how pressure differentials are converted into mechanical torque in gear motors, just as it is in the hydraulic gear pump where mechanical energy is converted into hydraulic pressure. The relationship between volume changes and angular displacement forms the basis for calculating both torque and flow characteristics in these hydraulic components.

1.1 Volume Displacement Analysis

To further analyze the torque characteristics, we must first understand the volume displacement relationship with angular rotation. In gear motors, as in the hydraulic gear pump, the volume of fluid displaced is directly related to the gear geometry and the angle of rotation. For internal meshing gears, the volume displacement dV_i corresponding to a small rotation angle dθ_i can be derived based on the gear parameters and meshing characteristics.

The displacement volume per unit angle for the internal gear, often referred to as the displacement gradient, is a critical parameter that influences both torque characteristics and overall efficiency. This parameter is analogous to the displacement per revolution in a hydraulic gear pump, which determines its flow rate at a given speed. For the internal gear, this relationship can be expressed as:

Where R_bi is the base circle radius of the internal gear. Similarly, for the external gear, the displacement gradient is:

These relationships show that the volume displacement per unit angle depends on the geometric properties of the gears, specifically the difference between the squares of the addendum and base circle radii. This is analogous to how a hydraulic gear pump's flow rate depends on its geometric parameters and rotational speed.

Figure 2-4: Volume displacement characteristics as a function of angular rotation, demonstrating the periodic nature similar to certain hydraulic gear pump designs

The periodic nature of the volume displacement, as illustrated in Figure 2-4, directly contributes to the transient torque characteristics of the motor. As the gears rotate and mesh, the displacement volume changes in a cyclical pattern, leading to corresponding fluctuations in torque output. This periodic behavior is a key consideration in both motor and hydraulic gear pump design, as it affects system vibration, noise, and overall stability.

1.2 Torque Calculation from Displacement

Substituting the volume displacement relationships into the energy equation allows us to derive expressions for the torque on each gear. From equations (2-9), (2-10), and (2-11), we can solve for the torques T_i and T_e:

These equations demonstrate that the torque on each gear is directly proportional to the pressure difference across the motor and to the geometric factor (R²_a - R²_b). This relationship highlights the importance of both hydraulic factors (pressure differential) and mechanical factors (gear geometry) in determining torque output, a principle that also applies to the hydraulic gear pump in its conversion of mechanical input to hydraulic output.

However, these equations represent idealized conditions without considering the meshing characteristics and the changing number of teeth in contact during rotation. In reality, as gears rotate, the number of teeth engaged in meshing changes, leading to variations in the effective radius and thus in the torque output. This is similar to how a hydraulic gear pump experiences pressure fluctuations due to changing volumes between meshing teeth.

Figure 2-5: Gear meshing phases illustrating changing tooth contact patterns, which affect transient torque characteristics in both motors and the hydraulic gear pump

Figure 2-5 illustrates different phases of gear meshing, showing how the number and position of contacting teeth change during rotation. These variations cause the effective radius (and thus torque) to fluctuate periodically. The period of this fluctuation is related to the gear tooth spacing and the rotational speed, creating a characteristic frequency that must be considered in system design. This same periodic behavior is responsible for the pressure ripple observed in hydraulic gear pump operation.

1.3 Transient Torque Characteristics

To fully understand the transient behavior, we must account for the changing meshing conditions throughout the gear rotation cycle. The instantaneous torque can be expressed as a function of the angular position, incorporating the varying number of teeth in contact and their respective lever arms.

For a gear pair with N teeth, the transient torque will exhibit periodic behavior with a fundamental frequency corresponding to N times the rotational frequency. This means that for each full rotation of the gear, there will be N torque fluctuations. This characteristic is analogous to the flow ripple frequency in a hydraulic gear pump, which is also related to the number of teeth and rotational speed.

The amplitude of these torque fluctuations depends on several factors:

• The pressure differential (p_in - p_out)
• The gear geometry (tooth profile, size, number of teeth)
• The meshing characteristics (contact ratio, backlash)
• Operating conditions (speed, fluid properties)

In practical applications, these torque fluctuations can lead to vibration and noise, which may affect system performance and longevity. Therefore, understanding and minimizing these fluctuations is crucial for optimizing motor design, just as it is for improving the performance of the hydraulic gear pump.

Figure 2-6: Transient torque characteristics showing periodic fluctuations around the mean value, comparable to pressure fluctuations in a hydraulic gear pump

Figure 2-6 shows a typical transient torque profile, with the instantaneous torque fluctuating around the mean value calculated from the idealized equations. The amplitude and frequency of these fluctuations provide important information for system design, particularly for applications requiring smooth torque output.

The relationship between torque and angular velocity, as established in equation (2-5), means that these torque fluctuations will be accompanied by corresponding fluctuations in rotational speed. This coupling between torque and speed transients must be considered in the design of control systems for hydraulic machinery, whether they incorporate motors, the hydraulic gear pump, or both.

1.4 Combined Torque from Internal and External Motors

As previously mentioned, the total output torque of the combined internal and external meshing gear motor is the sum of the torques from each component. However, due to their different gear ratios and meshing characteristics, the torque fluctuations from each component may not be in phase, leading to complex interaction effects.

The total instantaneous torque T_total can be expressed as:

Where θ is the angular position of the output shaft. The specific forms of T_i(θ) and T_e(θ) depend on the respective gear geometries and meshing characteristics.

By carefully designing the gear parameters, it is possible to achieve partial cancellation of the torque fluctuations from the internal and external components. This approach, known as "torque ripple cancellation," can significantly reduce overall torque fluctuations, resulting in smoother operation. A similar principle is applied in some advanced hydraulic gear pump designs to minimize pressure ripple.

The effectiveness of this approach depends on the proper matching of gear parameters, including the number of teeth, tooth profile, and phase relationship between the internal and external gear sets. Computational methods, such as finite element analysis and multi-body dynamics simulation, are typically used to optimize these parameters for minimal torque fluctuation.

1.5 Efficiency Considerations

While the previous analysis has focused on ideal conditions without losses, real-world performance must account for both volumetric and mechanical efficiency factors. These efficiency factors, introduced in equation (2-5) as η_v and η_m, affect both the magnitude and characteristics of the transient torque.

Volumetric efficiency accounts for fluid leakage within the motor, which reduces the effective volume displacement. This leakage tends to increase with pressure and can vary with operating conditions, introducing additional variability in the torque output. Similarly, in a hydraulic gear pump, volumetric efficiency is reduced by internal leakage, affecting the pump's output flow.

Mechanical efficiency accounts for frictional losses within the motor, including those in the gear meshing, bearings, and seals. These losses convert some of the hydraulic energy into heat, reducing the mechanical output torque for a given pressure differential. Frictional losses can vary with speed, load, and temperature, adding further complexity to the transient torque characteristics.

Both efficiency factors are important considerations in motor design and application. While higher efficiency is generally desirable, there may be trade-offs between efficiency and other performance characteristics such as torque smoothness, noise, and cost. Similar trade-offs exist in the design and selection of a hydraulic gear pump for a specific application.

Figure 2-7: Efficiency characteristics of a typical gear motor, showing how volumetric and mechanical efficiency vary with operating conditions, similar to efficiency curves for a hydraulic gear pump

Figure 2-7 illustrates how both volumetric and mechanical efficiency typically vary with operating pressure and speed. These efficiency curves are crucial for selecting the appropriate operating range for a motor to achieve optimal performance, just as they are for selecting the right operating parameters for a hydraulic gear pump.

1.6 Practical Implications for System Design

The transient torque characteristics of gear motors have significant implications for the design and operation of hydraulic systems. Understanding these characteristics allows engineers to:

• Select appropriate motors for specific applications based on torque smoothness requirements
• Design effective damping and isolation systems to minimize vibration and noise
• Develop control strategies that account for torque fluctuations
• Optimize gear geometries for improved performance
• Match motor characteristics with other system components, including the hydraulic gear pump, for overall system efficiency

For applications requiring precise speed control or smooth operation, such as in robotics or precision manufacturing, minimizing torque fluctuations is particularly important. In these cases, motors with lower torque ripple may be specified, even if they come at a higher cost or lower efficiency. Similarly, the hydraulic gear pump selected for such systems should have low pressure ripple to complement the motor performance.

In other applications where cost and robustness are primary concerns, some level of torque fluctuation may be acceptable. In these cases, simpler motor designs can be used, with appropriate system design to accommodate the fluctuations through damping or other means.

The interaction between the motor and other system components, particularly the hydraulic gear pump, is also critical. The pump must be sized to provide the required flow and pressure for the motor under all operating conditions, while the motor's transient characteristics must be compatible with the load requirements. Proper matching of these components ensures efficient, reliable system operation.

1.7 Advanced Analysis Techniques

Modern engineering practice employs various advanced analysis techniques to study and predict the transient torque characteristics of gear motors. These include:

• Computational Fluid Dynamics (CFD) for analyzing the flow patterns and pressure distributions within the motor
• Finite Element Analysis (FEA) for studying the structural behavior and stress distribution in the gears
• Multi-body Dynamics (MBD) simulation for analyzing the dynamic interactions between moving components
• System-level simulation for evaluating the motor's performance within the context of the entire hydraulic system

These advanced techniques allow engineers to optimize motor designs before physical prototypes are built, reducing development time and cost. They also provide insights into the detailed mechanisms contributing to torque fluctuations, enabling more targeted design improvements. Similar advanced techniques are employed in the design and optimization of the hydraulic gear pump.

Experimental validation remains important, however, as numerical models always involve simplifying assumptions. Test rigs instrumented with torque transducers, pressure sensors, and high-speed data acquisition systems are used to measure transient torque characteristics and validate simulation results.

Figure 2-8: Experimental setup for measuring transient torque characteristics, including instrumentation for pressure, torque, and speed measurement, with a complementary hydraulic gear pump test stand visible in the background

Figure 2-8 shows a typical experimental setup for measuring transient torque characteristics. Such test facilities allow for detailed investigation of how design parameters, operating conditions, and fluid properties affect torque transients, providing valuable data for both motor development and application engineering.

1.8 Comparison with Other Motor Types

While this analysis has focused on internal and external meshing gear motors, it is informative to compare their transient torque characteristics with other types of hydraulic motors, such as vane motors, piston motors, and gerotor motors. Each motor type has distinct torque characteristics that make it suitable for specific applications.

Gear motors, including those with internal and external meshing configurations, generally exhibit higher torque ripple than piston motors but offer advantages in terms of simplicity, cost, and robustness. This makes them well-suited for applications where cost and reliability are more important than absolute smoothness. The hydraulic gear pump shares these characteristics, offering simplicity and reliability with somewhat higher pressure ripple than more complex pump designs.

Understanding the relative advantages and disadvantages of different motor types allows engineers to make informed selections based on application requirements. When combined with a properly selected hydraulic gear pump and other system components, the result is a hydraulic system optimized for its intended purpose.

In summary, the transient torque analysis of internal and external meshing gear motors is a complex but essential aspect of hydraulic system design. By understanding the fundamental relationships between pressure, volume displacement, and torque, and by accounting for the periodic nature of gear meshing, engineers can optimize motor performance for specific applications. The principles discussed here also find parallels in the analysis of the hydraulic gear pump, highlighting the interconnected nature of hydraulic system components.

As hydraulic technology continues to advance, ongoing research into torque characteristics and fluctuation reduction will lead to improved motor designs with better performance, efficiency, and reliability. These improvements, combined with advances in the hydraulic gear pump and other system components, will continue to expand the capabilities and applications of hydraulic power transmission.