Analysis of Output Torque in New Gear Motors

3.2.2.1 Instantaneous Output Torque Analysis of New Gear Motors

The new gear motor essentially consists of three external meshing gear motors and three internal meshing gear motors working independently or in combination within a single housing. The central large gear (z₁, m₁) forms three external meshing gear motors with three common-gear external teeth (z₂, m₂), while three common-gear internal teeth (z₃, m₃) form three internal meshing gear motors with three small gears (z₄, m₄). The torque output can be derived using the torque output formulas for external and internal meshing gear motors established earlier, similar to how external gear pumps operate with their own torque characteristics.

From previous derivations, the instantaneous output torque of each external meshing gear motor relative to the center of the central large gear is:

Where:

• hₐ₁ — Addendum height of the motor's central large gear, m
• hₐ₂ — Addendum height of the common-gear external teeth, m
• R₁ — Pitch circle radius of the motor's central large gear, m
• R₂ — Pitch circle radius of the common-gear external teeth, m
• f — Distance from the gear meshing point to the pitch point, m

The instantaneous output torque of each internal meshing gear motor relative to the center of the common gear is:

Where:

• hₐ₃ — Addendum height of the common-gear internal teeth, m
• hₐ₄ — Addendum height of the small gear, m
• R₃ — Pitch circle radius of the common-gear internal teeth, m
• R₄ — Pitch circle radius of the small gear, m
• f — Distance from the gear meshing point to the pitch point, m

Since the internal motor outputs torque and rotational speed through the large gear shaft, the combined torque calculation becomes more complex, involving the interaction of multiple components similar to how external gear pumps combine their outputs in parallel configurations:

The new gear motor contains multiple internal and external motors within a single housing. These internal and external motors operate independently, with their torque output ripples shown in Figures 3-8 and 3-5 respectively, both exhibiting periodic waveform distributions. Similar to the behavior observed in external gear pumps, when multiple different waves propagate through the same physical medium, they remain independent and do not affect each other. The displacement of the medium at any point is the vector sum of the displacements caused by all waves at that point.

The pulsation periods of the instantaneous torque output by the external and internal meshing gear motors are 2π/z₂ and 2π/z₄ respectively. When z₂ = z₄/3, the periods of the two waves are the same. By adjusting the initial phase angles of the external and internal motors so that their wave peaks and troughs can superimpose and cancel each other out, the pulsation of the motor's instantaneous output torque can be minimized. This minimum pulsation condition is achieved when the angular difference between the two waves follows a specific pattern: Δφ = k₁π/z₂ (k₁ = 1, 3, 5, ...). This means that when a pair of meshing gears in the external motor just enters or exits meshing, a pair of meshing gears in the internal motor should have their meshing point coinciding with the pitch point, or vice versa. This torque output curve is shown in Figure 3-9, demonstrating significantly reduced pulsation compared to conventional external gear pumps.

Figure 3-9: Gear motor output combined torque ripple curve 1
- Combined operation of internal and external motors
- External motor operating alone
- Internal motor operating alone

When the difference in initial phase angles between the internal and external motors is an even multiple of half the waveform period (Δφ = k₂π/z₂ where k₂ = 0, 2, 4, ...), the meshing gears of both the external and internal motors enter or exit meshing simultaneously, or their meshing points coincide with their respective pitch points at the same time. This results in wave peaks叠加 upon wave peaks and wave troughs upon wave troughs, producing maximum torque pulsation. The corresponding output torque curve is shown in Figure 3-10, illustrating a phenomenon also observed in less optimized external gear pumps under certain operating conditions.

When the initial phase angle difference Δφ is neither an odd nor even multiple of half the waveform period, meaning the gear meshing conditions of the external and internal motors do not match either of the aforementioned scenarios, the two waves superimpose into an irregular waveform. The resulting instantaneous output torque pulsation of the motor is neither maximum nor minimum, as shown in Figure 3-11. This intermediate condition represents the typical operating state of many standard external gear pumps in industrial applications.

3.2.2.2 Differential Connection Output Torque Analysis of New Gear Motors

One of the most significant differences between the new gear motor and traditional gear motors is the ability of its internal and external motors to operate independently. The displacements of the internal and external meshing gear motors differ, and when high-pressure oil is supplied to them in opposite directions, both torques are output through the same shaft. Since the torque generated by the external motor exceeds that of the internal motor, the internal motor is forced to rotate in the reverse direction, effectively functioning as a pump. In this configuration, the torque output direction of both motors remains the same as when the external motor operates alone, but with reduced torque and increased rotational speed compared to external gear pumps in standard configurations.

From equations (3-24) and (3-37), the instantaneous torque outputs of the external and internal meshing gear motors are:

Therefore, the instantaneous output torque when the motor is in differential connection is:

When the difference in initial phase angles between the internal and external meshing gear motors is an even multiple of half the waveform period (Δφ' = k'π/z₂ where k' = 0, 2, 4, ...), the meshing gears of both motors enter or exit meshing simultaneously, or their meshing points coincide with their respective pitch points at the same time. This causes wave peaks to coincide with wave peaks and wave troughs to coincide with wave troughs, resulting in minimal pulsation of the instantaneous output torque during differential connection. The corresponding torque curve is shown in Figure 3-12, representing a significant improvement over traditional external gear pumps in similar configurations.

Figure 3-12: Gear motor output torque ripple curve during differential connection 1
- Differential connection of internal and external motors
- External motor operating alone
- Internal motor operating alone

When the difference in initial phase angles between the internal and external meshing gear motors is an odd multiple of half the waveform period (Δφ' = k'π/z₂ where k' = 1, 3, 5, ...), a pair of meshing gears in the external motor just enters or exits meshing while a pair in the internal motor has their meshing point coinciding with the pitch point, or vice versa. This causes wave peaks to coincide with wave troughs, resulting in maximum pulsation of the instantaneous output torque during differential connection. The corresponding torque curve is shown in Figure 3-13, a phenomenon that external gear pumps designers work to minimize through careful engineering.

When the initial phase angle difference Δφ' is neither an odd nor even multiple of half the waveform period, meaning the gear meshing conditions do not match either of the aforementioned scenarios, the two waves superimpose into an irregular waveform. The resulting instantaneous output torque pulsation during differential connection is neither maximum nor minimum, as shown in Figure 3-14. This represents the typical performance envelope for many industrial external gear pumps under varying operating conditions.

Understanding these torque characteristics is crucial for optimizing the performance of the new gear motor across various applications. By carefully controlling the phase relationships between internal and external components, engineers can tailor the torque output characteristics to specific application requirements, much like how external gear pumps are optimized for particular pressure and flow conditions. The ability to configure the motor for minimum or controlled pulsation enables smoother operation, reduced noise, and extended service life in critical applications.

Further research into materials science and manufacturing techniques promises to enhance the performance of these new gear motors even further. Advanced materials can reduce internal friction and wear, while precision manufacturing can ensure tighter tolerances that maintain optimal meshing conditions. These improvements, combined with the fundamental torque characteristics analyzed here, position the new gear motor as a versatile and efficient alternative to both traditional gear motors and external gear pumps in many industrial applications.

The differential connection capability represents a significant advancement in gear motor technology, offering increased flexibility in system design. By allowing the same motor to operate in multiple configurations with varying torque and speed characteristics, engineers can reduce system complexity and improve overall efficiency. This flexibility, combined with the ability to control torque pulsation through phase angle adjustment, makes the new gear motor an attractive option for applications ranging from industrial machinery to mobile equipment, where it can often outperform both traditional gear motors and external gear pumps.

In conclusion, the torque analysis presented here provides a comprehensive framework for understanding and optimizing the performance of the new gear motor design. By leveraging the independent operation of internal and external meshing components and carefully controlling their phase relationships, engineers can achieve significant improvements in torque output characteristics compared to conventional designs. As with the ongoing development of external gear pumps, continued refinement of this technology will likely lead to even greater efficiency, reliability, and performance in the future.