1. Introduction
Permanent magnet gears have gained significant attention in recent years due to their unique advantages such as contact – less transmission, high torque density, and high efficiency. Among them, the multi – shaft ring – plate permanent magnet gear (MRMG) is a novel type of gear that combines the characteristics of mechanical and magnetic gears. This paper aims to provide a comprehensive analysis of the MRMG, including its structure, operating mechanism, magnetic field and torque calculation, performance evaluation through finite – element analysis and experiments, and a comparison with other types of permanent magnet gears.
2. Structure and Operating Mechanism of MRMG
2.1 Structure of MRMG
The MRMG consists of an inner permanent – magnet ring, an outer permanent – magnet ring (also known as the ring plate), a central shaft, and multiple eccentric shafts (\(n\geq3\)). The inner permanent – magnet ring is composed of an inner permanent magnet, a yoke, and a central shaft. The outer permanent – magnet ring is made up of an outer permanent magnet, a yoke, and eccentric shafts. The centers of the inner and outer permanent – magnet rings are \(O_i\) and \(O_o\) respectively, and the distance between them is the eccentricity \(\delta\). The magnetic poles of the inner and outer permanent – magnet rings are distributed with different pole pairs \(P_i\) and \(P_o\), as shown in Table 1.
| Component | Description |
|---|---|
| Inner Permanent – magnet Ring | Composed of inner permanent magnet, yoke, and central shaft; Inner permanent magnet with inner radius \(R_{i2}\), outer radius \(R_{i1}\); Yoke with inner radius \(R_{i3}\) |
| Outer Permanent – magnet Ring | Composed of outer permanent magnet, yoke, and eccentric shafts; Outer permanent magnet with inner radius \(R_{o1}\), outer radius \(R_{o2}\); Yoke with outer radius \(R_{o3}\) |
| Eccentric Shafts | Uniformly distributed on the yoke of the outer permanent – magnet ring, with the same eccentricity as the inner and outer permanent – magnet rings |
[Insert Figure 1: Schematic diagram of MRMG mechanical structure]
2.2 Operating Mechanism
When the inner permanent – magnet ring rotates, the magnetic field interaction between the inner and outer permanent – magnet rings causes the outer permanent – magnet ring to perform a translational – oscillatory motion around the center of the inner permanent – magnet ring. The eccentric shafts on the outer permanent – magnet ring act as cranks, enabling the outer permanent – magnet ring to maintain a constant minimum air – gap length during its motion.
The motion relationship between the inner and outer permanent – magnet rings can be described by the following equations. Let the angular velocity of the inner permanent – magnet ring be \(\omega_i\) and that of the outer permanent – magnet ring be \(\omega_o\). The magnetic deflection angle \(\beta\) is given by \(\beta=\omega_{i}t-\frac{P_{i}-P_{o}}{P_{i}}\omega_{o}t\) (\(-90^{\circ}/P_{i}\leq\beta\leq90^{\circ}/P_{i}\)). The electromagnetic torque \(T_{i}(R_{i1},\beta)\) generated on the outer boundary \(R_{i1}\) of the inner permanent – magnet ring due to \(\beta\) is related to the torques \(T_{o}\) and \(T_{s}\) of the outer permanent – magnet ring and the eccentric shaft, as well as their moments of inertia \(J_{o}\) and \(J_{s}\). The equations of motion are: \(\frac{J_{o}}{P_{i}-P_{o}}\frac{d\omega_{o}}{dt}=\frac{T_{i}(R_{i1},\beta)}{P_{i}}-\frac{1}{P_{i}-P_{o}}(T_{o}-\frac{P_{e}}{\omega_{o}})\) \(n\cdot J_{s}\frac{d\omega_{s}}{dt}=T_{o}-T_{s}-\frac{P_{m}}{\omega_{s}}\) Since \(\omega_{s}=\omega_{o}\), by combining these two equations, we can obtain the speed ratio \(i_{\omega}=\frac{\omega_{o}}{\omega_{i}}\) and torque ratio \(i_{T}=\frac{T_{i}(R_{i1},\beta)}{T_{s}}\) of the MRMG.
During the start – up process of the MRMG, due to the speed difference between the inner and outer permanent – magnet rings, a magnetic deflection angle \(\beta\) and an electromagnetic torque \(T_{i}\) greater than the load are generated, causing the outer permanent – magnet ring to perform a planar accelerating oscillation. When the speeds of the outer permanent – magnet ring and the eccentric shaft gradually reach the synchronous speed of the eccentric magnetic field, the MRMG enters a stable operating state.
3. Air – gap Magnetic Field and Torque Calculation of MRMG
3.1 Air – gap Magnetic Field Calculation
To calculate the air – gap magnetic field of the MRMG, we first establish fixed coordinate systems \(X_{i}O_{i}Y_{i}\) and \(X_{o}O_{o}Y_{o}\) with the centers \(O_{i}\) and \(O_{o}\) of the inner and outer permanent – magnet rings as the origins respectively. The motion relationships between the radial and circumferential coordinates of the two coordinate systems are as follows: \(\left\{\begin{array}{l}r_{i}=\sqrt{r_{o}^{2}-\delta^{2}\sin^{2}(\theta_{i}-\omega_{o}t)}-\delta\cos(\theta_{i}-\omega_{o}t)\\\theta_{i}=\omega_{o}t + \arccos\left(\frac{r_{i}^{2}+\delta^{2}-r_{o}^{2}}{2\delta r_{i}}\right)\\r_{o}=\sqrt{r_{i}^{2}+\delta^{2}+2\delta r_{i}\cos(\theta_{i}-\omega_{o}t)}\\ \theta_{o}=\omega_{o}t+\arccos\left(\frac{r_{o}^{2}+\delta^{2}-r_{i}^{2}}{2\delta r_{o}}\right)\end{array}\right.\)
Assuming that the inner and outer permanent magnets are radially magnetized with magnetization intensities \(M_{i}\) and \(M_{o}\), the magnetization intensities can be expressed as Fourier series. The vector magnetic potentials \(A_{I}\), \(A_{III}\) in the regions of the inner and outer permanent magnets satisfy the Poisson equations, and the vector magnetic potentials \(A_{II}^{i}\), \(A_{II}^{o}\) in the air – gap (passive region) satisfy the Laplace equations. By applying the boundary conditions at the interface between the permanent magnets and the air – gap, we can obtain the relationships between the radial and tangential air – gap magnetic densities \(B_{IIr}^{i}\), \(B_{II\theta}^{i}\), \(B_{IIr}^{o}\), \(B_{II\theta}^{o}\) and the magnetic deflection angle \(\beta\) in the non – eccentric case.
However, the obtained air – gap magnetic densities are based on different coordinate systems and cannot be directly used to calculate the eccentric air – gap magnetic density of the MRMG. Therefore, we use the fractional – linear transformation method to transform the magnetic densities in different coordinate systems to the complex – plane of the same coordinate system. After transformation, the inner and outer permanent – magnet rings in the complex – plane have a concentric magnetic field. By calculating the concentric magnetic field in the complex – plane, we can obtain the radial and tangential air – gap magnetic densities \(B_{IIr}(r_{i},\theta_{i})\) and \(B_{II\theta}(r_{i},\theta_{i})\) in the original plane.
3.2 Electromagnetic Torque and Force Calculation
The electromagnetic torque and force of the MRMG can be calculated using the Maxwell stress – tensor method. The radial and tangential electromagnetic forces \(F_{ir}\) and \(F_{i\theta}\) on the inner permanent – magnet ring are given by: \(F_{ir}=\frac{Lr_{i}}{2\mu_{0}}\int_{0}^{2\pi}[B_{IIr}^{2}(r_{i},\theta_{i}) – B_{II\theta}^{2}(r_{i},\theta_{i})]d\theta_{i}\) \(F_{i\theta}=\frac{Lr_{i}}{\mu_{0}}\int_{0}^{2\pi}[B_{IIr}(r_{i},\theta_{i})B_{II\theta}(r_{i},\theta_{i})]d\theta_{i}\) The electromagnetic torque \(T_{i}\) on the inner permanent – magnet ring is obtained by taking the moment of \(F_{i\theta}\): \(T_{i}=\frac{Lr_{i}^{2}}{\mu_{0}}\int_{0}^{2\pi}[B_{IIr}(r_{i},\theta_{i})B_{II\theta}(r_{i},\theta_{i})]d\theta_{i}\)
The electromagnetic force balance equations for each eccentric shaft are: \(\sum_{k = 1}^{n}F_{k\theta}=F_{i\theta}\) \(\sum_{k = 1}^{n}F_{kr}=F_{ir}+m_{o}\omega_{o}^{2}\delta\) \(\frac{F_{i\theta}R_{o1}}{h}=\sum_{k = 1}^{n}F_{k\theta}\sin(N_{k})-\sum_{k = 1}^{n}F_{kr}\cos(N_{k})\) where \(N_{k}=\omega_{o}t-(2k – 1)\frac{180^{\circ}}{n}\), \(m_{o}\) is the mass of the outer permanent – magnet ring, and h is the distance between the center O and the eccentric shaft.
3.3 Transmission Loss Calculation
The transmission losses of the MRMG mainly include eddy – current losses \(P_{e}\) and mechanical losses \(P_{m}\). The eddy – current losses in the permanent magnets and yokes of the inner and outer permanent – magnet rings are calculated based on the induced voltage and resistance of the materials. The mechanical losses mainly come from the rolling and sliding friction torques of the rotating – arm bearings.
The eddy – current loss \(P_{e}\) is the sum of the eddy – current losses in the inner and outer permanent – magnet rings: \(P_{e}=P_{i1}+P_{i2}+P_{o1}+P_{o2}=\frac{(B_{IIr}^{i})^{2}\omega_{i}^{2}L}{4}\left[S_{i1}\frac{(R_{i1}+R_{i2})^{2}}{\rho_{1}}+S_{i2}\frac{(R_{i2}+R_{i3})^{2}}{\rho_{2}}\right]+\frac{(B_{IIr}^{o})^{2}\omega_{o}^{2}L}{4P_{i}}\left[S_{o1}\frac{(R_{o1}+R_{o2})^{2}}{\rho_{1}}+S_{o2}\frac{(R_{o2}+R_{o3})^{2}}{\rho_{2}}\right]\) The mechanical loss \(P_{m}\) of the rotating – arm bearing is calculated as \(P_{m}=3.10\pi\cdot10^{-4}\omega_{o}(M_{rr}+M_{sl})\)
The transmission efficiency \(\eta\) of the MRMG is given by \(\eta=\frac{i_{\omega}T_{s}}{T_{i}}=\frac{T_{s}\omega_{o}}{T_{s}\omega_{o}+P_{loss}}=\frac{T_{s}\omega_{o}}{T_{s}\omega_{o}+P_{e}+P_{m}}\)
4. Finite – Element Analysis of MRMG
4.1 Steady – state Analysis
To verify the correctness of the air – gap magnetic field and torque calculation models, we perform finite – element analysis on the MRMG. The model parameters are shown in Table 2.
| Model Parameter | Value |
|---|---|
| Inner Permanent – magnet Pole Pairs \(P_{i}\) | 22 |
| Outer Permanent – magnet Pole Pairs \(P_{o}\) | 23 |
| Outer Yoke (Ring Plate) Inner Diameter \(R_{o3}\)/mm | 96 |
| Outer Permanent – magnet Outer Radius \(R_{o2}\)/mm | 100 |
| Outer Permanent – magnet Inner Radius \(R_{o1}\)/mm | 94 |
| Inner Permanent – magnet Outer Radius \(R_{i1}\)/mm | 90 |
| Inner Permanent – magnet Inner Radius \(R_{i2}\)/mm | 84 |
| Inner Yoke Inner Diameter \(R_{i3}\)/mm | 88 |
| Inner and Outer Permanent – magnet Ring Eccentricity \(\delta\)/mm | 3 |
| Axial Length L/mm | 30 |
| Permanent – magnet Remanence \(B_{r}\)/T | 1.25 |
| Number of Eccentric Shafts | 3 |
| Permanent – magnet Resistivity \(\rho_{1}\)/(Ω·m) | \(1.44\times10^{-6}\) |
| Yoke Resistivity \(\rho_{2}\)/(Ω·m) | \(2\times10^{-7}\) |
| Rotating – arm Bearing Model | 6005 – 2RS |
We calculate the radial and tangential air – gap magnetic densities at the center of the air – gap (\(\beta = 0^{\circ}\) and \(r_{i}=\frac{R_{o1}+R_{i1}-\delta\cos\theta_{i}}{2}\)) using both the finite – element method and the theoretical model. The comparison results show that the theoretical calculation and the finite – element simulation results of \(B_{IIr}\) and \(B_{II\theta}\) are basically the same, with an average error of less than 4%, indicating the correctness of the air – gap magnetic field model.
We also compare the electromagnetic torque \(T_{i}\) and radial electromagnetic force \(F_{ir}\) of the inner permanent – magnet ring obtained from the finite – element method and the theoretical model when the inner permanent – magnet ring rotates through a pair of magnetic poles (\(0^{\circ}\leq\beta\leq16.36^{\circ}\)). The results show that the average error of the torque and force is less than 5%, verifying the correctness of the torque and force calculation models.
[Insert Figure 2: Comparison curves of \(B_{IIr}\) and \(B_{II\theta}\) between finite – element and theoretical calculations] [Insert Figure 3: Comparison curves of \(T_{i}\) and \(F_{ir}\) between finite – element and theoretical calculations]
4.2 Dynamic Analysis
We perform dynamic analysis on the MRMG using the finite – element method and compare the results with the dynamic model established in this paper. The inner permanent – magnet ring rotates at a speed of \(\omega_{i}=20r/min\), and the outer permanent – magnet ring is in a linear loading mode with a load torque increased to \(4.36N\cdot m\) within \(0.1s\).
The results show that during the start – up process of the MRMG, the speed and torque fluctuate greatly. As the operation time increases, the fluctuation amplitude gradually decreases and finally converges to the target set value. The finite – element simulation curve is basically consistent with the curve of the established model, with an average error of less than 7%, verifying the correctness of the dynamic model.
[Insert Figure 4: Comparison curves of MRMG dynamic speed and torque between finite – element simulation and theoretical calculation]
4.3 Eddy – Current Loss Analysis
We compare the eddy – current loss \(P_{e}\) of the MRMG obtained from the finite – element simulation and the theoretical calculation. The results show that the \(P_{e}\) value of the MRMG increases gradually with the increase of \(\omega_{o}\). The \(P_{e}\) value obtained from the theoretical calculation is basically the same as that of the finite – element simulation, with an average error of less than 5%, verifying the correctness of the eddy – current loss calculation formula.
[Insert Figure 5: Comparison curves of \(P_{e}\) between finite – element and theoretical calculations]
5. Dynamic Loading Experiment of MRMG
5.1 Experimental Setup
We conduct dynamic loading experiments on the MRMG with the minimum number of eccentric shafts (\(n = 3\)) to study its power – splitting characteristics and dynamic redundant capabilities. The experimental setup consists of a driving motor, an input – end speed and torque sensor, an experimental prototype, an output – end speed and torque sensor, three permanent – magnet synchronous generators, and three sets of Y – type symmetric load resistors.
The driving motor provides input speed for the central shaft of the inner permanent – magnet ring. The three generators are connected to the eccentric shafts through couplings and output voltage in a power – splitting manner. Each generator’s three – phase stator windings are connected to a set of Y – type symmetric load resistors to consume the output power and provide load torque for the eccentric shafts.
[Insert Figure 6: MRMG experimental prototype and dynamic loading experimental device]
5.2 Experimental Results and Analysis
5.2.1 Verification of \(i_{\omega}\) and \(i_{T}\) in Steady – state Operation
We study the \(i_{\omega}\) and \(i_{T}\) of the MRMG in the “single – input/multi – output” mode with the same load on each eccentric shaft. The experimental data, theoretical calculation data, and finite – element simulation data are shown in Table 3.
| Inner Permanent – magnet Ring | Outer Permanent – magnet Ring | Eccentric Shaft Total Output Torque \(T_{s}\)/(N·m) | Transmission Ratio \(i_{\omega}\) | Torque Ratio \(i_{T}\) | ||
|---|---|---|---|---|---|---|
| Input Speed \(\omega_{i}\)/(r/min) | Input Torque |
| Inner Permanent – magnet Ring | Outer Permanent – magnet Ring | Eccentric Shaft Total Output Torque \(T_{s}\)/(N·m) | Transmission Ratio \(i_{\omega}\) | Torque Ratio \(i_{T}\) | ||
|---|---|---|---|---|---|---|
| Input Speed \(\omega_{i}\)/(r/min) | Input Torque \(T_{i}\)/(N·m) | Output Speed \(\omega_{o}\)/(r/min) | ||||
| Experiment | – | – 27.70 | – 36.11 | 1.10 | – 21.75 | – 25.18 |
| Theory | 1.66 | – 26.92 | – 36.51 | – 21.99 | – 24.47 | |
| Simulation | – | – 25.98 | – 36.01 | – 21.69 | – 23.62 | |
| Experiment | 2.43 | – 37.28 | – 53.64 | – 22.07 | – 24.53 | |
| Theory | – 36.45 | – 53.64 | 1.52 | – 22.07 | – 23.98 | |
| Simulation | – 33.84 | – 53.17 | – 21.88 | – 22.26 | ||
| Experiment | 4.04 | – 56.29 | – 89.81 | – 22.23 | – 23.95 | |
| Theory | – 55.33 | – 88.92 | 2.35 | – 22.01 | – 23.54 | |
| Simulation | – 53.82 | – 90.17 | – 22.32 | – 22.90 | ||
| Experiment | 5.06 | – 67.22 | – 111.81 | 2.82 | – 22.10 | – 23.84 |
| Theory | – 66.06 | – 111.27 | – 21.99 | – 23.43 | ||
| Simulation | – 64.79 | – 109.35 | – 21.61 | – 22.98 | ||
| Experiment | 6.07 | – 77.46 | – 133.86 | – 22.05 | – 23.76 | |
| Theory | – 76.10 | – 133.47 | 3.26 | – 21.99 | – 23.34 | |
| Simulation | – 74.45 | – 132.09 | – 21.76 | – 22.84 |
The results show that the \(i_{\omega}\) and \(i_{T}\) curves are basically straight lines with constant slopes. The relative errors of the average transmission ratio \(i_{\omega}\) and torque ratio \(i_{T}\) among the experimental, theoretical, and simulation results are within 1% and 6% respectively, which meet the design requirements.
5.2.2 Verification of Transmission Losses
We measure the relationship between \(\omega_{i}\), \(P_{loss}\), and \(\eta\) of the MRMG in two operation modes through experiments. The experimental data are shown in Table 4.
| Operation Mode | Inner Permanent – magnet Ring | Outer Permanent – magnet Ring | Shaft 1 \(T_{s1}\)/(N·m) | Shaft 2/3 \(T_{s2/3}\)/(N·m) | MRMG | ||
|---|---|---|---|---|---|---|---|
| \(\omega_{i}\)/(r/min) | \(T_{i}\)/(N·m) | \(\omega_{o}\)/(r/min) | \(P_{loss}\)/(W) | \(\eta\) (%) | |||
| Single – input/multi – output mode with equal loads on each eccentric shaft | 1.66 | – 27.70 | – 36.11 | 0.37 | 0.61 | 87.17 | |
| 2.43 | – 37.28 | – 53.64 | 0.51 | 0.89 | 90.59 | ||
| 4.04 | – 56.29 | – 89.80 | 0.78 | 1.81 | 92.40 | ||
| 5.06 | – 67.22 | – 111.81 | 0.94 | 2.60 | 92.70 | ||
| 6.07 | – 77.46 | – 133.86 | 1.08 | 3.82 | 92.24 | ||
| Single – input/multi – output mode with different loads on each eccentric shaft | 1.59 | – 18.12 | – 34.74 | 0.25 | 0.18 | 0.79 | 73.55 |
| 2.53 | – 23.15 | – 54.93 | 0.33 | 0.24 | 1.47 | 75.97 | |
| 3.92 | – 30.87 | – 86.34 | 0.46 | 0.34 | 2.36 | 81.34 | |
| 4.94 | – 36.35 | – 108.30 | 0.55 | 0.40 | 3.49 | 81.42 | |
| 6.06 | – 42.30 | – 133.42 | 0.65 | 0.48 | 4.35 | 83.80 | |
| Single – input/single – output mode | 1.46 | – 13.14 | – 31.62 | 0.33 | 0.92 | 54.39 | |
| 2.50 | – 19.09 | – 54.80 | 0.51 | 2.07 | 58.56 | ||
| 4.09 | – 26.57 | – 90.07 | 0.78 | 4.02 | 64.65 | ||
| 4.93 | – 30.28 | – 108.63 | 0.91 | 5.28 | 66.22 | ||
| 6.06 | – 34.14 | – 132.07 | 1.07 | 6.87 | 68.30 |
It can be seen from the table that \(\eta\) increases with the increase of \(i_{T}\) and \(\omega_{i}\). The “single – input/multi – output” mode with equal loads on each eccentric shaft has a much higher \(\eta\) value than the “single – input/single – output” mode. Reducing the load difference between the eccentric shafts can improve the \(\eta\) of the MRMG. The eddy – current loss \(P_{e}\) accounts for a small proportion (\(< 1\%\)) in \(P_{loss}\). The average errors of \(P_{loss}\) and \(\eta\) between the experimental results and the theoretical model in different operation modes are within an acceptable range, indicating that the established model is suitable for analyzing the full – condition dynamic operation process of the MRMG.
6. Comparison with Other Permanent Magnet Gears
6.1 Comparison with Concentric Permanent Magnet Gears (CPMG)
CPMGs, mainly represented by magnetic – field – modulated gears, have advantages such as high torque density and high transmission efficiency. However, their magnetic – modulation devices are complex, and the prototype manufacturing is difficult due to the structure of three – layer rotors and two – layer air – gaps. In contrast, the MRMG has a relatively simple structure. Although it also needs to deal with the problem of eccentric magnetic field, the use of multiple eccentric shafts can effectively distribute the electromagnetic force, improving the reliability and service life.
6.2 Comparison with Cycloid Permanent Magnet Gears (CMG)
CMGs use the eccentric air – gap magnetic field formed by the inner and outer permanent – magnet rings and have characteristics of high torque density and large transmission ratio. But the rotating – arm bearing of CMGs is located inside the cycloid gear, resulting in a small center distance between the rotating – arm bearing and the output shaft. The large radial load caused by the eccentric magnetic field seriously reduces the reliability and service life of the rotating – arm bearing. The MRMG overcomes this shortcoming by placing the eccentric shafts and their associated rotating – arm bearings outside the center of the ring plate. It can increase the number of eccentric shafts and rotating – arm bearings to disperse the non – uniform electromagnetic force, and the transmission losses in different load conditions are higher than those of CMGs, which is beneficial for reducing the load on the bearing and improving the overall performance.
7. Conclusion
The multi – shaft ring – plate permanent magnet gear (MRMG) is a novel type of permanent – magnet gear with unique structure and performance.
- The non – uniform air – gap length of the MRMG has time – varying cosine characteristics, which can modulate the magnetic fields of the permanent magnets with different pole pairs of the inner and outer permanent – magnet rings into the same pole – harmonic number for speed and torque transmission. The magnetic deflection angle \(\beta\) plays a crucial role in regulating the air – gap magnetic density and promoting the re – balance of the air – gap magnetic field. All the quantifiable mathematical models and transmission characteristics are functions of the \(\beta\) angle.
- By dividing the MRMG into a passive region and two active regions and using the continuity of the magnetic field in the air – gap, the eccentric air – gap magnetic field equations of the inner and outer permanent – magnet rings can be obtained. The fractional – linear transformation method can transform the eccentric magnetic field into a concentric magnetic field, enabling the calculation of the air – gap magnetic density, electromagnetic torque, and the electromagnetic force balance equations of each eccentric shaft.
- The MRMG has a high torque – density characteristic (about \(138 kN·m/m^{3}\)) due to the large effective coupling area of the inner and outer permanent – magnet rings, which accounts for more than 2/3 of the total magnetic – ring area, and each permanent – magnet can participate in the generation of the air – gap magnetic field and torque transmission. The eddy – current loss in the permanent magnets and yokes is small (\(< 1\%\) of \(P_{loss}\)) during stable operation because of the low relative speed between the inner and outer permanent – magnet rings.
- The MRMG has two transmission modes: “single – input/single – output” and “single – input/multi – output”. The “single – input/single – output” mode has large power losses and low transmission efficiency because only one eccentric shaft bears the torque, resulting in unbalanced force on the outer permanent – magnet ring and large radial – force fluctuations of the eccentric shafts. The “single – input/multi – output” mode has a more uniform force distribution on the outer permanent – magnet ring, smaller force fluctuations, and higher transmission efficiency.
- In the “single – input/multi – output” mode, reducing the load difference between the eccentric shafts can reduce the internal – branch power loss. When the load difference cannot be changed, increasing the total load torque can improve the transmission efficiency. The “single – input/multi – output” mode with equal loads on each eccentric shaft is the optimal operation mode, followed by the mode with load – bearing but load – deviation on each eccentric shaft, and the “single – input/single – output” mode has the lowest transmission efficiency, with the highest measured efficiency being only 73.68% of the optimal mode.
