Analysis of structure and starting characteristics of three-shaft ring-plate permanent magnet gear

== Abstract ==
Combined with transmission mode of cycloid permanent magnet gear and ring plate mechanical gear, a three-shaft ring-plate permanent magnet gear transmission structure (TRMG) is proposed and the operating mechanism of structure is given. TRMG simulation model is established and the correctness is verified by transient finite element method. Based on momentum moment theorem, TRMG start-up process and start-up position of are analyzed theoretically. By calculating TRMG torque impulse, the optimal start-up range and start-up position of TRMG are determined.

'''Keywords:''' Multi-axle ring-plate permanent magnet gear, start-up characteristics, start-up position, dynamic characteristics

== 1. Introduction ==

Start-up characteristics refer to the dynamic characteristics of driven rotor at rated load condition from stationary to stable operation stage. Mechanical gearbox, as an important core component of electromechanical transmission system, bears a large impact load in start-up process. With the increase of service frequency, a series of faults such as oil leakage, shaft channeling and tooth damage often occur in mechanical gearbox [1,2].

The non-contact transmission of permanent magnet gear using air gap magnetic field solve the damage caused by starting process of mechanical gear, and realize the conversion requirements between low speed and high speed and low torque which has a broad application prospect [3-5].

Literature [6] based on the principle of mechanical Cycloid Gear transmission, a transmission structure of Cycloid permanent Magnetic Gear, CMG is proposed, which has high transmission ratio and high torque density (Transmission ratio≥20, Torque density≥180kNm/m<sup>3</sup>). Since then, scholars have studied the transmission structure, magnetization, bearing capacity and eddy current loss of CMG, and further improved the transmission ratio and torque density of structure (Transmission ratio≥100, Torque density≥280kNm/m<sup>3</sup>) [7-10].

Although CMG has large transmission ratio and torque density, its service life is relatively short due to a large unbalanced magnetic pull by internal rotor shaft. In addition, the start-up characteristics of CMG have not been studied in the above literatures [11,12].

In order to improve the service life of CMG rotating arm bearing and solve start-up characteristics, this paper proposes a three-shaft ring-plate permanent magnet gear transmission structure (TRMG) based on ring-plate mechanical gear transmission mode. Compared with CMG, TRMG moves the rotating arm bearing to the outside of the cycloid wheel to increase the number of rotating arm bearing, disperse the payload of rotating arm bearing and prolong the service life of rotating arm bearing. In addition, this paper uses theorem of momentum moment to analyze the start-up position of TRMG and determine the optimal start-up range and start-up position of TRMG.

== 2. TRMG Operation mechanism ==

[[#img-1|Figure 1]] shows the mechanical structure of TRMG. In [[#img-1|Figure 1]], three eccentric high-speed shafts are connected with ring plate permanent magnet ring through rotating arm bearings. The ring plate permanent magnet ring consists of outer yoke iron and outer ring permanent magnet, and is located outside the low-speed permanent magnet ring. The low-speed permanent magnet ring is composed of inner yoke iron and inner ring permanent magnet, and a low-speed shaft is installed in the center.

<div id='img-1'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"|  [[File:Draft_Dongning_226371700-image1.jpg|600x600px]] 
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figure 1'''. TRMG mechanical structure 
|}


In [[#img-1|Figure 1]], the center distances of three eccentric high-speed shafts to low-speed shaft are equal to the eccentricity between ring plate permanent magnet ring and low-speed permanent magnet ring. The low-speed permanent magnet ring rotates around the center of low-speed shaft.

When the eccentric high-speed shaft moves the ring plate permanent magnet ring, the outer ring permanent magnet revolves around the inner ring permanent magnet. As the relative position of two permanent magnets changes, the permanent magnet embedded in low-speed permanent magnet ring is driven to rotate low-speed shaft by the change of magnetic field force.

Due to the large difference between the eccentricity and the radius of the low-speed permanent magnet ring, a large transmission ratio can be achieved.

Set transmission ratio of TRMG as  <math display="inline">i</math>, and pole pairs of inner ring permanent magnet and outer ring permanent magnet as <math display="inline">p_\mbox{i}</math>  and  <math display="inline">p_\mbox{o}</math>, respectively, then:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>i=-\frac{p_\mbox{o}-p_\mbox{i}}{p_\mbox{i}}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
|}

Since the pole pairs of outer ring permanent magnet is always one more than that of inner ring permanent magnet, when the ring plate permanent magnet ring revolves around the low-speed permanent magnet ring once, the low-speed permanent magnet ring rotates a pair of magnetic poles in reverse.

==3. TRMG start-up position analysis==

Set the eccentric high-speed shaft be driving shaft (the active motion of ring plate permanent magnet ring), and the low-speed shaft be driven shaft (the connection load of low-speed permanent magnet ring), and the starting angle between ring plate permanent magnet ring and low-speed permanent magnet ring is <math display="inline">\beta </math>, and the output torque of low-speed permanent magnet ring is  <math>M_{out}</math>. It can be seen from literature [13-15] that:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>M_{out}(\beta )=M_{max}sin(p_\mbox{i}\beta )</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|}


In Equation (2),  <math>M_{max}</math> is the static maximum torque. [[#img-2|Figure 2]] shows the relationship between  <math>M_{out}</math> and <math display="inline">\beta </math>.

<div id='img-1'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"|  [[File:Draft_Dongning_226371700-image10.png|391x391px]] 
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figure 2'''. ''M''<sub>ou</sub><sub>t</sub> and ''β'' curve 
|}


In [[#img-2|Figure 2]], when starting angle is  <math display="inline">{\beta }_1</math>,  <math display="inline">{\beta }_3</math>,  <math display="inline">{\beta }_6</math> or  <math display="inline">{\beta }_7</math> the output torque of point Q, V, K or Z is equal to the load torque  <math>M_{load}</math>. When the starting angle is <math display="inline">{\beta }_2</math>, the output torque of point U is the static maximum torque <math>M_{max}</math>. When the starting angle is 0,  <math display="inline">{\beta }_4</math> or <math display="inline">{\beta }_5</math> , the structures of point P, W or I are in start-up equilibrium position.

Set the difference between output torque and load torque as <math display="inline">\Delta M</math>, then:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math display="inline">\Delta M\mbox{=}M-M_{load}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Delta M\mbox{=}J\frac{d\sigma }{dt}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
|}

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math display="inline">d\beta =\Delta \sigma dt</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
|}


In Equations (4) and (5),  <math>J</math> is the inertia of low-speed permanent magnet ring,  <math>\sigma </math> is its angular velocity and  <math>\Delta \sigma </math> is the change of angular velocity.

When the relative rotation angle between the ring plate permanent magnet ring and the low-speed permanent magnet ring increases from 0 to  <math display="inline">{\beta }_1</math>, set the work done by  <math display="inline">\Delta M</math> on the rotor be <math display="inline">G_1</math>, then:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>G_1=\int_{t_0}^{t_1}\Delta M\sigma dt</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|}

Substituting Equations (4) and (5) into Equation (6), and the equation (6) is transformed by equality, we can get:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>G_1={\int }_0^{{\beta }_1}\Delta Md\beta +{\int }_{t_0}^{t_1}J\frac{d\sigma }{dt}{\sigma }_0dt{=}{\int }_0^{{\beta }_1}\Delta Md\beta +J{\sigma }_0{\int }_{{\sigma }_0}^{{\sigma }_1}d\sigma</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
|}


It can be seen from [[#img-2|Figure 2]] that the first integral in Equation (7) can be approximately regarded as the area of  <math display="inline">\Delta \mbox{PSQ}</math>. When TRMG runs at curve PQ, the forward electromagnetic torque of low-speed permanent magnet ring is less than load torque, and the reverse acceleration can be obtained. Therefore, Equation (7) can be written as:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>G_1=J{\sigma }_0\left({\sigma }_1-{\sigma }_0\right)-\Delta \mbox{PSQ}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
|}


Substitute Equation (3) into Equation (6), then:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>G_1={\int }_{t_0}^{t_1}J\frac{d\sigma }{dt}{\sigma }dt{=}{\frac{1}{2}}{J({\sigma }_1^2}-{\sigma }_0^2)</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
|}


Equation (8) can be simplified as follows:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Delta \mbox{PSQ}=\frac{J{\left({\sigma }_1-{\sigma }_0\right)}^2}{2}\mbox{=}{\frac{J{\sigma }_1^2}{2}\vert }_{{\sigma }_0\mbox{=}0}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
|}

In Equation (10), TRMG starts from static state, so  <math>{\sigma }_0\mbox{=}0</math>.

According to Equation (10), if  <math display="inline">\Delta \mbox{PSQ}</math> is larger, the reverse acceleration kinetic energy and reverse impulse obtained by low-speed permanent magnet ring are larger at this stage, and TRMG is more difficult to start, so it is called deceleration area.

Similarly, when the relative angle between the ring-plate permanent magnet ring and the low-speed permanent magnet ring increases from  <math display="inline">{\beta }_1</math> to <math display="inline">{\beta }_3</math>, set the work done by  <math display="inline">{\beta }_3</math> on rotor be <math display="inline">G_2</math>, then:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>G_2= \int_{{\beta }_1}^{{\beta }_3}\Delta Md\beta +J{\sigma }_1{\int }_{{\sigma }_1}^{{\sigma }_3}d\sigma ={\int }_{{\beta }_1}^{{\beta }_3}\Delta Md\beta +J{\sigma }_1\left({\sigma }_3-{\sigma }_1\right)</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
|}

The integral of first term in Equation (11) can be approximated as  <math display="inline">\Delta \mbox{QUV}</math> area. As TRMG runs at curve QV, the forward electromagnetic torque received by low-speed permanent magnet ring is greater than load torque, and the forward acceleration can be obtained. Therefore, Equation (11) can be expressed as:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>G_2=J{\sigma }_1\left({\sigma }_3-{\sigma }_1\right)+\Delta \mbox{QUV}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
|}

Equation (12) can be simplified as follows:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Delta \mbox{QUV}=\frac{J{\left({\sigma }_3-{\sigma }_1\right)}^2}{2}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
|}

According to Equation (13), if  <math display="inline">\Delta \mbox{QUV}</math> is larger, the forward acceleration kinetic energy and forward torque impulse obtained by rotor can more easily offset the reverse impulse when  <math display="inline">\beta \in [0,{\beta }_1]</math>, and TRMG can more easily start-up, so it is called acceleration area.

== 4. TRMG start-up process analysis ==

Set the relative rotation angle of ring plate permanent magnet ring and low-speed permanent magnet ring is <math display="inline">\alpha </math>.

According to the operation mechanism of TRMG, due to TRMG is in equilibrium position before start-up, the output torque of TRMG is 0 as the force between air gap magnetic fields of the ring plate permanent magnet ring and the low-speed permanent magnet ring is balanced. At this point, the equilibrium position of TRMG can be divided into two kinds: one is that when TRMG is start-up,  <math display="inline">M</math> increases positively with <math display="inline">\alpha </math> increase. The other is that when TRMG is start-up,  <math display="inline">M</math> reverse increases with α increase.

Set the starting point of TRMG  <math display="inline">\left(\beta =0\right)</math> is started at equilibrium position of forward torque period, and the active rotor is generated a forward rotating magnetic field by the action of magnetic field air gap.

When  <math display="inline">\alpha \in [0,{\beta }_1]</math>, there is a slip between ring plate permanent magnet ring and low-speed permanent magnet ring, and the forward electromagnetic torque of low-speed permanent magnet ring is less than load torque, the low-speed permanent magnet ring speeds up in reverse rotation and generates a reverse impulse of  <math display="inline">\Delta \mbox{PSQ}</math>. TRMG does not start properly at this stage.

When  <math display="inline">\alpha \in [{\beta }_1,{\beta }_3]</math>, the electromagnetic torque of low-speed permanent magnet ring is always larger than load torque, and the forward angular acceleration can be obtained and the forward impulse of  <math display="inline">\Delta \mbox{QUV}</math> can be generated. Since the low-speed permanent magnet ring has acquired the reverse impulse of  <math display="inline">\Delta \mbox{PSQ}</math> to accelerate in reverse, the low-speed permanent magnet ring decelerates in reverse and then accelerate in forward direction by the action of forward torque impulse.

If  <math display="inline">\Delta \mbox{PSQ}<\Delta \mbox{QUV}</math>, before  <math display="inline">\alpha </math>  increases to  <math display="inline">{\beta }_3</math>, the low-speed permanent magnet ring reaches synchronous speed of rotating magnetic field, and TRMG enters adjustment stage of speed and torque. By combined action of magnetic and mechanical damping, TRMG oscillation gradually attenuates, and the rotational speed and torque gradually become stable.

If  <math display="inline">\Delta \mbox{PSQ}>\Delta \mbox{QUV}</math>,  <math display="inline">\alpha </math>  enters interval <math display="inline">\left[{\beta }_3,{\beta }_4\right]</math>, the forward electromagnetic torque of low-speed permanent magnet ring is less than load torque, which makes it decelerate forward first and then accelerate backward. Due to the speed of low-speed permanent magnet ring is still less than synchronous speed of rotating magnetic field, and TRMG accelerates in reverse by combined action of reverse electromagnetic torque and load torque, which eventually leads to start-up failure.

The above analysis shows that:

(1) If TRMG start-up smoothly, the low-speed permanent magnet ring should reach synchronous speed by sufficient forward electromagnetic torque impulse in  <math display="inline">\left[0,{\beta }_4\right]</math> interval.

(2) If  <math display="inline">\Delta \mbox{QUV}</math> (acceleration area) remains same, adjust  <math display="inline">\beta </math> to reduce  <math display="inline">\Delta \mbox{PSQ}</math> (deceleration area), so as to reduce the reverse torque impulse of low-speed permanent magnet ring, easy to start-up.

When  <math display="inline">\beta \in [{\beta }_1,{\beta }_2]</math>, TRMG can start-up normally. When  <math display="inline">\beta \in [0,{\beta }_1]\cup [{\beta }_2,{\beta }_3]</math>, TRMG may not start-up. When  <math display="inline">\beta \in [{\beta }_3,{\beta }_4]</math>, TRMG cannot start-up, so the start-up position determines whether TRMG starts smoothly.

== 5. TRMG finite element simulation analysis ==

Due to ANSYS Maxwell simulation software cannot dynamically simulate the object rotating around eccentric axis, in order to verify the start-up position theory, the operation mode of TRMG is changed, and the TRMG operation process is judged by verifying the dynamic characteristics of the changed model.

===5.1 TRMG model transformation===

Referring to the various operation modes of cycloidal permanent magnet gear, TRMG can be changed into five operation modes:

(1) The ring plate permanent magnet ring rotates only around the axis of low-speed permanent magnet ring, and the low-speed permanent magnet ring rotates around its own axis. At this time, the direction of input and output speed is opposite.

(2) The low-speed permanent magnet ring rotates only around the axis of ring plate permanent magnet ring, and the ring plate permanent magnet ring rotates around its own axis. At this time, the direction of input and output speed is the same.

(3) The ring plate permanent magnet ring is fixed, and the low-speed permanent magnet ring rotates around its axis and revolves around the axis of the ring plate permanent magnet ring. At this time, the direction of input and output speed is opposite.

(4) The low-speed permanent magnet ring is fixed, and the ring plate permanent magnet ring rotates around its axis and revolves around the axis of the low-speed permanent magnet ring. At this time, the direction of input and output speed is the same.

(5) Both the ring plate and the low-speed permanent magnet ring rotate around their own shafts. At this time, the transmission mode is similar to that of the traditional internal meshing gear, and the input and output speed directions are the same.

Among the 5 types mentioned above, (1) is the motion mode of TRMG, and (2)~(5) are the motion mode of TRMG after fixing different TRMG components. For (1), (2), (3) and (4), there are rotation movement around the eccentric axis, so Ansys Maxwell cannot be used for dynamic simulation. For the motion mode (5), the ring plate and the low-speed permanent magnet ring both move around their own shafts in a fixed axis, which can be accurately simulated dynamically. Therefore, we finally changed TRMG into model (5). If the dynamic characteristics of model (5) meet the design requirements of transmission ratio and torque density after modification, the dynamic characteristics of TRMG can also meet the same requirements under the same structural size, rated power and rated load conditions of the model. The mechanism diagram of TRMG and its changed model (5) is shown in [[#img-3|Figure 3]].

<div id='img-1'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"|  [[File:Draft Dongning 226371700-image88.png.jpg|495x495px]] 
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figure 3'''. Simplified TRMG and changed model structure 
|}


According to the operation principle of planetary gear with small tooth difference, the simplified structure of TRMG and changed model are shown in [[#img-3|Figure 3]].

Set the transmission ratio of model (5) is <math>i_1</math>, the speed of low-speed permanent magnet ring is <math display="inline">n_\mbox{o}</math>, the speed of ring-plate permanent magnet ring is <math display="inline">n_\mbox{i}</math>, and its revolution speed is <math display="inline">n_\mbox{h}</math>. For transmission scheme (1), adding a rotational speed <math display="inline">-n_\mbox{h}</math> around the axis of low-speed permanent magnet ring to the whole mechanism can be converted into the changed structure (5) as shown in Figure 3 (b). The changed mechanism transmission ratio is:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>i_1=\frac{n_\mbox{i}^\mbox{h}}{n_\mbox{o}^\mbox{h}}=\frac{n_\mbox{i}-n_\mbox{h}}{n_\mbox{o}-n_\mbox{h}}=\frac{p_\mbox{i}}{p_\mbox{o}}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
|}

Since the plate permanent magnet ring does not rotate around its axis, <math display="inline">n_i=0</math>.

=== 5.2 Dynamic characteristics of TRMG equivalent model ===

Set TRMG transfer power  <math display="inline">P=1\mbox{kw}</math>, output speed  <math display="inline">67\mbox{r/}min</math>, transmission ratio  <math display="inline">i=22:1</math>, TRMG structure parameters as shown in [[#tab-1|Table 1]] can be obtained.

<div class="center" style="font-size: 75%;">'''Table 1'''. Values of TRMG model parameters</div>

<div id='tab-1'></div>
{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;" 
|-style="text-align:center"
! Symbol !! Description !! Value (Unit)
|-style="text-align:center"
|  <math>\pi  </math> 
|  Pole pairs of inner permanent magnet
|  22
|-style="text-align:center"
|  <math> p_0 </math>
|  Pole pairs of outer permanent magnet
|  23
|-style="text-align:center"
| <math>R_1  </math>  
|  Inner radius of outer yoke iron
|  96(mm)
|-style="text-align:center"
|  <math>R_2 </math>  
|  Outer radius of ring plate permanent magnet ring
|  100(mm)
|-style="text-align:center"
|  <math>R_3 </math>  
|  Inner radius of the ring plate permanent magnet ring
|  94(mm)
|-style="text-align:center"
|  <math>R_4 </math>  
|  Outer radius of low-speed permanent magnet ring
|  90(mm)
|-style="text-align:center"
|  <math>R_5 </math>  
|  Inner radius of low-speed permanent magnet ring
|  84(mm)
|-style="text-align:center"
|  <math>R_6 </math>
|  Outer radius of inner yoke iron
|  88(mm)
|-style="text-align:center"
|  <math>h_i  </math> 
|  Inner yoke iron thickness
|  15(mm)
|-style="text-align:center"
|  <math>h_0 </math> 
|  Outer yoke iron thickness
|  15(mm)
|-style="text-align:center"
|  <math> e</math> 
|  Eccentricity
|  3(mm)
|-style="text-align:center"
|  <math> M </math>
|  Magnetization
|  -890000(A/m)
|-style="text-align:center"
|  <math>\mu_0  </math>
|  Vacuum permeability
|  <math>4\pi\times 10^{-7} </math> 
|-style="text-align:center"
|  <math>\mu  </math>
|  Relative permeability
|  1.0997
|-style="text-align:center"
|  <math>B_r  </math>
|  Permanent magnet remanence
|  1.21(M)
|}


According to the changed model transmission ratio, the input speed of ring plate permanent magnet ring is  <math display="inline">70\mbox{r/}min</math>, and the load torque of low-speed permanent magnet ring is  <math display="inline">124\mbox{N}\cdot \mbox{m}</math>. The dynamic characteristic curve of changed model is shown in [[#img-4|Figure 4]].

<div id='img-1'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"| [[File:Draft_Dongning_226371700-image65-c.png|564x564px]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figure 4'''. TRMG characteristic curve
|}


From [[#img-4|Figure 4]], the electromagnetic torque and output speed of low-speed permanent magnet ring change regularly with time, finally converging to  <math display="inline">67\mbox{r/}min</math>, around  <math display="inline">124\mbox{N}\cdot \mbox{m}</math>, and  <math display="inline">M_{max}=130\mbox{N}\cdot \mbox{m}</math>, meeting the design requirements. The ratio of input to output of speed and torque is about 22/23, which meets the transmission ratio design requirements in Equation (14).

=== 5.3 TRMG start-up characteristic analysis ===

Changing the starting angle between ring plate permanent magnet ring and low-speed permanent magnet ring  <math display="inline">\beta </math>. In [[#img-2|Figure 2]], point P, Q, U and V as starting point which are the left of interval  <math display="inline">\left[0,{\beta }_1\right]</math> 、 <math display="inline">\left[{\beta }_1,{\beta }_2\right]</math> 、 <math display="inline">\left[{\beta }_2,{\beta }_3\right]</math> and  <math display="inline">\left[{\beta }_3,{\beta }_4\right]</math>.

Figure 5 shows the speed and torque characteristic curve with starting point P.

<div id='img-5'></div>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"| [[File:Draft_Dongning_226371700-image74.png|717x717px]] 
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figure 5'''. Speed and Torque characteristic curve Start with starting point P 
|}


From [[#img-5|Figure 5]], it can be seen that from moment 0, the electromagnetic torque of low-speed permanent magnet ring gradually increases from 0 and changes periodically. Its rotating speed accelerates in opposite direction. So TRMG fails to start normally at point P. Because when  <math display="inline">\alpha \in [0,{\beta }_1]</math> , the forward electromagnetic torque of low-speed permanent magnet ring is smaller than load torque, and the low-speed permanent magnet ring accelerates to rotate in opposite direction. When <math display="inline">\alpha \in [{\beta }_1,{\beta }_3]</math> , the forward impulse on the low-speed permanent magnet ring cannot offset the reverse impulse generated in interval  <math display="inline">\left[0,{\beta }_1\right]</math>  <math display="inline">(\Delta \mbox{PSQ}>\Delta \mbox{QUV)}</math> , which prevents it from accelerating to the synchronous speed of rotating magnetic field. When'' '' <math display="inline">\alpha \in [{\beta }_3,{\beta }_4]</math> , the reverse acceleration continues under the combined action of reverse electromagnetic torque and the load torque, which eventually leads to start-up failure.

<span id='_Hlk78752918'></span>Figure 6 shows the speed and torque <span style="text-align: center; font-size: 75%;">characteristic</span> curve with starting point Q.
[[File:Draft_Dongning_226371700-image79.png|centre|thumb|719x719px|<span style="text-align: center; font-size: 75%;">Figure 6. Speed and Torque characteristic curve Start </span>with starting point Q]]

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div>

From Figure 6, it can be seen that from moment 0, the low-speed permanent magnet ring gradually increases and periodically changes from electromagnetic torque equal to load torque, and the speed gradually increases from 0 and stabilizes gradually after vibration. TRMG starts smoothly. This is because: when  <math display="inline">\alpha \in [{\beta }_1,{\beta }_3]</math> , the forward electromagnetic torque impulse of the low-speed permanent magnet ring is greater than the load torque from moment 0, and the deceleration area  <math display="inline">\Delta \mbox{PSQ}</math> is avoided. So that the speed of the low-speed permanent magnet ring reaches synchronous speed of rotating magnetic field smoothly, and the speed and torque curve gradually converge.

<span id='_Hlk78752983'></span>Figure 7 shows the speed and torque <span style="text-align: center; font-size: 75%;">characteristic</span> curve with starting point U.
[[File:Draft_Dongning_226371700-image81.png|centre|thumb|730x730px|<span style="text-align: center; font-size: 75%;">Figure 7. Speed and Torque characteristic curve Start </span>with starting point U]]

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div>

From Figure 7, it can be seen that from moment 0, the electromagnetic torque of the low-speed permanent magnet ring which is equal to the maximum torque, starts to decrease gradually and shows periodic changes. The speed first rotates forward and then decreases in reverse direction. This is because: when  <math display="inline">\alpha \in [{\beta }_2,{\beta }_3]</math> , the forward torque impulse of the low-speed permanent magnet ring is larger than the load torque and causing it rotates forward. But its forward impulse is not sufficient to reach the synchronous speed of rotating magnetic field. Therefore,  when ''<math display="inline">\alpha </math>'' continues to increase and enters interval<math display="inline">\left[{\beta }_3,{\beta }_4\right]</math>, the low-speed permanent magnet ring continues to accelerate in opposite direction, which ultimately leads to the failure of TRMG start-up.

<span id='_Hlk78753021'></span>Figure 8 shows the speed and torque <span style="text-align: center; font-size: 75%;">characteristic</span> curve with starting point V.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Draft_Dongning_226371700-image84.png|centre|thumb|728x728px|<span style="text-align: center; font-size: 75%;"> Figure 8. Speed and Torque characteristic curve Start </span>with starting point V]]<span style="text-align: center; font-size: 75%;"> </span></div>

From Figure 8, it can be seen that from moment 0, the electromagnetic torque of low-speed permanent magnet ring shows periodic changes and starts to decrease gradually from equal to the load torque. The speed accelerates reversely from 0. This is because: when'' '' <math display="inline">\alpha \in [{\beta }_3,{\beta }_4]</math> , the electromagnetic torque of the low-speed permanent magnet ring is less than the load torque, and the torque difference gradually increases with α increasing, which makes the low-speed permanent magnet ring reverse speed up rotation. After TRMG enters negative electromagnetic torque period, TRMG accelerates reverse rotates under the action of load torque and negative electromagnetic torque, which eventually leads to start-up failure.

== 6. Conclusions ==
(1) Based on structure mechanism of mechanical ring-plate drive mechanism, TRMG magnetic structure is proposed, and the operation mechanism and changed model of the model are elaborated. The dynamic verification model by finite element software shows that the simulation results are equivalent to the theoretical design output, which proves the established model is correct.

(2) Through the analysis of the starting position, it can be seen that  <math display="inline">\left[{\beta }_1,{\beta }_2\right]</math> interval in Figure2 is the optimal starting interval of TRMG, and point Q is the optimal starting point.

(3) If TRMG can be started successfully, the deceleration interval  <math display="inline">\Delta \mbox{PSQ}</math> of low-speed permanent magnet ring should be minimized. and increase acceleration interval  <math display="inline">\Delta \mbox{QUV}</math> to achieve synchronous speed of coupled magnetic field.

== Acknowledgement ==
This work was funded by the National Natural Science Foundation of China (Grant No.51375063), and also sponsored by the Natural Science Research Project of Liaoning Province Education Department (Grant No.JDL2020001) and partly funded by the Technological Innovation Research Foundation Project of Dalian (Grant No.2018J12SN071).

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