You do not have permission to edit this page, for the following reason:

You are not allowed to execute the action you have requested.


You can view and copy the source of this page.

x
 
1
== Abstract ==
2
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.
3
4
'''Keywords:''' Multi-axle ring-plate permanent magnet gear, Start-up characteristics, Start-up position, Dynamic characteristics
5
6
== 1. Introduction ==
7
<span id='_Hlk77463873'></span>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].
8
9
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].
10
11
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].
12
13
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].
14
15
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.
16
17
== 2. TRMG Operation mechanism ==
18
<span id='_Hlk78752063'></span>Figure 1 shows the mechanical structure of TRMG. In 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.
19
20
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Draft_Dongning_226371700-image1.jpg|centre|thumb|600x600px|<span style="text-align: center; font-size: 75%;">Figure 1. TRMG mechanical structure</span>]]</div>
21
22
In 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.
23
24
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.
25
26
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.
27
28
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:
29
30
{| class="formulaSCP" style="width: 100%; text-align: center;" 
31
|-
32
| 
33
{| style="text-align: center; margin:auto;" 
34
|-
35
| <math>i=-\frac{p_\mbox{o}-p_\mbox{i}}{p_\mbox{i}}</math>
36
|}
37
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
38
|}
39
40
41
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.
42
43
== 3. TRMG start-up position analysis ==
44
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:
45
46
{| class="formulaSCP" style="width: 100%; text-align: center;" 
47
|-
48
| 
49
{| style="text-align: center; margin:auto;" 
50
|-
51
| <math>M_{out}(\beta )=M_{max}sin(p_\mbox{i}\beta )</math>
52
|}
53
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
54
|}
55
56
57
In Equation (2),  <math>M_{max}</math> is the static maximum torque. Figure 2 shows the relationship between  <math>M_{out}</math> and '' <math display="inline">\beta </math> ''.
58
59
<div id="Figure2" class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Draft_Dongning_226371700-image10.png|centre|thumb|391x391px|<span style="text-align: center; font-size: 75%;">Figure 2. ''M''<sub>ou</sub><sub>t</sub>'' ''and ''β ''curve</span>]]</div>
60
61
<span id='_Hlk78752251'></span><span id='_Hlk73217278'></span>In 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.
62
63
Set the difference between output torque and load torque as <math display="inline">\Delta M</math> , then:
64
65
{| class="formulaSCP" style="width: 100%; text-align: center;" 
66
|-
67
| 
68
{| style="text-align: center; margin:auto;" 
69
|-
70
| <math display="inline">\Delta M\mbox{=}M-M_{load}</math>
71
|}
72
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
73
|}
74
75
{| class="formulaSCP" style="width: 100%; text-align: center;" 
76
|-
77
| 
78
{| style="text-align: center; margin:auto;" 
79
|-
80
| <math>\Delta M\mbox{=}J\frac{d\sigma }{dt}</math>
81
|}
82
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
83
|}
84
85
{| class="formulaSCP" style="width: 100%; text-align: center;" 
86
|-
87
| 
88
{| style="text-align: center; margin:auto;" 
89
|-
90
| <math display="inline">d\beta =\Delta \sigma dt</math>
91
|}
92
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
93
|}
94
95
96
In Equations (4) and (5),  <math>J</math> is the inertia of low-speed permanent magnet ring;  <math>\sigma </math> is its angular velocity;  <math>\Delta \sigma </math> is the change of angular velocity.
97
98
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:
99
100
{| class="formulaSCP" style="width: 100%; text-align: center;" 
101
|-
102
| 
103
{| style="text-align: center; margin:auto;" 
104
|-
105
| <math display="inline">G_1={\int }_{t_0}^{t_1}\Delta M\sigma dt</math>
106
|}
107
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
108
|}
109
110
111
Substituting Equations (4) and (5) into Equation (6), and the equation (6) is transformed by equality, we can get:
112
113
{| class="formulaSCP" style="width: 100%; text-align: center;" 
114
|-
115
| 
116
{| style="text-align: center; margin:auto;" 
117
|-
118
| <math display="inline">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>
119
|}
120
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
121
|}
122
123
124
It can be seen from 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:
125
126
{| class="formulaSCP" style="width: 100%; text-align: center;" 
127
|-
128
| 
129
{| style="text-align: center; margin:auto;" 
130
|-
131
| <math display="inline">G_1=J{\sigma }_0\left({\sigma }_1-{\sigma }_0\right)-\Delta \mbox{PSQ}</math>
132
|}
133
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
134
|}
135
136
137
Equation (8) can be simplified as follows:
138
139
{| class="formulaSCP" style="width: 100%; text-align: center;" 
140
|-
141
| 
142
{| style="text-align: center; margin:auto;" 
143
|-
144
| <math display="inline">\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>
145
|}
146
| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
147
|}
148
149
150
In Equation (9), TRMG starts from static state, so  <math>{\sigma }_0\mbox{=}0</math> .
151
152
According to Equation (9), 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.
153
154
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:
155
156
{| class="formulaSCP" style="width: 100%; text-align: center;" 
157
|-
158
| 
159
{| style="text-align: center; margin:auto;" 
160
|-
161
| <math>G_2\mbox{=}{\int }_{{\beta }_1}^{{\beta }_3}\Delta Md\beta +J{\sigma }_1{\int }_{{\sigma }_1}^{{\sigma }_3}d\sigma \mbox{=}{\int }_{{\beta }_1}^{{\beta }_3}\Delta Md\beta +J{\sigma }_1\left({\sigma }_3-{\sigma }_1\right)</math>
162
|}
163
| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
164
|}
165
166
167
The integral of first term in Equation (10) 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 (10) can be expressed as:
168
169
{| class="formulaSCP" style="width: 100%; text-align: center;" 
170
|-
171
| 
172
{| style="text-align: center; margin:auto;" 
173
|-
174
| <math display="inline">G_2=J{\sigma }_1\left({\sigma }_3-{\sigma }_1\right)+\Delta \mbox{QUV}</math>
175
|}
176
| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
177
|}
178
179
180
Equation (11) can be simplified as follows:
181
182
{| class="formulaSCP" style="width: 100%; text-align: center;" 
183
|-
184
| 
185
{| style="text-align: center; margin:auto;" 
186
|-
187
| <math display="inline">\Delta \mbox{QUV}=\frac{J{\left({\sigma }_3-{\sigma }_1\right)}^2}{2}</math>
188
|}
189
| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
190
|}
191
192
193
According to Equation (12), 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.
194
195
== 4. TRMG start-up process analysis ==
196
Set the relative rotation angle of ring plate permanent magnet ring and low-speed permanent magnet ring is '' <math display="inline">\alpha </math> ''.
197
198
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.
199
200
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.
201
202
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.
203
204
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.
205
206
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.
207
208
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.
209
210
The above analysis shows that:
211
212
(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.
213
214
(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.
215
216
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.
217
218
== 5. TRMG finite element simulation analysis ==
219
Due to ANSYS Maxwell simulation software cannot dynamically simulate the object rotating around eccentric axis, in order to verify the start-up position theory, TRMG should be equivalent to a model, and the operation process of TRMG can be judged by verifying the dynamic characteristics of the equivalent model.
220
221
=== 5.1 Dynamic characteristics of TRMG equivalent model ===
222
Set TRMG transfer power  <math display="inline">P=1\mbox{kw}</math> , output speed  <math display="inline">n_\mbox{o}\mbox{=}67\mbox{r/}min</math> , transmission ratio  <math display="inline">G=22:1</math> , TRMG structure parameters as shown in Table 1 can be obtained.
223
224
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
225
<span style="text-align: center; font-size: 75%;">Table1 Values of TRMG model parameters</span></div>
226
227
{| style="width: 69%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
228
|-
229
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Symbol</span>
230
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Description</span>
231
|  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Value (Unit)</span>
232
|-
233
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''pi''</span>
234
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Pole pairs of inner permanent magnet</span>
235
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">22</span>
236
|-
237
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''p''<sub>o</sub></span>
238
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Pole pairs of outer permanent magnet</span>
239
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">23</span>
240
|-
241
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''R''<sub>1</sub></span>
242
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Inner radius of outer yoke iron</span>
243
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">96(mm)</span>
244
|-
245
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''R''<sub>2</sub></span>
246
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Outer radius of ring plate permanent magnet ring</span>
247
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">100(mm)</span>
248
|-
249
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''R''<sub>3</sub></span>
250
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Inner radius of the ring plate permanent magnet ring</span>
251
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">94(mm)</span>
252
|-
253
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''R''<sub>4</sub></span>
254
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Outer radius of low-speed permanent magnet ring</span>
255
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">90(mm)</span>
256
|-
257
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''R''<sub>5</sub></span>
258
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Inner radius of low-speed permanent magnet ring</span>
259
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">84(mm)</span>
260
|-
261
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''R''<sub>6</sub></span>
262
|  style="border: 1pt solid black;text-align: center;"|<span id='_Hlk68261478'></span><span style="text-align: center; font-size: 75%;">Outer radius of inner yoke iron</span>
263
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">88(mm)</span>
264
|-
265
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''h''<sub>i</sub></span>
266
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Inner yoke iron thickness</span>
267
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">15(mm)</span>
268
|-
269
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''h''<sub>o</sub></span>
270
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Outer yoke iron thickness</span>
271
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">15(mm)</span>
272
|-
273
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''e''</span>
274
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Eccentricity</span>
275
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">3(mm)</span>
276
|-
277
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''M''</span>
278
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Magnetization</span>
279
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">-890000(A/m)</span>
280
|-
281
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''μ''<sub>0</sub></span>
282
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Vacuum permeability</span>
283
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">4π×10<sup>-7</sup></span>
284
|-
285
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''μ''</span>
286
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Relative permeability</span>
287
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">1.0997</span>
288
|-
289
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''B''<sub>r</sub></span>
290
|  style="border: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">Permanent magnet remanence</span>
291
|  colspan='2'  style="border-top: 1pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">1.21(M)</span>
292
|}
293
294
295
<span id='_Hlk78752609'></span>According to the operation principle of planetary gear with small tooth difference, the simplified structure of TRMG and equivalent model are shown in Figure 3.
296
[[File:Draft_Dongning_226371700-image60.png|centre|thumb|600x600px|<span style="text-align: center; font-size: 75%;">Figure 3. Simplified TRMG and equivalent model structure</span>]]
297
298
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div>
299
300
By adding a speed around the axis of low-speed permanent magnet ring (contrary to the revolution speed of ring plate permanent magnet ring) to the whole mechanism, it can be converted into a conversion structure in which both ring plate permanent magnet ring and low-speed permanent magnet ring rotate on a fixed axis around their own axis, as shown in Figure 3 (b). The transmission ratio of the conversion structure is:
301
302
{| class="formulaSCP" style="width: 100%; text-align: center;" 
303
|-
304
| 
305
{| style="text-align: center; margin:auto;" 
306
|-
307
| <math>i_1=\frac{p_\mbox{i}}{p_\mbox{o}}</math>
308
|}
309
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
310
|}
311
312
313
Where,  <math>i_1</math> is equivalent model transmission ratio.
314
315
<span id='_Hlk78752709'></span>According to the equivalent model transmission ratio, the input speed of ring plate permanent magnet ring is  <math display="inline">n_\mbox{i}\mbox{=}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 equivalent model is shown in Figure 4.
316
317
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Draft_Dongning_226371700-image65-c.png|centre|thumb|564x564px|<span style="text-align: center; font-size: 75%;">Figure 4. TRMG characteristic curve</span>]]</div>
318
319
From 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 (13).
320
321
=== 5.2 TRMG start-up characteristic analysis ===
322
Changing the starting angle between ring plate permanent magnet ring and low-speed permanent magnet ring  <math display="inline">\beta </math> . In 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> .
323
324
<span id='_Hlk78752810'></span>Figure 5 shows the speed and torque <span style="text-align: center; font-size: 75%;">characteristic</span> curve with starting point P.
325
[[File:Draft_Dongning_226371700-image74.png|centre|thumb|717x717px|<span style="text-align: center; font-size: 75%;">Figure 5. Speed and Torque characteristic curve Start with starting point P</span>]]
326
From 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.
327
328
<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.
329
[[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]]
330
331
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div>
332
333
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.
334
335
<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.
336
[[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]]
337
338
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div>
339
340
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. After '' <math display="inline">\alpha </math> '' increasing entry interval <math display="inline">\left[{\beta }_3,{\beta }_4\right]</math> , the low-speed permanent magnet ring continues to reverse accelerate rotation, eventually leading to TRMG start-up failure.
341
342
<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.
343
344
<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>
345
346
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.
347
348
== 6. Conclusions ==
349
(1) Based on structure mechanism of mechanical ring-plate drive mechanism, TRMG magnetic structure is proposed, and the operation mechanism and equivalent 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.
350
351
(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.
352
353
(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.
354
355
== Acknowledgement ==
356
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).
357
358
== References ==
359
[1] Chen X.F., LI J.M., Cheng H., Li B., He Z.J. Research and Application of Condition Monitoring and Fault Diagnosis Technology in Wind Turbines. Journal of Mechanical Engineering, 47(9):45-52,2011.
360
361
[2] Kang Y.F., Luan J.Y., Tian Y., Zheng H.Q. Cao J.H., Application of the Order Tracking Analysis in Gear Wearing. Journal of Vibration and Shock. 25(4):112-113+118+180,2006.
362
363
[3] Jing L.B., Zhang Y., Huang Z., Chen J.L. Development Course and Research Question of Magnetic Gear. Journal of Mechanical Transmission. 43(1):165-170,2019.
364
365
[4] Atallah K., Rens J., Mezani S. Howe D., A novel “pseudo” direct-drive brushless permanent magnet machine. IEEE Transactions on Magnetics. 44(11): 4349-4352, 2008.
366
367
[5] Rens J., Atallah K., Calverley S.D., Howe D. A novel magnetic harmonic gear. IEEE Transactions on Industry Applications, 46(1): 206-212,2010.
368
369
[6] Jorgensen F., Andersen T., Rasmussen P. The Cycloid Permanent Magnetic Gear. IEEE Transactions on Industry Applications. 6(44): 1659-1665,2008.
370
371
[7] Rens J., Clark R., Calverley S., Howe D. Design, analysis and realization of a novel magnetic harmonic gear. IEEE Transactions on Industry Applications. 46(1): 206-212,2009.
372
373
[8] Davey K., Hutson T., Mcdonald L., Hutson G. The design and construction of cycloidal magnetic gears. Inernational Conference on Electrical Machines, 2008.
374
375
[9] Huang H., Qu R., Bird J. Performance of Halbach Cycloidal Magnetic Gears with Respect to Torque Density and Gear Ratio. 2019 IEEE International Electric Machines & Drives Conference (IEMDC). 2019: 1977-1984.
376
377
[10] Hao W.N., Zhou J. The Cycloid Magnetic Gear Based on Halbach Array. Journal of Mechanical Strength. 39(1):226-229,2017.
378
379
[11] Wu S.Z., He W.D., Zhang Y.H. Analysis of Force and Contact Characteristics of Rotating Arm Bearings for RV Transmissions Mechanism. Journal of South China University of Technology (Natural Science Edition). 48(6):25-33,2020.
380
381
[12] Zhao D.L., Dang W.J., Sun W.P., Guo P.C. Influence of unbalanced magnetic pull on rub-impact rotor-bearing system. Journal of Huazhong University of Science and Technology(Natural Science Edition). 47(10):34-39,2019.
382
383
[13] Jian L., Chau K.T. Analytical calculation of magnetic field distribution in coaxial magnetic gears. Progress In Electromagnetics Research. 92: 1-16,2009.
384
385
[14] Lubin T., Mezani S., Rezzoug A. Analytical computation of the magnetic field distribution in a magnetic gear. IEEE Transactions on magnetics. 46(7): 2611-2621,2010.
386
387
[15] Deng Z., Nas I., Dapino M.J. Torque Analysis in Coaxial Magnetic Gears Considering Nonlinear Magnetic Properties and Spatial Harmonics. IEEE Transactions on Magnetics. PP(2): 1-11,2019.
388

Return to Ge Liu 2021a.

Back to Top