Ge et al 2022a - Revision history

Rimni: /* 6. TRMG multibody dynamics analysis */

2022-09-09T10:12:28Z

‎6. TRMG multibody dynamics analysis

Rimni: /* References */

2022-09-01T10:43:06Z

‎References

Rimni at 10:39, 1 September 2022

2022-09-01T10:39:32Z

Rimni at 10:15, 1 September 2022

2022-09-01T10:15:00Z

Rimni: /* 5.1 Force analysis of TRMG with multi-shafts drive */

2022-09-01T10:14:24Z

‎5.1 Force analysis of TRMG with multi-shafts drive

Rimni at 10:13, 1 September 2022

2022-09-01T10:13:38Z

Rimni at 10:07, 1 September 2022

2022-09-01T10:07:28Z

Rimni: /* 4. TRMG dynamic model simulation */

2022-09-01T10:06:14Z

‎4. TRMG dynamic model simulation

Rimni at 09:59, 1 September 2022

2022-09-01T09:59:22Z

Rimni at 08:49, 1 September 2022

2022-09-01T08:49:00Z

@@ Line 581: / Line 581: @@
 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
-! Part - Part !! Constraint
+! style="text-align: left;"|Part - Part !! Constraint
 |-style="text-align:center"
-|  Input shaft A- Ground
+| style="text-align: left;"|Input shaft A- Ground
 |  Rotating
 |-style="text-align:center"
-|  Input shaft B- Ground
+|  style="text-align: left;"|Input shaft B- Ground
 |  Rotating
 |-style="text-align:center"
-|  Input shaft C- Ground
+|  style="text-align: left;"|Input shaft C- Ground
 |  Rotating
 |-style="text-align:center"
-|  Input shaft A- Ring-plate permanent magnet ring
+|  style="text-align: left;"|Input shaft A- Ring-plate permanent magnet ring
 |  Rotating
 |-style="text-align:center"
-|  style="text-align: center;"|Input shaft B- Ring-plate permanent magnet ring
+|  style="text-align: left;"|Input shaft B- Ring-plate permanent magnet ring
 |  style="text-align: center;"|Rotating
 |-
-|  style="text-align: center;"|Input shaft C- Ring-plate permanent magnet ring
+|  style="text-align: left;"|Input shaft C- Ring-plate permanent magnet ring
 |  style="text-align: center;"|Rotating
 |-
-|  style="text-align: center;"|Low-speed permanent magnet ring - Output shaft
+|  style="text-align: left;"|Low-speed permanent magnet ring - Output shaft
 |  style="text-align: center;"|Fixed
 |-
-|  style="text-align: center;"|Output shaft - Ground
+|  style="text-align: left;"|Output shaft - Ground
 |  style="text-align: center;"|Rotating
 |}

@@ Line 663: / Line 663: @@
 [11]  Lei Z.C.L.L.Z.,  Bin Y. Study on double-crank reducer with external gear board. Transactions of the Chinese Society for Agricultural Machinery, 39(8):149-152, 2008.
-[12] Ge  Y.,  Liu D.,   Wang D. Research on three-shaft ring-plate permanent magnetic gear variable speed transmission device. 2021 IEEE International Conference on Electrical Engineering and Mechatronics Technology (ICEEMT), 2021.
+[12] Ge  Y.,  Liu D.,   Wang D. Research on three-shaft ring-plate permanent magnetic gear variable speed transmission device. 2021 IEEE International Conference on Electrical Engineering and Mechatronics Technology (ICEEMT),  pp. 66-70, 2021.
 [13] Ge  Y.,  Liu D. Analysis of structure and starting characteristics of three-shaft ring-plate permanent magnet gear. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 37(4), 44, 2021.

@@ Line 647: / Line 647: @@
 [3] Yang B., Liu Y.,  Zhong Z. Experimental research on noise source identification of ring-plate pin-cycloid planetary reducer. ICLEM 2010: Logistics for Sustained Economic Development, 1596-1601, 2010.
-[4]. M. LIU, Y. SONG, C. ZHANG, and G. TIAN, Force Analysis of Rollers of Ring-plate-type Planetary Indexing Cam Mechanism, China Mechanical Engineering, 19(16),1912-1915,2008.
+[4]. Liu M., Song Y., Zhang C., Tian G. Force analysis of rollers of ring-plate-type planetary indexing cam mechanism. China Mechanical Engineering, 19(16):1912-1915, 2008.
-[5] M. A. Jafarizadeha, R. Hassannejadb, M. M. Ettefaghb, and S. Chitsazb, Asynchronous input gear damage diagnosis using time averaging and wavelet filtering,Mechanical Systems and Signal Processing, 22,172-201,2008.
+[5]  Jafarizadeha M.A.,  Hassannejadb R.,  Ettefaghb M.M.,  Chitsazb S. Asynchronous input gear damage diagnosis using time averaging and wavelet filtering. Mechanical Systems and Signal Processing, 22:172-201, 2008.
-[6] W. Q. Wang, F. Ismail, and M. F. Golnaraghi, Assessment of Gear Damage Monitoring Techniques Using Vibration Measurements, Mechanical Systems and Signal Processing, 15 (5), 905-922, 2001.
+[6]  Wang W.Q.,  Ismail F., Golnaraghi M.F. Assessment of gear damage monitoring techniques using vibration measurements. Mechanical Systems and Signal Processing, 15(5):905-922, 2001.
-[7] J. H. Shin, and S. M. Kwon, On the lobe profile design in a cycloid reducer using instant velocity center, Mechanism and Machine Theory, 41,596-616,2006.
+[7]  Shin J.H.,   Kwon S.M. On the lobe profile design in a cycloid reducer using instant velocity center. Mechanism and Machine Theory, 41:596-616, 2006.
-[8] W. Liu, T. Lin, and Y. Xie, Vibration Characteristic Analysis and Experimental Research of Double-ring Gear Reducer, China Mechanical Engineering, 20(10),1192-1196,2009.
+[8]  Liu W.,  Lin T.,  Xie Y. Vibration characteristic analysis and experimental research of double-ring gear reducer. China Mechanical Engineering, 20(10):1192-1196, 2009.
-[9] ZJSYZ. Ce, and M. Xianju, Statics Analysis of Three-ring Gear Reducer by Finite Element Method, Transactions of the Chinese Society for Agricultural Machinery, 38(3),141-143, 2007.
+[9] Ce Z.J.S.Y.Z., Xianju M.  Statics analysis of three-ring gear reducer by finite element method. Transactions of the Chinese Society for Agricultural Machinery, 38(3):141-143, 2007.
-[10] J. Zhang, Y. M. Song, and C. Zhang, Elasto-dynamic Analysis of Ring-plate Gear Reducer with Small Tooth Number Difference, Journal of Mechanical Engineering, 44 (2), 118-123, 2008.
+[10]  Zhang J.,  Song Y.M.,  Zhang C. Elasto-dynamic analysis of ring-plate gear reducer with small tooth number difference. Journal of Mechanical Engineering, 44(2):118-123, 2008.
-[11] ZCLLZ. Lei, and Y. Bin, Study on Double-crank Reducer with External Gear Board, Transactions of the Chinese Society for Agricultural Machinery, 39(8), 149-152, 2008.
+[11]  Lei Z.C.L.L.Z.,  Bin Y. Study on double-crank reducer with external gear board. Transactions of the Chinese Society for Agricultural Machinery, 39(8):149-152, 2008.
-[12] Y. Ge, D. Liu, and D. Wang, Research on Three-Shaft Ring-Plate Permanent Magnetic Gear Variable Speed Transmission Device, 2021 IEEE International Conference on Electrical Engineering and Mechatronics Technology (ICEEMT), 2021.
+[12] Ge  Y.,  Liu D.,   Wang D. Research on three-shaft ring-plate permanent magnetic gear variable speed transmission device. 2021 IEEE International Conference on Electrical Engineering and Mechatronics Technology (ICEEMT), 2021.
-[13] Y. Ge, and D. Liu, Analysis of structure and starting characteristics of three-shaft ring-plate permanent magnet gear, Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 34(4), 1-11, 2021.
+[13] Ge  Y.,  Liu D. Analysis of structure and starting characteristics of three-shaft ring-plate permanent magnet gear. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 37(4), 44, 2021.
-[14] O. Molokanov, P. Dergachev, S. Dergachev, E. Kuznetsova, and P. Kurbatov, A Novel Double-Rotor Planetary Magnetic Gear, IEEE transactions on magnetics, 54 (11),1-5, 2018.
+[14]  Molokanov O.,  Dergachev P.,  Dergachev S., Kuznetsova E., Kurbatov P. A novel double-rotor planetary magnetic gear. IEEE transactions on magnetics, 54(11):1-5, 2018.
-[15] Y. J. Ge, C. Y. Nie, and Q. Xin, A Three Dimensional Analytical Calculation of the Air-Gap Magnetic Field and Torque of Coaxial Magnetic Gears, Progress In Electromagnetics Researc, 131,391-407,2012.
+[15]  Ge Y.J.,  Nie C.Y.,  Xin Q. A three dimensional analytical calculation of the air-gap magnetic field and torque of coaxial magnetic gears. Progress In Electromagnetics Researc, 131:391-407, 2012.
-[16] K. Davey, L. McDonald, and T. Hutson, Axial flux cycloidal magnetic gears, IEEE transactions on magnetics, 50 (4), 2014.
+[16]  Davey K.,  McDonald L.,  Hutson T. Axial flux cycloidal magnetic gears. IEEE Transactions on Magnetics, 50(4):1-7, 2014.
-[17] Z. Deng, I. Nas, and M. J. Dapino, Torque Analysis in Coaxial Magnetic Gears Considering Nonlinear Magnetic Properties and Spatial Harmonics, IEEE Transactions on Magnetics, (2),1-11,2019.
+[17]  Deng Z.,  Nas I.,   Dapino M.J. Torque analysis in coaxial magnetic gears considering nonlinear magnetic properties and spatial harmonics. IEEE Transactions on Magnetics, 55(2):1-11, 2019.
 </div>

@@ Line 486: / Line 486: @@
 |-
 | <math>\begin{array}{c} F_{{Ar}}{=}\displaystyle\frac{r_1L}{6{\mu }_0}{\int }_0^{2\pi }\left[B_{r}^2\left(\beta ,t\right)- B_{t}^2\left(\beta ,t\right)\right]d\beta -\displaystyle\frac{1}{3}I_{g}+\displaystyle\frac{\sqrt{3}M}{108r}-\displaystyle\frac{M}{54r}\sin 2\omega t-\displaystyle\frac{M}{9r}\sin(\omega t-30\mbox{°})\sin\omega t-\\
-\displaystyle\frac{M}{18r}\sin(2\omega t-60\mbox{°})- \displaystyle\frac{\sqrt{3}r_2r_1L}{3l{\mu }_0}\sin(\varphi -30\mbox{°}){\int }_0^{2\pi }\left[B_{r}^{}\left(\beta ,t\right)B_{t}\left(\beta ,t\right)\right]d\beta \end{array}</math>
+-\displaystyle\frac{M}{18r}\sin(2\omega t-60\mbox{°})- \displaystyle\frac{\sqrt{3}r_2r_1L}{3l{\mu }_0}\sin(\varphi -30\mbox{°}){\int }_0^{2\pi }\left[B_{r}^{}\left(\beta ,t\right)B_{t}\left(\beta ,t\right)\right]d\beta \end{array}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (16)
@@ Line 509: / Line 509: @@
 | <math>\begin{array}{c}
 F_{{Cr}}{=}\displaystyle\frac{r_1L}{6{\mu }_0}{\int }_0^{2\pi }\left[B_{r}^2\left(\beta ,t\right)-B_{t}^2\left(\beta ,t\right)\right]d\beta -\displaystyle\frac{1}{3}I_{g}+\displaystyle\frac{\sqrt{3}M}{9r}{\cos}^2\omega t+\displaystyle\frac{M}{18r}\sin 2\omega t+\\
-\displaystyle\frac{\sqrt{3}r_2r_1L}{3l{\mu }_0} \cdot \cos\omega t{\int }_0^{2\pi }\left[B_{r}^{}\left(\beta ,t\right)B_{t}\left(\beta ,t\right)\right]d\beta
++\displaystyle\frac{\sqrt{3}r_2r_1L}{3l{\mu }_0} \cdot \cos\omega t{\int }_0^{2\pi }\left[B_{r}^{}\left(\beta ,t\right)B_{t}\left(\beta ,t\right)\right]d\beta
 \end{array}</math>
 |}

@@ Line 485: / Line 485: @@
 {| style="text-align: center; margin:auto;"
 |-
-| \begin{array}{c} <math>F_{{Ar}}{=}\displaystyle\frac{r_1L}{6{\mu }_0}{\int }_0^{2\pi }\left[B_{r}^2\left(\beta ,t\right)- B_{t}^2\left(\beta ,t\right)\right]d\beta -\displaystyle\frac{1}{3}I_{g}+\displaystyle\frac{\sqrt{3}M}{108r}-\displaystyle\frac{M}{54r}\sin 2\omega t-\displaystyle\frac{M}{9r}\sin(\omega t-30\mbox{°})\sin\omega t-\\
+| <math>\begin{array}{c} F_{{Ar}}{=}\displaystyle\frac{r_1L}{6{\mu }_0}{\int }_0^{2\pi }\left[B_{r}^2\left(\beta ,t\right)- B_{t}^2\left(\beta ,t\right)\right]d\beta -\displaystyle\frac{1}{3}I_{g}+\displaystyle\frac{\sqrt{3}M}{108r}-\displaystyle\frac{M}{54r}\sin 2\omega t-\displaystyle\frac{M}{9r}\sin(\omega t-30\mbox{°})\sin\omega t-\\
 \displaystyle\frac{M}{18r}\sin(2\omega t-60\mbox{°})- \displaystyle\frac{\sqrt{3}r_2r_1L}{3l{\mu }_0}\sin(\varphi -30\mbox{°}){\int }_0^{2\pi }\left[B_{r}^{}\left(\beta ,t\right)B_{t}\left(\beta ,t\right)\right]d\beta \end{array}</math>
 |}

← Older revision		Revision as of 10:07, 1 September 2022
Line 430:		Line 430:


−	According to TRMG mechanical analysis model, TRMG dynamic force is analyzed in MATLAB, and <math>F_{t}=1378</math> N and <math>F_{r}=441</math> N are replaced into the equation to obtain the change curves of force of input shaft and support shafts in [[#img-9\|Figure 9]].	+	According to TRMG mechanical analysis model, TRMG dynamic force is analyzed in MATLAB, and <math>F_{t}=1378</math>N and <math>F_{r}=441</math>N are replaced into the equation to obtain the change curves of force of input shaft and support shafts in [[#img-9\|Figure 9]].

	<div id='img-9'></div>		<div id='img-9'></div>

← Older revision		Revision as of 10:06, 1 September 2022
Line 409:		Line 409:


−	According to the transmission ratio and Maxwell stress tensor method, the input torque <math display="inline">M=5.64</math> N·m, and the tangential and radial electromagnetic force received by the low-speed permanent magnet ring is <math>F_{t}=1378</math> N and <math>F_{r}=441</math> N, respectively.	+	According to the transmission ratio and Maxwell stress tensor method, the input torque <math display="inline">M=5.64</math>N·m, and the tangential and radial electromagnetic force received by the low-speed permanent magnet ring is <math>F_{t}=1378</math>N and <math>F_{r}=441</math>N, respectively.

	===4.2 TRMG each shaft force analysis===		===4.2 TRMG each shaft force analysis===

@@ Line 294: / Line 294: @@
 |   <math display="inline">p_{i}</math>
 |  Pole pairs of inner permanent magnet
-|  22
+|  <math display="inline">22</math>
 |-style="text-align:center"
 |  <math display="inline">p_{o}</math>
 | Pole pairs of outer permanent magnet
-| 23
+| <math display="inline">23</math>
 |-style="text-align:center"
 | <math display="inline">r_1</math>
 | Inner radius of outer yoke iron
-| 96(mm)
+| <math display="inline">96</math>(mm)
 |-style="text-align:center"
 | <math display="inline">r_2</math>
 | Outer radius of outer permanent magnet ring
-| 100(mm)
+| <math display="inline">100</math>(mm)
 |-style="text-align:center"
 | <math display="inline">r_3</math>
 | Inner radius of outer permanent magnet ring
-| 94(mm)
+| <math display="inline">94</math>(mm)
 |-style="text-align:center"
 | <math display="inline">r_4</math>
 | Outer radius of inner permanent magnet ring
-| 90(mm)
+| <math display="inline">90</math>(mm)
 |-style="text-align:center"
 | <math display="inline">r_5</math>
 | Inner radius of inner permanent magnet ring
-| 84(mm)
+| <math display="inline">84</math>(mm)
 |-style="text-align:center"
 | <math display="inline">r_6</math>
 | Outer radius of inner yoke iron
-| 88(mm)
+| <math display="inline">88</math>(mm)
 |-style="text-align:center"
 | <math display="inline">h_{i}</math>
 | Inner yoke iron thickness
-| 15(mm)
+| <math display="inline">15</math>(mm)
 |-style="text-align:center"
 | <math display="inline">h_{o}</math>
 | Outer yoke iron thickness
-| 15(mm)
+| <math display="inline">15</math>(mm)
 |-style="text-align:center"
 | <math display="inline">r</math>
 | Eccentricity
-| 3(mm)
+| <math display="inline">3</math>(mm)
 |-style="text-align:center"
 | <math display="inline">L</math>
 | Axial length
-| 30(mm)
+| <math display="inline">30</math>(mm)
 |-style="text-align:center"
 | <math display="inline">M_{m}</math>
 | Magnetization
-| -890(kA/m)
+| -<math display="inline">890</math>(kA/m)
 |-style="text-align:center"
 | <math display="inline">{\mu }_0</math>

@@ Line 183: / Line 183: @@
-In Eq. (6), <math>F{'}_{Ar}</math>,  <math>F{'}_{Br}</math>, and  <math>F{'}_{Cr}</math> are respectively the radial force of support shafts A, C and input shaft B acting on the ring-plate permanent magnet ring through the rolling bearing under the ideal condition, and:
+In Eq. (6), <math>F'_{Ar}</math>,  <math>F'_{Br}</math>, and  <math>F'_{Cr}</math> are respectively the radial force of support shafts A, C and input shaft B acting on the ring-plate permanent magnet ring through the rolling bearing under the ideal condition, and:
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 190: / Line 190: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>{F{ { '}}}_{{Ar}}={F{{ '}}}_{{Br}}={F{{ '}}}_{{Cr}}=\frac{r_1L}{6{\mu }_0}{\int }_0^{2\pi }\left[B_{r}^2\left(\beta ,t\right)- B_{t}^2\left(\beta ,t\right)\right]d\beta -</math><math>\frac{1}{3}I_{g}</math>
+| <math>F'_{Ar}=F'_{Br}=F'_{Cr}=\frac{r_1L}{6{\mu }_0}{\int }_0^{2\pi }\left[B_{r}^2\left(\beta ,t\right)- B_{t}^2\left(\beta ,t\right)\right]d\beta -</math><math>\frac{1}{3}I_{g}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (7)
@@ Line 222: / Line 222: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>F_{{At}}=F_{{Ct}}{=}0</math>
+| <math>F_{At}=F_{Ct}=0</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (9)
@@ Line 262: / Line 262: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>F_{{Cr}}{=}\frac{r_1L}{6{\mu }_0}{\int }_0^{2\pi }\left[B_{r}^2\left(\beta ,t\right)- B_{t}^2\left(\beta ,t\right)\right]d\beta -</math><math>\frac{1}{3}I_{g}+\frac{1}{3}F_{{Bt}}\sin(\omega t-</math><math>60\mbox{°})cos\omega t-\frac{\sqrt{3}r_1r_2L}{3l{\mu }_0}\cos\omega t{\int }_0^{2\pi }\left[B_{r}^{}\left(\beta ,t\right)B_{t}^{}\left(\beta ,t\right)\right]d\beta </math>
+| <math>F_{Cr}=\frac{r_1L}{6{\mu }_0}{\int }_0^{2\pi }\left[B_{r}^2\left(\beta ,t\right)- B_{t}^2\left(\beta ,t\right)\right]d\beta -</math><math>\frac{1}{3}I_{g}+\frac{1}{3}F_{{Bt}}\sin(\omega t-</math><math>60\mbox{°})cos\omega t-\frac{\sqrt{3}r_1r_2L}{3l{\mu }_0}\cos\omega t{\int }_0^{2\pi }\left[B_{r}^{}\left(\beta ,t\right)B_{t}^{}\left(\beta ,t\right)\right]d\beta </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (13)