Ni et al 2024a - Revision history

Rimni at 09:56, 12 July 2024

2024-07-12T09:56:29Z

Rimni at 13:18, 9 July 2024

2024-07-09T13:18:38Z

Rimni at 13:05, 9 July 2024

2024-07-09T13:05:04Z

Rimni: /* 5、Conclusions */

2024-07-09T12:39:57Z

‎5、Conclusions

Rimni at 12:38, 9 July 2024

2024-07-09T12:38:20Z

Rimni: /* 4.2 Structural nonlinear aeroelastic analysis */

2024-07-09T12:34:36Z

‎4.2 Structural nonlinear aeroelastic analysis

Rimni at 12:31, 9 July 2024

2024-07-09T12:31:57Z

Rimni at 12:16, 9 July 2024

2024-07-09T12:16:01Z

Rimni at 12:15, 9 July 2024

2024-07-09T12:15:10Z

Rimni: /* 4.2 Structural Nonlinear Aeroelastic Analysis */

2024-07-09T12:14:20Z

‎4.2 Structural Nonlinear Aeroelastic Analysis

@@ Line 745: / Line 745: @@
 ==References==
 [1] Chen S.S., Jia M.L., Liu Y.X., et al. Development status on deformation modes and key technologies of aerodynamic layout design about morphing aircraft (in Chinese). Acta Aeronautica et Astronautica Sinica, 45(5), 629595, 2024.
@@ Line 774: / Line 775: @@
 [14] Yang N., Wu Zh.G., Yang Ch., Cao Q.K. Flutter analysis of a folding wing with structural nonlinearity. Engineeing Mechanics, 29(2):197-204, 2012.
-[15] Zhao Y.. H.. Aeroelasticity and Control [M]. Beijing: Science Press. 2007.
+[15] Zhao Y.H. Aeroelasticity and control. Science Press, Beijing, 2007.
-[16] Verstraete M. L., Preidikman S., Bruno A. Roccia, Dean T. Mook. A numerical nodel to study the nonlinear and unsteady aerodynamics of bioinspired morphing-wing comcepts. International Journal of Micro Air Vehicles. 2015,7(3):327-345.
+[16] Verstraete M.L., Preidikman S.,  Roccia B.A., Mook D.T. A numerical nodel to study the nonlinear and unsteady aerodynamics of bioinspired morphing-wing comcepts. International Journal of Micro Air Vehicles, 7(3):327-345, 2015.
-[17] Albano E., Rodden W. P.. A doublet-lattice method for calculating lift distributions on oscillating surfaces in subsonic flows . AIAA Journal. 1969,7(2): 279-285.
+[17] Albano E., Rodden W.P. A doublet-lattice method for calculating lift distributions on oscillating surfaces in subsonic flows. AIAA Journal, 7(2):279-285, 1969.
-[18] Landabl M. T..Kernel function for nonplanar oscillating surfaces in a subsonic flow . AIAA Journal. 1967, 5(5): 1045-1046.
+[18] Landabl M.T. Kernel function for nonplanar oscillating surfaces in a subsonic flow. AIAA Journal, 5(5):1045-1046,  1967.
-[19] Rodden W. P., Giesing J. P., Kalman T. P.. Refinement of the nonplanar aspects of the subsonic doublet-lattice lifting surface method. Journal of Aircraft. 1972,9(1):69-73.
+[19] Rodden W.P., Giesing J.P., Kalman T.P. Refinement of the nonplanar aspects of the subsonic doublet-lattice lifting surface method. Journal of Aircraft, 9(1):69-73, 1972.
-[20] Rodden W. P., Taylor P. F., Mcintosh S. C.. Further refinement of the subsonic doublet-dattice method. Journal of Aircraft. 1998,35(5):720-727.
+[20] Rodden W.P., Taylor P.F., Mcintosh S.C. Further refinement of the subsonic doublet-dattice method. Journal of Aircraft, 35(5):720-727, 1998.
-[21] Li P. CH., Ni Y.G., Hou Ch., Wan X. P., Zhao M. Y. Nonlinear aeroelastic aodeling of a folding wing structure . Journal of Physics: Conf. Series 1215 (2019) 012009
+[21] Li P.CH., Ni Y.G., Hou Ch., Wan X.P., Zhao M.Y. Nonlinear aeroelastic aodeling of a folding wing structure. Journal of Physics: Conf. Series, 1215, 012009, 2019
-[22] Jones R. T.. The unsteady lift of a wing of finite aspect ratio[R]. NACA Report-681, 1941.
+[22] Jones R.T. The unsteady lift of a wing of finite aspect ratio. NACA Report-681, 1941.
-[23] Roger K. L.. Airplane math modeling methods for active control design . AGARD-CP-228. 1977:4.1-4.11.
+[23] Roger K.L. Airplane math modeling methods for active control design. AGARD-CP, 228:4.1-4.11, 1977.
-[24] Karpel M.. Extensions to the minimum-state aeroelastic modeling method . AIAA Journal. 1991,29(11):2007-2009.
+[24] Karpel M. Extensions to the minimum-state aeroelastic modeling method. AIAA Journal, 29(11):2007-2009, 1991.
-[25] Vepa R.. Finite state modeling of aeroelastic systems[R]. NASA CR-2779, 1977.
+[25] Vepa R. Finite state modeling of aeroelastic systems. NASA CR-2779, 1977.
-[26] Bae J.S., Inman D.J., Lee I.. Effects of structural nonlinearity on subsonic aeroelastic characteristcs of an aircraft wing with control surface .Journal of Fluids and Structures. 2004(19):747-763.
+[26] Bae J.S., Inman D.J., Lee I. Effects of structural nonlinearity on subsonic aeroelastic characteristcs of an aircraft wing with control surface. Journal of Fluids and Structures, 19(6):747-763, 2004.
-[27] Sinha A., Griffin J. H.. Effects of friction dampers on aerodynamically unstable rotor stages .AIAA Journal. 1985,23(2):262-270.
+[27] Sinha A., Griffin J.H. Effects of friction dampers on aerodynamically unstable rotor stages. AIAA Journal, 23(2):262-270, 1985.
-[28] Liu B. H., Li M., Tan T. C.. Flutter study of a two-dimensional airfoil with structural nonlinearities . Engineering Mechanics. 2013,30(4):448-454.
+[28] Liu B.H., Li M., Tan T.C. Flutter study of a two-dimensional airfoil with structural nonlinearities. Engineering Mechanics, 30(4):448-454, 2013.
-[29] Qian Y. J.. Aerodynamics [M]. Beijing: Beijing University of Aeronautics and Astronautics Press.2004
+[29] Qian Y.J. Aerodynamics. Beijing University of Aeronautics and Astronautics Press, Beijing, 2004.

@@ Line 764: / Line 764: @@
 [9] Zhao Y.H., Hu H.Y. Parameterized aeroelastic modeling and flutter analysis for a folding wing. Journal of Sound Vibration, 331:208-324, 2012.
-[10] Tang D., Dowell E. H.. Theoretical and experimental aeroelastic study for folding wing structures. Journal of Aircraft. 2008,45(4):1136-1147.
+[10] Tang D., Dowell E.H. Theoretical and experimental aeroelastic study for folding wing structures. Journal of Aircraft, 45(4):1136-1147,  2008.
-[11] Hu Wei, Yang Zhichun, Gu Yingsong, Wang Xiaochen. The nonlinear aeroleastic characteristics of a folding wing with cubic stiffness .Journal of Sound and Vibration. 2017,400:22-39.
+[11] Hu W., Yang Z., Gu Y., Wang X. The nonlinear aeroleastic characteristics of a folding wing with cubic stiffness. Journal of Sound and Vibration, 400:22-39, 2017.
-[12] Ni Y.G., Zhang W., Lv Y.. Fast structural dynamic modeling and analysis for horizontally folding wing . Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería. 2021,37(35):35.
+[12] Ni Y.G., Zhang W., Lv Y. Fast structural dynamic modeling and analysis for horizontally folding wing. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 37(3), 35, 2021.
-[13] Ni Y.G., Zhang W., Lv Y.. Multi-body dynamics modeling and transient characteristics analysis for a folding wing . Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería. 2021,37(1):4.
+[13] Ni Y.G., Zhang W., Lv Y.. Multi-body dynamics modeling and transient characteristics analysis for a folding wing. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 37(1), 4, 2021.
-[14] Yang N, Wu Zh. G., Yang Ch., Cao Q.K.. Flutter Analysis of a folding wing with structural nonlinearity . Engineeing Mechanics. 2012, 29(2):197-204.
+[14] Yang N., Wu Zh.G., Yang Ch., Cao Q.K. Flutter analysis of a folding wing with structural nonlinearity. Engineeing Mechanics, 29(2):197-204, 2012.
 [15] Zhao Y.. H.. Aeroelasticity and Control [M]. Beijing: Science Press. 2007.

@@ Line 744: / Line 744: @@
 '''Funding:''' This research is supported by Natural Science Basic Research Plan in Shaanxi Province of China (2021JQ-847 and 2023-JC-YB-076).
-===References===
+==References==
-[1] Chen S S, Jia M L, Liu Y X,et al. Development status on deformation modes and key Technologies of Aerodynamic Layout Design about Morphing Aircraft .Acta Aeronautica et Astronautica Sinica [J]. 2024,45(5):629595. (in Chinese).
+[1] Chen S.S., Jia M.L., Liu Y.X., et al. Development status on deformation modes and key technologies of aerodynamic layout design about morphing aircraft (in Chinese). Acta Aeronautica et Astronautica Sinica, 45(5), 629595, 2024.
-[2] Ma H，Song B F，Zhou Z B，Zhang D S H．Analysis of the development status of multi-section morphing wing technology [J/OL]．Tactical Missile Technology. [https://doi.org/10.16358/j.issn.1009-1300.20230121 https://doi.org/10.16358/j.issn.1009-1300.20230121].[2024-02-08]. (in Chinese).
+[2] Ma H., Song B.F., Zhou Z.B., Zhang D.S.H．Analysis of the development status of multi-section morphing wing technology (in Chinese)．Tactical Missile Technology, 2024. [https://doi.org/10.16358/j.issn.1009-1300.20230121 https://doi.org/10.16358/j.issn.1009-1300.20230121, 2024-02-08].
-[3] Snyder M P，Sanders B，Eastep F E，et al. Vibration and flutter characteristics of a folding wing [J]. Journal of Aircraft. 2009，46(3)：791-799.
+[3] Snyder M.P., Sanders B., Eastep F.E., Frank G.J. Vibration and flutter characteristics of a folding wing. Journal of Aircraft, 46(3):791-799, 2009.
-[4]Wang I，Gibbs S C，Dowell E H. Aeroelastic model of multisegmented folding wings：Theory and experiment[J]. Journal of Aircraft. 2012，49(3)：911-921.
+[4] Wang I., Gibbs S.C., Dowell E.H. Aeroelastic model of multisegmented folding wings：Theory and experiment. Journal of Aircraft, 49(3)911-921, 2012.
-[5] Lee D H，Weisshaar T A. Aeroelastic studies on a folding wing configuration[C]. 46th AIAA/ ASME/ASCE/AHS/ASC Structures ， Structural Dynamics and Materials Conference，Austin，Texas，2005.
+[5] Lee D.H., Weisshaar T.A. Aeroelastic studies on a folding wing configuration. 46th AIAA/ ASME/ASCE/AHS/ASC Structures,  Structural Dynamics and Materials Conference, Austin, Texas, 2005.
-[6] Lee D H，Chen P C. Nonlinear aeroelastic studies on a folding wing configuration with free play hinge nonlinearity[C]. 47th AIAA/ASME/ASCE/AHS/ASCStructures，Structural Dynamics and Materials Conference，Newport，Rhode Island，2006.
+[6] Lee D.H., Chen P.C. Nonlinear aeroelastic studies on a folding wing configuration with free play hinge nonlinearity. 47th AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics and Materials Conference, Newport, Rhode Island, 2006.
-[7] Attar P，Tang D，Dowell E H.. Nonlinear aeroelastic study for folding wing structures[J]. AIAA Journal. 2010，48(10)：2187-2195.
+[7] Attar P., Tang D., Dowell E.H. Nonlinear aeroelastic study for folding wing structures. AIAA Journal, 48(10)2187-2195, 2010.
-[8] Reich G W，Bowman J C，Sanders B，et al. Development of an integrated aeroelastic multibody morphing simulation tool[C]. 47th AIAA/ASME/ASCE/AHS/ASC Structures ， Structural Dynamics and Materials Conference，Newport，Rhode Island，2006.
+[8] Reich G.W., Bowman J.C., Sanders B., et al. Development of an integrated aeroelastic multibody morphing simulation tool. 47th AIAA/ASME/ASCE/AHS/ASC Structures,  Structural Dynamics and Materials Conference, Newport, Rhode Island, 2006.
-[9] Zhao Y. H., Hu H.Y.. Parameterized aeroelastic modeling and flutter analysis for a folding wing[J]. Journal of Sound Vibration. 2012,(331):208-324.
+[9] Zhao Y.H., Hu H.Y. Parameterized aeroelastic modeling and flutter analysis for a folding wing. Journal of Sound Vibration, 331:208-324, 2012.
-[10] Tang D., Dowell E. H.. Theoretical and experimental aeroelastic study for folding wing structures[J]. Journal of Aircraft. 2008,45(4):1136-1147.
+[10] Tang D., Dowell E. H.. Theoretical and experimental aeroelastic study for folding wing structures. Journal of Aircraft. 2008,45(4):1136-1147.
-[11] Hu Wei, Yang Zhichun, Gu Yingsong, Wang Xiaochen. The nonlinear aeroleastic characteristics of a folding wing with cubic stiffness [J].Journal of Sound and Vibration. 2017,400:22-39.
+[11] Hu Wei, Yang Zhichun, Gu Yingsong, Wang Xiaochen. The nonlinear aeroleastic characteristics of a folding wing with cubic stiffness .Journal of Sound and Vibration. 2017,400:22-39.
-[12] Ni Y.G., Zhang W., Lv Y.. Fast structural dynamic modeling and analysis for horizontally folding wing [J]. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería. 2021,37(35):35.
+[12] Ni Y.G., Zhang W., Lv Y.. Fast structural dynamic modeling and analysis for horizontally folding wing . Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería. 2021,37(35):35.
-[13] Ni Y.G., Zhang W., Lv Y.. Multi-body dynamics modeling and transient characteristics analysis for a folding wing [J]. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería. 2021,37(1):4.
+[13] Ni Y.G., Zhang W., Lv Y.. Multi-body dynamics modeling and transient characteristics analysis for a folding wing . Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería. 2021,37(1):4.
-[14] Yang N, Wu Zh. G., Yang Ch., Cao Q.K.. Flutter Analysis of a folding wing with structural nonlinearity [J]. Engineeing Mechanics. 2012, 29(2):197-204.
+[14] Yang N, Wu Zh. G., Yang Ch., Cao Q.K.. Flutter Analysis of a folding wing with structural nonlinearity . Engineeing Mechanics. 2012, 29(2):197-204.
 [15] Zhao Y.. H.. Aeroelasticity and Control [M]. Beijing: Science Press. 2007.
-[16] Verstraete M. L., Preidikman S., Bruno A. Roccia, Dean T. Mook. A numerical nodel to study the nonlinear and unsteady aerodynamics of bioinspired morphing-wing comcepts[J]. International Journal of Micro Air Vehicles. 2015,7(3):327-345.
+[16] Verstraete M. L., Preidikman S., Bruno A. Roccia, Dean T. Mook. A numerical nodel to study the nonlinear and unsteady aerodynamics of bioinspired morphing-wing comcepts. International Journal of Micro Air Vehicles. 2015,7(3):327-345.
-[17] Albano E., Rodden W. P.. A doublet-lattice method for calculating lift distributions on oscillating surfaces in subsonic flows [J]. AIAA Journal. 1969,7(2): 279-285.
+[17] Albano E., Rodden W. P.. A doublet-lattice method for calculating lift distributions on oscillating surfaces in subsonic flows . AIAA Journal. 1969,7(2): 279-285.
-[18] Landabl M. T..Kernel function for nonplanar oscillating surfaces in a subsonic flow [J]. AIAA Journal. 1967, 5(5): 1045-1046.
+[18] Landabl M. T..Kernel function for nonplanar oscillating surfaces in a subsonic flow . AIAA Journal. 1967, 5(5): 1045-1046.
-[19] Rodden W. P., Giesing J. P., Kalman T. P.. Refinement of the nonplanar aspects of the subsonic doublet-lattice lifting surface method[J]. Journal of Aircraft. 1972,9(1):69-73.
+[19] Rodden W. P., Giesing J. P., Kalman T. P.. Refinement of the nonplanar aspects of the subsonic doublet-lattice lifting surface method. Journal of Aircraft. 1972,9(1):69-73.
-[20] Rodden W. P., Taylor P. F., Mcintosh S. C.. Further refinement of the subsonic doublet-dattice method[J]. Journal of Aircraft. 1998,35(5):720-727.
+[20] Rodden W. P., Taylor P. F., Mcintosh S. C.. Further refinement of the subsonic doublet-dattice method. Journal of Aircraft. 1998,35(5):720-727.
-[21] Li P. CH., Ni Y.G., Hou Ch., Wan X. P., Zhao M. Y. Nonlinear aeroelastic aodeling of a folding wing structure [J]. Journal of Physics: Conf. Series 1215 (2019) 012009
+[21] Li P. CH., Ni Y.G., Hou Ch., Wan X. P., Zhao M. Y. Nonlinear aeroelastic aodeling of a folding wing structure . Journal of Physics: Conf. Series 1215 (2019) 012009
 [22] Jones R. T.. The unsteady lift of a wing of finite aspect ratio[R]. NACA Report-681, 1941.
-[23] Roger K. L.. Airplane math modeling methods for active control design [J]. AGARD-CP-228. 1977:4.1-4.11.
+[23] Roger K. L.. Airplane math modeling methods for active control design . AGARD-CP-228. 1977:4.1-4.11.
-[24] Karpel M.. Extensions to the minimum-state aeroelastic modeling method [J]. AIAA Journal. 1991,29(11):2007-2009.
+[24] Karpel M.. Extensions to the minimum-state aeroelastic modeling method . AIAA Journal. 1991,29(11):2007-2009.
 [25] Vepa R.. Finite state modeling of aeroelastic systems[R]. NASA CR-2779, 1977.
-[26] Bae J.S., Inman D.J., Lee I.. Effects of structural nonlinearity on subsonic aeroelastic characteristcs of an aircraft wing with control surface [J].Journal of Fluids and Structures. 2004(19):747-763.
+[26] Bae J.S., Inman D.J., Lee I.. Effects of structural nonlinearity on subsonic aeroelastic characteristcs of an aircraft wing with control surface .Journal of Fluids and Structures. 2004(19):747-763.
-[27] Sinha A., Griffin J. H.. Effects of friction dampers on aerodynamically unstable rotor stages [J].AIAA Journal. 1985,23(2):262-270.
+[27] Sinha A., Griffin J. H.. Effects of friction dampers on aerodynamically unstable rotor stages .AIAA Journal. 1985,23(2):262-270.
-[28] Liu B. H., Li M., Tan T. C.. Flutter study of a two-dimensional airfoil with structural nonlinearities [J]. Engineering Mechanics. 2013,30(4):448-454.
+[28] Liu B. H., Li M., Tan T. C.. Flutter study of a two-dimensional airfoil with structural nonlinearities . Engineering Mechanics. 2013,30(4):448-454.
 [29] Qian Y. J.. Aerodynamics [M]. Beijing: Beijing University of Aeronautics and Astronautics Press.2004

@@ Line 731: / Line 731: @@
 From [[#img-11|Figure 11]], it can be seen that as the initial moment  <math>f_0</math> increases, the amplitude of the wingtip normal displacement decreases. As the friction stiffness <math>K_{\theta '}</math> increases, the amplitude of normal displacement at the leading edge of the wing tip does not change significantly. In [[#img-11|Figure 11]](a),  <math>K_{\theta '}=1Nm/rad</math>, as the initial moment  <math>f_0</math> increases,  <math>\delta {}'</math> increases. When  <math>\delta {}'</math> reaches a certain degree, the effect of the initial moment and the external moment caused by free-play nonlinearity is opposite, weakening the free-play nonlinearity. In [[#img-11|Figure 11]](b),  <math>f_0=0.05Nm</math>, as the friction stiffness  <math>K_{\theta '}</math> increases,  <math>\delta {}'</math> decreases. In a smaller displacement range,  <math>K_{\theta '}\theta </math> plays a major role, so the initial moment  <math>f_0</math> still plays a major role in a larger displacement range. This phenomenon indicates that compared to frictional stiffness, frictional moment plays an important role in reducing response amplitude. At the same time, when friction nonlinearity is added to the free-play nonlinearity at the outer hinge, the normal amplitude of the wing tip is always larger than when friction nonlinearity is added to the free-play nonlinearity at the inner hinge. In summary, to a certain extent, frictional nonlinearity can effectively reduce vibration amplitude, which is beneficial nonlinearity.
-===5、Conclusions===
+==5. Conclusions==
 The structural nonlinear aeroelastic analysis of folding wings is conducted by the modified Doublet Lattice Method. Firstly, a modified Doublet Lattice Method is studied to relax the aspect ratio limitation of aerodynamic elements, and its correctness is verified through numerical examples. Secondly, the rational function approximation method is discussed, and the modified Doublet Lattice Method is approximated using the minimum state method to obtain the unsteady aerodynamic forces in the time domain. Finally, the aeroelastic characteristics of a folding wing with free-play and friction nonlinearities are studied by combining time-domain unsteady aerodynamics and nonlinear structural dynamics models. The following conclusions can be drawn:
@@ Line 741: / Line 741: @@
 (3) Friction moment plays an important role in reducing response amplitude and is a beneficial nonlinearity.
-<big>'''Funding: '''</big>This research is supported by Natural Science Basic Research Plan in Shaanxi Province of China (2021JQ-847 and 2023-JC-YB-076).
 ===References===

@@ Line 698: / Line 698: @@
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|(b) <math>V=0.97V_f</math>, <math>\delta =0.2\mbox{°}</math>, <math>f_0=0.01Nm</math>
 |-
-|colspan="2" style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image193.png|312px]]
+|colspan="2" style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image193.png|332px]]
 |-
 |colspan="2" style="text-align: center;font-size: 75%;padding-bottom:10px;"|(c) <math>V=0.95V_f</math>, <math>\delta =1\mbox{°}</math>, <math>f_0=0.05Nm</math>
@@ Line 718: / Line 718: @@
 |+
 |-
-|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image199.png|282px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image199.png|302px]]
-|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image200.png|center|312px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image200.png|center|332px]]
 |-
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|(a) Influence of initial moment

@@ Line 692: / Line 692: @@
 |+
 |-
-|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image185.png|382px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image185.png|352px]]
-|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image186.png|center|382px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image186.png|center|352px]]
 |-
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"| (a) <math>V=1.0V_f</math>, <math>\delta =0.02\mbox{°}</math>, <math>f_0=0.007Nm</math>
@@ Line 718: / Line 718: @@
 |+
 |-
-|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image199.png|252px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image199.png|282px]]
-|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image200.png|center|282px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image200.png|center|312px]]
 |-
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|(a) Influence of initial moment
@@ Line 729: / Line 729: @@
-From [[#img-11|Figure 11]], it can be seen that as the initial moment  <math>f_0</math> increases, the amplitude of the wingtip normal displacement decreases. As the friction stiffness <math>K_{\theta '}</math> increases, the amplitude of normal displacement at the leading edge of the wing tip does not change significantly. For (a) in [[#img-11|Figure 11]],  <math>K_{\theta '}=1Nm/rad</math>, as the initial moment  <math>f_0</math> increases,  <math>\delta {}'</math> increases. When  <math>\delta {}'</math> reaches a certain degree, the effect of the initial moment and the external moment caused by free-play nonlinearity is opposite, weakening the free-play nonlinearity. For (b) in [[#img-11|Figure 11]],  <math>f_0=0.05Nm</math>, as the friction stiffness  <math>K_{\theta '}</math> increases,  <math>\delta {}'</math> decreases. In a smaller displacement range,  <math>K_{\theta '}\theta </math> plays a major role, so the initial moment  <math>f_0</math> still plays a major role in a larger displacement range. This phenomenon indicates that compared to frictional stiffness, frictional moment plays an important role in reducing response amplitude. At the same time, when friction nonlinearity is added to the free-play nonlinearity at the outer hinge, the normal amplitude of the wing tip is always larger than when friction nonlinearity is added to the free-play nonlinearity at the inner hinge. In summary, to a certain extent, frictional nonlinearity can effectively reduce vibration amplitude, which is beneficial nonlinearity.
+From [[#img-11|Figure 11]], it can be seen that as the initial moment  <math>f_0</math> increases, the amplitude of the wingtip normal displacement decreases. As the friction stiffness <math>K_{\theta '}</math> increases, the amplitude of normal displacement at the leading edge of the wing tip does not change significantly. In [[#img-11|Figure 11]](a),  <math>K_{\theta '}=1Nm/rad</math>, as the initial moment  <math>f_0</math> increases,  <math>\delta {}'</math> increases. When  <math>\delta {}'</math> reaches a certain degree, the effect of the initial moment and the external moment caused by free-play nonlinearity is opposite, weakening the free-play nonlinearity. In [[#img-11|Figure 11]](b),  <math>f_0=0.05Nm</math>, as the friction stiffness  <math>K_{\theta '}</math> increases,  <math>\delta {}'</math> decreases. In a smaller displacement range,  <math>K_{\theta '}\theta </math> plays a major role, so the initial moment  <math>f_0</math> still plays a major role in a larger displacement range. This phenomenon indicates that compared to frictional stiffness, frictional moment plays an important role in reducing response amplitude. At the same time, when friction nonlinearity is added to the free-play nonlinearity at the outer hinge, the normal amplitude of the wing tip is always larger than when friction nonlinearity is added to the free-play nonlinearity at the inner hinge. In summary, to a certain extent, frictional nonlinearity can effectively reduce vibration amplitude, which is beneficial nonlinearity.
 ===5、Conclusions===

@@ Line 685: / Line 685: @@
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+|-style="background:white;"
+|align="center" |
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+{|style="margin: 0em auto 0.1em auto;width:auto;"
 |-
-| [[Image:Review_423043765546-image185.png|282px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image185.png|382px]]
-| [[Image:Review_423043765546-image186.png|center|282px]]
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 |}
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+From [[#img-9|Figures 9]]  and [[#img-10|10]], it can be seen that when the friction stiffness is constant, an appropriate friction moment can suppress the divergence of displacement. Friction nonlinearity can effectively weaken the divergent motion of the normal displacement of the leading edge node of the wing tip, transforming it from an unstable motion tending towards divergence to a limit cycle motion with limited amplitude. It can be observed that the normal displacement response of the wingtip is centered around the zero position. This is because both free-play nonlinearity and friction nonlinearity are centrally symmetric, and after superposition, they remain centrally symmetric. This result also verifies the correctness of the model proposed in this paper.
-  [[Image:Review_423043765546-image193.png|312px]] </div>
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+In order to further clarify the influence of physical parameters of friction nonlinearity on the nonlinear aeroelasticity of free-play, the effects of moment  <math>f_0</math> and frictional stiffness  <math>K_{\theta '}</math> on the response of free-play nonlinear aeroelastic systems are analyzed. Taking the free-play  <math>\delta =1\mbox{°}</math> between the inner and outer hinges as an example, the displacement tends to diverge at a speed of  <math>V=0.96V_f</math> and <math>V=0.95V_f</math>, the effect of frictional nonlinearity on the aeroelasticity of the free-play is shown in [[#img-11|Figure 11]].
-(c)  <math>V=0.95V_f</math> , <math>\delta =1\mbox{°}</math> , <math>f_0=0.05Nm</math> </div>
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div id='img-11'></div>
-Fig.10 Friction at the outer hinge</div>
+{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
+|-style="background:white;"
-From Figs. 9 and 10, it can be seen that when the friction stiffness is constant, an appropriate friction moment can suppress the divergence of displacement. Friction nonlinearity can effectively weaken the divergent motion of the normal displacement of the leading edge node of the wing tip, transforming it from an unstable motion tending towards divergence to a limit cycle motion with limited amplitude. It can be observed that the normal displacement response of the wingtip is centered around the zero position. This is because both free-play nonlinearity and friction nonlinearity are centrally symmetric, and after superposition, they remain centrally symmetric. This result also verifies the correctness of the model proposed in this paper.
+|align="center" |
+{|style="margin: 0em auto 0.1em auto;width:auto;"
-In order to further clarify the influence of physical parameters of friction nonlinearity on the nonlinear aeroelasticity of free-play, the effects of moment  <math>f_0</math> and frictional stiffness  <math>K_{\theta '}</math> on the response of free-play nonlinear aeroelastic systems are analyzed. Taking the free-play  <math>\delta =1\mbox{°}</math> between the inner and outer hinges as an example, the displacement tends to diverge at a speed of  <math>V=0.96V_f</math> and <math>V=0.95V_f</math> , the effect of frictional nonlinearity on the aeroelasticity of the free-play is shown in Fig.11.
+|+
-−
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-−
-{|
 |-
-| [[Image:Review_423043765546-image199.png|252px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image199.png|252px]]
-| [[Image:Review_423043765546-image200.png|center|282px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image200.png|center|282px]]
 |}
-From Fig.11, it can be seen that as the initial moment  <math>f_0</math> increases, the amplitude of the wingtip normal displacement decreases. As the friction stiffness <math>K_{\theta '}</math> increases, the amplitude of normal displacement at the leading edge of the wing tip does not change significantly. For (a) in Fig.11,  <math>K_{\theta '}=1Nm/rad</math> , as the initial moment  <math>f_0</math> increases,  <math>\delta {}'</math> increases. When  <math>\delta {}'</math> reaches a certain degree, the effect of the initial moment and the external moment caused by free-play nonlinearity is opposite, weakening the free-play nonlinearity. For (b) in Fig.11,  <math>f_0=0.05Nm</math> , as the friction stiffness  <math>K_{\theta '}</math> increases,  <math>\delta {}'</math> decreases. In a smaller displacement range,  <math>K_{\theta '}\theta </math> plays a major role, so the initial moment  <math>f_0</math> still plays a major role in a larger displacement range. This phenomenon indicates that compared to frictional stiffness, frictional moment plays an important role in reducing response amplitude. At the same time, when friction nonlinearity is added to the free-play nonlinearity at the outer hinge, the normal amplitude of the wing tip is always larger than when friction nonlinearity is added to the free-play nonlinearity at the inner hinge. In summary, to a certain extent, frictional nonlinearity can effectively reduce vibration amplitude, which is beneficial nonlinearity.
+From [[#img-11|Figure 11]], it can be seen that as the initial moment  <math>f_0</math> increases, the amplitude of the wingtip normal displacement decreases. As the friction stiffness <math>K_{\theta '}</math> increases, the amplitude of normal displacement at the leading edge of the wing tip does not change significantly. For (a) in [[#img-11|Figure 11]],  <math>K_{\theta '}=1Nm/rad</math>, as the initial moment  <math>f_0</math> increases,  <math>\delta {}'</math> increases. When  <math>\delta {}'</math> reaches a certain degree, the effect of the initial moment and the external moment caused by free-play nonlinearity is opposite, weakening the free-play nonlinearity. For (b) in [[#img-11|Figure 11]],  <math>f_0=0.05Nm</math>, as the friction stiffness  <math>K_{\theta '}</math> increases,  <math>\delta {}'</math> decreases. In a smaller displacement range,  <math>K_{\theta '}\theta </math> plays a major role, so the initial moment  <math>f_0</math> still plays a major role in a larger displacement range. This phenomenon indicates that compared to frictional stiffness, frictional moment plays an important role in reducing response amplitude. At the same time, when friction nonlinearity is added to the free-play nonlinearity at the outer hinge, the normal amplitude of the wing tip is always larger than when friction nonlinearity is added to the free-play nonlinearity at the inner hinge. In summary, to a certain extent, frictional nonlinearity can effectively reduce vibration amplitude, which is beneficial nonlinearity.
 ===5、Conclusions===

@@ Line 670: / Line 670: @@
 |+
 |-
-|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image173.png|306px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image173.png|326px]]
-|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image174.png|center|282px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image174.png|center|302px]]
 |-
-|style="text-align: center;font-size: 75%;padding-bottom:10px;"|<math>V=1.01V_f</math>, <math>\delta =0.02\mbox{°}</math>, <math>f_0=0.005Nm</math>
+|style="text-align: center;font-size: 75%;padding-bottom:10px;"| (a) <math>V=1.01V_f</math>, <math>\delta =0.02\mbox{°}</math>, <math>f_0=0.005Nm</math>
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|(b) <math>V=0.99V_f</math>, <math>\delta =0.2\mbox{°}</math>, <math>f_0=0.02Nm</math>
 |-
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 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

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 Only the nonlinear aeroelastic response of the folding wing in the fully unfolded state has been analyzed here. The mass matrix and stiffness matrix of each substructure of the folding wing are obtained through DMAP language. The modified Doublet Lattice Method is used to obtain the unsteady aerodynamic forces of a folding wing in the frequency domain, and the expression of the aeroelastic equation in the state space is obtained through the minimum state approximation. For each substructure, six vibration modes are retained, and 10 degrees of freedom are coupled to each other through MPC and torsion springs at the interface, while 2 independent nonlinear degrees of freedom are retained. The final dynamic equation of the folding wing obtained is 22. After introducing 4 aerodynamic state variables, the aeroelastic equation in the state space is finally obtained, that is, Eq.(24) is 48, Finally, the time-domain aeroelastic response of the folding wing is obtained using the Runge-Kutta method in MATLAB.
-===4.1 Aerodynamic Model Validation===
+===4.1 Aerodynamic model validation===
 To verify the correctness of the modified Doublet Lattice Method, two examples are presented here.
@@ Line 658: / Line 658: @@
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 8'''. Minimum state approximation results for unsteady aerodynamics of folding wings
 |}
 ===4.2 Structural nonlinear aeroelastic analysis===

@@ Line 660: / Line 660: @@
-===4.2 Structural Nonlinear Aeroelastic Analysis===
+===4.2 Structural nonlinear aeroelastic analysis===
 According to Li et al. [21], it can be seen that when  <math>\delta =0</math> , the model is equal to the linear model. And the linear flutter velocity  <math>V_f=31.25m/s</math> . The  <math>\delta =0.02\mbox{°}</math>,  <math>\delta =0.2\mbox{°}</math>,  <math>\delta =1\mbox{°}</math> of the inner and outer hinges are taken as examples. An initial disturbance of 0.01mm is added to the normal direction of the trailing edge node of the wing tip to obtain the velocity at which the normal displacement response of the leading edge of the folding wing tip begins to diverge. Assuming  <math>K_{\theta '}=1Nm/rad</math>, the different moments are introduced separately, the displacement response obtained are shown in [[#img-9|Figures 9]]  and [[#img-10|10]].
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 |+
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-|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image173.png|276px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image173.png|306px]]
-|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image174.png|center|252px]]
+|style="text-align: center;padding:10px;"| [[Image:Review_423043765546-image174.png|center|282px]]
 |-
 |style="text-align: center;font-size: 75%;padding-bottom:10px;"|<math>V=1.01V_f</math>, <math>\delta =0.02\mbox{°}</math>, <math>f_0=0.005Nm</math>