NI et al 2021a - Revision history

Rimni at 14:43, 14 December 2021

2021-12-14T14:43:16Z

Rimni: /* 1．Introduction */

2021-12-14T14:42:53Z

‎1．Introduction

Rimni at 10:49, 22 September 2021

2021-09-22T10:49:53Z

Rimni: /* 3.2 Dynamic characteristics of folding wing with rigid connection */

2021-09-22T10:46:30Z

‎3.2 Dynamic characteristics of folding wing with rigid connection

Rimni at 10:43, 22 September 2021

2021-09-22T10:43:31Z

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Rimni at 10:34, 22 September 2021

2021-09-22T10:34:02Z

Rimni: /* 1．Introduction */

2021-09-22T10:29:28Z

‎1．Introduction

Rimni at 10:21, 22 September 2021

2021-09-22T10:21:48Z

Rimni: /* 3.2 Dynamic characteristics of folding wing with rigid connection */

2021-09-22T10:05:00Z

‎3.2 Dynamic characteristics of folding wing with rigid connection

Rimni at 10:03, 22 September 2021

2021-09-22T10:03:22Z

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 A fast structural dynamic modeling approach is proposed to investigate the structural dynamic characteristics of a folding wing effectively. For the folding wing with the elastic connection, the interface compatible relationship of the traditional fixed interface component modal synthesis method is modified, and the internal force of the interface is completely expressed in the structural dynamic equation, so the influence of the connection stiffness on the wing structure dynamics can be considered. For the folding wing with the rigid connection, a mixed-loaded interface component mode synthesis of a folding wing is proposed based on fixed interface and loaded interface method, which makes it feasible to accurately reflect the influence of elasticity and inertia of fuselage and outer wing on inner wing. The dynamic characteristics of the folding wing with different folding angles, different connections and different connection positions are investigated. The final dynamic equation has a highly reduced size and more concise. To verify the accuracy of this method, the results are compared with those from MSC.NASTRAN.
-==2．Fast structural dynamics modeling of folding wing==
+==2. Fast structural dynamics modeling of folding wing==
 The folding wing is composed of three components, namely fuselage, inner wing and outer wing, denoted <math> A </math>, <math> B </math> and <math> C </math>, respectively. The geometric diagram is shown in [[#img-1|Figure 1]]. The coordinate system with subscript represents the local coordinate system of each component, while the coordinate system without subscript represents the global coordinate system. In order to verify the feasibility of the modeling method, the folding wing model is simplified as aluminum plate with equal thickness. All components are modeled by plates with the Yong’s modulus of  <math>7.1\times {10}^{10}Pa</math>, Poisson’s coefficient of 0.33 and a density of  <math>2.7\times {10}^3kg/m^3</math>, which are discretized using CQUAD4 elements with 2mm thickness in MSC.NASTRAN. Each node of this element has six degrees of freedom: three translations, namely,  <math>u_x,u_y,u_z</math> and three rotations, i.e.,  <math>{\theta }_x,{\theta }_y,{\theta }_z</math>. For each component, interface nodes are used at the connection positions of the folding axis to coordinate the boundary displacement. For the folding wing with rigid connection, MPC (Multi - Point Constraint) is used to couple the six degrees of freedom of interface nodes. For the folding wing with elastic connection, a set of torsion spring with a rotational degree of freedom  <math>{\theta }_x</math> are modeled and the other five degrees of freedom are defined via MPC. Among them, the torsional spring stiffness between components <math> A </math> and <math> B </math> is  <math>K_A</math>, and that between components <math> B </math> and <math> C </math> is  <math>K_B</math>. In this paper the structural dynamics model with two different connection types and different connection positions is established. The simulation method of two different connection types is shown in [[#img-2|Figure 2]].

@@ Line 18: / Line 18: @@
 '''Keywords''': Folding wing, component modal synthesis, dynamic analysis
-==1．Introduction==
+==1. Introduction==
 During the morphing process of morphing aircraft, the wing shape and geometric size change significantly leads to the redistribution of mass and stiffness of aircraft structure, resulting in different dynamic characteristics under different configurations [1]. The analysis of structural dynamic characteristics is the basis of flutter analysis and nonlinear aeroelastic analysis. In addition, the analysis of structural dynamic characteristics is also very important in the dynamic modeling of flexible multi-body system [2]. Therefore, it is particularly important to analyze the dynamic characteristics of the morphing aircraft, which can obtain the natural frequency and modal information of the wing structure and lay the foundation for the subsequent dynamic analysis [3].

@@ Line 32: / Line 32: @@
 ==2．Fast structural dynamics modeling of folding wing==
-The folding wing is composed of three components, namely fuselage, inner wing and outer wing, denoted A, B and C, respectively. The geometric diagram is shown in [[#img-1|Figure 1]]. The coordinate system with subscript represents the local coordinate system of each component, while the coordinate system without subscript represents the global coordinate system. In order to verify the feasibility of the modeling method, the folding wing model is simplified as aluminum plate with equal thickness. All components are modeled by plates with the Yong’s modulus of  <math>7.1\times {10}^{10}Pa</math>, Poisson’s coefficient of 0.33 and a density of  <math>2.7\times {10}^3kg/m^3</math>, which are discretized using CQUAD4 elements with 2mm thickness in MSC.NASTRAN. Each node of this element has six degrees of freedom: three translations, namely,  <math>u_x,u_y,u_z</math> and three rotations, i.e.,  <math>{\theta }_x,{\theta }_y,{\theta }_z</math>. For each component, interface nodes are used at the connection positions of the folding axis to coordinate the boundary displacement. For the folding wing with rigid connection, MPC (Multi - Point Constraint) is used to couple the six degrees of freedom of interface nodes. For the folding wing with elastic connection, a set of torsion spring with a rotational degree of freedom  <math>{\theta }_x</math> are modeled and the other five degrees of freedom are defined via MPC. Among them, the torsional spring stiffness between components <math> A </math> and <math> B </math> is  <math>K_A</math>, and that between components <math> B </math> and <math> C </math> is  <math>K_B</math>. In this paper the structural dynamics model with two different connection types and different connection positions is established. The simulation method of two different connection types is shown in [[#img-2|Figure 2]].
+The folding wing is composed of three components, namely fuselage, inner wing and outer wing, denoted <math> A </math>, <math> B </math> and <math> C </math>, respectively. The geometric diagram is shown in [[#img-1|Figure 1]]. The coordinate system with subscript represents the local coordinate system of each component, while the coordinate system without subscript represents the global coordinate system. In order to verify the feasibility of the modeling method, the folding wing model is simplified as aluminum plate with equal thickness. All components are modeled by plates with the Yong’s modulus of  <math>7.1\times {10}^{10}Pa</math>, Poisson’s coefficient of 0.33 and a density of  <math>2.7\times {10}^3kg/m^3</math>, which are discretized using CQUAD4 elements with 2mm thickness in MSC.NASTRAN. Each node of this element has six degrees of freedom: three translations, namely,  <math>u_x,u_y,u_z</math> and three rotations, i.e.,  <math>{\theta }_x,{\theta }_y,{\theta }_z</math>. For each component, interface nodes are used at the connection positions of the folding axis to coordinate the boundary displacement. For the folding wing with rigid connection, MPC (Multi - Point Constraint) is used to couple the six degrees of freedom of interface nodes. For the folding wing with elastic connection, a set of torsion spring with a rotational degree of freedom  <math>{\theta }_x</math> are modeled and the other five degrees of freedom are defined via MPC. Among them, the torsional spring stiffness between components <math> A </math> and <math> B </math> is  <math>K_A</math>, and that between components <math> B </math> and <math> C </math> is  <math>K_B</math>. In this paper the structural dynamics model with two different connection types and different connection positions is established. The simulation method of two different connection types is shown in [[#img-2|Figure 2]].
 <div id='img-1'></div>

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 |-
 |style="padding:10px;"|  [[Image:Draft_NI_373325365-image148.png|294px]]
 |-
 |  style="text-align: center;font-size: 75%;"|(a) Connection position 1
 |  style="text-align: center;font-size: 75%;"|(b) Connection position 2
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding:10px;"| '''Figure 7'''. Variation of frequency with stiffness for folding wing with rigid connection
+| colspan="2" style="padding:10px;"| '''Figure 7'''. Variation of frequency with stiffness for folding wing with rigid connection
 |}

@@ Line 1,764: / Line 1,764: @@
 [9] Wang I., Gibbs S.C., Dowell E.H. Aeroelastic model of multi segmented folding wings: theory and experiment．Journal of Aircraft, 49(3):911-921, 2012．
-[10] 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] Zhao Y.H., Hu H.Y. Parameterized aeroelastic modeling and flutter analysis for a folding wing. Journal of Sound Vibration, 331(2):308-324, 2012.
-[11] Hu W., Yang Z.C., Gu Y.S., et al．The nonlinear aeroelastic characteristics of a folding wing with cubic stiffness．Journal of Sound and Vibration ,400(21):22-39, 2017．
+[11] Hu W., Yang Z.C., Gu Y.S., Wang X．The nonlinear aeroelastic characteristics of a folding wing with cubic stiffness．Journal of Sound and Vibration, 400(21):22-39, 2017．
 [12] Lee D.H., Chen P.C. Nonlinear aeroelastic studies on a folding wing configuration with free-play hinge nonlinearity. AIAA-2006-1734, 2006．

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 During the morphing process of morphing aircraft, the wing shape and geometric size change significantly leads to the redistribution of mass and stiffness of aircraft structure, resulting in different dynamic characteristics under different configurations [1]. The analysis of structural dynamic characteristics is the basis of flutter analysis and nonlinear aeroelastic analysis. In addition, the analysis of structural dynamic characteristics is also very important in the dynamic modeling of flexible multi-body system [2]. Therefore, it is particularly important to analyze the dynamic characteristics of the morphing aircraft, which can obtain the natural frequency and modal information of the wing structure and lay the foundation for the subsequent dynamic analysis [3].
-Lee et al. [4] used Matlab to generate a high fidelity folding wing model proposed by Lockheed Martin, which is composed of rod, plate and other elements. The dynamic characteristics of the folding wing are analyzed in MSC.NASTRAN. However the effect of the fuselage on the structure is not taken into consideration in Ref.[4]. Snyder et al. [5] used MSC.PATRAN to establish the simplified structure of folding wing with different folding angles. The sensitivity of natural frequency to hinge stiffness and folding angle is analyzed in MSC.NASTRAN. The above research did not verify the correctness of the structural model. Tang et al. [6] established the theoretical model of folding wing structure using linear plate theory. Firstly, the modal analysis of each substructure is carried out by ANSYS, and then the Lagrange equation is used to synthesize to establish the dynamic model of the whole structure, and the correctness of the model is verified. At the same time, the variation rule of natural frequency with folding angle and hinge stiffness is analyzed. Andersen et al. [7] analyzed the natural frequency of the folding wing in MSC.NASTRAN and verified by the ground vibration test. Attar et al. [8] further considered the geometric nonlinearity of the structure and established the dynamic equation of the structure by using Kirchhoff plate theory and virtual work principle. Wang et al. [9] proposed a more general structural dynamic model for predicting the natural frequencies of folding wings, which is based on the beam theory. The natural frequencies of three test models in the whole folding angle range are measured. Zhao et al. [10] established the parametric structural dynamic model of the plate folding wing by the component synthesis method, and compared the results with those by MSC.NASTRAN. The correctness of the structural model is verified, and the influence factors of natural frequency are analyzed. In consideration of the nonlinear factors at the hinge of the folding wing, Hu et al. [11] established the dynamic model of the two wing segments by Lagrange equation, in which the cubic stiffness is considered. Lee et al. [12] used fictitious mass method to establish the structure with hinge free-play for the folding wing. The dynamic response characteristics are studied. Yang et al. [13] proposed a method for modeling multiple degrees of freedom in chord direction with structural nonlinearity by the fixed interface component modal synthesis method. The method introduces the angle difference between the inner and outer wing at the interface. The model synthesis of the two wing segments are achieved. Guo et al. [14-15] assumed that the wing was composed of three carbon fiber composite plates, and established the nonlinear dynamic equation by Hamilton principle and von Karman large deformation theory. The vibration modal function is solved according to the modal analysis, the results of harmonic response by ANSYS and the boundary conditions. The simulation results are compared with those by ANSYS, and the nonlinear dynamic characteristics of 1:2 resonance are further analyzed.
+Lee et al. [4] used Matlab to generate a high fidelity folding wing model proposed by Lockheed Martin, which is composed of rod, plate and other elements. The dynamic characteristics of the folding wing are analyzed in MSC.NASTRAN. However the effect of the fuselage on the structure is not taken into consideration in [4]. Snyder et al. [5] used MSC.PATRAN to establish the simplified structure of folding wing with different folding angles. The sensitivity of natural frequency to hinge stiffness and folding angle is analyzed in MSC.NASTRAN. The above research did not verify the correctness of the structural model. Tang et al. [6] established the theoretical model of folding wing structure using linear plate theory. Firstly, the modal analysis of each substructure is carried out by ANSYS, and then the Lagrange equation is used to synthesize to establish the dynamic model of the whole structure, and the correctness of the model is verified. At the same time, the variation rule of natural frequency with folding angle and hinge stiffness is analyzed. Andersen et al. [7] analyzed the natural frequency of the folding wing in MSC.NASTRAN and verified by the ground vibration test. Attar et al. [8] further considered the geometric nonlinearity of the structure and established the dynamic equation of the structure by using Kirchhoff plate theory and virtual work principle. Wang et al. [9] proposed a more general structural dynamic model for predicting the natural frequencies of folding wings, which is based on the beam theory. The natural frequencies of three test models in the whole folding angle range are measured. Zhao et al. [10] established the parametric structural dynamic model of the plate folding wing by the component synthesis method, and compared the results with those by MSC.NASTRAN. The correctness of the structural model is verified, and the influence factors of natural frequency are analyzed. In consideration of the nonlinear factors at the hinge of the folding wing, Hu et al. [11] established the dynamic model of the two wing segments by Lagrange equation, in which the cubic stiffness is considered. Lee et al. [12] used fictitious mass method to establish the structure with hinge free-play for the folding wing. The dynamic response characteristics are studied. Yang et al. [13] proposed a method for modeling multiple degrees of freedom in chord direction with structural nonlinearity by the fixed interface component modal synthesis method. The method introduces the angle difference between the inner and outer wing at the interface. The model synthesis of the two wing segments are achieved. Guo et al. [14-15] assumed that the wing was composed of three carbon fiber composite plates, and established the nonlinear dynamic equation by Hamilton principle and von Karman large deformation theory. The vibration modal function is solved according to the modal analysis, the results of harmonic response by ANSYS and the boundary conditions. The simulation results are compared with those by ANSYS, and the nonlinear dynamic characteristics of 1:2 resonance are further analyzed.
 At present, the structural simulation method of folding wing is mainly based on finite element method, and the analytical method is used in a small number of literatures. This is because the analytical method is only suitable for simple engineering structures, while the finite element method has obvious advantages in dealing with various complex engineering problems. In most literatures, the continuous change process of folding wing of variant aircraft from unfolded state to folded state is usually decomposed into different configurations of wing, and then the repeated modeling of wing structure under different configurations is not conducive to the rapid analysis of structural dynamic characteristics of folding wing.

@@ Line 47: / Line 47: @@
-<div id='img-1'></div>
+<div id='img-2'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 1,277: / Line 1,277: @@
 ===3.2 Dynamic characteristics of folding wing with rigid connection===
-When the folding wing is rigidly connected, that is to say, the six degrees of freedom of interface nodes are defined by MPC, the mixed-loaded interface component modal synthesis method can be directly used for rapid modeling and dynamic characteristics analysis. For the two connection points in [[#img-3|Figure 3]] , the results in this paper are compared with those by MSC.NASTRAN, as shown in [[#tab-3|Tables 3]] and [[#tab-4|4]]. According to the number of main modes reserved by different components, different mode combinations are formed. For example, (4,4,4) represents that components <math> A </math>, <math>B </math> and <math> C </math> retain the first four order main modes, and "-" represents that the relative error is greater than 15%.
+When the folding wing is rigidly connected, that is to say, the six degrees of freedom of interface nodes are defined by MPC, the mixed-loaded interface component modal synthesis method can be directly used for rapid modeling and dynamic characteristics analysis. For the two connection points in [[#img-3|Figure 3]], the results in this paper are compared with those by MSC.NASTRAN, as shown in [[#tab-3|Tables 3]] and [[#tab-4|4]]. According to the number of main modes reserved by different components, different mode combinations are formed. For example, (4,4,4) represents that components <math> A </math>, <math>B </math> and <math> C </math> retain the first four order main modes, and "-" represents that the relative error is greater than 15%.
 <div class="center" style="font-size: 75%;">'''Table 3'''. Natural frequency of folding wing with rigid connection at connection positions 1 (<math>\theta =0^{\circ }</math>)</div>
@@ Line 1,686: / Line 1,686: @@
-It can be seen from [[#img-7|Figure 7]] that the effect of folding angle on the frequency of folding wing is obvious, and the frequency of two different connection positions is quite different. The first frequency is not very sensitive to the folding angle when the rigid connection is used at the connection position 1. The second frequency increases with the increase of folding angle. When the folding angle is less than <math>{90}^{\circ }</math> , the third, fourth and fifth order frequencies tend to decrease. The sixth order frequency first increases and then decreases. More importantly, the second and third frequencies, as well as the fourth and fifth frequencies, tend to couple with each other with the increase of folding angle, but there is no mode transition phenomenon. When the rigid connection is adopted at the connection position 2, the change of the first frequency is not obvious with the increase of the folding angle. The second, third and sixth order frequencies first decrease and then increase. The fourth and fifth order frequencies are basically reduced. When the rigid connection is used at connection position 2, the fifth and sixth frequencies are coupled with each other. Compared with the frequency of elastic connection, the frequency of rigid connection is higher. Compared with Figure5, it can be seen that the frequency of rigid connection is greater than that of elastic connection. This is mainly due to the large stiffness of the structure at the interface when it is rigidly connected, which makes the folding wing structure more rigid. When the mass is constant, the eigenvalue of the structure increases, so the frequency is larger.
+It can be seen from [[#img-7|Figure 7]] that the effect of folding angle on the frequency of folding wing is obvious, and the frequency of two different connection positions is quite different. The first frequency is not very sensitive to the folding angle when the rigid connection is used at the connection position 1. The second frequency increases with the increase of folding angle. When the folding angle is less than <math>{90}^{\circ }</math> , the third, fourth and fifth order frequencies tend to decrease. The sixth order frequency first increases and then decreases. More importantly, the second and third frequencies, as well as the fourth and fifth frequencies, tend to couple with each other with the increase of folding angle, but there is no mode transition phenomenon. When the rigid connection is adopted at the connection position 2, the change of the first frequency is not obvious with the increase of the folding angle. The second, third and sixth order frequencies first decrease and then increase. The fourth and fifth order frequencies are basically reduced. When the rigid connection is used at connection position 2, the fifth and sixth frequencies are coupled with each other. Compared with the frequency of elastic connection, the frequency of rigid connection is higher. Compared with [[#img-5|Figure 5]], it can be seen that the frequency of rigid connection is greater than that of elastic connection. This is mainly due to the large stiffness of the structure at the interface when it is rigidly connected, which makes the folding wing structure more rigid. When the mass is constant, the eigenvalue of the structure increases, so the frequency is larger.
 It can be seen from the comparisons that the frequency obtained by the mixed-loaded interface component modal synthesis method are consistent with those obtained by MSC.NASTRAN. The actual modal shape of the folding wing can be obtained by the eigenvectors in the eigenvalue analysis of the final folding wing dynamic equation. Taking the folding wing fully unfolded with rigid connection as an example, the main mode combination is still (6,8,10), and the actual mode comparison is shown in [[#img-8|Figure 8]].
@@ Line 1,744: / Line 1,744: @@
 ==References==
-[1] Wilson J R. Morphing UAVs Change the Shape of Warfare.Aerospace America ,2004(42):28~29.
+[1] Wilson J.R. Morphing UAVs change the shape of warfare. Aerospace America, 42(2):28-29, 2004.
-[2] Sofla A Y N , Meguid S A, Tan K T, Yeo W K.Shape morphing of aircraft wing:Status and challenges.Materials and Design. 2010:1284~1292.
+[2] Sofla A.Y.N., Meguid S.A., Tan K.T., Yeo W.K. Shape morphing of aircraft wing: Status and challenges. Materials and Design, 31(3):1284-1292, 2010.
-[3] Ni Y. G., Zhang W., Lv Y., Zhao H.. Application and modeling analysis of wing folding technology[J]. Flight Dynamics. 2020,38(6): 1-7.
+[3] Ni Y.G., Zhang W., Lv Y., Zhao H. Application and modeling analysis of wing folding technology. Flight Dynamics, 38(6):1-7, 2020.
-[4] 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. April,2005, AIAA 2005-1996.
+[4] 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, AIAA 2005-1996, April 2005.
-[5] Snyder M. P., Sanders B.. Vibration and Flutter Characteristics of a Folding Wing[J]. Journal of Aircraft. 2009,46(3):791-799
+[5] Snyder M.P., Sanders B. Vibration and flutter characteristics of a folding wing. Journal of Aircraft, 6(3):791-799, 2009.
-[6] Tang D., Dowell E. H.. Theoretical and Experimental Aeroelastic Study for Folding Wing Structures[J]. Journal of Aircraft.2008,45(4):1136-1147.
+[6] Tang D., Dowell E.H. Theoretical and experimental aeroelastic study for folding wing structures. Journal of Aircraft, 45(4):1136-1147, 2008.
-[7] Andersen G. R., Cowan D. L.. Aeroelastic Modeling, Analysis and Testing of a Morphing Wing Structure[C]. 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conference. Honolulu, Haiwaii. April, 2007, AIAA 2007-1734.
+[7] Andersen G.R., Cowan D.L. Aeroelastic modeling, analysis and testing of a morphing wing structure. 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conference, Honolulu, Haiwaii, AIAA 2007-1734, April 2007.
-[8] Attar P. J., Tang D., Dowell E. H.. Nonlinear aeroelastic study for folding wing structures[J]．AIAA Journal. 2010, 48(10): 2187-2195．
+[8] Attar P.J., Tang D., Dowell E.H. Nonlinear aeroelastic study for folding wing structures．AIAA Journal, 48(10):2187-2195, 2010．
-[9] Wang I.，Gibbs S. C.，Dowell E. H.． Aeroelastic model of multi segmented folding wings: theory and experiment[J]．Journal of Aircraft. 2012,49( 3): 911-921．
+[9] Wang I., Gibbs S.C., Dowell E.H. Aeroelastic model of multi segmented folding wings: theory and experiment．Journal of Aircraft, 49(3):911-921, 2012．
-[10] 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.
+[10] Zhao Y.H., Hu H.Y. Parameterized aeroelastic modeling and flutter analysis for a folding wing. Journal of Sound Vibration, (331):208-324, 2012.
-[11] Hu W., Yang Z. C.，Gu Y. S., et al．The nonlinear aeroelastic characteristics of a folding wing with cubic stiffness[J]．Journal of Sound and Vibration. 2017,400( 21) : 22-39．
+[11] Hu W., Yang Z.C., Gu Y.S., et al．The nonlinear aeroelastic characteristics of a folding wing with cubic stiffness．Journal of Sound and Vibration ,400(21):22-39, 2017．
-[12] Lee D. H.，Chen P. C.. Nonlinear aeroelastic studies on a folding wing configuration with free-play hinge nonlinearity [R]. AIAA-2006-1734, 2006．
+[12] Lee D.H., Chen P.C. Nonlinear aeroelastic studies on a folding wing configuration with free-play hinge nonlinearity. AIAA-2006-1734, 2006．
-[13] Yang N., Wu Z. G.,Yang C., Cao Q. K.. Flutter analysis of a folding wing with structural nonlinearity[J]. Engineering Mechanics. 2012,29(2):197-196.
+[13] Yang N., Wu Z.G., Yang C., Cao Q.K. Flutter analysis of a folding wing with structural nonlinearity. Engineering Mechanics, 29(2):197-196, 2012.
-[14] Piao J. L., Guo X. Y., Zhang W., Zu W. Z.. Nonlinear dynamics and mode analysis for a Z-type folding wings [J]. Journal of dynamics and control. 2017,15(1):29-38.
+[14] Piao J.L., Guo X.Y., Zhang W., Zu W.Z. Nonlinear dynamics and mode analysis for a Z-type folding wings. Journal of Dynamics and Control, 15(1):29-38, 2017.
-[15] Guo X. Y., Zhang Y., Zhang W.. Nonlinear vibration characteristics of Z-type folding plates with internal resonance[J]. Journal of Vibration Engineering. 2018,31(2):183-197.
+[15] Guo X.Y., Zhang Y., Zhang W. Nonlinear vibration characteristics of Z-type folding plates with internal resonance. Journal of Vibration Engineering, 31(2):183-197, 2018.
-[16] Roy R. Craig. Coupling of substructures for dynamic analysis: an overview. 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Atlanta. (2000)2000~1573.
+[16] Craig Jr. R.R. Coupling of substructures for dynamic analysis: an overview. 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Atlanta, 2000-1573, April 2000.
-[17] Benfield W. A., Hruda R. F.. Vibration Analysis of Structures by Component Substitution[J].AIAA Journal. 1971, 9(7):1255-1261.
+[17] Benfield W.A., Hruda R.F. Vibration analysis of structures by component substitution. AIAA Journal, 9(7):1255-1261, 1971.

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−	~~<span style="text-align: center; font-size: 75%;">(e) Comparison of fifth mode shape</span></div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−
−	{\|
−	\|-
−	~~\| [[Image:Draft_NI_373325365-image161.png\|192px]]~~
−	~~\| [[Image:Draft_NI_373325365-image162.png\|center\|186px]]~~
−	\|}
−	~~</div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<span style="text-align: center; font-size: 75%;">(f) Comparison of sixth mode shape</span></div>~~
−
−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<span style="text-align: center; font-size: 75%;">'''Figure 8. '''Modal shape comparison (left: fixed interface component modal synthesis method; right: MSC.NASTRAN )</span></div>~~

	It can be seen from [[#img-8\|Figure 8]] that the modal shapes by the mixed-loaded interface component modal synthesis method are consistent with those by MSC.NASTRAN. And the modal information can be obtained accurately, which is mainly due to the correct selection of the master-slave component and the correct reflection of the connection type between components. Therefore, the accurate displacements of related nodes can be obtained for further dynamic analysis.		It can be seen from [[#img-8\|Figure 8]] that the modal shapes by the mixed-loaded interface component modal synthesis method are consistent with those by MSC.NASTRAN. And the modal information can be obtained accurately, which is mainly due to the correct selection of the master-slave component and the correct reflection of the connection type between components. Therefore, the accurate displacements of related nodes can be obtained for further dynamic analysis.

@@ Line 1,236: / Line 1,236: @@
 It can be seen from [[#img-6|Figure 6]]  that with the increase of stiffness, the frequency basically increases. Even if the stiffness changes in a relatively large range, the fixed interface component modal synthesis can be used to well simulate the dynamic characteristics of the structure. When the connection position 1 is elastic connection, the mode shapes by the main mode combination of (6,8,10) are compared those by MSC.NASTRAN, as shown in [[#img-6|Figure 6]].
-<div id='img-1'></div>
+<div id='img-6'></div>
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;"
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@@ Line 1,474: / Line 1,474: @@
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="font-size: 75%;">'''Table 4''' Natural frequency of folding wing with rigid connection at connection positions 2 (<math>\theta =0^{\circ }</math>)</div>
-<span style="text-align: center; font-size: 75%;">'''Table 4''' Natural frequency of folding wing with rigid connection at connection positions 2 ( <math>\theta =0^{\circ }</math> )</span></div>
 <div id='tab-4'></div>
@@ Line 1,670: / Line 1,669: @@
-It can be seen from Table 3 and Table 4 that with the increase of the number of retained main modes, the accuracy of frequency is improved, especially for higher-order frequencies. For different modal combinations, the natural frequencies of the first six orders are consistent with those by MSC.NASTRAN and the relative error is within the acceptable range. For example, for the main mode combination of (4,4,4) and (6,6,6), lower order frequencies can be well obtained. This is mainly because the influence of inertia and elasticity of adjacent components is considered in the mixed-loaded interface component modal synthesis method, which improves the accuracy. With the increase of the number of main modes reserved, the accuracy of higher-order frequencies is improved. It can be seen that when the number of reserved modes of component is too small, some information of overall modes will be lost. For example, when the main mode combination is (4,4,4), the error of the first six frequencies of the two connection positions are within the acceptable range, but the error of the seventh to tenth frequencies is more than 15%. It is necessary to retain more information of the main mode to reduce the error caused by high-order mode truncation. Therefore, in order to improve the accuracy of dynamic analysis, it is very important to retain the appropriate modal information. Compared with Table 1 and Table 4, it can be found that the frequency of rigid connection is higher than that of elastic connection at the same connection position, which is mainly because the rigid connection increases the overall stiffness of the wing structure. For different folding angles, the main mode combination of (6,8,10) is used to calculate the natural frequency of the folding wing. Figure 7 shows the change of the frequency with the folding angle at different connection positions.
+It can be seen from [[#tab-3|Tables 3]] and [[#tab-4|4]] that with the increase of the number of retained main modes, the accuracy of frequency is improved, especially for higher-order frequencies. For different modal combinations, the natural frequencies of the first six orders are consistent with those by MSC.NASTRAN and the relative error is within the acceptable range. For example, for the main mode combination of (4,4,4) and (6,6,6), lower order frequencies can be well obtained. This is mainly because the influence of inertia and elasticity of adjacent components is considered in the mixed-loaded interface component modal synthesis method, which improves the accuracy. With the increase of the number of main modes reserved, the accuracy of higher-order frequencies is improved. It can be seen that when the number of reserved modes of component is too small, some information of overall modes will be lost. For example, when the main mode combination is (4,4,4), the error of the first six frequencies of the two connection positions are within the acceptable range, but the error of the seventh to tenth frequencies is more than 15%. It is necessary to retain more information of the main mode to reduce the error caused by high-order mode truncation. Therefore, in order to improve the accuracy of dynamic analysis, it is very important to retain the appropriate modal information. Compared with [[#tab-1|Tables 1]] and [[#tab-4|4]], it can be found that the frequency of rigid connection is higher than that of elastic connection at the same connection position, which is mainly because the rigid connection increases the overall stiffness of the wing structure. For different folding angles, the main mode combination of (6,8,10) is used to calculate the natural frequency of the folding wing. [[#img-7|Figure 7]] shows the change of the frequency with the folding angle at different connection positions.
+<div id='img-7'></div>
-{|
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
-| [[Image:Draft_NI_373325365-image148.png|294px]]
+|style="padding:10px;"|  [[Image:Draft_NI_373325365-image148.png|294px]]
-| [[Image:Draft_NI_373325365-image149.png|center|282px]]
+|-
 |}
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+It can be seen from [[#img-7|Figure 7]] that the effect of folding angle on the frequency of folding wing is obvious, and the frequency of two different connection positions is quite different. The first frequency is not very sensitive to the folding angle when the rigid connection is used at the connection position 1. The second frequency increases with the increase of folding angle. When the folding angle is less than <math>{90}^{\circ }</math> , the third, fourth and fifth order frequencies tend to decrease. The sixth order frequency first increases and then decreases. More importantly, the second and third frequencies, as well as the fourth and fifth frequencies, tend to couple with each other with the increase of folding angle, but there is no mode transition phenomenon. When the rigid connection is adopted at the connection position 2, the change of the first frequency is not obvious with the increase of the folding angle. The second, third and sixth order frequencies first decrease and then increase. The fourth and fifth order frequencies are basically reduced. When the rigid connection is used at connection position 2, the fifth and sixth frequencies are coupled with each other. Compared with the frequency of elastic connection, the frequency of rigid connection is higher. Compared with Figure5, it can be seen that the frequency of rigid connection is greater than that of elastic connection. This is mainly due to the large stiffness of the structure at the interface when it is rigidly connected, which makes the folding wing structure more rigid. When the mass is constant, the eigenvalue of the structure increases, so the frequency is larger.
-<span style="text-align: center; font-size: 75%;">(a) Connection position 1                 (b) Connection position 2</span></div>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-−
 {|
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@@ Line 1,764: / Line 1,803: @@
 <span style="text-align: center; font-size: 75%;">'''Figure 8. '''Modal shape comparison (left: fixed interface component modal synthesis method; right: MSC.NASTRAN )</span></div>
-It can be seen from Figure 8 that the modal shapes by the mixed-loaded interface component modal synthesis method are consistent with those by MSC.NASTRAN. And the modal information can be obtained accurately, which is mainly due to the correct selection of the master-slave component and the correct reflection of the connection type between components. Therefore, the accurate displacements of related nodes can be obtained for further dynamic analysis.
+It can be seen from [[#img-8|Figure 8]] that the modal shapes by the mixed-loaded interface component modal synthesis method are consistent with those by MSC.NASTRAN. And the modal information can be obtained accurately, which is mainly due to the correct selection of the master-slave component and the correct reflection of the connection type between components. Therefore, the accurate displacements of related nodes can be obtained for further dynamic analysis.
 ==4. Conclusions==