NI et al 2020a - Revision history

Rimni at 11:32, 22 October 2020

2020-10-22T11:32:30Z

Rimni at 11:30, 22 October 2020

2020-10-22T11:30:34Z

Rimni at 11:27, 22 October 2020

2020-10-22T11:27:42Z

Rimni at 11:05, 22 October 2020

2020-10-22T11:05:57Z

Rimni at 11:04, 22 October 2020

2020-10-22T11:04:15Z

Rimni at 10:55, 22 October 2020

2020-10-22T10:55:39Z

Rimni at 10:54, 22 October 2020

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Rimni at 10:54, 22 October 2020

2020-10-22T10:54:00Z

Rimni at 10:52, 22 October 2020

2020-10-22T10:52:32Z

Rimni at 10:51, 22 October 2020

2020-10-22T10:51:34Z

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 ==2. Multi-body dynamic equations for folding wing==
-As shown in [[#img-1|Figure 1]], the folding wing consists of three separate components: the fuselage, the inboard wing and the outboard wing, denoted A, B, C, respectively.  <math>oxyz</math> is the inertial frame, which is used to describe the motion of the system.  <math>o_Ax_Ay_Az_A</math>， <math>o_Bx_By_Bz_B</math>， <math>o_Cx_Cy_Cz_C</math> are the floating frame to describe the elastic deform.  <math>{\theta }^A</math>, <math>{\theta }^B</math>, <math>{\theta }^C</math> are the folding angles of component A, B, C, respectively, which are positive in counter clockwise direction. All components are modeled by plates with the Young’s modulus of <math>7.1\times {10}^{10}</math> Pa, Poisson’s coefficient of 0.33 and a density of <math>2.7\times {10}^3</math>kg/m<math>^3</math>, which are discretized using CQUAD4 elements with  <math>2</math>mm 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, all nodes at the interface are used to coordinate the boundary displacement. In this paper, the interfaces are assumed to be rigid. The finite element model for a folding wing is shown in [[#img-2|Figure 2]] .
+As shown in [[#img-1|Figure 1]], the folding wing consists of three separate components: the fuselage, the inboard wing and the outboard wing, denoted A, B, C, respectively.  <math>oxyz</math> is the inertial frame, which is used to describe the motion of the system.  <math>o_Ax_Ay_Az_A</math>， <math>o_Bx_By_Bz_B</math>， <math>o_Cx_Cy_Cz_C</math> are the floating frame to describe the elastic deform.  <math>{\theta }^A</math>, <math>{\theta }^B</math>, <math>{\theta }^C</math> are the folding angles of component A, B, C, respectively, which are positive in counter clockwise direction. All components are modeled by plates with the Young’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}^3 \,kg/m^3</math>, which are discretized using CQUAD4 elements with  <math>2 \, mm</math> 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, all nodes at the interface are used to coordinate the boundary displacement. In this paper, the interfaces are assumed to be rigid. The finite element model for a folding wing is shown in [[#img-2|Figure 2]] .
 <div id='img-1'></div>

@@ Line 261: / Line 261: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\theta }^B=\left\{ \begin{array}{c}
+| style="text-align: center;" | <math>{\theta }^B=\left\{ \begin{array}{l}
 {120}^{\mbox{°}}\left( \displaystyle\frac{t}{T_s}-\displaystyle\frac{1}{2\pi }\sin\displaystyle\frac{2\pi t}{T_s}\right),(0\leq t\leq T_s)\\
 {120}^{\mbox{°}},(t>T_s)
@@ Line 269: / Line 269: @@
 |}
-where  <math>{120}^{\mbox{°}}</math> stands for the steady state angle,  <math>T_s</math> is the time that the folding angle  <math>{\theta }^B</math> goes from  <math>0^{\mbox{°}}</math> to  <math>{120}^{\mbox{°}}</math>. The whole program is established in MATLAB, which can be solved by the computational algorithm in [[#img-3|Figure 3]] . The first six modes of each component are used and the modal damping ratio is 0.02. Therefore, there are 46 equations in Equation (9).  <math>c_1</math> and  <math>c_2</math> can be set to 5 and 10, respectively. To determine the effect of aerodynamic force, the transient response of the rigid and flexible wing with gravity can be performed first, then the one with gravity and aerodynamic force.
+where  <math>{120}^{\mbox{°}}</math> stands for the steady state angle,  <math>T_s</math> is the time that the folding angle  <math>{\theta }^B</math> goes from  <math>0^{\mbox{°}}</math> to  <math>{120}^{\mbox{°}}</math>. The whole program is established in MATLAB, which can be solved by the computational algorithm in [[#img-3|Figure 3]]. The first six modes of each component are used and the modal damping ratio is 0.02. Therefore, there are 46 equations in Equation (9).  <math>c_1</math> and  <math>c_2</math> can be set to 5 and 10, respectively. To determine the effect of aerodynamic force, the transient response of the rigid and flexible wing with gravity can be performed first, then the one with gravity and aerodynamic force.
 <div id='img-3'></div>
@@ Line 281: / Line 281: @@
 ===5.1. Transient response of rigid folding wing with gravity===
-To determine the effect of aerodynamic force on the transient response, the response of rigid folding wing with gravity can be carried out firstly. [[#img-4|Figure 4]]  shows the vertical displacement of the wingtip. The final displacement is 118.2mm for different <math>T_s</math>. When  <math>T_s</math> is set to 2s, 4s, 6s,  <math>{\theta }^B</math> equals <math>{90}^{\mbox{°}}</math>, implying that component B is perpendicular to components A and C and the displacement is the maximum. The greater <math>T_s</math>, the slower the morphing process.
+To determine the effect of aerodynamic force on the transient response, the response of rigid folding wing with gravity can be carried out firstly. [[#img-4|Figure 4]]  shows the vertical displacement of the wingtip. The final displacement is 118.2mm for different <math>T_s</math>. When  <math>T_s</math> is set to 2 <math>s</math>, 4 <math>s</math>, 6 <math>s</math>,  <math>{\theta }^B</math> equals <math>{90}^{\mbox{°}}</math>, implying that component B is perpendicular to components A and C and the displacement is the maximum. The greater <math>T_s</math>, the slower the morphing process.
 <div id='img-4'></div>
@@ Line 292: / Line 292: @@
-During the morphing process, the inertia tensors of each component vary relative to the inertial coordinate. Superimposing the tensor of each part, which is obtained in the inertial coordinate, the inertia tensor of the folding wing can be obtained. As shown in [[#img-5|Figure 5]], it can be seen that for different <math>T_s</math> , the inertia tensor remains constant at the end of the folding motion as the time changes. And the rotational inertia Izz, Iyy, Ixx in the three directions sequentially decrease and remain positive, which is consistent with the definition of the moment of inertia, while the inertial product Ixy, Ixz, Iyz are negative. The more slowly the folding motion is, the slower each component of the inertia tensors varies.
+During the morphing process, the inertia tensors of each component vary relative to the inertial coordinate. Superimposing the tensor of each part, which is obtained in the inertial coordinate, the inertia tensor of the folding wing can be obtained. As shown in [[#img-5|Figure 5]], it can be seen that for different <math>T_s</math>, the inertia tensor remains constant at the end of the folding motion as the time changes. And the rotational inertia Izz, Iyy, Ixx in the three directions sequentially decrease and remain positive, which is consistent with the definition of the moment of inertia, while the inertial product Ixy, Ixz, Iyz are negative. The more slowly the folding motion is, the slower each component of the inertia tensors varies.
 <div id='img-5'></div>
@@ Line 332: / Line 332: @@
 ===5.2 Transient response of rigid folding wing with gravity and aerodynamic===
-Then, the transient responses of the rigid folding wing with gravity and aerodynamic force is compared with ones in Section 5.1 to reveal the effect of aerodynamic force on the transient response. When the morphing process is relatively slow, the unsteady effect of the aerodynamic force can be ignored. Hence, here <math>T_s</math> is set to  <math>6s</math> , indicating that the process is relatively slow. Since the folding wing is rigid, the displacement response of the wingtip is consistent with one of the rigid folding wing with the gravity, implying that the vertical displacement response of the wingtip with gravity and aerodynamic force is shown in [[#img-4|Figure 4]], the corresponding displacement response corresponds to <math>T_s=6s</math>. Therefore, only the mechanical response is analyzed at different velocities and angles of attack, respectively, as shown in Figures [[#img-7|7]] and [[#img-8|8]]. The free flow velocity is set to 20 <math>m/s</math>, 25 <math>m/s</math> and 30 <math>m/s</math>, and the angle of attack equals to <math>1^{\circ }</math>,  <math>2^{\circ }</math> and <math>3^{\circ }</math>. Compared with the mechanical responses of the rigid folding wing with gravity, namely, the corresponding curves in [[#img-6|Figure 6]], it can be found that the aerodynamic force changes the mechanical response.
+Then, the transient responses of the rigid folding wing with gravity and aerodynamic force is compared with ones in Section 5.1 to reveal the effect of aerodynamic force on the transient response. When the morphing process is relatively slow, the unsteady effect of the aerodynamic force can be ignored. Hence, here <math>T_s</math> is set to  <math>6s</math>, indicating that the process is relatively slow. Since the folding wing is rigid, the displacement response of the wingtip is consistent with one of the rigid folding wing with the gravity, implying that the vertical displacement response of the wingtip with gravity and aerodynamic force is shown in [[#img-4|Figure 4]], the corresponding displacement response corresponds to <math>T_s=6s</math>. Therefore, only the mechanical response is analyzed at different velocities and angles of attack, respectively, as shown in Figures [[#img-7|7]] and [[#img-8|8]]. The free flow velocity is set to 20 <math>m/s</math>, 25 <math>m/s</math> and 30 <math>m/s</math>, and the angle of attack equals to <math>1^{\circ }</math>,  <math>2^{\circ }</math> and <math>3^{\circ }</math>. Compared with the mechanical responses of the rigid folding wing with gravity, namely, the corresponding curves in [[#img-6|Figure 6]], it can be found that the aerodynamic force changes the mechanical response.
 <div id='img-7'></div>

@@ Line 261: / Line 261: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{\theta }^B=\left\{  \begin{array}{c}
+| style="text-align: center;" | <math>{\theta }^B=\left\{ \begin{array}{c}
-{120}^{\mbox{°}}\left(\displaystyle\frac{t}{T_s}-\frac{1}{2\pi }sin\displaystyle\\frac{2\pi t}{T_s}\right),(0\leq t\leq T_s)\\
+{120}^{\mbox{°}}\left( \displaystyle\frac{t}{T_s}-\displaystyle\frac{1}{2\pi }\sin\displaystyle\frac{2\pi t}{T_s}\right),(0\leq t\leq T_s)\\
 {120}^{\mbox{°}},(t>T_s)
 \end{array}\right.</math>

@@ Line 262: / Line 262: @@
 |-
 | style="text-align: center;" | <math>{\theta }^B=\left\{  \begin{array}{c}
-{120}^{\mbox{°}}(\frac{t}{T_s}-\frac{1}{2\pi }sin\frac{2\pi t}{T_s}),(0\leq t\leq T_s)\\
+{120}^{\mbox{°}}\left(\displaystyle\frac{t}{T_s}-\frac{1}{2\pi }sin\displaystyle\\frac{2\pi t}{T_s}\right),(0\leq t\leq T_s)\\
 {120}^{\mbox{°}},(t>T_s)
 \end{array}\right.</math>

@@ Line 228: / Line 228: @@
 |}
-Combining Equations (8) and (9),One can rewrite Equation(7):
+Combining Equations (8) and (9),One can rewrite Equation (7):
 {| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 66: / Line 66: @@
 |}
-where  <math>\textbf{M}^i， \textbf{D}^i， \textbf{K}^i</math> （<math>i=A,B,C</math>） are the generalized mass matrix, generalized damping matrix and generalized stiffness matrix, respectively,  <math>\textbf{Q}_c^i</math> is the vector of constraint force,  <math>\textbf{Q}_g^i</math> refers to the vector related to the gravity, <math>\textbf{Q}_v^i</math> represents the quadratic velocity vector that arises from the differentiating the kinetic energy with respect to time and with respect to the generalized coordinates, <math>\textbf{Q}_a^i</math> denotes the generalized aerodynamic force and  <math>\textbf{q}^i</math> is the generalized displacement of each component and it is written in detail.
+where  <math>\textbf{M}^i</math>， <math>\textbf{D}^i</math>， <math>\textbf{K}^i</math> （<math>i=A,B,C</math>） are the generalized mass matrix, generalized damping matrix and generalized stiffness matrix, respectively,  <math>\textbf{Q}_c^i</math> is the vector of constraint force,  <math>\textbf{Q}_g^i</math> refers to the vector related to the gravity, <math>\textbf{Q}_v^i</math> represents the quadratic velocity vector that arises from the differentiating the kinetic energy with respect to time and with respect to the generalized coordinates, <math>\textbf{Q}_a^i</math> denotes the generalized aerodynamic force and  <math>\textbf{q}^i</math> is the generalized displacement of each component and it is written in detail.
 {| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 80: / Line 80: @@
 |}
-where  <math>\textbf{R}^i</math> is the position vector of the origin of each floating frame.  <math>\textbf{q}_f^i</math> is the modal displacement.  <math>\textbf{D}_{ff}</math> and  <math>\textbf{K}_{ff}</math>, which are the sub matrices in generalized damping matrix and generalized stiffness matrix, will be different when taken the aerodynamic force into consideration. The displacement of each component consists of large rigid body motions and small elastic deformations. Therefore, the mass matrices are function of the generalized displacement.
+where  <math>\textbf{R}^i</math> is the position vector of the origin of each floating frame,  <math>\textbf{q}_f^i</math> is the modal displacement,  <math>\textbf{D}_{ff}</math> and  <math>\textbf{K}_{ff}</math>, which are the sub matrices in generalized damping matrix and generalized stiffness matrix, will be different when taken the aerodynamic force into consideration. The displacement of each component consists of large rigid body motions and small elastic deformations. Therefore, the mass matrices are function of the generalized displacement.
 ==3. Unsteady aerodynamic forces ==