Yuan 2018a - Revision history

Rimni at 12:01, 4 May 2020

2020-05-04T12:01:19Z

Rimni at 12:00, 4 May 2020

2020-05-04T12:00:06Z

Rimni at 11:58, 4 May 2020

2020-05-04T11:58:56Z

Rimni at 11:58, 4 May 2020

2020-05-04T11:58:01Z

Rimni at 11:56, 4 May 2020

2020-05-04T11:56:48Z

Rimni at 11:55, 4 May 2020

2020-05-04T11:55:14Z

Rimni at 11:54, 4 May 2020

2020-05-04T11:54:15Z

Rimni at 11:51, 4 May 2020

2020-05-04T11:51:12Z

Rimni at 11:49, 4 May 2020

2020-05-04T11:49:52Z

Rimni at 11:48, 4 May 2020

2020-05-04T11:48:53Z

← Older revision		Revision as of 12:01, 4 May 2020
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−	When the denominator in Equation (16) equals to zero, torsion angle drives to infinite. By now, the wing is in a divergent state. And divergent velocity <math>V_D</math> can be written as	+	When the denominator in Equation (12) equals to zero, torsion angle drives to infinite. By now, the wing is in a divergent state. And divergent velocity <math>V_D</math> can be written as

	{\| class="formulaSCP" style="width: 100%; text-align: center;"		{\| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 204: / Line 204: @@
 |}
+where  <math>q_d</math> is dynamic pressure,  <math>S</math> is airfoil area,  <math>c</math> is cross section chord,  <math>C_{MAC}</math> is moment coefficient,  <math>\frac{\partial C_L}{\partial \alpha }</math> is slope of lift curve, while  <math>e</math> is the distance between stiffness center and aerodynamic center.
-Where  <math>q_d</math> is dynamic pressure,  <math>S</math> is airfoil area,  <math>c</math> is cross section chord,  <math>C_{MAC}</math> is moment coefficient,  <math>\frac{\partial C_L}{\partial \alpha }</math> is slope of lift curve, while  <math>e</math> is the distance between stiffness center and aerodynamic center.
 Constraints of the airfoil are applied by two springs. Equilibrium equation of aerodynamic moment and spring elastic moment is written as
@@ Line 219: / Line 218: @@
 |}
+where  <math>k_{\alpha }</math> is spring elastic coefficient. Torsion angle  <math>\theta </math> can be obtained and written as
-Where  <math>k_{\alpha }</math> is spring elastic coefficient. Torsion angle  <math>\theta </math> can be obtained and written as
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 233: / Line 231: @@
-When the denominator of in Equation (16) equals to zero, torsion angle drives to infinite. By now, the wing is in a divergent state. And divergent velocity  <math>V_D</math> can be written as
+When the denominator in Equation (16) equals to zero, torsion angle drives to infinite. By now, the wing is in a divergent state. And divergent velocity  <math>V_D</math> can be written as
 {| class="formulaSCP" style="width: 100%; text-align: center;"

@@ Line 199: / Line 199: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>M_e=q_dScC_{MAC}+q_dS\left[\frac{\partial C_L}{\partial \alpha }({\alpha }_0+ \theta )\right]e</math>
+| <math>M_e=q_dScC_{MAC}+q_dS\left[\displaystyle\frac{\partial C_L}{\partial \alpha }({\alpha }_0+ \theta )\right]e</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (10)
@@ Line 227: / Line 227: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\theta =\frac{\frac{q_dS}{k_{\alpha }}(e\frac{\partial C_L}{\partial \alpha }{\alpha }_0+cC_{MAC})}{1-\frac{q_deS}{k_{\alpha }}\frac{\partial C_L}{\partial \alpha }}</math>
+| <math>\theta =\displaystyle\frac{\displaystyle\frac{q_dS}{k_{\alpha }}(e\displaystyle\frac{\partial C_L}{\partial \alpha }{\alpha }_0+cC_{MAC})}{1-\displaystyle\frac{q_deS}{k_{\alpha }}\displaystyle\frac{\partial C_L}{\partial \alpha }}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (12)

@@ Line 281: / Line 281: @@
 </div>
-and design variable  <math>X_L\leq X\leq X_U</math>,  <math>X={\left(X_1,X_2,\cdots ,X_n\right)}^T</math>.
+and design variable
 '''''(1) Objective function'''''

@@ Line 87: / Line 87: @@
 | <math>A=\oint {\varphi }_it_ids+\sum_i{\varphi }_iA_i</math>
 |-
-| <math>{\xi }_0=\frac{\oint {\varphi }_it_ix_ids+\sum_i{\varphi }_iA_ix_i}{A}</math>
+| <math>{\xi }_0=\displaystyle\frac{\displaystyle\oint {\varphi }_it_ix_ids+\displaystyle\sum_i{\varphi }_iA_ix_i}{A}</math>
 |-
-|<math>{\eta }_0=\frac{\oint {\varphi }_it_iy_ids+\sum_i{\varphi }_iA_iy_i}{A}</math>
+|<math>{\eta }_0=\displaystyle\frac{\displaystyle\oint {\varphi }_it_iy_ids+\displaystyle\sum_i{\varphi }_iA_iy_i}{A}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (4)

@@ Line 78: / Line 78: @@
 where  <math>E_i</math> is Young’s modulus of practical material and  <math>E_0</math> is Young’s modulus of material sample.
-The equivalent area of each wing cross section component and cross section centroid can be obtained from Equation (4).
+The equivalent area of each wing cross section component and cross section centroid can be obtained from Equation (4)
-−
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<math>A=\oint {\varphi }_it_ids+\sum_i{\varphi }_iA_i</math> </div>
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 87: / Line 84: @@
+|
 {| style="text-align: center; margin:auto;"
 |-
 | <math>{\xi }_0=\frac{\oint {\varphi }_it_ix_ids+\sum_i{\varphi }_iA_ix_i}{A}</math>
@@ Line 92: / Line 91: @@
 | style="width: 5px;text-align: right;white-space: nowrap;" | (4)
 |}
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
@@ Line 99: / Line 97: @@
 where  <math>{\varphi }_i</math> is reduction factor,  <math>t_i</math> is the thickness of skin which number is  <math>i</math>,  <math>A</math> is the sum area of all component cross section,  <math>{\xi }_0</math> and  <math>{\eta }_0</math> are  <math>x</math> and  <math>y</math> coordinate value of centroid, respectively.
-Stiffness center computation is carried out in inertia axis, so it is needed to transfer coordinate, which can be implement through subtraction the value of centroid. Inertia moment of reduced cross section to inertia axis can be obtained from Equation (5).
+Stiffness center computation is carried out in inertia axis, so it is needed to transfer coordinate, which can be implement through subtraction the value of centroid. Inertia moment of reduced cross section to inertia axis can be obtained from Equation (5)
-−
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<math>J_x=\oint y_i{}^2{\varphi }_it_ids+\sum_i{\varphi }_iA_iy_i{}^2</math> </div>
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 108: / Line 103: @@
+|
 {| style="text-align: center; margin:auto;"
 |-
 | <math>J_y=\oint x_i{}^2{\varphi }_it_ids+\sum_i{\varphi }_iA_ix_i{}^2</math>
@@ Line 114: / Line 111: @@
 |}
-And moment of area of reduced cross section to inertia axis can be obtained from Equation (6).
+And moment of area of reduced cross section to inertia axis can be obtained from Equation (6)
-−
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<math>S_x=\oint y_i{\varphi }_it_ids+\sum_i{\varphi }_iA_iy_i</math> </div>
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 123: / Line 117: @@
+|
 {| style="text-align: center; margin:auto;"
 |-
 | <math>S_y=\oint x_i{\varphi }_it_ids+\sum_i{\varphi }_iA_ix_i</math>
@@ Line 131: / Line 127: @@
 Based on structural mechanics theory, stiffness center of cross section can be written as
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 139: / Line 132: @@
+|
 {| style="text-align: center; margin:auto;"
 |-
 | <math>Y=\displaystyle\frac{1}{J_y}\left(\int_sS_y\rho ds-\Omega \displaystyle\frac{\displaystyle\oint \displaystyle\frac{S_y}{Gt}ds}{\displaystyle\oint \displaystyle\frac{ds}{Gt}}\right)</math>

@@ Line 133: / Line 133: @@
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<math>X=\frac{1}{J_x}\left(\int_sS_x\rho ds-\Omega \frac{\oint \frac{S_x}{Gt}ds}{\oint \frac{ds}{Gt}}\right)</math> </div>
+<math>X=\displaystyle\frac{1}{J_x}\left(\int_sS_x\rho ds-\Omega \displaystyle\frac{\displaystyle\oint \displaystyle\frac{S_x}{Gt}ds}{\displaystyle\oint \displaystyle\frac{ds}{Gt}}\right)</math> </div>
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 140: / Line 140: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>Y=\displaystyle\frac{1}{J_y}\left(\int_sS_y\rho ds-\Omega \displaystyle\frac{\oint \displaystyle\frac{S_y}{Gt}ds}{\oint \displaystyle\frac{ds}{Gt}}\right)</math>
+| <math>Y=\displaystyle\frac{1}{J_y}\left(\int_sS_y\rho ds-\Omega \displaystyle\frac{\displaystyle\oint \displaystyle\frac{S_y}{Gt}ds}{\displaystyle\oint \displaystyle\frac{ds}{Gt}}\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (7)

@@ Line 168: / Line 168: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>C_{tor}=GJ=\displaystyle\frac{\Omega^2}{\oint \displaystyle\frac{ds}{G_i t_i}}</math>
+| <math>C_{tor}=GJ=\displaystyle\frac{\Omega^2}{\displaystyle\oint \displaystyle\frac{ds}{G_i t_i}}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (9)