Appendix
Considering the two Euler angles, the transformation matrix between the deformed and undeformed local coordinate systems is as follows [1]:
$$\begin{aligned} T=\left( {{\begin{array}{ccc} {1-\frac{1}{2}({v}'^{2}+{w}'^{2})}&{} {{v}'}&{} {{w}'} \\ {-{v}'-{w}'\phi +\frac{{v}'\phi ^{2}}{2}}&{} {1-\frac{\phi ^{2}}{2}-\frac{{v}'^{2}}{2}-\frac{{v}'{w}'\phi }{2}}&{} {\phi -\frac{{v}'{w}'}{2}-\frac{\phi ^{3}}{6}-\frac{{w}'^{2}\phi }{2}} \\ {-{w}'+{v}'\phi +\frac{{w}'\phi ^{2}}{2}}&{} {-\phi -\frac{{v}'{w}'}{2}+\frac{\phi ^{3}}{6}+\frac{{v}'^{2}\phi }{2}}&{} {1-\frac{\phi ^{2}}{2}-\frac{{w}'^{2}}{2}+\frac{{v}'{w}'\phi }{2}} \\ \end{array} }} \right) \nonumber \\ \end{aligned}$$
(A.1)
Matrices \(\left[ {H_1 } \right] \) and \(\left[ {H_2 } \right] \) in Eq. (4) are defined as [21]:
$$\begin{aligned} \left[ {H_1 } \right] =\left( {{\begin{array}{ccc} 1&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 1&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 1 \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ \end{array} }} \right) , \left[ {H_2 } \right] =\left( {{\begin{array}{ccc} 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ {\frac{1}{2}{w}'+\frac{1}{4}{w}'^{3}-\frac{1}{4}{w}'\phi ^{2}+\frac{1}{2}{v}'^{2}{w}'}&{}\quad {\phi -\frac{1}{2}{v}'{w}'+\frac{1}{2}{v}'^{2}\phi -\frac{1}{6}\phi ^{3}}&{}\quad {1+\frac{1}{2}{v}'^{2}-\frac{1}{2}\phi ^{2}+\frac{1}{2}{w}'{v}'\phi } \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ {-\frac{1}{2}{v}'-\frac{1}{4}{v}'\phi ^{2}+\frac{1}{4}{v}'{w}'^{2}}&{}\quad {-1-\frac{1}{2}{w}'^{2}+\frac{1}{2}\phi ^{2}+\frac{1}{2}{v}'{w}'\phi }&{}\quad {\phi +\frac{1}{2}{w}'{v}'+\frac{1}{2}{w}'^{2}\phi -\frac{1}{6}\phi ^{3}} \\ 0&{}\quad 0&{}\quad 0 \\ {1-\frac{1}{2}{v}'{w}'\phi }&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ \end{array} }} \right) \nonumber \\ \end{aligned}$$
(A.2)
By applying the Green–Lagrange strain theory, matrix \(\left[ {H_3 } \right] \) in Eq. (8) is obtained as [21]:
$$\begin{aligned} \left[ {H_3 } \right] =\left( {{\begin{array}{ccc} 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0 \\ {-{v}''{v}'y-\frac{1}{2}{v}''{w}'z-\frac{1}{2}{w}''{w}'y-\frac{1}{2}{\phi }'{w}''(y^{2}+z^{2})}&{}\quad {\frac{1}{2}{w}''z}&{}\quad {-\frac{1}{2}{w}''y} \\ {-y+\phi z+y{w}''z-\frac{1}{2}{v}'{w}'z-\frac{1}{2}{v}'^{2}y+\frac{1}{2}\phi ^{2}y-2\phi {v}''yz+{v}''y^{2}+\frac{1}{2}{w}'{\phi }'(y^{2}+z^{2})+{w}''\phi (y^{2}-z^{2})}&{}\quad {-\frac{1}{2}{w}'z}&{}\quad {\frac{1}{2}{w}'y} \\ 0&{}\quad 0&{}\quad 0 \\ {-{w}''{w}'z-\frac{1}{2}{v}''{v}'z-\frac{1}{2}{w}''{v}'y+\frac{1}{2}{\phi }'{v}''(y^{2}+z^{2})}&{}\quad {-\frac{1}{2}{v}''z}&{}\quad {\frac{1}{2}{v}''y} \\ {-z-\phi y-\frac{1}{2}{w}'^{2}z+\frac{1}{2}\phi ^{2}z-\frac{1}{2}{w}'{v}'y+{w}''z^{2}+{v}''yz+2{w}''\phi yz+\phi {v}''(y^{2}-z^{2})-\frac{1}{2}{v}'{\phi }'(y^{2}+z^{2})}&{}\quad {\frac{1}{2}{v}'z}&{}\quad {-\frac{1}{2}{v}'y} \\ {-{w}''y+{v}''z+{w}''\phi z+{v}''\phi y+{w}''^{2}zy+{w}''{v}''(y^{2}-z^{2})-{v}''^{2}yz}&{}\quad 0&{}\quad 0 \\ {{\phi }'(y^{2}+z^{2})+\frac{1}{2}({w}'{v}''-{v}'{w}'')(y^{2}+z^{2})}&{}\quad -z&{}\quad y \\ \end{array} }} \right) \nonumber \\ \end{aligned}$$
(A.3)
Blade acceleration vectors \(\left\{ {Q_1 } \right\} \)and \(\left\{ {Q_1 } \right\} \) in Eq. (6) are expressed as [21]:
$$\begin{aligned} \left\{ {Q_1 } \right\}= & {} \left[ {P(\left\{ {s,0,0} \right\} ^{t})} \right] \left\{ {\vec {\alpha }_{{\textit{XYZ}}} } \right\} \nonumber \\&+\left[ {P(\vec {\omega }_{{\textit{xyz}}} )} \right] ^{t}\left[ {P(\left\{ {s,0,0} \right\} ^{t})} \right] \left\{ {\vec {\omega }_{{\textit{xyz}}} } \right\} \nonumber \\ \left\{ {Q_2 } \right\}= & {} \left\{ {\ddot{u},\ddot{v},\ddot{w}} \right\} +2\left[ {P(\left\{ {\dot{u},\dot{v},\dot{w}} \right\} ^{t})} \right] \left\{ {\vec {\omega }_{{\textit{xyz}}} } \right\} \nonumber \\&+\left[ {P(\left\{ {u,v,w} \right\} ^{t})} \right] \left\{ {\vec {\alpha }_{{\textit{xyz}}} } \right\} \nonumber \\&+\left\{ {\vec {\omega }_{{\textit{xyz}}} } \right\} ^{t}\left[ {P(\left\{ {u,v,w} \right\} ^{t})} \right] \left\{ {\vec {\omega }_{{\textit{xyz}}} } \right\} \end{aligned}$$
(A.4)
where antisymmetric matrices \(\left[ {P(\vec {\omega }_{{\textit{xyz}}} )} \right] \) and \( \left[ {P(\vec {\omega }_{\xi \eta \zeta } )} \right] \), which represent the corresponding angular velocity of each coordinate system, are written as:
$$\begin{aligned} \left[ {P(\vec {\omega }_{{\textit{xyz}}} )} \right] =\dot{T}_r T_r^t ; \left[ {P(\vec {\omega }_{\xi \eta \zeta } )} \right] =T\left[ {P(\vec {\omega }_{{\textit{xyz}}} )} \right] T^{t}+\dot{T}T^{t}\nonumber \\ \end{aligned}$$
(A.5)
In Eq. (A.4), the angular acceleration of each coordinate system is obtained directly by taking the partial derivative of each angular velocity component of Eq. (A.5), as follows:
$$\begin{aligned} \left[ {P(\vec {\alpha }_{{\textit{xyz}}} )} \right] =\frac{\partial }{\partial t}\left[ {P(\vec {\omega }_{{\textit{xyz}}} )} \right] ; \left[ {P(\vec {\alpha }_{\xi \eta \zeta } )} \right] =\frac{\partial }{\partial t}\left[ {P(\vec {\omega }_{\xi \eta \zeta } )} \right] \nonumber \\ \end{aligned}$$
(A.6)
Moreover, \(\left[ {J_1 } \right] \) and \(\left[ {J_2 } \right] \) in Eq. (6) are as follows:
$$\begin{aligned} \left[ {J_1 } \right] =\int _A {\left[ {P(\vec {r})} \right] \mathrm{d}A,} \left[ {J_2 } \right] =\int _A {\left[ {P(\vec {r})} \right] ^{t}\left[ {P(\vec {r})} \right] \mathrm{d}A}\nonumber \\ \end{aligned}$$
(A.7)
The variational vector of Eq. (23), constructed by using the modal coordinate variables, is expressed as [21]
$$\begin{aligned}&\left\{ {\delta \psi } \right\} _{11\times 1}^t =\left[ Q \right] _{11\times N} \left\{ {\delta q_i } \right\} _{N\times 1}^t\nonumber \\&\quad =\left( {{\begin{array}{ccccc} {2\sum _{i=1}^N {\mathfrak {R}{a}_{1i} q_i } }&{}\quad {2\sum _{i=1}^N {\mathfrak {R}{a}_{2i} q_i } }&{}\quad {2\sum _{i=1}^N {\mathfrak {R}{a}_{3i} q_i } }&{}\quad {\ldots }&{}\quad {2\sum _{i=1}^N {\mathfrak {R}{a}_{Ni} q_i } } \\ {2\sum _{i=1}^N {\mathfrak {R}{b}_{1i} q_i } }&{}\quad {2\sum _{i=1}^N {\mathfrak {R}{b}_{2i} q_i } }&{}\quad {2\sum _{i=1}^N {\mathfrak {R}{b}_{3i} q_i } }&{}\quad {\ldots }&{}\quad {2\sum _{i=1}^N {\mathfrak {R}{b}_{Ni} q_i } } \\ {2\sum _{i=1}^N {\mathfrak {R}{c}_{1i} q_i } }&{}\quad {2\sum _{i=1}^N {\mathfrak {R}{c}_{2i} q_i } }&{}\quad {2\sum _{i=1}^N {\mathfrak {R}{c}_{3i} q_i } }&{}\quad {\ldots }&{}\quad {2\sum _{i=1}^N {\mathfrak {R}{c}_{Ni} q_i } } \\ {Q_{1v} }&{}\quad {Q_{2v} }&{}\quad {Q_{3v} }&{}\quad {\ldots }&{}\quad {Q_{Nv} } \\ {{Q}'_{1v} }&{}\quad {{Q}'_{2v} }&{}\quad {{Q}'_{3v} }&{}\quad {\ldots }&{}\quad {{Q}'_{Nv} } \\ {{Q}''_{1v} }&{}\quad {{Q}''_{2v} }&{}\quad {{Q}''_{3v} }&{}\quad {\ldots }&{}\quad {{Q}''_{Nv} } \\ {Q_{1w} }&{}\quad {Q_{2w} }&{}\quad {Q_{3w} }&{}\quad {\ldots }&{}\quad {Q_{Nw} } \\ {{Q}'_{1w} }&{}\quad {{Q}'_{2w} }&{}\quad {{Q}'_{3w} }&{}\quad {\ldots }&{}\quad {{Q}'_{Nw} } \\ {{Q}''_{1w} }&{}\quad {{Q}''_{2w} }&{}\quad {{Q}''_{3w} }&{}\quad {\ldots }&{}\quad {{Q}''_{Nw} } \\ {Q_{1\phi } }&{}\quad {Q_{2\phi } }&{}\quad {Q_{2\phi } }&{}\quad {\ldots }&{}\quad {Q_{N\phi } } \\ {{Q}'_{1\phi } }&{}\quad {{Q}'_{2\phi } }&{}\quad {{Q}'_{3\phi } }&{}\quad {\ldots }&{}\quad {{Q}'_{N\phi } } \\ \end{array} }} \right) \left\{ {\begin{array}{l} \delta q_1 \\ \delta q_2 \\ \delta q_3 \\ \cdot \\ \cdot \\ \cdot \\ \delta q_N \\ \end{array}} \right\} \end{aligned}$$
(A.8)
where \(\mathfrak {R}{a}_{ji} \), \(\mathfrak {R}{b}_{ji} \) and \(\mathfrak {R}{c}_{ji} \) are the components of matrices \([\mathfrak {R}{a}]\), \([\mathfrak {R}{b}]\) and \([\mathfrak {R}{c}]\), which are represented as follows:
$$\begin{aligned} \left\{ {\begin{array}{l} {}[\mathfrak {R}{a}]_{N\times N} =-\frac{1}{2}\int \limits _0^s {\left[ {\left\{ {{Q}'_v } \right\} ^{t}\left\{ {{Q}'_v } \right\} +\left\{ {{Q}'_w } \right\} ^{t}\left\{ {{Q}'_w } \right\} } \right] _{N\times N} \mathrm{d}s} \\ {}[\mathfrak {R}{b}]_{N\times N} =-\frac{1}{2}\left[ {\left\{ {{Q}'_v } \right\} ^{t}\left\{ {{Q}'_v } \right\} +\left\{ {{Q}'_w } \right\} ^{t}\left\{ {{Q}'_w } \right\} } \right] _{N\times N} \\ {}[\mathfrak {R}{c}]_{N\times N} =-\frac{1}{2}\left[ \left\{ {{Q}''_v } \right\} ^{t}\left\{ {{Q}'_v } \right\} +\left\{ {{Q}'_v } \right\} ^{t}\left\{ {{Q}''_v } \right\} \right. \\ \left. +\left\{ {{Q}''_w } \right\} ^{t}\left\{ {{Q}'_w } \right\} +\left\{ {{Q}'_w } \right\} ^{t}\left\{ {{Q}''_w } \right\} \right] _{N\times N} \\ \end{array}} \right. \nonumber \\ \end{aligned}$$
(A.9)
The vectors of variational multipliers for the kinetic energy \(\left\{ {R^{\tilde{T}}} \right\} \), potential energy \(\left\{ {R^{\varPi }} \right\} \) and the external work performed by the aerodynamic forces \(\left\{ {R^{W_{\mathrm{aero}} }} \right\} \) are obtained as follows:
$$\begin{aligned}&\left\{ {R^{\tilde{T}}} \right\} :\left\{ {\begin{array}{ll} R_2^{\tilde{T}} =R_3^{\tilde{T}} =R_6^{\tilde{T}} =R_9^{\tilde{T}} =R_{11}^{\tilde{T}} =0; \\ R_1^{\tilde{T}} =\rho A\left( {\begin{array}{ll} s\varOmega ^{2}+2\varOmega \dot{v}\epsilon +\left( {-\ddot{u}+u\varOmega ^{2}-s\varOmega ^{2}\beta _\mathrm{c}^2 -\beta _\mathrm{c} w\varOmega ^{2}+z_G \left( {\ddot{w}^{\prime }-\varOmega ^{2}(w^{\prime }+\beta _\mathrm{c} )-2\dot{\phi }\varOmega } \right) +y_G \left( {\ddot{v}^{\prime }-\varOmega ^{2}v^{\prime }} \right) } \right) \epsilon ^{2} \\ +\,\left( {-\dot{v}\varOmega \beta _\mathrm{c}^2 -y_G \varOmega \left( \phi ^{2}+v^{\prime 2} \right) ^{\cdot }-z_G \varOmega w^{\prime }\dot{v}^{\prime }} \right) \epsilon ^{3} \\ \end{array}} \right) ; \\ R_4^{\tilde{T}} =\rho A\left( {\begin{array}{ll} (-\ddot{v}+\varOmega ^{2}v+y_G \varOmega ^{2})\epsilon +(-2\varOmega \dot{u}+2\beta _\mathrm{c} \varOmega \dot{w}+z_G (\ddot{\phi }-\phi \varOmega ^{2}+2\varOmega \dot{w}^{\prime })+2y_G \varOmega \dot{v}^{\prime })\epsilon ^{2} \\ +\,(-2z_G \varOmega (\phi v^{\prime })^{.}+2y_G \varOmega (w^{\prime }+\beta _\mathrm{c} )\dot{\phi })\epsilon ^{3} \\ \end{array}} \right) ; \\ R_5^{\tilde{T}} =(-y_G \rho As\varOmega ^{2}\epsilon +\rho A(z_G s\varOmega ^{2}\phi -2y_G \varOmega \dot{v})\epsilon ^{2}+(2z_G \rho A\varOmega \dot{v}\phi +2\rho J_{yz} \dot{\phi }\varOmega )\epsilon ^{3}); \\ R_7^{\tilde{T}} =\rho A((-\ddot{w}-s\varOmega ^{2}\beta _\mathrm{c} )\epsilon +(-2\beta _\mathrm{c} \varOmega \dot{v}-y_G \ddot{\phi })\epsilon ^{2}+2z_G \beta _\mathrm{c} \varOmega \dot{\phi }\epsilon ^{3}); \\ R_8^{\tilde{T}} =(-z_G \rho As\varOmega ^{2}\epsilon +\rho A(-y_G s\varOmega ^{2}\phi -2z_G \varOmega \dot{v})\epsilon ^{2}+(-2\rho Ay_G \varOmega \dot{v}\phi +2\rho J_{yy} \dot{\phi }\varOmega )\epsilon ^{3}); \\ R_{10}^{\tilde{T}} =\left( {\begin{array}{ll} -\varOmega ^{2}J_{yz} \rho +(-\rho J_{xx} \ddot{\phi }+\rho (J_{yy} -J_{zz} )\phi \varOmega ^{2}-2\rho J_{yz} \varOmega \dot{v}^{\prime }-2\rho J_{yy} \varOmega \dot{w}^{\prime })\epsilon \\ +\,\rho A(y_G (-\ddot{w}-s\varOmega ^{2}(w^{\prime }+\beta _\mathrm{c} ))+z_G (\ddot{v}-\varOmega ^{2}(sv{)}'))\epsilon ^{2}+\rho A\varOmega \dot{v}(-2y_G (w^{\prime }+\beta _\mathrm{c} )+2z_G v^{\prime })\epsilon ^{3} \\ \end{array}} \right) ; \\ \end{array}} \right. \nonumber \\ \end{aligned}$$
(A.10)
$$\begin{aligned}&\left\{ {R^{\varPi }} \right\} :\left\{ {\begin{array}{l} R_1^\varPi =R_2^\varPi =R_3^\varPi =R_4^\varPi =R_7^\varPi =0; \\ R_5^\varPi =\epsilon ^{2}\left( {-.5GI_{xx} \phi ^{\prime }w^{\prime \prime }} \right) +\epsilon ^{3}\left( {\begin{array}{l} EI_{zz} v^{\prime \prime 2}v^{\prime }+.25GI_{xx} w^{\prime \prime 2}v^{\prime }+.5EI_{yz} (w^{\prime }v^{\prime \prime 2}+w^{\prime }w^{\prime \prime 2}+w^{\prime \prime }v^{\prime }v^{\prime \prime }) \\ +(.5E-.25G)I_{xx} w^{\prime }v^{\prime \prime }w^{\prime \prime } \\ \end{array}} \right) ; \\ R_6^\varPi =\epsilon \left( {EI_{zz} v^{\prime \prime }+EI_{yz} w^{\prime \prime }} \right) +\epsilon ^{2}\left( {-E(I_{yy} -I_{zz} )w^{\prime \prime }\phi +.5GI_{xx} w^{\prime }\phi ^{\prime }-2EI_{yz} \phi v^{\prime \prime }} \right) \\ +\,\epsilon ^{3}\left( {\begin{array}{l} EI_{zz} v^{\prime \prime }v^{\prime 2}+E(I_{yy} -I_{zz} )v^{\prime \prime }\phi ^{2}+(.5E-.25G)I_{xx} v^{\prime }w^{\prime }w^{\prime \prime }+.25GI_{xx} w^{\prime 2}v^{\prime \prime } \\ +EI_{yz} (.5w^{\prime \prime }w^{\prime 2}+.5w^{\prime \prime }v^{\prime 2}-2w^{\prime \prime }\phi ^{2}+v^{\prime }v^{\prime \prime }w^{\prime }) \\ \end{array}} \right) ; \\ R_8^\varPi =\epsilon ^{2}(.5GI_{xx} \phi ^{\prime }v^{\prime \prime })+\epsilon ^{3}\left( {\begin{array}{l} EI_{yy} w^{\prime \prime 2}w^{\prime }+.25GI_{xx} v^{\prime \prime 2}w^{\prime }+.5EI_{yz} (v^{\prime }w^{\prime \prime 2}+v^{\prime }v^{\prime \prime 2}+w^{\prime \prime }w^{\prime }v^{\prime \prime }) \\ +(.5E-.25G)I_{xx} v^{\prime \prime }v^{\prime }w^{\prime \prime } \\ \end{array}} \right) ; \\ R_9^\varPi =(EI_{yy} w^{\prime \prime }+EI_{yz} v^{\prime \prime })\epsilon +\epsilon ^{2}(-.5GI_{xx} v^{\prime }\phi ^{\prime }-E(I_{yy} -I_{zz} )v^{\prime \prime }\phi +2EI_{yz} w^{\prime \prime }\phi ) \\ +\,\epsilon ^{3}\left( {\begin{array}{l} .25GI_{xx} v^{\prime 2}w^{\prime \prime }+EI_{yz} (.5v^{\prime \prime }v^{\prime 2}+.5w^{\prime 2}v^{\prime \prime }+v^{\prime }w^{\prime \prime }w^{\prime }-2v^{\prime \prime }\phi ^{2}) \\ -E(I_{yy} -I_{zz} )\phi ^{2}w^{\prime \prime }+(.5E-.25G)I_{xx} v^{\prime }v^{\prime \prime }w^{\prime }+EI_{yy} w^{\prime \prime }w^{\prime 2} \\ \end{array}} \right) ; \\ R_{10}^\varPi =\epsilon ^{2}(-E(I_{yy} -I_{zz} )w^{\prime \prime }v^{\prime \prime }+EI_{yz} (-v^{\prime \prime 2}+w^{\prime \prime 2}))+\epsilon ^{3}(E(I_{yy} -I_{zz} )(v^{\prime \prime 2}-w^{\prime \prime 2})\phi -4EI_{yz} w^{\prime \prime }\phi v^{\prime \prime }); \\ R_{11}^\varPi =\epsilon (GI_{xx} \phi ^{\prime })+\epsilon ^{2}(-.5GI_{xx} (v^{\prime \prime }w^{\prime }-v^{\prime }w^{\prime \prime })); \\ \end{array}} \right. \nonumber \\ \end{aligned}$$
(A.11)
$$\begin{aligned}&\left\{ {R_{\mathrm{C}}^{W_{\mathrm{aero}} } } \right\} :\left\{ {\begin{array}{l} R_{\mathrm{C},1}^{W_{\mathrm{aero}} } ,R_{\mathrm{C},2}^{W_{\mathrm{aero}} } ,R_{\mathrm{C},3}^{W_{\mathrm{aero}} } =0; R_{\mathrm{C},4}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {\begin{array}{l} \left( {(\dot{w}+W_i )^{2}-s\varOmega (\phi +\phi _\mathrm{p} )(\dot{w}+W_i )} \right) \epsilon ^{2} \\ +\left( {\begin{array}{l} -.25c(1-2a)(\dot{\phi }+\varOmega (w^{\prime }+\beta _\mathrm{c} ))W_i +2.W_i \varOmega v(w^{\prime }+\beta _\mathrm{c} ) \\ -\dot{v}W_i (\phi +\phi _\mathrm{p} )-2s\varOmega W_i \chi -W_i \varOmega w^{\prime }v^{\prime }s \\ +\left( {s\varOmega (\phi +\phi _\mathrm{p} )-2W_i } \right) (w^{\prime }+\beta _\mathrm{c} )V_i \mathrm{sin}(\varOmega t)\\ -V_i W_i (\phi +\phi _\mathrm{p} )\mathrm{cos}(\varOmega t) \\ \end{array}} \right) \\ \end{array}} \right) \epsilon ^{3}; \\ R_{\mathrm{C},5}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {cs\varOmega w^{\prime }W_i (1+2a)/8} \right) ; R_{\mathrm{C},8}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {-\frac{1}{8}cv^{\prime }\varOmega sW_i (2a+1)} \right) \epsilon ^{3};R_{\mathrm{C},6}^{W_{\mathrm{aero}} },R_{\mathrm{C},9}^{W_{\mathrm{aero}} } =0 \\ R_{\mathrm{C},7}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {\begin{array}{l} -s\varOmega \left( {\begin{array}{l} \dot{w}+W_i \\ -s\varOmega (\phi +\phi _\mathrm{p} +\epsilon \chi ) \\ \end{array}} \right) \epsilon +\left( {\begin{array}{l} s\varOmega \dot{v}(\phi +\phi _\mathrm{p} +\epsilon \chi )-\dot{v}((\dot{w}+W_i ) \\ -s\varOmega (\phi +\phi _\mathrm{p} ))+.25c\dot{\phi }s\varOmega (1-2a) \\ +.25cs\varOmega ^{2}(1-2a)(w^{\prime }+\beta _\mathrm{c} ) \\ -s\varOmega ^{2}v(w^{\prime }+\beta _\mathrm{c} )+.5\varOmega ^{2}s^{2}w^{\prime }v^{\prime } \\ +s\varOmega (w^{\prime }+\beta _\mathrm{c} )V_i \mathrm{sin}(\varOmega t) \\ +\left( {2s\varOmega (\phi +\phi _\mathrm{p} +\epsilon \chi )-(\dot{w}+W_i )} \right) V_i \mathrm{cos}(\varOmega t) \\ \end{array}} \right) \epsilon ^{2} \\ +\left( {\begin{array}{l} 0.5s\varOmega W_i (w^{\prime }+\beta _\mathrm{c} )^{2}+.5s\varOmega \dot{v}w^{\prime }v^{\prime }-\varOmega \dot{v}v(w^{\prime }+\beta _\mathrm{c} )+.25c(1-2a)\varOmega \dot{v}(w^{\prime }+\beta _\mathrm{c} ) \\ +\left( {v^{\prime }W_i +\dot{v}(w^{\prime }+\beta _\mathrm{c} )-2.s\varOmega v^{\prime }(\phi +\phi _\mathrm{p} )} \right) V_i \mathrm{sin}(\varOmega t) \\ +\left( {\begin{array}{l} 2(\phi +\phi _\mathrm{p} )\dot{v}+.25c(1-2a)\dot{\phi }-\varOmega v(w^{\prime }+\beta _\mathrm{c} ) \\ +s\varOmega w^{\prime }v^{\prime }+0.25(1-2a)c\varOmega (w^{\prime }+\beta _\mathrm{c} ) \\ \end{array}} \right) V_i \mathrm{cos}(\varOmega t) \\ +V_i^2 (w^{\prime }+\beta _\mathrm{c} )\mathrm{sin}(\varOmega t)\mathrm{cos}(\varOmega t)+\mathrm{cos}(\varOmega t)^{2}V_i^2 (\phi +\phi _\mathrm{p} ) \\ \end{array}} \right) \epsilon ^{3} \\ \end{array}} \right) ; \\ R_{\mathrm{C},10}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {\begin{array}{l} 0.25c(1+2a)\left( {\varOmega s(\phi +\phi _\mathrm{p} )-(\dot{w}+W_i )} \right) \epsilon ^{2} \\ +\left( {\begin{array}{l} c(1+2a)\dot{v}\left( {-.25W_i +s\varOmega (\phi +\phi _\mathrm{p} )} \right) +c^{2}s\varOmega a(1-2a)\dot{\phi }/8 \\ +.25c(1+2a)s\varOmega (w^{\prime }+\beta _\mathrm{c} )V_i \mathrm{sin}(\varOmega t) \\ +\left( {-W_i +2\varOmega s(\phi +\phi _\mathrm{p} )} \right) c(1+2a)V_i \mathrm{cos}(\varOmega t)/4 \\ \end{array}} \right) \epsilon ^{3}; \\ \end{array}} \right) ; \\ \end{array}} \right. \nonumber \\&\left\{ {R_{\mathrm{NC}}^{W_{\mathrm{aero}} } } \right\} :\left\{ {\begin{array}{l} R_{\mathrm{NC},1}^{W_{\mathrm{aero}} } ,R_{\mathrm{NC},2}^{W_{\mathrm{aero}} } ,R_{\mathrm{NC},3}^{W_{\mathrm{aero}} },R_{\mathrm{NC},5}^{W_{\mathrm{aero}} },R_{\mathrm{NC},6}^{W_{\mathrm{aero}} },R_{\mathrm{NC},8}^{W_{\mathrm{aero}} },R_{\mathrm{NC},9}^{W_{\mathrm{aero}} },R_{\mathrm{NC},11}^{W_{\mathrm{aero}} } =0; \\ R_{\mathrm{NC},4}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {\left( {-s^{2}\varOmega ^{2}c_{\mathrm{D}} /c_{\mathrm{L}\alpha } } \right) \epsilon ^{2}+\left( {\begin{array}{l} -.25c\dot{\phi }s\varOmega (\phi +\phi _\mathrm{p} )-2s\varOmega \dot{v}c_{\mathrm{D}} /c_{\mathrm{L}\alpha } -1/c_{\mathrm{L}\alpha } c_{\mathrm{L}0} \varOmega sW_i \\ -2s\varOmega c_{\mathrm{D}} /c_{\mathrm{L}\alpha } V_i \mathrm{cos}(\varOmega t) \\ \end{array}} \right) \epsilon ^{3}} \right) ; \\ R_{\mathrm{NC},7}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {\begin{array}{l} \left( {-.25c\ddot{w}+.25cs\varOmega ^{2}(w^{\prime }+\beta _\mathrm{c} )+.25c\dot{\phi }s\varOmega +s^{2}\varOmega ^{2}c_{\mathrm{L}0} /c_{\mathrm{L}\alpha } } \right) \epsilon ^{2} \\ +\left( {\begin{array}{l} -s\varOmega W_i c_{\mathrm{D}} /c_{\mathrm{L}\alpha } +2\varOmega s\dot{v}c_{\mathrm{L}0} /c_{\mathrm{L}\alpha } +.25s\varOmega c\dot{v}^{\prime }w^{\prime } \\ -.25c\varOmega (\phi +\phi _\mathrm{p} )V_i \mathrm{sin}(\varOmega t) \\ \left( {.25c\dot{\phi }+.5c\varOmega (w^{\prime }+\beta _\mathrm{c} )+2s\varOmega c_{\mathrm{L}0} /c_{\mathrm{L}\alpha } } \right) V_i \mathrm{cos} (\varOmega t) \\ \end{array}} \right) \epsilon ^{3} \\ \end{array}} \right) ; \\ R_{\mathrm{NC},10}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {-c^{2}s\varOmega a(1-2a)\dot{\phi }/16} \right) \epsilon ^{3}; \\ \end{array}} \right. \nonumber \\ \end{aligned}$$
(A.12)
$$\begin{aligned}&\left\{ {R_{\mathrm{C}}^{W_{\mathrm{aero}} } } \right\} :\left\{ {\begin{array}{l} R_{\mathrm{C},1}^{W_{\mathrm{aero}} } ,R_{\mathrm{C},2}^{W_{\mathrm{aero}} },R_{\mathrm{C},3}^{W_{\mathrm{aero}} } R_{\mathrm{C},5}^{W_{\mathrm{aero}} },R_{\mathrm{C},6}^{W_{\mathrm{aero}} } , R_{\mathrm{C},8}^{W_{\mathrm{aero}} } ,R_{\mathrm{C},9}^{W_{\mathrm{aero}} } =0; \\ R_{\mathrm{C},4}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {\left( {W^{2}-s\varOmega W_i \phi _\mathrm{p} } \right) \epsilon ^{2}+\left( {\begin{array}{l} -.25c\beta _\mathrm{c} \varOmega (1-2a)W_i -V_i W_i \phi _\mathrm{p} \mathrm{cos}(\varOmega t) \\ +\left( {s\varOmega \phi _\mathrm{p} -2W_i } \right) \beta _\mathrm{c} V_i \mathrm{sin}(\varOmega t) \\ \end{array}} \right) } \right) \epsilon ^{3}; \\ R_{\mathrm{C},7}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {\begin{array}{l} -s\varOmega \left( {W_i -s\varOmega \phi _\mathrm{p} } \right) \epsilon +\left( {\begin{array}{l} +.25\beta _\mathrm{c} cs\varOmega ^{2}(1-2a)+s\varOmega \beta _\mathrm{c} V_i \mathrm{sin}(\varOmega t) \\ +\left( {2s\varOmega \phi _\mathrm{p} -W_i } \right) V_i \mathrm{cos}(\varOmega t) \\ \end{array}} \right) \epsilon ^{2} \\ +\left( {\begin{array}{l} 0.5s\varOmega W_i \beta _\mathrm{c} ^{2}+0.25\beta _\mathrm{c} (1-2a)c\varOmega V_i \mathrm{cos}(\varOmega t) \\ +V_i^2 \beta _\mathrm{c} \mathrm{sin}(\varOmega t)\mathrm{cos}(\varOmega t)+\mathrm{cos}(\varOmega t)^{2}V_i^2 \phi _\mathrm{p} \\ \end{array}} \right) \epsilon ^{3} \\ \end{array}} \right) ; \\ R_{\mathrm{C},10}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {\left( {c(1+2a)\left( {s\varOmega \phi _\mathrm{p} -W_i } \right) /4} \right) \epsilon ^{2}+\left( {\begin{array}{l} +.25\beta _\mathrm{c} c(1+2a)s\varOmega V_i \mathrm{sin}(\varOmega t) \\ +\left( {-W_i +2s\varOmega \phi _\mathrm{p} } \right) c(1+2a)V_i \mathrm{cos}(\varOmega t)/4 \\ \end{array}} \right) \epsilon ^{3};} \right) ; \\ \end{array}} \right. \nonumber \\&\left\{ {R_{\mathrm{NC}}^{W_{\mathrm{aero}} } } \right\} :\left\{ {\begin{array}{l} R_{\mathrm{NC},1}^{W_{\mathrm{aero}} } ,R_{\mathrm{NC},2}^{W_{\mathrm{aero}} } ,R_{\mathrm{NC},3}^{W_{\mathrm{aero}} } ,R_{\mathrm{NC},5}^{W_{\mathrm{aero}} } ,R_{\mathrm{NC},6}^{W_{\mathrm{aero}} } ,R_{\mathrm{NC},8}^{W_{\mathrm{aero}} } ,R_{\mathrm{NC},9}^{W_{\mathrm{aero}} } ,R_{\mathrm{NC},10}^{W_{\mathrm{aero}} } ,R_{\mathrm{NC},11}^{W_{\mathrm{aero}} } =0; \\ R_{\mathrm{NC},4}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {\left( {-s^{2}\varOmega ^{2}c_{\mathrm{D}} /c_{\mathrm{L}\alpha } } \right) \epsilon ^{2}-2s\varOmega c_{\mathrm{D}} V_i \mathrm{cos}(\varOmega t)\epsilon ^{3}/c_{\mathrm{L}\alpha } } \right) ; \\ R_{\mathrm{NC},7}^{W_{\mathrm{aero}} } =\frac{\rho _\mathrm{a} c_{\mathrm{L}\alpha } c}{2}\left( {\left( {.25cs\varOmega ^{2}\beta _\mathrm{c} +s^{2}\varOmega ^{2}c_{\mathrm{L}0} /c_{\mathrm{L}\alpha } } \right) \epsilon ^{2}+\left( {\begin{array}{l} -s\varOmega W_i c_{\mathrm{D}} /c_{\mathrm{L}\alpha } -.25c\varOmega \phi _\mathrm{p} V_i \mathrm{sin}(\varOmega t) \\ \left( {.5c\varOmega \beta _\mathrm{c} +2s\varOmega c_{\mathrm{L}0} /c_{\mathrm{L}\alpha } } \right) V_i \mathrm{cos} (\varOmega t) \\ \end{array}} \right) \epsilon ^{3}} \right) ; \\ \end{array}} \right. \nonumber \\ \end{aligned}$$
(A.13)
In Eqs. (A.10) through (A.13), the magnitude orders of the terms are indicated by an arbitrary parameter of \(\epsilon =1\), and the components of free-stream velocity \(V_{{\mathrm{wind}}} \) and \(W_{{\mathrm{wind}}} \) are denoted by \(V_i \) and \(W_i \), respectively. Moreover, the related circulatory and non-circulatory terms of the aerodynamic equation are defined by Eqs. (A.12) and (A.13), respectively, through which the overall aerodynamic equation is expressed as \(\left\{ {R^{W_{\mathrm{aero}} }} \right\} =C\mathrm{(k)}\left\{ {R_{\mathrm{C}}^{W_{\mathrm{aero}} } } \right\} +\left\{ {R_{\mathrm{NC}}^{W_{\mathrm{aero}} } } \right\} \).
