The following are derivations of the output motions available in FAST for a 2-bladed turbine configuration. The motions for a 3-bladed turbine are very similar. Note that some of the motions are given multiple names in order to support variation among the user’s preferences. 以下是针对双叶片风电机组中可获得的输出运动的推导结果。三叶片风电机组的运动非常相似。请注意，某些运动被赋予了多个名称，以适应用户的偏好差异。 Blade 1 Tip Motions: OoPDefl1=TipDxc1=rQS1(BldFlexL)−TipRadj3B1⋅i1B1 Blade 1 OoP tip deflection (relative to rotor) (directed along the xc1-axis), (m) IPDefl1=TipDyc1=rQS1(BldFlexL)−TipRadj3B1⋅i2B1 Blade 1 IP tip deflection (relative to rotor) (directed along the yc1-axis), (m) TipDxb1=rQS1(BldFlexL)−TipRadj3B1⋅j1B1 Blade 1 flapwise tip deflection (relative to rotor) (directed along the xb1-axis), (m) TipDyb1=rQS1(BldFlexL)−TipRadj3B1⋅j2B1 Blade 1 edgewise tip deflection (relative to rotor) (directed along the yb1-axis), (m) $\begin{array}{r}{T i p D z c I=T i p D z b I=\Big[r^{\varrho s I}\left(B l d F l e x L\right)-T i p R a d j_{3}^{B I}}\end{array}$  $\cdot\dot{\pmb{i}}_{3}^{B I}=\left[r^{Q S I}\left(B l d F l e x L\right)\!-T i p R a d\pmb{j}_{3}^{B I}\right.$ ⋅j3B1 Blade 1 axial tip deflection (relative to rotor) (directed along the zc1-/zb1-axis), (m) $T i p A L x b I=\mathbf{\nabla}^{E}a^{S I}\left(B l d F l e x L\right)\cdot\mathbf{\boldsymbol{n}}_{I}^{B I}\left(B l d F l e x L\right)$ Blade 1 flapwise tip acceleration (absolute) (directed along the xb1-axis), (m/sec2) $T i p A L y b I=\mathbf{\nabla}^{E}a^{S I}\left(B l d F l e x L\right)\cdot\mathbf{\mathbf{\cdot}}\mathbf{\mathbf{}}n_{2}^{B I}\left(B l d F l e x L\right)$ Blade 1 edgewise tip acceleration (absolute) (directed along the yb1-axis), (m/sec2) $T i p A L z b I=\mathbf{\nabla}^{E}a^{S I}\left(B l d F l e x L\right)\cdot\mathbf{\boldsymbol{n}}_{3}^{B I}\left(B l d F l e x L\right)$ Blade 1 axial tip acceleration (absolute) (directed along the zc1-/zb1-axis), (m/sec2) $R o l l D e j l I=T i p R D x b I=\left(\frac{I\delta\theta}{\pi}\right)^{H}\!\pmb{\theta}^{M I}\left(B l d F l e x L\right)\cdot\pmb{j}_{I}^{B I}$ Blade 1 roll tip deflection (relative to the undeflected position), (about the xb1-axis), (deg) $P t c h D e j l-T i p R D y b I=\left(\frac{I\,\delta\theta}{\pi}\right)^{H}\!\theta^{M I}\left(B l d F l e x L\right)\cdot\dot{J}_{2}^{B I}$ Blade 1 pitch tip deflection (relative to the undeflected position), (about the yb1-axis), (deg) where: $^{H}\!\theta^{M I}\left(B l d F l e x L\right)=^{E}\!\omega_{B I F I}^{M I}\left(B l d F l e x L\right)q_{B I F I}+^{E}\!\omega_{B I E I}^{M I}\left(B l d F l e x L\right)q_{B I E I}+^{E}\!\omega_{I}^{I}\,,$ BΜ11F2(BldFlexL) qB1F 2 where: $r^{~o s t}\left(B l d F l e x L\right)\!=\!r^{~o V}+r^{~V P}+r^{P Q}+r^{Q S I}\left(B l d F l e x L\right)$ ![](ff3b6c808f53eed30fb804e400aefda635080690456a987aad15c16d5a0b6883.jpg) $$ {\begin{array}{l}{S p n i A L x b{I}{\,=\,}^{E}a^{S t}\left(R^{S p a n i}\right)\cdot{n}_{I}^{B I}\left(R^{S p a n i}\right)}\\ {1,2,...{\,,5)},\,(\mathrm{m/sec}^{2})}\\ {S p n i A L y b{I}{\,=\,}^{E}a^{S t}\left(R^{S p a n i}\right)\cdot{n}_{2}^{B I}\left(R^{S p a n i}\right)}\\ {1,2,...{\,,5)},\,(\mathrm{m/sec}^{2})}\\ {S p n i A L z b{I}{\,=\,}^{E}a^{S t}\left(R^{S p a n i}\right)\cdot{n}_{3}^{B I}\left(R^{S p a n i}\right)}\\ {1,2,...{\,,5)},\,(\mathrm{m/sec}^{2})}\end{array}} $$ Blade 1 local flapwise acceleration (absolute) of span station $i$ (directed along the local xb1-axis) ${\bf\nabla}^{i}={\bf\nabla}$ Blade 1 local edgewise acceleration (absolute) of span station $i$ (directed along the local yb1-axis) ${\boldsymbol{\mathbf{\mathit{i}}}}=$ Blade 1 axial acceleration (absolute) of span station $i$ (directed along the zc1-/zb1-/local zb1-axis) $(i=$ Blade 2 Tip Motions: The output motions of blade 2 are similar to those of blade 1. ![](2057057aaac1479c30036b6175385a20823e1b62a204561d7dd450dc0ac8fb7a.jpg) Blade 1 pitch angle (position) (positive towards feather / about the minus zc1- /minus zb2-axis), (deg) Blade 2 pitch angle (position) (positive towards feather / about the minus zc2- Teeter Motions: $$ T e e t D e f l=R o t T e e t P=T e e t P y a=\left(\frac{I\,\delta\theta}{\pi}\right)q_{T e e t} $$ Rotor teeter angle (position) (about the ya-axis), (deg) Rotor teeter angular velocity (about the ya-axis), (deg/sec) Rotor teeter angular acceleration (about the ya-axis), (deg/sec) ![](5ae9ae923e89b7ee52899a62a43cb9af6327bdd5839b12f492c4132f898ce7f4.jpg) ![](e1db52774350e87a4ba12810c360d943d26c454ca95528c4ecf6eee0cda1fa87.jpg) Rotor-Furl Motions: 180 RotFurl=RotFurlP RF Nacelle IMU translational velocity (directed along the xs-axis), (m/sec) Nacelle IMU translational velocity (directed along the ys-axis), (m/sec) Nacelle IMU translational velocity (directed along the zs-axis), (m/sec) Nacelle IMU translational acceleration (directed along the xs-axis), $(\mathrm{m/sec}^{2})$ Nacelle IMU translational acceleration (directed along the ys-axis), $(\mathrm{m/sec}^{2})$ Nacelle IMU translational acceleration (directed along the zs-axis), $(\mathrm{m/sec}^{2})$ Nacelle IMU angular (rotational) velocity (about the xs-axis), (deg/sec) Nacelle IMU angular (rotational) velocity (about the zs-axis), (deg/sec) Nacelle IMU angular (rotational) acceleration (about the xs-axis), (deg/sec2) Nacelle IMU angular (rotational) acceleration (about the ys-axis), (deg/sec2) Nacelle IMU angular (rotational) acceleration (about the zs-axis), (deg/sec2) Rotor-furl angle (position) (about the rotor-furl axis), (deg) Rotor-furl angular velocity (about the rotor-furl axis), (deg/sec) Rotor-furl angular acceleration (about the rotor-furl axis), (deg/sec2) ![](c80ac1f42a7cf2cd3aa056fddb53af0ebc8b8cdb5e5d6fbd15630e314107e179.jpg) Nacelle yaw angle (position) (about the zn-/zp-axis), (deg) Nacelle yaw angular velocity (about the zn-/zp-axis), (deg/sec) Nacelle yaw angular acceleration (about the zn-/zp-axis), (deg/sec2) Tower-Top Motions: Nacelle yaw error (about the zt-axis), (deg) $$ \ Y a w B r T D x p=\Big[\pmb{r}^{z o}-\big(T o w e r H t+P t f m\,R e\,f\big)\pmb{a}_{2}\Big]\cdot\pmb{b}_{I} $$ Tower-top / yaw bearing translational deflection (relative to undeflected position) (directed along the xp-axis), (m) $\ Y a w B r T D y p=-\Big[{r}^{Z O}-\big(T o w e r H t+P t f m\,R e\,f\big)\,{a}_{2}\Big]\cdot{b}_{3}$ (directed along the yp-axis), (m) $\ Y a w B r T D z p=\Big[\pmb{r}^{z o}-\big(T o w e r H t+P t f m\,R e\,f\big)\pmb{a}_{2}\Big]\cdot\pmb{b}_{2}$ (directed along the zp-axis), (m) $\begin{array}{r}{T T D s p F A=Y a w B r T D x t=\Big[\underline{{r}}^{Z O}-\big(T o w e r H t+P t f m\,R e\,f\big)a_{2}\Big]\cdot a_{I}}\end{array}$ position) (directed along the xt-axis), (m) $T T D s p S S=Y a w B r T D y t=-\Big[r^{Z O}-\big(T o w e r H t+P t f m\,R e\,f\big)a_{2}\Big]\cdot a_{3}$ undeflected position) (directed along the yt-axis), (m) $T T D s p A x=Y a w B r T D z t=\left[r^{z o}-\left(T o w e r H t+P t f m\,R e\,f\right)a_{2}\right]$ ⋅a2 position) (directed along the zt-axis), (m) Tower-top / yaw bearing translational deflection (relative to undeflected position) Tower-top / yaw bearing translational deflection (relative to undeflected position) Tower-top / yaw bearing fore-aft (translational) deflection (relative to undeflected Tower-top / yaw bearing side-to-side (translational) deflection (relative to Tower-top / yaw bearing axial (translational) deflection (relative to undeflected $Y a w B r T A x p={}^{E}a^{o}\cdot b_{_{I}}$ $Y a w B r T A y p=-\,^{E}a^{o}\cdot b_{s}$ $Y a w B r T A z p={}^{E}a^{o}\cdot b_{2}$ Tower-top / yaw bearing translational acceleration (directed along the xp-axis), (m/sec) Tower-top / yaw bearing translational acceleration (directed along the yp-axis), (m/sec) Tower-top / yaw bearing translational acceleration (directed along the zp-axis), (m/sec2) $$ T T D s p R o l l=Y a w B r R D x t=\left(\frac{l\,\delta\boldsymbol{\theta}}{\pi}\right)^{\,x}\!\!\!\boldsymbol{\theta}^{B}\cdot\!\boldsymbol{a}_{l} $$ Tower-top / yaw bearing roll deflection (relative to the undeflected position) (about the xt-axis), (deg) ![](b38c4988ac1e5719ff698fcbf8f729662bca9a2a8b26838cf362115222c35539.jpg) ![](0c218afa5aa05229840608cb8327fe4452c4006430a61cd378e0c48bb48f3d00.jpg) Tower Local Gage Motions: Tower local fore-aft translational acceleration (absolute) of node $i$ (directed along the local xt-axis) $(i=$ Tower local side-to-side translational acceleration (absolute) of node $i$ (directed along the local yt-axis) # Tail-Furl Motions: TailFurl=TailFurlP Tail-furl angle (position) (about the tail-furl axis), (deg) $T a i l F u r l V=\left(\frac{I\,\!\delta O}{\pi}\right)\dot{q}_{\scriptscriptstyle T F r l}$ $T a i l F u r l A=\left(\frac{I\,\delta0}{\pi}\right)\ddot{q}_{\scriptscriptstyle T F r l}$ Tail-furl angular velocity (about the tail-furl axis), (deg/sec) Tail-furl angular acceleration (about the tail-furl axis), (deg/sec2) Platform Motions: $P t f m T D x t=r^{Z}\cdot{\pmb a}_{I}$ $P t f m T D y t=-r^{Z}\cdot{\bf{\boldsymbol{a}}}_{3}$ $P t f m T D z t=r^{Z}\cdot{\bf{a}}_{2}$ PtfmSurge=PtfmTDxi=qS g $P t f m S w a y=P t f m T D y i=q_{S w}$ $P t f m H e a\nu e=P t f m T D z i=q_{H\nu}$ $P t f m T V x t={}^{E}\nu^{Z}\cdot{\bf{a}}_{I}$ $P t f m T V y t=-\,^{E}\nu^{Z}\cdot{\bf{a}}_{3}$ $P t f m T V z t={}^{E}\nu^{Z}\cdot{\bf{a}}_{2}$ PtfmTVxi= qSg $P t\/m T\o V\!y i=\dot{q}_{S w}$ PtfmTVzi= qHv $P t f m T A x t={}^{E}{\pmb{a}}^{Z}\cdot{\pmb{a}}_{I}$ PtfmTAyt= −EaZ PtfmTAzt= a a PtfmTAxi= qSg PtfmTAyi= qSw PtfmTAzi= qHv Platform horizontal surge displacement (directed along the xt-axis), (m) Platform horizontal sway displacement (directed along the yt-axis), (m) Platform vertical heave displacement (directed along the zt-axis), (m) Platform horizontal surge displacement (directed along the xi-axis), (m) Platform horizontal sway displacement (directed along the yi-axis), (m) Platform vertical heave displacement (directed along the zi-axis), (m) Platform horizontal surge velocity (directed along the xt-axis), (m/sec) Platform horizontal sway velocity (directed along the yt-axis), (m/sec) Platform vertical heave velocity (directed along the zt-axis), (m/sec) Platform horizontal surge velocity (directed along the xi-axis), (m/sec) Platform horizontal sway velocity (directed along the yi-axis), (m/sec) Platform vertical heave velocity (directed along the zi-axis), (m/sec) Platform horizontal surge acceleration (directed along the xt-axis), $(\mathrm{m/sec}^{2})$ Platform horizontal sway acceleration (directed along the yt-axis), $(\mathrm{m/sec}^{2})$ Platform vertical heave acceleration (directed along the zt-axis), $(\mathrm{m/sec}^{2})$ Platform horizontal surge acceleration (directed along the xi-axis), $(\mathrm{m/sec}^{2})$ Platform horizontal sway acceleration (directed along the yi-axis), $(\mathrm{m/sec}^{2})$ Platform vertical heave acceleration (directed along the zi-axis), $(\mathrm{m/sec}^{2})$
180 PtfmRoll=PtfmRDxi= 4R π 180 PtfmPitch = PtfmRDyi qp πPlatform roll tilt displacement (about the xi-axis), (deg) Platform pitch tilt displacement (about the yi-axis), (deg)
180 PtfmYaw = PtfmRDzi = πPlatform yaw displacement (about the zi-axis), (deg) qy
180 E a
PtfmRVxt = π 180Platform roll tilt velocity (about the xt-axis), (deg/sec)
PtfmRVyt = π 180 EPlatform pitch tilt velocity (about the yt-axis), (deg/sec)
PtfmRVzt = π 180Platform yaw velocity (about the zt-axis), (deg/sec)
PtfmRVxi = πPlatform roll tilt velocity (about the xi-axis), (deg/sec)
180 PtfmRVyi = qp πPlatform pitch tilt velocity (about the yi-axis), (deg/sec)
180 PtfmRVzi = qy πPlatform yaw velocity (about the zi-axis), (deg/sec)
180 PtfmRAxt = πPlatform roll tilt acceleration (about the xt-axis), (deg/sec2)
180 PtfmRAyt = πPlatform pitch tilt acceleration (about the yt-axis), (deg/sec?)
180 PtfmRAzt = πPlatform yaw acceleration (about the zt-axis), (deg/sec²)
180 PtfmRAxi = YR πPlatform roll tilt acceleration (about the xi-axis), (deg/sec²)
180 PtfmRAyi = dip πPlatform pitch tilt acceleration (about the yi-axis), (deg/sec2)
$P t j m R A z i=\left(\frac{l\,{\delta}O}{\pi}\right)\ddot{q}_{_Y}$ Platform yaw acceleration (about the zi-axis), (deg/sec) Tail-Furl Motions: $T F i n A l p h a=\Bigg(\frac{I\,\!\!\delta\boldsymbol{\theta}}{\pi}\Bigg)T F i n A O A$ $T F i n C L i f t=T F i n C L$ $T F i n C D r a g=T F i n C D$ $T F i n D n\,P r\,s=T F i n Q$ $T F i n C P F x=T F i n K F x\,/\,l,000$ $T F i n C P F y=T F i n K F y\ /\ l,O O O$ Tail fin lift coefficient, (-) Tail fin drag coefficient, (-) Tail fin dynamic pressure, (Pa) Tail fin tangential force, (kN) Tail fin normal force, (kN) # Wind Motions: Wind $V x i=u$ Wind $W i n d V y i=\nu W i n d$ Wind $V z i=w$ Wind Nominal hub-height wind velocity (directed along the xi-axis), $(\mathrm{m/s})$ Cross-wind hub-height velocity (directed along the yi-axis), $(\mathrm{m/s})$ Vertical hub-height wind velocity (directed along the zi-axis), $(\mathrm{m/s})$ $$ T o t W i n d V=\sqrt{W i n d V x i^{2}+W i n d V y i^{2}+W i n d V z i^{2}} $$ Total hub-height wind velocity magnitude, $(\mathrm{m/s})$ ) $$ H o r W i n d V=\sqrt{W i n d V\!x i^{2}+W i n d V\!y i^{2}} $$ Horizontal hub-height wind velocity magnitude (in the xi-/yi-plane), (m/s) HorWndDir Horizontal hub-height wind direction (about the zi-axis), (deg) VerWndDir Vertical hub-height wind direction (about an axis orthogonal to the zi-axis and the horizontal wind vector), (deg)