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.
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: $r^{~o s t}\left(B l d F l e x L\right)\!=\!r^{~o V}+r^{~V P}+r^{P Q}+r^{Q S I}\left(B l d F l e x L\right)$
{\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.
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)
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= qSg
$P t\/m T\o V\!y i=\dot{q}_{S w}$
PtfmTVzi= qHv
$P t f m T A x t={}^{E}{\pmb{a}}^{Z}\cdot{\pmb{a}}_{I}$
PtfmTAyt= −EaZ
PtfmTAzt= a a
PtfmTAxi= qSg
PtfmTAyi= qSw
PtfmTAzi= qHv
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})$
<html><body><table><tr><td>180 PtfmRoll=PtfmRDxi= 4R π 180 PtfmPitch = PtfmRDyi qp π</td><td>Platform roll tilt displacement (about the xi-axis), (deg) Platform pitch tilt displacement (about the yi-axis), (deg)</td></tr><tr><td>180 PtfmYaw = PtfmRDzi = π</td><td>Platform yaw displacement (about the zi-axis), (deg) qy</td></tr><tr><td>180 E a</td><td></td></tr><tr><td>PtfmRVxt = π 180</td><td>Platform roll tilt velocity (about the xt-axis), (deg/sec)</td></tr><tr><td>PtfmRVyt = π 180 E</td><td>Platform pitch tilt velocity (about the yt-axis), (deg/sec)</td></tr><tr><td>PtfmRVzt = π 180</td><td>Platform yaw velocity (about the zt-axis), (deg/sec)</td></tr><tr><td>PtfmRVxi = π</td><td>Platform roll tilt velocity (about the xi-axis), (deg/sec)</td></tr><tr><td>180 PtfmRVyi = qp π</td><td>Platform pitch tilt velocity (about the yi-axis), (deg/sec)</td></tr><tr><td>180 PtfmRVzi = qy π</td><td>Platform yaw velocity (about the zi-axis), (deg/sec)</td></tr><tr><td>180 PtfmRAxt = π</td><td>Platform roll tilt acceleration (about the xt-axis), (deg/sec2)</td></tr><tr><td>180 PtfmRAyt = π</td><td>Platform pitch tilt acceleration (about the yt-axis), (deg/sec?)</td></tr><tr><td>180 PtfmRAzt = π</td><td>Platform yaw acceleration (about the zt-axis), (deg/sec²)</td></tr><tr><td>180 PtfmRAxi = YR π</td><td>Platform roll tilt acceleration (about the xi-axis), (deg/sec²)</td></tr><tr><td>180 PtfmRAyi = dip π</td><td>Platform pitch tilt acceleration (about the yi-axis), (deg/sec2)</td></tr></table></body></html>
$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)
