| Variable | Direction/ Rotation Axis | Description |
| q1 | localizedblade coordinates | Blade 1 flapwise tip displacement for natural mode 1 |
| q2 | Blade 2 flapwise tip displacement for natural mode 1 |
| q3 | e2 =f2 | Teeter angle |
| q4 | c1=e1 | Azimuth angle, rotor side of drive train |
| q5 | d3=C3 | Nacelle tilt angle |
| q6 | b2=d2 | Nacelle yaw angle |
| q7 | a1 | Longitudinal tower top displacement for natural mode 1 |
| q8 | a3 | Latitudinal tower top displacement for natural mode 1 |
| q9 | a1 | Longitudinal tower top displacement for natural mode 2 |
| q10 | a3 | Latitudinal tower top displacement for natural mode 2 |
| q 11 | localizedblade coordinates | Blade1 flapwise tip displacement for natural mode 2 |
| q12 | Blade 2 flapwise tip displacement for natural mode 2 |
| q13 | localizedblade coordinates | Blade 1 edgewise tip displacement for natural mode 1 |
| q14 | Blade 2 edgewise tip displacement for natural mode 1 |
| q15 | C1=e1 | Azimuth angle, generator side of drive train, low speed shaft side of gearbox |
+
+Table 3.4: Other Angle Variables
+
+
+| Variable | Direction | Description |
| Hs | a2 | Height of the rigid base of the tower |
| H | a2 | Length of the flexible portion of the tower |
| HH | a2 | Elevation of the hub (hub height) relative to the Earth's surface assuming negligible nacelle tilt and tower deflection |
| TWRHTOFFSET | C2 | Distancebetween thehub and the tower-top base plate |
| u7 | a1 | Total tower-top displacement in the longitudinal direction |
| u8 | a3 | Total tower-top displacement in the lateral direction |
| DN | C1=e1 | Distance between the tower-topbase plate and the teeter pin |
| DNM1 | C1=e1 | Distance between the tower-top base plate and the nacelle center of mass |
| DNM2 | C2 | Distance between the origin of c and the nacelle center of mass |
| Ru | -f1=-g1 | Distance from the teeter pin to the blade axes intersection (undersling) |
| R UM | -f1 =-g 1 | Distance from the teeter pin to the hub center of mass |
| RH | i3 | Hub radius; radius of the rigid portion of the rotor |
| R | i3 | Total radius of the rotor |
+
+Relationships between the various coordinate systems, points, DOFs, and other angles and distances are illustrated graphically in Fig. 3.1. FAST_AD employs the convention that downwind displacements are positive displacements.
+
+In FAST_AD, the bottom part of the tower can be modeled as rigid to a height $H_{S}$ ; thus, the length of the flexible part of the tower, $H.$ is defined as:
+
+$$
+H=H_{H}-T W R H T O F F S E T-H_{S}
+$$
+
+where $H_{H}$ is the elevation of the hub (hub height) relative to the Earth’s surface and TWRHTOFFSET is the vertical distance between the hub and the tower-top base plate, both specified while assuming that the tower deflection and nacelle tilt are negligible.
+
+The sums of the tip deflections for both natural modes in the longitudinal and lateral deflections form the total longitudinal and lateral displacements of the tower-top base plate, $u_{7}$ and $u_{\delta.}$ , respectively:
+
+$$
+u_{7}=q_{7}+q_{9}
+$$
+
+and
+
+$$
+u_{\delta}=q_{\delta}+q_{I\partial}
+$$
+
+
+Figure 3.1: FAST_AD coordinate system illustrations
+
+Since the generalized coordinates associated with the tip deflections are functions of time, so too are the total longitudinal and lateral displacements of the tower-top base plate.
+
+In section 3.2, dealing with deflections of the tower and blades, the assumption is employed that the deflections are small. With this assumption, the tower-top rotations in both the longitudinal and lateral directions, $\theta_{7}$ and $\theta_{8}$ respectively, can be approximated:
+
+$$
+\theta_{7}=-\Bigg(\frac{d\phi_{_{I T}}(h)}{d h}_{_{h=H}}q_{7}+\frac{d\phi_{_{2T}}(h)}{d h}\biggl|_{h=H}q_{9}\Bigg)
+$$
+
+$$
+\theta_{\vartheta}=\frac{d\phi_{_{I T}}(h)}{d h}\bigg\vert_{h=H}q_{\vartheta}+\frac{d\phi_{_{2T}}(h)}{d h}\bigg\vert_{h=H}q_{I\0}
+$$
+
+where $\phi_{l T}(h)$ and $\phi_{2T}(h)$ are the first and second natural mode shapes of the tower, respectively. In these expressions, the elevation along the flexible part of the tower, $h$ , ranges from zero to $H.$ . Note that $h$ equals zero at an elevation of $H_{S}$ relative to the Earth’s surface. Also, the derivatives of the mode shapes are evaluated at an elevation of $h=H$ as indicated. The derivation of these natural mode shapes of the tower is presented in section 3.2 where the tower is assumed to deflect in the longitudinal and lateral directions independently; yet, the natural mode shapes in each direction are assumed to be identical in each direction. The negative sign is present in Eq. (3.4) since positive longitudinal displacements of the tower-top base plate tend to rotate the base plate about the negative ${\pmb a}_{3}$ -axis. The generalized coordinates associated with the tip deflections are functions of time; so are the longitudinal and lateral tower-top rotations of the plate.
+
+Attached to the tower-top base plate is a yaw bearing (O). The yaw bearing allows everything atop the tower to rotate $\left(q_{\delta}\right)$ as winds change direction. The yaw bearing also has the flexibility to allow everything atop the tower to tilt $(q_{\cal S})$ when responding to wind loads. The nacelle houses the generator and gearbox and supports the rotor. The center of mass of the nacelle (D) is related to the tower-top base plate by the position vector $r^{O D10}$ :
+
+$$
+\pmb{r}^{O D}=D_{N M I}\pmb{c}_{I}+\left(D_{N M2}+T W R H T O F F S E T\right)\pmb{c}_{2}
+$$
+
+Blade 1 is at an azimuth angle of $q_{4}$ . The zero-azimuth reference position can be located by the azimuth offset parameter $z(4)$ . Blade 2 is naturally $180^{\circ}$ out of phase with blade 1. Drive train flexibility allows the induction generator to see an angular velocity that is different than $n$ times the angular velocity of the rotor, where $n$ is the gearbox ratio. The twist of the low-speed shaft is, because of its torsional flexibility, modeled with the $q_{I5}$ parameter. The azimuth angle $q_{I5}$ , is essentially the sum of the azimuth angle, $q_{4}$ , and the twist of the low-speed shaft.
+
+If applicable to the wind turbine under consideration, teeter motion of the rotor is about a pin (P) fixed on the low-speed shaft. The position vector connecting the teeter pin to the tower-top base plate, $r^{O P}$ , is10:
+
+$$
+\pmb{r}^{O P}=D_{N}\pmb{c}_{I}+T W R H T O F F S E T\pmb{c}_{2}
+$$
+
+For an upwind turbine configuration, the distance between the tower-top base plate and the teeter pin in the $c_{I}=e_{I}$ direction, $D_{N},$ must be less than zero.
+
+The delta-3 angle, $\delta_{3}$ , orients the axis of the unconed rotor blades so it is no longer perpendicular to the axis of the teeter pin. In this case, the teeter motion has both a flapping and a pitching component. If the delta-3 angle is zero, teeter motion is purely flapping motion.
+
+Each blade can be coned a different amount ( $\beta_{I}$ for blade 1 and $\beta_{2}$ for blade 2), though the coning angles are constant, not changing with time. Coning begins in the very center of the hub at point Q, which is offset of the teeter pin (P) by a distance $R_{U}$ along the central axis of the hub (undersling length). The position vector connecting the blade axes intersection point and the teeter pin, $\bar{\mathbf{r}}^{P Q}$ , is:
+
+$$
+{\pmb r}^{P Q}=-R_{U}{\pmb g}_{I}
+$$
+
+Located between point $\mathrm{P}$ and point Q, along the central axis of the hub, is the hub center of mass (C) (not shown in Fig. 3.1). The position vector connecting the hub center of mass and the teeter pin, $\bar{\pmb{r}}^{P C}$ , is:
+
+$$
+{\pmb r}^{P C}=-R_{\phantom{}_{U M}}{\pmb g}_{I}
+$$
+
+Similar to the tower, the root of each rotor blade can be considered rigid to some radius $R_{H}$ representing the robustness of the hub (hub radius). The length of the flexible part of each blade is thus $R-R_{H},$ where $R$ is the total radius of the rotor (also not shown in Fig. 3.1). The flexible part of each blade is assumed to deflect in the local flapwise (out-of-plane of rotor if pitch and twist distribution equal zero) and local edgewise (in-plane of rotor if pitch and twist distribution equal zero) directions independently. Local means that the flapwise and edgewise directions are unique to each blade element as defined by the sum of the distributed structural pretwist angle, $\theta_{S}(\bar{r^{\flat}})^{9}$ , and the blade collective pitch angle. Unlike the tower, the natural mode shapes in each direction are permitted to be different. The natural mode shapes for each blade are assumed to be identical.
+
+Because each blade can have some distributed structural pretwist, defining the deflection in two directions is complicated. The most viable method is to define the total blade curvature as the combination of the local curvature in each local blade element direction (flapwise or edgewise), resolved into in-plane and out-of-plane components by orienting them with the structural pretwist and blade collective pitch angles. This curvature can then be integrated twice to get the total deflection shape. Assuming that the blade deflections are small, the local curvatures in the flapwise and edgewise directions at a span of $r$ , and time $t$ , $\kappa_{\/F}(r_{\/,}t)$ and $\kappa_{E}(r,t)$ respectively, for blade 1, are11:
+
+$$
+\kappa_{\scriptscriptstyle F}(r,t)\!=\!q_{\scriptscriptstyle I}\frac{d^{\,2}\phi_{\scriptscriptstyle I B F}(r)}{d r^{2}}\!+\!q_{\scriptscriptstyle I I}\frac{d^{\,2}\phi_{\scriptscriptstyle2B F}(r)}{d r^{2}}
+$$
+
+and
+
+$$
+\kappa_{\scriptscriptstyle E}(r,t)\!=\!q_{\scriptscriptstyle I3}\frac{d^{2}\phi_{\scriptscriptstyle I B E}(r)}{d r^{2}}
+$$
+
+where $\phi_{I B F}(r)$ and $\phi_{2B F}(r)$ are the first and second natural mode shapes, respectively, of the blades in the flapwise direction and $\phi_{I B E}(r)$ is first natural mode shape of the blades in the edgewise direction. The dependency on these curvatures with time is inherent in the generalized coordinates $q_{I},\,q_{I I}$ , and $q_{I3}$ . In these expressions, the radius along the flexible part of the blade,
+
+$$
+\kappa(x)\!=\!\frac{\displaystyle{\bigg|\frac{d^{\,2}y(x)}{d x^{\,2}}\bigg|}}{\displaystyle{\bigg\{\!I\!+\!\!\left[\frac{d y(x)}{d x}\right]^{2}\!\bigg\}^{\,3/2}}}
+$$
+
+If the curve is composed only of small deflections, then:
+
+$$
+\frac{\mathit{d y}(x)}{\mathit{d x}}<