Model bodies and constraints for a HAWT
In this section we describe out approach to modeling a HAWT, including the specific “bodies” that include massless nodes with six DOF (position and orientation), 6-DOF rigid bodies, rigid connections, and flexible beams. In the model descirbed here, the tower and blades are flexible beams, with all other components being rigid. Table 1 defines the subscripts associated with coordinate systems used to define the model, which are illustrated in Figure 1. For example, the tower top is defined by \(\underline{x}^\mathrm{r}_\mathrm{tt}\), \(\underline{u}_\mathrm{tt}\), \(\underline{\underline{R}}^\mathrm{r}_\mathrm{tt}\), and \(\underline{\underline{R}}_\mathrm{tt}\), which is the reference position, displacement, reference orientation, and relative rotation, respectively. The current orientation, denoted by a c-superscript, is \(\underline{\underline{R}}^\mathrm{c}_\mathrm{tt} = \underline{\underline{R}}_\mathrm{tt} \underline{\underline{R}}^\mathrm{r}_\mathrm{tt}\).
subscript
description
g
global/inertial
tb
tower base; first node in tower finite element
tt
tower top; last node in tower finite element
yb
yaw-base; bearing center of mass
sb
shaft base (does not rotate with rotor)
a
azimuth (rotates with rotor)
h
hub center of mass (rotates with rotor)
c\(i\)
coned CS for blade \(i\) (rotates with rotor; does not pitch)
b\(i\)
blade CS for blade \(i\) (rotates with rotor; pitches)
nm
Nacelle center of mass
3 Schematic showing the coordinate systems and elements for a horizontal axis wind turbine. Only one blade is shown.
Constraints
In this section we define the constraints for our HAWT model. We start at the base and move up through the connections. We assume the tower base is fixed, for which the constraints can be written
(12)\[\begin{split}\underline{\Phi}_\mathrm{tb} =
\begin{bmatrix}
\underline{u}_\mathrm{tb}
\\ \mathrm{axial}\left({ \underline{\underline{R}}_\mathrm{tb} }\right)
\end{bmatrix}\end{split}\]
\(\underline{\Phi}_\mathrm{tb} \in \mathbb{R}^6\), where we have used the axial vector to enforce zero rotation following [@Sonneville-Bruls:2012]. The tower is represented by a flexible beam. The constraint between the tower top and the yaw base is complicated by the yaw controller and the motion of the beam model for the tower top. We apply the yaw controller to the yaw base plate in the reference configuration, resulting in the total rotation for the yaw:
\[\underline{\underline{R}}^\mathrm{c}_\mathrm{yb} = \underline{\underline{R}}_\mathrm{tt} \underline{\underline{R}}_\mathrm{yc} \underline{\underline{R}}_\mathrm{yb}^\mathrm{r}\]
which can be written
\[\underline{\underline{R}}_\mathrm{yb} \underline{\underline{R}}_\mathrm{yb}^\mathrm{r} = \underline{\underline{R}}_\mathrm{tt} \underline{\underline{R}}_\mathrm{yc} \underline{\underline{R}}_\mathrm{yb}^\mathrm{r}\]
and simplified to
\[\underline{\underline{R}}_\mathrm{yb} = \underline{\underline{R}}_\mathrm{tt} \underline{\underline{R}}_\mathrm{yc},\]
Keeping the yaw displacement tied to tower top displacement, the constraints between the tower-top and yaw baseplate are
\[\begin{split}\underline{\Phi}_\mathrm{yb-tt} =
\begin{Bmatrix}
\underline{u}_\mathrm{yb}
-\underline{u}_\mathrm{tt}
\\ \mathrm{axial}\left({ \underline{\underline{R}}_\mathrm{yb} \underline{\underline{R}}_\mathrm{yc}^T \underline{\underline{R}}_\mathrm{tt}^T }\right)
\end{Bmatrix}\end{split}\]
\(\underline{\Phi}_\mathrm{yb-tt} \in \mathbb{R}^6\).
The shaft base and yaw CS’s are assumed to be rigidly connected. Here, shaft-base relative rotation is the same as the yaw relative rotation:
\[\begin{split}\underline{\Phi}_\mathrm{sb-yb} =
\begin{Bmatrix}
\underline{u}_\mathrm{sb} + \underline{x}^\mathrm{r}_\mathrm{sb} -\underline{u}_\mathrm{yb} -\underline{x}^\mathrm{r}_\mathrm{yb}
- \underline{\underline{R}}_\mathrm{yb} \left(\underline{x}^\mathrm{r}_\mathrm{sb}-\underline{x}^\mathrm{r}_\mathrm{yb}\right)
\\ \mathrm{axial}\left({ \underline{\underline{R}}_\mathrm{sb} \underline{\underline{R}}_\mathrm{yb}^T}\right)
\end{Bmatrix}\end{split}\]
\(\underline{\Phi}_\mathrm{sb-yb} \in \mathbb{R}^6\).
The azimuth CS is free to rotate about the \(\widehat{x}_\mathrm{sb}\) axis and is tied to the location of the shaft-base CS.
\[\begin{split}\underline{\Phi}_\mathrm{a-sb} =
\begin{Bmatrix}
\underline{u}_\mathrm{a}-\underline{u}_\mathrm{sb}
\\ \widehat{z}_\mathrm{a}^T \widehat{x}_\mathrm{sb} \\
\widehat{y}_\mathrm{a}^T \widehat{x}_\mathrm{sb}
\end{Bmatrix} =
\begin{Bmatrix}
\underline{u}_\mathrm{a}-\underline{u}_\mathrm{sb}
\\ \widehat{z}_\mathrm{a}^{\mathrm{r}T} \underline{\underline{R}}_\mathrm{a}^T \underline{\underline{R}}_\mathrm{sb} \widehat{x}_\mathrm{sb}^\mathrm{r} \\
\widehat{y}_\mathrm{a}^{\mathrm{r}T} \underline{\underline{R}}_\mathrm{a}^T \underline{\underline{R}}_\mathrm{sb} \widehat{x}_\mathrm{sb}^\mathrm{r}
\end{Bmatrix}\end{split}\]
\(\underline{\Phi}_\mathrm{a-sb} \in \mathbb{R}^5\).
The hub coordinate system (for a rigid shaft) has the same relative rotation as the azimuth CS
\[\begin{split}\underline{\Phi}_\mathrm{h-a} =
\begin{Bmatrix}
\underline{u}_\mathrm{h}+\underline{x}^\mathrm{r}_\mathrm{h} -\underline{u}_\mathrm{a} - \underline{x}^\mathrm{r}_\mathrm{a}-
\underline{\underline{R}}_\mathrm{a} \left(\underline{x}^\mathrm{r}_\mathrm{h}-\underline{x}^\mathrm{r}_\mathrm{a}\right)
\\ \mathrm{axial}\left({ \underline{\underline{R}}_\mathrm{h} \underline{\underline{R}}_\mathrm{a}^T}\right)
\end{Bmatrix}\end{split}\]
\(\underline{\Phi}_\mathrm{h-a} \in \mathbb{R}^6\).
The cone CS for blade \(i\) rotates with the hub and the cone CS displacement is equal to that of the hub, but does not pitch.
\[\begin{split}\underline{\Phi}_{\mathrm{c}i\mathrm{-h}} =
\begin{Bmatrix}
\underline{u}_{\mathrm{c}i}-\underline{u}_\mathrm{h}
\\ \mathrm{axial}\left({ \underline{\underline{R}}_{\mathrm{c}i} \underline{\underline{R}}_\mathrm{h}^T}\right)
\end{Bmatrix}\end{split}\]
\(\underline{\Phi}_{\mathrm{c}i-h} \in \mathbb{R}^6\).
The blade-root CS for blade \(i\) is offset from the cone CS and it pitches.
\[\begin{split}\underline{\Phi}_{\mathrm{b}i-\mathrm{c}i} =
\begin{Bmatrix}
\underline{u}_{\mathrm{b}i} + \underline{x}^\mathrm{r}_{\mathrm{b}i}-\underline{u}_{\mathrm{c}i}
- \underline{x}^\mathrm{r}_{\mathrm{c}i}
- \underline{\underline{R}}_{\mathrm{c}i}\left(\underline{x}^\mathrm{r}_{\mathrm{b}i} - \underline{x}^\mathrm{r}_{\mathrm{c}i}\right)
\\ \mathrm{axial}\left({ \underline{\underline{R}}_{\mathrm{b}i} \underline{\underline{R}}_{\mathrm{pc}i}^T \underline{\underline{R}}_{\mathrm{c}i}^T }\right)
\end{Bmatrix}\end{split}\]
\(\underline{\Phi}_{\mathrm{b}i-h} \in \mathbb{R}^6\).
The nacelle-mass and yaw-base CS’s are assumed to be rigidly connected.
\[\begin{split}\underline{\Phi}_\mathrm{nm-yb} =
\begin{Bmatrix}
\underline{u}_\mathrm{nm} + \underline{x}^\mathrm{r}_\mathrm{nm} -\underline{u}_\mathrm{yb} -\underline{x}^\mathrm{r}_\mathrm{yb}
- \underline{\underline{R}}_\mathrm{yb} \left(\underline{x}^\mathrm{r}_\mathrm{nm}-\underline{x}^\mathrm{r}_\mathrm{yb}\right)
\\ \mathrm{axial}\left({ \underline{\underline{R}}_\mathrm{nm} \underline{\underline{R}}_\mathrm{yb}^T}\right)
\end{Bmatrix}\end{split}\]
\(\underline{\Phi}_\mathrm{nb-sb} \in \mathbb{R}^6\).
Constraint Gradient
In this section we derive the constraint gradient matrices associated with each of the constraints defined in the previous section. Consider the constraint given by Eq. (12), which after taking the variation can be written
\[\begin{split}\underline{\underline{B}}_\mathrm{tb}
\begin{Bmatrix}
\delta \underline{u}_\mathrm{tb}\\
\underline{\delta \theta}_\mathrm{tb}
\end{Bmatrix}
= \begin{bmatrix}
\underline{\underline{I}} & \underline{\underline{0}} \\
\underline{\underline{0}} & \mathrm{AX}\left( \underline{\underline{R}}_\mathrm{tb} \right)
\end{bmatrix}
\begin{Bmatrix}
\delta \underline{u}_\mathrm{tb}\\
\underline{\delta \theta}_\mathrm{tb}
\end{Bmatrix}\end{split}\]
where we leveraged the following identity
\[\mathrm{axial}\left({ \underline{\underline{A}} \widetilde{a}}\right) = \frac{1}{2} \left(\mathrm{tr} \left( \underline{\underline{A^T}} \right) \underline{\underline{I}} - \underline{\underline{A}}^T \right) \underline{a}\]
and we introduced the operator
\[\mathrm{AX}\left( \underline{\underline{A}} \right) = \frac{1}{2} \left(\mathrm{tr} \left( \underline{\underline{A}} \right) \underline{\underline{I}} - \underline{\underline{A}} \right)\]
Proceeding in a similar fashion for the remaining contraints, we find the following contraint gradient matrices:
\[\begin{split}\underline{\underline{B}}_\mathrm{yb-tt}
\begin{Bmatrix}
\delta \underline{u}_\mathrm{yb}\\
\underline{\delta \theta}_\mathrm{yb}\\
\delta \underline{u}_\mathrm{tt}\\
\underline{\delta \theta}_\mathrm{tt}
\end{Bmatrix}
= \begin{bmatrix}
\underline{\underline{I}} & \underline{\underline{0}} & -\underline{\underline{I}} & \underline{\underline{0}} \\
\underline{\underline{0}} & \mathrm{AX}\left( \underline{\underline{R}}_\mathrm{yb}
\underline{\underline{R}}_\mathrm{yc}^T
\underline{\underline{R}}_\mathrm{tt}^T \right) &
\underline{\underline{0}} & -\mathrm{AX}\left( \underline{\underline{R}}_\mathrm{tt}
\underline{\underline{R}}_\mathrm{yc}
\underline{\underline{R}}_\mathrm{yb}^T\right)
\end{bmatrix}
\begin{Bmatrix}
\delta \underline{u}_\mathrm{yb}\\
\underline{\delta \theta}_\mathrm{yb}\\
\delta \underline{u}_\mathrm{tt}\\
\underline{\delta \theta}_\mathrm{tt}
\end{Bmatrix}\end{split}\]
\[\begin{split}\underline{\underline{B}}_\mathrm{sb-yb}
\begin{Bmatrix}
\delta \underline{u}_\mathrm{sb}\\
\underline{\delta \theta}_\mathrm{sb}\\
\delta \underline{u}_\mathrm{yb}\\
\underline{\delta \theta}_\mathrm{yb}
\end{Bmatrix}
= \begin{bmatrix}
\underline{\underline{I}} & \underline{\underline{0}} & -\underline{\underline{I}} & \widetilde{\underline{\underline{R}}_\mathrm{yb}
\left( \underline{x}_\mathrm{sb}^\mathrm{r}-\underline{x}_\mathrm{yb}^\mathrm{r} \right)} \\
\underline{\underline{0}} & \mathrm{AX}\left( \underline{\underline{R}}_\mathrm{sb} \underline{\underline{R}}_\mathrm{yb}^T \right) &
\underline{\underline{0}} & -\mathrm{AX}\left( \underline{\underline{R}}_\mathrm{yb} \underline{\underline{R}}_\mathrm{sb}^T \right)
\end{bmatrix}
\begin{Bmatrix}
\delta \underline{u}_\mathrm{sb}\\
\underline{\delta \theta}_\mathrm{sb}\\
\delta \underline{u}_\mathrm{yb}\\
\underline{\delta \theta}_\mathrm{yb}
\end{Bmatrix}\end{split}\]
\[\begin{split}\underline{\underline{B}}_\mathrm{a-sb}
\begin{Bmatrix}
\delta \underline{u}_\mathrm{a}\\
\underline{\delta \theta}_\mathrm{a}\\
\delta \underline{u}_\mathrm{sb}\\
\underline{\delta \theta}_\mathrm{sb}
\end{Bmatrix}
= \begin{bmatrix}
\underline{\underline{I}} & \underline{\underline{0}} & -\underline{\underline{I}} & \underline{\underline{0}} \\
\underline{0}^T & \widehat{z}_\mathrm{a}^T \widetilde{x}_\mathrm{sb} &
\underline{0}^T & -\widehat{z}_\mathrm{a}^T \widetilde{x}_\mathrm{sb} \\
\underline{0}^T & \widehat{yb}_\mathrm{a}^T \widetilde{x}_\mathrm{sb} &
\underline{0}^T & -\widehat{yb}_\mathrm{a}^T \widetilde{x}_\mathrm{sb}
\end{bmatrix}
\begin{Bmatrix}
\delta \underline{u}_\mathrm{a}\\
\underline{\delta \theta}_\mathrm{a}\\
\delta \underline{u}_\mathrm{sb}\\
\underline{\delta \theta}_\mathrm{sb}
\end{Bmatrix}\end{split}\]
\[\begin{split}\underline{\underline{B}}_\mathrm{h-a}
\begin{Bmatrix}
\delta \underline{u}_\mathrm{h}\\
\underline{\delta \theta}_\mathrm{h}\\
\delta \underline{u}_\mathrm{a}\\
\underline{\delta \theta}_\mathrm{a}
\end{Bmatrix}
= \begin{bmatrix}
\underline{\underline{I}} & \underline{\underline{0}} & -\underline{\underline{I}} &
\widetilde{\underline{\underline{R}}_\mathrm{a}
\left( \underline{x}_\mathrm{h}^\mathrm{r}-\underline{x}_\mathrm{a}^\mathrm{r} \right)} \\
\underline{\underline{0}} & \mathrm{AX}\left( \underline{\underline{R}}_\mathrm{h} \underline{\underline{R}}_\mathrm{a}^T \right) &
\underline{\underline{0}} & -\mathrm{AX}\left( \underline{\underline{R}}_\mathrm{a} \underline{\underline{R}}_\mathrm{h}^T \right)
\end{bmatrix}
\begin{Bmatrix}
\delta \underline{u}_\mathrm{h}\\
\underline{\delta \theta}_\mathrm{h}\\
\delta \underline{u}_\mathrm{a}\\
\underline{\delta \theta}_\mathrm{a}
\end{Bmatrix}\end{split}\]
\[\begin{split}\underline{\underline{B}}_{\mathrm{c}i-\mathrm{h}}
\begin{Bmatrix}
\delta \underline{u}_{\mathrm{c}i}\\
\underline{\delta \theta}_{\mathrm{c}i}\\
\delta \underline{u}_\mathrm{h}\\
\underline{\delta \theta}_\mathrm{h}
\end{Bmatrix}
= \begin{bmatrix}
\underline{\underline{I}} & \underline{\underline{0}} & -\underline{\underline{I}} & \underline{\underline{0}} \\
\underline{\underline{0}} & \mathrm{AX}\left( \underline{\underline{R}}_{\mathrm{c}i} \underline{\underline{R}}_\mathrm{h}^T \right) &
\underline{\underline{0}} & -\mathrm{AX}\left( \underline{\underline{R}}_\mathrm{h} \underline{\underline{R}}_{\mathrm{c}i}^T \right)
\end{bmatrix}
\begin{Bmatrix}
\delta \underline{u}_{\mathrm{c}i}\\
\underline{\delta \theta}_{\mathrm{c}i}\\
\delta \underline{u}_\mathrm{h}\\
\underline{\delta \theta}_\mathrm{h}
\end{Bmatrix}\end{split}\]
\[\begin{split}\underline{\underline{B}}_{\mathrm{b}i-\mathrm{c}i}
\begin{Bmatrix}
\delta \underline{u}_{\mathrm{b}i}\\
\underline{\delta \theta}_{\mathrm{b}i}\\
\delta \underline{u}_{\mathrm{c}i}\\
\underline{\delta \theta}_{\mathrm{c}i}
\end{Bmatrix}
= \begin{bmatrix}
\underline{\underline{I}} & \underline{\underline{0}} & -\underline{\underline{I}} &
\widetilde{\underline{\underline{R}}_{\mathrm{c}i}
\left( \underline{x}_{\mathrm{b}i}^\mathrm{r}-\underline{x}_{\mathrm{c}i}^0 \right)} \\
\underline{\underline{0}} & \mathrm{AX}\left( \underline{\underline{R}}_{\mathrm{b}i}
\underline{\underline{R}}_{\mathrm{pc}i}^T
\underline{\underline{R}}_{\mathrm{c}i}^T \right) &
\underline{\underline{0}} & -\mathrm{AX}\left( \underline{\underline{R}}_{\mathrm{c}i}
\underline{\underline{R}}_{\mathrm{pc}i}
\underline{\underline{R}}_{\mathrm{b}i}^T\right)
\end{bmatrix}
\begin{Bmatrix}
\delta \underline{u}_{\mathrm{b}i}\\
\underline{\delta \theta}_{\mathrm{b}i}\\
\delta \underline{u}_{\mathrm{c}i}\\
\underline{\delta \theta}_{\mathrm{c}i}
\end{Bmatrix}\end{split}\]
\[\begin{split}\underline{\underline{B}}_\mathrm{nm-yb}
\begin{Bmatrix}
\delta \underline{u}_\mathrm{nm}\\
\underline{\delta \theta}_\mathrm{nm}\\
\delta \underline{u}_\mathrm{yb}\\
\underline{\delta \theta}_\mathrm{yb}
\end{Bmatrix}
= \begin{bmatrix}
\underline{\underline{I}} & \underline{\underline{0}} & -\underline{\underline{I}} & \widetilde{\underline{\underline{R}}_\mathrm{yb}
\left( \underline{x}_\mathrm{nm}^\mathrm{r}-\underline{x}_\mathrm{yb}^\mathrm{r} \right)} \\
\underline{\underline{0}} & \mathrm{AX}\left( \underline{\underline{R}}_\mathrm{nm} \underline{\underline{R}}_\mathrm{yb}^T \right) &
\underline{\underline{0}} & -\mathrm{AX}\left( \underline{\underline{R}}_\mathrm{yb} \underline{\underline{R}}_\mathrm{nm}^T \right)
\end{bmatrix}
\begin{Bmatrix}
\delta \underline{u}_\mathrm{nm}\\
\underline{\delta \theta}_\mathrm{nm}\\
\delta \underline{u}_\mathrm{yb}\\
\underline{\delta \theta}_\mathrm{yb}
\end{Bmatrix}\end{split}\]
Additional iteration matrix terms due to constraints
For a residual expressed as in Eq. (1), there may be additional terms that arise in the \(\underline{\underline{K}}\) term in Eq. (4) if the constraint-gradient matrix is not constant with respect to the depdendent variables, e.g., \(\underline{\underline{B}}=\underline{\underline{B}}(\underline{q})\). Consider, for example, \(\underline{\underline{B}}_\mathrm{tb}\), for which the associated residual force can be written
\[\begin{aligned}
\underline{F}_\mathrm{tb}=\underline{\underline{B}}_\mathrm{tb}^T \underline{\lambda}_\mathrm{tb}
\end{aligned}\]
where \(\underline{F}_\mathrm{tb}\,,\underline{\lambda}_\mathrm{tb} \in \mathbb{R}^6\). Taking the variation we obtain
(13)\[\begin{split}\delta \underline{F}_\mathrm{tb}=
\underline{\underline{K}}_\mathrm{tb,c}
\begin{bmatrix}
\delta \underline{u}_\mathrm{tb} \\
\delta \underline{\theta}_\mathrm{tb}
\end{bmatrix}
+ \underline{\underline{B}}_\mathrm{tb}^T \delta \underline{\lambda}_\mathrm{tb}\end{split}\]
where
\[\begin{split}\begin{aligned}
\underline{\underline{K}}_\mathrm{tb,c}
\begin{bmatrix}
\delta \underline{u}_\mathrm{tb} \\
\delta \underline{\theta}_\mathrm{tb}
\end{bmatrix}
=
\begin{bmatrix}
\underline{\underline{0}} & \underline{\underline{0}} \\
\underline{\underline{0}} & \mathrm{AX} \left( - \underline{\underline{R}}_\mathrm{tb} \widetilde{ \lambda_{\mathrm{tb},2} } \right)
\end{bmatrix}
\begin{bmatrix}
\delta \underline{u}_\mathrm{tb} \\
\delta \underline{\theta}_\mathrm{tb}
\end{bmatrix}
\end{aligned}\end{split}\]
and
\[\begin{split}\begin{aligned}
\underline{\lambda}_\mathrm{tb} =
\begin{bmatrix}
\underline{\lambda}_{\mathrm{tb},1}\\
\underline{\lambda}_{\mathrm{tb},2}
\end{bmatrix}
\end{aligned}\end{split}\]
The \(\underline{\underline{B}}^T_\mathrm{tb}\) term in Eq. (13) goes in the appropriate location in the upper right quadrant of Eq. (4) and the \(\underline{\underline{K}}_\mathrm{tb,c}\) term is added to the \(\underline{\underline{K}}\) matrix in the upper left quadrant.
Proceeding in a similar fashion for the remaining constraint forces, we find:
\[\begin{split}\begin{aligned}
\underline{\underline{K}}_\mathrm{yb-tt,c}
\begin{bmatrix}
\delta \underline{u}_\mathrm{yb} \\
\delta \underline{\theta}_\mathrm{yb}\\
\delta \underline{u}_\mathrm{tt} \\
\delta \underline{\theta}_\mathrm{tt}
\end{bmatrix}
=
\begin{bmatrix}
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & \mathrm{AX} \left(\underline{\underline{R}}_\mathrm{yb} \underline{\underline{R}}_\mathrm{yc}^T \underline{\underline{R}}_\mathrm{tt}^T \widetilde{\lambda}_{\mathrm{yb-tt},2} \right) &
\underline{\underline{0}} & \mathrm{AX2}\left( \underline{\lambda}_{\mathrm{yb-tt},2}, \underline{\underline{R}}_\mathrm{tt} \underline{\underline{R}}_\mathrm{yc} \underline{\underline{R}}_\mathrm{yb}^T\right) \\
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & -\mathrm{AX2} \left( \underline{\lambda}_{\mathrm{yb-tt},2}, \underline{\underline{R}}_\mathrm{yb} \underline{\underline{R}}_\mathrm{yc}^T \underline{\underline{R}}_\mathrm{tt}^T\right)
& \underline{\underline{0}} & -\mathrm{AX} \left( \underline{\underline{R}}_\mathrm{tt} \underline{\underline{R}}_\mathrm{yc} \underline{\underline{R}}_\mathrm{yb}^T \widetilde{\lambda}_{\mathrm{yb-tt},2} \right) &
\end{bmatrix}
\begin{bmatrix}
\delta \underline{u}_\mathrm{yb} \\
\delta \underline{\theta}_\mathrm{yb}\\
\delta \underline{u}_\mathrm{tt} \\
\delta \underline{\theta}_\mathrm{tt}
\end{bmatrix}
\end{aligned}\end{split}\]
where
\[\begin{split}\begin{aligned}
\mathrm{AX2}\left(\underline{a},\underline{\underline{A}}\right) = \frac{1}{2}
\begin{bmatrix}
a_1 (A_{23}- A_{32}) & -a_1 A_{13}-a_2 A_{32}-a_3 A_{33} & a_1 A_{12}+a_2 A_{22} + a_3 A_{23} \\
a_2 A_{23} + a_1 A_{31} + a_3 A_{33} & a_2 (A_{31}-A_{13}) & -a_1 A_{11}-a_2 A_{21} - a_3 A_{13} \\
-a_1 A_{21}-a_2 A_{22}-a_3 A_{32} & a_1 A_{11}+a_2 A_{12}+a_3 A_{31} & a_3 (A_{12}-A_{21})
\end{bmatrix}
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
\underline{\underline{K}}_\mathrm{sb-yb,c}
\begin{bmatrix}
\delta \underline{u}_\mathrm{sb} \\
\delta \underline{\theta}_\mathrm{sb}\\
\delta \underline{u}_\mathrm{yb} \\
\delta \underline{\theta}_\mathrm{yb}
\end{bmatrix}
=
\begin{bmatrix}
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & \mathrm{AX} \left(\underline{\underline{R}}_\mathrm{sb} \underline{\underline{R}}_\mathrm{yb}^T \widetilde{\lambda}_{\mathrm{sb-yb},2} \right) &
\underline{\underline{0}} & \mathrm{AX2}\left( \underline{\lambda}_{\mathrm{sb-yb},2}, \underline{\underline{R}}_\mathrm{yb} \underline{\underline{R}}_\mathrm{sb}^T\right) \\
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & -\mathrm{AX2} \left( \underline{\lambda}_{\mathrm{sb-yb},2}, \underline{\underline{R}}_\mathrm{sb} \underline{\underline{R}}_\mathrm{yb}^T\right)
& \underline{\underline{0}} & -\mathrm{AX} \left( \underline{\underline{R}}_\mathrm{yb} \underline{\underline{R}}_\mathrm{sb}^T \widetilde{\lambda}_{\mathrm{sb-yb},2} \right)
-\widetilde{\lambda}_\mathrm{sb-yb,1} \widetilde{ \underline{\underline{R}}_\mathrm{yb} \left( \underline{x}^\mathrm{r}_\mathrm{sb}-\underline{x}^\mathrm{r}_\mathrm{yb} \right) }
\end{bmatrix}
\begin{bmatrix}
\delta \underline{u}_\mathrm{sb} \\
\delta \underline{\theta}_\mathrm{sb}\\
\delta \underline{u}_\mathrm{yb} \\
\delta \underline{\theta}_\mathrm{yb}
\end{bmatrix}
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
\underline{\underline{K}}_\mathrm{a-sb,c}
\begin{bmatrix}
\delta \underline{u}_\mathrm{a} \\
\delta \underline{\theta}_\mathrm{a}\\
\delta \underline{u}_\mathrm{sb} \\
\delta \underline{\theta}_\mathrm{sb}
\end{bmatrix}
=
\begin{bmatrix}
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & \lambda_\mathrm{a-sb,2} \widetilde{x}_\mathrm{sb} \widetilde{z}_\mathrm{a}
+ \lambda_\mathrm{a-sb,3} \widetilde{x}_\mathrm{sb} \widetilde{y}_\mathrm{a}
& \underline{\underline{0}} &
-\lambda_\mathrm{a-sb,2} \widetilde{z}_\mathrm{a} \widetilde{x}_\mathrm{sb}
- \lambda_\mathrm{a-sb,3} \widetilde{y}_\mathrm{a} \widetilde{x}_\mathrm{sb}
\\
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & -\lambda_\mathrm{a-sb,2} \widetilde{x}_\mathrm{sb} \widetilde{z}_\mathrm{a}
- \lambda_\mathrm{a-sb,3} \widetilde{x}_\mathrm{sb} \widetilde{y}_\mathrm{a}
& \underline{\underline{0}} &
\lambda_\mathrm{a-sb,2} \widetilde{z}_\mathrm{a} \widetilde{x}_\mathrm{sb}
+ \lambda_\mathrm{a-sb,3} \widetilde{y}_\mathrm{a} \widetilde{x}_\mathrm{sb}
\end{bmatrix}
\begin{bmatrix}
\delta \underline{u}_\mathrm{a} \\
\delta \underline{\theta}_\mathrm{a}\\
\delta \underline{u}_\mathrm{sb} \\
\delta \underline{\theta}_\mathrm{sb}
\end{bmatrix}
\end{aligned}\end{split}\]
where
\[\begin{split}\begin{aligned}
\lambda_\mathrm{a-sb} =
\begin{bmatrix}
\underline{\lambda}_\mathrm{a-sb,1} \\
\lambda_\mathrm{a-sb,2} \\
\lambda_\mathrm{a-sb,3} \\
\end{bmatrix} \in \mathbb{R}^5
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
\underline{\underline{K}}_\mathrm{h-a,c}
\begin{bmatrix}
\delta \underline{u}_\mathrm{h} \\
\delta \underline{\theta}_\mathrm{h}\\
\delta \underline{u}_\mathrm{a} \\
\delta \underline{\theta}_\mathrm{a}
\end{bmatrix}
=
\begin{bmatrix}
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & \mathrm{AX} \left(\underline{\underline{R}}_\mathrm{h} \underline{\underline{R}}_\mathrm{a}^T \widetilde{\lambda}_{\mathrm{h-a},2} \right) &
\underline{\underline{0}} & \mathrm{AX2}\left( \underline{\lambda}_{\mathrm{h-a},2}, \underline{\underline{R}}_\mathrm{a} \underline{\underline{R}}_\mathrm{h}^T\right) \\
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & -\mathrm{AX2} \left( \underline{\lambda}_{\mathrm{h-a},2}, \underline{\underline{R}}_\mathrm{h} \underline{\underline{R}}_\mathrm{a}^T\right)
& \underline{\underline{0}} & -\mathrm{AX} \left( \underline{\underline{R}}_\mathrm{a} \underline{\underline{R}}_\mathrm{h}^T \widetilde{\lambda}_{\mathrm{h-a},2} \right)
-\widetilde{\lambda}_\mathrm{h-a,1} \widetilde{ \underline{\underline{R}}_\mathrm{a} \left( \underline{x}^\mathrm{r}_\mathrm{h}-\underline{x}^\mathrm{r}_\mathrm{a} \right) }
\end{bmatrix}
\begin{bmatrix}
\delta \underline{u}_\mathrm{h} \\
\delta \underline{\theta}_\mathrm{h}\\
\delta \underline{u}_\mathrm{a} \\
\delta \underline{\theta}_\mathrm{a}
\end{bmatrix}
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
\underline{\underline{K}}_{\mathrm{c}i\mathrm{-h,c}}
\begin{bmatrix}
\delta \underline{u}_{\mathrm{c}i} \\
\delta \underline{\theta}_{\mathrm{c}i}\\
\delta \underline{u}_\mathrm{h} \\
\delta \underline{\theta}_\mathrm{h}
\end{bmatrix}
=
\begin{bmatrix}
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & \mathrm{AX} \left(\underline{\underline{R}}_{\mathrm{c}i} \underline{\underline{R}}_\mathrm{h}^T \widetilde{\lambda}_{\mathrm{c}i\mathrm{-h,2}} \right) &
\underline{\underline{0}} & \mathrm{AX2}\left(\underline{\lambda}_{\mathrm{c}i\mathrm{-h,2}}, \underline{\underline{R}}_\mathrm{h} \underline{\underline{R}}_{\mathrm{c}i}^T\right) \\
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & -\mathrm{AX2} \left( \underline{\lambda}_{\mathrm{c}i\mathrm{-h,2}}, \underline{\underline{R}}_{\mathrm{c}i} \underline{\underline{R}}_\mathrm{h}^T\right)
& \underline{\underline{0}} & -\mathrm{AX} \left( \underline{\underline{R}}_\mathrm{h} \underline{\underline{R}}_{\mathrm{c}i}^T \widetilde{\lambda}_{\mathrm{c}i\mathrm{-h,2}} \right)
\end{bmatrix}
\begin{bmatrix}
\delta \underline{u}_{\mathrm{c}i} \\
\delta \underline{\theta}_{\mathrm{c}i}\\
\delta \underline{u}_\mathrm{h} \\
\delta \underline{\theta}_\mathrm{h}
\end{bmatrix}
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
\underline{\underline{K}}_{\mathrm{b}i-\mathrm{c}i\mathrm{,c}}
\begin{bmatrix}
\delta \underline{u}_{\mathrm{b}i} \\
\delta \underline{\theta}_{\mathrm{b}i}\\
\delta \underline{u}_{\mathrm{c}i} \\
\delta \underline{\theta}_{\mathrm{c}i}
\end{bmatrix}
= \nonumber \\
\begin{bmatrix}
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & \mathrm{AX} \left(\underline{\underline{R}}_\mathrm{bi} \underline{\underline{R}}_\mathrm{pci}^T \underline{\underline{R}}_\mathrm{ci}^T \widetilde{\lambda}_{\mathrm{bi-ci},2} \right) &
\underline{\underline{0}} & \mathrm{AX2}\left( \underline{\lambda}_{\mathrm{bi-ci},2}, \underline{\underline{R}}_\mathrm{ci} \underline{\underline{R}}_\mathrm{pci} \underline{\underline{R}}_\mathrm{bi}^T\right) \\
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & -\mathrm{AX2} \left( \underline{\lambda}_{\mathrm{bi-ci},2}, \underline{\underline{R}}_\mathrm{bi} \underline{\underline{R}}_\mathrm{pci}^T \underline{\underline{R}}_\mathrm{ci}^T\right)
& \underline{\underline{0}} & -\mathrm{AX} \left( \underline{\underline{R}}_\mathrm{ci} \underline{\underline{R}}_\mathrm{pci} \underline{\underline{R}}_\mathrm{bi}^T \widetilde{\lambda}_{\mathrm{bi-ci},2} \right)
-\widetilde{\lambda}_\mathrm{ci-bi,1} \widetilde{ \underline{\underline{R}}_\mathrm{ci} \left( \underline{x}^\mathrm{r}_\mathrm{bi}-\underline{x}^\mathrm{r}_\mathrm{ci} \right) }
\end{bmatrix}
\begin{bmatrix}
\delta \underline{u}_\mathrm{bi} \\
\delta \underline{\theta}_\mathrm{bi}\\
\delta \underline{u}_\mathrm{ci} \\
\delta \underline{\theta}_\mathrm{ci}
\end{bmatrix}
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
\underline{\underline{K}}_\mathrm{nm-yb,c}
\begin{bmatrix}
\delta \underline{u}_\mathrm{nm} \\
\delta \underline{\theta}_\mathrm{nm}\\
\delta \underline{u}_\mathrm{yb} \\
\delta \underline{\theta}_\mathrm{yb}
\end{bmatrix}
= \nonumber \\
\begin{bmatrix}
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & \mathrm{AX} \left(\underline{\underline{R}}_\mathrm{nm} \underline{\underline{R}}_\mathrm{yb}^T \widetilde{\lambda}_{\mathrm{nm-yb},2} \right) &
\underline{\underline{0}} & \mathrm{AX2}\left( \underline{\lambda}_{\mathrm{nm-yb},2}, \underline{\underline{R}}_\mathrm{yb} \underline{\underline{R}}_\mathrm{nm}^T\right) \\
\underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}} & \underline{\underline{0}}\\
\underline{\underline{0}} & -\mathrm{AX2} \left( \underline{\lambda}_{\mathrm{nm-yb},2}, \underline{\underline{R}}_\mathrm{nm} \underline{\underline{R}}_\mathrm{yb}^T\right)
& \underline{\underline{0}} & -\mathrm{AX} \left( \underline{\underline{R}}_\mathrm{yb} \underline{\underline{R}}_\mathrm{nm}^T \widetilde{\lambda}_{\mathrm{nm-yb},2} \right)
-\widetilde{\lambda}_\mathrm{nm-yb,1} \widetilde{ \underline{\underline{R}}_\mathrm{yb} \left( \underline{x}^\mathrm{r}_\mathrm{nm}-\underline{x}^\mathrm{r}_\mathrm{yb} \right) }
\end{bmatrix}
\begin{bmatrix}
\delta \underline{u}_\mathrm{nm} \\
\delta \underline{\theta}_\mathrm{nm}\\
\delta \underline{u}_\mathrm{yb} \\
\delta \underline{\theta}_\mathrm{yb}
\end{bmatrix}
\end{aligned}\end{split}\]
Data layout
Tangent stiffness matrix
4 Schematic illustrating the nonzero entries of the tangent stiffness matriix.
5 Schematic illustrating the nonzero entries of the constraint-gradient matriix.
Sonneville, V., and O. Brüls. 2012. “Formulation of Kinematic Joints and Rigidity Constraints in Multibody Dynamics Using a Lie Group Approach.” In Proceedings of the 2nd Joint International Conference on Multibody System Dynamics. Stuttgart, Germany.