4.14.2. Theory Manual for the Tuned Mass Damper Module in OpenFAST

Author: William La Cava & Matthew A. Lackner Department of Mechanical and Industrial Engineering University of Massachusetts Amherst Amherst, MA 01003 wlacava@umass.edu, lackner@ecs.umass.edu

This document was edited by Jason M. Jonkman of NREL to include an independent vertically oriented TMD in OpenFAST. jason.jonkman@nrel.gov

This manual describes updated functionality in OpenFAST that simulates the addition of tuned mass dampers (TMDs) for structural control. The dampers can be added to the blades, nacelle, tower, or substructure. For application studies of these systems, refer to [stc-LR11a, stc-LR11b, stc-NRL13, stc-SL13, stc-SL11, stc-SL14]. The TMDs are three independent, 1 DOF, linear mass spring damping elements that act in the local \(x\), \(y\), and \(z\) coordinate systems of each component. The other functionality of the structural control (StC) module, including an omnidirectional TMD and TLCD are not documented herein. We first present the theoretical background and then describe the code changes.

4.14.2.1. Theoretical Background

4.14.2.1.1. Definitions

Table 4.7 Definitions
Variable	Description
\(O\)	origin point of global inertial reference frame
\(P\)	origin point of non-inertial reference frame fixed to component (blade, nacelle, tower, substructure) where TMDs are at rest
\(TMD\)	location point of a TMD
\(G\)	axis orientation of global reference frame
\(N\)	axis orientation of local component reference frame with unit vectors \(\hat{\imath}, \hat{\jmath}, \hat{k}\)
\(\vec{r}_{_{_{TMD/O_G}}} = \left[ \begin{array}{c} x \\ y\\ z \end{array} \right]_{_{TMD/O_G}}\)	position of a TMD with respect to (w.r.t.) \(O\) with orientation \(G\)
\(\vec{r}_{_{_{TMD/P_N}}} = \left[ \begin{array}{c} x \\ y\\ z \end{array} \right]_{_{TMD/P_N}}\)	position of a TMD w.r.t. \(P_N\)
\(\vec{r}_{_{_{TMD_X}}}\)	position vector for \(TMD_X\)
\(\vec{r}_{_{_{TMD_Y}}}\)	position vector for \(TMD_Y\)
\(\vec{r}_{_{_{TMD_Z}}}\)	position vector for \(TMD_Z\)
\(\vec{r}_{_{P/O_G}} =\left[ \begin{array}{c} x \\ y\\ z \end{array} \right]_{_{P/O_G}}\)	position vector of component w.r.t. \(O_G\)
\(R_{_{N/G}}\)	3 x 3 rotation matrix transforming orientation \(G\) to \(N\)
\(R_{_{G/N}} = R_{_{N/G}}^T\)	transformation from \(N\) to \(G\)
\(\vec{\omega}_{_{N/O_N}} = \dot{\left[ \begin{array}{c} \theta \\ \phi \\ \psi \end{array} \right]}_{_{N/O_N}}\)	angular velocity of component in orientation \(N\); defined likewise for \(G\)
\(\dot{\vec{\omega}}_{_{N/O_N}} = \vec{\alpha}_{_{N/O_N}}\)	angular acceleration of component
\(\vec{a}_{G/O_G} = \left[ \begin{array}{c}0 \\ 0\\ -g \end{array} \right]_{/O_G}\)	gravitational acceleration in global coordinates
\(\vec{a}_{G/O_N} = R_{_{N/G}} \vec{a}_{G/O_G} = \left[ \begin{array}{c}a_{_{G_X}} \\ a_{_{G_Y}}\\ a_{_{G_Z}} \end{array} \right]_{/O_N}\)	gravity w.r.t. \(O_N\)

4.14.2.1.2. Equations of motion

The position vectors of the TMDs in the two reference frames \(O\) and \(P\) are related by

\[\vec{r}_{_{TMD/O_G}} = \vec{r}_{_{P/O_G}} + \vec{r}_{_{TMD/P_G}}\]

Expressed in orientation \(N\),

\[\vec{r}_{_{TMD/O_N}} = \vec{r}_{_{P/O_N}} + \vec{r}_{_{TMD/P_N}}\]

\[\Rightarrow \vec{r}_{_{TMD/P_N}} = \vec{r}_{_{TMD/O_N}} - \vec{r}_{_{P/O_N}}\]

Differentiating, 1

\[\dot{\vec{r}}_{_{TMD/P_N}}= \dot{\vec{r}}_{_{TMD/O_N}} - \dot{\vec{r}}_{_{P/O_N}} - \vec{\omega}_{_{N/O_N}} \times \vec{r}_{_{TMD/P_N}}\]

differentiating again gives the acceleration of the TMD w.r.t. \(P\) (the nacelle position), oriented with \(N\):

(4.201)\[\begin{split}\begin{array}{cc} \ddot{\vec{r}}_{_{TMD/P_N}} = & \ddot{\vec{r}}_{_{TMD/O_N}} - \ddot{\vec{r}}_{_{P/O_N}} - \vec{\omega}_{_{N/O_N}} \times (\vec{\omega}_{_{N/O_N}} \times \vec{r}_{_{TMD/P_N}}) \\[1.1em] &- \vec{\alpha}_{_{N/O_N}} \times \vec{r}_{_{TMD/P_N}} - 2 \vec{\omega}_{_{N/O_N}} \times \dot{\vec{r}}_{_{TMD/P_N}} \end{array}\end{split}\]

The right-hand side contains the following terms:

Table 4.8 RHS terms
\(\ddot{\vec{r}}_{_{TMD/O_N}}\)	acceleration of the TMD in the inertial frame \(O_N\)
\(\ddot{\vec{r}}_{_{P/O_N}} = R_{_{N/G}} \ddot{\vec{r}}_{_{P/O_G}}\)	acceleration of the Nacelle origin \(P\) w.r.t. \(O_N\)
\(\vec{\omega}_{_{N/O_N}} = R_{_{N/G}} \vec{\omega}_{_{N/O_G}}\)	angular velocity of nacelle w.r.t. \(O_N\)
\(\vec{\omega}_{_{N/O_N}} \times (\vec{\omega}_{_{N/O_N}} \times \vec{r}_{_{TMD/P_N}})\)	Centripetal acceleration
\(\vec{\alpha}_{_{N/O_N}} \times \vec{r}_{_{TMD/P_N}}\)	Tangential acceleration
\(2\vec{\omega}_{_{N/O_N}} \times \dot{\vec{r}}_{_{TMD/P_N}}\)	Coriolis acceleration

The acceleration in the inertial frame \(\ddot{\vec{r}}_{_{TMD/O_N}}\) can be replaced with a force balance

\[\begin{split}\begin{aligned} \ddot{\vec{r}}_{_{TMD/O_N}} = \left[ \begin{array}{c} \ddot{x} \\ \ddot{y} \\ \ddot{z} \end{array} \right]_{_{TMD/O_N}} = \frac{1}{m} \left[ \begin{array}{c} \sum{F_X} \\ \sum{F_Y} \\ \sum{F_Z} \end{array} \right]_{_{TMD/O_N}} = \frac{1}{m} \vec{F}_{_{TMD/O_N}} \end{aligned}\end{split}\]

Substituting the force balance into Equation (4.201) gives the general equation of motion for a TMD:

(4.202)\[\begin{split}\begin{array}{cc} \ddot{\vec{r}}_{_{TMD/P_N}} = & \frac{1}{m} \vec{F}_{_{TMD/O_N}} - \ddot{\vec{r}}_{_{P/O_N}} - \vec{\omega}_{_{N/O_N}} \times (\vec{\omega}_{_{N/O_N}} \times \vec{r}_{_{TMD/P_N}}) \\[1.1em] & - \vec{\alpha}_{_{N/O_N}} \times \vec{r}_{_{TMD/P_N}} - 2 \vec{\omega}_{_{N/O_N}} \times \dot{\vec{r}}_{_{TMD/P_N}} \end{array}\end{split}\]

We will now solve the equations of motion for \(TMD_X\), \(TMD_Y\), and \(TMD_Z\).

4.14.2.1.2.1. TMD_X :

The external forces \(\vec{F}_{_{TMD_X/O_N}}\) are given by

\[\begin{split}\vec{F}_{_{TMD_X/O_N}} = \left[ \begin{array}{c} - c_x \dot{x}_{_{TMD_X/P_N}} - k_x x_{_{TMD_X/P_N}} + m_x a_{_{G_X/O_N}} + F_{ext_x} + F_{StopFrc_{X}} \\ F_{Y_{_{TMD_X/O_N}}} + m_x a_{_{G_Y/O_N}} \\ F_{Z_{_{TMD_X/O_N}}} + m_x a_{_{G_Z/O_N}} \end{array} \right]\end{split}\]

\(TMD_X\) is fixed to frame \(N\) in the \(y\) and \(z\) directions so that

\[\begin{split}{r}_{_{TMD_X/P_N}} = \left[ \begin{array}{c} x_{_{TMD_X/P_N}} \\ 0 \\ 0 \end{array} \right]\end{split}\]

The other components of Eqn. (4.202) are:

\[\begin{split}\vec{\omega}_{_{N/O_N}} \times (\vec{\omega}_{_{N/O_N}} \times \vec{r}_{_{TMD_X/P_N}}) = x_{_{TMD_X/P_N}} \left[ \begin{array}{c} - (\dot{\phi}_{_{N/O_N}}^2 + \dot{\psi}_{_{N/O_N}}^2) \\ \dot{\theta}_{_{N/O_N}}\dot{\phi}_{_{N/O_N}} \\ \dot{\theta}_{_{N/O_N}}\dot{\psi}_{_{N/O_N}} \end{array} \right]\end{split}\]

\[\begin{split}2\vec{\omega}_{_{N/O_N}} \times \dot{\vec{r}}_{_{TMD_X/P_N}} = \dot{x}_{_{TMD_X/P_N}} \left[ \begin{array}{c} 0 \\ 2\dot{\psi}_{_{N/O_N}} \\ -2\dot{\phi}_{_{N/O_N}} \end{array} \right]\end{split}\]

\[\begin{split}\vec{\alpha}_{_{N/O_N}} \times \vec{r}_{_{TMD_X/P_N}} = x_{_{TMD_X/P_N}} \left[ \begin{array}{c} 0 \\ \ddot{\psi}_{_{N/O_N}} \\ -\ddot{\phi}_{_{N/O_N}}\end{array} \right]\end{split}\]

Therefore \(\ddot{x}_{_{TMD_X/P_N}}\) is governed by the equations

(4.203)\[\begin{split}\begin{aligned} \ddot{x}_{_{TMD_X/P_N}} =& (\dot{\phi}_{_{N/O_N}}^2 + \dot{\psi}_{_{N/O_N}}^2-\frac{k_x}{m_x}) x_{_{TMD_X/P_N}} - (\frac{c_x}{m_x}) \dot{x}_{_{TMD_X/P_N}} -\ddot{x}_{_{P/O_N}}+a_{_{G_X/O_N}} \\ &+ \frac{1}{m_x} ( F_{ext_X} + F_{StopFrc_{X}}) \end{aligned}\end{split}\]

The forces \(F_{Y_{_{TMD_X/O_N}}}\) and \(F_{Z_{_{TMD_X/O_N}}}\) are solved noting \(\ddot{y}_{_{TMD_X/P_N}} = \ddot{z}_{_{TMD_X/P_N}} = 0\):

(4.204)\[F_{Y_{_{TMD_X/O_N}}} = m_x \left( - a_{_{G_Y/O_N}} +\ddot{y}_{_{P/O_N}} + (\ddot{\psi}_{_{N/O_N}} + \dot{\theta}_{_{N/O_N}}\dot{\phi}_{_{N/O_N}} ) x_{_{TMD_X/P_N}} + 2\dot{\psi}_{_{N/O_N}} \dot{x}_{_{TMD_X/P_N}} \right)\]

(4.205)\[F_{Z_{_{TMD_X/O_N}}} = m_x \left( - a_{_{G_Z/O_N}} +\ddot{z}_{_{P/O_N}} - (\ddot{\phi}_{_{N/O_N}} - \dot{\theta}_{_{N/O_N}}\dot{\psi}_{_{N/O_N}} ) x_{_{TMD_X/P_N}} - 2\dot{\phi}_{_{N/O_N}} \dot{x}_{_{TMD_X/P_N}} \right)\]

4.14.2.1.2.2. TMD_Y:

The external forces \(\vec{F}_{_{TMD_Y/P_N}}\) on \(TMD_Y\) are given by

\[\begin{split}\vec{F}_{_{TMD_Y/P_N}} = \left[ \begin{array}{c} F_{X_{_{TMD_Y/O_N}}} + m_y a_{_{G_X/O_N}}\\ - c_y \dot{y}_{_{TMD_Y/P_N}} - k_y y_{_{TMD_Y/P_N}} + m_y a_{_{G_Y/O_N}} + F_{ext_y} + F_{StopFrc_{Y}} \\ F_{Z_{_{TMD_Y/O_N}}}+ m_y a_{_{G_Z/O_N}} \end{array} \right]\end{split}\]

\(TMD_Y\) is fixed to frame \(N\) in the \(x\) and \(z\) directions so that

\[\begin{split}{r}_{_{TMDYX/P_N}} = \left[ \begin{array}{c} 0 \\ y_{_{TMD_Y/P_N}} \\ 0 \end{array} \right]\end{split}\]

The other components of Eqn. (4.202) are:

\[\begin{split}\vec{\omega}_{_{N/O_N}} \times (\vec{\omega}_{_{N/O_N}} \times \vec{r}_{_{TMD_Y/P_N}}) = y_{_{TMD_Y/P_N}} \left[ \begin{array}{c} \dot{\theta}_{_{N/O_N}}\dot{\phi}_{_{N/O_N}} \\ -(\dot{\theta}_{_{N/O_N}}^2 + \dot{\psi}_{_{N/O_N}}^2) \\ \dot{\phi}_{_{N/O_N}}\dot{\psi}_{_{N/O_N}} \end{array} \right]\end{split}\]

\[\begin{split}2\vec{\omega}_{_{N/O_N}} \times \dot{\vec{r}}_{_{TMD_Y/P_N}} = \dot{y}_{_{TMD_Y/P_N}} \left[ \begin{array}{c} - 2 \dot{\psi}_{_{N/O_N}} \\ 0 \\ 2 \dot{\theta}_{_{N/O_N}} \end{array} \right]\end{split}\]

\[\begin{split}\vec{\alpha}_{_{N/O_N}} \times \vec{r}_{_{TMD_Y/P_N}} = y_{_{TMD_Y/P_N}} \left[ \begin{array}{c} - \ddot{\psi}_{_{N/O_N}} \\ 0 \\ \ddot{\theta}_{_{N/O_N}} \end{array} \right]\end{split}\]

Therefore \(\ddot{y}_{_{TMD_Y/P_N}}\) is governed by the equations

(4.206)\[\begin{split}\begin{aligned} \ddot{y}_{_{TMD_Y/P_N}} = & (\dot{\theta}_{_{N/O_N}}^2 + \dot{\psi}_{_{N/O_N}}^2-\frac{k_y}{m_y}) y_{_{TMD_Y/P_N}} - (\frac{c_y}{m_y}) \dot{y}_{_{TMD_Y/P_N}} -\ddot{y}_{_{P/O_N}} + a_{_{G_Y/O_N}}\\ &+ \frac{1}{m_y} (F_{ext_Y} + F_{StopFrc_{Y}}) \end{aligned}\end{split}\]

The forces \(F_{X_{_{TMD_Y/O_N}}}\) and \(F_{Z_{_{TMD_Y/O_N}}}\) are solved noting \(\ddot{x}_{_{TMD_Y/P_N}} = \ddot{z}_{_{TMD_Y/P_N}} = 0\):

(4.207)\[F_{X_{_{TMD_Y/O_N}}} = m_y \left( - a_{_{G_X/O_N}} + \ddot{x}_{_{P/O_N}} - (\ddot{\psi}_{_{N/O_N}} - \dot{\theta}_{_{N/O_N}}\dot{\phi}_{_{N/O_N}}) y_{_{TMD_Y/P_N}} - 2\dot{\psi}_{_{N/O_N}} \dot{y}_{_{TMD_Y/P_N}} \right)\]

(4.208)\[F_{Z_{_{TMD_Y/O_N}}} = m_y \left( - a_{_{G_Z/O_N}} + \ddot{z}_{_{P/O_N}} + (\ddot{\theta}_{_{N/O_N}} + \dot{\phi}_{_{N/O_N}}\dot{\psi}_{_{N/O_N}}) y_{_{TMD_Y/P_N}} + 2\dot{\theta}_{_{N/O_N}} \dot{y}_{_{TMD_Y/P_N}} \right)\]

4.14.2.1.2.3. TMD_Z :

The external forces \(\vec{F}_{_{TMD_Z/O_N}}\) are given by

\[\begin{split}\vec{F}_{_{TMD_Z/O_N}} = \left[ \begin{array}{c} F_{X_{_{TMD_Z/O_N}}} + m_z a_{_{G_X/O_N}} \\ F_{Y_{_{TMD_Z/O_N}}} + m_z a_{_{G_Y/O_N}} \\ - c_z \dot{z}_{_{TMD_Z/P_N}} - k_z z_{_{TMD_Z/P_N}} + m_z a_{_{G_Z/O_N}} + F_{ext_z} + F_{StopFrc_{Z}} + F_{Z_{PreLoad}} \end{array} \right]\end{split}\]

where \(F_{Z_{PreLoad}}\) is a spring pre-load to shift the neutral position when gravity acts upon the mass for the \(TMD_Z\). \(TMD_Z\) is fixed to frame \(N\) in the \(x\) and \(y\) directions so that

\[\begin{split}{r}_{_{TMD_Z/P_N}} = \left[ \begin{array}{c} 0 \\ 0 \\ z_{_{TMD_Z/P_N}} \end{array} \right]\end{split}\]

The other components of Eqn. (4.202) are:

\[\begin{split}\vec{\omega}_{_{N/O_N}} \times (\vec{\omega}_{_{N/O_N}} \times \vec{r}_{_{TMD_Z/P_N}}) = z_{_{TMD_Z/P_N}} \left[ \begin{array}{c} \dot{\theta}_{_{N/O_N}}\dot{\psi}_{_{N/O_N}} \\ \dot{\phi}_{_{N/O_N}}\dot{\psi}_{_{N/O_N}} \\ -(\dot{\theta}_{_{N/O_N}}^2 + \dot{\phi}_{_{N/O_N}}^2) \end{array} \right]\end{split}\]

\[\begin{split}2\vec{\omega}_{_{N/O_N}} \times \dot{\vec{r}}_{_{TMD_Z/P_N}} = \dot{z}_{_{TMD_Z/P_N}} \left[ \begin{array}{c} 2\dot{\phi}_{_{N/O_N}} \\ -2\dot{\theta}_{_{N/O_N}} \\ 0 \end{array} \right]\end{split}\]

\[\begin{split}\vec{\alpha}_{_{N/O_N}} \times \vec{r}_{_{TMD_Z/P_N}} = z_{_{TMD_Z/P_N}} \left[ \begin{array}{c} \ddot{\phi}_{_{N/O_N}} \\ -\ddot{\theta}_{_{N/O_N}} \\ 0 \end{array} \right]\end{split}\]

Therefore \(\ddot{z}_{_{TMD_Z/P_N}}\) is governed by the equations

(4.209)\[\begin{split}\begin{aligned} \ddot{z}_{_{TMD_Z/P_N}} = & (\dot{\theta}_{_{N/O_N}}^2 + \dot{\phi}_{_{N/O_N}}^2-\frac{k_z}{m_z}) z_{_{TMD_Z/P_N}} - (\frac{c_z}{m_z}) \dot{z}_{_{TMD_Z/P_N}} -\ddot{z}_{_{P/O_N}} + a_{_{G_Z/O_N}}\\ &+ \frac{1}{m_z} (F_{ext_Z} + F_{StopFrc_{Z}} + F_{Z_{PreLoad}}) \end{aligned}\end{split}\]

The forces \(F_{X_{_{TMD_Z/O_N}}}\) and \(F_{Z_{_{TMD_Z/O_N}}}\) are solved noting \(\ddot{x}_{_{TMD_Z/P_N}} = \ddot{y}_{_{TMD_Z/P_N}} = 0\):

(4.210)\[F_{X_{_{TMD_Z/O_N}}} = m_z \left( - a_{_{G_X/O_N}} + \ddot{x}_{_{P/O_N}} + (\ddot{\phi}_{_{N/O_N}} + \dot{\theta}_{_{N/O_N}}\dot{\psi}_{_{N/O_N}}) z_{_{TMD_Z/P_N}} + 2\dot{\phi}_{_{N/O_N}} \dot{z}_{_{TMD_Z/P_N}} \right)\]

(4.211)\[F_{Y_{_{TMD_Z/O_N}}} = m_z \left( - a_{_{G_Y/O_N}} + \ddot{y}_{_{P/O_N}} - (\ddot{\theta}_{_{N/O_N}} - \dot{\phi}_{_{N/O_N}}\dot{\psi}_{_{N/O_N}}) z_{_{TMD_Z/P_N}} - 2\dot{\theta}_{_{N/O_N}} \dot{z}_{_{TMD_Z/P_N}} \right)\]

4.14.2.1.3. State Equations

4.14.2.1.3.1. Inputs:

The inputs are the component linear acceleration and angular position, velocity and acceleration:

\[\begin{split}\vec{u} = \left[ \begin{array}{c} \ddot{\vec{r}}_{_{P/O_G}} \\ \vec{R}_{_{N/G}} \\ \vec{\omega}_{_{N/O_G}} \\ \vec{\alpha}_{_{N/O_G}} \end{array} \right] \Rightarrow \left[ \begin{array}{c} \ddot{\vec{r}}_{_{P/O_N}} \\ \vec{\omega}_{_{N/O_N}} \\ \vec{\alpha}_{_{N/O_N}} \end{array} \right] = \left[ \begin{array}{c} \vec{R}_{_{N/G}} \ddot{\vec{r}}_{_{P/O_G}} \\ \vec{R}_{_{N/G}} \vec{\omega}_{_{N/O_G}} \\ \vec{R}_{_{N/G}} \vec{\alpha}_{_{N/O_G} }\end{array} \right]\end{split}\]

4.14.2.1.3.2. States:

The states are the position and velocity of the TMDs along their respective DOFs in the component reference frame:

\[\begin{split}\vec{R}_{_{TMD/P_N}} = \left[ \begin{array}{c} x \\ \dot{x} \\ y \\ \dot{y} \\ z \\ \dot{z} \end{array} \right]_{_{TMD/P_N}} = \left[ \begin{array}{c} {x}_{_{TMD_X/P_N}} \\ \dot{x}_{_{TMD_X/P_N}} \\ {y}_{_{TMD_Y/P_N}} \\ \dot{y}_{_{TMD_Y/P_N}} \\ {z}_{_{TMD_Z/P_N}} \\ \dot{z}_{_{TMD_Z/P_N}} \end{array} \right]\end{split}\]

The equations of motion can be re-written as a system of non-linear first-order equations of the form

\[\dot{\vec{R}}_{_{TMD}} = A \vec{R}_{_{TMD}} + B\]

where

\[\begin{split}A(\vec{u}) = \left[ \begin{array}{cccccc} 0& 1 &0&0&0&0 \\ (\dot{\phi}_{_{P/O_N}}^2 + \dot{\psi}_{_{P/O_N}}^2-\frac{k_x}{m_x}) & - (\frac{c_x}{m_x}) &0&0&0&0 \\ 0&0&0& 1 &0&0 \\ 0&0& (\dot{\theta}_{_{P/O_N}}^2 + \dot{\psi}_{_{P/O_N}}^2-\frac{k_y}{m_y}) & - (\frac{c_y}{m_y}) &0&0 \\ 0&0&0&0&0& 1 \\ 0&0&0&0& (\dot{\theta}_{_{P/O_N}}^2 + \dot{\phi}_{_{P/O_N}}^2-\frac{k_z}{m_z}) & - (\frac{c_z}{m_z}) \\ \end{array} \right]\end{split}\]

and

(4.212)\[\begin{split}B(\vec{u}) = \left[ \begin{array}{l} 0 \\ -\ddot{x}_{_{P/O_N}}+a_{_{G_X/O_N}} + \frac{1}{m_x} ( F_{ext_X} + F_{StopFrc_{X}}) \\ 0 \\ -\ddot{y}_{_{P/O_N}}+a_{_{G_Y/O_N}} + \frac{1}{m_y} (F_{ext_Y}+ F_{StopFrc_{Y}}) \\ 0 \\ -\ddot{z}_{_{P/O_N}}+a_{_{G_Z/O_N}} + \frac{1}{m_z} (F_{ext_Z}+ F_{StopFrc_{Z}} + F_{Z_{PreLoad}}) \end{array} \right]\end{split}\]

The inputs are coupled to the state variables, resulting in A and B as \(f(\vec{u})\).

4.14.2.1.4. Outputs

The output vector \(\vec{Y}\) is

\[\begin{split}\vec{Y} = \left[ \begin{array}{c} \vec{F}_{_{P_G}} \\ \vec{M}_{_{P_G}} \end{array} \right]\end{split}\]

The output includes reaction forces corresponding to \(F_{Y_{_{TMD_X/O_N}}}\), \(F_{Z_{_{TMD_X/O_N}}}\), \(F_{X_{_{TMD_Y/O_N}}}\), \(F_{Z_{_{TMD_Y/O_N}}}\), \(F_{X_{_{TMD_Z/O_N}}}\), and \(F_{Y_{_{TMD_Z/O_N}}}\) from Eqns. (4.204), (4.205), (4.207), (4.208), (4.210), and (4.211). The resulting forces \(\vec{F}_{_{P_G}}\) and moments \(\vec{M}_{_{P_G}}\) acting on the component are

(4.213)\[\begin{split}\begin{aligned} \vec{F}_{_{P_G}} = R^T_{_{N/G}} & \left[ \begin{array}{l} k_x {x}_{_{TMD/P_N}} + c_x \dot{x}_{_{TMD/P_N}} - F_{StopFrc_{X}} - F_{ext_x} - F_{X_{_{TMD_Y/O_N}}} - F_{X_{_{TMD_Z/O_N}}} \\ k_y {y}_{_{TMD/P_N}} + c_y \dot{y}_{_{TMD/P_N}} - F_{StopFrc_{Y}} - F_{ext_y} - F_{Y_{_{TMD_X/O_N}}} - F_{Y_{_{TMD_Z/O_N}}} \\ k_z {z}_{_{TMD/P_N}} + c_z \dot{z}_{_{TMD/P_N}} - F_{StopFrc_{Z}} - F_{ext_z} - F_{Z_{_{TMD_X/O_N}}} - F_{Z_{_{TMD_Y/O_N}}} - F_{Z_{PreLoad}} \end{array} \right] \end{aligned}\end{split}\]

and

\[\begin{split}\vec{M}_{_{P_G}} = R^T_{_{N/G}} \left[ \begin{array}{c} M_{_X} \\ M_{_Y} \\ M_{_Z} \end{array} \right]_{_{N/N}} = R^T_{_{N/G}} \left[ \begin{array}{c} -(F_{Z_{_{TMD_Y/O_N}}}) y_{_{TMD/P_N}} + (F_{Y_{_{TMD_Z/O_N}}} ) z_{_{TMD/P_N}} \\ (F_{Z_{_{TMD_X/O_N}}}) x_{_{TMD/P_N}} - (F_{X_{_{TMD_Z/O_N}}} ) z_{_{TMD/P_N}} \\ -(F_{Y_{_{TMD_X/O_N}}}) x_{_{TMD/P_N}} + ( F_{X_{_{TMD_Y/O_N}}}) y_{_{TMD/P_N}} \end{array} \right]\end{split}\]

4.14.2.1.4.1. Stop Forces

The extra forces \(F_{StopFrc_{X}}\), \(F_{StopFrc_{Y}}\), and \(F_{StopFrc_{Z}}\) are added to output forces in the case that the movement of TMD_X, TMD_Y, or TMD_Z exceeds the maximum track length for the mass. Otherwise, they equal zero. The track length has limits on the positive and negative ends in the TMD direction (X_PSP and X_NSP, Y_PSP and Y_NSP, and Z_PSP and Z_NSP). If we define a general maximum and minimum displacements as \(x_{max}\) and \(x_{min}\), respectively, the stop forces have the form

\[\begin{split}F_{StopFrc} = -\left\{ \begin{array}{lr} \begin{aligned} k_S \Delta x & \quad : ( x > x_{max} \wedge \dot{x}<=0) \vee ( x < x_{min} \wedge \dot{x}>=0)\\ k_S \Delta x + c_S \dot{x} & \quad : ( x > x_{max} \wedge \dot{x}>0) \vee ( x < x_{min} \wedge \dot{x}<0)\\ 0 & \quad : \text{otherwise} \end{aligned} \end{array} \right.\end{split}\]

where \(\Delta x\) is the distance the mass has traveled beyond the stop position and \(k_S\) and \(c_S\) are large stiffness and damping constants.

4.14.2.1.4.2. Pre-Load Forces

The extra force \(F_{Z_{PreLoad}}\) is added to the output forces as a method to shift the at rest position of the TMD_Z when gravity is acting on it. This is particularly useful for substructure mounted StCs when very large masses with soft spring constants are used. This appears in the term \(\vec{F}_{_{TMD_Z/O_N}}\) and in eq equations of motion given by (4.212) and resulting forces in (4.213).

4.14.2.2. Code Modifications

The Structural Control (StC) function is a submodule linked into ServoDyn. In addition to references in ServoDyn.f90 and ServoDyn.txt, new files that contain the StC module are listed below.

4.14.2.2.1. New Files

StrucCtrl.f90 : the structural control module
StrucCtrl.txt : registry file include files, inputs, states, parameters, and outputs shown in Tables 1 and 2
StrucCtrl_Types.f90: automatically generated

4.14.2.2.2. Variables

Table 4.9 Summary of field definitions in the StC registry. Note that state vector \(\vec{tmd_x}\) corresponds to \(\vec{R}_{_{TMD/P_N}}\), and that the outputs \(\vec{F}_{_{P_G}}\) and \(\vec{M}_{_{P_G}}\) are contained in the MeshType object (y.Mesh). \(X_{DSP}\), \(Y_{DSP}\), and \(Z_{DSP}\) are initial displacements of the TMDs.
DataType	Variable name
DataType	Variable name
InitInput
	InputFile
	Gravity
	\(\vec{r}_{_{N/O_G}}\)
Input u
	\(\ddot{\vec{r}}_{_{P/O_G}}\)
	\(\vec{R}_{_{N/O_G}}\)
	\(\vec{\omega}_{_{N/O_G}}\)
	\(\vec{\alpha}_{_{N/O_G}}\)
Parameter p
	\(m_x\)
	\(c_x\)
	\(k_x\)
	\(m_y\)
	\(c_y\)
	\(k_y\)
	\(m_z\)
	\(c_z\)
	\(k_z\)
	\(K_S = \left[ k_{SX}\hspace{1em}k_{SY}\hspace{1em}k_{SZ}\right]\)
	\(C_S = \left[c_{SX}\hspace{1em}c_{SY}\hspace{1em}c_{SZ}\right]\)
	\(P_{SP}=\left[X_{PSP}\hspace{1em}Y_{PSP}\hspace{1em}Z_{PSP}\right]\)
	\(P_{SP}=\left[X_{NSP}\hspace{1em}Y_{NSP}\hspace{1em}Z_{NSP}\right]\)
	\(F{ext}\)
	\(Gravity\)
	TMDX_DOF
	TMDY_DOF
	TMDZ_DOF
	\(X_{DSP}\)
	\(Y_{DSP}\)
	\(Z_{DSP}\)
State x
	\(\vec{tmd_x}\)
Output y
	Mesh

The input, parameter, state and output definitions are summarized in Table 1. The inputs from file are listed in Table 2.

Table 4.10 Data read in from TMDInputFile.
Field Name	Field Type	Description
TMD_CMODE	int	Control Mode (1:passive, 2:active)
TMD_X_DOF	logical	DOF on or off
TMD_Y_DOF	logical	DOF on or off
TMD_Z_DOF	logical	DOF on or off
TMD_X_DSP	real	TMD_X initial displacement
TMD_Y_DSP	real	TMD_Y initial displacement
TMD_Z_DSP	real	TMD_Z initial displacement
TMD_X_M	real	TMD mass
TMD_X_K	real	TMD stiffness
TMD_X_C	real	TMD damping
TMD_Y_M	real	TMD mass
TMD_Y_K	real	TMD stiffness
TMD_Y_C	real	TMD damping
TMD_Z_M	real	TMD mass
TMD_Z_K	real	TMD stiffness
TMD_Z_C	real	TMD damping
TMD_X_PSP	real	positive stop position (maximum X mass displacement)
TMD_X_NSP	real	negative stop position (minimum X mass displacement)
TMD_X_K_SX	real	stop spring stiffness
TMD_X_C_SX	real	stop spring damping
TMD_Y_PSP	real	positive stop position (maximum Y mass displacement)
TMD_Y_NSP	real	negative stop position (minimum Y mass displacement)
TMD_Y_K_S	real	stop spring stiffness
TMD_Y_C_S	real	stop spring damping
TMD_Z_PSP	real	positive stop position (maximum Z mass displacement)
TMD_Z_NSP	real	negative stop position (minimum Z mass displacement)
TMD_Z_K_S	real	stop spring stiffness
TMD_Z_C_S	real	stop spring damping
TMD_P_X	real	x origin of P in nacelle coordinate system
TMD_P_Y	real	y origin of P in nacelle coordinate system
TMD_P_Z	real	z origin of P in nacelle coordinate system

4.14.2.3. Acknowledgements

The authors would like to thank Dr. Jason Jonkman for reviewing this manual.

1: Note that \(( R a ) \times ( Rb ) = R( a \times b )\).