1 port flexible body
One-port (6x6) flexible body linear model
It computes the $(6 \times 6)$ linear dynamic model of a flexible body $\mathcal{A}$ between acceleration twist and wrench applied by this body to another (parent) body
- in its body frame $\mathcal{R}_a = (O,\mathbf{x}_a,\mathbf{y}_a,\mathbf{z}_a)$, $O$ is the reference point of $\mathcal{A}$,
- at the point $P$ : the connection point with another (parent) body
See also: 1-port NASTRAN flexible body, Multi-Port_rigid_body
Contents
Description
If the body was rigid, its direct dynamics model at point $P$ would read as (for notations, see the help of the block Multi-Port_rigid_body):
$$ \mathbf D^{\mathcal{A}}_P = \boldsymbol \tau_{GP}^T \left[\begin{array}{cc} m^{\mathcal{A}} \mathbf{1}_3 & \mathbf{0}_{3 \times 3} \\ \mathbf{0}_{3 \times 3} & \mathbf{I}^{\mathcal{A}}_G \end{array}\right]\boldsymbol \tau_{GP} $$
In case of a flexible body, the direct dynamic model $\mathbf{M}^{\mathcal{A}}_P(\mathrm{s})$ at point $P$ is described by hybrid-cantilever model:
$$ \left\{ \begin{array}{l} \left[\begin{array}{c} \mathbf{F}_{./\mathcal{A}} \\ \mathbf{T}_{./\mathcal{A},P} \end{array}\right] = \mathbf D^{\mathcal{A}}_P \left[\begin{array}{c} \mathbf{a}_{P} \\ \dot{\boldsymbol{\omega}} \end{array}\right] + \mathbf L_P^T \ddot{\boldsymbol \eta} \\ \ddot{\boldsymbol\eta} + \mbox{diag} (2\xi_i \omega_i)\dot{\boldsymbol\eta} + \mbox{diag} (\omega_i^2)\boldsymbol\eta = -\mathbf L_P \left[\begin{array}{c} \mathbf{a}_{P} \\ \dot{\boldsymbol{\omega}} \end{array}\right] \end{array} \right. $$ This model depends on:
- the frequencies $\omega_i$ and damping ration $\xi_i$ ($i=1,\cdots,N$) of the $N$ flexible modes,
- the $N \times 6$ matrix of modal participation factor $\mathbf{L}_P$,
- the "rigid" dynamic model $\mathbf D^{\mathcal{A}}_{P}$ of the body $\mathcal{A}$ expressed at point $P$.
A state-space representation of the direct dynamic model $\mathbf{M}^{\mathcal{A}}_P(\mathrm{s})$ is:
$$ \left\{\begin{array}{lcl} \left[\begin{array}{c} \dot{\boldsymbol\eta} \\ \ddot{\boldsymbol\eta} \end{array}\right] & = & \left[\begin{array}{cc} \mathbf{0}_{n \times n} & \mathbf{1}_n \\ -\mbox{diag}(\omega_i^2) & -\mbox{diag}(2 \xi_i \omega_i) \end{array} \right] \left[\begin{array}{c} \boldsymbol\eta \\ \dot{\boldsymbol\eta} \end{array}\right] + \left[\begin{array}{c} \mathbf{0}_{n \times 6} \\ -\mathbf L_P \end{array}\right] \left[\begin{array}{c} \mathbf{a}_{P} \\ \dot{\boldsymbol{\omega}} \end{array}\right] \\ \left[\begin{array}{c} \mathbf{F}_{./\mathcal{A}} \\ \mathbf{T}_{./\mathcal{A},P} \end{array}\right] & = & \left[\begin{array}{cc} -\mathbf L_P^T \mbox{diag}(\omega_i^2) & -\mathbf L_P^T \mbox{diag}(2 \xi_i \omega_i) \end{array} \right] \left[\begin{array}{c} \boldsymbol\eta \\ \dot{\boldsymbol\eta} \end{array}\right] + \underbrace{\left(\mathbf D^{\mathcal{A}}_P - \mathbf L_P^T \mathbf L_P\right)}_{\mathbf D^{\mathcal{A}}_{P0}}\left[\begin{array}{c} \mathbf{a}_{P} \\ \dot{\boldsymbol{\omega}} \end{array}\right] \end{array}\right. $$
with $\mathbf D^{\mathcal{A}}_{P0}$ the residual mass of the body $\mathcal{A}$ rigidly cantilevered to the other body at point $P$.
The block diagram, implementing $-\mathbf{M}^{\mathcal{A}}_P(\mathrm{s})$, is:
Ports
Input:
Output:
Parameters
All parameters are expressed in the inherit body frame $\mathcal{R}_a$. Each parameter can be an uncertain parameter declared with "ureal".
The main parameters are:
- the body mass (unit: $kg$)
- the $3 \times 3$ inertia matrix of the body expressed at its center of mass $G$ (unit of each term: $kg.m^2$)
- the $3\times 1$ vector $\overrightarrow{OG}$, the position of the center of mass $G$ in $\mathcal{R}_a$
- the $3\times 1$ vector $\overrightarrow{OP}$, the position of the connection point $P$ in $\mathcal{R}_a$
- the number $n$ of flexible modes (with $n \leq 10$)
- the damping ratio $\xi_i, \, i=\{1,... n\}$ of flexible modes (common for all flexible modes)
- the frequency $\omega_i, \, i=\{1,... n\}$ of each flexible mode (unit: $rd \, s^{-1}$)
- the modal participation factors at point $P$: $l_{ij,P}, \, i=\{1,... n\}, \, j= \{1,... 6\}$ both in translations (unit: $kg^{-1/2}$) and rotations (unit: $m.kg^{-1/2}$), expressed at point $P$ and in frame $\mathcal{R}_a$.
REMARK:
The block initialization function checks that the residual mass $\mathbf D^{\mathcal{A}}_{P0}$ is define positive for all the parametric configurations. That may be time consuming when the number of parametric uncertainties is high.
See also:
Simulink diagram under the mask