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

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:

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:

  • Xddot: the acceleration twist $\left[\begin{array}{c} \mathbf{a}_P \\ \boldsymbol{\dot{\omega}}\end{array}\right]_{\mathcal{R}_a}$ (units: $m/s^2$ and $rd/s^2$) at point $P$ in the inherit body frame $\mathcal{R}_a$.
  • Output:

  • W_body/.: the wrench $\left[\begin{array}{c} \mathbf{F}_{\mathcal{A/.}} \\ \mathbf{T}_{\mathcal{A/.},P} \end{array}\right]_{\mathcal{R}_a}$ (units: $N$ and $N.m$) applied by the body $\mathcal{A}$ at point $P$ to another body.
  • 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:

    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