Port inversions of parametric Two-Input Two-Output Port models of flexible substructures

Abstract

In the context of multi-body modeling techniques, this paper details the port inversion operations, which can be performed on the Two-Input-Two-Output Port (TITOP) model of a flexible substructure to take into account various boundary conditions and to assemble it with the other substructures of the whole system. From the model obtained considering the body is clamped at a parent point and free at some others child points, it is shown that the dynamics obtained under the free-free, free-clamped, clamped-clamped boundary conditions can be characterized analytically. The clamped-clamped condition and its link with the Craig-Bampton formulation are more particularly investigated, since it is required to model closed-kinematic chains of flexible bodies, like in truss structures, using a sub-structuring approach. Furthermore, the channel inversion operations are extended to the case of parameter varying models. The model of a triangular mechanism, fully parameterized according to the cross section parameters of each mechanism beams, is developed and validated. This model is used to illustrate the capabilities of this parametric approach to import the model in multi-domain simulation software and the possibility to optimize directly the mechanism design.

Appendix: 6 DOFs TITOP model of a flexible beam

Let us consider the flexible beam $\mathcal{A}$ depicted in Fig. 12, characterized by its length $l$, width $b$, height $h$, Young modulus $E$, Poisson coefficient $\nu$ and mass density $\rho$.

Fig. 12

An homogeneous flexible beam $\mathcal{A}$

The 6 DOFs TITOP model summarized in the section is the direct extension of the 3 DOFs (planar) TITOP model presented [23]. It is composed of:

• the model ${\mathbf{D}}_{\mathit{PC}}^{\mathcal{A},t_y{r}_z}(\mathrm{s})$ of the bending dynamics in the $(\mathbf{x},\mathbf{y})$ plane with 4 flexible modes $\boldsymbol{\eta}_{t_{y}r_{z}}$ detailed in equation (30),

• the model ${\mathbf{D}}_{\mathit{PC}}^{\mathcal{A},t_z{r}_y}(\mathrm{s})$ of the bending dynamics in the $(\mathbf{x},\mathbf{z})$ plane with 4 flexible modes $\boldsymbol{\eta}_{t_{z}r_{z}}$ detailed in equation (33),

• the model ${\mathbf{D}}_{\mathit{PC}}^{\mathcal{A},t_x}(\mathrm{s})$ of the traction dynamics along the $\mathbf{x}$-axis with 1 flexible mode $\eta _{t_{x}}$ detailed in equation (34),

• the model ${\mathbf{D}}_{\mathit{PC}}^{\mathcal{A},r_x}(\mathrm{s})$ of the torsion dynamics around the $\mathbf{x}$-axis with 1 flexible mode $\eta _{r_{x}}$ detailed in equation (35).

The damping ratios of the flexible modes are ignored to lighten the following equations.

$$\begin{array}{rcl}\left[\begin{array}{c}{\dot{\mathit{\eta }}}_{t_y{r}_z}\\ {\ddot{\mathit{\eta }}}_{t_y{r}_z}\\ {\ddot{\mathbf{u}}}_C(2)\\ {\ddot{\mathbf{u}}}_C(6)\\ {\mathbf{W}}_{P/}(2)\\ {\mathbf{W}}_{P/}(6)\end{array}\right]& =& \left[\begin{array}{cccc}{\mathbf{0}}_4& {\mathbf{1}}_4& {\mathbf{0}}_{4\times 2}& {\mathbf{0}}_{4\times 2}\\ -I_z{\mathbf{M}}_{ff}^{-1}\mathbf{K}& {\mathbf{0}}_4& {\mathbf{M}}_{ff}^{-1}{\mathbf{\Phi }}^T& -{\mathbf{M}}_{ff}^{-1}{\mathbf{M}}_{rf}\\ -I_z\mathbf{\Phi }{\mathbf{M}}_{ff}^{-1}\mathbf{K}& {\mathbf{0}}_4& \mathbf{\Phi }{\mathbf{M}}_{ff}^{-1}{\mathbf{\Phi }}^T& \mathit{\tau }-\mathbf{\Phi }{\mathbf{M}}_{ff}^{-1}{\mathbf{M}}_{rf}\\ I_z{\mathbf{M}}_{rf}^T{\mathbf{M}}_{ff}^{-1}\mathbf{K}& {\mathbf{0}}_4& {(\mathit{\tau }-\mathbf{\Phi }{\mathbf{M}}_{ff}^{-1}{\mathbf{M}}_{rf})}^T& {\mathbf{M}}_{rf}^T{\mathbf{M}}_{ff}^{-1}{\mathbf{M}}_{rf}-{\mathbf{M}}_{rr}\end{array}\right]\\ & & \times \left[\begin{array}{c}{\mathit{\eta }}_{t_y{r}_z}\\ {\dot{\mathit{\eta }}}_{t_y{r}_z}\\ {\mathbf{W}}_{/C}(2)\\ {\mathbf{W}}_{/C}(6)\\ {\ddot{\mathbf{u}}}_P(2)\\ {\ddot{\mathbf{u}}}_P(6)\end{array}\right]\end{array}$$

(30)

with:

$$\left[\begin{array}{cc}{\mathbf{M}}_{rr}& {\mathbf{M}}_{rf}^T\\ {\mathbf{M}}_{rf}& {\mathbf{M}}_{ff}\end{array}\right]=\frac{\rho bhl}{55440}\left[\begin{array}{cccccc}55440& 27720l& 462l^2& 27720& -5544l& 462l^2\\ 27720l& 18480l^2& 198l^3& 19800l& -3432l^2& 264l^3\\ 462l^2& 198l^3& 6l^4& 181l^2& -52l^3& 5l^4\\ 27720& 19800l& 181l^2& 21720& -3732l& 281l^2\\ -5544l& -3432l^2& -52l^3& -3732l& 832l^2& -69l^3\\ 462l^2& 264l^3& 5l^4& 281l^2& -69l^3& 6l^4\end{array}\right]$$

(31)

$$I_z=h{b}^3/12,\mathbf{K}=\frac{E}{70l^3}\left[\begin{array}{cccc}6l^4& -30l^2& 8l^3& l^4\\ -30l^2& 1200& -600l& 30l^2\\ 8l^3& -600l& 384l^2& -22l^3\\ l^4& 30l^2& -22l^3& 6l^4\end{array}\right],\left[\begin{array}{c}{\mathit{\tau }}^T\\ {\mathbf{\Phi }}^T\end{array}\right]=\left[\begin{array}{cc}1& 0\\ l& 0\\ 0& 0\\ 1& 0\\ 0& 1\\ 0& 0\end{array}\right].$$

(32)

$$\begin{array}{rcl}\left[\begin{array}{c}{\dot{\mathit{\eta }}}_{t_z{r}_y}\\ {\ddot{\mathit{\eta }}}_{t_z{r}_y}\\ {\ddot{\mathbf{u}}}_C(3)\\ -{\ddot{\mathbf{u}}}_C(5)\\ {\mathbf{W}}_{P/}(3)\\ -{\mathbf{W}}_{P/}(5)\end{array}\right]& =& \left[\begin{array}{cccc}{\mathbf{0}}_4& {\mathbf{1}}_4& {\mathbf{0}}_{4\times 2}& {\mathbf{0}}_{4\times 2}\\ -I_y{\mathbf{M}}_{ff}^{-1}\mathbf{K}& {\mathbf{0}}_4& {\mathbf{M}}_{ff}^{-1}{\mathbf{\Phi }}^T& -{\mathbf{M}}_{ff}^{-1}{\mathbf{M}}_{rf}\\ -I_y\mathbf{\Phi }{\mathbf{M}}_{ff}^{-1}\mathbf{K}& {\mathbf{0}}_4& \mathbf{\Phi }{\mathbf{M}}_{ff}^{-1}{\mathbf{\Phi }}^T& \mathit{\tau }-\mathbf{\Phi }{\mathbf{M}}_{ff}^{-1}{\mathbf{M}}_{rf}\\ I_y{\mathbf{M}}_{rf}^T{\mathbf{M}}_{ff}^{-1}\mathbf{K}& {\mathbf{0}}_4& {(\mathit{\tau }-\mathbf{\Phi }{\mathbf{M}}_{ff}^{-1}{\mathbf{M}}_{rf})}^T& {\mathbf{M}}_{rf}^T{\mathbf{M}}_{ff}^{-1}{\mathbf{M}}_{rf}-{\mathbf{M}}_{rr}\end{array}\right]\\ & & \times \left[\begin{array}{c}{\mathit{\eta }}_{t_z{r}_y}\\ {\dot{\mathit{\eta }}}_{t_z{r}_y}\\ {\mathbf{W}}_{/C}(3)\\ -{\mathbf{W}}_{/C}(5)\\ {\ddot{\mathbf{u}}}_P(3)\\ -{\ddot{\mathbf{u}}}_P(5)\end{array}\right]\end{array}$$

(33)

with $I_{y}=bh^{3}/12$. The minus sign on inputs and outputs 2 and 4 takes into account that $\mathbf{x}\times \mathbf{z}=-\mathbf{y}$.

$$\left[\begin{array}{c}{\dot{\eta }}_{t_x}\\ {\ddot{\eta }}_{t_x}\\ {\ddot{\mathbf{u}}}_C(1)\\ {\mathbf{W}}_{P/}(1)\end{array}\right]=\left[\begin{array}{cccc}0& 1& 0& 0\\ -\frac{3E}{\rho l^2}& 0& \frac{3}{\rho bhl}& -\frac{3}{2}\\ -\frac{3E}{\rho l^2}& 0& \frac{3}{\rho bhl}& -\frac{1}{2}\\ \frac{3Ebh}{2}& 0& -\frac{1}{2}& -\frac{\rho bhl}{4}\end{array}\right]\left[\begin{array}{c}{\eta }_{t_x}\\ {\dot{\eta }}_{t_x}\\ {\mathbf{W}}_{/C}(1)\\ {\ddot{\mathbf{u}}}_P(1)\end{array}\right],$$

(34)

$$\left[\begin{array}{c}{\dot{\eta }}_{r_x}\\ {\ddot{\eta }}_{r_x}\\ {\ddot{\mathbf{u}}}_C(4)\\ {\mathbf{W}}_{P/}(4)\end{array}\right]=\left[\begin{array}{cccc}0& 1& 0& 0\\ -\frac{3G}{\rho l^2}& 0& \frac{3}{\rho I_{p}l}& -\frac{3}{2}\\ -\frac{3G}{\rho l^2}& 0& \frac{3}{\rho I_{p}l}& -\frac{1}{2}\\ \frac{3G{I}_p}{2l}& 0& -\frac{1}{2}& -\frac{\rho I_{p}l}{4}\end{array}\right]\left[\begin{array}{c}{\eta }_{r_x}\\ {\dot{\eta }}_{r_x}\\ {\mathbf{W}}_{/C}(4)\\ {\ddot{\mathbf{u}}}_P(4)\end{array}\right]$$

(35)

with $G=\frac{E}{2(1+\nu )}$ and $\displaystyle I_{p}=I_{z}+I_{y}$.

Then the $12\times 12$ TITOP model ${\mathbf{D}}_{\mathit{PC}}^{\mathcal{A}}(\mathrm{s})$ of the flexible beam is assembled according to Fig. 13.

Fig. 13

Block-diagram model of ${\mathbf{D}}_{\mathit{PC}}^{\mathcal{A}}(\mathrm{s})$ of a flexible beam