MDKCs and Scattering Networks for Mindlin's System

We now introduce scaled dependent variables

$$(i_{1},i_{2},i_{3},i_{4},i_{5},i_{6},i_{7},i_{8}) = (r_{1}v,q_{x},q_{y},\omega_{x},\omega_{y},r_{2}m_{x},r_{2}m_{y}, r_{3}m_{xy})$$

where again, $r_{1}$, $r_{2}$ and $r_{3}$ are positive constants, as well as the scaled time variable $t' = v_{0}t$. The MD-passive circuit representation of Mindlin's system is shown in Figure 5.16, where we have used the coordinates defined by (3.21) with the transformation matrix (3.22). Here, the port with terminals $A$ and $A'$ is assumed to be short-circuited (we will return to this port in §5.5.2). It is best to view this as a three-loop network (on the left, in Figure 5.16) corresponding to the subsystem (5.31), coupled to a five-loop network on the right (subsystem (5.32)).

Because ${\bf P}_{M}$ from (5.33) is not diagonal, the coupling between the loops with currents $i_{6}$ and $i_{7}$ (corresponding to the moments $m_{x}$ and $m_{y}$) is of a type not previously encountered in the systems examined in this thesis. It can be interpreted in terms a coupled inductance between the loops (see §2.3.7); in Figure 5.16, self-inductances are indicated by directed arrows, and mutual inductance by bidirectional arrows. The element values are as indicated in the figure.

Optimal choices of $r_{1}$, $r_{2}$ and $r_{3}$, and the optimal stability bound on $v_{0}$ are a little more difficult to find in this case. As before, however, they follow from a positivity requirement on the inductance values defined in Figure 5.16. This requirement is simply applied to $L_{1}$, $L_{2}$, $L_{3}$, $L_{4}$, $L_{5}$ and $L_{8}$, but $L_{6}$ and $L_{7}$ define a coupled inductance between the loops with currents $i_{6}$ and $i_{7}$. The coupling matrix will be

$$\begin{bmatrix}L_{6}&L_{7}\\ L_{7}&L_{6}\\ \end{bmatrix}$$

and is required to be positive semi-definite for passivity. This is true if

$$L_{6}\geq \vert L_{7}\vert$$ (5.38)

An optimal choice for $r_{1}$ is easily shown to be

$$r_{1} = \sqrt{\frac{2}{\min_{x,y}(\rho h)\max_{x,y}(\kappa^{2}Gh)}}$$ (5.39)

and gives a first bound on $v_{0}$, which is

$$v_{0}\geq v_{M^{+}}\triangleq\sqrt{\frac{2\max_{x,y}(\kappa^{2}Gh)}{\min_{x,y}(\rho h)}}$$ (5.40)

Figure 5.16: MDKC and MDWD network for Mindlin's system.

Figure 5.17: Modified MDKC and multidimensional DWN for Mindlin's system.

From the positivity requirement on $L_{4}$, $L_{5}$, $L_{8}$, as well as condition (5.36), we have a second bound on $v_{0}$,

$$v_{0}\geq \max_{r2,r3>0}\left(\frac{12(r_{2}+r_{3})}{\min_{x,y}(\... ...{3}}{1-\nu})}{12r_{2}}, \frac{\max_{x,y}(\frac{Eh^{3}}{1+\nu})}{12r_{3}}\right)$$ (5.41)

This is a simple minimax-type problem--we would like to minimize the bound on $v_{0}$, which is the maximum of three quantities as per (5.39), with respect the parameters $r_{2}$ and $r_{3}$. The solution is

$$r_{2} = \frac{1}{12}\max_{x,y}\left(\frac{Eh^{3}}{1-\nu}\right)\sqrt{\fra... ...ax_{x,y}(\frac{Eh^{3}}{1-\nu})+\max_{x,y}(\frac{Eh^{3}}{1+\nu})}}\hspace{0.4in}$$ (5.42)

$$r_{3} = \frac{1}{12}\max_{x,y}\left(\frac{Eh^{3}}{1+\nu}\right)\sqrt{\fra... ...rho h^{3})}{\max_{x,y}(\frac{Eh^{3}}{1-\nu})+\max_{x,y}(\frac{Eh^{3}}{1+\nu})}}$$ (5.43)

which gives the second bound

$$v_{0}\geq v_{M^{-}}\triangleq \sqrt{\frac{\max_{x,y}(\frac{Eh^{3}}{1-\nu})+\max_{x,y}(\frac{Eh^{3}}{1-\nu})}{\min_{x,y}(\rho h^{3})}}$$ (5.44)

and the overall stability bound for the combined network will be

$$v_{0}\geq v_{M}\triangleq\max(v_{M^{-}}, v_{M^{+}})$$ (5.45)

When the material parameters and the thickness are constant, this bound reduces to

$$v_{0}\geq \sqrt{2}\max\left(\sqrt{\frac{G\kappa}{\rho}},\sqrt{\frac{E}{\rho(1-\nu^{2})}}\right) = \sqrt{2}\gamma_{M, max}^{g}$$

The MDWD network, shown at bottom in Figure 5.16, follows immediately from the MDKC; here, as for the parallel-plate problem discussed in §3.8.1, we have used step sizes $T_{j} = \Delta$, $j=1,\hdots,5$. Recall that because coordinate $t_{5} = t' = v_{0}t$, a step size of $T_{5} = \Delta$ implies a time step of $\Delta/v_{0} = T$, and we have indicated pure time delays of duration $T$ by ${\bf T}$. As for the Timoshenko network of Figure 5.6, reflection-free ports will be necessary due to the memoryless gyrator couplings between the loops with currents $i_{2}$ and $i_{4}$, and $i_{3}$ and $i_{5}$ in the MDKC. The coupled inductance has been treated as a vector scattering junction terminated on a vector inductor, as discussed in §2.3.7. We also note in passing that this network may be balanced in the same way as the Timoshenko system (see §5.2.6) in order to obtain a much better bound on $v_{0}$ (at the expense of increased network complexity).

It is also of course possible to put the MDKC into a form which yields, upon discretization, a DWN. This new form is shown in Figure 5.17; now the transverse velocity $v$ and bending moments $m_{x}$, $m_{y}$ and $m_{xy}$ are treated as voltages, and inductors in these loops are replaced by gyrators terminated on capacitances. In particular, the coupled inductance in Figure 5.16 is replaced by a coupled capacitance. In order to discretize this MDKC, we apply the trapezoid rule to all the inductances and capacitances with direction $t_{5}$ (using a step size of $T_{5} = \Delta$), and to the Jaumann two-ports, we make use of the alternative spectral mappings defined by (4.109), with step sizes $T_{j} = \Delta/2$, $j=1,\hdots,4$. We have chosen these step sizes such that an interleaved algorithm results; the computational grid is shown in Figure 5.18. Grid quantities (capitalized) are shown next to the points at which they are to be calculated. The grid on the right, which operates on grid functions $V$, $Q_{x}$ and $Q_{y}$ is identical to the grid for the DWN for the (2+1)D parallel-plate problem (see Figure 4.18), which is to be expected, since the related subnetwork of the MDKC shown in Figure 5.17 is the same as that for the parallel-plate problem (see Figure 4.49). It is coupled via gyrators (these couplings are indicated by curved arrows) to a second grid, over which grid functions $\Omega_{x}$, $\Omega_{y}$, $M_{x}$, $M_{y}$ and $M_{xy}$ are calculated. In particular, $M_{x}$ and $M_{y}$ are calculated together as a vector quantity at a vector parallel junction--this vector is written as ${\bf M}$ in Figure 5.18. Waveguide connections (of delay $T/2$) are represented by solid lines, and self-loops and sign-inversions in the signal paths are not shown. Note that at the grey dots, we will have parallel scattering junctions, and at the white dots we will have series junctions; junction quantities are calculated at alternating multiples of $T/2$.

Figure 5.18: Computational grid for the multidimensional DWN shown in Figure 5.17.