## 6 Numerical examples
### Example 2  
This is the case of a rectangular laminated plate with 12 plies, designed to be uncoupled and orthotropic in bending, with axes of orthotropy at \(0^{\circ}\) and \(90^{\circ}\); the plate is simply supported, with side lengths along the axes \(x\) and \(y\) respectively equal to \(a\) and \(b\). The additional requirement is that the frequency of the first mode of the bending vibrations must be greater than a minimum specified value \(\omega_{0}\). So, the objective function is still \(I(\mathbf{P})\), but now \(H_{ii}=1,\ i=9,10,17,18\), so that  
\[I(\mathbf{P})=P_{9}^{2}+P_{10}^{2}+P_{17}^{2}+P_{18}^{2}=0, \tag{26}\]  
and the problem is constrained by the condition  
\[g(\mathbf{P})=\omega_{0}-\omega_{pq}(\mathbf{P})\leq 0 \tag{27}\]  
where the frequency \(\omega_{pq}(\mathbf{P})\) of the mode having \(p\) and \(q\) half waves respectively along \(x\) and \(y\), is given by, see [23],  
\[\omega_{pq}^{2}=\frac{\pi^{4}\,p^{4}h^{3}}{12\sigma\ a^{4}}\sqrt{R_{0}^{2}+R_{ 1}^{2}}\left(1+\chi^{2}\right)^{2}\varphi(\xi_{0},\xi_{1})\,, \tag{28}\]  
with  
\[\varphi(\xi_{0},\xi_{1})=\tau+\frac{1}{\sqrt{1+\rho^{2}}}\Bigg{[}\left(-1 \right)^{K}\rho\xi_{0}\,\frac{\chi^{4}-6\chi^{2}+1}{\left(1+\chi^{2}\right)^{2 }}+4\xi_{1}\frac{1-\chi^{2}}{1+\chi^{2}}\Bigg{]}. \tag{30}\]  
\(h\) is the total thickness of the plate, \(\sigma\) its mass per unit area and the three dimensionless quantities \(\chi\), \(\rho\) and \(\tau\) represent respectively the geometric/modal, anisotropic and isotropic parts of the problem:  
Figure 5: Directional diagrams of \(A_{xx}\)*, \(D_{xx}\)*, almost superposed, and \(B_{xx}\)*, reduced to a dot in the centre (GPa).  
_Algorithms_**2009**, 2**  
\[\chi=\frac{qa}{pb},\quad\rho=\frac{R_{0}}{R_{1}},\quad\tau=\frac{T_{0}+2T_{1}}{ \sqrt{R_{0}^{2}+R_{1}^{2}}}. \tag{31}\]  
Moreover,  
\[\xi_{0}=\frac{1}{np^{3}}\sum\
olimits_{j=1}^{np}d_{j}\cos 4\delta_{j},\quad\xi_{1 }=\frac{1}{np^{3}}\sum\
olimits_{j=1}^{np}d_{j}\cos 2\delta_{j}, \tag{32}\]  
are the classical bending lamination parameters, the coefficients \(d_{j}\) being  
\[d_{j}=12j(j-np-1)+4+3 np(np+2). \tag{33}\]  
The results of the numerical tests for this example are shown in Table 4; they refer to the case of a plate with sides \(a\)=1 m, \(b\)=0.5 m, designed to have \(\omega_{11}\)\(\geq\)\(\omega_{0}\)= 150 s\({}^{-1}\); the mass of the plate per unit of area is \(\sigma\)=2.4 kg/m\({}^{2}\) (12 plies of carbon-epoxy T300-5208). The best result is obtained with the trajectory T3: [-66.15\({}^{\circ}\), 66.53\({}^{\circ}\), 44.83\({}^{\circ}\), -36.44\({}^{\circ}\), 48.52\({}^{\circ}\), -5.68\({}^{\circ}\), 77.57\({}^{\circ}\), -89.94\({}^{\circ}\), -84.48\({}^{\circ}\), 38.45\({}^{\circ}\), 57.48\({}^{\circ}\), -46.03\({}^{\circ}\)]. For this sequence, the objective function is very close to zero (7.20\(\times\)10\({}^{-35}\)) and the frequency \(\omega_{11}\)=162.72 s\({}^{-1}\). The elastic properties of this laminate are shown in Table 5 and in Fig. 6; they show that the laminate is really uncoupled (all the polar components of **B** are practically null) and \(K\)=1 orthotropic in bending, with orthotropy axes at 0\({}^{\circ}\) and 90\({}^{\circ}\) (\(\varPhi_{0}\)=45\({}^{\circ}\), \(\varPhi_{1}\)=90\({}^{\circ}\)); in addition, the extension behaviour is completely anisotropic, as a result of the fact that no requirement on **A** has been formulated.  
It is worth to recall that bending orthotropy together with uncoupling is a hard requirement to be obtained, cfr. [24]; this example shows that ALE-PSO can easily find uncoupled laminates that are orthotropic in bending.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{Test} & Run’s & Mean & Standard & Best & Worst & \multirow{2}{*}{\(n_{eval}\)} \\  & best & value & deviation & value & value & \\ \hline \multirow{3}{*}{T1} & average & 9.78\(\times\)10\({}^{-3}\) & 4.69\(\times\)10\({}^{-3}\) & 4.50\(\times\)10\({}^{-3}\) & 1.80\(\times\)10\({}^{-2}\) & 25510 \\  & particle & 1.04\(\times\)10\