# Frequency Response Analysis of Perforated Shells with Uncertain Materials and Damage
## 4 Frequency Response Analysis under Uncertainty
### The Stochastic Eigenproblem  
As before, let \(D\subset\mathbb{R}^{2}\) denote the computational domain. Assume that the Young's modulus is a random field expressed in a series of the form  
\[E(x,y,\xi)=E_{0}(x,y)+\sum_{m=1}^{M}E_{m}(x,y)\xi_{m},\quad(x,y)\in D, \tag{25}\]  
where \(M\in\mathbb{N}\) and \(\xi=(\xi_{1},\xi_{2},\ldots,\xi_{M})\) is a vector of mutually independent random variables. For simplicity, we assume here that each \(\xi_{m}\) is uniformly distributed in some interval of \(\mathbb{R}\) and, therefore, after suitable rescaling, the random vector \(\xi\) takes values in the domain \(\Gamma:=[-1,1]^{M}\). A parametrization of the form (25) is standard in the literature and may be seen to result from a Karhunen-Loeve expansion of the underlying random field \(E(x,y,\xi)\), see e.g., [18]. We assume that \(E(x,y,\xi)\) is strictly uniformly positive and uniformly bounded, i.e., there exist \(E_{\min},E_{\max}>0\) such that  
\[E_{\min}\leq\operatorname*{ess\,inf}_{(x,y)\in D}E(x,y,\xi)\leq\operatorname* {ess\,sup}_{(x,y)\in D}E(x,y,\xi)\leq E_{\max}\quad\forall\xi\in\Gamma. \tag{26}\]  
Given an integrable function \(v=v(\xi)\), we denote by  
\[\mathbb{E}_{\Gamma}[v]=2^{-M}\int_{\Gamma}v(\xi)\;d\xi\]  
its expectation taken over the parameter space \(\Gamma\).  
Assuming the stochastic model (25) for the [PERSON]'s modulus, the eigenpairs of the free vibration problem now depend on \(\xi\in\Gamma\). The stochastic version of the eigenvalue problem is obtained simply by replacing the differential operators in Equation (7) with their stochastic counterparts and assuming that the resulting equation holds for every realization of \(\xi\in\Gamma\). In variational form, the problem reads: find functions \(\mathbf{u}(t,\cdot)\colon\Gamma\to V\) and \(\omega^{2}(t,\cdot)\colon\Gamma\to\mathbb{R}\) such that, for all \(\xi\in\Gamma\), we have  
\[ta_{m}(\xi;\mathbf{u}(t,\xi), \mathbf{v})+ta_{s}(\xi;\mathbf{u}(t,\xi),\mathbf{v})+t^{3}a_{b}( \xi;\mathbf{u}(t,\xi),\mathbf{v})\] \[=\omega^{2}(t,\xi)m(t;\mathbf{u}(t,\xi),\mathbf{v})\quad\forall \mathbf{v}\in V. \tag{27}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{a Regular} & \multicolumn{3}{c}{b Halton: \(H(7,37)\)} \\ \hline \(k\) & \(\lambda\) & \(K_{x}\) & \(K_{y}\) & \(k\) & \(\lambda\) & \(K_{x}\) & \(K_{y}\) \\ \hline
1 & 0.108012 & 3/2 & 5 & 1 & 0.106164 & 3/2 & 5 \\
2 & 0.108026 & 3/2 & 5 & 2 & 0.106785 & 3/2 & 5 \\
3 & 0.112789 & 2 & 5 & 3 & 0.110631 & 1 & 4 \\
4 & 0.112798 & 2 & 5 & 4 & 0.110981 & 1 & 4 \\
5 & 0.112905 & 1 & 4 & 5 & 0.113495 & 2 & 5 \\
6 & 0.112909 & 1 & 4 & 6 & 0.113902 & 2 & 5 \\
7 & 0.114934 & 3/2 & 4 & 7 & 0.114367 & 3/2 & 4 \\
8 & 0.114939 & 3/2 & 4 & 8 & 0.114711 & 3/2 & 4 \\
9 & 0.132725 & 2 & 6 & 9 & 0.131692 & 2 & 6 \\
10 & 0.132736 & 2 & 6 & 10 & 0.132256 & 2 & 6 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Hyperbolic. Parameters: \(\epsilon=2\pi/30\), \(t=1/100\); No material scaling.  
Here, \(a_{m}(\zeta;\cdot,\cdot)\), \(a_{s}(\zeta;\cdot,\cdot)\) and \(a_{b}(\zeta;\cdot,\cdot)\) are stochastic equivalents of the deterministic bilinear forms in Equation (8).  
After integration and assembly, we obtain the stochastic stiffness matrix \(\mathbf{S}(\xi)\), which can now be expressed as  
\[\mathbf{S}(\xi)=\mathbf{S}^{(0)}+\sum_{m=1}^{M}\mathbf{S}^{(m)}\xi_{m},\quad \xi\in\Gamma.\]  
Here, each matrix \(\mathbf{S}^{(m)}\) corresponds to a term in the series (25). Observe that the mass matrix \(\mathbf{M}\) does not depend on the parameter vector \(\xi\in\Gamma\).  
From the free vibration examples of the previous section, it should be clear that special care must be taken in matching of eigenmodes. When the problem is cast into the stochastic setting, the eigenvalues are bound to cross within the parameter space, and therefore it makes sense to match eigenmodes according to their axial and angular wavenumbers \(K_{x}\) and \(K_{y}\) rather than by simple increasing enumeration. This has previously been illustrated in [8; 9].