# Frequency Response Analysis of Perforated Shells with Uncertain Materials and Damage
## 1 Introduction  
Vibration based structural health monitoring remains a vibrant research area with an extensive body of literature. A review article from 2004 by [PERSON] and others [1] alone cites over 150 publications. A more recent (2011) review article is by [PERSON] and [PERSON] [2] concentrating on damage identification methods. To use the language of inverse problems, modelling of damage identification methods requires the forward part, i.e., simulation of the mechanical systems we are interested in, and the inverse part or identification of the structure or changes within. In this paper, we give an overview of the issues one must consider when designing methods for vibration based health monitoring systems for perforated thin shells, especially in relation to frequency response analysis. Engineering constructions made of perforated materials are challenging from the computational mechanics point of view. Some examples of relevant industrial constructions are drying drums and Trommel screens. However, the literature on perforated materials is somewhat sparse. The introduction in [PERSON] et al. [3] serves as an excellent overview of the complexities of perforated shell designs--our interest--and their applications. It is natural to seek a simplified problem via homogenization, a process where the material properties of a non-perforated structure are adjusted to match those of the perforated one. Recently, [PERSON] and his collaborators have published a number of papers within the nuclear reactor design context, see [4; 5; 6]. For a detailed and exhaustive case study for a single cylindrical shell submerged in fluid, see [7]. In this paper, however, we do not advocate such an approach. The homogenization of shell structures under loading and especially in dynamic settings is not well understood. The main culprit is that this class of engineering problems is a parameter-dependent problem, where theparameter is the (dimensionless) thickness. It is characteristic of such problems that there are local features in the solution that vary in size and amplitude depending on the parameter and one can always define a critical value of the parameter where the local features become dominant, and homogenization fails. In frequency response analysis, there is an added complication. Let us consider shells of revolution. It follows immediately that in free vibration the eigenmodes must have integer valued wave numbers in the angular part of the mode. This wave number in turn depends on the parameters of the problem, including the shell geometry and the material parameters; see [8]. Thus, successful homogenization would require matching of modes which would be a more difficult problem than simply associating the first ten modes, say, in both cases. In addition, the dependence on the shell geometry means that it is not feasible to homogenize a perforated plate (a sheet of metal) and then ignore the effects of the curvature changes. Here, we are interested in the frequency response of perforated shells when either the material parameters such as Young's modulus or the structure or both are random in a well-defined way. The structural randomness is introduced via a simple damage model where the perforation patterns are perturbed with uniform intensity over the whole domain. We study two structures, a cylindrical shell which is of practical importance, and an example of different shell geometry, a perforated hyperbolic shell (cooling tower), which is more of an academic interest but serves to underline the central issues. It follows naturally that inclusion of uncertainty in the simulations leads to distribution of frequency responses that require statistical analysis [9]. The main contribution of this work can be divided into two interconnected parts: First, through understanding and analysis of the model problems, we can design a set of experiments that illustrate the expected phenomena, and, second, the experiments can be computed and the results analyzed within the standard stochastic framework.