# Vibrational Analysis of Building Structures with Irregularities
## 1 Introduction
The concept of using power spectral density (PSD) to characterize randomness draws from multiple scattering studies [17; 18; 19], where fluctuations due to scatterers are described through auto-correlation, indicating the scattering cross-section. When the randomness is wide-sense stationary, the PSD is the Fourier transform of the auto-correlation, a result called the Wiener-Khintchine theorem [20; 21]. Specifically, [PERSON] [19] used the PSD of irregularities in an infinite elastic beam to estimate mean energy transfer through irregular regions, decomposing the beam's deflection into coherent and incoherent parts. [PERSON] then wrote the elastic beam equation as follows:  
\[B\frac{\partial^{4}w}{\partial x^{4}}+m(1+\xi(x))\frac{\partial^{2}w}{\partial t ^{2}}=0 \tag{1}\]  
where \(\xi\) is the fluctuation in the mass density, with the solution split into the following:  
\[w=\tilde{w}+w^{1}\]  
where \(\tilde{w}\) and \(w^{1}\) are the coherent and incoherent solutions, respectively. Figure 1: Schematic drawing of an example of the floor/ceiling structure. The upper plate (floor) and the ceiling are joined together by parallel joists. The cavity space is filled with sound-absorbing material, and the ceiling is connected by the steel batten s running perpendicular to the joists. In contrast to those previous methods, this paper constructs the total solution \(w\) using Fourier basis functions with evenly spaced spatial wave modes. Incorporating the Fourier transform of structural irregularities directly into the calculation corresponds, mathematically, to a convolution of the transformed irregularities with the beam or plate Fourier coefficients. This method allows us to quantify how the width of the spectral distribution influences the accuracy of our numerical solution, revealing a clear relationship between the spectral width and approximation precision. Traditional small perturbation theory is often applied to study structural irregularities, but it proves insufficient when structural elements interact with waves of similar length scales. As [PERSON] and [PERSON] noted [PERSON] and [PERSON] (1958), models that rely solely on averaged parameter values fail to accurately represent behavior at higher frequencies; these solutions simply do not reflect the actual structural dynamics. Statistical energy analysis (SEA) has become popular for high-frequency ranges, effectively handling thousands of modes. However, its applicability to the mid-frequency range remains uncertain, where the SEA's underlying assumption of a diffuse energy field are known not to hold; see e.g., [PERSON] and [PERSON] (1958). In this study, we extend the deterministic modeling in [PERSON] and [PERSON] (1958) to account for random irregularities in individual components and junctions, in contrast to SEA's approach of treating structures as interconnected elements subject to energy excitation and coupling. Our method aligns with hybrid approaches, such as those by [PERSON], [PERSON], and [PERSON] (1958); [PERSON], and [PERSON] (1958); [PERSON] and [PERSON] (1958), which express irregularities as modifications to the deterministic stiffness matrix. However, in our approach, the focus is on modeling structural deformation rather than energy flow. Consequently, the effects of random variations within structural components are derived from the Lagrangian, which captures the system's behavior in terms of deformation dynamics. This allows us to account for irregularities in the structure's form, grounding our analysis in the direct mechanics of the total structural deformation. The computational model underpinning this work is the specialized variation of the FEM introduced in [PERSON] (1958). Traditional FEM discretizes structural volumes into a mesh, formulating the system Lagrangian as a sum over the mesh elements. That process generates a matrix system to solve, with nodal motions across the mesh as the primary variables.