# Vibrational Analysis of Building Structures with Irregularities
## 3 Method of Solution  
### Deterministic Modeling  
This section presents our deterministic model, where structural deflections are then obtained by minimizing the first variation of the structure's Lagrangian. Fourier expansions for the deflection of individual components simplifies computations by diagonalizing differential operators and representing deflections across components within a unified framework. Structural irregularities can then be effectively represented in a Fourier basis. By expressing these irregularities--such as variations in geometry, stiffness, and mass density--through their Fourier components, we can incorporate them seamlessly into our model. Analysis using the finite element method (FEM) traditionally approximates structural deformation through stiffness matrices derived from the Lagrangian's minimization, which requires solving a system of linear equations such as the following:  
\[M\mathbf{c}=\mathbf{f} \tag{2}\]  
where \(\mathbf{c}\) is the deformation vector, \(\mathbf{f}\) the excitation vector, and \(M\) the matrix determined by the strain and kinetic energy integrals. As frequency increases, the degrees of freedom grow, requiring a larger \(M\). In contrast, our approach uses spectral elements in the Fourier basis, making the Fourier coefficients themselves the elements of \(\mathbf{c}\) rather than the values at spatial grid points. This allows for the straightforward inclusion of structural irregularities within \(M\) by adding off-diagonal matrices derived from their Fourier transforms. For the concreteness of the discussion that will follow, we consider a floor/ceiling double leaf structure, with the coefficient \(\mathbf{c}\) made up of coefficients \(\mathbf{c}_{1}\) for the floor, \(\mathbf{c}_{2}\) for joists, and \(\mathbf{c}_{3}\) for the ceiling below. Similar indexing is used for the displacements \(w\). Using Euler beam and Kirchhoff plate assumptions (as in [32]), our model primarily captures vertical bending motion, justified for low- to mid-frequency ranges where bending dominates over shear deformation. The Lagrangian \(\mathcal{L}(w)\), with \(w(x,y,t)\) as the deflection, is defined by the following:  
\[\mathcal{L}(w)=\int_{0}^{T}[\mathcal{K}(t)+\mathcal{W}(t)-\mathcal{P}(t)]dt \tag{3}\]  
where \(\mathcal{P}\), \(\mathcal{K}\), and \(\mathcal{W}\) are the structure's instantaneous potential energy, kinetic energy, and external work, respectively. A stationary Lagrangian yields the true structural motion, that is as follows:  
\[\delta\mathcal{L}(w)=0.\]  
The time-harmonic vibrations at the frequency \(\omega\), strain, and kinetic energies of the Kirchhoff plate (of dimensions \(A\times B\)) are given by the following equation:  
\[\int_{0}^{A}\int_{0}^{B}\biggl{\{}\frac{D}{2}\biggl{[}\Bigl{(}\
abla^{2}w \Bigr{)}^{2}+2(1-\
u)\Bigl{(}w_{xy}^{2}-w_{xx}w_{yy}\Bigr{)}\biggr{]}-\frac{1 }{2}m\omega^{2}w^{2}\biggr{\}}dxdy, \tag{4}\]  
where \(D\) is the flexural rigidity, \(\
u\) Poisson's ratio, and \(m\) mass density. The Euler beam is similar, with \(y\)-derivatives set to zero in (4), as well as with flexural rigidity written as \(D=EI\) for Young's modulus \(E\) and the moment of inertia \(I\). The Fourier basis for simply supported components simplifies these expressions, with displacements of the structure expressed by Fourier sine functions, as well as mode summation truncated to \(N\) for computational purposes. The Fourier modes are as follows:  
\[\phi_{m}(x)=\sqrt{\frac{2}{A}}\sin k_{m}x\quad\text{and}\quad\psi_{n}(y)= \sqrt{\frac{2}{B}}\sin\kappa_{n}y,\;\;m,n=1,2,\ldots,N,\]  
where \(k_{m}=\pi m/A\) and \(\kappa_{n}=\pi n/B\) satisfy orthogonality, which streamlines the calculations and leads to diagonal stiffness matrices when the materials are homogeneous. Deflection of each joint (in the \(x\) direction) is a linear combination of the \(\phi_{m}(x)\), while deflection of each plate is a linear combination of \(\phi_{m}(x)\psi_{n}(y)\).