## 2 Materials and Methods  
### Non-Destructive Modal Analysis  
Modal analysis is the process of determining modal parameters of mechanical systems that describe their dynamic behavior. It can be approached theoretically, numerically, or experimentally. Theoretical modal analysis involves solving the eigenvalue problem of the mathematical model of a mechanical system. The equations that describe the linearized motion of a general form in a particular configuration are provided by  
\[\mathbf{M}\mathbf{\ddot{q}}(t)+\mathbf{D}\mathbf{\dot{q}}(t)+\mathbf{K}\mathbf{q}(t)=\mathbf{f}(t) \tag{1}\]where **M**, **D** and **K** are the mass matrix, the damping matrix and the stiffness matrix, respectively. The vector \(\textbf{q}(t)\) contains the degrees of freedom (DOF) of the system. Equation (1) is a set of differential equations in matrix form for the dynamic response of a structure modelled with a finite number of degrees of freedom. An eigenvalue analysis, commonly known as the most prevalent form of dynamic analysis, is often performed. Apart from determining the frequencies, it is also essential to investigate the mode shapes that occur at the natural frequencies. These mode shapes represent the undamped free vibration response of the structure resulting from an initial disturbance from its static equilibrium position. To obtain this solution, the damping and applied force terms in the general equation are neglected. Subsequently, it is assumed that each node experiences sinusoidal functions (\(\textbf{q}=\textbf{A}\sin(\omega t)\)) with peak amplitudes specific to that node. The number of eigenvalues or natural frequencies in the model is equal to the total number of DOFs. Each eigenvalue or frequency is associated with a corresponding eigenvector or mode shape. Since none of the eigenvectors can be zero vectors (\(\textbf{A}\
eq 0\)), the equation that needs to be solved is as follows:  
\[(\textbf{K}-\omega^{2}\textbf{M})=0 \tag{2}\]  
It is important to mention that our typical focus lies on the first few eigenvalues of the model. This is because the finite element model, being an approximation of the structure, tends to yield inaccurate higher eigenvalues and vectors. According to the theoretical solution, the structure exhibits indefinite vibrations in various mode shapes. However, due to the presence of damping in all structures, these vibrations ultimately decay over time. Modal parameters are determined through measurements, which is an experimental approach. This involves recording the time histories of excitation force and corresponding response of the system. The system's dynamic behavior can be described in the frequency domain using the frequency response function (FRF) or in the time domain using the impulse response function (IRF). These functions are calculated from measurement data using the Inverse Fourier Transform. Impact testing has become the most popular method for experimental modal analysis as FRF measurements can be computed using a FFT analyzer. Impact testing is a fast, convenient, and low-cost way of finding the modes of machines and structures. The FRF is defined as the ratio of output signal to input signal. There are three primary types of FRFs based on the type of response parameter being analyzed. These response parameters can be displacement, velocity, or acceleration in relation to frequency. A set of all the FRFs is called frequency response function matrix. Furthermore, the Laplace transform (\(s=i\omega\)) and neglecting initial conditions in the frequency domain of Equation (1) are given by  
\[\textbf{Q}(s)=\textbf{H}(s)\textbf{F}(s) \tag{3}\]  
where \(\textbf{H}(\omega)=\left(\textbf{K}+i\omega\textbf{C}-\omega^{2}\textbf{M} \right)^{-1}\) is called the transfer function matrix.