## 2 Materials and Methods
### Numerical Simulation by FEM
#### 2.3.1 Elasto-Dynamic Equilibrium Equations and Approximation with Finite Elements
And if it is desired to take into account the dampening as well, then from the Caughey series [27], only the first two terms and the resulting damping, called Rayleigh damping, will be considered, and the damping matrix will have the form \(C=b1\,M+b2\,K\). The general form of the motion equation will be as follows:  
\[\varepsilon(u(t,x))=(\
abla_{x}u(t,x)+(\
abla_{x}u(t,x))^{T})/2, \tag{8}\]0  
The most useful methods for solving the matrix system of differential equations of the second order of the form given in (15) or (17) are the step-by-step integration method in relation to time, more precisely the Newmark method, and the modal analysis method. The modal analysis method is preferred to the Newmark method both in terms of precision and in terms of working time. 3.2 Obtaining the Natural Frequencies and the Natural Modes of Vibration with the Method of Modal Analysis  
The systems (15) and (17) are systems of second-order differential equations with constant coefficients without damping, respectively, with damping and with a time-varying force \(F(t)\). We propose to determine the eigenfrequencies (natural) and eigenmodes (modal forms) of Equation (15) or (17), in the case of free movement (\(F(t)=0\)) and undamped (C = 0 in the case of (17)). For this purpose, the first step will consist in obtaining from (16) a system of ordinary differential equations explicitly in \(\ddot{q}\), making the substitution \(q=M^{1/2}U\); the system gives the following results:  
\[\varepsilon(u(t,x))=(\
abla_{x}u(t,x)+(\
abla_{x}u(t,x))^{T})/2, \tag{8}\]1  
We noted with \(H=M^{-1/2}KM^{-1/2}\) the matrix resulting from this substitution which is positive definite, and therefore, through a spectral decomposition, can be obtained: \(H=Q\Lambda Q^{T}\), where Q is an orthogonal matrix (\(Q^{T}Q=I\)) and \(\Lambda=\text{diag}(\lambda_{i})\), \(i=1\),\(\ldots\), \(m\). Through the spectral decomposition, in addition to the eigenvalues \(\lambda_{i}\), the normalized eigenvectors \(r^{(i)}\), \((r^{(i)}r^{(j)}=\delta_{ij})\) of the H matrix are also obtained, and the Q matrix has the normalized eigenvectors \(r^{(i)}\) as columns. To obtain the modal forms or eigenmodes, the substitution is made with \(w=Q^{T}q\), and from Equation (18), we have the following:  
\[\varepsilon(u(t,x))=(\
abla_{x}u(t,x)+(\
abla_{x}u(t,x))^{T})/2, \tag{8}\]2This is a second-order differential system uncoupled in \(w\), because the matrix \(\Lambda\) is diagonal. To solve the system (19), we need the initial conditions (16), and using the two substitutions above, we have \(w=Q^{T}q=Q^{T}M^{1/2}U\), and for \(t=0\), we have \(w(0)=Q^{T}M^{1/2}U(0)\) and \(\dot{w}(0)=Q^{T}M^{1/2}U(0)\). In this stage we can determine the displacements \(U(t)\) from (19):  
\[\varepsilon(u(t,x))=(\
abla_{x}u(t,x)+(\
abla_{x}u(t,x))^{T})/2, \tag{8}\]3  
This procedure reduces the general forced motion problem obtained with FEM to an equivalence that involves solving the uncoupled forced motion equations for \(m\) spring-mass elements. If we consider free motion, i.e., \(F(t)=0\), and excite only the i-th spring-mass element (by choosing appropriate initial conditions), the solution will be a harmonic vibration at the natural frequency of the i-th spring-mass element. The natural frequencies of the n decoupled systems are therefore the natural vibration frequencies of the structure. The corresponding eigenvectors \(r^{(i)}\) are associated with the deformations \(U^{(i)}\) corresponding to the i-th mode of vibration \(U^{(i)}=M^{-1/2}r^{(i)}\).