## 3 Macro-Element Application
### Linear Range
#### 3.1.2 Vibration Analysis  
Free vibration analysis is conducted in a simply supported squared plate. The plate is isotropic and shows uniform thickness, t, and mass density values, \(\rho\). The free harmonic vibration of a thin plate is governed by the following differential equation:  
\[D\
abla^{4}w(x,y)-\omega^{2}\rho tw(x,y)=0 \tag{25}\]  
Here, \(w(x,y)\) is a given mode in the cartesian space, \(\
abla^{4}\) is the biharmonic differential operator in Cartesian coordinates, i.e., \(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}\), and D is the bending stiffness given as \(Et^{3}/12/\left(1-v^{2}\right)\). For the present example, a squared plate simply supported at its four edges, the solution of the vibration mode shapes has been formulated by [PERSON] [102]:  
\[w(x,y)=A_{nm}\sin\Bigl{(}\frac{m\pi x}{a}\Bigr{)}\sin\Bigl{(}\frac{n\pi y}{a} \Bigr{)}\,\ (m,n=1,2,\ldots) \tag{26}\]  
in which, \(A_{nm}\) is the amplitude of vibration (which are the unknown coefficients), \(m\) is the number of half-waves in the x-direction, \(n\) is the number of half-waves in the y-direction, and \(a\) is equal to the length, L, of the squared plate side. By replacing Equation (26) in the differential equation, one obtains the exact solution for the vibration of frequencies in terms of the parameters \(m\) and \(n\):  
\[\omega=\sqrt{\frac{D}{\rho t}}\ \Bigl{[}\Bigl{(}\frac{m\pi}{L}\Bigr{)}^{2}+ \Bigl{(}\frac{n\pi}{L}\Bigr{)}^{2}\Bigr{]} \tag{27}\]  
Figure 5: Convergence study conducted for a squared plate test. By adopting different values for the integer parameters m and n, the theoretical and exact frequencies of the first eight modes have been found. For the discrete model, since it relies on an FE-based strategy, the natural frequencies and mode shapes are obtained through the following matrix form equation:  
\[\left([K]-\omega^{2}[M]\right)\{w\}=0 \tag{28}\]  
Here, \([K]\) and \([M]\) are the global stiffness and mass matrices. The mass of the discrete system is carried by the rigid plate elements by providing a representative density value, \(\rho_{system}\). Uniform mass distribution is assumed to assemble the mass matrix either through a lumped or consistent strategy. The former admits that the mass of the rigid plate is lumped in its nodes, this allowing to obtain a diagonal mass matrix. In the consistent strategy, mass is distributed in the rigid plates following a linear interpolation rule, as when computing the local stiffness matrix, and both translational and rotational inertias are accounted. Towards a convergence study, four mesh refinements have been considered: \(N=4,\ N=8,\ N=16\) and \(N=30\). A thickness value of L/10 and a gap spacing between the discrete cells of \(e=20\) mm have been admitted. Furthermore, the analysis is complemented with the results from a standard continuum FE model (with a mapped mesh, element size of \(L/30\) and with a consistent mass matrix) designated as FEA. Results gathered in Table 1 are given as normalised angular frequencies (with the exact solution). According to the h-refinement dependency verified in the elasticity problems considered before, a higher mesh refinement lead to lower errors. Yet, a mesh size of L/16 provides solutions with an error lower than 5% (lumped approach), which seems to constitute a sufficient refinement choice for larger case studies. A consistent mass matrix strategy yields more accurate eigenfrequency contents than the lumped one; that is especially clear for coarser meshes in which significant differences have been found (higher than 20%). On the other hand, the modal deformed shapes depicted in Figure 6 indicate that all the discretisation fit the FEA results up to the first four modes. After that, coarser meshes are unable to catch higher modal responses, due to the implicit mechanical arrangement of the quadrilateral rigid elements. Nonetheless, the higher refinement (\(L/30\)) reproduces well all the modal displacements.