## 2 Materials and Methods  
### Numerical Modal Analysis  
Static equilibrium Equation (1) in a dynamic domain needs to be modified to account for inertia and damping terms (2) [9].  
\[K\,q(t)=F(t) \tag{1}\]  
\[M\ddot{q}(t)+C\dot{q}(t)+Kq(t)=F(t) \tag{2}\]  
where:  
\(M\), \(C\), \(K\) are mass, damping and stiffness matrices,  
\(\ddot{q}(t),\,\,\,\dot{q}(t)\) are vectors of acceleration, damping and displacements,  
\(F(t)\) is an external load vector,  
In case of free undamped vibration, the equation is simplified to form 3:  
\[M\ddot{q}(t)+Kq(t)=0 \tag{3}\]  
The displacement vector for harmonic motion is reported in Equation (4):  
\[q=q_{a}\sin(\omega t) \tag{4}\]  
where:  
\(q_{a}\)--amplitude vector, in  
\(\omega\)--angular frequency, rad  
\(f=\omega/2\pi\)--frequency, Hz  
\(T=1/f\)--period, s  
Figure 3: Notional Goodman diagram.  
By differentiating Equation (4) by time, we obtain \(\ddot{q}\)  
\[\ddot{q}=-\omega^{2}q_{a}\sin(\omega t) \tag{5}\]  
Generalized equation form after reordering (3) and substituting vectors with (4) and (5) is given as:  
\[(K-\lambda M)q_{a}=0 \tag{6}\]  
where:  
\(\lambda=\omega^{2}\)  
Equation (6) is a fundamental equation for free undamped vibrations for linear system and linear-elastic material behavior (where the Hook law is applied). The reported relationship is crucial in understanding of system eigenvalues called natural frequencies and eigenvectors representing modal forms. In order to find \(\lambda_{i}\) (natural frequencies) and associated vectors \(q_{a,i}\) (modal forms), the following equation needs to be solved:  
\[det|K-\lambda\ M|=0 \tag{7}\]  
The particular eigen vector \(q_{a,i}\) describes the movement of the system for given \(\lambda_{i}\) and needs to respect Equation (8).  
\[q_{a,i}{}^{T}\ M\ q_{a,i}=I \tag{8}\]  
where:  
_I_--unitary mass  
Modal analysis results due to normalization reported in (8) are artificial in terms of magnitude; however, the pattern of displacement or concentration of strain and stresses are true. The results need to be correlated with experimental tests or with a more complex analysis that includes damping and excitation items. In a complex environment, uncertainty about the damping and forcing function may lead to unreliable and questionable results, therefore an experimental test is always recommended.  
The stiffness matrix used in Equation (7) is expressed by Equation (9)  
\[k_{e}=\int_{e}B^{T}CBdx \tag{9}\]  
where:  
_B_--matrix of shape function derivatives  
C_--material coefficient matrix  
The material coefficient matrix during the optimization process remained unchanged since it is associated with material properties. The matrix of shape function derivatives is affected by the change of dimensional characteristics.  
To better understand the dependency between stiffness (K) and mass (M) matrices to natural frequency, the harmonic oscillator (mass-spring system) model is presented in Figure 4.  
The natural frequency of the mass spring system is described by Equation (10) derived from 6.  
\[f=\frac{\sqrt{\frac{k}{m}}}{2\ \pi} \tag{10}\]  
Figure 4: Harmonic oscillator.  
Stiffness is directly proportional to the natural frequency and is affected not only by geometrical features but also by temperature effects that reduce Young's modulus in material properties. Mass is inversely proportional to the natural frequency.