## 3 Mathematical Modelling, Theoretical Aspects
### Modelling of the Moment Inertia  
Modelling the moment of inertia of the cross-section presumes to assume a variation law. The variation law can be formed through interpolation functions between reference values in form of piecewise, multilinear, or polynomial functions. For the first two types of function, results are satisfying, although in the points where the function is not continuous the iterative process may be divergent, and the precision decreases by two orders of magnitudes. In the particular case of polynomial functions, the passing through the critical values is continuous, and the desired precision is in close correlation with the polynomial function degree. The modelling of the moment of inertia in this study regardless of the kinematic constraints of the joints is performed with cubic functions, a piecewise polynomial formula known as the spline function. The natural boundary conditions are shown in Figure 1. Be\(f\) a function defined on \([a\ b]\) which is interpolated in \(m+1\) nodes on \(n\) sub-intervals through a cubic interpolant \(S(x)\) defined in Figure 1.  
Cubic interpolant \(S(x)\), is a cubic polynomial piecewise function expressed in a general form as in Expression (2) [21]. \[S(x)=\underset{j=0}{\overset{m-1}{\underset{j=0}{\overset{1}{\underset{j=0}{ \overset{1}{\underset{j=0}{\overset{1}{\underset{j=0}{\overset{1}{\underset{j}}{ \overset{1}{\underset  
every \(g\) function defined on an interval \([a\ b]\), which approximates function \(f\) through nods \(x_{j}\)[21].  
\[I=\int_{x_{0}}^{x_{a}}\bigl{(}f^{\pi}(x)\bigr{)}^{2}dx \tag{5}\]  
One can introduce notation \(h_{j}=\bigl{(}x_{j+1}-x_{j}\bigr{)}\), with respect to the general form presented in Expression (4), thus the boundary conditions from Relation (3a)-(3c) can be defined as recursive sequences, presented in Relation (6a)-(6c) [21]. \[S_{j+1}\bigl{(}x_{j+1}\bigr{)}=S_{j}\bigl{(}x_{j+1}\ \bigr{)}\ \ \to a_{j+1}=a_{j}+b_{j}h_{j}+c_{j}{h_{j}}^{2}+d_{j}{h_{j}}^{3} \tag{6a}\] \[S_{j}^{{}^{\prime}}\bigl{(}x_{j+1}\bigr{)}=S_{j+1}{}^{{}^{\prime}}\bigl{(}x_{j +1}\bigr{)}\ \ \to b_{j+1}=b_{j}+2c_{j}h_{j}+3d_{j}{h_{j}}^{2}\] (6b) \[S_{j}^{{}^{\prime\prime}}\bigl{(}x_{j+1}\bigr{)}=S_{j+1}{}^{{}^{\prime\prime}} \bigl{(}x_{j+1}\bigr{)}\ \ \to c_{j+1}=c_{j}+3d_{j}h_{j} \tag{6c}\]  
The numerical value of coefficient \(a_{j}\) results from the function evaluation of \(f\) at the input value of \(x_{j}\), denoted with \(f\bigl{(}x_{j}\ \bigr{)}\), as for \(a_{j+1}\) results from the function evaluation of \(f\) at the input value of \(x_{j+1}\), denoted with \(f\bigl{(}x_{j+1}\ \bigr{)}\). For the recursive sequences from Relations (6a)-(6c) it is required the cubic interpolants coefficients as expressions which depends only on two specific arguments, thus coefficients \(a_{j}\) and \(c_{j}\) are chosen. Coefficient \(d_{j}\), can be expressed from Relation (6c), presented in Expression (7) [21]. \[d_{j}=\frac{c_{j+1}-c_{j}}{3h_{j}} \tag{7}\]  
Substituting Expression (7) in the recursive Sequence (6a,b), Expression (8a,b) can be expressed [21]. \[a_{j+1}=a_{j}+b_{j}h_{j}+\frac{{h_{j}}^{2}}{3}\bigl{(}2c_{j}+c_{j+1}\bigr{)} \tag{8a}\] \[b_{j+1}=b_{j}+h_{j}\bigl{(}c_{j}+c_{j+1}\bigr{)} \tag{8b}\]  
Substituting Expression (8a) in Expression (8b) it is possible to express coefficient \(b_{j}\) with arguments \(a_{j}\) and \(c_{j}\), as in Expression (9) [21]. \[b_{j}=\frac{1}{h_{j}}(a_{j+1}-a_{j})-\frac{h_{j}}{3}\bigl{(}2c_{j}+c_{j+1} \bigr{)} \tag{9}\]  
The recursive Sequences (8b) and (9) can be expressed also for sub-interval \(\bigl{[}x_{j-1}\ x_{j}\bigr{]}\), therefor Expression (8b) can be written as in Equation (10) [21]. Figure 1.0  
Equation (10) represents \(m-1\) equations, and in order to solve a linear system of equations, two additional relations are required in the form of the natural boundary conditions expressed in Relation (11) [21].