## 2 Basic Concepts  
The proposed modeling methodology is based on the classical Bernoulli-Euler theory. A CNC _router_ operating properly functions within the elastic range. Any operation in the plastic range might lead to catastrophic failure. The proposed model forecasts the behavior of a CNC _router_ geometry under normal operating conditions. Basic concepts and notation are presented in this section.  
According to the Bernoulli-Euler theory, the behavior of vertical displacements \(w(x,t)\) along a thin beam of uniform cross-section without damping, subjected to a load distribution \(f(x,t)\), is described by  
\[EI\frac{\partial^{4}w(x,t)}{\partial x^{4}}+\rho A\frac{\partial^{2}w(x,t)}{ \partial t^{2}}=f(x,t) \tag{1}\]  
where \(E\) is the Young modulus, \(I\) is the second moment of area, \(\rho\) is the density and \(A\) is the beam cross section area. We consider the classical solution of split variables, in which the vertical displacement is expressed as  
\[w(x,t)=W(x)T(t), \tag{2}\]  
Considering the unloaded case (i.e., \(f(x,t)=0\)), previous expressions lead to  
\[\frac{EI}{\rho A}\frac{1}{W(x)}\frac{\partial^{4}W(x)}{\partial x^{4}}=-\frac {1}{T(t)}\frac{\partial^{2}T(t)}{\partial t^{2}} \tag{3}\]Since the left term is a function of \(x\) and the right term is a function of \(t\), the only solution is that both terms are equal to a constant value, usually denoted as \(\omega^{2}\). The expression (3) is then split in two equations:  
\[\frac{\partial^{4}W(x)}{\partial x^{4}}=\beta^{4}W(x) \tag{4}\]  
\[\frac{\partial^{2}T(t)}{\partial t^{2}}=-\omega^{2}T(t) \tag{5}\]  
where  
\[\beta^{4}=\frac{\rho A}{EI}\omega^{2} \tag{6}\]  
The general solution of (4) is given by  
\[\begin{array}{ll}W(x)&=c_{1}cosh(\beta x)+c_{2}sinh(\beta x)+c_{3}cos(\beta x )+c_{4}sin(\beta x)\\ &=g(\beta,x)\mathbf{c}\end{array} \tag{7}\]  
where  
\[\begin{array}{ll}g(\beta,x)=\left[\begin{array}{cc}cosh(\beta x)&sinh( \beta x)&cos(\beta x)&sin(\beta x)\end{array}\right]\\ \mathbf{c}=\left[\begin{array}{cc}c_{1}&c_{2}&c_{3}&c_{4}\end{array}\right] ^{T}\end{array} \tag{8}\]  
The coefficients \(c_{1}\), \(c_{2}\), \(c_{3}\) and \(c_{4}\) depend on the boundary conditions. In fact, given a set of boundary conditions, there exist an infinite number of solutions \(\mathbf{c}\) and \(\beta\) for (4). In particular, the solutions for \(\beta\) is a countable set. If the positive solutions for \(\beta\) are ordered as \(\beta^{1}<\beta^{2}<\beta^{3}...\), for each \(\beta^{k}\) the corresponding value \(\omega^{k}\) can be computed by (6). Such values \(\omega^{1}<\omega^{2}<\omega^{3}...\) are named _natural frequencies_. In this way, for each \(k\)-th natural frequency \(\omega^{k}\), we have a parameter \(\beta^{k}\), and a particular vector of coefficients \(\mathbf{c}^{k}\) can be computed such that the resulting function \(W^{k}(x)=g(\beta^{k},x)\mathbf{c}^{k}\) is as a solution of (4). This solution \(W^{k}(x)\) is called the \(k\)-th _mode_.  
On the other hand, given a natural frequency \(\omega^{k}\) and particular initial conditions, the solution of (5) is denoted as \(T^{k}(t)\), called the \(k\)-th _modal amplitude_, which can be generally described as  
\[T^{k}(t)=a^{k}sin(\omega^{k}t)+b^{k}cos(\omega^{k}t) \tag{9}\]  
where the coefficients \(a^{k}\) and \(b^{k}\) depend on the initial conditions.  
In this way, the general solution for the displacement \(w(x,t)\) can be expressed as  
\[w(x,t)=\sum_{k=1}^{\infty}W^{k}(x)T^{k}(t) \tag{10}\]