## 2 Theoretical Formulation
### Rayleigh-Ritz Solution and Response  
From Equations (7)-(9), (11) and (16), it can be seen that only five unknown parameters are needed to obtain the total energy of the cutting system, namely, \(w\), \(u_{\rm c}\), \(u_{\rm b}\), \(v_{\rm c}\), \(v_{\rm b}\). In this paper, the Rayleigh-Ritz method (RRM) is used to approximately express the displacement components. The \(N^{*}N\)-terms Rayleigh-Ritz solutions for the problem are of the following form:  
\[ \begin{array}{l}w=\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}w_{ij}(t)\varphi _{i}(x)\psi_{j}(y),\\ u_{\rm b}=\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}p_{ij}(t)\varphi_{i}(x) \psi_{j}(y),v_{\rm b}=\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}q_{ij}(t) \varphi_{i}(x)\psi_{j}(y)\\ u_{\rm c}=\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}r_{ij}(t)\varphi_{i}(x) \psi_{j}(y),v_{\rm c}=\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}s_{ij}(t) \varphi_{i}(x)\psi_{j}(y)\end{array}  \tag{17}\]  
where \(w_{ij}\), \(p_{ij}\), \(q_{ij}\), \(r_{ij}\), and \(s_{ij}\) are Ritz coefficients (or generalized modal coordinates) and \(\varphi_{i}(x)\) and \(\psi_{j}(y)\) are admissible functions. It should be noted that, unlike conventional Rayleigh-Ritz solutions, the admissible functions adopted here just satisfy a totally unconstrained condition and [PERSON]'s penalty method is used to handle constraints. As shown in Figure 1, the clamped constraint in \(y=0\) is realized by some springs with high stiffness, where translational springs are used to limit the transverse displacement, \(w\), in \(y=0\), and torsional springs are applied in \(y=0\) to limit the rotation, \(w_{y}\). To improve the numerical stability and convergent rate following admissible functions, permitting non-zero displacement and translation at both ends for a free-free beam are adopted. \[\begin{array}{l}x-\text{direction}:\quad\varphi_{i}(x)=\left\{\begin{array}[ ]{rl}\left(\frac{x}{L}\right)^{i-1},&i=1,2,3\\ \cos\frac{(i-3)\pi x}{L},&i=4,5,\ldots,N\\ \left(\frac{y}{W}\right)^{j-1}&j=1,2,3\\ \cos\frac{(j-3)\  
where \(k_{\rm t}\) and \(k_{\rm r}\) are stiffness coefficients of the translational and rotational springs, respectively, and both of them take high value. Therefore, for this system, the Lagrangian is \(L=T-U-V\) and the work done by nonconservative forces is \(W_{F}\). Using the Lagrange equation, the governing equation of motion of the system can be given by:  
\[\left[ \begin{array}{cccc}{\bf M}_{11}+{\bf M}_{11}^{*}&0&0&0&0\\ 0&0&0&0&0\\ \vdots&&0&0&0\\ &&0&0&0\\ {\rm sym.}&&\ldots&0\end{array} \right]\left\{ \begin{array}{c}{\bf\tilde{w}} \\ {\bf\tilde{p}}\\ {\bf\tilde{q}}\\ {\bf\tilde{r}}\\ {\rm sym.}&&\ldots\end{array} \right\}+\left[ \begin{array}{c}{\bf C}_{11}&0 &0&0&0\\ 0&0&0\\ \vdots&&0&0&0\\ {\rm sym.}&&\ldots&0\end{array} \right]\left\{ \begin{array}{c}{\bf\tilde{w}} \\ {\bf\tilde{p}}\\ {\bf\tilde{q}}\\ {\bf\tilde{r}}\\ {\rm\bf\tilde{s}}\end{array} \right\}\right. \tag{21}\]  
\[+\left[ \begin{array}{cccc}{\bf K}_{11}+{\bf K}_{11}^{*}+{\bf K}_{11}^{ \prime}&{\bf K}_{12}&{\bf K}_{13}&{\bf K}_{14}&{\bf K}_{15}\\ {\bf K}_{22}+{\bf K}_{22}^{\prime}&{\bf K}_{23}&{\bf K}_{24}&0\\ \vdots&&{\bf K}_{33}+{\bf K}_{33}^{\prime}&0&{\bf K}_{35}\\ {\rm sym.}&&\ldots&{\bf K}_{44}&{\bf K}_{45}\\ \end{array} \right]\left\{ \begin{array}{c}{\bf w}\\ {\bf p}\\ {\bf q}\\ {\bf r}\\ {\rm\bf s}\end{array} \right\}=\left\{ \begin{array}{c}{\bf F}_{z}(t)\\ {\bf F}_{x}(t)\\ {\bf F}_{y}(t)\\ 0\\ 0\end{array}