## 1 Introduction
[20] presented a method to evaluate the superficial quality in the machining process in a CNC _router_. [PERSON] et al. [21] proposed a complex vibration model for the linear displacement mechanism of the power screw, a mechanism widely used in CNC machines. Experimental measurements of the vibrations in similar configurations as a CNC _router_ are reported by different authors [22; 23; 24; 25]. These investigations deal with the tool usage at different conditions including adding damping to the machine. This work proposes a new methodology for modeling vibration systems of coupled thin beams under vertical vibration, which is later applied to a CNC _router_ type geometry. The main contribution of this research is to propose and validate a method that models the vertical vibrations of a system of collinear and perpendicular beams, which are located in a plane, and the applied load is perpendicular to this plane. This arrangement is equivalent to the structure of a CNC _router_. Furthermore, since this method is an analytical solution to the model, it does not require computational effort and can be used in design stages to identify critical operating conditions of equipment subjected to vertical loads that produce vibration. Additionally, it can be used to monitor and control the mechanical vibrations of operating equipment, taking advantage of its low computational cost. The first step consists of decomposing the structure geometry into a set of coupled beams with attached masses. Next, boundary conditions are defined for the beams, considering the couplings. These conditions lead to the computation of modes for the complete structure and natural frequencies. The modal amplitudes are represented as a state-space control system. Different scenarios with different loads (e.g., constant, harmonic, time-dependent)can be efficiently simulated with the resulting model. It is shown through numerical experiments that the obtained models are accurate when compared with a Finite Element Method (FEM) simulation, whenever the cross-section of the beams is small. Moreover, a validation based on an experimental setting with a real structure is also performed. The obtained state-space model is more suitable for control purposes in comparison to computationally expensive FEM models, particularly for CNC structures in which the model has to be updated when the tool moves. The methodology introduced here is related to other works that proposed analytical models for beams with attached masses. For instance, collinear beams with concentrated masses and coupled boundary conditions were investigated in [26; 27]. [PERSON] et al. [28] studied a general solution procedure for time-dependent and nonlinear boundary conditions. They consider the concentrated masses as internal boundary conditions, and their developed method focuses on non-linear subsystems in 1-d resulting from the non-linear time-dependent boundary conditions. [PERSON] and [PERSON] [29] studied a system of two beams with elastic constraints at one end and free at the other end. The beams are coupled through a mass-spring device. The modes are analyzed for different parameters such as the spring constant of the coupling. [PERSON] [30] presented an analysis of a Timoshenko beam with several endorsed masses for a collinear structure. An interesting analytical method for a collinear array of beams with coupled masses was developed by [PERSON] et al. [31], the novelty of their method is that many rigid bodies can be connected to many beams in any position and orientation. [PERSON] et al. [32] presented an analysis of transverse vibrations of one beam with several masses added. They used the classical Bernoulli-Euler theory and calculated the fundamental frequencies and vibration modes of the assembly. An extension of this work adding the considerations of the beams theory of [PERSON] and [PERSON], is discussed by the same group in [33]. Notice that all these works analyzed arrays of collinear beams, while in our approach we consider 2-d arrays. The article is organized as follows.