A method to measure the damping introduced by linear guides in large milling machines

1 A METHOD TO MEASURE THE DAMPING INTRODUCED BY LINEAR GUIDES IN LARGE MILLING MACHINES I. Oleagaa,*,J.J. Zulaikaa,, F.J. Campab and J. Hernandoc. a Tecnalia, Paseo Mikeletegi 7, 20009 Donostia-San Sebastian, Spain {ibone.oleaga, juanjo.zulaika}@tecnalia.com b University of the Basque Country, Department of Mechanical Engineering Alameda de Urquijo s/n, 48013 Bilbao, Spain [email protected] c Nicolás Correa, calle Alcalde Martín Cobos, 16ª, 09007 Burgos, Spain [email protected] Abstract. Simulation of the dynamic behavior of a milling machine requires accurate stiffness, inertia, and damping values. Unlike the properties of stiffness and inertia, damping values are not generally available in the bibliography, which contains only reference values. In the present work, a method is proposed to calculate and to identify the damping introduced by the linear guides assembled in large-scale milling machines. One immediate advantage of this method would be to improve the dynamic frequency responses of milling machines, and as a consequence, their productivity, wherever the conditions for chatter arise that limit productivity. The method is based on a combination of experimental frequency responses and theoretical modes obtained from a finite element model, where damping is represented by means of ideal discrete dampers. Hence, energy dissipation can be accurately represented by associating the damping of each linear guide in a discrete way, at the specific points where the dissipation of energy occurs. The method has been satisfactorily validated on a real large machine. Keywords: Machine tool dynamics; damping; linear guides; modeling. *Corresponding author. Tel.: +34 946 430 850; fax: +34 946 460 900. This is the accepted manuscript of the article that appeared in final form in Mechanical Systems and Signal Processing 171 : (2022) // Article ID 108908, which has been published in final form at https://doi.org/10.1016/j.ymssp.2022.108908. © 2022 Elsevier under CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 2 NOMENCLATURE B Number of nodes of finite element model c Damping constant cm Modal damping constant ckl Damping value of a k node in the structure in the l direction ckflq Damping value of an l node in the q direction when a k node is exited in the f direction [𝐶𝐶] Damping matrix [𝐶𝐶𝑚𝑚] Modal damping matrix {𝐹𝐹(𝑡𝑡)} External force vector {Fm} External force vector in modal coordinates [I] Identity matrix k Spring constant stiffness [𝐾𝐾] Stiffness matrix m Mass 𝑚𝑚𝑖𝑖𝑖𝑖 Mass value of j degrees of freedom in i mode 𝑚𝑚𝑚𝑚𝑖𝑖𝑖𝑖 Modal mass value of j degrees of freedom in i mode [𝑀𝑀] Mass matrix [𝑀𝑀𝑚𝑚] Modal mass matrix N Number of degrees of freedom of finite element model {p},{ 𝑝𝑝󰇗},{𝑝𝑝󰇘} Displacement, velocity, and acceleration vectors in modal coordinates. P External force in the Laplace domain P0 Maximum force 𝑃𝑃0𝑚𝑚 Modal maximum force s Laplace variable x, x󰇗, x󰇘 Displacement, velocity, and acceleration {x},{ x󰇗},{x󰇘} Displacement, velocity and acceleration vector in natural coordinates X, X 󰇗 Displacement, velocity in Laplace domain X0 Maximum displacement amplitude 𝑋𝑋0𝑚𝑚 Modal maximum displacement [𝑋𝑋] Orthonormal modal vectors ξ ij Modal damping coefficient in the i direction in j mode. ϕij Mass normalized modal vector at point i, in the j direction ωn Natural frequency 3 1. INTRODUCTION The productivity of milling operations is frequently restricted by the generation of self-excited vibrations, especially regenerative chatter vibrations [1, 2]. The modes excited in chatter vibrations can be from the structural components of the machine [3, 4], from the spindle/toolholder/tool system [5], or from the workpiece, when it has flexible thin walls or floors [6, 7]. Chatter is nowadays a major limitation to higher productivity levels in many machining processes [8]. For online detection, one alternative is the estimation of the dominant chatter frequency by designing algorithm-based adaptive filters [9]. After detection, several authors have proposed methods as the suppression of the vibration through the combination of online variation of the spindle speed changing the periodicity of tooth impacts [10, 11, 12, 13], the use of a two-link robotic arm [14], the use of tuned mass dampers (TMD) [15,16, 17, 18], or by using active damping devices (ADD) [19, 20]. For offline avoidance, over the past fifty years, various authors have been working to understand, model and predict the onset of machine tool chatter by calculating the stability lobe diagrams [21, 22, 23]. In machining, the inertia, stiffness and damping of the whole machining system, machinetool-workpiece-tooling, determines the dynamic performance of the process, thus affecting the dimensional accuracy and surface finish of the part and the integrity of the tool and machine tool components [24-25]. The problem under study becomes even more complex when certain effects are considered, such as multi-point contact, process damping, and the variation of the modal parameters, due to machine tool motion and material removal from a workpiece [27-29]. From the point of view of the machine tool builder, the identification of the most limiting modes regarding chatter and their modal parameters helps to improve both the machine design and its productivity. To do so, a modal analysis using finite elements must be performed to calculate the influence of the modes at the Tool Center Point (TCP), but the damping data is also required and determinant for a good prediction of the machine dynamic stability. Milling machine damping can be classified in two main types: inherent damping that occurs naturally, such as material damping or joint-induced damping in a functional machine, and additional damping, which results from specially constructed devices added to the machine structure as TMDs or ADDs [33]. In this paper, only inherent damping is considered. The concept of damping involves the physical mechanisms that dissipate mechanical energy [8, 30], which, in turn, enhance the dynamic stability of mechanical systems. At present, the bibliography contains only reference values. Schlecht [31] provided an overview of typical damping values for rolling bearings, ranging between 2 N s mm-1 and 1000 N s mm-1, for bearings with inner diameters of 55 mm and 160 mm, respectively. Groche and Hofmann [32] gave an overview of typical damping 4 values for linear roller and ball guideways from 4 N s mm-1 for linear guides of 35 mm in size to 10.5 N s mm-1 for linear guides of 45 mm in size. Even if the damping in a machine tool cannot be predicted accurately, according to Beads [34] and Weck [35], approximately 90% of the total vibratory energy dissipation of the entire machine tool structure originates at the joints. The remaining 10%, known as material damping, originates in material during cyclic deformation. In general, as long as the stress fatigue limit, the material melting point temperature, and the audio frequency range are not exceeded, the material damping will be independent of temperature, vibration amplitude, and frequency [36]. Within machine tool joints, linear guides can be highlighted because of their high capacity to dissipate energy due to their large contact surface. It is therefore of interest to focus on the study of linear guide damping and material damping to develop accurate models of machine tool damping. Machine damping in finite element modelling (FEM) can be represented in different forms. Modal damping consists of associating a modal damping coefficient with each mode or natural frequency. When damping is characterized in that way, a global value in a continuous form is associated with the whole structure for each natural frequency and a damping coefficient of 0.02 is usually considered [37]. Different modal damping coefficient measurement methods can be found in the bibliography. For example, Montalvão [38] presented a novel method for the identification of the modal damping coefficients for each natural frequency from FRFs, based on dissipation energy per vibration cycle. Zhang [39] analyzed the dynamic characteristics of a guideway and proposed a systematic procedure to predict the dynamic behavior of a whole machine-tool structure with respect to natural frequencies. However, no absolute real damping values are associated with the machine elements in terms of modal damping. Although it is a good way of analyzing the global behavior of a machine, this method cannot identify the sources of damping in a given machine, and in the case of a redesign, it cannot help identifying the structural joints where a change in damping will have most influence. Another way to model damping by FEM is to represent the viscous damping of the linear guide through a distribution of discrete dampers together with material hysteretic damping. There are many mechanisms whereby vibrational energy can be dissipated within the volume of a material element as it is cyclically deformed [40]. Such mechanisms are associated with internal reconstructions of the micro and macro structure. Accurate material damping values can be found in the bibliography [41], as the majority of published information is of empirical nature. On the other hand, the dynamic characteristics of linear guides in a machine-tool structure are determined by many factors [39], such as the size of the linear guide, excitation force frequencies and natural machine frequencies, the value of modal vectors of linear guides at those frequencies, distributed 5 pressure on the joint surfaces, the types of guides, the guide material, lubrication on the linear guide surfaces, and the surface finishing method, among others. A few reference values of the discrete damping can be found in the bibliography. Several works have been completed for the identification of universal damping parameters of roller linear guide elements. Deng [42], Popov [43], De Vicente [44], and Al-Bender [45] studied the interface behavior at rolling contact. A source of uncertainty of these studies is that the elements are studied in isolation from the other elements, rather than when assembled in the milling machine. Brecher, Fey and Bäumer [46] described a methodology to identify discrete damper values for machine tool components of linear axes. To do so, they used a specially designed test bench. Albert [47], Neugebauer [48], Rossteuscher [49] and Wu [50] also measured linear guide damping, although their measurements were also done on linear guides assembled on a test bench. However, the position of the linear guide in a machine and the loads it carries will change from one machine to another, which means different damping values for each machine and position. In the experience of the authors, even guides with the same characteristics on the same machine, but joining different components, can dissipate different amounts of energy. Therefore, the tests performed on a guide assembled on a test bench will not always be representative, as can easily be seen when analyzing a machine with the same linear guides joining different components [51]. If these linear guides are replaced one by one with higher damping linear guides, the influence of the new linear guide at the TCP will not be always the same. Hence, there is a need to develop a method that can calculate the damping introduced by a guide once assembled in the machine. The study presented in this paper will introduce a theoretical-experimental methodology to measure absolute damping values of linear guides assembled in the milling machine. The mathematical basis of the method will be first presented. Then, it will be applied and validated in an operational machine and, finally, the conclusions will be presented. 2. PROCEDURE TO MEASURE THE DAMPING OF THE ASSEMBLED LINEAR GUIDES Linear guide damping in a milling machine can be discretely modeled using FEM software to understand its effect and predict the machine dynamic behavior. But to do so, a method is needed to measure damping once the guides have been assembled in the milling machine, as their damping capability depends on their position along the machine tool structure, thus identifying the main sources of damping in the machine. In this study, linear guide damping was calculated from experimental data gathered from experimental impact tests at the TCP. To do so, two points around each linear guide were tested: 6 one on the structure before the linear guide, near the carriage, and the other on the structure on the other side of the linear guide, see Figure 1. The red base of Figure 1 is a magnetic base for the accelerometer. Magnetic bases can introduce damping in the measurements, but as it is used on both sides of the linear guide, when the difference of both points is calculated, that effect should cancel itself out. More points can be tested for each linear guide, although if the points are located near the carriages, then similar information is obtained in the experience of the authors. Using the experimentally measured frequency responses, the damping at each point can be calculated, and the difference between both points will be associated with the damping introduced by the linear guide assembled in the structure. Fig. 1 Measurements points before and after a linear guide assembled in the test machine. In the case of a model with one degree of freedom (dof), the resonance measurement method can be used to measure damping from experimental data [38, 52]. Large-scale milling machines require N dof systems to represent the milling machine, and the resonance measurement method cannot be applied. However, if a change to modal coordinates is applied to the N dof system, instead of an N dof system, the sum of N systems of 1 dof is obtained by means of modal superposition, and the resonance method can then be applied to each 1 dof system. Hence, the damping value of the point for each dof can be obtained in modal coordinates. Finally, the modal coordinates change can be undone and the damping of the point can be obtained in natural coordinates, in their various directions: Cxx, Cyy, Czz, Cxy, Cxz, Cyx, Cyz, Czx, and Czy. This approach is explained step by step below. 7 2.1 Resonance measurement method in 1 dof system The equation of motion of a forced 1 dof system with viscous damping is: 𝑚𝑚𝑥𝑥󰇘+𝑐𝑐𝑥𝑥󰇗+𝑘𝑘𝑥𝑥=𝐹𝐹(𝑡𝑡) (1) where x, x󰇗,and x󰇘 represent the displacement, velocity, and acceleration of the dof respectively; m, c, and k represent the mass, damping, and stiffness; and, F is the force. Assuming a harmonic solution: 𝐹𝐹(𝑡𝑡)=𝐹𝐹0 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖 (2) x = 𝑋𝑋0 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖, 𝑥𝑥󰇗 =𝑖𝑖𝑖𝑖𝑋𝑋0 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖,𝑥𝑥󰇘 =−𝑖𝑖2𝑋𝑋0 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖 (3) Substituting equations (2) and (3) in equation (1), and applying resonance condition 𝑖𝑖1= �𝑘𝑘 𝑚𝑚 , inertial and stiffness terms are cancelled: −𝑖𝑖1 2𝑚𝑚𝑋𝑋0 + 𝑐𝑐𝑖𝑖𝑖𝑖1𝑋𝑋0 + 𝑘𝑘𝑋𝑋0 = 𝐹𝐹0 (4) −𝑘𝑘 𝑚𝑚𝑚𝑚𝑋𝑋0 + 𝑐𝑐𝑖𝑖𝑖𝑖1𝑋𝑋0 + 𝑘𝑘𝑋𝑋0 = 𝐹𝐹0 (5) 𝑐𝑐𝑖𝑖𝑖𝑖1𝑋𝑋0 = 𝐹𝐹0 (6) From equation (6), it is concluded that at resonance, the external force F0 will be equal to the damping force. With hammer impact experimental test, the damping of a one dof system can be easily obtained using the value of the measured mobility at the natural frequency. 𝑐𝑐 = 𝐹𝐹0 𝑖𝑖𝑖𝑖1𝑋𝑋0=F 𝑋𝑋󰇗0 (7) The same is not so for the N dof system. In the following section, this method will be extended to an N dof system with some coordinate changes. 2.2 Resonance measurement method in an N dof system In systems with N dof, the equation of motion is a system with N coupled equations: 8 [M]∗{x󰇘}+[C]∗{x󰇗}+[K]∗{x}={F(t)} (10) where [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, {𝐹𝐹(𝑡𝑡)} is the external force vector, and {x} is the displacement vector in natural coordinates. Milling machines can be represented by linear and proportional systems, so if a change to modal coordinates, {p}, is applied to the N dof system pre-multiplying by the transpose of the modal matrix [X]T, a decoupled system of equations of the model can be obtained: {x}=[X]∗{p} (11) [X]T[M][X]∗{p󰇘}+[X]T[C][X]∗{p󰇗}+[X]T[K][X]∗{p}=[X]T{F(t)} (12) [Mm]∗{p󰇘}+[Cm]∗{p󰇗}+[Km]∗{p}={𝐹𝐹𝑚𝑚} (13) where, [Mm] is the modal mass matrix, [Cm] is the modal damping matrix, [Km] is the modal stiffness matrix, and {𝐹𝐹𝑚𝑚} is the modal external force vector. In this way, it is possible to obtain N decoupled equations of 1 dof, and the damping of each equation can therefore be obtained in modal coordinates applying equation (7): cm=𝐹𝐹0m X0m∗𝑖𝑖n (14) Now, as damping values are needed in natural coordinates to introduce them in the numerical model, the base coordinate change must be undone. [C]= [[X]T]−1[Cm][X]−1 (15) In a finite element model, different kinds of elements can be used: mass, beams, dampers, bush, tria, quads, tetras, etc. Mass, beam, damper, bush, tria and quad elements nodes have 6 dof, 3 translational dof and a further 3 rotational dof, and tetra elements nodes only have 3 translational dof. Some solvers, such as NASTRAN, apply 6 dof to every node, and those that are not used, like rotational dof of tetra elements nodes, are automatically grounded. Thus, a B node finite element model requires 6B *6B modal matrix. It is advisable to avoid the inverse of this matrix, because of the large dimensions of the modal vector matrix, as a large-scale milling machine can be usually represented by between 300000 9 nodes and 700000 nodes. Instead, [X]−1 and [[X]T]−1 can be obtained for the base change without performing a matrix inversion, in the following way: [X]T[M][X]=[X]T�mij�[X]=[Mm]=�mmij� (16) where, mij and mmij are the mass and the modal mass values, respectively, of j dof in the ith mode. The mass matrix used in the finite element model is a lumped matrix. As it is a diagonal matrix, only the diagonal components will have a value different from zero. This property will help to avoid calculating the inverse of the previous matrix, because if modal vectors are normalized to unity mass, the previous equation can be rewritten as: [X]T[M][X]=[X]T�mij�[X]=[I] (17) Therefore, the inverse of [[X]T]−1 can be obtained as: [[X]T]−1 =[M][X] (18) And the inverse of [X]−1 as: [X]−1 =[X]T[M] (19) Now, the value of the mass matrix with natural coordinates is needed for the substitution of the modal coordinates in Eq. (18) and Eq. (19). If the modal vector matrix is normalized to unity mass and multiplied by its transpose, then the inverse of the mass matrix in natural coordinates is obtained: [M]−1 =[X][X]T (20) And as [M] is a diagonal matrix, the inverse of the diagonal mass matrix will also be a diagonal matrix with values only in the diagonal components: [M]−1 =�1 mij� (21) From Eq. (20), the value of the inverse mass matrix in natural coordinates is obtained: 16 to the TCP for each FRF measurement, which were then averaged to yield the final result. The coherence between these three measurements was monitored to ensure good coherence especially in natural frequencies. Each FRF measured 3200 lines. Six points on the machine were selected to measure the linear-guide damping values. The machine was divided into different modules for the selection of these points, joined by linear guides: ram, frame, column and bed. Figure 8 shows these six measured points in the finite elements model of the milling machine. Fig. 8 The six measured points on the FEM model of the milling machine. The damping generated on the carriages of each guide can be obtained with the method presented in this study. The same carriage with the same damping capacities, will dissipate different amount of energy depending on the structural joint that is measured, because the mass and stiffness distribution of the joint is different. The same happens with different modes of a single joint, because each mode will have different mass and stiffness distributions, which will have a direct influence on the amount of vibratory energy that the guide will dissipate. The mobility of one point is dominated by mass and stiffness distribution, and damping. The natural frequencies will depend on the distribution of both stiffness and mass, but the amplitude of the modes at natural frequencies are influenced by stiffness, mass, and damping. In this method, it was assumed that the damping introduced by all the carriages of each linear guide was the same. Figure 9 shows the mobilities in the Z direction of points at both sides of the linear guide that joins the column to the bed, as shown in Figure 8. In this figure, the carriage at that structural joint dissipates energy in the first mode at 13 Hz, but dissipates no energy in the fourth mode, at 27.8 Hz. This result is because the distributed mass and stiffness of the points before and after the linear guide favor the damping of the linear guide at 13 Hz, but the same is not true at 27.8 Hz. An analysis of the first mode in Figure 7 shows that the modal shape permits the two points, before and after the linear guide, to move in different ways, which means that their contribution to linear guide damping is likely to be higher. The mode shape of the fourth mode in Figure 7 at 27.8 Hz, shows that both points move together, so there will therefore be no energy dissipation at 17 that joint. Hence, it can be concluded from Figure 9 that the carriages that join the column to the bed dissipate energy in the first mode at 13 Hz, but dissipate no energy in the fourth mode at 27.8 Hz. Fig. 9 Dynamic mobilities before and after the linear guide joining the column/bed in the Z direction. The mobility axis is blind for confidentiality reasons. As shown in Table 1, the damping values obtained for the carriages are between 0-128 N·s·mm-1. These values are for carriages of size 55 and size 65, which are therefore bigger than the reference carriage values of size 35 and size 45 presented in the introduction [31, 32]; nevertheless, they are of the same order of magnitude. Table 1. Measured damping values at the linear guides in N·s·mm-1. CXX CYY CZZ CXY CXZ CYX CYZ CzX CZY Ram/Frame 4 4 12 8 0 0 0 4 16 Frame/Column 4 4 92 8 0 0 0 4 28 Column/Bed 76 4 16 56 128 0 12 4 24 The damping values shown in Table 1 indicate which linear guide dissipates greater energy. The milling machine has the same type and size of linear guides that join the ram to the frame and the column to the bed, but they introduce very different damping values: the biggest difference between these guides is when forces are introduced in the X direction, as the guides that join the column to the bed will dissipate more energy than those that join the ram to the frame, specifically, 20 times more in the X direction and 7 times more in the Y direction. However, the biggest difference is when the TCP is also excited in the X direction and the energy is dissipated in the Z direction: the carriages joining the ram to the frame dissipate no energy as their flexibility on both 18 sides of the linear guides is the same, and instead, the carriages joining the column to the bed introduce the biggest damping value in the machine. Carriages and guides joining the frame to the column are smaller than the other linear guides, although they introduce more damping in the Z direction when the machine TCP is excited in the Z direction than the carriages and guides joining the ram to the frame and the column to the bed. These points therefore confirm the importance of measuring damping values under real working conditions. As previously mentioned, the damping values obtained for each carriage in Table 1 are used to define the distributed discrete dampers in the model. A carriage is modeled with 2D elements and 1D elements representing carriage stiffness and damping. The stiffness and damping in each carriage are represented by four springs and four damper elements, one for each direction. In Figure 10, a linear guide is represented: a 1D spring element represents the stiffness and, in parallel, a 1D damper represents the damping introduced in the milling machine. Fig. 10 Finite element model of slide way and carriage. 3.2 Validation of damping values at linear guides The methodology presented in this paper has been validated by comparing the dynamic flexibility of the machine at the TCP obtained with the finite element model and the experimental dynamic flexibilities measured at the TCP, see Figure 11. It can be concluded that the finite element model of the machine, with material damping and discrete damping values at the linear guides, accurately represents the real milling machine for chatter prediction purposes. These figures represent real dynamic flexibility curves together with theoretical dynamic flexibility curves. The dynamic flexibility curve values and shapes at resonance are in both cases very similar, bearing in mind the simplifications done: 1D elements defining linear guide stiffness and damping Carriages Guide 19 Fig. 11 Dynamic flexibility at the TCP of the machine measured experimentally (EXP) and calculated with FEM in the X, Y, and Z directions. The flexibility axis is blind for confidentiality reasons. The aim of the method was to measure the damping in each guide of the milling machine to obtain accurate frequency response functions at the TCP to predict the dynamic behavior of the machine. In Figure 11, the similarity although not exact, may be seen between the experimentally measured amplitude values and the amplitude values obtained with FEM due to simplifications and model assumptions. Some systems required to adjust FRF outputs. For example, in the study of a torsional vibration damper the outputs need to be adjusted because viscoelastic materials are used, and these are the aim of the study, so a framework is implemented in the model to adjust viscoelastic materials properties [54]. The difference in values could be because any of the three is not exactly represented due to the simplifications in the finite element model, or because other simplifications. For example, no clearances have been modeled. For example, in FRFXX, 1 Hz difference can be seen in the first mode, and 3 Hz in the second mode. So, the stiffness idealization will not only have an effect on the natural frequency values, it will also have a direct effect on the compliance frequency response amplitude values. On the other hand, in Fig. 11, the width of the modes is very similar. There are different methods for the numerical calculation of mode modal damping from the widths of compliance 20 responses. The half-power bandwidth method is a procedure commonly employed to extract damping ratios ξ from the FRF estimates and has been proved to be sufficiently accurate for a number of practical cases in which damping is less than 10% [55]. Table 2 shows each mode modal damping in each direction, calculated with the Half-Power Bandwidth Method. As the width of the modes depends only on the damping, the values in Table 2 suggest that the estimation of the damping of linear-guides assembled in the machine using the proposed method yields good results. Table 2. Modal damping values in each direction obtained with the experimental Half-Power Bandwidth Method (EXP) and with material damping and calculated linear guide damping values (FEM). Modal damping value EXP (%) FEM (%) ξx1 4.2 3.9 ξx2 3.0 2.4 ξy1 2.1 2.3 ξz1 2.0 2.0 ξz2 6.2 9.2 4. CONCLUSIONS A theoretical-experimental methodology has been presented in this paper to measure absolute damping values of linear guides assembled in a milling machine. These values are of great interest to predict the machine dynamic behavior. Here, the damping is obtained with the guides integrated in the machine because, as it has been seen in the case study, the same type and size of linear guides can dissipate different amounts of vibratory energy when joined to different structural components. Consequently, modelling machine tool damping with discrete distributed damping values is a basic requirement, in order to ascertain where damping originates in the machine tool. In terms of damping, linear-guide damping values will help for milling machine redesign. These values will show where the vibratory energy dissipates and will help the milling machine designers to decide where linear guides should be replaced with greater or lesser damping capacity. For example, after a dynamic stability study, linear guides with higher damping values could be introduced where they have a greater effect on chatter limiting modes. It is worth highlighting that this method also allows to identify the linear guides that introduce less damping than their capacities in machine tools, to identify certain linear guides that can be replaced by other linear guides with less damping capacities, and thereby to optimize the milling 21 machine and its cost ratios. In the machine of the case study, improvements in the damping capacity of the linear guides between the ram and the frame are not recommended, because these linear guides dissipate less vibratory energy than their capacity, while the same linear guides between the column and the bed dissipate more energy. Finally, it must be underlined that this theoretical-experimental method is designed to improve the vibratory energy dissipation of fully installed and assembled machines. 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