Effects of nonlinear behaviour of linear ball guideway on chatter frequency of lathe machine tool

Linear ball guideways (LBGs) have several advantages for the precise positioning control of machine tools. However, the ball-groove contact behaviour leads to poor dynamic compliance, which induces a significant nonlinear behaviour of the guideway and a poor chatter stability. The nonlinearity complicates the chatter prediction due to the dependence of the contact stiffness. This approach aims to offer an industrially orientated method of estimation of chatter stability for structures based on LBG. Consequently, the present study first employs Hertzian contact theory to describe the ball-groove contact behaviour in the LBG of a lathe machine. Two static models are presented, both predict the LBG stiffness under different cutting loads. The first model describes is simplified, ignores the effects of the contact force on the deformation of the guideway components, even with that limitation depicts the main behaviour of the system. The second model is more precise and adopts a dynamic substructuring cosimulation method to incorporate the effects of structural deformation into the analysis. For both models, a linearisation approach is employed to analyse the dynamic behaviour of the system under static loading and define the dynamic compliance. The chatter frequencies of the LBG are estimated using the second model and are shown to be in good agreement with the experimental results. In general, the results of the analysis show that fundamental shifts in the frequency response of the LBG occur under specific values of the cutting force and gravity load. In other words, the results confirm that this method provides sufficient estimation of nonlinear behaviour of machine tool based on the LBG.


A, B
Positive constants of the elliptical ball-groove contact area b lim Limit chip width

Introduction
Modern CNC lathes increasingly employ a linear ball guideway (LBG) for positioning purposes rather than a traditional dovetail slide mechanism. However, a high instability of the guideway structure is commonly observed, even in the theoretically stable region. The resulting chatter seriously degrades the machining performance. Hence, it is essential that the dynamic behaviour of the LBG be properly understood such that its frequency response under different cutting conditions can be reliably predicted and its design and the machining conditions tailored accordingly. Therefore, it is necessary to find an effective way to analyse the impact of LBG on machine tool dynamics. Wang et al. [1] investigated the nonlinear dynamics of a machine tool positioning table using a three-degreeof-freedom dynamic model and found that the dynamic compliance of the system varied dramatically with the contact deformation and force. Similarly, Xu and Kong [2,3] showed that the dynamic nonlinear stiffness of a LBG results in a chaotic behaviour, which varies in accordance with the excitation frequency and magnitude of the cutting load. Several studies have examined the stiffness behaviour of LBGs under the effects of a vertical load [4,5]. However, in practical machine tools, the dynamic behaviour of the LBG is governed by multi-dimensional torsion and bending loads [6,7]. Moreover, the preload applied to the LBG has a fundamental effect on the dynamic properties of the entire system and hence impacts the chatter stability of the tool [8,9]. Yang et al. [10] showed that the contact stiffness of the groove in linear rolling guideways may vary by as much as 68-85% in different directions, depending on the type of load. Several studies have shown that the ballgroove contact behaviour, which represents the main source of nonlinearity in LBGs, can be well described by Hertzian contact theory [11,12]. Sun et al. [13] used a piecewise finite element modelling approach to simulate the dynamic contact behaviour of the rolling balls and grooves in a linear rolling guideway. The validity of the proposed approach was demonstrated by comparing the simulation results with the analytical and experimental solutions.
The mentioned models aim to describe the behaviour as faithfully as possible, corresponding to their primary purpose of analysing the load distribution. However, the use of these models for the global analysis of a machine tool, where LBGs are only one of many components, is limited by computational complexity. This work aims to find an efficient approach to solving LBG-based structures. The goal is to use an effective model that describes non-linear behaviour yet, at the same time, will be computationally undemanding. A model based on the Hertzian contact theory meets these needs; it describes the fundamental behaviour of LGB nonlinearity with low computational demand.
To properly design a nonlinear LBG, it is necessary to divide the system into its linear and nonlinear parts and to treat both parts separately [14]. The dynamic substructuring of machine tools is generally performed as a means of analysing the position-dynamic dependence of the tool [15,16], or to examine the variability of the tool behaviour [17]. However, it was shown in [18] that dynamic substructuring also provides an efficient approach for analysing joint contact. In a previous study [19], the present group employed a linearisation approach to describe the dynamic behaviour of a simple planar nonlinear system. In the present study, a similar approach is adopted to examine the complex dynamic behaviour of the LBG structure in a CNC lathe machine.
This approach aims to offer an industrially oriented method of chatter stability estimation for structures based on LBG. The well-known methods are used in a new computational scheme. That enables effective analysis of chatter behaviour, avoiding nonlinear solving dynamics. The four steps are used, first estimate the reaction force in bounds, next linearise the nonlinear reaction of LBG, then are calculated linearised frequency response function with estimated stability lobe diagrams, and the last step to match the linearised solution with the cutting condition load.
Two static models are proposed to estimate the frequency response and chatter behaviour of the guideway structure under different cutting conditions. In the first model, the guideway components are assumed to be perfectly rigid, so the effects of load-induced deformation can be ignored. This model enables the estimation of fast initial solutions and trend verification in a more complex second model. By contrast, in the second model, the effects of structural deformation on the dynamic ballgroove contact behaviour are taken into account using a substructuring and cosimulation technique. In both models, the dynamic compliance of the guideway structure is analysed using a linearisation approach. The chatter stability of the linear guideway is estimated using the second model and is shown to be in good qualitative agreement with the experimental observations under typical cutting conditions.

Regenerative chatter
Regenerative chatter is a form of self-excited vibration in machining operations which occurs when the tool comes into contact with the surface cut in previous rounds. Gegg et al. [20] presented a simplified model of the self-excited vibration effect, in which the tool was represented as a mechanical oscillator, as shown in Fig. 1. This system can be described as follows: In the proposed model, it was assumed that the waves left on the material surface in previous rounds of the cutting process interacted with the deflection of the tool in subsequent rounds and lead to a variable cutting depth as a result. The variable depth then produced force pulses, which supplied energy to the vibrating tool. The magnitude of the pulses varied in accordance with the phase shift between the previous turn tool path and the current tool deflection, with the strongest pulses occurring at a shift of 180 • and the weakest pulses (zero force) occurring at 0 • . The instability of the cutting system was thus determined by the energy flow. In particular, the system became unstable when the pulsation energy produced by the variable cutting depth exceeded the ability of the system to dissipate the energy.
Analysing vibration during machining requires knowledge of the chatter phenomenon. Regenerative chatter is a kind of self-excited vibration caused during machining operation where the tool is in contact with the surface from the previous round of cuts. Gegg presents a simplified model of self-excited vibration [20]. The simple model represents  Fig. 1. The dynamic behaviour of the cutting process through the material is affected by the previous cut, which left waves on the material surface. These waves interact with the actual deflection of the tool, and the result is a variable depth. Varying depth causes force pulses that supply energy to the vibrating tool. The magnitude of pulsations is given by phase shifts -the strongest pulsations occur at a shift of 180 • ; in contrast, none happen at a 0 • shift, where the instability of the cutting system is determined by energy flow. If the system's pulsation energy supplies outweigh the system's ability to dissipate the energy, then the system is unstable.
When evaluating the cutting stability of machine tools, one of the most critical parameters is the chip width. According to Tlusty [21], the limiting chip width, b lim , varies inversely with the total specific cutting force (K s ), the angle of the cutting force (β), and the negative real part of the system transfer function Re [FRF] which is solution of Eq.1, i.e.
According to the adopted chatter model, the stability boundary is independent of the feed. In general, the chatter frequency, f c , of a machine tool can be characterised as follows: where is the spindle speed and N is the harmonic order.
Equations (3) and (4) show that the chatter frequency is insensitive to the damping effect. Hence, when estimating the chatter frequency, the effects of process damping can be ignored, and the frequency response shift can be used to estimate changes in the system stiffness and clarify changes in the machine tool dynamics directly. However, process damping has a critical effect on the b lim stability limit in the low-speed region. Consequently, if the estimated chatter stability conditions lie in the low-speed area, the analysis should be extended to include a process damping model, as described in [22]. Figure 2 presents a simple schematic illustration of the LBG structure considered in the present study. As shown, the system consists of two LBG structures to facilitate positioning in the z-and x-axis directions, respectively, and a ball screw and nut mechanism to constrain motion in the z-axis direction. The present analysis focuses on the case of a grooving operation, for which the force vector, F c , is dominant in the X-Y plane. Consequently, the effects of the nut and ball screw behaviour, which act perpendicular to the main force input, can be neglected in a simple static analysis.

Regenerative chatter model for lathe system
Drawing an analogy between the system shown in Fig. 2 and the simple grooving model shown in Fig. 1, the stiffness term k in Eq. (1) represents the stiffness of the guideway structure, while the frequency response function, FRF, is determined by the deflection of the tool in the x-axis direction under the effects of the cutting force load, F c . In analysing the regenerative chatter of the guideway structure, the following assumptions are made: • The LBGs are the key source of nonlinear behaviour in the considered system and are the main determinants of the total system stiffness. • The nature enables replacement of the LBG with point stiffness because the bending components are neglectable and a similar load of each ball in a single row is expected. • The cast iron structures of the linear guideways have linear stiffness. • The effects of gravity must be taken into account due to the nonlinear characteristic of the guideway behaviour. • The ball screw mechanism has only a minor effect on the system response during grooving and can thus be neglected in a simplified static analysis, or considered as linear in modal analyses and detailed static analyses. • The deflection of the guideway in the y-axis direction has only a minor effect on the chatter behaviour during grooving and can be neglected. • The chatter frequency is determined mainly by the system stiffness and mass, and the process damping effect is sufficiently small to be ignored.

Ball-groove contact
The ball-groove contact is the main source of nonlinear stiffness in LBGs and determines the overall rigidity and nonlinear behaviour of the entire guideway structure. It was shown in [23] that the behaviour of the ball-groove contact can be described using Hertzian contact theory, in which the basic ball contact force is defined by the equivalent radius and equivalent Young's modulii of the contact area and contact bodies, respectively. According to Johnson [23] (eqs. 5-9), for an ideal circular contact area between the balls and the groove, the contact force can be formulated as Meanwhile, the equivalent elastic modulus, E * , can be expressed in terms of the elastic modulus, E, and Poisson ratio, ν , of each contact body as However, for the LBG problem considered in the present study, the ball-groove contact has an elliptical rather than circular geometry, as shown in Fig. 3. Hence, Eq. (5) should be rewritten as where μ varies as a function of the eccentricity of the contact ellipse, e, which is defined in terms of two constants,

Fig. 3 Ball-groove contact model
A and B, as follows: The elliptical constants, A and B, can also be used to define the relative radius, R e , of the contact area, i.e.
Assuming that the ball and groove undergo no deformation at the contact point between them, the contact force in Eq. (7) can be rewritten in terms of the non-linear stiffness parameter, k n , as follows: The analysis above considers only a single point of contact between the groove and ball (see Fig. 3). However, in practice, the rail and carriage in the LBG are connected through two contact points at the top and bottom of each ball, respectively. For the case of a LBG with two rows of balls on either side of the carriage (see Fig. 4), the stiffness of each row should thus be expressed as where k n is the stiffness of the contact between each ball and the groove and n b is the number of balls in contact in the row. For simplicity, it is assumed that the individual rows have an identical stiffness behaviour, i.e. k A = k B = k C = k D . Each rows (A,B,C,D) can then be described by the reaction force F A which depends on the local reaction x A , four of these equations then describe LGB behaviour as

Static model of preloaded linear guideway
As shown in Fig. 4, the individual contacts in the LBG are assumed to form a perfect square. Substituting the parameters given in Table 1 for a commercial R1671-212-20 LBG into Eqs. (5)-(11) yields the nonlinear stiffness of The initial contact deformation, x p , of each ball in row k A can then be determined as Substituting the total preload force of 580 N in Table 1 into Eq, 13, the preload force acting on row k A is found to be 410.1 N. The contact deformation, x p , is then obtained as 1.95 μm. Figure 5 shows the force-deflection characteristic of each row of balls in the considered LBG. A clear nonlinearity in the stiffness is observed, where the border between the two stiffness regimes corresponds to the value of the preload deformation (x p ). At deflection values lower than x p , contact between the balls and grooves is lost, and hence the load is not transferred between them.

Single direction loading model
The nonlinear behaviour of LBGs is usually evaluated for the case of single-direction loading [24]. Such an approach has the advantage of simplicity and enables the loading characteristics provided by the manufacturer to be employed directly. However, the model is unable to reflect the typical loading conditions encountered in real-world machining operations, in which the guideway is subjected to a combination of directional loads and torques. The static model described in Section 3.2 considers the case where the guideway is subject to combined twodirectional loading and torque. For the case of singledirectional loading, the model can be simplified as follows: The LBG has geometric symmetry, and hence the nonlinear stiffness can be represented arbitrarily by k A . Furthermore, x 0 is the initial deflection of the guideway in the x-axis direction under the effects of the preload force, and x is the deflection of the guideway in the xaxis direction during the subsequent machining process. The preload force, F pr , can be defined as The preload effect is lost when the deflection is equal to twice the value of that produced under the initial preload force. The preload lost force, F l , the loading force at which the preload effect is lost, can then be defined simply as Figure 6 shows the relationship between LGB's deflection and loading force, the preload area and the area with loss of contact are not very distinguishable. It does not seem that this nonlinearity has some crucial role; however, it has. Figure 7 shows that the total system stiffness reduces by around 20% when the loading force reach the pointwhere is the preload effect lost. Furthermore, a distinct difference in the stiffness behaviours of the different rows of balls is observed. In particular, the stiffness of rows A and D reduces, whereas that of rows B and C increases. As shown in Table 1, the considered R1671-212-20 guideway has a preload force, F p , of 580 N. The preload lost force is thus calculated from Eq. (17) to be 1641 N, as shown in Fig. 7. 4 Nonlinear analysis of machine tool structure with linear ball guideway Figure 8 shows the machine tool slide structure considered in the present study consisting of a central rectangular beam and two wide LBGs (see Table 1) for the related parameters). To facilitate the analysis, the structure is simplified to a single axis in order to reduce the number of unknown parameters and highlight the behaviour of the LBG. In particular, the analysis considers the central beam to have a fixed position in the z-axis direction (i.e. the servo brake is activated).
The present study proposes two static models based on the simplified structure shown on the right-hand side of

Analysis model based on simplified static contact joint load
In developing the simplified static solution for the considered problem, it is first necessary to define the boundary conditions and external loads. For simplicity, the radial reaction in the ball screw mechanism of the guideway and the deflection of the guideway in the z-axis direction are both ignored. Furthermore, all of the guideway components (with the exception of the central beam) are assumed to be perfectly rigid. Two loading forces are considered, namely the weight, F g , and the cutting force, F c According to the static scheme shown in Fig. 8, the force reactions in each LBG can be described by Eqs. (18)-(22), while the torque reaction between the two LBGs is given by Eq. (23). These equations define the load of both LBGs, using the model in Fig. 4, the force reactions LBG's rows are numerically solved. Note that the model parameters for the static analysis are listed in Table 2.
The results obtained for each eight ball grove contact reaction (A 1 -D 2 ) are then transformed into a linearised stiffness model. Figure 9 shows the linearised stiffness results obtained for each row of balls as a function of the cutting force, F c . The front LBG contacts have suffixes 1 and the rear 2. Note that under a force of F c = 0 N, the structure is loaded only by the weight, F g . For the rows subject to a tensile load (D 1 , B 2 ), the stiffness reduces as the cutting force increases due to the loss of contact between the balls and grooves. By contrast, for the rows under a compressive stress (A 1 , B 1 , C 1 , A 2 , C 2 , D 2 ), the contact area between the balls and grooves increases, and hence the stiffness also increases. The minimal stiffness of the system coincides with the point at which contact is lost between the balls and grooves, i.e. F c = 380 N.
Having defined the boundary conditions and external loads, the model for modal analysis then assumes all the structural components to be flexible and replaces the ball rows with linear springs. Note that the spring stiffness parameters are taken as the linearised stiffness parameters (A 1 -D 2 ) obtained from the previous nonlinear static model. A modal analysis is then performed to determine the  Fig. 8).
Finally, the modal shapes are determined under different static loads. Figure 10 shows the variation of the first three natural frequencies of the experimental structure with the cutting force, F c . The results show that the second mode experiences a high frequency drop of around 10 Hz under a load of 350 N. The first mode also exhibits a frequency drop at around the same value of the loading force. However, in this case, the frequency drop is relatively small (∼ 2 Hz). The frequency of the third mode, by contrast, shows no obvious drop and increases progressively by around 2 Hz as the cutting force increases. Overall, the results suggest that the chatter stability of the guideway is dominated by the second natural frequency of the guideway structure. Figure 11 shows the first three mode shapes of the linearised guideway structures under the effects of the gravity load and combined gravity and cutting load,  respectively. The first and third modes show no obvious change in shape under the effects of the cutting load, although a slight change in frequency is observed. However, the second mode shows a more significant change in shape as a result of the loss of contact between the balls and grooves, and hence a greater reduction in the frequency occurs.
In performing the harmonic analysis, a unit force was applied to the tool tip of the experimental structure in the x-direction and the corresponding tool deflection was evaluated. Figure 12 shows the results obtained for the dynamic compliance of the tool tip in the x-axis direction given a gravity loading force only and a combined gravity force and cutting force of F c = 375 N, respectively. The results show that the natural frequency reduces, while the amplitude increases, under the effects of the cutting force

Nonlinear static analysis using substructuring approach
The analysis in Section 4.1 ignores the flexibility of the guideway structure components on the contact stiffness. To obtain a more accurate prediction of the stiffness behaviour of the guideway, it is necessary to integrate an FEM model with the nonlinear model of the LBG. However, most FEM packages support nonlinear elements only without preload, and hence it is impossible to apply such models directly to the considered problem, in which the LBG is subject to a preload force before the machining process commences.
Accordingly, in the second model proposed in this study, the guideway system is divided into a linear part and a nonlinear part, respectively. For the linear part, modal reduction is applied, which reduces the number of modes of the linear structure, whilst preserving the static behaviour of the defined contact nodes. Meanwhile, in the nonlinear part, is represented by eight nonlinear equations for each contact row (see Eq. 12). The nonlinear and linear are then joined in the Simulink environment using forcedeflection relations in contact nodes. Figure 13 shows the results obtained for the stiffness of the four rows of balls in the guideway structure as a function of the loading force. It is observed that the stiffness at contact B 2 reduces far more rapidly than in the rigid body analysis due to the platform flexibility below the linear ball carriage, which increases the ball-contact deformation. However, other than this tendency, the results presented in Fig. 13 are qualitatively similar to those in Fig. 9. Hence, it is inferred that the simplified static model in Section 4.1 Fig. 12 Dynamic compliance of tool tip under gravity force and combined gravity force and cutting load provides an adequate description of the primary behaviour of the considered structure.
As shown in Fig. 14, the first two frequency modes of the guideway both show a two-step drop in frequency as the loading force increases as a result of the structure compliance below LBG. Compared to results of the simplified static analysis 4.1, the minimum frequency drop moves to the higher frequency to 73 Hz this is caused by that the minimal stiffness of both LBG do not meet in the same load. The third frequency mode shifts to the higher frequency -160 Hz; however, it has no significant trend in dependency on load; therefore, Fig. 14 shows only first two mode trends which differ from Fig. 10. It is noted that the minimum frequency occurs at the same load as in the simplified system (350 N).   Fig. 15. The compliance for zero cutting load differs from the simplified model (see Fig. 12) mainly by the third mode position. Significant changes are for loads of 240 N and 380 N; the compliance of the first mode increases dramatically and becomes more significant for the behaviour. Unlike in the simplified model, the second mode decreases its compliance. Above 380 N load, the trends change and the first mode decreases and the second mode increases. Since the static substructuring analysis considers the guideway structure to provide a more reliable load distribution, it will be used for the following

Chatter stability and frequency estimation
Chatter stability is an essential property of machine tools. However, it is difficult to predict the chatter behaviour of structures with nonlinear joint stiffness since the dynamic compliance of the overall system is load dependent. Hence, before the chatter can be analysed, it is first necessary to estimate the dynamic behaviour of the system under different loads. Linearised systems are applied to equations 2-4; the solution of these equations gives the stability boundaries for the LBG under different cutting loads (see Fig. 16). In other words, for each static load that acts on the tool, there exists a particular linearised dynamic characteristic of the system and a unique stability diagram exists for the load of each static system. Figure 16 shows a significant discontinuity in stability at the minimum system of F c = 375 N. However, a more minor discontinuity is also observed at F c = 218 N, corresponding to the loss of contact at B 2 .
The cross-sections of the surface in Fig. 16 represent the stability lobes for different cutting forces. Figure 17 shows four illustrative cross-sections corresponding to loads of 0, 200, 400, and 900 N, respectively. Note that the force of 0 N corresponds to the case where the structure is loaded only by gravity and is an important value since the boundary thus corresponds to that obtained in impulse hammer measurements when the structure is unloaded.
It is noted that in this condition, stability prediction based on impulse hammer measurement is likely to be highly unreliable for the higher load -with increasing load the stable area between the two lobes is filled with new instability, and the stable area for the unloaded system becomes highly unstable after loading. Paradoxically, the highest criterion of the unconditional stability band has a load in the case of minimum rigidity, and with the lower system compliance the lobe shifts to a lower spindle speed. As the load increases, the area between the two lobes decreases slightly, and the lobes move toward higher spindle speeds (as evidenced by the curves for 400 N and 900 N, respectively).
In the lathe machining process, the cutting load depends on the chip width, feed, etc. Therefore, for each loaf variable would be a different stability diagram which could be compiled using linearisation synthesis. The method used to create such a synthesis is described in [19]. The algorithm arranges a stability diagram corresponding to the load. For illustration, stability diagrams were created for the feed of 0.1 and 0.05 mm/rev. However, it is possible to create any feed corresponding to a linearised range. The resulting diagram is then unique for every feed and cutting force. Figure 18 shows an illustrative stability chatter diagram for feeds of 0.1 mm/rev and 0.05 mm/rev, respectively, given a specific cutting force of Ks = 2500 MPa. For a feed of 0.1 mm/rev, the lobe diagram has noticeable regions corresponding to a loss of contact at B 2 and D 1 are observed, resulting in notches in the range of 10000-12000 RPM at loads of 218 N and 375 N. The loss of contact in B 2 also affects the speed range of 3500-5000 RPM causes; with increasing load, the first mode becomes significant and reduces the stable area. For a lower feed of 0.05 mm/rev, the system behaves more like a linear system. In particular, the changes in the lobe diagram shape are less significant since due to lower load the more considerable changes shift The stepped effect in the chatter stability diagram is given by the linearisation interval used for the nonlinear static analysis of the contacts in the bonds. Overall, the results presented in Fig. 18 show that the required fineness of linearisation increases with a reducing feed. Thus, when using consistent linearisation, this effect of linearisation intervals is more pronounced in the case of slower feeds. Part of the chatter stability prediction is also the frequency response analysis. Applying the linearised solution to Eq. (4) yields the predicted chatter frequency for specific cutting conditions. By monitoring the frequency response, the influence of both structural damping and process damping can be eliminated, thereby allowing the effects of stiffness nonlinearities to be reduced.

Comparison of simulated impulse with impulse hammer measurements
The cosimulation model enables time-domain simulation. It is thus feasible to simulate the structure response for a particular impact and compare the results with the corresponding experimental observations and linearised solutions. In the present study, the simulation input was taken as the force signals acquired in real-world impulse hammer tests. The hammer impact points and sensor locations used in the experimental and simulation processes are shown in Fig. 19.  Figure 20 compares the experimental and simulation results for the dynamic compliance of the test structure under the effects of a 160-g impulse. It is noted that the measurement results correspond to the modelled values. In general, the results show that the second mode has the lowest dynamic stiffness in the x-direction. In addition, the simulation results for the third mode frequency are around 30 Hz higher than the measured data. The higher modes in comparison then approximately correspond in frequency position but differ in peak size. The model assumes a constant damping parameter, and higher frequencies are thus attenuated compared to the experimental results. However, the stability of the considered system depends mainly on the frequency shifts which occur at low frequencies (i.e. the first two modes), and hence discrepancies between the measurement and model are insignificant. In addition, its significance is enhanced by the logarithmic scale of Fig. 20.   Fig. 20 Comparison of simulated and measured dynamic compliance of test structure in x-axis direction given impulse force of 160 g

Experimental machining trials for chatter behaviour verification
In practice, experimental verification of the chatter behaviour faces several key challenges, including the low stability of the system itself and the fact that there is no stable area between the lobes in the stability diagram at lower speeds, the area practically filled with harmonic lobes without any windows of higher chip width stability.
Due to the dimensions of the tested structure and the requirements for independent movement, the verification trials were performed on a large horizontal milling machine (TOS WHN 13 A), which allowed both the spindle function and table movement to change the chip width and ensure feed in the cut. The milling machine provided several advantages for the verification process, including most notably its large dimensions and sliding guide mounting, which allowed its dynamics to be isolated from those of the tested LBG system. However, the machine offered only a very limited choice of spindle speeds for the cutting process (i.e. 430, 465, 600, and 765 RPM). Consequently, the acquisition of an experimental lobe diagram was almost impossible, and hence the only feasible option for evaluating the chatter behaviour of the guideway was to verify the frequency response under unstable machining conditions. Figure 21 presents a photograph of the experimental setup. As shown, the guideway was mounted on a milling machine table. Strain gauges were attached to the tool holder to reconstruct the force load, and accelerometers were applied to the body of the system to measure vibration in different directions during the machining (Fig. 19). The signals generated by the strain gauges and accelerometers were collected by a DAQ module and interfaced to a PC for subsequent processing. The workpiece was a steel cylinder (C45) with a diameter of 70 mm, and it was mounted on the spindle of the horizontal milling machine. In the machining Fig. 21 Photograph of experimental setup used to verify chatter behaviour of LBG trials, grooves were cut into the end face of the workpiece with a grooving width in the range of 1 to 4.5 mm with increments of 0.5 mm.
As the width was increased, the depth was gradually reduced from 3 to 1.4 mm in steps of 0.2 mm. For each machining condition (i.e. groove width and depth), two trials were performed in order to ensure the reliability of the measurement results. The signals acquired from the strain gauges and accelerometers were used to construct a corresponding power spectral density (PSD) diagram from which the frequency response of the system was then determined.

Results and discussion
The PSD maps in Fig. 22a to d show the dependence of the response frequency of the experimental structure in the x-axis direction on the width of the chip at spindle speeds of 430, 465, 600, and 765 RPM, respectively. The most significant spectral changes are observed at a spindle speed of 765 RPM, for which a frequency component of 74 Hz, which is not predicted by linear theory, is detected at machining depths greater than 4 mm. A similar phenomenon also occurred at the other spindle speeds; however, the signal strength was far weaker and is thus not easily seen in the corresponding PSDs. For spindle speeds of 430 RPM and 465 RPM, the PSD changes in the area above 3 mm, multiple new spectra occur, and the original peaks are expanding. The least noticeable is the spectral change in PSD at 600 RPM where these changes in PSD are the weakest and when there are small changes for data above 4 mm. These changes can also be observed at higher harmonic frequencies.
The spectral changes observed in the PSDs can be attributed most feasibly to the nonlinearity of the system. The nonlinear model presented in Section 4.2 accurately depicts the behaviour of the system and the aforementioned spectra, whereas a simple linear approach cannot. Figure 23a and b compare the chatter frequencies predicted by the nonlinear model and a linear model for chip widths of 1 mm and 4.5 mm, respectively. It will be recalled that the linear model does not consider the change in system stiffness caused by the load and represents the results obtained in impulse hammer measurements. For reference purposes, the estimated spectra are compared with the experimental data acquired by the strain gauges, which more accurately describe the self-excited frequency responses and eliminate the surrounding influences of random excitation. The size of the plotted points corresponds to the signal strength of the corresponding PSD and hence provides an indication of the measured instability. In other words, the points indicate the most dominant chatter frequencies.
It should be noted that the results correspond to unstable cutting conditions. Although the chatter has an impulse character, during its initialisation, the static component of the cutting force is dominant and the force oscillates around the value of the static force. Therefore, the static force component is expected to load the LBGs, resulting in initialising the corresponding chatter frequency. Consequently, the linearised estimates for frequency prediction are not as accurate as those for the machining process performed at the edge of stability, since, during unstable cutting, the tool passes through many different stiffness states.
Nonetheless, despite this limitation, the estimates obtained using the cosimulation model are in good qualitative agreement with the measurements results for the self-excited vibrations. In most cases, the estimates deviate from the measured frequencies by no more than 2 Hz. However, more significant deviations are observed for the largest chip width of 4.5 mm and lower spindle speeds (430~465 RPM). For example, an estimation error of 5 Hz occurs at 465 RPM, while an error of 3 Hz occurs at 430 RPM. Notably, these findings confirm the nonlinear behaviour of the system since, according to linear theory and impulse hammer measurements, no vibration should occur in the spectrum band of 70~80 Hz.
Overall, the results confirm that the nonlinearity of the LBG structure must be taken into account especially when the external load causes preload loss. Furthermore, the measurements obtained by a modal hammer for such structures provide only limited information about the structure response. In future work, the linearisation method applied in the present study for stability estimation will be integrated with a process damping model to better describe the system response in the low-speed region. Furthermore, the estimation performance will be enhanced through the application of a stochastic approach to the dispersion of the specific cutting force and the definition of the probability band of the stability lobe diagram.

Conclusion
This study has provided an industrial-orientated analysis of the nonlinear vibration behaviour of LBGs on the chatter stability. Having constructed a ball-groove contact model based on Hertzian contact theory, two models for nonlinear static analysis have been derived, namely (1) a simplified model in which the contact-induced deformation of the solid machine parts is ignored, and (2) a dynamic cosimulation The results obtained from the two models have elucidated the manner in which a nonlinear contact between the balls and grooves in the LBG structure influences the entire dynamics of the system. The difference between the two models lies in the faster progression of the preload loss in the cosimulation model due to the deformation of the surrounding structure. However, both models show a frequency decrease as a result of the contact lost in the LBG, and thus properly explain the observed behaviour of the system. Due to the limitation of the simplified model, only the results of the cosimulation model have been applied to the synthesis of linearisation at the operating points to predict the chatter behaviour of the system under different load conditions. The validity of the simulation results has been investigated by experimental trials. Due to the low damping performance of the considered system, the experimental verification of the nonlinear chatter predictions has been limited to a simple comparison of the predicted and measured chatter frequencies. However, the two sets of results have been shown to be in good general agreement (within 3 Hz).
In general, the results obtained in this study have confirmed the importance of analysing the nonlinearity of the LBG joints in machine tools, such as lathes and milling machines. The presented method offers an effective way to predict the chatter behaviour of structures based on LGB. The results have shown that the nonlinear contact behaviour of the balls and grooves has a dramatic effect on the behaviour of the entire system and must therefore be taken into account during the design process. The methods presented can also be used in many fields for load distribution analysis and to explain the frequency change caused by loading.