Experimental Investigation of Dynamic Chip Formation in Orthogonal Cutting

Tamas G. Molnara,∗, Szabolcs Berezvaia, Adam K. Kissa, Daniel Bachrathya, Gabor Stepana

aDepartment of Applied Mechanics, Budapest University of Technology and Economics, H-1111 Budapest, Hungary

Abstract Orthogonal planing experiments are conducted with high-speed camera recordings and simultaneous cutting force measurements in order to analyze chip formation during machine tool vibrations. By means of carefully designed periodic forcing of the cutting tool, non-stationary cutting experiments are performed including wave generation, wave removal and wave-on-wave cutting, which are then compared to stationary cutting tests. The experiments are performed to address how the cutting force is affected by the chip thickness variation, the surface waviness and the fluctuation of the cutting direction. The results are used to assess some theoretical models that involve shear angle variation and chip segmentation. Keywords: metal cutting, chip formation, chatter, cutting experiments, high-speed camera

1. Introduction geometric concepts take into account that chatter gives rise to chip thickness variation, surface waviness and cutting direction Chatter vibrations may arise in a wide range of metal cut- fluctuation, which affect the geometry of the chip deformation ting processes including turning [1, 2, 3], milling [4, 5, 6, 7], zones and hence the cutting force. drilling [8], grinding [9], boring [10] and turn-milling [11]. These machine tool vibrations are known for their harmful na- ture that reduces the efficiency of the cutting process, the tool The focus of this paper is the experimental investigation of life and the quality of the surface finish. Although chatter has stationary and time-varying chip formation. Orthogonal plan- been studied for several decades, its accurate prediction based ing experiments are conducted to make comparison with the on dynamical models is still a challenge for researchers and in chip formation theory in [12, 13, 14], and the results are inter- industry. preted from machine tool vibrations point of view. Although The difficulties in the prediction of chatter originate in the the planing process operates with small cutting speeds, it en- intricate nature of chip formation and the complexity of the un- ables the accurate realization of various tool paths. This way, derlying physics. A large number of factors influence the mate- cutting force fluctuations during time-varying chip removal can rial removal process such as cutting conditions, tool and work- be studied thoroughly. piece geometry and material properties. During chatter, further complications arise from the fact that chip formation is time- Apart from the inspection of cutting force signals, special at- varying and is affected by the dynamics of the machine, which tention is devoted to the assessment of shear angle models and varies from one cutting setup to another. the observation of chip segmentation. These are supported by Consequently, researchers typically simplify the analysis of high-speed camera recordings of the chip formation process, chatter by introducing elementary models with few parameters which are rarely used by quantitative studies in the literature that are usually identified by experiments. In order to con- (for some examples, see [15, 16, 17, 18]). Although experimen- struct these fundamental models of chip formation, stationary tal results on chip formation have been published for more than (chatter-free) orthogonal cutting experiments are performed. half a century (see e.g. wave generation and wave cutting exper- Then, time-varying chip formation is predicted (extrapolated) iments on a lathe in [19, 20, 21, 22, 23]), today’s sophisticated from the relationships obtained for stationary cutting. instruments can still provide new insights into the mechanics The extension of stationary chip formation models to the of metal cutting. Some recent works about high-speed camera case of dynamic (time-varying) cutting processes is not trivial. recordings of machining processes can be found in [18, 24, 25]. Hereinafter, the works of [12, 13, 14] are followed and sim- ple geometric concepts are investigated to extend the stationary relationships for the study of machine tool vibrations. These The paper is organized as follows. Section 2 describes a the- oretical model based on [12, 13, 14], which uses a geometrical explanation for the cutting force fluctuations. The experimental ∗Corresponding author setup compiled for the planing tests is introduced in Section 3. Email addresses: [email protected] (Tamas G. Molnar), Sections 4-6 present the results of various stationary and non- [email protected] (Szabolcs Berezvai), [email protected] (Adam K. Kiss), [email protected] (Daniel Bachrathy), [email protected] stationary cutting experiments, while conclusions are summa- (Gabor Stepan) rized in Section 7.

Preprint submitted to International Journal of Machine Tools and Manufacture July 22, 2019 2. Basic Model of Chip Formation z In order to address the effect of machine tool vibrations on normal to the y the cutting force, a basic model of dynamic (time-varying) chip effectiv e cutting formation is formulated following [12, 13, 14]. This model is direction used later to explain most of the experimental observations. In effectiv e cutting shear the model, orthogonal cutting is investigated and vibrations are direction plane tool considered in the feed direction only since vibrations in the ar nominal cutting direction are typically less significant in the . Da flank models of regenerative machine tool chatter [26]. First, the tool F a face F s rak e is considered to be sharp, then remarks are given regarding the n F

t a face

D .

. v effect of the tool edge radius. The behavior of the workpiece . F F c . material is approximated as elastic-perfectly plastic. Built-up f ba
r a F . z ba F v edge formation is not taken into account by the model, although ba
r a z a Da it may occur at small cutting speeds. Furthermore, it is assumed r F F y u that there is no contact between the tool’s flank face and the F workpiece. v . w orkpiece The model considers the effect of fluctuations in the cutting direction due to machine tool vibrations and the waviness of the initial surface to be cut. These have significant influence on the Figure 1: Illustration of the cutting velocity (blue) and the cutting force com- geometrical parameters of chip formation such as the effective ponents (brown) via Merchant’s circle [14]. Notice that the effective cutting cutting direction and the effective rake angle (see definitions be- velocity v determines the effective cutting direction (dashed line), opposed to low) and the size of deformation zones in the chip. Since these the nominal cutting velocity vc. The cutting force can be decomposed into four parameters strongly affect the cutting force, the term geomet- pairs of components: (i) F = Fu + Fv, (ii) F = Fy + Fz, (iii) F = Ft + Ff and (iv) F = Fs + Fn according to alignments to (i) the rake face, (ii) the nominal rical explanation is used when the cutting direction fluctuation cutting direction, (iii) the effective cutting direction and (iv) the shear plane, and the surface waviness are taken into account to address cut- respectively. ting force variations. Due to its geometrical nature, this expla- nation could be applied to various workpiece and tool material combinations, however, the experiments of this paper are re- Note that the effective chip thickness is of less importance in stricted to aluminum workpiece and carbide tool. this model as the shear plane length will be used instead later.

2.1. Fluctuation of the cutting direction 2.2. Components of the cutting force

Let vc denote the nominal cutting velocity vector (that is par- First, the idealized case of a perfectly sharp tool with zero allel to the y axis in Fig. 1). When machine tool vibrations oc- tool edge radius is considered. The cutting force is decomposed cur, it is modified by the vibration velocityz ˙, and the effective into various component pairs which are then used to build a cutting velocity becomes v as indicated in Fig. 1. The direc- cutting force model for the sharp tool. The effect of tool edge tion of the effective cutting velocity, i.e., the effective cutting radius is addressed later in a separate subsection. direction is given by the angle ∆α where Let F denote the resultant cutting force acting on the tool. The direction and the components of the cutting force are il- z˙ tan ∆α = , (1) lustrated in Fig. 1 via Merchant’s circle [14]. On one hand, v c the cutting force can be decomposed into components Fu and cf. Fig. 1. In what follows, the effective cutting direction is used Fv being parallel with and perpendicular to the rake face, re- instead of the nominal one in order to describe the dynamics of spectively. The ratio of these components is a kind of average chip formation. friction coefficient µa that is associated with the average friction As a first step, the direction perpendicular to the effective angle βa: cutting direction is used to determine the rake angle of the tool. µa = tan βa . (4) Thus, the nominal rake angle αr is modified by the effective The direction of the cutting force is characterized by the angle cutting angle ∆α, and the effective rake angle becomes βa. This angle can be determined by measuring the y- and z- directional cutting force components Fy and Fz and by using α = α − ∆α , (2) r ! Fz as shown in Fig. 1. In addition, the chip thickness should also βa = arctan + αr , (5) Fy be measured according to the effective cutting direction. Thus, cf. Fig. 1. the effective uncut chip thickness heff can be calculated as the projection of the nominal uncut chip thickness h perpendicular Apart from the decompositions F = Fu +Fv and F = Fy + Fz, to the effective cutting direction: the cutting force can be decomposed into tangential and feed components Ft and Ff, which are aligned with and perpendicu- heff = h cos ∆α . (3) lar to the effective cutting direction, respectively. 2 2.3. Cutting force model for sharp tools (a)stationary cutting (b) wave removal During chip formation in orthogonal cutting, the material a h*≠h a h*=h h r h r in front of the cutting edge is sheared over a primary shear ltool l tool zone [14]. This zone is approximated by the shear plane, whose F F length is denoted by l and whose width is the chip width w. The v=v v=v shear plane is located at shear angle Φ, which is measured from c w orkpiece c w orkpiece . . the effective cutting direction. Based on the shear angle Φ, the h l h h v v h l h h v v =0 ~ = * = c ≠0 ~ *≠ = c cutting force F can be calculated in the following manner [14]. (c)wave generation (d) wave-on-wave cutting The cutting force is decomposed into components Fs and Fn, a a a a D r D r which are aligned with and perpendicular to the shear plane, F+Da a a respectively. The component Fs is calculated from the shear l tool l tool h v stress τs in the shear plane. According to [20, 27, 28, 29], this v F F shear stress is constant for a large variety of cutting conditions Da w orkpiece Da w orkpiece and is uniformly distributed over the shear plane. Thus, the . . shear plane-directional cutting force component becomes h l h h v v h h h v v ≠0 ~ ≠ * ≠ c =0 = * ≠ c

Fs = τswl , (6) Figure 2: Illustration of (a) stationary cutting, (b) wave removal, (c) wave gen- eration and (d) synchronous wave-on-wave cutting. where wl is the shear plane area. According to Fig. 1, the mag- nitude of the resultant cutting force becomes F Using this angle, the formulas in Eqs. (6)-(8) can be used to F = s . (7) cos(Φ + βa − αr + ∆α) obtain the tool face force (instead of the total cutting force). By adding ploughing force, the components of the total cutting The y- and z-directional cutting force components that can be force become measured during experiments are τswl Fy = cos(βa − αr) + Py , F = F cos(β − α ) , cos(Φ + β − α + ∆α) y a r a r (11) (8) τ wl Fz = F sin(βa − αr) , s Fz = sin(βa − αr) + Pz . cos(Φ + βa − αr + ∆α) while the tangential and feed components become Both for zero and nonzero tool edge radius, there are two pa- Ft = F cos(βa − αr + ∆α) , rameters in the cutting force model, which have key importance (9) in the subsequent analysis: the shear angle Φ and the shear Ff = F sin(βa − αr + ∆α) . plane length l, as explained in the subsequent subsection. 2.4. Correction by considering the tool edge radius 2.5. Shear angle models For a real cutting tool with nonzero tool edge radius, the cut- ting force is the resultant of the force exerted on the tool’s rake There exist various models to characterize the shear angle face (called tool face force) and the force exerted on the tool Φ. The two most popular models are the maximum shear edge radius (called ploughing force) [30, 19, 29]. Hereinafter, stress principle (MSSP) and the minimum energy principle the y- and z-directional components of the ploughing force are (MEP) [14]. The MSSP states that the shear plane lies in the direction of maximum shear stress, which is 45◦ (or π/4) from denoted by Py and Pz, respectively. The components of the the cutting force vector F [14]: tool face force become Fy − Py and Fz − Pz after subtracting the ploughing force from the total cutting force. π Φ + β − (α − ∆α) = . (12) The cutting force model constructed above neglects the tool a r 4 edge radius and the ploughing force, which makes the total cut- ting force equal to the tool face force. However, for nonzero Meanwhile, according to the MEP, the shear angle Φ satisfies tool edge radius, the total cutting force and the tool face force that the power −Ftv of the cutting force is minimal [12, 14], are different and must be distinguished. Since the model above which leads to is based on determining the force direction from the friction β − (α − ∆α) π Φ + a r = . (13) along the rake face, it is related rather to the tool face force than 2 4 to the total cutting force. In order to obtain the total cutting force, corrections must be made to the model. Notice that for both the MSSP and the MEP, the shear angle is determined by the effective rake angle α = αr − ∆α. Since the average friction angle βa is related to the direction of the tool face force, Eq. (5) must be corrected to [30] In the literature, several other shear angle models can be found [31, 28, 29]. Some of them use direct empirical relation- ! Fz − Pz ships [32], others relate the shear angle to the (empirically de- βa = arctan + αr . (10) Fy − Py termined) chip compression ratio [27, 22, 32, 33], while further 3 models use slip-line field theory [34]. Most models claim that 1.6 qMSSP=45° the shear angle fluctuates if the initial surface is wavy and/or the lMSSP tool vibrates during cutting [31, 20, 13, 21, 23]. Note that the 1.2 MSSP and the MEP lead to a fluctuating shear angle Φ if the l q =22° cutting angle ∆α fluctuates, thus the shear angle Φ depends on 0.8 MEP MEP * the vibration velocityz ˙ through Eq. (1). In this paper, we inves- hMEP * hMSSP tigate whether these shear angle models – which were originally 0.4 h introduced for stationary cutting – give accurate cutting force shear plane length [mm] uncut chip thickness and thickness uncut chip predictions for non-stationary cutting. Our assumption is that 0 the formation of the shear plane is a much faster process than 40 45 50 55 60 workpiece position y [mm] the tool oscillation, thus a quasi-stationary model is adequate. Figure 3: (a) Illustration of the nominal uncut chip thickness h at the tool 2.6. Shear plane length tip, the shifted uncut chip thickness h∗ at the end of the shear plane and the The shear plane length l can be calculated from the locations shear plane length l assuming the MSSP and the MEP shear angle models for the case of wave removal (see Fig. 2(b)). The figure corresponds to an or- of the end points of the shear plane, which are affected by ma- thogonal planing experiment (see details in Sec. 5) where wave removal took ◦ chine tool vibrations. This is illustrated in Fig. 2, where four place with rake angle αr = 2.5 , cutting speed vc = 5 m/min, mean chip thick- cases are considered following [12, 19, 13, 27, 21, 23, 28, 29]. ness hmean = 200 µm, chip thickness amplitude hamp = 150 µm and wavelength λ = 10 mm. Notice the phase shift θ between h and h∗, h and l. Cases (a)-(b) illustrate ideal cutting without vibrations (v = vc, ∆α = 0) where the tool path is a straight line and the machined surface (the lower surface of the chip) is flat without any wavi- ness. Case (a) is called stationary cutting where the initial sur- The cutting angle ∆α should be taken into account when cal- face (the upper surface of the chip) is also flat, thus the nominal culating the shear plane length, which is related to the nominal uncut chip thickness h is constant (h˙ = 0). Meanwhile, a wavy uncut chip thickness h at the tool tip: initial surface is cut in case (b) resulting in a time-varying nom- inal uncut chip thickness (h˙ 0). This case is called wave re- h , l = . (16) moval [13, 27, 21, 23, 28, 29]. Cases (c)-(d) correspond to ma- sin(Φ + ∆α) chine tool vibrations with wavy tool path (v , vc, ∆α , 0) and wavy machined surface. Case (c), where a flat initial surface is For the wave-on-wave cutting case in Fig. 2(d), there is no di- ˙ rect relationship between the shear plane length l and the chip machined and the nominal uncut chip thickness varies (h , 0), ∗ is called wave generation [19, 13, 27, 21, 23, 29]. Whereas in thicknesses h or h . The length of the shear plane can be calcu- case (d), a wavy initial surface is cut with a wavy tool path, lated numerically from the coordinates of its end points. thus this case is referred to as wave-on-wave cutting [19, 13]. When the waviness on the initial surface is the same as the one 2.7. Phase shift between cutting force and chip thickness on the machined surface (i.e., they have the same shape, ampli- According to theoretical models [19, 13, 28, 29, 33] and ex- tude, wavelength and phase), the nominal uncut chip thickness periments [35, 19, 27, 21] (see later also in Sec. 5), there is a is constant (h˙ = 0) and this case is called synchronous wave-on- noticeable phase shift between the cutting force and the nomi- wave cutting. nal uncut chip thickness if the latter is time-varying. Typically, When at least one of the surfaces of the chip is flat, the shear a geometrical explanation is given to address this phase shift, al- plane length is related to the nominal uncut chip thickness at a though some papers indicate that geometric relations alone are certain position. The nominal uncut chip thickness at the tool insufficient to describe dynamic cutting due to the complicated tip is denoted by h, while the nominal uncut chip thickness at underlying physics [34]. the end of the shear plane is indicated by h∗. For the stationary For wave generation, the geometrical explanation [12, 13, cutting case in Fig. 2(a), these two chip thickness values are the 14] is based on the fact that the cutting direction fluctuates (see same (h∗ = h), as the chip thickness is constant in time. Thus, Eq. (1)). This causes fluctuations in the effective rake angle, the chip thickness is directly related to the shear plane length l: the effective chip thickness, the shear angle and the shear plane length [27] as taken into account by Eqs. (2)-(3), (12)-(13) h h∗ l = = , (14) and (16). Fluctuations may also occur in the effective clear- sin Φ sin Φ ance angle, in the cutting speed [27, 21, 23] and even in the cf. Fig. 2(a). For the wave removal case in Fig. 2(b), the shear shear stress in the shear plane [33], which are neglected here. plane length l is related to the shifted uncut chip thickness h∗ In the end, the cutting force depends on the vibration velocity at the end of the shear plane (and not to the nominal uncut chip z˙(t) through Eq. (11). Since for a harmonic tool position z(t), thickness h at the tool tip): the velocityz ˙(t) is in 90◦ phase shift with respect to the posi- tion, the velocity dependency causes a phase shift between the h∗ l = , (15) cutting force and the chip thickness variation, where the force sin Φ leads the chip thickness. In [13], a formula can also be found cf. Fig 2(b). For the wave generation case in Fig. 2(c), the cut- for the phase shift. Note that the velocity dependency drops ting velocity v fluctuates as it is tangent to the wavy tool path. when considering the MSSP shear angle expression, since the 4 . z high-speed camera typically larger for wave removal than for wave generation, and typically increases for increasing chip thickness variation and tool decreasing rake angle. The observation of this phase shift will y z play an important role in the experiments described in Sec. 5. x w 3. Experimental Setup h In the rest of the paper, the cutting force is investigated workpiece experimentally for dynamic (time-varying) chip formation. v c dynamometer Cases (a)-(d) of Fig. 2 are studied systematically in order to investigate the effects of chip thickness variation (occurring in high-speed cases (b) and (c)), surface waviness (cases (b) and (d)) and fluc- camera tuation of the cutting direction (cases (c) and (d)) on the cutting force. Note that all of these effects occur simultaneously in ac- y z tool x tual machining processes in the presence of chatter. Here, the early work of Albrecht [12] is followed, where the theoretical analysis of cases (a)-(d) was discussed similarly to Sec. 2. The ribs present work contributes to the literature, on the one hand, by workpiece the direct analysis of cutting force signals with special attention to the chip segmentation phenomenon and to the analysis and explanation of the cutting force characteristics during dynamic dynamometer cutting. On the other hand, emphasis is put on the assessment of shear angle models and the results are also supplemented by high-speed camera recordings of the cutting process. In order to investigate chip formation, dry orthogonal plan- Figure 4: Schematic representation and photo of the experimental setup for ing tests were performed. The experimental setup is shown orthogonal planing tests. in Fig. 4. Ribs were prepared on an aluminum (A2024-T351) workpiece and were machined using carbide cutting tools. Tool coating and lubricant were not used in these experiments. The angle Φ + ∆α is constant (velocity-independent) according to length of the ribs was 100 mm, the width of the ribs (i.e., the Eq. (12). Thus, in this approach, the MSSP is not able to explain chip width) was w = 2 mm, while the width of the cutting tool ◦ ◦ ◦ the phase shift between the cutting force and the chip thickness was 5 mm. Four tools with rake angle αr = 2.5 , 5 , 10 and in wave generation, while the MEP has no such deficiency. 15◦ were used, while the flank angle was 10◦ for each tool. The As for wave removal, the geometric explanation for the phase tool edge radius was measured by microscope and was found to shift is different [35, 13]. In this case, the cutting direction is be 13, 15, 26 and 34 µm, respectively. not fluctuating (∆α = 0), the effective chip thickness is the same Note that the initial fast wearing of the tools had already as the nominal one (heff = h), there are no velocity-dependent taken place before measuring the tool edge radius and starting fluctuations in the rake angle (α = αr) or in the shear angle (Φ is the experiments of this work. The presented set of experiments constant according to Eqs. (12)-(13)). However, the shear plane involves about 100 cutting tests for each tool. Since in the plan- length is now given by Eq. (15) and it is related to the shifted ing process the total length of the workpiece material cut by uncut chip thickness h∗ (see Fig. 2(b)). Since the shifted uncut a single tool is small (for example, much smaller compared to chip thickness h∗ is ahead of the nominal one h with respect turning experiments), we did not experience any noticeable tool to the tool’s motion, h∗ has a phase lead compared to h (see wear during the experiments. Accumulated built-up edge for- the corresponding formula given in [13]). This is illustrated mation was also avoided by cleaning the tool and removing the in Fig. 3, where the uncut chip thickness values of an actual remains of the chip from the tool after each cut. If there was cutting test are depicted (details of the experiment are given in built-up edge formation during a single cut, it could be observed ∗ ∗ Sec. 5). Here, the shifted uncut chip thicknesses hMSSP and hMEP by high-speed camera or by significant changes in the cutting were calculated numerically from the prescribed tool path using force signal. Biased experimental results affected by significant the shear angle from the MSSP and MEP models, respectively. built-up edge formation were disregarded. Figure 3 shows the corresponding shear plane lengths lMSSP and For the cutting tests, the workpiece was mounted on the ta- lMEP as well. The shear plane lengths – and the cutting forces ble of an NCT EmR-610Ms CNC milling machine as shown related to them – also exhibit a phase shift with respect to the in Fig. 4. The nominal cutting speed was provided by the feed nominal uncut chip thickness h and the relationship of l and h motion of the machine via moving the workpiece in the y di- is nonlinear. As shown later amongst the experimental results, rection. The nominal cutting speed was set to vc = 5, 10 and the MSSP typically predicts a larger shear plane length and a 30 m/min. These low cutting speed values were chosen due larger phase shift than the MEP does in wave removal. to the speed constraints of the machine and in order to investi- The phase shift between cutting force and chip thickness is gate the chip segmentation phenomenon discussed later. Before 5 1000 each cutting test, 120 mm space was provided for accelerating (a) the workpiece, while 60 mm space was used for deceleration. 800

The tool was mounted on (and moving together with) the house [N] of the main spindle of the CNC milling machine. Vibrations F 600 during cutting were imitated by moving the tool in the z direc- 400 tion, which enabled the generation of various tool paths (with a 200 command rate of 500 Hz). Sinusoidal tool paths were used to generate fluctuating cutting force in cases (b)-(d) of Fig. 2. cutting force 0 In order to verify the accurate realization of the prescribed -200 Fy tool motion, the tool position was queried from the CNC 1000 Fz milling machine with a sampling frequency of 1000 Hz. The Fx cutting force components were measured in the (x, y, z) coordi- 800 nate system (see Fig. 4) by mounting the workpiece on a Kistler [N] (b) F 600 transient 9129AA multicomponent dynamometer, and the data were ac- process quired using a 5080A charge amplifier and four NI-9234 In- 400 put Modules in a NI cDAQ-9178 Chassis at 51200 Hz sam- 200 pling frequency. The chip formation process was recorded by cutting force a Photron FASTCAM SA5 High-Speed Camera System with 0 7000 frames per second and resolution of 1024 × 1024 pix- -200 0 0.2 0.4 0.6 0.8 1.0 1.2 els. Canon MP-E 65 mm macro lens with 5:1 magnifica- time t [s] tion were used to magnify the observed area during record- ing. In order to meet the strict lighting requirements of the Figure 5: Cutting force signals for stationary planing processes (case (a) ◦ high-speed camera, two Vision Device VD7000 LED lamps of Fig. 2). The rake angle was αr = 10 , the nominal cutting speed was and an endoscope-like Hayashi HDF7010 Fiber Optic LED sys- vc = 5 m/min and the nominal uncut chip thickness was (a) h = 400 µm, (b) h = 100 µm. The light gray shaded region indicates the transient process be- tem were applied. For more details on the high-speed camera fore reaching the steady state cutting process. The typical shapes of a continu- measurement setup, the reader is referred to [18, 36, 37]. A ous and a segmented chip are shown in panels (c) and (d), respectively. video about the high-speed camera recordings can be found un- der http://siren.mm.bme.hu/Videoblog.html.

force component and a repeated abrupt increase in z-directional 4. Experimental Results for Stationary Cutting (feed) component during segmentation. That is, the frequent First, stationary cutting (case (a) of Fig. 2) was considered, peaks in the cutting force signal do not indicate the noise of where the initial surface is flat and only the workpiece is mov- the measurement, but are related to the physics of chip forma- ing along the y direction, which generates a straight line tool tion. Note that the dynamics of the dynamometer also causes path and a constant nominal uncut chip thickness. In this case, high frequency components in the force signal, but these were certain cutting force components coincide: the y-directional Fy removed using the method introduced in [38]. and the tangential Ft, as well as the z-directional Fz and the feed The segmentation-induced cutting force fluctuation is neg- directional Ff. A sequence of experiments were performed for ligible for the small chip thickness value in Fig. 5(b). The ◦ ◦ ◦ ◦ rake angles αr = 2.5 , 5 , 10 and 15 , nominal cutting speeds explanation of chip segmentation is out of scope of this pa- vc = 5, 10 and 30 m/min, and 22 different nominal uncut chip per, although there exist several physical models in the liter- thickness values ranging from 10 µm to 500 µm. ature [39, 40, 41, 42, 43]. Here, the occurrence of segmentation Figure 5(a-b) show an example for the measured cutting is investigated based on the inspection of the cutting force sig- force signals that act on the tool as a function of time for a nal and the high-speed camera recordings only. According to large (h = 400 µm) and a small (h = 100 µm) nominal uncut the sequence of cutting tests, segmentation is more likely to ◦ chip thickness value (with αr = 10 and vc = 5 m/min). In both occur (and has larger effect on the cutting force) when the nom- cases, the cutting force components undergo a short transition inal uncut chip thickness is large. Slight segmentation takes (see shaded region) until reaching their steady state, which may place for uncut chip thickness values h & 150 µm. This is as- be caused by thermal processes or by the flexibility (slight de- sociated with small variations in the deformed chip thickness formation) of the tool holder structure. and small, less abrupt fluctuations in the cutting force. For According to the experiments, the phenomenon of chip seg- larger chip thickness values, h & 300 µm, strong segmentation mentation occurs for large chip thickness values: the chip thick- evolves, which is accompanied by peculiar sharp peaks in the ness fluctuates although its nominal value is constant. The typ- cutting force signal. The tangential cutting force and the de- ical shapes of a continuous and a segmented chip are shown in formed chip thickness repeatedly drops in this case, sometimes the high-speed camera pictures of Fig. 5(c-d). The segmenta- even nearly to zero. The chip thickness associated with the on- tion phenomenon is also clearly indicated by the sharp peaks set of segmentation typically increases with increasing cutting of the cutting force signal in Fig. 5(a). Namely, there is a re- speed and rake angle for the cutting parameter ranges consid- peated abrupt decrease in the y-directional (tangential) cutting ered in the experiments. Furthermore, it was also observed that 6 1000 (a) ar=15° (b) ar=10° F F t t 800 v v c=5 m/min c=5 m/min v v c=10 m/min c=10 m/min

[N] v v F 600 c=30 m/min c=30 m/min strong strong segmentation segmentation effect of effect of 400 edge radius edge radius cutting force

200 F F f f

0 1000 (c) ar=5° (d) ar=2.5° F F t t 800 v v c=5 m/min c=5 m/min v v c=10 m/min c=10 m/min

[N] v v F 600 c=30 m/min c=30 m/min strong strong segmentation segmentation effect of effect of 400 edge radius edge radius cutting force

200 F F f f

0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 nominal uncut chip thickness h [mm] nominal uncut chip thickness h [mm]

Figure 6: Average tangential and feed cutting force components as a function of the nominal uncut chip thickness for nominal cutting speeds vc = 5, 10 and ◦ ◦ ◦ ◦ 30 m/min and for rake angle (a) αr = 15 , (b) αr = 10 , (c) αr = 5 and (d) αr = 2.5 . Gray shading indicates the regions where the cutting force characteristics qualitatively change and deviate from linear due to the effect of the tool edge radius (light gray) or due to strong chip segmentation (dark gray).

the higher the cutting speed is, the higher the frequency of seg- force components were determined by this method for each of mentation is: the corresponding wavelength is roughly the same the cutting force characteristics in Fig. 6 separately. The partic- but the cutting takes less time for higher cutting speeds. ular ploughing force values were the following: ◦ Py = 61, 49, 44 N and Pz = 51, 39, 38 N for αr = 15 , For each stationary cutting test, the middle third of the force ◦ Py = 58, 65, 47 N and Pz = 53, 59, 43 N for αr = 10 , signals was averaged. The characteristics of the average cutting ◦ Py = 65, 60, 49 N and Pz = 60, 58, 40 N for αr = 5 , force components as a function of the nominal uncut chip thick- ◦ ◦ ◦ ◦ ◦ Py = 61, 61, 49 N and Pz = 58, 55, 41 N for αr = 2.5 ness are shown in Fig. 6(a-d) for αr = 15 , 10 , 5 and 2.5 , re- spectively. The tangential and feed force characteristics can be for cutting speeds vc = 5, 10, 30 m/min, respectively. Note approximated by a (shifted) linear function in a wide range of that the ploughing force strongly depends on the tool edge ra- chip thicknesses (50 µm < h < 300 µm). This linear tendency dius and the choice of the tool. The ploughing force values is assumed in most cutting force models [20, 14] and it is also were used later to apply the model of Sec. 2 to the case of time- captured by the model of Sec. 2. Namely, the cutting force com- varying cutting. It is important that the cutting force predictions of this model are accurate in the range of medium uncut chip ponents Fy and Fz given by Eqs. (11) and (14) consist of a term proportional to the nominal uncut chip thickness h (which is the thickness (50 µm < h < 300 µm) only, where the cutting force tool face force) and a constant term (the ploughigh force). This versus chip thickness relationship is linear. allows one to determine the values of the ploughing force com- In other ranges of the nominal uncut chip thickness h, the cut- ponents Py and Pz by fitting straight lines to the linear range ting force characteristics are nonlinear. For small chip thickness (50 µm < h < 300 µm) of the cutting force characteristics and values (h ≤ 50 µm), the characteristics decrease in an exponen- finding their intersection with the vertical axis. The ploughing tial fashion [44] due to the effect of the tool edge radius (see 7 1000 0.8 45° ar=2.5° ar=10° (a) (b) ar=2.5° ar=10° MEP (c) 0.7 40° ar=5° ar=15° a ar=5° ar=15° 800 m 0.6 35° icient icient

[N] 30° F ff

F F 0.5 600 t effect of 25° edge radius 0.4 20° strong 400 F f 0.3 segmentation shear angle 15° MSSP cutting force 0.2 200 10°

average friction coe average a =2.5° a =10° strong 0.1 effect of strong 5° effect of r r a a segmentation edge radius segmentation edge radius r=5° r=15° 0 0 0° 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 nominal uncut chip thickness h [mm] nominal uncut chip thickness h [mm] nominal uncut chip thickness h [mm]

Figure 7: Characteristics of (a) the cutting force, (b) the friction coefficient and (c) the shear angle as a function of the nominal uncut chip thickness for nominal ◦ ◦ ◦ ◦ cutting speed vc = 10 m/min and for rake angles αr = 2.5 , 5 , 10 and 15 . The regions affected by the tool edge radius and strong chip segmentation are indicated by light and dark gray shadings, respectively.

the light gray shaded region). The cutting force drops to zero Fig. 7(a). Note that the cutting force components typically de- for zero chip thickness, since the tool is no longer in contact crease with increasing rake angle. Figure 7(a) also shows the with the workpiece. It is important to note that in this region standard deviation of the cutting force components during each ◦ the ploughing force strongly depends on the value of the chip cutting test with αr = 10 (see the error bars). The increase thickness, therefore the theoretical model of Sec. 2 (which as- of the standard deviation at large uncut chip thickness values sumes a constant ploughing force) is not applicable. Modeling (h ≥ 300 µm) clearly reflects the occurrence of strong chip seg- cutting forces for chip thicknesses in the range of the tool edge mentation. radius is out of scope of this paper. Meanwhile, the average friction coefficient was also calcu- For large chip thicknesses (h ≥ 300 µm), a slight degressive lated from the average of the measured cutting force compo- tendency can be observed on the characteristics of the tangential nents using Eqs. (4) and (10). According to Fig. 7(b), the fric- force, while the feed force characteristics may break down and tion coefficient is approximately constant in the medium range decrease (see the dark gray shaded region). This break down of uncut chip thicknesses (50 µm < h < 300 µm) and its value (around h = 300 µm) can be predicted and explained by theo- is around 0.5 ≤ µa ≤ 0.6. Similar results were reported in [30]. retical shear zone models [39, 40, 41, 42, 43], and a similar This justifies that using a constant friction coefficient is a good break down was observed directly in [43]. The break down is approximation in this range of parameters. Note, however, that more pronounced for small cutting speeds and small rake an- the friction coefficient is in fact varying along the tool-chip gles, since it is associated with chip segmentation, which be- interface [46]. For large chip thickness values (h ≥ 300 µm), comes noticeably more significant at these large chip thickness the friction coefficient decreases with increasing chip thick- values. Note that the characteristics of the tangential and the ness. For small chip thickness values in the range of the tool feed forces are qualitatively different [45], which is often ne- edge radius (h ≤ 50 µm), the calculated friction coefficient is glected in cutting force models used for analyzing machine tool not meaningful, since Eq. (10) describes the case where the chatter [14]. tool face force is present. Also note that the friction coeffi- Analyzing experimental results from point of view of ex- cient increases with the rake angle. Further friction properties plaining the physics of chip segmentation is out scope of this and empirical formulas for the friction coefficient can be found paper. Still, for the sake of completeness, some experimen- in [30, 31, 32, 28, 29, 46]. In what follows, the oscillations of tal results are presented for large uncut chip thickness values the friction coefficient [20] are neglected and the mean friction subject to segmentation (h ≥ 300 µm, see the dark gray shaded angle βa is used even in non-stationary cutting. region in Fig. 6). These experiments show the domain of appli- Furthermore, the average shear angle Φ was calculated from cability of the theoretical model of Sec. 2, which is restricted the average friction angle βa using the MSSP model (12) and the to medium chip thickness values (50 µm < h < 300 µm, see the MEP model (13) as depicted in Fig. 7(c). Again, these results white region in Fig. 6). are meaningful outside the range of the tool edge radius only. According to Fig. 6, the cutting speed has moderate effect on The average shear angle is approximately a constant function of the cutting force magnitude (at least in the investigated range the uncut chip thickness in the range 50 µm < h < 300 µm. The of parameters), while the effects of the rake angle and the chip shear angle increases with the rake angle, and the MEP predicts thickness are more pronounced [23]. Thus, the cutting speed larger shear angle than the MSSP does. Meanwhile in the range vc = 10 m/min is selected for further analysis. For this cut- h ≥ 300 µm, the average shear angle increases with the chip ting speed, the cutting force characteristics are summarized in thickness and there are large shear angle fluctuations around the 8 45° 14 (a) (b) 40° 12 MSSP MSSP 35° 10 MEP MEP

F 30° 8 25° 6 20° 4 15° shear angle 2

MSSP of measurements number 10° slight MEP 0 5° Image processing segmentation 0100200 300 400 500 0100 200 300 400 500 t t 0° shear stress s [MPa] shear stress s [MPa]

nominal uncut chip thickness h [mm] Figure 9: Distribution of the shear stress over the set of experiments with ◦ αr = 2.5 , vc = 5 m/min and (a) 10 µm ≤ h ≤ 500 µm (22 measurements), Figure 8: Comparison of the shear angle detected from high-speed camera (b) 50 µm < h < 300 µm (12 measurements). measurements via image processing techniques and the shear angle calculated from the cutting force signals via the MSSP and the MEP shear angle mod- ◦ els [18, 36]. The cutting conditions are αr = 15 and vc = 10 m/min. workpiece position y [mm] 40 45 50 55 60 0 prescribed

[mm] -0.1 realized average due to strong chip segmentation. Note that the shear z angle predictions based on the MSSP and the MEP gradually -0.2 lose validity as segmentation becomes more intensive. In order to test the applicability of the MSSP and the -0.3 tool position MEP, the shear angle was determined also from high- -0.4 speed camera recordings via image processing techniques. Planing experiments with α = 15◦, v = 10 m/min and 0

r c [mm] h = 50, 100, 150, 200, 250 µm were recorded. The shear an- z gle was determined by calculating the displacement field of -0.2 the chip’s motion from two subsequent frames and applying Wiener-filtering. The details and results of this analysis are -0.4 published in [18, 36]. The mean value and the standard de- tool position -120 -80 -40 0 40 80 120 160 viation of the shear angle during the particular cutting tests are workpiece position y [mm] depicted in Fig. 8. For comparison, the shear angle values cal- Figure 10: Prescribed and realized tool paths in wave generation for α = 2.5◦, culated from the cutting force signals via the MSSP and the r vc = 5 m/min, hmean = 200 µm, hamp = 150 µm, λ = 10 mm. MEP are also shown. The results of the MEP and the image pro- cessing are in excellent agreement for small uncut chip thick- ness values. For larger chip thicknesses, the accuracy of the MEP decreases due to the gradual onset of slight chip segmen- be attributed to the effects of tool edge radius and strong chip tation. In contrast, the MSSP underestimates the shear angle segmentation by which the model of Sec. 2 loses validity. Be- for the whole range of chip thicknesses. Note that the chip seg- sides, the MSSP gives a smaller mean shear stress value than mentation phenomenon is well-observable in high-speed cam- the MEP does. era recordings, see Fig. 5(c,d).

Finally, the shear stress τs in the shear plane can also be ob- tained from Eq. (11) using the MSSP or the MEP. The distri- 5. Experimental Results for Sinusoidal Tool Path bution of the shear stress over the different measurements is ◦ shown in the histogram of Fig. 9 for αr = 2.5 , vc = 5 m/min. After the stationary cutting tests, cases (b)-(c) of Fig. 2 were In Fig. 9(a), results are indicated for the whole set of exper- considered by performing wave generation and wave removal iments (10 µm ≤ h ≤ 500 µm, 22 measurements), while in pairs of planing experiments. During wave generation, the ini- Fig. 9(b), experiments are considered in the medium chip thick- tial surface was flat, the workpiece was moving in the y direc- ness range only without including the effect of tool edge radius tion with constant speed, while the tool was moving up and and strong chip segmentation (50 µm < h < 300 µm, 12 mea- down sinusoidally in the z direction (see Fig. 2(c)). This gen- surements). According to Fig. 9, the assumption that the shear erated sinusoidal tool path and sinusoidal nominal uncut chip stress is constant regardless the cutting conditions yields a good thickness. During wave removal, the sinusoidal wavy surface approximation, since the distribution of the shear stress is nar- was cut by generating a straight line tool path with workpiece row with a well-defined mean value. Based on the comparison motion only (see Fig. 2(b)). The nominal uncut chip thickness of the two panels, deviations in the calculated shear stress can was again sinusoidal but the tool was not moving in this case. 9 iuodluctci hcns cluae rmtemeasured the from (calculated thickness non- actual chip the uncut Therefore, sinusoidal direction. when friction changed discrepancy to motion slight due tool’s a paths the was tool realized there and that desired seen the between be can It lines. dashed in at shown were are cut machine of the the by) for and 10 queried path Fig. (and tool sinusoidal by desired generated The when one path moving. tool was (prescribed) tool desired the the of realization so- accurate the 53]. 52, to 51, 50, related 14, is 49, 48, contact 47, [35, flank ture e the damping that a process strongly Note called have flank would the force. which and ting tool, surface the machined of the face between contact 10 ditional was angle flank the ∆ cor- amplitudes to these respond generation, wave In thickness. chip uncut h values mean wavelength of ations realized The 2(b)). Fig. (see removal wave for (b) 2(c)), Fig. (see generation wave for (a) variation are thickness conditions chip sinusoidal thickness for chip signals uncut force Cutting 11: Figure amp α h utn pe a etat kept was speed cutting The eut r rsne o iuodlci hcns vari- thickness chip sinusoidal for presented are Results amp =

≈ cutting force F [N] cutting force F [N] 1000 1000 1200 1200 50 -200 -200 200 400 600 800 200 400 600 800 tan , 0 0 0 0 100 h ± α (b) ∆ (a) mean r ∆ h = α y α , mean amp 2 amp = h 150 . = 5 sas hw yprl ie ntodi two on lines purple by shown also is ◦ and 0 , = 200 = utaino h utn ieto,where direction, cutting the of fluctuation v F F , c 02 04 06 08 90 80 70 60 50 40 30 20 10 02 04 06 08 90 80 70 60 50 40 30 20 10 h x x 100 200 = ◦ λ amp hsflcuto i o as n ad- any cause not did fluctuation this , µ m 5 = y m, , ff 2 µ = 0m.Svrlcmiain of combinations Several mm. 10 / π/λ 150 c,wihhsa xesv litera- extensive an has which ect, eecniee o h nominal the for considered were m F F min, h 0 ma niae ytethick the by indicated as mm 100 y y amp , = h 200 mean = 1 v . F F c 150 8 , z z ◦ = = , 250 200 m 5 3 µ . .Tesatadend and start The m. 6 µ h h µ ◦ / m, , n amplitudes and m i oesr the ensure to min 5 h . amp ff 4 ◦ ce h cut- the ected ff , rn clsfrcmaio ihtectigfrecomponents force cutting the with comparison for scales erent = 7 150 . 2 workpiece position workpiece position workpiece position workpiece position ◦ Since . µ m, λ = 0mm. 10 10 h ek ftefre,wihi oelkl o ag chip large for likely more is which force), (see the occur of may segmentation peaks chip the Besides, thicknesses. chip (especially signals force cutting measured conclusions The following the drawn. tests, be cutting could of series the for 11) Fig. shift. phase their sig- of thickness observation chip the and and force This nals cutting figure. of the in comparison lines the purple enables by shown also are thickness position chip tool’s uncut the of values Scaled amplitude with was around sinusoidally angle varying rake was thickness the chip measurements, both (correspond- For removal α wave 2(b)). shows Fig. to (b) ing panel with while (associated test 2(c)), generation wave Fig. a shows (a) Panel 11. Fig. di numerical data. the measured from in of arising one tion noise non-sinusoidal sinu- the measured avoid prescribed the to the of order of instead tangent path tool the soidal by de- e approximated the the still of However, was instead analysis one. further sinusoidal for sired used was position) tool F r z eoedsotdadnnsnsia o ag enuncut mean large for non-sinusoidal and distorted become ) ae nteiseto ffresgas(uha h n in one the as (such signals force of inspection the on Based in shown are signal force cutting measured the for Examples = y y [mm] [mm] 2 . 5 ◦ h oia utn pe was speed cutting nominal the , shift phase q ≈ 24° h amp = segmentation 150 shift phase q ≈ µ 12° n wavelength and m F y and wave generation ff wave removal F ciectigdirection cutting ective z h orsodn cutting corresponding The . v h c acltdfo the from calculated = m 5 h mean / 100 100 λ i n the and min 0.4 0.4 0 0 = = 0.4 ff 0.4 0 0 200 0mm. 10 erentia- uncut chip thickness h [mm] uncut chip thickness h [mm] µ m hc ie nFg 2adaedntdby denoted are and 12 by Fig. indicated in are lines which thick characteristics, 10 gives force into segments cutting 10 separated average the an be of can average signals The con- force segments. thickness thus corresponding chip periods, uncut 10 nominal of the sists case, this the In mul- periodically, characteristics times. force varies tiple cutting thickness the across chip sweep signals the force As uncut thickness. decreasing and chip increasing to corresponding wave components and plotted generation wave Di are for respectively. in removal, components thickness shown chip force also uncut cutting the is the against shift where is phase shift 12(a,b), This phase Fig. the significant. cutting 11(b)), the (Fig. more removal to typically wave respect For with delayed signal. is force thickness chip uncut the hysteresis. which with chatter), place tool takes machine segmen- and chip behavior chatter-free and between or formation tation occurs chip switching (continuous states behavior: stable bistable two a to and lead (nonlinear earities the by e caused occurrence flyover is the nonsmooth) chip which in chatter, of tool hysteresis hysteresis machine the the of to sense, non- of this analogous presence In is the segmentation process. indicate 41, the may [40, in hysteresis models linearities the is, zone That shear nonlinear 42]. of analysis bifurcation better is amplitudes. hysteresis thickness chip the large and for generation, visible wave wave for for pronounced than Usually more removal are 11(b). hysteresis Fig. of its region can and shaded hysteresis segmentation the This in observed values. be thickness clearly chip smaller segmentation for decreases, thickness persists chip uncut the when val- and thickness ues, chip larger thick- at chip begins uncut segmentation the increases, when ness Namely, variation thickness. the chip uncut to the respect of with segmen- hysteresis chip exhibits of cut- typically occurrence tation the stationary cutting, to non-stationary similarly In angles, ting. rake small and thicknesses 11. Fig. experi- in cutting as stationary same the refer- corresponding are For the conditions removal. cutting of wave The results for ments. the (b) indicate time-varying generation dots the wave of for ence, function (a) a thickness as force chip cutting uncut the of Characteristics 12: Figure y and - nadto,asih hs hf a eosre nFg 11: Fig. in observed be can shift phase slight a addition, In the by predicted be can segmentation chip of hysteresis The

cutting force F [N] 1000 1200 200 400 600 800 0 z drcinlcmoet,rsetvl.Teetiklines thick These respectively. components, -directional 0.4 0 based on tool position [mm] position tool based on wave generation uncut chip thickness F F F F . . 0.3 0.2 0.1 z y z y ,avg ,avg ff c 5] nbt ae,poesnonlin- process cases, both In [54]. ect ff rn oosdsigihctigforce cutting distinguish colors erent h a (b) (a) 0.4 0 based on tool position [mm] position tool based on wave removal uncut chip thickness F F F F F . . 0.3 0.2 0.1 z y z y ,avg ,avg y , avg and F z , avg o the for h 11 ok rprywt h E ha nl oe,teMS is MSSP the model, angle shear explanation MEP this the to While with due direction. properly velocity works cutting vibration the the of on fluctuation the force the de- cutting generation, the on the wave based of for was pendency shift that phase Recall the of explanation shift. geometrical phase this for tion cutting distributed force models. and cutting force models relating of damping terms process in to crucial expressions is shift phase the ori- of the the gin understanding and Therefore, force force thickness. cutting cutting chip Distributed the uncut between nominal 55]. shift 59, phase show [58, also tool along models the force of cutting face the rake of the distribution This the cap- be considering 53]. also by 50, can tured sev- 57, phenomenon 49, improvement in 56, stability observed speed [47, the low also literature the as was in larger which experiments cut eral decreased, of is depth speed (chatter-free) spindle stable maximum the n 4,5,4,4,5,5,5,5] rcs apn strongly damping Process 53]. 52, 51, 50, damp- a 49, process 48, called propor- 56, is inversely and [47, speed ing be cutting models, to nominal the theoretical considered to many tional is In damping chatter. additional of the onset the an a and to which chining leads force, expressions Linearization damping force additional cutting chatter. in velocity-dependent tool significance a machine great of of veloc- has models dependency the Velocity dynamical with the the associated tool. time-derivative the is indicate of which the may ity thickness, on curves chip force closed uncut cutting the Such the of 55]. dependence [12, in reported large and removal wave amplitudes. of thickness case curves chip in closed area The larger thickness. enclose chip typically uncut the the leads arrows), the force (see cutting around direction clockwise uncut circle the signals and in curves force force closed measured the cutting the between Since shift thickness. phase chip a indicates of clearly presence which the functions, characteris- single-valued force of cutting instead the in tics appear curves closed that show Note 11. Fig. in as same the are conditions di wave the cutting stationary for corresponding The model the angle of experiments. plane results shear cutting shear the MEP indicate the the dots of (b) reference, function For MSSP a removal. the as (a) force from cutting calculated the length of Characteristics 13: Figure ff sdsusdi e.2 hr xs emtia explana- geometrical a exist there 2, Sec. in discussed As iia lsdcre ntectigfrecaatrsiswere characteristics force cutting the in curves closed Similar csteocrec fcatra o utn pes tmakes it speeds: cutting low at chatter of occurrence the ects

cutting force F [N] 1000 1200 200 400 600 800 ff rn clso h oiotlae ntetopanels. two the in axes horizontal the of scales erent 0 1.6 0 wave removal based on MSSP [mm] MSSP based on shear plane length shear plane . . 1.2 0.8 0.4 F F F F z y z y ,avg ,avg MSSP l a (b) (a) 0.8 0 wave removal ff cstesaiiyo ma- of stability the ects based on MEP [mm] MEP based on shear plane length shear plane . . 0.6 0.4 0.2 F F F F z y z y ,avg ,avg MEP l h ˙ of xeiet hnteMS ohfrwv eeainadwave and generation the wave with for agreement both better concluded MSSP in the be is than can model experiments angle it shear Therefore, MEP the meaningful that more models. is damping which process direction, for clockwise but wider, the slightly in are of curves run case closed the they the to 13(a), comparison Fig. In in MSSP 12(b). terms the Fig. in in than thickness narrower charac- chip much average the are of the they in but present lines), still (thick teristics are curves closed the Here, qualitatively a is which additional models, di negative damping a process to in in curve damping correspond the closed would counter-clockwise and characteristics a force the the that cutting and Note the length between thickness. plane one chip shear the the than results between thickness which shift chip angle, phase shear overcompensates larger small MSSP a too predicts in the it is, cut- shift: That phase measured the indicates the This leads components. length force 12. plane ting di- shear Fig. opposite calculated in the the previously that shown i.e., was direction, that curves counter-clockwise closed rection These the still lines). in are (thick run curves ones averaged closed the force narrow in but cutting visible measured lines), the (thin characteristics from disappear seemingly curves (13) and components. (12) force Eqs. cutting from average calculated the using was via which the path angle, plane tool and shear shear realized MSSP the the the from the Here, numerically for obtained was respectively. 13(a,b) length models, Fig. explanation. angle in this shear found of can MEP validity be lines the can straight justifies plot if it This then (i.e., only), length seen plane be character- shear force closed the cutting if against the cutting Thus, plotting istics when the length. appear between not plane do shift shear curves the phase and no components is force there explanation, this length plane thickness shear the between shift phase dependency case. velocity that the in since dropped shift is phase the explain to unable position tool corresponding The 2(d)). Fig. are conditions (see cutting experiment The cutting comparison. wave-on-wave for a lines of green and signals purple force Cutting 14: Figure ff enhl,tecs fteMPi hw nFg 13(b). Fig. in shown is MEP the of case the Meanwhile, closed the Here, MSSP. the of case the shows 13(a) Figure the uses explanation geometrical the removal, wave For rn eairfo h n hti expected. is that one the from behavior erent

cutting force F [N] 1000 -200 200 400 600 800 0 0 h hc a eosrtdi i.3 codn to According 3. Fig. in demonstrated was which , F x 02 04 06 08 90 80 70 60 50 40 30 20 10 F y F z z l z . n h nu chip uncut the and workpiece position workpiece position α r = 2 . 5 ◦ , v c = 12 m 5 h e the au fteto position tool the of value pecially ( constant was the z of angle rake was nominal tool the where experiment, cutting wave values thickness was chip speed cutting The wavelength ∆ of paths amplitudes tool and sinusoidal with ducted uncut nominal 13]. [19, the thickness of chip variations involving without observed be wavi- time in the constant was is, thickness ( the chip (nominally) That uncut were nominal surfaces the chip and analyzed. lower same and was upper the pre- one performed on was second are ness motion the 2) tool and Fig. sinusoidal twice same of the (d) Here, panel sented. to (corresponding experiments Paths Tool Sinusoidal Double for Results Experimental 6. ones. stationary of set a of single instead a experiment good from determined non-stationary the be can on characteristics 13(b), force Based Fig. cutting MSSP. in the results the non-stationary and over stationary chosen of agreement is MEP sta- better the slightly the when again, between is, agreement experiments The non-stationary in and dots 13(a,b). tionary by Fig. indicated also and are 12(a,b) 6) Fig. Fig. 4, Sec. and in cutting. (discussed previously experiments stationary shown cutting in stationary the characteristics of results force The cutting the the qualitatively as become characteristics same length plane shear 4. sus Sec. in conclusion same the to lead dur- cutting measurements stationary camera ing high-speed the that Recall removal. h amp ˙ / y α min, [mm] iue1 rsnstectigfresga fawave-on- a of signal force cutting the presents 14 Figure con- were experiments cutting wave-on-wave of sequence A cutting wave-on-wave synchronous for results the Finally, ver- force cutting the MEP, the applying when Furthermore, = amp = ff ) steto a oigu n ondrn cutting, during down and up moving was tool the As 0). h c fflcutosi h utn ieto ol clearly could direction cutting the in fluctuations of ect = 200 = F 3 α 200 . y r 6 µ hwsrn aitos o eeec,ascaled a reference, For variations. strong show ) ◦ = .Atog h oia nu hpthickness chip uncut nominal the Although m. µ , m, h 5 2 z . = z . amp 4 5 amp ◦ ◦ 200 , hl h mltd fteto ahwas path tool the of amplitude the while , = = 7 h . 200 2 100 µ = ◦ ) h utn oc opnns(es- components force cutting the m), v c 100 utain ntectigdirection. cutting the in fluctuations µ , and m = z 150 wave-on-wave cutting n hfe n cldvalue scaled a and shifted a and m 5 µ z n 200 and m , n olvlct ˙ velocity tool and λ / 200 = i n h oia uncut nominal the and min 0mm. 10 µ ,wihcrepn to correspond which m, µ eeprescribed. were m z 100 r losonby shown also are -10 10 λ . = tool velocity z [mm/s] 0.4 0mm 10 0 tool position z [mm] 1000 Φ, the shear plane length l was calculated from the prescribed MSSP MEP Fy Fy Fy Fy ,avg tool paths. Besides, since the value of the shear stress τs in MSSP MEP Fz Fz,avg Fz Fz Eq. (11) was unknown, the average shear stress obtained from 800 the stationary cutting experiments with 50 µm < h < 300 µm was used (that is, the effects of tool edge radius and strong segmentation were disregarded). The average shear stress val- [N]

F 600 ues were τs = 262 MPa for the MSSP and τs = 313 MPa for the MEP. Similarly, the ploughing force was obtained from sta- tionary cutting data, resulting in Py = 61.2 N and Pz = 57.7 N. 400 Finally, two different cutting forces were calculated: one for the

cutting force MSSP and one for the MEP, whose components are depicted in Fig. 15 by thick dashed and thick solid lines, respectively. 200 Note that the shear plane length l is different for increasing and decreasing effective rake angles α (and their relationship wave removal is nonlinear). Thus, the theory predicts ellipse-shaped closed 0 -5° 0° 5° 10° curves in Fig. 15 when plotting the cutting force components effective rake angle a (that depend on the shear plane length) as a function of the fluc- tuating rake angle. Comparison of theory (thick lines) and ex- Figure 15: Cutting force components as a function of the fluctuating effective periment (thin lines) shows that the MEP shear angle model is rake angle for a wave-on-wave cutting experiment. The cutting conditions are the same as in Fig. 14. able to accurately explain the tendency of cutting force fluctua- tions due to the variations of the effective rake angle. However, the cutting force is slightly underestimated by the model, possi- bly due to the discrepancy between the actual value of the shear of the vibration velocityz ˙ are also shown in the figure. The stress τs and its value calculated from stationary cutting data. ◦ cutting force component Fy is in approximately 90 phase shift Meanwhile, the MSSP does not capture the tendency of the cut- with respect to the tool position and is almost in phase with ting force accurately: it gives a too wide closed curve with too the vibration velocity. The correlation of the cutting force with large negative slope. Thus, even in the case of wave-on-wave the vibration velocity shows that fluctuations in the cutting di- cutting, the geometrical explanation provides a more accurate rection strongly affect the cutting force during dynamic chip description of dynamic chip formation when it is supplemented formation. by the MEP shear angle model rather than by the MSSP. It is shown below that primarily the effective rake angle fluc- tuations are responsible for the cutting force variations as pre- 7. Conclusions dicted by the geometrical explanation in Sec. 2. Recall that in stationary cutting, the cutting force components typically de- A set of orthogonal planing experiments was performed in crease with increasing nominal rake angle (see Fig. 7(a) and order to investigate chip formation during machine tool vibra- Sec. 4). A similar tendency is expected for time-varying (fluc- tions. The cutting force was measured and the chip forma- tuating) effective rake angles. Thin lines in Fig. 15 show the tion process was recorded via high-speed camera for station- components Fy and Fz of the cutting force versus the effective ary cutting and non-stationary (wave generation, wave removal rake angle α for a wave-on-wave cutting experiment. Differ- and wave-on-wave cutting) experiments. The measured cutting ent colors distinguish cutting force components corresponding force signals were compared to theoretical predictions. The the- to increasing and decreasing effective rake angle. The nomi- ory uses a geometrical explanation: the cutting direction fluctu- ◦ nal rake angle was αr = 2.5 , the amplitude of the tool path ates for a vibrating tool, which affects the effective rake angle, was zamp = 200 µm, thus the effective rake angle was fluctuat- the shear angle, the shear plane length and the effective chip ◦ ◦ ◦ ing with amplitude 7.2 between −4.7 and 9.7 . Apparently, thickness, while the surface waviness also causes variations in the cutting force components decrease with increasing rake an- the shear plane length. gle, similarly to stationary cutting. Experiments with different The main contributions of the paper are: (i) the application nominal rake angles have shown that the cutting force fluctua- of the geometrical explanation to address cutting force fluctua- tions are typically larger when the effective rake angle fluctuates tions under dynamic cutting conditions; (ii) the assessment of around a smaller nominal value. shear angle models; (iii) the experimental observation of some The decreasing tendency in Fig. 15 can be captured by the key features of chip segmentation; (iv) the high-speed camera model of Sec. 2. Again, a key point is the geometrical explana- recording of the chip formation process. The experiments lead tion that the rake and shear angles fluctuate for a vibrating tool. to the following conclusions for the investigated range of cut- In order to compare the theory to experiments, the cutting force ting parameters. components were calculated from Eq. (11) as a function of the effective rake angle for −4.7◦ ≤ α ≤ 9.7◦. To this end, the shear • In stationary cutting, the cutting force components typically angle Φ was obtained from the MSSP and the MEP shear angle increase with increasing uncut chip thickness. There is a models (12) and (13), respectively. Based on the shear angle break down in the feed force characteristics for large chip 13 thickness values due to the appearance of strong chip seg- alone cannot explain the improvement of stability at low cutting mentation. speeds for certain machining processes (such as low radial im- • The cutting force components typically decrease with in- mersion milling). In these processes, other phenomena such as creasing rake angle. the flank contact [47, 56, 49, 57, 50, 53] or the distribution of the cutting force along the rake face [58, 59, 55] can be respon- • The cutting speed has moderate effect on the cutting force (in sible for the low speed stability improvement. In this sense, the parameter ranges investigated). the experimental results of this paper may help understanding • Chip segmentation is more likely to occur and is stronger for the cause of the low speed stability improvement phenomenon, large uncut chip thickness, small cutting speed and small rake which facilitates the design of chatter-free cutting processes. angle values. Its frequency increases with the cutting speed, The experimental results can further be used to validate more but the corresponding wavelength is approximately constant. sophisticated analytical models of chip formation [43] or the fi- • The occurrence of chip segmentation typically exhibits hys- nite element analysis of orthogonal cutting processes. Our fu- teresis with respect to the variation of the nominal uncut chip ture research involves building high-fidelity finite element mod- thickness in non-stationary cutting. els of metal cutting, which are validated by the low speed exper- iments of this paper. The advantage of low speed cutting tests • The MSSP predicts smaller shear angle than the MEP does. is that various wavy tool paths can be realized accurately in In stationary cutting, the prediction of the MEP is the one a well-controlled clean experimental environment. Such well- that is closer to high-speed camera observations. controlled tool motion is hard to achieve at high speeds. How- • In non-stationary cutting, there is a phase shift between the ever, the high-fidelity finite element models enable the inves- nominal uncut chip thickness and cutting force signals. The tigation of orthogonal cutting at high cutting speeds and with role of this phase shift is critical for understanding additional more severe oscillations of the tool, which is a more realistic damping forces arising during machine tool vibrations (such representation of chatter in industrial practice. This method en- as the process damping). The phase shift is typically larger ables justifying the results of this paper at high speeds, which for wave removal than for wave generation. is an important step of future research work. • The phase shift can well be explained geometrically by the fluctuation of the cutting direction in wave generation and by Acknowledgments the lag between the shear plane length and the nominal uncut chip thickness in wave removal. The research leading to these results has received fund- • The MEP shear angle model performs better than the MSSP ing from the European Research Council under the European does in predicting the phase shift. Union’s Seventh Framework Programme (FP/2007-2013) / • In non-stationary cutting, effective rake angle fluctuations ERC Advanced Grant Agreement n. 340889. This work ´ cause variations in the cutting force. The variations are larger has been supported by the UNKP-18-3-I. New National Ex- for small nominal rake angles, and the cutting force tends cellence Program of the Ministry of Human Capacities. The to decrease with increasing effective rake angle (similarly to research leading to these results was supported by the Hun- stationary cutting). garian Scientific Research Fund (OTKA FK-124462 and PD- 124646). The research reported in this paper has been sup- The most important conclusion of this paper is that cutting ported by the National Research, Development and Innovation force fluctuations can be accurately predicted by the geomet- Fund (TUDFO/51757/2019-ITM, Thematic Excellence Pro- rical explanation using the MEP shear angle model. Accord- gram). ing to this explanation, the cutting force during dynamic chip formation is determined rather by the effective cutting velocity References than the nominal one, while the shear plane length is in stronger correlation with the cutting force than the nominal uncut chip [1] M. Wiercigroch, E. 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