Experimental observations on unsafe zones in milling processes

1. Introduction

Milling processes (figure 1a) are subjected to a so-called surface regeneration effect due to the rotating motion of the tool: one of the cutting edges leaves a pattern on the machined surface (figure 2a), which is then cut by the subsequent cutting edge. This effect was recognized and modelled via delayed vibratory states by the pioneers of the field in the 1950s [1,2], while the mathematical theory of the governing delay differential equations (DDEs) was established later [3]. The phenomenon has been in the focus of many studies (e.g. [4–8]) due to the practical relevance of avoiding chatter. Chatter is an undesired vibratory motion of the cutting tool relative to the machined workpiece; its sound is clearly distinguishable from that of a chatter-free cutting process. If the relatively low-frequency vibration modes of machine tool structures are involved in chatter then fatigue may damage the machines [9], while high-frequency chatter increases cutting edge wear [10] and reduces machined surface quality [11].

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Owing to the periodic cutter/workpiece engagement (CWE), the milling process is also subjected to both external forcing and parametric excitation, which have time-period that is identical to the tooth pass period, i.e. to the delay [12]. The corresponding time-periodic response of the system is actually the desired stationary (or steady-state) cutting process. The onset of chatter is related to the loss of stability of this desired periodic motion. Stability can be predicted by means of linearization around this stationary periodic motion, which leads to a non-trivial problem when both parametric excitation and time delay are present [13–16]. In practice, the results of these calculations are usually summarized and visualized in the so-called stability lobe diagrams (SLDs) [17] where the stability boundaries are depicted with respect to the two most relevant cutting parameters: the spindle speed Ω and the axial depth of cut a. Along the stability boundaries, two types of vibrations occur related to generalized Hopf (H) and period doubling (PD) bifurcations also called in discrete systems Neimark–Sacker and flip bifurcations, respectively [18].

If SLDs are calculated in manufacturing industry, the most commonly used method is the so-called zeroth-order approximation (ZOA) [19], which is based on time averaging that neglects most of the effects of parametric excitation. This can be considered as a modified D-subdivision method [20], which directly uses the frequency response functions (FRFs) of the machine tool/tool/workpiece system. More accurate SLDs can be constructed by applying Floquet theory [13] to extend ZOA with further frequency modulations; these are often referred to as the multi-frequency (MF) solutions in the literature [16,21–24].

Accurate SLDs can also be produced with the so-called time-domain-based methods. These include weighted residual methods such as collocation techniques, different polynomial-based methods and least-squares-based solutions [25]. The most straightforward method is the semidiscretization method (SDM), which is based on the discretization of the delayed state only [26–29], while the time derivatives are kept without discretization. Most of the roughing operations can be accurately analysed by fast ZOA algorithms. However, especially for finishing operations that typically involve interrupted milling processes [30], MF or time domain-based solutions are needed to describe mode interactions [31] and to identify all the stability boundaries with PD bifurcations [32].

The cutting force acting at the cutting edge is an essential part in the mathematical models of milling processes. Its dependence on the state variables is influenced by many underlying physical phenomena that are still unexplored in their full complexity. All these effects are combined in the so-called cutting force characteristic, which is still determined empirically by extensive laboratory tests [17]. It is typical to consider the cutting force proportional to the chip width w and as a function of the local chip thickness h described by a specific cutting force characteristic f(h). For this, there are several simplified mathematical expressions: linear (L), shifted linear (SL), power (P), cubic polynomial (C) and exponential (E) functions (figure 1c, [33]). Note that many of these expressions involve nonlinearity, which has key importance in what follows. Moreover, the cutting force is ‘switched off’ when the chip thickness becomes zero or ‘negative’. This introduces also a kind of non-smooth behaviour [34] in the system.

In autonomous systems, Hopf bifurcation means that, at the stability boundary of the steady-state (trivial) solution, a periodic motion (limit cycle) emerges, which is stable or unstable around the unstable or stable steady-state solution, then referred to as super- or subcritical Hopf bifurcations, respectively [35]. This applies directly for the autonomous models of turning processes. The theory was extended for time-periodic [13] and for delayed systems [3] using infinite-dimensional state spaces. In time-periodic systems, the role of the trivial solution is taken by periodic ones, and the role of limit cycles are taken by quasi-periodic oscillations with two essential frequencies, one of them related to the time-periodicity of the system.

There are numerous studies on the ‘criticality’ (in other words, on the sense) of the bifurcations in manufacturing processes. Most of them report subcritical behaviour, which implies the ‘unsafe’ nature of the cutting processes as explained later in detail [30]. These calculations are analytically demanding: normal form theory can be combined with centre manifold reduction [36–40]; alternatively, multiple scales [41] or Galerkin methods [42] can be applied. Numerical solutions are also available by solving the related boundary value problem and calculating the emerging invariant limit cycle in autonomous systems (like turning) or invariant torus in the time-periodic case (like milling) [33,40,43,44]. The subcriticality induces the coexistence of the stable stationary cutting (SSC) solution and the chattering solution in a bistable [45] or unsafe zone [46] in the SLDs close to the stability boundaries. Subcritical phenomena are usually identified experimentally by means of the so-called hysteresis measurement [46] (figure 2d), when one of the key cutting parameters (usually the chip width or the axial depth of cut) is selected as the bifurcation parameter, and it is first increased and then decreased sweeping through the critical domain around the stability limit. The unstable object is assumed to exist between the separated stable branches of the hysteresis loop. There exist more advanced techniques to trace unstable periodic orbits experimentally by means of feedback [47], but this kind of control was not realizable due to the lack of appropriate actuators.

There is a critical point of this type of experiment: the slowly varied bifurcation parameter may slightly modify the bifurcation phenomenon, which is also called dynamic bifurcation effect [48]. This means that due to the continuous variation of the bifurcation parameter, the transition between the stable and unstable states occurs beyond the transition point for the constant parameters case. This shift can also be taken as hysteresis by mistake.

The goal of this study is to confirm the existence of the unsafe (bistable) zones in milling processes by means of extensive experimental work that excludes the effect of slowly variable parameters. In order to achieve this goal, the measurements have to be planned carefully by taking into account the theoretical linear and nonlinear behaviour of the regenerative phenomenon occurring in different milling operations (figure 1a,c).

(c) Linear stability

In order to analyse stability, one can introduce a so-called variational equation [13] ‘around’ the period-one stationary solution $\mathrm{col}({\mathbf{x}}_{\mathrm{P}},{\dot{\mathbf{x}}}_{\mathrm{P}})=\mathrm{col}(\mathbf{U},\mathbf{U}{[{\mathrm{\lambda }}_l]}_{l=1}^{2n}){\mathbf{q}}_{\mathrm{P}}$ with time period T_Z = τ

$$\dot{\mathbf{u}}(t)=\mathbf{D}(t)\mathbf{u}(t)+\mathbf{E}(t)\mathbf{u}(t-\tau )+\mathbf{g}(t,\mathbf{u}(t),\mathbf{u}(t-\tau )),$$1.7

where q = q_P + u. The time-periodic coefficient matrices are

$$\mathbf{D}(t)={\left[{\mathrm{\lambda }}_k\right]}_k^{2n}+{\mathbf{U}}^{⊺}\frac{\mathrm{\partial }\mathbf{F}}{\mathrm{\partial }\mathbf{x}(t)}(t,{\mathbf{x}}_{\mathrm{P}}(t),{\mathbf{x}}_{\mathrm{P}}(t))\mathbf{U}\mathrm{and}\mathbf{E}(t)={\mathbf{U}}^{⊺}\frac{\mathrm{\partial }\mathbf{F}}{\mathrm{\partial }\mathbf{x}(t-\tau )}(t,{\mathbf{x}}_{\mathrm{P}}(t),{\mathbf{x}}_{\mathrm{P}}(t))\mathbf{U},$$1.8

respectively, while g(t, •, •) contains the higher-order terms. Note that the stability of x_P or, equivalently, the stability of the trivial solution of (1.7) is a general requirement for a good quality milling operation, where vibrations occur due to periodic forcing only. First, the linear part of (1.7) is considered for stability analysis. According to the Floquet theory [13], the stability of the linear part of system (1.7) is determined by the monodromy operator U(T)

$${\mathbf{u}}_T(\theta )=(U(T){\mathbf{u}}_0)(\theta ).$$1.9

The spectrum of the linear operator U(T) consists of the so-called characteristic multipliers denoted by μ. If all multipliers lie inside the unit-circle of the complex plane, the time-periodic stationary solution is asymptotically stable.

There exist several semi-analytical methods to discretize the operator U(T) and approximate it with a matrix A whose eigenvalues approximate the dominant characteristic multipliers [25]. These methods lead to the following form

$${\mathbf{z}}_p=\mathbf{A}{\mathbf{z}}_0,$$1.10

where $p\in \mathbb{N}$ is the resolution of the time period, z_j = col^r_l=0 u_tj( − lΔθ) is the discretized state with t₀ = 0, t_p = T, Δθ = Δt = T/p and r = ⌊τ/Δθ⌋ is the delay resolution. The SDM [27] is one of the simplest among these methods, which serves sufficiently accurate results within reasonable computational time. In this paper, we used SDM to predict SLDs (figure 3).

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2. Revealing unsafe zone in milling processes

In this section, different measurement techniques capturing the unsafe zone phenomenon in milling processes are summarized. Special attention is paid to the existence and sensitivity of this phenomenon. While its existence can be investigated relatively easily (for example with a harsh entrance into the workpiece (figure 2a,b) an exact quantitative description of its sensitivity is an extremely complicated task.

To establish an effective measurement technique for the experimental investigation of the unsafe zone, one needs to distinguish between SSC and stable chattering (limiting) motion (CH). These can exist simultaneously for the same set of technological parameters, that is, mathematically speaking, bistability can occur. Although much larger amplitude is expected for chatter than for the stationary cutting state, this property cannot always be used for the direct detection of chatter. In measurement circumstances, it may be impossible to measure oscillations directly at the tool tip and sensors must be further away from the cutting zone. In such cases, chatter may be detected by the inspection of its distinctive spectrum and/or the surface pattern left on the workpiece.

As it was shown in [68], an impact can induce a transition from stationary cutting process to chattering motion. This phenomenon is also shown in figure 2c, where a prepared laboratory experiment was used to demonstrate the existence of the unsafe zone. Although this hitting methodology appears simple and promising, the quantification of the perturbation caused by the impact is not trivial. Since the impulse is scattered all over the bandwidth, it might result in an unsatisfactory excitation around the sensitive (chatter) frequency. In addition, one might hit the process in the ‘wrong direction’ that does not induce chatter. Since the phase space of a DDE is infinite-dimensional, the perturbation may completely bypass the unstable invariant torus that repels the motion towards the chattering state [33].

One possible solution to avoid the quantification issues is the exploitation of hysteresis effects caused by the subcritical behaviour of cutting, which is a common approach in nonlinear dynamics. In milling, this can be realized by using a symmetric roof-shaped workpiece, by which a regular increase and decrease of the axial depth of cut is ensured (figure 2d, [46]). Although the existence of the bistability phenomenon is clearly revealed by this method, the quantification of the bistable region leads to difficulties again. The slope of the roof-shaped workpiece must be small in order to improve measurement quality, since the theoretical prediction is typically performed by assuming constant bifurcation parameter in time. This assumption does not hold, since the finite (small) length of the workpiece results in a relatively steep and highly tilted upper surface causing significant rate of change in the axial depth of cut. Variation of the bifurcation parameter in time generates a dynamic bifurcation effect [48,69], which shifts even the linear stability boundaries depending on the sign and the rate of the parameter change. This shift may appear to be hysteresis while the real mechanisms behind bistability remain hidden.

To avoid the dynamic bifurcation effect, one can manufacture steps (figure 4b) on top of the workpiece. The steps result in a discrete increase and decrease of the axial depth of cut. However, due to the limited number of the steps that can be machined within the finite length of the workpiece, the resolution of this method is insufficient to determine the size of the unsafe zone accurately. This experimental method is suitable only for showing the existence of the unsafe zone phenomenon without involving a dynamic change of the bifurcation parameter, and for providing a rough estimation of the size of the unsafe zone. In the paper, this methodology is referred to as ‘step measurement’ (SM).

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As a final approach, the idea of harmonic excitation is considered. The vibratory system is excited by a burst harmonic perturbation that mimics the chatter frequency. This experimental method needs sophisticated instrumentation. It requires a built-in actuator that is capable of providing the force needed to move the machine tool structure with a given amplitude and frequency. This was realized by a built-in inertial drive (ID) in a large heavy-duty milling machine as shown in figure 5f . This measurement technique is referred to as ‘harmonic test’ (HT).

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This paper focuses on the dynamics of the SM and HT methodologies, in order demonstrate the existence of the unsafe zone, and to quantify the effect of unstable invariant tori without any effects from dynamic bifurcations. The conclusions use experimental observations that are then related to the analysis of a mechanical model of the milling process.

3. Preliminary experiments

In order to conduct a series of measurements demonstrating the effect of unsafe zone, preliminary tests must be performed to characterize the workpiece material and the dynamic behaviour of the machine tool. In this section, cutting force characteristics, machine tool dynamics and linear stability are investigated experimentally and summarized for both the SM and HT cases. Note that the SM was performed in a laboratory while the HT took place in an industrial environment.

A series of down milling (DM) tests were carried out for different chip thickness values using workpiece materials AL2024-T351 in SM and C45 in HT. For AL2024-T351, the nonlinear cutting force parameters in (1.6) were directly identified from a dense equidistant measurement set [17] resulting in K_c, j = (1049.8, 532.9, 0) MPa, K_e, j = (13.4, 15.2, 0) kN m⁻¹ and E_j = (66.9, 45.4, 0) · 10³/m. In the case of C45, the linear cutting force parameters K_e, j and K_c, j were measured only, since there was no possibility to perform extensive measurement series in the industrial environment of the HT: K_c, j = (1889, 734, 387) MPa, K_e, j = (69.7, 83.2, 87.6) kN m⁻¹. The exponents E_j = 200 · 10³/m in the HT case were fit heuristically such that they do not contradict the linear stability observations discussed below. Its value needs to be large enough to mimic the industrial shifted linear characteristics around the average chip thickness and small enough to capture finite tangent when the chip thickness vanishes.

Modal tap tests were also performed on the milling tool-tip to determine the overall dynamic behaviour of the milling machine (black lines in figure 3a,d,g). The modal parameters were determined by assuming linear non-proportionally damped dynamics (red lines in figure 3a,d,g [63,71]). The parameters were later used for linear stability analysis (figure 3c,f ,j). A smaller dominant set of proportional modes (blue lines and tables in figure 3, where ${\mathbf{U}}_k={\mathbf{p}}_k/\sqrt{m_k}$ and ∥p∥ = 1) were also used later for performing nonlinear SDM.

Experimental stability (chatter) tests were performed for both the SM and the HT in order to see the discrepancies from theoretical linear stability predictions. The results are summarized in figure 3. In the laboratory environment (SM), chatter tests were performed with machining properties MP₁:{Z = 2, a_e = 10 mm, DM, D = 16 mm, κ = 90 °, η = 30 °} (figure 3) on a dense grid of technological parameters, see the green circles (SSC), the red crosses (chatter, CH) and the blue diamonds (undetermined case) in figure 3a. It is important to note that there is a difference from the theoretical prediction (black line) in terms of both elevation (axial depth of cut) and ‘resonances’ (natural frequencies). Both can be explained by the change of the dynamics during cutting compared to the non-operating (tap tested) case. In fact, for the SM set-up a dummy tool was shot by softball and measured by a laser displacement sensor in order to estimate varying dynamics during spindle rotation (without cutting). Modifying functions were used for ω_n, k(n) and ζ_k(n) accordingly [53] (figure 3c).

In the industrial (HT) case, up milling (UM) chatter tests were performed at a chosen spindle speed only, in order to provide validation of theoretical stability analysis for each ram position. The samples corresponding to MP₂:{Z = 8, a_e = 100 mm, UM, D = 125 mm, κ = 45 °, η = 0 °} and MP₃:{Z = 8, a_e = 60 mm, UM, D = 125 mm, κ = 45 °, η = 0 °} are presented in figure 3fj, where some differences from theoretical results (shown by black and blue lines) were again experienced.

After the chatter tests, the nonlinear experiments were prepared based on the measured SLDs. In the SM case, spindle speeds are selected from the regions where the predicted and measured stability limits are close to each other. For the HT cases, the two validated spindle speeds are accepted for later nonlinear analysis.

4. Step measurement

The SM is one of the simplest approach to demonstrate the phenomenon of bistability while excluding the dynamic bifurcation effect. Based on the results of experimental chatter tests (figure 3c), the workpiece is prepared such that steps are machined onto it in increasing and then in decreasing order (figure 4b). Since the length of the workpiece is given, a trade-off must be made between the number and the length of steps. It is important at all steps to provide enough space (time) for the tool to reach its stationary state. Obviously, there is smaller possibility to capture the phenomenon of bistability in the SM than in the HT due to the limited resolution of axial depth of cut steps on the workpiece. When the measurements do not capture bistability, the test results cannot be distinguished from supposingly linear chatter tests where instability should arise in symmetric arrangement along the steps.

For the measurements, we also use the industrial jargon to label vibration states as ‘stable’ as SSC and CH. In fact, all these states are related to a stable invariant object of the dynamical system (1.4). Chatter CH state involves both the evolved threshold vibration and the onset of amplifying vibrations originated, on the one hand, from the USC period one solution x_P. The stability loss is caused by an orbital Hopf bifurcation [13], which introduces dominant (chatter) frequency (figure 3i) and its modulations [72]. The Hopf bifurcation in the nonlinear milling model is subcritical and gives rise to an unstable invariant torus branch (ITB) of quasi-periodic solutions x _QP with Ω_Z and ω_c. The arising subcritical branches can be found by, for example, nonlinear SDM [65] or path-following methods based, for example, collocation [33].

On the other hand, this unstable torus can also repel the process from the expected SSC to an amplifying solution toward the threshold (CH) state that we hear and measure in the long term. The threshold vibration state can be quasi-periodic or chaotic resembling the frequency content of the linear solution [74]. Large amplitude severe chatter is typically more chaotic in nature and deviates more from the linear spectrum.

It is important to note that cutting is not a ‘clean’ process; specifically, for parameter values near the linear stability boundary, noise-induced coherence resonance activates oscillations with frequency content similar to that of chatter [75]. We denote this weak frequency content as S(n), which appears in the otherwise stable region. We distinguish between this coherence resonance and the threshold state of chatter using a heuristic approach based on the higher frequency modulation of the dominant chatter frequency ω_c (see A in figure 4c).

The results of the SMs are discussed below. The SMs that show an asymmetric pattern of SSC/S(n) and CH states indicates the existence of the bistable zone. Experiments with a symmetric pattern could either imply linear behaviour or indicate that the resolution of the steps was insufficient to reveal the asymmetry. Since we experienced not only symmetric but asymmetric patterns, too, we believe the second is true. The performed SMs (A–F) are depicted in figure 4c–h. The corresponding axial depth of cut values span through the experimental stability limit shown by ‘combs’ in figure 3c(ii). In all tests, the feed was f_Z = 0.1 mm/tooth, that is the tool was rotating roughly 150 times at each step. Lateral acceleration signals were acquired in two directions on the spindle and in one on the workpiece, while sound pressure was also measured (figure 4a).

Wavelet transformation was performed for all signals to obtain a good quality time-frequency diagram. Among all recorded signals, the y-directional acceleration of the spindle carried the richest frequency spectrum (figure 4). In all time-frequency plots, harmonics of Ω and Ω_Z are indicated by white and black side-ticks. The instances where the axial depth of cut changed are indicated with thin ‘horizontal’ black lines. The dominant vibration (chatter) frequency is indicated by black arrows. Note that frequency contents, which do not have a relation with the actual cutting process can be identified at the approaching phase (AP) parts. According to figure 4, all measurements start with a defined SSC or S(n) state. Only three measurements (D,E,F) exhibit the bistability phenomenon, while the others are symmetric (measurements (A,B,C).

In the symmetric case A, strong harmonics of Ω_Z = 200 Hz appear, while harmonics of Ω = 100 Hz are also visible due to tool run-out. Weak noise-induced coherence resonance appears in the critical dominant frequencies before and after the CH, identified as S(n). Measurements A and B contain brief but strong CH state in the middle of the tool path with weak modulations of slightly changing frequency. Clear chatter regions with strong base and modulation frequencies are arranged symmetrically in measurement C, where strong changes of the chatter frequencies can be followed. Measurement D starts with an SSC, followed by coherence resonance S(n) of the frequency associated with the critical eigenvalue, then transitioning to clear chatter case, which later appears as an asymmetric measurement case with strong chatter followed by S(n) in the end. This asymmetric arrangement is forming an evidence for bistability. The next measurement E starts in the coherence resonance state S(n), transitioning to strong asymmetric chatter till the end of the workpiece. Obviously, some doubt can appear in this case since the whole measurement started with an almost unstable case, but the chatter stays so strong in the end, without any sign of settle, we identified this case as a weak evidence for the bistability. Measurement F is the most clear proof of bistability; it illustrates a slight noise in the increasing depth cut part and strong chatter in the decreasing depth of cut part. Even if one considers the noisy S(n) state as CH (which is not the case), the following CH states still strongly support the existence of bistability.

The HT experiment was designed to overcome the resolution problems of the SM and to show that bistable zones can appear even for a large industrial machine with a different workpiece material.

5. Burst harmonic perturbation (harmonic test)

This form of excitation can be considered as the controlled version of the hammer blow figure 2c, where the aim is to push the system out from its local SSC state by some form of disturbance. Note that HT avoids the potential confusion of the dynamic bifurcation effect, as shown in figure 2d. Here, according to the initial linear calculations presented in figure 3, we excite the system with a burst of harmonic force set to the calculated dominant vibration (chatter) frequency figure 3i. By gradually increasing the excitation amplitude of the built in exciter, theoretically the unstable object that separates the bistability can be found. It is important to emphasize that the exciter ID is mounted on the ram (figure 5f ), which means the excitation is really unlocalized. Without any characterization of the ID and the ram structural mechanics, the real actual amplitude of the tool cannot be given.

In this manner, the only aim is to show that there is a limit excitation amplitude that separates the SSC from CH. By avoiding any characterization of the ID, this limiting value is a voltage value, which refers to the driving voltage amplitude of the injected harmonic excitation. A new vibratory state is introduced referred to this excitation burst denoted by E. During the E part, the system is forced both by the cutting and the excitation forces resulting in a quasi-periodic forced state. It is believed that this quasi-periodic forced state can push out the system to its threshold CH state and staying on after strictly ending the E part.

Owing to the flexibility of the ram, the motion is also copied to the face surface of the workpiece, in which the different zones can be identified (figure 5l). The SSC creates the usual ‘feed pattern’, which is considered as the desired milling process. When the excitation switched on the system responses with a quasi-periodic forced state E. Then, by the vanishing external forcing from ID, the mechanical system has its ‘free motion’ (only excited by the regenerative cutting force) and the threshold motion CH might arise.

In this section, two measurement sets are presented related to the initial linear calculations (figure 3f ,j) for two different ram extensions. Among the measurement sets one of them can be considered as a clear evidence of bistability (HT, MP₂, figure 3f , figure 5a–e), while the other one (HT, MP₃, figure 3j, figure 5g–k) can be considered as a weak but valid proof.

As an initial measurement HT sets were performed at Ω = 500 r.p.m.; however, the bistable effect was weak and except some successful trials the 500 r.p.m. was considered as a failed attempt. Then, the spindle speed was set to Ω = 510 r.p.m., where it was possible to see and measure the phenomenon. The results presented in figure 5 are ordered from the top to the bottom with decreasing axial depth of cut a. The cutting process was completely CH at a = 2.3 mm. The first presented result at the top are for a = 2.2 mm, where the asymmetric (SSC/E/CH) behaviour can be realized for drive voltage amplitude V = 1 V (figure 5b), which vanishes at V = 0.5 V (figure 5a). The spindle and tooth passing harmonics are denoted in the same manner as it was explained in §4. This measurement was considered as a successful strong evidence of bistability, even though using V = 1 V (figure 5b) initially some self-excitation had appeared due to the entrance, which died out (figure 2a,b). By decreasing the axial depth of cut to a = 2 mm, the limit voltage increased between V = 2 V (figure 5c) and V = 3 V (figure 5d). In this case, the V = 3 V case is strongly bistable, while in the case of V = 2 V some leftover vibration remained after E part, but it died out. During the experiment, it was verified that a further decrease in the voltage would result in a more rapid ‘die-out’ of this leftover vibration, consequently, it has considered as the limit for the selected a. In the last case, when a = 1.8 mm no bistability was found (SSC/E/SSC) even using V = 10 V (figure 5e) drive voltage amplitude for the ID. According to this result, it can be concluded that the Ω = 510 r.p.m. measurement shows the effect of subcritical Hopf bifurcation, namely, the appearance of the bistabile state ‘below’ the stability limit in a.

The next set of measurement was performed with lower ram extension for Ω = 350 r.p.m. with MP₃. Again, the panels are ordered in a decreasing axial depth of cut manner from the top to the bottom in figure 5g–k. This set of measurement was more sensitive for perturbation and most of the cases the predicted dominant chatter frequency was a bit apparent in the time signals, even though it does not leave any recognizable mark on the surface (figure 5l). Notwithstanding, there was significant difference between the apparent chatter frequency and the really chattering case, distinguished with S(n) and CH. First, completely unstable (CH/E/CH, not presented here) case was measured at a = 2.1 mm. Then S(n)/E/S(n) and S(n)/E/CH types were experienced introducing V = 0.1 V (figure 5g) and V = 0.2 V (figure 5h) voltages, respectively, for a = 2.0 mm. By decreasing the depth of cut to first a = 1.9 mm, then to a = 1.8 mm the limit voltage remained between V = 0.1 V (figure 5i) and V = 0.2 V (figure 5j). The weak bistability was demonstrated, although dominant (chatter) frequency was apparent during all tests, somewhat as a sign the operation was extremely close to the stability boundary. In the end at a = 1.7 mm, the system remained clearly stable (SSC/E/SSC) even setting V = 10 V drive amplitude, as presented in figure 5k.

According to the result presented in this section, the existence of the bistability is clearly demonstrated, although it was not apparent clearly in every trial. It can be considered as a quite remarkable achievement to experience this highly nonlinear phenomenon in this industrial case. Also, the difference in a does not seem significant; however, it reaches 10%, which can be considered as a small but noticeable difference.

6. Reconstruction of bifurcation diagrams

In this section, we construct the empirical bifurcation diagrams and compare them to the theory. In all examples, the transmissibility between the sensor locations and the tool tip was measured by using experimental modal analysis [63] to overcome the unlocalized nature of the signal acquisition (figures 4a and 5f ). The stationary period one solution (SSC/USC) was predicted numerically, while the subcritical ITBs [33] were estimated by using nonlinear SDM (§1.12) and path-following collocation method [33].

The comparison is presented in figure 6 both for the SM and HT cases. The bifurcation diagrams are constructed by adding the vibration amplitudes A_st and A_c corresponding to the tooth passing frequency Ω_Z of the stationary solution (forced vibration) and to the dominant chatter frequency ω_c, respectively, for various depths of cut a. In all cases presented in figure 6, the deviation between theoretical and experimental results can be attributed, on the one hand, to the significant discrepancies in linear stability predictions (figure 3a(ii)). This causes a shift in the diagrams with respect to a. On the other hand, there is a shift in the amplitude of the stationary solution (SSC/USC), too. This is due to the experienced runout of the two-fluted carbide milling tool, which affects the tooth passing harmonics Ω_Z by transferring kinetic energy to the spindle harmonics Ω. In figure 6, ‘hand’-fit experimental branches are depicted with thick lines, in which a similar trend can be recognized as in the theoretical predictions (thin lines). For example, it can be explained why measurement A shows symmetric and why measurement F shows highly asymmetric arrangements in terms of a in figure 4. This is caused by the relative steepness of the quasi-periodic branch in figure 6 [33].

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A much better correlation of theory and experiments can be observed in the HT (figure 6e,f ) case, where both bifurcation diagrams are better justified in terms of the Hopf point (H) and the vibration amplitudes. Note that only the measurement at n = 510 r.p.m. in figure 6e can be considered as a clean test, due to the continuous slight appearance of the chatter frequency in the case n = 350 r.p.m. presented in figure 6f . Regardless of the expected discrepancies in the stability limit (H), the curvature of the emerging unstable quasi-periodic branch (ITB) is well given by the theory, which results in lower vibration amplitude for the large amplitude chatter branch.

In all cases, significant difference can be recognized between the ITBs constructed by nonlinear SDM and the path-following technique. This is due to the low-order approximation of the SDM, which cannot follow the real curvature of the emerging ITB branches.

The first fly-over moment can be realized by determining the local chip thicknesses (1.2). Theoretically, this would refer to the size of the unsafe zone, but in milling this connection is questionable due to the multiple edges participating in the cutting process. Even by the consideration of minimum possible chip thickness [76], in SM cases, the fly-over point comes ‘early’ after the Hopf (H) point, which can cause the fold of these branches, rather experiencing supercritical nature right after the subcritical emerging. This would be opposite to the experienced really strong subcritical nature of, for example, test F in figure 4. In the HT cases, the fly-over appears much ‘further’ from H points, due to the quite stiff y directional modes, which prevents ‘early’ fly over in the process. Moreover, due to the high edge number (Z = 8) the fold would appear after much more significant subcritical behaviour as it was experienced in the tests.

7. Conclusion

Milling processes in metal cutting are modelled by delay differential equations with parametric and external excitation. The desired cutting process corresponds to a stable periodic solution of these equations. When this periodic motion loses stability, undesired chatter occurs. Unfortunately, chatter may appear also due to large enough perturbations in parameter domains where the desired cutting process is stable. The existence of these so-called unsafe (or bistable) parameter domains has been shown theoretically and experimentally.

The bifurcation analysis of the periodic delay differential equation models predicted the existence of unstable quasi-periodic solutions emerging from a subcritical generalized Hopf (or Neimark–Sacker) bifurcation of the desired periodic solution. The nonlinear extension of the semidiscretization and the path-following collocation methods made it possible to carry out these calculations. The parameter points where the fly over effect appears first on the unstable quasi-periodic solutions were also determined.

In order to check the results, two different experimental techniques were developed, both of which avoid the dynamic bifurcation effect related to slowly varying bifurcation parameters. While the SM is straightforward, the harmonic burst excitation (HT) needs expensive instrumentation built in the machine tool. The SM technique was strongly influenced by the dynamics of the rotating spindle, which was taken into account by the method presented in [53]. Qualitative agreement was shown between theory and experiments with theoretical results being somewhat shifted compared to the experimental ones. The HT case was presented with less number of measurements, but it can still be considered as an appropriate industrial demonstration of the phenomenon.

In some cases, the SM was not able to capture the phenomenon due to the insufficient resolution of the parameter steps. Meanwhile, especially in the HT case, noise-induced vibrations caused resonance of the nearby multipliers, which can be taken mistakenly as chatter vibration. Still, it is now confirmed even in case of an industrial case study that the presented theory is capable of predicting the qualitative behaviour of milling processes for small and large perturbations, that is both on linear and nonlinear levels. This mostly experimental work proved that subcritical bifurcation, that is bistability phenomenon exists in milling processes, although it is extremely difficult to obtain accurate quantified agreement between theory and experiments.

The bistable behaviour is experienced by the machinist through the asymmetric behaviour of certain machining operations. These observations are explained by the intricate delayed and nonlinear dynamics that originates in the milling process itself. This phenomenon appears already without the consideration of the fly-over effect, which limits the growing nonlinear vibrations. In milling processes, the fly-over appears in strange rhythms due to the multiple edge arrangements, which in extreme cases can cause even a full one-revolution escape of the tool. Many effects including run-out and rubbing/cutting transitions [76] influence this complex but still pure geometric picture, which all need to be considered for an improved approach. From an industrial view-point, it would also be useful if those chatter states were selected and characterized, which are still acceptable due to their moderate vibration amplitudes.

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Competing interests

We declare we have no competing interests.

Funding

This work was partially supported by European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013) ERC Advanced grant agreement, no. 340889, Hungarian EMMI ÚNKP Bolyai+ (ÚNKP-18-4-BME-193) and MTA Bolyai Research Scholarship (BO/00497/18/6), Hungarian NKFI FK 124361, EU MC-Suite project (H2020-FoF-2015-680478), EU Marie Skłodowska-Curie IF 704133 Piezomach, NSERC Discovery Accelerator Supplement grant no. 477876-2015.