1 Introduction

The occurrence of high vibrations in machining limits the achievable Material Removal Rate MRR [1], the surface quality and the tool life TL [2]. Vibrations are typically due to regenerative effects that bring cutting process to instability. Spindle speed variation (SSV ) is one of the available techniques for suppressing chatter vibrations [3]. It is based on a continuous modulation of the spindle speed that aims at repressing the regenerative effect and thus the growth of undesired vibrations. Over the years, SSV has been developed and tested both in turning and milling applications. First, studies assessed the vibration mitigation properties of the spindle speed variation SSV. Radulescu et al. [4] developed a time domain model for this purpose. Insperger et al. [5, 6] developed an analytical formulation for predicting the stability map when the SSV is adopted. The model was validated through simulations. The same approach was used by Kapoor et al. [7]. Albertelli et al. [8] developed a time domain simulation model for studying SSV in turning. The model was validated through experimental tests. Urbikain et al. [9] studied the effect of SSV on the stability map in a real turning application through the perturbation theory. Yamato et al. [10] studied the stability of variable delay turning through an energy approach. Zhang and Ni [11], adopting a similar method, provided useful indications for selecting the SSV parameters. The approach was tested through numerical simulations. Otto and Radons [12] studied the effectiveness of SSV in turning according to different eigenfrequencies and damping ratios. Wu and Chen [13] extended the application of SSV to non-circular turning. Ding et al. [14] developed a control strategy based on the simultaneous adjusting of the parameters governing the SSV. They demonstrated the validity of the proposed approach through simulations and executing some preliminary cutting tests. The same authors in [15] developed a close-loop implementation that combined monitor and control strategies suitable for turning applications. Meng et al. [16] developed a state feedback control for assuring the steady state stability in SSV turning. They tested the control strategy through simulations. Zatarain et al. [17] carried out milling simulations and experimentally validated the numerical findings. Totis et al. [18] developed an analytical formulation for fast estimating the stability maps when the SSV is adopted in milling. A proper experimental validation was not provided. Sinusoidal spindle speed variation SSSV is the most studied approach for modulating the spindle speed. Alternative approaches were developed. For instance, Yilmaz et al. [19] conceived an approach based on a pseudo-random spindle speed variation. Nam et al. [20] recently developed a SSV strategy characterized by a constant acceleration rate that allowed achieving higher MRR than conventional SSV. The same authors in [21] defined novel chatter indices for assessing the chatter growth. Albertelli et al. [22] developed a generalized algorithm based on cyclostationary theory for chatter detection when the spindle speed variation is used.

From the quantitative perspective, it was proven that stability enhancement due to SSV is effective and robust especially in the high-order lobes region of the stability maps, both in turning [8] and milling [17]. The SSV, thanks to the the stabilization capabilities, reduces the risk of tool chipping and too early failures that typically occur when vibrations affect the cutting [23]. Although several research works have been published on SSV so far, few of them focused on potential secondary detrimental effects of the technique. For instance, Albertelli [8] studied the thermal load of the spindle motor due to the spindle modulation and performed a feasibility study with respect to industrial turning applications. Urbikain et al. [9] found that SSV slightly increased (about 20%) the power consumption of the lathe. For what concerns the tool duration, Albertelli et al. [24] found that sinusoidal spindle speed variation SSSV has a negative impact on the life of the tool, fostering the formation of cracks that tend to progressively detach the coating and thus increasing the wear rate. In this study, the achieved results were obtained comparing SSSV machining to constant speed machining CSM, both in stable conditions. Chiappini et al. in [25] used a finite element model FEM for simulating SSSV cutting. It was found that the modulation of the spindle speed is the responsible of an additional mechanical-thermal load on the cutting edge that could bring to the cracks formation. Although both the research works gave an interesting interpretation of the involved phenomena, a quantification of the tool life TL reduction, according to the adopted cutting parameters, was not provided. Being able to quantify the TL reduction would be extremely useful for assessing the potentialities and the limitations of the SSSV, especially in terms of industrial applicability. Although variable speed machining VSM was first conceived in the 1970s, its wide diffusion in real applications has not been registered so far. For instance, the use of such technique to stabilize machining operations that otherwise could be regularly carried out with stiffer, but even more expensive machines, needs to be properly analyzed. In such way, a trade off between tooling and machine costs could be investigated. In order to bridge this gap, in this paper a generalization of the Taylor’s model for estimating the TL in turning, when both CSM and SSSV are indiscriminately used, was developed.

The literature on the study of the wear of the tool is huge. Some of the works focused on the performance assessment of new tool materials and coatings. For instance, Wojciechowski et al. [26] assessed the performance of Boron Nitride Dispersed Cemented Carbide on a specific spheroidal cast iron focusing on the main wear phenomena. Wojciechowski and Twardowski [27] compared the duration of sintered carbide and cubic boron nitride in hardened steel milling without developing a proper tool life modelling. Even the effects of the adopted cooling lubrication strategy on the tribological behaviour [28] and on energy consumption have been subject of several studies [29]. Cryogenic cutting allows reducing the specific energy involved in cutting. Comparative studies on the life of the tool were carried out. For instance, Albertelli et al. [30] developed a tool life modelling considering conventional and cryogenic Ti6Al4V milling. Cryogenic cutting showed increments in tool duration only if high cutting speeds were adopted. Although Wong et al. [31] put into evidence the limitation of the empirical approaches, Johansson et al. [32], performing an interesting assessment of different tool life modelling formulations, found that Taylor’s model assured a tool life estimation error that ranged from 8% to 21% and that the error was not material dependant.

So far, the literature that has been dealt with the tool wear under unstationary cutting conditions is rather poor. For instance, Galante et al. [33] developed a new tool life modelling approach based on Gaussian probability distribution that allows developing more flexible models with respect to the Taylor’s model. One of the first studies that presented the difficulties related to the wear modelling when different cutting conditions are set was carried out by Jemielniak et al. in [34]. Lin in [35] and Pálmai in [36] proposed a cumulative wear model for taking into considerations different spindle speed steps. Since, to the authors’ knowledge, no specific modelling approaches for estimating the TL when the cutting speed is continuously modulated have been developed, a preliminary formulation is first presented in this study. The paper was structured as follows. In Section 2, a more exhaustive description of the research goals together with the explanation of the conceived approach were provided. The experimental set-up, the designed experimental campaign and the preliminary tests were also presented. In Section 3, the results of the experimental tests, the development of the modelling formulations and their validation were reported and critically analysed. In Section 4, the conclusions were also outlined.

Nomenclature

β :

regression coefficients

\(\boldsymbol {\hat {\beta }}\) :

regression coefficient estimations

L :

least square function

X :

regression variables matrix

y :

response variables vector

χ :

primary tool lead angle

Δ A Ω % :

percentage tracking error with respect to the set AΩ set

ΔRV A%:

percentage tracking error with respect to the set RV Aset

\(\hat {TL}_{s}\) :

TL estimation carried out with the s th formulation

Ω :

Spindle speed

Ω 0 :

Nominal spindle speed

Ω 0set :

nominal imposed spindle speed

\({{\varOmega }}_{meas}\left (t\right )\) :

measured spindle speed through the spindle encoder

\({{\varOmega }}_{set} \left (t\right )\) :

imposed spindle speed set-point

a p :

radial depth of cut

A Ω :

amplitude of the sinusoidal modulation of the spindle speed

\(A_{B_{ik}}\) :

area of the wear land on the tool flank

CI :

confidence interval

\(CT_{i \left (p+1 \right )}\) :

cutting time associated to the average flank width \(VBB_{i \left (p+1 \right )}\)

C T i p :

cutting time associated to the average flank width V BBip

D F j :

degrees of freedom of the factor j

f :

feed per revolution

F − valuej:

Fisher tests for the factor j

freq :

frequency of the sinusoidal spindle variation SSSV

H 0j :

null hypothesis for the jth paired CSM − SSSV test

\(l_{B_{ik}}\) :

length of the wear area used for the V BBik computation

m :

number of considered factors in the 2m experimental plans

M S j :

Mean Squares of the factor j

n r c :

number of replicates of the central points

n r :

number of replicates of tests carried out at the corner points

P − valuej:

P value of the test for the factor j

R 2 :

coefficient of determination of the regression

\(R^{2}_{adj}\) :

adjusted coefficient of determination of the regression

r 𝜖 :

insert corner radius

RVA :

non-dimensional amplitude variation of the sinusoidal spindle variation SSSV: sinusoidal amplitude/Ω0

SE :

standard error

\(SS^{III}_{j} \equiv SS_{\left (j \mid k,{\ldots } ,r \right )}\) :

adjusted sum of squares (type III) of the factor j

\(SS_{\left (j,k,\ldots ,r \right )}\) :

sequential sum of square (type I) considering the factors j,k,…,r in the model

t :

time

TL :

tool life

T L i :

Tool Life TL of the i th tested cutting edge

\(TL_{s-error_{\%}}\) :

percentage errors in the TL estimation adopting the s th formulation

v c :

cutting speed

VBB :

flank wear average width

V B B t :

flank wear width threshold

\(VBB_{ik_{j}}\) :

local measurement of the flank width VBB after the k th stop of the i th wear test

V B B i k :

is the k th measurement of the average flank width associated to the i th wear test

X n :

radial coordinate—distance from the external surface (micro-hardness measurements)

2 Material and methods

According to [8], even in this research experimental tool wear tests (ISO 3685, [37]) were performed both in constant speed CSM and variable speed VSM machining. The definition of the tests was carried out adopting a Design of Experiments DOE approach. More specifically, in both the cases a full factorial scheme was used and properly motivated. Additional information were provided in Section 2.1. The tool life TL data were analyzed and used for the generalized regression model development. A proper validation was even carried out.

2.1 Design of experiments

A steel turning application was selected. Indeed, for the cutting speeds typically adopted in steel machining and considering the limiting eigenmodes generally associated to tool-holder systems (70–160 Hz), it was demonstrated that SSSV can assure effective chatter suppression properties [8]. Moreover, such application is rather widespread in most of the shop-floors. In this research, two levels full factorial designs 2m (m is the number of the considered factors) were conceived for the tests performed at CSM and for the tests that involved the sinusoidal spindle speed modulation SSSV. Although other design of experiments schemes allow to reduce the experimental effort (i.e. Taguchi, Box-Behnken), it was decided to use a full factorial scheme since it provides a more flexible approach especially if it is necessary to combine and analyze the results of the two separate experimental sessions. Moreover, Tsui [38] found that Taguchi approach can lead to non-optimal solutions, information loss and efficiency loss. Indeed, Medan et al. [39] obtained better estimation errors adopting a full factorial design instead of using Taguchi. Box-Behnken is a factorial scheme with an incomplete block design that can lead to regions of poor prediction quality (corners), Montgomery [40].

For both the experimental plans, the cutting velocity vc and the feed per revolution f were the main analyzed factors. Since the SSSV can be described by Eq. 1, two additional factors were considered: RVA and freq. RVA is the non-dimensional amplitude variation parameter while freq is the frequency parameter. It is worth noting that, since the spindle modulation makes the chip thickness to continuously vary [8, 25], the f parameter was considered in the experimentation at CSM although, according to the Taylor’s theory, the effect of such parameter should be less relevant than vc.

$$ \begin{array}{@{}rcl@{}} {{\varOmega}} (t)&=&{{\varOmega}}_{0} \left( 1+ RVA \cdot \sin{\left( 2\pi\cdot freq \cdot t \right)} \right);\\ A_{{{\varOmega}}}&=&RVA\cdot {{\varOmega}}_{0} \end{array} $$
(1)

For what concerns the radial depth of cut ap, since it is well-known in literature (i.e. Johansson et al. [32] and Hägglund [41]) that its effect on the wear of the tool is less relevant than f and considering the need to limit the experimental resources, this parameter was not varied in the experimentation. For this purpose, a radial depth of cut ap = 2mm was set. This choice simultaneously took into consideration some aspects:

  • high ap values should be used since SSV is typically used for rough machining

  • high ap values should be used in order to avoid the effect of the radius of the tool in the flank wear measurement (see Fig. 5).

  • ap should be lower than the active part of the cutting edge (6mm)

  • should be limited in order to consider the lathe limitations in terms of maximum spindle torque and power.

  • ap should be limited to avoid chatter vibrations [1]. Preliminary cutting tests were performed in order verify the absence of any dangerous vibrations (see Section 2.3).

For all the analyzed factors (vc, f, RVA, freq) two levels were considered. Moreover, in order to track possible deviations from the linearity (i.e. curvatures due to second order effects with respect to the considered factors) or to enhance the model adequacy as well, center points were added to the 2m factorial design scheme. In case of curvature effects (assessment performed in Section 3.1 and Section 3.2) , additional test conditions can be added. To restrict the experimental effort/budget, it was decided to carry out one single test nr = 1 for each corner condition [40] while nrc = 5 replicates were set for the intermediate cutting conditions, left side of Fig. 10. Finally, it was decided to use the described approach for the following reasons:

  • Since it was found [24, 42] that the wear mechanisms in variable speed machining VSM are rather common in tool wear tests, the expected deviations from the Taylor’s model (linear with respect to the factors in logarithmic coordinates) were supposed to be limited

  • The Analysis of Variance (ANOVA) theoretical framework (Montgomery [40]) provides suitable tools for statistically assessing the curvature effects

  • The adequacy of the proposed approach can be further verified selecting, for the model validation phase, different cutting conditions from the ones used for the modelling step (Section 3.4).

The analyzed factors and the corresponding selected values, both for CSM and SSSV, were resumed in Tables 1 and 2, respectively. The values of the analyzed factors were chosen according to [42] and considering the technological limitations associated to the adopted tool (vc and f ) and lathe. Indeed, the set parameters allow continuously changing the cutting speed in the SSSV tests. Preliminary turning tests were carried out in order to check the feasibility of both the DOE plans (Section 2.3). Moreover, it was demonstrated in [8] that the selected parameters (RVA and freq) assured relevant chatter suppression properties if unstable cut occurred. It is worth noting that all the tests in this experimental campaign were executed in stable conditions (Section 2.3). All the cutting tests (both at CSM and adopting SSSV ) were completely randomized. More details on the experimental set-up were reported in Section 2.2.

Table 1 Design of experiments—CSM, factors and values
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Table 2 Design of experiments—SSSV, factors and values
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2.2 Experimental set-up

The flank wear average width VBB was monitored during the cutting tests. The flank wear threshold V BBt = 0.15mm was used as the end − of − tool − life criteria. The cutting time TL that corresponds to the considered wear threshold was the main process response for all the tested conditions. Although the selected wear threshold V BBt = 0.15mm is lower than the ones typically used and suggested by the ISO 3685 [37], preliminary wear tests confirmed that the chosen threshold avoided a too high dispersion of the results in terms of insert duration TL. This choice was carried out to limit the experimental effort. Steel bars (material 39NiCrMo3, with hardness 255HB, ultimate tensile strength 1145MPa, Yield strength 1015MPa and an elongation at break 14.5% (UNI 7845 − 78 [43])) hardened and tempered were used to perform the wear tests. The tool life tests were performed following the standard ISO3685 [37]. A Stereomicroscope Optika SZN − T with a Motic SMZ − 168T was used to measure flank wear width VBB during the cutting tests. More details on the performed wear measurements can be found in Albertelli et al. [42]. A carbide tool with a lead angle χ = 95 ∘ was adopted (ISO code TNMG220404 − M5 5625 (tool radius equal to r𝜖 = 0.4mm, rake angle 13∘ and a relief angle equal to 0∘ with a Al2O3 − TiCN coating)) and fixed on tool holder, ISO code MTJNL2525M22. Cutting fluid (oil-water emulsion with 5% of HOCUT 795 SC) was used in order to reproduce realistic industrial machining conditions. The lubricant was injected to the cutting zone through a flexible and adjustable nozzle, visible also in Fig. 1. It is worth noting that a proper control unit was specifically developed (adopting National Instruments NI hardware and software) for performing the cutting tests in VSM. Specifically in this research, a Ω0set − RV Aset − freqset parameter combination can be set to the controller to perform cutting tests with the SSSV. The conceived solution was integrated with the drives and the numerical controller a SOMAB Unimab 400 lathe, refer to Fig. 1. Since it was not possible to use NC build-in functions for modulating the spindle speed, a tailored solution was developed. A circuit allowed to change the operating mode CSM/SSSV. If the SSSV mode was selected, the spindle speed set-point \({{\varOmega }}_{set} \left (t\right )\) was generated by the external control system. If the CSM mode was selected, the speed set-point was directly generated by the numerical controller NC of the lathe. In both the cases the speed set-point \({{\varOmega }}_{set} \left (t\right )\) and the measured spindle speed \({{\varOmega }}_{meas}\left (t\right )\) (through the spindle encoder) were acquired. The lathe was equipped with a spindle with a maximum power of 12.5kW that can rotate up to 3000rpm. A preliminary version of such control unit described in Albertelli et al. [42]. A Kistler dynamometer (9265B) with the associated charge amplifier (5070A) was used to measure the cutting force during the preliminary phases of the experimentation (see Section 2.3).

Fig. 1
figure 1

Experimental set-up: lathe and developed control unit

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2.3 Preliminary verifications

Before executing the wear tests, some preliminary checks were carried out:

  • lathe tracking performance verification

  • workpiece hardness analysis

  • verification of chatter free cutting conditions

  • dispersion of the tool life data according to the selected wear threshold V BBt = 0.15mm

The first verification was performed in order to exclude any side effects of the tracking performance of the developed control unit, especially when the SSSV was adopted, on the results of the experimentation. More specifically, since the spindle available torque is limited, the SSSV cannot be arbitrary implemented. Moreover, the maximum achievable speed modulation (in terms of combination of RVA and freq) depends on the nominal cutting speed. Several spindle speed tracking tests were carried out to assess the limitations of the adopted equipment (Fig. 1). All the tracking tests in the SSSV regime were executed with workpiece hold by the spindle in order to reproduce as much as possible the real cutting conditions (workpiece inertia). For each selected combination of SSSV parameters (RV Aset and freqset), the imposed spindle speed Ω0set was progressively increased. During each run, the actual spindle speed \({{\varOmega }}_{meas}\left (t\right )\) was acquired through the spindle encoder (see Fig. 1) and the developed acquisition system. A Fast Fourier Transform FFT was carried out in order to estimate the average spindle speed \(\bar {{{\varOmega }}}_{0~meas}\), the resulted modulation AΩ meas and the corresponding RV Ameas. In Table 3, some results were reported. The percentage tracking errors (ΔAΩ% and ΔRV A%) in terms of deviation from the nominal values (respectively AΩ set and RV Aset) was computed. Even in Fig. 2, it can be observed that up to 1000rpm the tracking errors are negligible while just starting from 1200rpm they become unacceptable. Several additional tests were carried out with different combinations of RV Aset-freqset.

Table 3 Tracking performance verification - example (RV A = 0.3, freq = 1Hz)
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Fig. 2
figure 2

Tracking errors RV A = 0.3 freq = 1Hz: AΩ set and AΩ meas

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The second verifications was carried out for investigating if the workpiece hardness changes according to the machined region. Indeed, a non-homogeneous material property could have affected the reliability of the tool life tests introducing a possible bias. In order to exclude such effect, several hardness measurements were performed. Both macro-hardness and micro-hardness measures were carried out on different workpiece locations as reported in Fig. 3. Macro-hardness measurements were executed on the lateral part of the workpiece (\(\varnothing =132 mm\)) both on the external raw surface (zone C) and on the internal turned part (zone B of \(\varnothing =130 mm\)). The macro-hardness data were statistically analyzed and it was confirmed that the hardness measured in the zone C cannot be be considered different from the one measured in zone B. An additional analysis was carried out investigating the dependence of the hardness on the radial coordinate X: three micro-hardness repetitions (durometer Future − Tech FM − 700, V ickers, loading 1kg, dwell time 15s) were performed increasing the distance Xn from the external surface zone B. The micro-hardness measurements were executed on the cross-section zone A, properly prepared through multiple polishing steps. The obtained results were reported in Fig. 4. It was statistically demonstrated that the hardness cannot be considered affected by the radial coordinate X.

Fig. 3
figure 3

Workpiece hardness analysis—effects of the position

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Fig. 4
figure 4

Micro-hardness measurements

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Both the performed verifications allowed to adequately plan the experimental campaign for the wear tests. Since the experimented tracking limitations and the fact that the bars to be machined can be considered homogeneous it was decide to partially randomize the tool life tests in order to limit the wasted material. Indeed, the cutting tests at high velocity vc were carried out machining the external parts of the bars as far as the tracking limitations occurred and the remaining part of the workpieces were used for the low cutting speed tests.

Before executing the whole experimentation (tool life tests), all the cutting conditions (combining the cutting parameters as reported in Tables 1 and 2) were tested in order to verify the absence of any undesired vibrations (i.e. due to regenerative chatter) that can negatively affect the life of the tool. For this purpose, the cutting forces were measured with the Kistler dynamometer and analyzed performing a FFT. No critical frequency components were found in the computed spectra for ap = 2mm. The performed tests confirmed that the selected radial depth of cut is far from the stability limit.

As anticipated, preliminary wear tests were executed in order to verify the adequacy of the selected wear threshold V BBt = 0.15mm. Repeated tests (wear tests carried out adopting the same cutting parameters) revealed that the wear rate is rather high when the flank shows an average wear land width close to 0.15 mm (see also Figs. 5 and 6). This allowed to proper discriminate the threshold overcoming and, as a consequence, to limit the dispersion of tool life TL data that typically occurs when the wear rate is low. The low dispersion of the tool life data TL can be even appreciated in Fig. 8 that describes the evolution of the flank wear for the repetitions of the tests executed with the center point conditions.

Fig. 5
figure 5

Flank wear land analysis: average flank width computation V BBik

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Fig. 6
figure 6

Example of the evolution of the average flank wear evolution V BBik of the i th test

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2.4 Tool wear analysis

For the i th tested insert, a set of n VBB measurements is available \(\overline {VBB}_{i}=\{VBB_{i1}, VBB_{i2}, \cdots ,VBB_{ik}, \cdots ,\) \( VBB_{ip},VBB_{i\left (p+1\right )},\cdots ,VBB_{in} \}\) where V BBik is the generic k th measurement of the average flank width (performed after the k th stop). According to ISO 3685 [37], the V BBik was computed using Eq. 2 where \(VBB_{ik_{j}}\) is the local measurement of the flank width (see Fig. 5). Integrating the \(VBB_{ik_{j}}\) over the length \(l_{B_{ik}}\) of the analyzed region, the area \(A_{B_{ik}}\) of the wear land and the V BBik can be computed. The flank width measurements \(VBB_{ik_{j}}\) were performed, as suggested by ISO 3685 [37], on the rectilinear portion of the insert avoiding the curvilinear part (length r𝜖) and the portion of the wear land affected by the notch (see Fig. 5). The TLi can be estimated linearly interpolating two subsequent cutting times (CTip and \(CT_{i \left (p+1 \right )}\)) that were associated respectively at the p th and the \({\left (p+1 \right )}\)th stops. Moreover, V BBip and \(VBB_{i \left (p+1 \right )}\) fulfill the relationship reported in Eq. 4. It was verified that the linear assumption between the wear and the tool life assured a proper accuracy since several VBB measurements for each wear test were available and the \(VBB_{i \left (p+1 \right )}\) and V BBip are rather closed. It was found that a higher order polynomial fitting of the \(VBB \left (TL \right ) \) curve brought to deviations in the estimated tool life TLi less than 1%.

$$ VBB_{ik}=A_{B_{ik}}/l_{B_{ik}}=1/l_{B_{ik}} \cdot {\int}_{0}^{l_{B_{ik}}} VBB_{ik_{j}} dl $$
(2)
$$ TL_{i}=CT_{ip}+\frac{CT_{i \left( p+1 \right)}-CT_{ip}}{VBB_{i \left( p+1 \right)}-VBB_{ip}} \cdot \left( VBB_{t}-VBB_{ip} \right) $$
(3)
$$ VBB_{ip}<VBB_{t}<VBB_{i \left( p+1 \right)} $$
(4)

2.5 Tool life modelling and validation

All the acquired data were statistically analyzed through the Analysis of Variance ANOVA. For what concerns the modelling approach, a multiple linear regression was adopted [40]. In the present research, the TL data or a transformation of them, were considered in the response variables vector \(\boldsymbol {y}=\left [y_{1}, y_{2},\cdots ,y_{n} \right ]\) while the main analyzed factors, a sub-combinations or a transformation of them (j, k, ⋯, r ), were considered as the regression variables matrix, (i.e. \(\boldsymbol {X}=\left [\boldsymbol {1}, \boldsymbol {x}_{j}, \boldsymbol {x}_{k},\cdots ,\boldsymbol {x}_{r} \right ]\)). \(\boldsymbol {\beta }=\left [ \beta _{0},\beta _{j},\beta _{k},\cdots ,\beta _{r} \right ]\) is the vector of the regression coefficients. The matrix notation was reported in Eq. 5.

$$ \boldsymbol{y}=\boldsymbol{X}\cdot \boldsymbol{\beta}+\boldsymbol{\epsilon} $$
(5)

The estimation of the unknown regression coefficients \(\boldsymbol {\hat {\beta }}\) was carried out through the least squares minimization. Indeed, Eq. 6 brings to Eq. 7.

$$ L=\boldsymbol{\epsilon}^{\boldsymbol{\prime}}\cdot \boldsymbol{\epsilon}=\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta} \right)^{\boldsymbol{\prime}} \cdot \left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta} \right) $$
(6)
$$ \boldsymbol{\hat{\beta}}=\left( \boldsymbol{X^{\prime}}\boldsymbol{X}\right)^{-1}\boldsymbol{X^{\prime}}\boldsymbol{y} $$
(7)

The developed models were also validated assessing the predicting capabilities on additional wear tests, executed for this specific purpose.

3 Results and discussion

In this section, the analysis of the performed wear tests was presented. The achieved tool life TL data were reported in Table 4. For sake of completeness, 30 total wear tests were carried out. 9 tests were executed at constant speed machining CSM: 4 tests considering the parameters vc − f combinations according to Table 1 and 5 central point tests. 21 tests were carried out in the sinusoidal spindle speed variation SSSV regime. 16 test conditions were originated combining the SSSV factors (vc, f, RVA, freq, see Table 2) while 5 tests were performed considering intermediate parameter values. A graphical representation of the initial experimental scheme is reported in Fig. 10 (left side). As already explained in the previous paper section, at the beginning, the results of the cutting tests performed at CSM and adopting the SSSV were separately analysed. The idea underpinning the conceived experimental campaigns was based on the suspicion that not all the parameters describing the speed modulating law really affect the life of the tool TL. If it is the case, the data coming from each plan could be grouped and can be analysed allowing the development of a general modelling. Before doing the analysis, a natural logarithmic transformation was performed on the following factors: TL (expressed in min), vc, f, RVA, freq.

Table 4 Tool life test results - CSM and SSSV factorial designs
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3.1 CSM modelling

The transformed tool life data lnTL, obtained in the constant speed machining CSM experimental session, were analyzed through the ANOVA [44]. For sake of generality, according to Eq. 8, the Fisher’s statistics F − valuej and the associated P − valuej were computed for each generic factor j considered in the model. \(SS^{III}_{j}\) is the adjusted (type III) sum of squares for the factor j, obtained removing any possible confounding due to the the remaining factors added to the model, Eq. 9. \(SS_{\left (j,k,\ldots ,r \right )}\) is the sequential sum of squares (type I) of the model considering the j,k,…,r factors while \(SS_{\left (k,\ldots ,r \right )}\) is the sum of squares (type I) considering a model with a subset of factors (k,…,r), thus excluding the one under investigation [44]. MSj is the resultant mean of squares (\(MS_{j}=SS^{III}_{j}/DF_{j}\)) while DFj is the correspondent degrees of freedom. As expected, the main affecting factors were lnvc and lnf while their interaction is not statistically relevant. The ANOVA results considering the main factors were reported in Table 5. For sake of completeness, the analyses performed on the residuals were summarize in Fig. 7. In addition, the regression equation (Eq. 10) was even outlined through the least square LCSM minimization, Eq. 6. All the estimated regression coefficients (\(\hat {\beta }_{j-CSM},\cdots \), \(\hat {\beta }_{r-CSM}\)) and the corresponding 95% confidence intervals CI were reported in Table 6. As suggested in [44], the CI were computed estimating the standard error \(\boldsymbol {SE_{CSM}}=\sqrt {diag\left (Cov\left (\boldsymbol {\hat {\beta }}_{CSM} \right ) \right )}\). The last two columns in Table 6 reported the t − student t − test statistic values and the corresponding P − values that measure how the model parameters are usefulness in the proposed formulation. The coefficient of determination \(R^{2}_{CSM}=99.26\%\) and the adjusted statistic \(R^{2}_{adj-CSM}=96.92\%\). It is worth noting that the statistical test on curvature (associated to the added center point) showed a quite high P − value, this means that the linearity in the factors effect cannot be confuted.

$$ F-value_{j}=\frac{SS^{III}_{j}/DF_{j}}{SS_{error}/DF_{error}} $$
(8)
$$ SS^{III}_{j}=SS_{\left( j \mid k,{\ldots} ,r \right)}=SS_{\left( j,k,\ldots,r \right)}-SS_{\left( k,\ldots,r \right)} $$
(9)
$$ \begin{array}{@{}rcl@{}} y_{CSM}&\equiv&\ln{TL_{CSM}}=\hat{\beta}_{0-CSM}+\hat{\beta}_{ln{v_{c}}-CSM} \cdot \ln{v_{c}}\\&&+\hat{\beta}_{ln{f}-CSM} \cdot \ln{f} \end{array} $$
(10)
Table 5 CSM: ANOVA results
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Fig. 7
figure 7

CSM: residuals analysis

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Table 6 CSM: regression results
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3.2 SSSV modelling

The described methodology (Section 3.1) was adopted even for analysing the transformed \(\ln {TL}\) data obtained from the experimental campaign performed in the sinusoidal spindle speed SSSV regime. The ANOVA results, considering the main factors (\(\ln {v_{c}}\), \(\ln {f}\), \(\ln {RVA}\) and \(\ln {freq}\) ) and the 2 − ways interactions, were reported in Table 7.

Table 7 SSSV: ANOVA results
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As can be observed, the relevant factors to be considered in the SSSV formulation are respectively \(\ln {v_{c}}\), \(\ln {f}\), \(\ln {RVA}\) while \(\ln {freq}\) seems not affecting the life of the tool. The effect of SSSV cutting can be also observed in Fig. 8 where the V BBik associated to the tests carried out in both the center points (Fig. 10, left side) were reported. Five repetitions for each cutting condition (CSM or SSSV ) were performed. Although the results were affected by the process variability, it is quite evident that the tools adopted for the SSSV cutting showed a reduced TL. The ANOV A residuals analysis was reported in Fig. 9. Even in this case, the regression equation was obtained (Eq. 11) as well as the \(R^{2}_{SSSV}=97.93\%\) and the adjusted statistic \(R^{2}_{adj-SSSV}=97.69\%\). For sake of simplicity, it was decided to consider in the model formulation the amplitude modulation RVA instead of its logarithmic transformation. Even in this case the linearity in the factors effect cannot be confuted (Table 7). Similarly to CSM modelling, the relevance of the model parameter was reported in Table 8.

$$ \begin{array}{@{}rcl@{}} y_{SSSV}&\equiv&\ln{TL_{SSSV}}\\ &=&\hat{\beta}_{0-SSSV}+\hat{\beta}_{\ln{v_{c}}-SSSV} \cdot \ln{v_{c}}\\&&+\hat{\beta}_{\ln{f}-SSSV} \cdot \ln{f}+\hat{\beta}_{RVA-SSSV} \cdot RVA \end{array} $$
(11)
Fig. 8
figure 8

V BBik evolution: effect of SSSV on TL in the intermediate cutting conditions

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Fig. 9
figure 9

SSSV formulation: residuals analysis

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Fig. 10
figure 10

General model scheme: initial experimental scheme (left) - resulted test representation and validation test conditions (right)

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Table 8 SSSV: regression results
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3.3 General model development

Since the frequency of the speed modulation freq resulted not to be a significant factor, therefore the unreplicated design in four factors conceived for performing the SSSV campaign can be considered a two times replicated plan with three factors: \(\ln {v_{c}}\), \(\ln {f}\) and RVA (Fig. 10). According to the CSM and SSSV models, the velocity and feed were found to be significant factors in both the cases, while for the SSSV case even RVA resulted to be significant. Since the experimentation performed at CSM can be considered a specific realization of the SSSV formulation (setting RV A = 0), it is interesting to develop a general model considering the whole tool life data-set. For sake of clarity, the resultant graphical representation was reported in Fig. 10. Moreover, the test conditions used for the model formulations validation (see Section 3.4) were put into evidence in Fig. 10, right side.

Before doing that, the test of equality of means was performed on the coefficients present in by both CSM (Section 3.1) and SSSV (Section 3.2) formulations: \(H_{0-\mathit {constant}}: \hat {\beta }_{0-CSM}\) \(=\hat {\beta }_{0-SSSV}\), \(H_{0-\ln {v_{c}}}:~\hat {\beta }_{ln{v_{c}}-CSM}=\hat {\beta }_{\ln {v_{c}}-SSSV} \), \( H_{0-\ln {f}}:\)\(\hat {\beta }_{\ln {f}-CSM}=\hat {\beta }_{\ln {f}-SSSV}\). All the tests confirmed that the two proposed formulations shared the fitted parameters (the generic null hypotheses H0k cannot be refused) and this confirmed the adequacy of a general model for interpreting all the experimental data. For sake of completeness, the ANOVA results and the residual analysis were respectively reported in Table 9 and Fig. 11.

Table 9 general: ANOVA results
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Fig. 11
figure 11

General residuals analysis

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The general model equation and the corresponding identified terms were reported respectively in Eq. 13 and Table 10.

$$ \begin{array}{@{}rcl@{}} y_{gen}&\equiv&\ln{TL_{gen}} \end{array} $$
(12)
$$ \begin{array}{@{}rcl@{}} &=&\hat{\beta}_{0-gen}+\hat{\beta}_{\ln{v_{c}}-gen} \cdot \ln{v_{c}}+\hat{\beta}_{\ln{f}-gen} \cdot \ln{f}\\&&+\hat{\beta}_{RVA-gen} \cdot RVA \end{array} $$
(13)
Table 10 General model: regression results
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3.4 Model validation and discussion

In order to validate the model, additional experimental tests were carried out. More specifically, new cutting conditions (both in CSM and SSSV ) were tested (Fig. 10). The adopted parameters and the new obtained tool life TL data were reported in Table 11.

Table 11 Tool life: tested conditions for the model validation
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In Table 12, the percentage errors \(TL_{s-error_{\%}}\) (Eq. 14) obtained adopting the s th formulation for the tool life estimation \(\hat {TL}_{s}\) were reported. For sake of clarity, as an example, \(TL_{gen-error_{\%}}\) is the percentage estimation error that results if the tool life is estimated through the general model formulation \(\hat {TL}_{gen}\) (Section 3.3). Other percentage errors can be analogously computed. It can be observed that the developed general model shows limited errors \(TL_{gen-error_{\%}}\) (less than 6%) and a lower standard error SEk−gen for all the shared \(\hat {\beta }_{k}\) with respect to the other formulations maybe because it was developed exploiting the full data set. SSSV formulation shows similar performances in terms of estimating errors (less than 5%) while CSM modelling exhibits a worse performance (although the error is limited to 10%) if compared to the other modelling approaches. The estimation errors would increased (ranging from − 9.4 to − 16%, see the values reported among brackets in Table 12 in the column \(TL_{CSM-error_{\%}}\)) if the formulation developed for CSM was used for predicting the tool duration performed with the sinusoidal spindle speed variation SSSV. As expected, the CSM model overestimates the tool life TL since it does not consider the detrimental effect of the SSSV. On the contrary, the SSSV formulation would behave well (error less than 5%) if it was used for estimating the TL of cutting tests performed at CSM. Indeed, although the SSSV formulation was developed just considering turning tests performed with RV A≠ 0 the model can be used for extrapolation (CSM). For sake of completeness, the values reported among brackets in Table 12 refer to a not proper exploitation of the developed model: for instance, the CSM model for predicting the tool duration in SSSV cutting or vice-versa. Moreover, referring to the specific literature, the maximum observed estimation error adopting the developed general formulation is in accordance with other advanced models suitable only for CSM [32]. This further confirms the adequacy of the proposed modelling approach. Since the limited prediction errors observed for the validation points and the statistical tests on curvature, the linear dependence of the tool life TL on the considered factors can be further confirmed. To the authors’ knowledge, no one developed specific formulation for modelling the effect of SSSV on the TL. Previous works of the same research group focused mainly on the phenomenological aspects and not on the quantification of the detrimental effects of SSSV when the cutting is stable.

$$ TL_{s-error_{\%}}=100\cdot\frac{TL-\hat{TL}_{s}}{\hat{TL}_{s}} $$
(14)
Table 12 predicting modelling errors
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Although the limitations of empirical approaches were underlined by Wong et al. [31], the authors believe that the proposed general formulations, in terms of the structure of the model, being developed from Taylor’s basic model, is suitable for considering the effects of SSSV on different materials, especially for other steels. The study carried by Johansson et al. [32] supports the previous considerations since they found that Taylor’s model works fine within a quite broad range of materials (steels, cast irons and stainless steels). Previous research works that focused on the phenomenological aspects that make the TL of SSSV cutting shorter than CSM can help understanding the potentialities of the developed formulation. Albertelli et al. [42] found that during SSSV turning a crack on the insert fosters the coating delamination and therefore a faster wear of the tool. This was observed considering the same insert-working material of the present study. Chiappini et al. [25] studied the mechanics of chip formation in SSSV turning of Ti6Al4V (heat resistant alloy HRA) and adopting a different insert geometry. It was found that, in view of the fact that the modulation of the cutting speed origins a fluctuation of the maximum cutting temperature and a peak of pressure on the inserts, thermal gradients and a subsequent thermal fatigue can affect the tool. According to other research works (i.e. Evans and Hutchinson [45]) that found that thermal gradients and thermal fatigue are the main relevant causes of coating delamination, it can be concluded that this was the cause of the observed detrimental effect of SSSV on the wear of the tool. Since the thermal fatigue is mainly associated to the speed modulation [25], it can be stated that an analogous TL reduction can be expected regardless the processed material. These considerations bring to conclude that the proposed formulation could be used for a wide range of materials especially if coated inserts are used. Moreover, since spindle speed variation is typically used for suppressing unstable cutting, its application is particularly suitable for roughing operations in which high depths of cut are involved.

4 Conclusions

A generalized model for the prediction of the tool duration in steel turning when the sinusoidal spindle speed SSSV is used was developed and presented. The proposed formulation is even suitable for modelling CSM constant speed machining. The model was outlined exploiting tool wear tests performed in different cutting conditions. Since the conceived tool life model takes into account the detrimental effects of spindle speed on tool life TL, it could be used to widely analyse the economic feasibility of the technique in different cutting scenarios and applications. The following results can therefore be summarised:

  • The statistical analysis performed on the experimental data confirmed that the continuous modulation of the spindle speed, while keeping the feed velocity constant, negatively affects the achievable useful life of the tool.

  • From the experimental session carried out with the sinusoidal spindle speed modulation it was found that the modulating frequency freq does not affect the tool duration.

  • A specific tool life model formulation for SSSV was developed exploiting the experimental test results. The normalized amplitude of the spindle modulation RVA, together with the other affecting parameters (cutting speed vc and feed f ), was included in the model.

  • According to the previous reported findings, it was possible to develop a generalized tool life model formulation that can be indifferently used both in regular cutting regime CSM and when the SSSV is adopted.

  • The generalized formulation was validated performing additional tool life tests. It was found that it capable of predicting the useful tool life within a maximum estimating error of 6%

Future research efforts will be surely focused on a broader validation and on a generalization of the developed model. Although there are some hints that bring to consider the proposed general formulation robust to changes in workpiece material and tool geometry, a proper validation would be extremely valuable. Moreover, even the model generalization considering the effect of the depth of cut could be developed.