Machining chatter in continuous facing under a constant nominal cutting speed: Tool vibration with time-varying delay

Abstract

This article studies regenerative chatter in single-point face-machining at nominally constant speed under continuous conditions. A temporal model for rotational speed was developed and experimentally verified. The resulting rotational time-delay was cast into the classical force feedback mechanism for chatter. A chatter model was formulated to allow slight spindle speed variation about the temporal model. The modified method of steps was then employed to solve the tool vibration in time-domain allowing one (single-degree-of-freedom) or two (2-degree-of-freedom model) vibrational modes. Exploratory facing experiments using a grooving tool were conducted on a nickel alloy workpiece. It was found that the tool was more susceptible to chatter at larger diameters. It appeared that the single-degree-of-freedom model captured the most relevant of the observed phenomena while cutting without spindle speed variation, however, neither the 1-degree-of-freedom nor the 2-degree-of-freedom models could effectively capture the experimentally observed chatter evolution characteristics while cutting with spindle speed variation.

Introduction

Face-turning (facing) operations are an important class of machining operations, especially in aero-engine components such as blade-discs and in automotive components such as brake-discs. An important sub-class of facing, found abundantly in industry, is constant speed facing (CSF) where the rotational speed of the workpiece is varied inversely with the radial distance of the tool keeping the peripheral speed constant. Productivity, tool temperature and power consumption are more stable with CSF than with constant rotational speed. Though papers could be found in the machining literature related to facing,^1–4 none of them treat the chatter phenomenon in CSF in such a manner as to capture both intrinsic and extrinsic spindle speed variation (SSV).

Consider a simple continuous facing operation as shown in Figure 1. For the subsequent descriptions, refer to Notation in Appendix 1. The instances of time-varying variables such as $y$ and $\tau$ are denoted using the subscript. Time derivatives are denoted using the over-dots.

$$\begin{array}{cc}\multicolumn{1}{c}{m{\stackrel{··}{y}}_t+c{\stackrel{·}{y}}_t+{\mathrm{ky}}_t}& =\kappa k(\stackrel{u_t}{\overbrace{u_0-y_t+\mu y_{t-\tau }}}-\frac{C{\stackrel{·}{y}}_t}{V})\hfill \\ =\kappa \mathrm{kF}(\tau ;y_t,{\stackrel{·}{y}}_t,y_{t-\tau })\hfill \end{array}$$

(1)

Figure 1.

Schematic of facing: SDOF and 2DOFa models.

Suppose the workpiece is rigidly rotating at constant angular speed ${\omega }_t=\omega =2\pi /\tau$ . Let the compliant tool be modeled as a single-degree-of-freedom (SDOF) system. With the tool fed radially inward or outward, let the constant tool advancement per rotation be $u_0$ . The archetypal model of chatter for this configuration can be given by equation (1) even though tool advancement and vibration direction are in perpendicular directions. ⁵

Conditions of chatter stability have been important since they enable a field practitioner to choose tool geometry and cutting conditions judiciously. Equation (1) affords a stability boundary in this sense only when $\tau$ is constant. In this regard, a convenient and interactive version for turning or boring incorporating process damping is now available on-line. ⁶

Now suppose angular speed ${\omega }_t$ is varied arbitrarily with $u_0$ held constant. For SDOF, equation (1) still describes the chatter phenomenon, but with a time-varying process delay ${\tau }_t$ . Calculation of ${\tau }_t$ is needed for the SDOF model and its enlargement into the two-degree-of-freedom model a (2DOFa; Figure 1) and two-degree-of-freedom model b (2DOFb; Figure 4); models to be elaborated later.

An implicit calculation of ${\tau }_t$ for an arbitrary ${\omega }_t$ is obtained by equating the total angle swept in one rotation to $2\pi$ . Assuming a fixed $u_0$ , small with respect to $D_i$ , the total cutting time $\mathbb{T}$ in a complete pass can be calculated by equating the area swept by the tool to the area of the workpiece. Equations (2) and (3) thus allow us to calculate ${\tau }_t$ and $\mathbb{T}$

$$\underset{t-{\tau }_t}{\overset{t}{\int }}{\omega }_{\xi }\mathrm{d}\xi =2\pi$$

(2)

$$\underset{0}{\overset{\mathbb{T}}{\int }}{u}_0{r}_{\xi }{\omega }_{\xi }\mathrm{d}\xi =\frac{\pi (D_o^2-D_i^2)}{4}$$

(3)

CSF

For CSF, in addition to $u_0$ , the instantaneous cutting speed $r_{\xi }{\omega }_{\xi }=V$ is also constant, whereupon a simple expression for $\mathbb{T}$ can be obtained from equation (3).

When the delay in equation (1) is time-varying as ${\tau }_t$ , equation (1) is a delay-differential equation with time-varying delay (DDE-TD). Tsao et al. ⁷ transformed the time-varying delay problem into a parametric vibration problem using angular position as the independent variable. In this work, time instant $t$ is retained as the independent variable and the requisite temporal variation of $r_t$ , ${\omega }_t$ and ${\tau }_t$ for CSF derived. For the ± and ∓ used in the following section, the upper symbol refers to the case of the tool moving outward $(i\to o)$ and the lower to the tool moving inward $(o\to i)$ . The symbol $D$ refers to $D_i$ and $D_o$ for the cases $(i\to o)$ and $(o\to i)$ , respectively.

For a given ${\mathcal{N}}_t$ , the radial position $r_t$ and the angular velocity ${\omega }_t$ are interrelated through equations (4) and (5)

$$r_t=\frac{D}{2}\pm {\mathcal{N}}_t{u}_0$$

(4)

$${\omega }_t=2\pi {\stackrel{·}{\mathcal{N}}}_t$$

(5)

Equation (4) holds when $u_0$ is constant. When $V$ is also constant, equation (6) is obtained and can be solved as an initial value problem (IVP)

$$\begin{array}{c}\hfill {\stackrel{·}{r}}_t\hfill \end{array}=\pm \frac{4u_{0}V}{\left(\pi r_t\right)}$$

(6)

Equations (7)–(10) give the final results for $r_t$ , ${\omega }_t$ , ${\tau }_t$ and $T_{\mathrm{n}}$ , respectively, the time instant of end of rotation $\mathrm{n}$ from start, where $\mathcal{T}=\pi D^2/(4u_{0}V)$ , $\Gamma =V/(\pi D)$ and ${\lambda }_t=\sqrt{1\pm t/\mathcal{T}}$

$$r_t=\frac{D{\lambda }_t}{2}$$

(7)

$${\omega }_t=\frac{2\pi \Gamma }{{\lambda }_t}$$

(8)

$${\tau }_t=\frac{(4\mathcal{T}\Gamma {\lambda }_t\mp 1)}{4\mathcal{T}{\Gamma }^2}$$

(9)

$$T_{\mathrm{n}}=\frac{\mathrm{n}(\pm \mathrm{n}+4\mathcal{T}\Gamma )}{4\mathcal{T}{\Gamma }^2}$$

(10)

Significantly, a closed-form expression of ${\tau }_t$ is available for CSF. ⁸ Figure 2(a) shows comparison of the experimentally obtained (to be described later) rotational speed ${\omega }_t$ data and the predictions of equation (7) with and without SSV. Here, by SSV, we mean superposing a small extrinsic rotational speed variation over the intrinsic variation for constant $V$ . Cases (1) and (2) show results without SSV where careful analysis showed the experimental data deviated from the model norm by at most 2.5% over the entire speed range. Cases (3) and (4) show data with SSV for $(i\to o)$ and $(o\to i)$ using the built-in computer numerical control (CNC) machine controller functionality. Figure 2(b) shows the corresponding delay function ${\tau }_t$ for cases (1)–(4) computed using equations (2) and (7).

Figure 2.

Experimental and modeled (a) angular speed ω_t and (b) delay function τ_t.

With this background, this article proceeds to investigate chatter in CSF with and without SSV for both $(i\to o)$ and $(o\to i)$ . In an attempt to fill the gap in literature,^9–11 where SSV only in turning has been studied, modeling and exploratory results are presented.

Method of steps

Referring to equation (2), as long as ${\tau }_t$ does not vanish, DDE-TD systems can be solved by the method of steps (ST-MOS) described by Shampine and Thompson (Balachandran et al. ¹² ). In essence, it consists of converting the DDE-TD system into a sequence of IVP systems over pieces of the total time interval of interest. For facing with a single-point tool, the delay corresponds exactly to the time duration for one rotation up to the current instant. Therefore, if the workpiece is spun arbitrarily, but in the same direction, ${\tau }_t$ does not vanish. Hence, ST-MOS adapts suitably to this circumstance, as and is now described for SDOF (Figure 1).

Figure 3(a) shows a graphical outline of the approach taken to solve the tool vibration for SDOF. The total duration of travel of the tool over the workpiece surface is divided by sequence of time-instants $0=T_0<T_1<\cdots <T_{\mathrm{n}}<\cdots <T_N=\mathbb{T}$ marking end of each rotation found using equation (2). The solution $y_t^{(\mathrm{n})}$ for the $\mathrm{n}\mathrm{th}$ rotation is computed over sequential time intervals $T_{\mathrm{n}-1}\le t\le T_{\mathrm{n}}$ by carrying over the values of the vibration position $Y_{\mathrm{n}-1}$ , velocity ${\stackrel{·}{Y}}_{\mathrm{n}-1}$ at $T_{\mathrm{n}-1}$ and $y_t^{(\mathrm{n}-1)}$ , and the solution history in the interval $T_{\mathrm{n}-2}\le t\le T_{\mathrm{n}-1}$ . Figure 3(b) shows the sequence of steps corresponding to this approach. Equation (1) has been recast in step C with purely kinematic quantities $\Omega$ , $\zeta$ , $C$ , $\mu$ and $\kappa$ as inputs to the system. The cells in green refer to the starting data with the corresponding one-time flows shown by arrows in broken lines. Cells in yellow are the intermediary steps with the solid arrows showing the loop structure needed to propagate the solution. The red cells show the end of the procedure where the individual solutions are accumulated. The system is initiated by providing the prior history $y_{t\le 0}^{(0)}$ . Though not physically evident as yet, the settling bias of $y_t$ designated as $\tilde{y}$ could be calculated by assigning ${\stackrel{··}{y}}_t={\stackrel{·}{y}}_t=0$ and $y_t=y_{t-\tau }=\tilde{y}$ in equation (1) resulting in $\begin{array}{c}\hfill \tilde{y}\hfill \end{array}=\kappa u_0/(1+\kappa (1-\mu ))$ .

Figure 3.

Tool vibration solution using modified method of steps: (a) graphical summary and (b) sequence of steps.

2DOFa and 2DOFb models

A similar procedure can be formulated when two modes are operational. In 2DOFa (Figure 1), the workpiece is assumed rigid as in SDOF and both modes are allotted to the tool. Alternatively, in 2DOFb (Figure 4), both the tool and the workpiece are compliant and one mode allotted each.

Figure 4.

Schematic for 2DOFb model.

Let the index $i$ take the value 2 for the mass close to the tool tip and 1 for any other mass. For 2DOFa, step C of Figure 3(b) is replaced by equation (12) in Appendix 2. The values for $m_i$ , $c_i$ and $k_i(i=1,2)$ can be arranged suitably from given modal frequencies $f_i$ , damping ratios ${\zeta }_{q_i}$ and stiffness $k_{q_i}$ . Only $y_{2,t-{\tau }_t}$ , the vibration time-history of mass $m_2$ from the preceding rotation enters the forcing function $F$ , defined in equation (1), whose structure is, however, unchanged. The settling biases ${\tilde{y}}_1^a$ and ${\tilde{y}}_2^a$ can be calculated as before by setting time derivatives to 0 and equating the current and delayed terms to the respective biases. The foregoing is detailed in equations (12)–(16) in Appendix 2.

In Figure 4, the 2DOFb model shown resembles that in Chung and Liu. ¹³ The derivation for $u_t$ is detailed in equation (17) in Appendix 2. Step C of Figure 3(b) is replaced by equation (18) in Appendix 2. The vibration history $y_{1,t-{\tau }_t}$ from the previous rotation and $y_{1,t}$ in the current rotation both enter the forcing function $F$ through $\Delta y_t=y_{2,t}-y_{1,t}$ . The biases ${\tilde{y}}_1^b$ and ${\tilde{y}}_2^b$ are calculated in equation (19) in Appendix 2 as before in 2DOFa.

Experimental

A nickel alloy workpiece (D_o = 152.0 mm, D_i = 8.0 mm) was machined in a Haas ST10 CNC lathe (150 mm chuck size, ∼300 mm swing over bed). All cuts were performed at a depth of cut d = 0.381 mm and tool advance of u₀ = 0.102 mm. SSV about the norm for CSF was achieved using machine controller parameters 165 (set to 100 rpm) and 166 (set to five corresponding to a 0.5-s period). Actual spindle speed was measured using a Compact Instruments (Model A2108/LSR) non-contact tachometer. The foregoing conditions were arrived upon after a number of trials to ensure that chatter was neither too feeble nor too violent.

A Sandvik RG123H079-16B-132BM grooving-type toolholder with a N123H2-0500-RO insert in the S05F grade was used. As shown in Figure 5(a), the toolholder was modified and inserted into the turret at a suitable length to excite a low (500–750 Hz) natural frequency as the primary mode, aided by finite element (FE) modeling on an exact computer-aided drafting (CAD) model of the tool. The modal tap-testing result of the tool workpiece system as assembled on the machine is shown in Figure 5(b).

Figure 5.

Tool/accelerometer (inset) arrangement and FRF: (a) assembly in turret and (b) FRF data.

A miniature PCB Electronics 352A72 accelerometer with a usable bandwidth of ∼8 kHz was glued firmly to the face of the toolholder behind the insert with Micro-Measurements M-Bond 200/200 and Catalyst C. Plumbing putty was applied all around in a ring and allowed to harden and seal the accelerometer from coolant. The guarded accelerometer cable was secured to the toolholder firmly using double-sided Velcro. Vibration data were collected at 100 kHz using a 12-bit data acquisition system simultaneously from the accelerometer and tachometer. During the cut, applying coolant through a nozzle directly on the top of the insert face from underneath was necessary to prevent premature insert breakage. The workpiece surface was cleaned with a rigid tool before taking the test cuts to prevent previous surface topography from interfering in the current cut.

In an attempt to confirm that $u_0$ was constant for CSF, a linear variable displacement transducer (LVDT) (Omega; Model LD620-2.5 PPTSC, linearity: 0.2% full scale output) was used to record the tool feed motion in the radial direction. However, since its travel was limited (2.5 mm), confirmation was found only for step changes in ${\omega }_t$ and not for CSF in the complete pass.

Results and discussions

Acceleration data up to 2 kHz are sufficient to understand machining chatter and only this range is considered hereafter. Figure 6 shows data collected with the experimental setup of Figure 5. Here, the X-axis is the time axis. The rotational speed data of the tachometer appearing as black dots are referred to the right Y-axis. Fast Fourier transform (FFT) amplitudes of 1 s time fragments of the accelerometer data are plotted as gray scale dB values with the frequency referred to the left Y-axis. While Figure 6(a), (c) and (d) shows data collected at a nominal cutting speed of 1.34 ms⁻¹, Figure 6(b) shows data collected at 0.67 ms⁻¹ where chatter was completely avoided. Distinct striations are seen in these data, but this does not correspond to consecutive rotations of the workpiece. Data in Figure 6(c) and (d) are cases where SSV was imposed, also evident from the fluctuating ${\omega }_t$ data. Figure 6(c) is the sole data corresponding to the tool moving from center to periphery.

Figure 6.

Experimental results: accelerometer data shown as a spectrogram with rotational speed data overlaid: (a) (o→i), w/o SSV; (b) (o→i), w/o SSV; (c) (i→o), w/SSV and (d) (o→i), w/SSV.

Table 1.

Parameters for simulations in Figure 7.

$\Omega$	$\kappa$	$C$	$\delta$	$\mu$	$V$	$u_0$	$\zeta$	$D_o$	$D_i$
2π650 s−1	0.08	50 µm	0.5 s	1.0	1.34 m s−1	0.102 mm	0.03	152.0 mm	8.0 mm

The authors were fully cognizant of the effect of tool wear and process parameters altering process damping^14,15. However, tool wear and gross loss of size on the part were not noticeable at the end of the cut. Moreover, the tool ceasing to vibrate due to edge wear would likely have been indicated by dark bands at the end of the cut in Figure 6, but this is not readily evident. Furthermore, considering that a fresh edge was used for every cut in the final reported data, the assumption of a constant process damping parameter $C=50\mu \mathrm{m}$ appears justified.

The experimental results were then rationalized using the models assembled in a Mathematica^® package. SSV was modeled as a sinusoidal fluctuation of ${\omega }_t$ for CSF as given in equation (11)

$${\omega }_t^{\mathrm{SSV}}={\omega }_t(1+\Delta \sin \left(2\pi \frac{t}{\delta }\right))$$

(11)

The goal of the model simulations was to replicate the chatter evolution characteristics seen in Figure 6. To enable direct comparison with experimental results, modeling results are also presented in a combined plot of modeled angular speed and acceleration spectrogram. However, instead of 1 s time-interval fragments, the spectrogram was assembled with modeled acceleration of each rotational period of the workpiece. The prior history of the acceleration input into the system did not affect the vibration solution for various combinations of parameters tested. There are three frequency regimes to be concerned with in understanding chatter here: the natural frequencies of the tool and/or workpiece (100–1000 Hz), workpiece rotational frequency (1–30 Hz) and SSV frequency (0–1 Hz). Since actual modal parameters were not available, multiple levels and combinations of parameters were simulated. In all the three models, the value of $\kappa$ impacted the magnitude of vibration as expected. Representative results are shown in Figures 7–9 where the tool natural frequency (597 Hz) and damping ratio (5.64%) as deduced from data in Figure 5(b) provided the initial values.

Figure 7.

Model results: SDOF (o→i) acceleration. Parameters given in Table 1. Increasing Δ: (a) Δ = 0%, (b) Δ = 10% and (c) Δ = 20%

Figure 8.

Model results: 2DOFa (o→i) acceleration. Parameters given in Table 2. Increasing Δ: (a) Δ = 0%, (b) Δ = 10% and (c) Δ = 20%.

Figure 9.

Model results: 2DOFb (o→i) acceleration. Parameters given in Table 3. Increasing Δ: (a) Δ = 0%, (b) Δ = 10% and (c) Δ = 20%.

Discussions

SDOF model

The data of Figure 6(a) suggest that the tool can be excited everywhere in the inspected bandwidth in the early part of the facing pass, and the dominant mode (DM) (∼650 Hz) can also be seen to become significant in the middle of the pass. Figure 7 shows results for SDOF (Figure 1), $(o\to i)$ w/SSV for three levels of $\Delta$ ; 0%, 10% and 20%. The single mode (650 Hz imposed) is more vigorous in the beginning of the cut at the workpiece periphery but progressively fades toward the center. Acceleration data and the workpiece finish of some initial trial cuts did corroborate this phenomenon. The tool is also seen to be excited sporadically over the whole frequency range with SSV. In Figure 6(b), DM is also visible in the early portion of before the first striation. However, while Figure 7 indicates effective suppression of chatter by SSV at all frequencies, Figure 6(d) only reveals a gradual cessation up to an increasing frequency starting from DM. This phenomenon seems to be mirrored in Figure 6(c). These observations suggest that the SDOF model may be adequate when machining without SSV. But with SSV, at least two modes need to be allowed and their complex interaction studied.

Table 2.

Parameters for simulations in Figure 8.

$f_1$	$f_2$	${\zeta }_{q_1}$	${\zeta }_{q_2}$	$k_{q_1}=k_{q_2}$	$\kappa$	$C$	$\delta$	$\mu$	$D_o$	$D_i$	$V$	$u_0$
650 s−1	1600 s−1	0.01	0.03	4.51 kN/mm	0.05	50 µm	0.5 s	1.0	152.0 mm	8.0 mm	1.34 m s−1	0.102 mm

Table 3.

Parameters for simulations in Figure 9.

$f_1$	$f_2$	${\zeta }_{q_1}$	${\zeta }_{q_2}$	$\kappa$	$k_{q_1}$	$k_{q_2}$	$C$	$\delta$	$\mu$	$D_o$	$D_i$	$V$	$u_0$
1600 s−1	650 s−1	0.01	0.03	0.10	9.02 kN/mm	4.51 kN/mm	50 µm	0.5 s	1.0	152.0 mm	8.0 mm	1.34 m s−1	0.102 mm

2DOFa and 2DOFb models

In 2DOFa, both vibrational modes are allotted to the tool, while in 2DOFb, one mode is allotted to the tool and workpiece each. These models also predicted greater propensity for chatter at larger diameters. A likely reason is that at larger diameters, a greater number of vibrational waves occur in each rotation of the workpiece and the mismatch in phase of $y_t$ from rotation to rotation is lower. The results for the two models are shown in Figures 8 and 9, again for representative values.

In the former, SSV suppresses only the lower frequency mode while suppressing both the lower and higher frequency modes in the latter. In the former, the sporadic cessation of vibration is more frequent than in the latter. In Figures 8(a) and 9(a), one can see the cessation and then reactivation of the low frequency mode in the early part of the cut indicating mode interaction is expected to occur in the early part of the cut and not in the middle of the cut as in Figure 6(a). These facts taken together indicate that the true process physics of SSV is not really captured through the 2DOFa and 2DOFb models.

Some reasons may be speculated for the disagreement. The model for ${\omega }_t^{\mathrm{SSV}}$ postulated in equation (11) and the constant tool advance $u_0$ assumed may be an oversimplification. Future work will need to fill these gaps along with elaborate experimentation where the DM frequency is also varied.

Conclusion

The article studies the process dynamics in CSF. The temporal model of SSV intrinsically needed and the resulting time-varying process delay was derived and experimentally validated. Exploratory facing tests in a nickel alloy were performed with and without extrinsic SSV to understand chatter evolution. A SDOF and 2DOFa and 2DOFb were formulated and solved using the modified MOS. Tentative agreement was reached for experimental data for SDOF without SSV. However, with SSV, the tool workpiece interaction was more complex that chatter evolution could not be completely captured with the SDOF or the 2DOFa and 2DOFb models.