On the chatter frequencies of milling processes with runout

doi:10.1016/j.ijmachtools.2008.02.002

International Journal of Machine Tools and Manufacture

Volume 48, Issue 10, August 2008, Pages 1081-1089

https://doi.org/10.1016/j.ijmachtools.2008.02.002 Get rights and content

Abstract

The detection of undesirable vibrations in milling operations is an important task for the manufacturing engineer. While monitoring the frequency spectra is usually an efficient approach for chatter detection, since these spectra typically have a clear and systematic structure, we show in this paper that the stability of the cutting process cannot always be determined from solely viewing the frequency spectra. Specifically, the disturbing effect of the tool runout can sometimes prevent the proper determination of stability. In this paper, we show these cases can be classified by alternative analysis of the vibration signal and the corresponding Poincaré section. Floquet theory for periodic systems is used to explore the influence of runout on the structure of milling chatter frequencies. Finally, the results from theoretical analysis are confirmed by a series of experimental cutting tests.

Introduction

Machine tool chatter is still a problem for the machining community. These violent vibrations of the machine tool are problematic since they result in a poor surface finish, cause large-amplitude acoustic emissions, and can sometimes lead to tool failure. Therefore, it is highly important to detect the onset of these vibrations.

Stability lobe diagrams for machining processes were developed by Tobias [1] and Tlusty et al. [2] in the sixties and the corresponding theory for interrupted cutting process, such as milling, has much attention over the past decade starting with the work of Altintas and Budak [3], [4]. Since then, several sophisticated models and analysis techniques have been developed to investigate the dynamic behavior of the milling process (see, e.g. [5], [6], [7], [8], [9]). More recent analytical investigations have led to the realization of a new instability behavior: in addition to quasi-periodic chatter (Hopf bifurcation), period doubling (period 2, flip bifurcation) has also been an observed instability behavior in milling [10], [11], [12].

Monitoring the vibration frequencies during machining is an efficient method for identifying machine tool chatter and distinguishing between different types of instabilities. In contrast to turning operations, which are characterized by a single chatter frequency, chatter vibrations in milling contain multiple frequencies due to the periodic nature of the process as it was shown in [13]. For ideal milling operations with symmetric tools, the cutting force is periodic at the tooth passing (TP) period. In this case, the vibration frequencies have a well defined, special structure, and stable and unstable machining cases can clearly be assessed based on the frequency spectra (see [13]). However, in practice, several other effects arise that influence and sometimes destroy the nice structure of the chatter frequencies. One of them is runout where the geometric axis of the milling cutter differs from the rotation axis. Runout causes the chip load to be distributed unevenly among the cutting teeth and shifts the frequency content of the cutting force signal away from the TP frequency and towards the spindle rotation (SR) frequency [14], [15], [16], [17]. In this case, the distinction between a stable and unstable cut is not always possible from the frequency spectrum. For instance, consider the case of a tool with an even number of cutting teeth which causes period-doubling chatter frequencies to coincide with the harmonics of the SR frequencies—thus these period-doubling cases cannot be distinguished from stable machining (see, e.g. [18], [19]).

In this paper, the results from Ref. [13] are generalized for a milling model with runout. The tool motion is analyzed with respect to the SR period, since this is the least period of the cutting force due to runout. The stability lobe diagrams and the corresponding frequency diagrams are compared for milling models with and without runout. The frequency components that arise due to runout are clearly identified. Some specific cases when the stability of the machining process cannot be assessed based on the frequency spectra are investigated. It is pointed out that for these problematical cases, alternative techniques for analyzing the vibration signals provide useful information to determine if chatter vibrations are present.

Section snippets

Mechanical model

The standard 2 DOF mechanical model of end milling is shown in Fig. 1. The tool is assumed to be flexible in comparison to the rigid workpiece. The 2 DOF oscillator is excited by the cutting force. The governing equation has the form $M \ddot{x} (t) + C \dot{x} (t) + Kx (t) = a_{p} H (t) (x (t - τ) - x (t)) + G (t),$ where the vector $x (t)$ contains the displacement of the tool tip in the x and y directions, the matrices $M$ , $C$ and $K$ are the modal mass, damping and stiffness matrices, and $a_{p}$ is the axial depth of cut. The time delay is

Stability analysis and vibration frequencies

Stability of the milling process can be estimated theoretically via the analysis of the governing Eq. (1). According to the Floquet theory of delayed differential equations [21], an infinite dimensional operator (monodromy operator) can be associated with the system that gives the connection between the current state of the cutter and the state one period earlier. This period is equal to the least period of the time-dependent coefficients in the equations and is often called principal period.

Zero runout vs. nonzero runout

In order to demonstrate the effect of runout, first, a theoretical study is presented. In Fig. 3, two stability charts and the corresponding chatter frequencies are presented for a 5% radial immersion down-milling operation with a two fluted cutting tool $(N = 2)$ . The parameters used in the computation are given in Section 5.

Stability charts were determined by both the semi-discretization method [22] and the time finite element method [23]. Both methods resulted exactly in the same stability

Experimental verification

Cutting tests were performed on a 5-axis linear motor Ingersol machining center with a Fischer 40,000 rpm, 40 kW spindle. A 12.75 mm diameter, 106 mm overhang, carbide end mill was used during all stability tests. The modal parameters of the compliant tool are $M = (\begin{matrix} 0.046 & 0 \\ 0 & 0.046 \end{matrix}) kg, C = (\begin{matrix} 4.32 & 0 \\ 0 & 4.32 \end{matrix}) Ns / m, K = (\begin{matrix} 9.57 & 0 \\ 0 & 9.57 \end{matrix}) \times 10^{5} N / m .$ Cutting coefficients in the tangential and normal directions were determined during separate cutting tests on a Kistler Model 9255B rigid dynamometer. The estimated cutting coefficient

Poincaré sections

In order to show the difference between period 1 chatter (point S) and stable machining (point P), $1 / τ$ sampled time history of the tool motion is analyzed both numerically and experimentally. Poincaré sections are obtained by plotting the stroboscopically sampled data in the $x$ – $y$ plane.

Fig. 5 shows the $1 / τ$ sampled time history and the corresponding Poincaré sections obtained by the numerical simulation of Eq. (1). The loss of contact between the tool and the workpiece was also incorporated into

Conclusions

The detection of machine tool chatter was investigated for the case of cutter runout based on frequency spectra and vibration signals. Due to the runout, the system is periodic at the SR period T, therefore this period was used to assess different types of chatter phenomena instead of the TP period $τ$ that is generally used in the literature. The analysis was performed according to the Floquet theory for periodic systems. A two fluted tool was investigated, and it was shown that the chatter that

Acknowledgments

This work was supported in part by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences (T.I.), by the Hungarian National Science Foundation under Grant nos. OTKA T068910 (G.S.) and K72911 (T.I.), by the US National Science Foundation CAREER Award CMS-0636641 (B.P.M.). The authors also acknowledge the useful discussions with Dr. Janez Gradišek.

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Citation Excerpt :
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On the chatter frequencies of milling processes with runout

Abstract

Introduction

Section snippets

Mechanical model

Stability analysis and vibration frequencies

Zero runout vs. nonzero runout

Experimental verification

Poincaré sections

Conclusions

Acknowledgments

Annals of the CIRP

International Journal of Machine Tools and Manufacture

Journal of Sound and Vibration

International Journal of Machine Tool Design and Research

International Journal of Machine Tools and Manufacture

International Journal of Mechanical Sciences

International Journal of Machine Tools and Manufacture

Annals of the CIRP

International Journal of Machine Tools and Manufacture

International Journal of Machine Tools and Manufacture

Journal of Sound and Vibration

International Journal of Machine Tools and Manufacture