Stability of up-milling and down-milling, part 1: alternative analytical methods
Introduction
Increased industrial competition has driven the need for manufacturers to reduce costs and increase dimensional accuracy. Machining operations are one of the most widely used manufacturing processes [1]. The efficiency of a machining operation is dictated by the metal removal rates, cycle time, machine down time and tool wear. Optimisation of these parameters without sacrificing part quality is of key importance. A primary factor that limits process optimisation in machining is a phenomenon called chatter. Chatter is a dynamic instability that can limit material removal rates, cause a poor surface finish and potentially damage the tool and the workpiece.
The history of machine tool chatter goes back almost 100 years, when Taylor [2] described machine tool chatter as the “most obscure and delicate of all problems facing the machinist”. After the extensive work of Tlusty et al. [3], Tobias [4] and Kudinov [5], [6], the so-called regenerative effect has become the most commonly accepted explanation for machine tool chatter [7], [8], [9], [10]. This effect is related to the wavy workpiece surface generated by the previous cutting tooth passage. The corresponding mathematical models are delay-differential equations (DDEs) with infinite dimensional state spaces.
For continuous cutting operations, like turning, the governing equation is autonomous, and stability conditions can be given in closed form [1], [11], [12]. The study of nonlinear phenomena in the cutting process showed that the chatter frequencies are related to unstable periodic motions about the stable stationary cutting, i.e. a so-called subcritical Hopf bifurcation occurs, as it was proved experimentally by Shi and Tobias [13] and later analytically by Stépán and Kalmár-Nagy [14].
In the case of milling, the direction of the cutting force is changing due to the tool rotation, and the cutting is also interrupted as each tooth enters and leaves the work-piece. Consequently, the resulting equation of motion is a DDE with a time periodic coefficient. The Floquet theory of periodic ordinary differential equations (ODEs) can be extended for these systems [15], [16], and the stability properties are determined by the eigenvalues of the monodromy operator of the systems. These eigenvalues are the so-called characteristic multipliers. The monodromy operator can be represented by an infinite dimensional matrix. This causes difficulties, of course, when trying to obtain closed form stability predictions. Usually, a finite dimensional approximate transition matrix is used to predict stability properties. Several analytical methods have been developed to determine the stability boundaries for milling [11], [17], [18], [19], [20], [21], [22], [23]. Numerical simulation can also be used to capture the interrupted nature of the milling process [24], [25], [26], [27], [28], but the exploration of parameter space via time domain simulation is inefficient.
Analytical investigations have predicted the occurrence of new bifurcation phenomena in interrupted cutting processes. In addition to Hopf bifurcations, period doubling bifurcations are also a typical form of instability, as it was shown analytically by Insperger and Stépán [21], Corpus and Endres [29], Bayly et al. [22], Davies et al. [20], via numerical simulation by Zhao and Balachandran [26], and confirmed experimentally by Bayly et al. [22] and Davies et al. [20]. The nonlinear analysis of Stépán and Szalai [30] showed that this period doubling bifurcation is also subcritical.
In this paper, two analytical methods are introduced for stability prediction of general milling operations: the finite element analysis in time (FEAT) method and the semi-discretization (SD) method. Both methods form a finite dimensional transition matrix as an approximation of the infinite dimensional monodromy operator. The FEAT method presented in this paper is an extension of the method developed by Bayly et al. [22] for an interrupted turning process. The current analysis is different because it models milling more closely by including the changing direction of cutting forces. The SD method, first introduced by Insperger and Stépán [31], is also applied to milling. Due to the complicated and not fully explored structure of the stability charts, the comparison of the results of the two basically different approximation methods gives validity to the calculations. The analyses are carried out for various radial immersions of up-milling and down-milling for a single degree of freedom (SDOF) mechanical model. Stability predictions show that the regions of instability for up-milling and down-milling are about reversed at low immersions. Experimental evidence is given to confirm stability predictions in Section 2 of the paper.
Section snippets
Mechanical model
A schematic diagram of the milling process is shown in Fig. 1. The structure is assumed to be flexible in the x-direction only, so the system can be treated as SDOF. A summation of cutting forces acting on the tool produces the following equation of motion:where m is the modal mass, ζ is the damping ratio, ωn is the natural angular frequency, and Fx is the cutting force in the x-direction for a zero helix cutter. According to Fig. 2, the x component of the
Up-milling and down-milling
The relationship between the direction of tool rotation and feed defines two types of partial immersion milling operations: the up-milling and the down-milling (see Fig. 3). Both operations essentially produce the same result, but the dynamics and stability properties are not the same. Partial immersion milling operations are characterized by the number N of teeth and the radial immersion ratio a/D, where a is the radial depth of cut, D the diameter of the tool.
Fig. 4 presents the specific
Finite element analysis in time
The stability of the milling process is dependent upon the perturbation growth or decay about the periodic motion determined by Eq. (13). Since this equation does not have a closed form solution, an approximate solution is sought to understand the behaviour of the system. One such approximation technique used for dynamic systems is time finite elements [32]. This method was first applied to an interrupted turning process by Halley [33] and Bayly et al. [34]. The authors matched an approximation
Semi-discretization method
Discretization techniques are important for differential equations for which the solution cannot be given in closed forms. The so-called semi-discretization is a well known technique in the finite element analysis of solid bodies, or in computational fluid mechanics. The basic idea is, that the corresponding partial differential equation (PDE) is discretized along the spatial coordinates only, while the time coordinates are unchanged. From a dynamical systems viewpoint, the PDE has an infinite
Stability charts, comparison of methods
Stability charts and chatter frequencies are determined by both methods for a series of milling processes. For the calculations, the experimentally identified parameters were used: m=2.573 kg, ζ=0.0032, fn=ωn/2π=146.5 Hz, Kn=2.0×108 N/m2 and Kt=5.5×108 N/m2 (see Part 2 of this report).
In Fig. 6, stability charts and the chatter frequencies f are presented for full immersion milling case. These frequencies are either fH or fPD determined by , , while the tooth pass excitation frequencies FTPE
Conclusions
In Fig. 7, the similarities and the differences between up-milling and down-milling cases can be clearly observed. The flip lobes, for example, vary in size, but they are located more or less at the same spindle speed ranges. This is not always true for the Hopf lobes. For low immersion up-milling, the Hopf lobes are located to the left of the flip lobes, while for down-milling, the Hopf lobes are positioned to the right of the flip lobes. Also, the frequency plots of up- and down-milling cases
Acknowledgements
This research was supported by the Hungarian National Science Foundation under grant no. OTKA T030762, the US National Science Foundation Grants DMI-9900108 (GOALI) and CMS-9625161 (CAREER) and the US National Defense Science and Engineering Graduate Fellowship.
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