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Click on the links below to find answers to frequently asked questions about high-speed machining. |
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| Click here to learn more about the History of chatter research. | ||||||||||
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Research in the area of milling stability has enjoyed a rich history. Taylor recognized the process limitations imposed by chatter, as well as the complexity in modeling its source, as early as 1906 when he stated that chatter is the “most obscure and delicate of all problems facing the machinist” [1]. Later, work by Arnold proposed the negative damping effect as the source of chatter [2], while research by Tlusty and Tobias led to a fundamental understanding of regeneration of waviness, or the overcutting of a machined surface by a vibrating cutter, as a primary feedback mechanism for the growth of self-excited vibrations (or chatter) due to the modulation of the instantaneous chip thickness, cutting force variation, and subsequent tool vibration [3-7]. Tlusty and Tobias also described the mode coupling effect as a second chatter mechanism. Efforts at modeling the process dynamics in order to select stable combinations of chip width, or axial depth of cut in peripheral milling operations, and spindle speed, can be loosely divided into 1) analytical; and 2) numerical techniques [8-34]. The most common output of these simulations is the stability lobe diagram [3, 7, 8], a graphical tool which identifies the boundary between stable and unstable cutting zones in a two-dimensional map of the primary control parameters: chip width, b, and spindle speed. Traditionally, spindle speed is varied along the abscissa (horizontal axis) and chip width along the ordinate (vertical axis) as demonstrated in Figure 1. The peaks of the intersecting lobes shown in this diagram occur approximately at spindle speeds where the tooth passing frequency is equal to an integer fraction of the natural frequency corresponding to the most flexible mode; these best spindle speeds can be estimated using Eq. 1, where fn is the natural frequency in Hz, m is the number of teeth on the cutter, j is an integer (j = 1, 2, 3…), and spindle speed is expressed in rev/min, or rpm. It should also be noted that an analog to the stability lobe diagram, the peak-to-peak or PTP diagram which identifies stability boundaries by abrupt discontinuities in the predicted peak-to-peak tool vibration or cutting force values, can be developed using time-domain numerical integration techniques [35]. (1) In general, stability lobe diagrams are developed by selecting the cutting parameters, which include the process-dependent specific cutting energy coefficients, radial immersion, and system dynamics (often selected as the tool point frequency response, although the workpiece dynamics must also be considered in some instances), then carrying out the selected simulation algorithm. In this case, the system dynamics are considered to be fixed and a new set of stability calculations must be completed if the system changes (e.g., a new tool is selected).
References
High-speed machining (HSM) may be defined in various ways. First, with regard to attainable cutting speeds, it is suggested that operating at cutting speeds significantly higher than those typically utilized for a particular material may be termed HSM (see Figure 1 [1]). Second, theoretical and experimental analyses have shown that increased local stability occurs when the tooth passing frequency of the cutter is equal to the natural frequency (or any of its fractional harmonics) of the most flexible system mode [2-3]. Selection of the maximum available spindle speed that corresponds to one of these stable tooth passing frequencies is also referred to as HSM. The latter definition of HSM relies on the concept of regeneration of waviness as a primary cause of instability (i.e., self-excited vibrations) in machining [3]. This waviness regeneration occurs when a cutter tooth encounters an undulating surface left by the previous tooth. The prediction of system stability depends on the phase relationship between the displacement of the current cutter tooth and the waviness it encounters. For certain phase relationships, the succeeding tool vibrations diminish, while for others they increase until either failure or a system non-linearity (e.g., the tool leaves the cut) limits the motion [4]. A typical stability lobe diagram, which predicts system stability as a function of spindle speed and machining parameters, is shown in Figure 2. Both stable and unstable regions are seen depending on the selected spindle speed and chip width, or axial depth of cut in peripheral end milling. This diagram may be calculated using analytic (see, for example, [5-9]) or numerical time-domain techniques [3, 10-12]. In either instance, knowledge of the machine (and sometimes workpiece) dynamics is required. In many cases, the system dynamics are obtained using impact testing and modal analysis. The direct frequency response function is measured at the tool point and multiple modes fit to the results. Modal parameters (i.e., mass, m, stiffness, k, and damping ratio, x) for each of the selected modes are extracted and used as input to the stability lobe analysis. Another possibility is milling experiments for direct stable speed selection. Here, machining tests are completed to locate chatter frequencies and select stable spindle speeds [13]. In all cases, however, the results are specific to the selected components (e.g., tool, holder, workpiece, spindle, and machine) and boundary conditions (e.g., holder force and drawbar force [14]).
References 1.
Schultz, H. and Moriwaki, T., 1992, High-Speed Machining, Annals
of the CIRP, 41/2: 637-643.
The onset of chatter during machining is primarily a function of the variation in chip thickness that occurs due to vibration of the tool, workpiece, or both. This situation is shown schematically in Fig. 1. In this figure, the flexible tool engages the workpiece and, due to the cutting force, begins vibrating. This vibration is imprinted on the machined surface. In milling, the next tooth on the rotating cutter overcuts this wavy surface produced by the previous tooth. This wavy surface varies the instantaneous chip thickness which, in turn, modulates the cutting force and the cutter vibration (i.e., a feedback mechanism is produced that can lead to self-excited vibrations, or chatter). Depending on the relationship between the wavy surface left by the previous tooth and the current cutter vibration, the resulting deflections and forces can grow very large (chatter) or diminish (stable cutting).
The tool point vibration response is typically obtained using impact testing (see Figure 1), where an instrumented hammer is used to excite the tool at its free end (i.e., the tool point) and the resulting vibration measured using an appropriate transducer, typically a low mass accelerometer mounted at the tool point. The complex ratio of the frequency domain vibration and force signals, or Frequency Response Function (FRF), is then used to generate the corresponding stability lobe diagram.
In order to analytically determine the tool point frequency response as a function of tool length and apply the method of tool tuning, the Receptance Coupling Substructure Analysis (RCSA) method was developed [1-3]. In this technique, based on earlier receptance coupling work by Duncan [4], Bishop and Johnson [5], and, later, Ferreira and Ewins [6], an experimental measurement of the holder/spindle substructure, or component, is coupled to an analytical model of the tool through two empirical complex stiffness vectors, which include linear and rotational stiffness and viscous damping terms that characterize the non-rigid behavior of the connection between the holder and tool (e.g., thermal shrink fit, collet, or elastic deformation interference fit [7]). The primary benefit of using receptance, rather than modal, coupling for this application is that no restrictions are placed on the number of modes included in either the holder/spindle experimental measurements or tool model and the holder/spindle frequency response data can be used directly without requiring a modal fit. Since the desired output is the tool point frequency response, the most straightforward approach is to directly couple component receptance terms and avoid the modal fitting step all together.
The model for the coupling
between the holder/spindle and tool components is shown in Figure 1.
There are three translational and three rotational assembly coordinates
identified, with spatial positions coincident with the coupling locations
(coordinates X2/Q2 on
the tool and X3/Q3 on the holder/spindle component) and
the point of interest (coordinates X1/Q1 at the free end of tool). The connection
between X2/Q2 and X3/Q3 is composed of a linear spring, kx, torsional spring, kq,
linear viscous damper, cx,
and rotational viscous damper, cq. In order to
determine the assembly direct, or driving point, frequency response at
the tool point, G11(w) = X1(w)/F1(w),
which is used as input to the selected process stability simulation,
the following steps must be completed: a) Use impact testing to measure the holder/spindle component (i.e., no tool inserted in holder) frequency response function (FRF), H33 = X3/F3, at the free end in two orthogonal directions in the plane of the cut, i.e., perpendicular to the spindle centerline. Typically, the measurement directions are selected to be coincident with the feed directions of the machine tool. b) Develop an analytic model of the free-free tool using the closed form receptance terms, which capture both the rigid body and transverse vibration behavior of the tool, developed by Bishop and Johnson [5]. We have selected to treat the tool as an Euler-Bernoulli beam with a constant cross-section, which requires that an effective diameter, deff, be determined for calculation of the 2nd area moment of inertia, I = pdeff4/64. The effective diameter is based on the tool overhang length, L, total length, LT, tool material density, r, shank diameter, d, and tool mass, M. See Eq. 1. Fundamentally, this equation calculates the diameter of a uniform cross-section beam with 1) a mass equal to the difference between the total tool mass and the mass of the tool shank inside the holder; and 2) a length equal to the overhang length of the tool, given the tool material density.
We
have also added structural, or hysteretic, damping to the tool model
by replacing
Young’s elastic modulus, E, for the
tool material with the complex modulus,
c) Measure the tool point response for the assembly in one direction at a known overhang. This data allows the determination of the connection parameters, kx, kq, cx, and cq, by nonlinear least squares best fit.
Mathematical
Derivation of Assembly FRF If harmonic external excitations of force F(t) = Feiwt and/or moment M(t) = Meiwt are applied to the assembly shown in Figure 1, the resulting displacements and rotations can be written as X(t) = Xeiwt and Q(t) = Qeiwt, respectively. In order to determine an analytical expression for the tool point frequency response G11(w), we apply the harmonic force F1(t) to coordinate X1 of the assembly. The resulting forces/moments and displacements/rotations for the individual components, represented in Figure 2, can then be expressed as shown in Eq. 3. The equilibrium conditions for this loading condition are shown in Eq. 4; the compatibility conditions are provided in Eq. 5. The latter conditions serve two purposes: 1) they define the relationship between the two component displacements/rotations and forces/moments; and 2) they specify that the component coordinates are at the same spatial locations as the assembly coordinates.
Because we have assumed
harmonic motion (due to harmonic excitation), the time derivative terms
in Eq. 5 can be rewritten in the form
To determine G11(w),
we first substitute the component displacements and rotations defined
in Eq. 3 into E
We can now make the substitution f1 = F1 from the equilibrium conditions and solve Eq. 8 for {f3 m3}T as shown in Eq. 9.
The relationships between the tool component displacement and rotation at coordinate 1, x1 and q1, and the component forces, f1 and f2, and moment, m2, were expressed in Eq. 3. These can be rewritten in matrix form as shown in Eq. 10. Substitution of x1 = X1, q1 = Q1, f1 = F1, f2 = -f3, m2 = -m3, and the result from Eq. 9 in Eq. 10 yields Eq. 11.
where and
The desired assembly receptance term G11(w) can now be determined from the top row in Eq. 11. This result is shown in Eq. 12. The receptance G41(w) = Q1/F1 is also available from the second row of Eq. 11. Although we are only interested in determining the assembly direct displacement to force frequency response at the tool point, the full receptance matrix (shown in Eq. 13) can be populated using this method.
References 1. Schmitz,
T.; Donaldson, R. Predicting High-Speed Machining Dynamics by Substructure Analysis.
Annals of the CIRP 2000, 49 (1), 303-308. Recent research by Winfough, Smith, Tlusty, Halley, and Davies et al. [1-3] has suggested that, rather than assuming fixed dynamics, the tool point frequency response can be varied by adjusting the tool overhang length in a method referred to as tool tuning. In this case, improved material removal rates can be obtained by 1) shifting the natural frequency corresponding to the most flexible mode (often the fundamental tool vibrational mode) and, therefore, the location of the peaks of the stability lobes as shown in Eq. 1, e.g., adjusting the tool length to move a lobe peak to the top available spindle speed; and/or 2) varying the tool length in order to obtain an overlap between the fundamental tool natural frequency and one of the spindle natural frequencies. This results in the dynamic absorber effect [4] where the matched natural frequencies lead to a dynamically stiffer system, similar to the result observed when adding the classic Frahm dynamic absorber [5] to a base structure in order to attenuate vibration at a particular excitation frequency. (1) References 1. Tlusty, J.; Smith, S.; Winfough, W. Techniques for the Use of
Long Slender End Mills in High-Speed Machining. Annals of the CIRP
1996,
45 (1), 393-396.
What is a stability lobe diagram? A stability lobe diagram is a plot that separates unstable combinations of chip width, or axial depth of cut in peripheral end milling, and spindle speed (i.e., those that produce chatter) from stable combinations. Stable cuts occur in the region below the stability boundary (or combination of all the stability ‘lobes’), while unstable cuts occur above the stability boundary. These stable and unstable regions are seen in Figure 1. As shown, it is often possible to increase the allowable axial depth of cut without chatter by increasing the spindle speed, rather than slowing down. This counterintuitive behavior is one reason that is important to characterize the dynamic response of the cutting tool and produce the corresponding stability lobe diagram.
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, where 
. To simplify notation, we drop the primes from E’ and 
, 
, 
, and 
.
and 
. Substitution in Eq. 5 yields Eq. 6. To simplify notation, we now define the
complex, frequency dependent stiffness terms 
and 
.
q. 6. This result is shown in Eq. 7. In Eq. 8, this equation
has been expressed in matrix form.
