Modeling and Analytical Solution of Chatter Stability for T-slot Milling

LI Zhongqun and LIU Qiang

1 School of Mechanical Engineering, Hunan University of Technology, Zhuzhou 412008, China

2 School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China

1 Introduction

Modeling and prediction of the stability in cutting process have been a focus area of manufacturing research since the pioneering chatter stability theories of Tlusty and Tobias during 1950’s[1–2]. Recently, the dynamic models of chatter stability in high speed machining(HSM), complex milling process such as plunge milling and different variety of end milling cutters have been developed and evaluated in the laboratory and industrial applications[3–5]. The main efforts of chatter vibration research are focused on the cutting force prediction, dynamic cutting coefficients identification, tool-spindle interfaces dynamics, HSM chatter avoidance, and stability lobes diagram(SLD) for various machining operations. There have been significant achievements on the chatter vibration research and applications during recent decades.

Chatter is an unstable vibration due to dynamic interactions between the cutting tool and workpiece. Under certain conditions, the amplitude of vibrations grows and the cutting system becomes unstable. TOBIAS[1]began the study of machining chatter with establishment of the basis of the regenerative chatter theory. In the early milling stability analysis, KOENIGSBERGER, et al[2], used the orthogonal chatter model considering an average direction and average number of teeth in cut. SRIDHAR, et al[3],firstly introduced the time-varying directional cutting force coefficients in modeling the chatter stability of milling.MINIS, et al[4], formulated and numerically solved the milling stability using the Nyquist criterion. BUDAK[5]presented and verified an analytical determination of stability limits, which has been applied to the stability of ball-end milling[6–8], and extended to 3D milling[9].

T-slot milling has found applications in aerospace,automotive and die machining industry. When applying a long and slender T-slot cutter to the milling process, the chatter vibration of the cutting process will lead to the poor surface finish, lower productivity and decreased tool life.However, there has been limited literature for the T-slot milling process[10].

Generally, a T-slot cutter can be regarded as a special case of inserted cutters. ENGIN, et al[11], presented a generalized mathematical model of inserted cutters for predication of cutting forces, vibrations and stability lobes in milling. Based on the model, CutPro?, developed in Manufacturing Automation Laboratory, can provide stability simulation for the inserted cutters. However,provides the classical SLD on the axial depth of cut, but it cannot provide the SLD on the radial depth of cut,which is more meaningful in T-slot milling operations.

Therefore, a dynamic model and its analytical solution of T-slot milling are presented in this paper. Originally based on the analytical milling stability model of BUDAK, the geometric features and milling operation of T-slot cutter are considered in modeling and analysis. In addition, the SLD on the radial depth of cut are developed theoretically and verified experimentally.

The paper is organized as follows. In section 2, the dynamic model of T-slot milling are presented and solved analytically. The stability lobes calculation and simulation on the radial depth of cut are discussed in section 3. The comparisons between theoretical and experimental results are given in section 4, followed by the conclusions.

2 Dynamics of T-slot Milling

As shown in Fig. 1, the T-slot cutter is made up of two rows of uniform-spaced inserts with the bottom surface of the cutter as the reference of the lower row inserts, and the top surface of the cutter as the reference of the upper row inserts. The inserts of upper row cut by using the top and side edges with the negative lead angles, and the inserts of lower row cut by using the bottom and side edges with the positive lead angles. The geometry above is the significant features different from other general insert end mill cutter.

Fig. 1. 3D digital model of a 6-flute T-slot cutter

2.1 Dynamic cutting forces of T-slot milling

Using the same principle of dynamic modeling of a flat end mill[5], a T-slot milling process with N-insert cutter can also be reduced to a 2-DOF vibration system in two orthogonal directions. The dynamics of the milling system can be given by the differential equations as follows:

where m, c, k are the mass, damping ratio and stiffness of the machine tool in the directions of x and y, respectively.Fxj, Fyjare the components of the cutting force applied on the jth tooth of the cutter in the directions of x and y.

The geometrical model of a T-slot cutter with 6 flutes is shown in Fig. 2. By slicing the cutting edges into M disks along the axial direction, the radial immersion angle at the lth plate on the jth cutting edge can be expressed as

where φ10is the angular displacement at the bottom of the first tooth, φpis the pitch angle and φp= 2π/N, dz is the height of each disk, Lfis the overall height of the cutting edge, δ1, δ2are the lead angle and axial rake angle of the inserts, respectively, and R is the radius of the cutter.

Fig. 2. Geometry of an insert of 6-flute T-slot cutter

By applying the linear-edge model[5]to each disk,through coordinate transformation, numerical integration along the axial direction and summation for the differential cutting forces of all the teeth, the overall instantaneous cutting forces acting on the whole cutter in the feed and normal directions can be expressed as follows:

where g(φjl) is a unit step function used to define whether the differential cutting edge is in or out of cut. Ktcand Krcare the tangential and radial cutting force coefficients,respectively. Δx, Δy are the dynamic displacement variation of the cutter and workpiece between the current and the previous tooth passes in the x and y directions, respectively.In matrix form, the above equations can be rewritten as follows:

where apis the axial depth of cut. Suppose Fr= Krc/ Ktc, the directional cutting force coefficients are given as follows:

The directional coefficients depend on the angular position of the cutter makes Eq. (4) time-varying as follows:

where A(t) is periodic at the tooth passing frequency ω=Nn/60, and n is the spindle speed. In general, Fourier series expansion of the periodic term is used for solution of the periodic systems. However, in chatter stability analysis,the inclusion of the higher harmonics in the solution may not be required for most cases as the response at the chatter limit is usually dominated with a single chatter frequency.Starting from this idea, Refs. [5–7] confirm that the higher harmonics do not affect the accuracy of the predictions unless the radial depth of cut is extremely small compared to the diameter of the cutting tool. Thus, it is sufficient to include only the average term in the Fourier series expansion of A(t), and the average directional cutting force coefficients take the following form:

where O is the discretized points number within a tool revolution. Substituting Eq. (7) into Eq. (6), we can obtain

2.2 Analytical solution of chatter stability for T-slot milling

In Eq. (8), A0does not vary with the time anymore and depends only on the immersion angle. The vibrations Δ(iω)are expressed in terms of the dynamic cutting forces F(iω)and the transfer function of the tool-workpiece engagement G(iω) as

The transfer function G(iω) can be given as

The dynamic cutting forces at chatter frequency ωcare obtained by substituting Eq. (9) into Eq. (8):

The stability turns into an eigenvalue problem, and it has a nontrivial solution only if its determinant is zero[5]as follows:

where G0=A0G is the oriented transfer function matrix, and the eigenvalue of the characteristic equation is

If the cross transfer functions are neglected, the analytical solution of the eigen-value can be obtained as

where a0=Gxx(iωc)Gyy(iωc)(αxxαyy?αxyαyx), a1=αxxGxx(iωc)?αyyGyy(iωc). By scanning the chatter frequency ωc, the critical depth of cut alimand the spindle speed n can be derived from the real and imaginary parts of the eigenvalues ΛI(xiàn), ΛRas follows:

where k is an integer corresponding to the number of vibration wave during a tooth period.

Therefore, for the given geometry of a T-slot cutter, the tool/part specific cutting forces, the transfer functions of the milling system and the chatter frequency ωc, ΛRand ΛI(xiàn)can be obtained by Eq. (14), and can be used in Eq. (15) to determine the stability limits aplimand the spindle speed n.When these procedures are repeated for the range of chatter frequency and vibration wave k, the stability lobes diagram of a T-slot milling system can be obtained.

3 Calculation of SLD on Radial Depth of Cut

The above-mentioned SLD on the axial depth of cut is useful in determining the chatter-free cutting conditions of an ordinary end mill. However, for a T-slot cutter, as the full length of its cutting edges has to be used in cutting process in some occasions, the SLD on the radial depth of cut should also be derived. The schematic diagram of obtaining this kind of SLD is shown in Fig. 3 and the procedures are as follows.

(1) To set the initial radial depth of cut ae= ae0and its increment Δae.

(2) To obtain the data of stability lobes under the given radial depth of cut ae. The data is represented by a two-dimensional array. It has two columns, the first one is the spindle speed, and the second one is the corresponding critical axial depth of cut.

(3) To let ae= ae+ Δaeand if ae≤2R, go back to (2).

(4) To interpolate with the simulated data to obtain the critical radial depth of cut for each axial depth of cut under the specified spindle speed.

(5) To increase the spindle speed and repeat (4) until it reaches the upper limit of the simulation spindle speed.

(6) To draw a figure with the spindle speed n as independent variable and the critical radial depth of cut aplimas dependent variable to obtain the SLD under the specified axial depth of cut.

Fig. 3. Schematic diagram of obtaining SLD on the radial depth of cut for T-slot milling

Using the analytical solution of chatter stability for T-slot milling, a Matlab-based simulation model was developed,which gathers the input data of cutting conditions, machine tool characteristics, workpiece material, tool geometry, and other related parameters in T-slot milling. The simulation interface is developed using the GUIDE of Matlab, and shown in Fig. 4.

Fig. 4. Interface of chatter stability simulation of T-slot milling

4 Experimental Verification and Discussion

Verification tests were conducted on a 5-axis vertical machining center JO’MACH143. The maximal spindle speed of the machine is 6 000 r/min. The cutting tool used is a 6-flute carbide solid T-slot cutter, the geometrical parameters of which are shown in Table 1. The material of the workpiece is Al7075/T6, the cutting force coefficients obtained by identification tests are Ktc=796.0 N/mm2,Krc=168.0 N/mm2.

Table 1. T-slot cutter used in the verification tests

The test system includes a Kistler 9722A500 impact hammer, sensitivity 10 mV/N, 500 N and a frequency range of 1–8 kHz; a Kistler 8775A50 accelerator, low impedance,sensitivity 100 mV/g, 50 g and a frequency range of 1–7 kHz; a National Instruments USB 9233 24-bit combined DAQ-Signal Conditioning unit; a Shure microphone; a tap testing software module CutPro? MALTF, a dynamic simulation Module for milling CutPro? Advanced Milling.

Hammer tests were conducted in both the feed (x) and the normal direction (y) to get the frequency response functions in these two directions. The obtained FRFs are shown in Fig. 5. The modal parameters of the milling system obtained by the Modal Analysis Module ofare listed in Table 2.

Fig. 5. Measured FRFs of the cutter

Table 2. Modal parameters of the machining system

To validate the presented analytical solution of stability limits for T-slot milling, the simulation result from this model was compared to that from CutPro?. As the SLD on the radial depth of cut cannot be obtained in CutPro?, only the SLD on the axial depth of cut was compared, which is shown in Fig. 6. The figure shows that the SLD from these two models are in good agreement in general, and the small discrepancy may be attributed to the different algorithms applied.

Fig. 6. SLD from T-slot milling model with that from CutPro?

The SLD on the radial depth of cut under full axial depth of cut (ap=20.0 mm) is predicated with the abovementioned program model and shown in Fig. 7. The experimental SLD was plotted in spindle speed increment of 400 r/min from 2 500 r/min to 5 700 r/min. Chatter was recorded using a Shure microphone and identified using fast Fourier transform(FFT). Results of chatter tests are also plotted in the same figure, which are in good agreement with the predictions.

Fig. 7. Experimental and predicated SLD

To reveal the time-domain properties of different points on the chatter stability lobes diagram, according to the method presented by author’s previous work[12], the dynamic simulations in the time-domain were conducted under cutting conditions corresponding to points A and B in Fig. 7. Point A is located in the chatter region, the spindle speed is 3 600 r/min, and the radial depth of cut is 1.5 mm.Point B is located in the stable region, the spindle speed is 4 450 r/min, and the radial depth of cut is 1.7 mm. In both cases, the feedrate is set as 2 000 mm/min. The tool’s vibration in x direction and its FFT are shown in Fig. 8 and Fig. 9, respectively.

Fig. 8. Tool vibrations in x direction

Fig. 9. FFT of tool vibrations in x direction between A and B

The vibration of the cutter in x direction shows that the milling process corresponding to point B is stable but that corresponding to point A is unstable. FFT of the simulated vibration of cutter in x direction shows that when milling under the cutting conditions corresponding to point B, the energy is almost concentrated at the tooth passing frequency (435 Hz) and its harmonics. However, when milling under the cutting conditions corresponding to point A, the energy is not all concentrated at the harmonics of the tooth passing frequency (360 Hz). The chatter is occurred at the frequency of 1 200 Hz which is around the natural frequency of the milling system.

5 Conclusions

(1) Based on the geometrical model of a T-slot cutter, the dynamic cutting force is modeled, in which a numerical method is employed to calculate the average directional cutting force coefficients which lead to an analytical solution of the chatter stability for T-slot milling.

(2) In order to determine the cutting conditions of T-slot milling, the stability lobes diagram is derived not only on the axial depth of cut but also on the radial depth of cut.

(3) The agreement of simulation result from T-slot model with that from CutPro?, as well as the agreement of predicated SLD with the experimental one has verified the proposed T-slot milling model.

(4) High efficient and chatter-free T-slot milling can be achieved with the cutting conditions determined by the SLD from the simulation model of T-slot milling.

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