Analysis of milling stability based on cutting force signal processing

A dynamic cutting force model of the end-milling process with tool runout error was established in this research to understand the underlying mechanism of chatter.. The accuracy of the c

機械工程系 碩士學位論文

中華民國 一零六 年 六月 三十日

Abstract

The milling operation is the most common form of machining Because the action of each cutting edge and workpiece is intermittent and periodical, the chip thickness varies periodically This could lead to self-excited vibrations and unstable cutting which is called chatter vibration Chatter causes machining instability and reduces productivity in the metal cutting process It has negative effects on the surface finish, dimensional accuracy, tool life and machine life Chatter identification is therefore necessary to control, prevent, or eliminate chatter and to identify the stable machining condition A dynamic cutting force model of the end-milling process with tool runout error was established in this research to understand the underlying mechanism of chatter The accuracy of the cutting force model in both time and frequency domains was evaluated by comparing to experimental force signals Time-frequency analysis approaches, specifically short time Fourier transform, continuous wavelet transform and Hilbert-Huang transform, were utilized to give an utterly different perspective

of chatter from the conventional Fourier spectrum which is insufficient in analyzing the signals of rich nonlinear characteristics By comparing the simulation with experimental result, chatter frequency was found to consist of two major components, frequency modulation alongside tooth passing frequency caused by the increased tool runout error and the non-stationary high frequency from the regenerative vibration Moreover, dimensionless chatter indicators, defined by the standard deviation and energy ratio of the specific intrinsic mode function, could identify the occurrence of chatter effectively The analysis result was then validated by the workpiece surface topography, surface roughness and the stability lobe diagram

Keywords: Milling process, Chatter detection, Time-frequency analysis, Wavelet transform,

Hilbert Huang transform

Acknowledgement

I would like to thank all the people who helped me to finish this thesis First of all, I would like to express my deep gratitude to my academic advisors: Professor Chun-Hui Chung and Professor Meng-Kun Liu for their valuable guidance, encouragement, and support throughout my work towards this thesis Without their help and guidance, this work would not be possible

I also would like to thank Mr Yi-Wen Qui who provided his experimental data which was used to verify my methodology in this thesis I would like to thank all of my labmates who have supported me a lot with laboratory facilities so that I could conduct my experiments

I thank my friends who always give me encouragements and supports during my research

Finally, I would like to thank my parents who always give me love, encouragement, and support throughout my life I would specially thank my wife and my son for their patience and support during my study I am very grateful for their love

dF differential cutting force in y direction

F experimental cutting force (N) s start angle (deg)

b axial depth of cut (mm) c chatter frequency (rad/s)

h instantaneous chip thickness (mm)  spindle speed (rpm)

s

K specific cutting force coefficient (N)  position angle (deg)

 cutting force angle (deg)  runout of cutting edge (mm)

,

K K radial and tangential cutting

coefficients (N/mm2)  time for one rotation (sec)

Contents

摘要 i

Abstract ii

Acknowledgement iii

Nomenclature iv

Contents v

Chapter 1 1

Introduction 1

1.1 Background 1

1.2 Objective and Scope 2

1.3 Outlines and Contribution of the Chapters 3

Chapter 2 4

Literature Review 4

2.1 Chatter Vibrations in Milling 4

2.2 Signal Analysis Approaches 6

Chapter 3 10

Dynamic Cutting Force Model 10

3.1 Regenerative Chatter Model 10

3.2 Dynamic Cutting Force Model 15

Chapter 4 17

Experimental Setup and Model Verification 17

4.1 Overview and Aim 17

4.2 Experimental Setup 17

4.2.1 Machine, Cutter and Workpiece 17

4.2.2 Cutting Force Measurement Equipment 19

4.2.3 Surface Topography Measurement Equipment 20

4.2.4 Surface Roughness Measurement Equipment 20

4.3 Cutting Force Coefficients 21

4 4 Experimental Design Parameters 25

4.5 Tool Tip Dynamics 27

4.5.1 Impact Testing 27

4.5.2 Modal Analysis 27

4.5.3 Stability Lobe Diagram 31

4.6 Simulation and Experimental Results 33

Chapter 5 36

Chatter Detection Methodology 36

5.1 Short-Time Fourier Transform Analysis 36

5.2 Continuous Wavelet Transform Analysis 38

5.3 Time-Frequency Analysis Based on HHT 41

5.3.1 Chatter Detection Methodology 41

5.3.2 Results and Discussions 44

5.3.3 Dimensionless Indexes for Chatter Identification 50

5.3.4 Method Verification 51

5.4 Chatter Identification in Small Size of End Mill 54

5.4.1 Overview and Aim 54

5.4.2 Experimental Setup 54

5.4.3 Chatter Identification by Time-Frequency Analysis 57

Chapter 6 60

Conclusions and Future Works 60

6.1 Conclusions 60

6.2 Future Works 60

Bibliography 61

List of Figures

Figure 2.1 Milling regenerative chatter 4

Figure 2.2 Stability lobe diagram 6

Figure 2.3 Time-frequency resolution of STFT [9] 8

Figure 3.1 The model of helical end mill geometry 10

Figure 3.2 Dynamic cutting force model 11

Figure 3.3 Flow chart of dynamic cutting force 16

Figure 4.1 The three- axis CNC milling machine at NTUST 18

Figure 4.2 A carbide end mill 18

Figure 4.3 Kistler Dynamometer 19

Figure 4.4 Experimental setup 19

Figure 4.5 Olympus BX51 at NUTST 20

Figure 4.6 A Mitutoyo portable surface roughness tester 21

Figure 4.7 The modeling of slot milling 21

Figure 4.8 Measured cutting forced signal at feed rate of 0.03 mm/tooth in (a) x direction and (b) y direction 23

Figure 4.9 Relationship between feet per tooth and mean force 25

Figure 4.10 Setup of the impact testing 27

Figure 4.11 Relative frequency response in x direction 28

Figure 4.12 Relative frequency response in y direction 29

Figure 4.13 Direct FRF in x direction 31

Figure 4.14 Direct FRF in y direction 32

Figure 4.15 Stability lobe diagram and experimental design parameters 33

Figure 4.16 Cutting force in (a) x direction and (b) y direction at spindle speed of 1500 rpm

and 1.4 mm depth of cut 34

Figure 4.17 Cutting force in x direction at spindle speed of 5250 rpm when (a) DOC = 0.8mm (b) DOC = 1.4mm 35

Figure 4.18 Fourier spectrum of the force signals at spindle speed of (a) 5250 rpm, DOC= 0.8 mm and (b) 5250 rpm, DOC =1.4 mm 35

Figure 5.1 Hanning window 36

Figure 5.2 Short time Fourier transform by Hanning window at spindle speed of 5250 rpm, DOC = 0.8 mm (a) Simulation and (b) experimental force signal 37

Figure 5.3 Short time Fourier transform by Hanning window at spindle speed of 5250 rpm, DOC = 1.4 mm (a) Simulation and (b) experimental force signal 37

Figure 5.4 Morlet Wavelet 39

Figure 5.5 Continuous Wavelet Transform 2-D in case of spindle speed of 5250 rpm and 0.8 mm depth of cut 40

Figure 5.6 Continuous Wavelet Transform 2-D in case of spindle speed of 5250 rpm and 1.4 mm depth of cut 40

Figure 5.7 Flow chart of the chatter identification method 41

Figure 5.8 Cutting force signal 42

Figure 5.9 Ensemble empirical mode decomposition 43

Figure 5.10 The first eight IMFs of the filtered signal at spindle speed of 5250 rpm with 0.8 mm depth of cut 45

Figure 5.11 The first eight IMFs of the filtered signal at spindle speed of 5250 rpm with 1.4 mm depth of cut 46

Figure 5.12 Instantaneous frequency of IMFs at spindle speed of (a) 5250 rpm, DOC= 0.8 mm (stable condition) and (b) 5250 rpm, DOC =1.4 mm (unstable condition) 48

Figure 5.13 Hilbert-Huang spectrum of IMF2 at (a) stable condition and (b) unstable

condition 49

Figure 5.14 Stability lobe diagram and the stability determined by proposed method 52

Figure 5.15 Microscopic of the surface machined with different spindle speed and constant depth of cut of 1.4 m 52

Figure 5.16 Five-axis CNC milling machine 55

Figure 5.17 A carbide end mill 56

Figure 5.18 Experimental setup 56

Figure 5.19 Machined surface topography 59

List of Tables

Table 4-1 Machine features 18

Table 4-2 Parameters of Tool 19

Table 4-3 Cutting parameters for cutting force coefficients 22

Table 4-4 Average cutting forces 23

Table 4-5 Specific cutting force coefficients 25

Table 4-6 Experimental design parameters 26

Table 4-7 Model parameters in x direction 30

Table 4-8 Model parameters in y direction 30

Table 5-1 IMF Property of force signal at spindle speed of 5250 rpm 47

Table 5-2 The energy ratio values of IMF2s 50

Table 5-3 The standard deviation values of IMF2s 51

Table 5-4 Surface roughness measured along the feed direction at different cutting conditions 53

Table 5-5 Machine’s features 55

Table 5-6 Parameters of Tool 56

Table 5-7 Experimental design parameters 57

Table 5-8 The standard deviation values of IMF2s 58

Table 5-9 The energy ratio values of IMF2s 58

in order to avoid machining instability and to improve the productivity Regenerative chatter

in milling is because the previous and current cuts are out of phase The instantaneous chip thickness is variable, it governs the cutting force which, in turn, affects subsequent tool vibrations [2] Therefore, dynamic cutting force model plays an important role in analyzing and predicting chatter vibration The accurate model could be used to analyze the machining process and optimize the cutting parameter

A stability lobe diagram based on regenerative chatter theory is generally a simple and useful way to predict and control chatter [2-4] This approach was however based on the restrictive assumption that the dynamics of spindle-tool system has not changed over the spindle speed range [5] The stability lobe diagram may inevitably misinterpret the important attributes of the machining processes which in fact always contain rich nonlinear characteristics [6, 7] Both time and frequency responses hence have to be monitored simultaneously

In order to distinguish whether chatter occurred, some machining process signals have

to be measured by using different techniques such as in-process real-time techniques based

on force and displacement signals or off-line machined surface characterization [8] The Fourier transform (FT) that can transform time domain signal into frequency domain signal

is suitable for the stationary signal process, especially when the feature components of signals are obvious in magnitude For non-stationary signal, the wavelet transform (WT) known as a flexible method is suitable for detection of a sudden frequency change in a signal [9] Another time-frequency approach, Hilbert-Huang transform (HHT) has been widely used to investigate machining vibration This method is a powerful tool for time -frequency analysis of nonlinear and non-stationary signals due to performing a time adaptive decomposition operation on signals and no uncertainty principle limitation on time or frequency resolution [10]

A dynamic cutting force model with tool runout error was established in this research

to fit the cutting force signal properly and analyze the chatter identification Time-frequency analysis approaches based on cutting force signal, specifically short time Fourier transform, continuous wavelet transform and Hilbert-Huang transform, were utilized to give an utterly different perspective of chatter from the conventional Fourier spectrum which is insufficient

in analyzing the signals of rich nonlinear characteristics

1.2 Objective and Scope

The stability of the process was investigated by the time-frequency analysis All three

of short time Fourier transform, WT and HHT approaches were used for processing the cutting force signals The difference between stable and unstable conditions was clearly distinguished from the analysis Most of the energy concentrated at tooth passing frequency

in stable cutting condition, while a large amount of energy emerged at high frequency when chatter occurs In order to detect chatter efficiently whether it is highly nonlinear behavior or not, dimensionless indexes such as standard deviation and energy ratio of intrinsic mode functions (IMF) of the cutting force signals were proposed and studied in this research

1.3 Outlines and Contribution of the Chapters

The thesis is organized into six chapters as follows Chapter 1 introduces background, objectives, methods, and contributions of the present work A literature review related to the research is described in Chapter 2 In chapter 3, dynamic cutting force model regarding runout error is presented Model verification is provided in chapter 4 in which experimental design parameters, tool tip dynamics, and comparison of simulated and experimental results are presented In chapter 5, chatter detection methodologies including short time Fourier transform, continuous wavelet transform analysis and time-frequency analysis based on HHT are given The results and discussions are also presented in this chapter Finally, the summarizations and future works of this thesis are carried out in chapter 6

Chapter 2

Literature Review

2.1 Chatter Vibrations in Milling

Regenerative chatter in milling is self-excited vibration This mechanism is associated with the phase shift between vibration waves on both sides of the chip left by previous cutting tooth and current tooth This phenomenon was firstly explained by Tlusty and Polacek (1963), and then by Schmitz and Smith, S (2008); Altintas and Yusuf (2012) Figure 2.1 shows that the wavy surface left by tooth 1 is removed by tooth 2 and so on This gain provides a feedback mechanism because the instantaneous chip thickness depends on both current vibration and surface left by the previous tooth passing When the vibrations from one cutting edge to the next are in phase, the chip thickness varies as the cycloidal path This causes forced vibration only, and the cutting is stable On the other hand, the out of phase shows a less favorable phase relationship, chip thickness variation is significant It results in chatter vibration in milling When chatter occurs, it could cause negative effects on the process such as producing noise, poor surface roughness, increasing tool wear, reducing tool life and productivity [2-4] Chatter prediction is therefore critically important for determining stable cutting parameters such as spindle speed, depth of cut (DOC), and feed rate

Dynamic cutting force model has played an important role in analyzing and predicting chatter vibration The accurate model could be used to analyze the machining process and optimize the cutting parameter The orthogonal chatter theory was more complicated to apply

in the milling process due to the changing thickness direction, the rotating cutting force, and the intermittent cutting periods [3, 4] The engagement between cutter and workpiece was referred to the instantaneous chip thickness and was considered as a function of cutting force

in the cutting force mechanism The force modeling for the cutting process was first presented

by Martellotti [11], in which the chip thickness was calculated by a simple expression of sin

t

h  f  A mechanistic model for the force system in end milling was developed by Kline et al [12] This model is based on chip load, cut geometry, and the relationship between cutting forces and chip load, the uncut chip thickness was reportedly proportional to the cutting force Tsai proposed a predictive force model in end-milling process based on geometrical analysis [13] A generalized geometric model of milling cutters was reported in [14] During milling operation, tool runout has negative effects on the performance of the cutting operations There are some possibilities leading to cutter runout [2]: axis of rotation errors of the spindle, an offset between the holder centerline and spindle axis of rotation an offset between the tool centerline and holder centerline, and radii variation between cutter teeth The average chip thickness was increased corresponding to the presence of runout for those teeth actually engaged in the cut, the effects of runout errors on dynamic chip thickness were investigated to improve the accuracy of the model [15, 16] The basics of chatter vibration in orthogonal cutting operations described by an accurate model were proposed by Tobias [17]

A stability lobe diagram based on regenerative chatter theory is generally a prevailing method to predict and control chatter Figure 2.2 represents an example of stability lobe diagram The diagram was divided into stable and unstable zones by a stability boundary generated by a series of intersected stability lobes [2, 3] Stable machining conditions with higher material removal rates could be thus achieved by using this method [18, 19] The development of stability lobe diagram was however based on the restrictive assumption that the dynamics of spindle-tool system has not changed over the spindle speed range [5]

Figure 2.2 Stability lobe diagram [2]

The stability lobe diagram may inevitably misinterpret the important attributes of the machining processes which in fact always contain rich nonlinear characteristics [6, 7] Both time and frequency responses hence have to be monitored simultaneously

2.2 Signal Analysis Approaches

The methods used generally for evaluating the vibration signals involved the transformation of the cutting force signals from time to frequency domain are Fast Fourier Transform (FFT), Short-Time Fourier Transform (STFT), and Continuous Wavelet Transform (CWT) Through these methods, it was possible to view properties or characteristic information of the cutting process hidden in the time domain [20] Fourier transform (FT)

transforms the signal f t ( ) from the time domain to the frequency domain fˆ( ) described

Fourier transform considered as the basis of modern signal processing is well known

to process the stationary signals, especially when the feature components of signals are obvious in magnitude [21, 22] There are a lot of studies using FFT to identify chatter vibration in metal cutting Chatter frequencies are often high non-integer multiples of frequency with the higher magnitudes appeared in the Fourier spectrum, which was analyzed and shown in the previous literature [7, 23] The Fourier-based methods are nonetheless based

on the assumption that the analyzed signal is stationary in the time domain The globally inclusive of information makes it difficult to provide any information about individual frequency event of the signal It is accordingly not suitable for many non-stationary systems when their transient responses are analyzed

Previous research was reported that the proper method for processing time-varying non-stationary signal is time–frequency analysis, which includes STFT, Wigner–Ville distribution, Gabor transform, WT, wavelet packet transform, and HHT They are effective time–frequency analysis methods To overcome this limitation of FFT, Gabor introduced a sliding window function to the FT and obtained a localized time–frequency [9] Equation 2-

2 represents the resultant transform named short-time Fourier transform

Figure 2.3 Time-frequency resolution of STFT [9]

Continuous and discrete wavelet transform (WT) were recognized by many researchers as flexible and effective tools for processing both stationary and non-stationary signals, particularly in machining [25] Wavelet is advantageous for the analysis of non-stationary time series data arisen from metal cutting according to Kunpeng et al [26] because

it possesses better resolution capabilities in both time and frequency domains Decomposed signals were usually obtained by choosing proper wavelet coefficients for feature extraction

in order to detect chatter [27] The sudden amplitude change of decomposed vibration signals could be considered as a valid evidence to identify chatter [7] By applying FT to the decomposed signals, their Fourier spectrum revealed clear presence of chatter [28] Support vector machine and wavelet transform were used by Peng et al [29] to detect chatter They reported that once chatter occurs, the corresponding vibration signal generally exhibited non-stationarity, and the frequency components of the signal varied over time The behavior of the detail coefficients from WT was used to determine the presence of chatter [30] Nevertheless, WT is still difficult to determine the suitable wavelet function and decomposition levels to carry out the chatter fingerprint In addition, the decomposed signal might be distorted and result in limited accuracy

Hilbert-Huang transform (HHT) has recently become a superior tool to investigate machining vibration This method is a powerful tool for time-frequency analysis of nonlinear and non-stationary signals due to a time adaptive decomposition operation on the signals It

is moreover not limited by uncertainty principle because of its precise time and frequency resolutions [10, 31] The complex time series signals were decomposed into a set of intrinsic mode functions (IMFs) which represent simple oscillatory modes [32] The instantaneous frequency (IF) could be consequently obtained by applying Hilbert transform to each IMF Fourier spectrum of IMFs and the marginal spectrum obtained by HHT could be used to observe the main frequencies of the signal among those moments with chatter, as reported by Wei et al [6] and Peng et al [33] The approach, however, was still burdensome to select IMFs that contain rich chatter information for analyzing signals Wavelet packets were utilized by Cao et al [34] to decompose the signal into a set of narrow bands in order to enhance the result of HHT The dimensionless chatter indicators defined by the mean value and standard deviation of Hilbert-Huang spectrum were reported as effective tools for chatter detection This method is nevertheless troublesome to determine the suitable wavelet decomposition level and to select optimal mother wavelet The nonlinear dimensionless indicators, complexity and power spectral entropy were utilized as chatter indicators after being processed with ensemble empirical mode decomposition (EEMD) [35]

A dynamic cutting force model with tool runout error in the end-milling process was proposed to generate the force signals in this study The stability of the process was then investigated by the time-frequency analysis All three of STFT, WT and HHT were used for processing the cutting force signals The difference between stable and unstable conditions was clearly distinguished from the analysis Most of the energy concentrated at tooth passing frequency in stable cutting condition, while a large amount of energy emerged at high frequency when chatter occurs The standard deviation and energy ratio of IMFs were defined

as dimensionless indicators for detecting the presence of chatter effectively, even though the cutting forces are known as non-stationary signals and changed in different states The proposed method was validated by the stability lobe diagram created by impact testing and the machined surface topography along with surface roughness

Chapter 3

Dynamic Cutting Force Model

3.1 Regenerative Chatter Model

This section presents a dynamic cutting force mode in which the cutting edges are decomposed into a set of elements by discretizing along tool axis The geometry of general end mill with its schematic representation is provided in Fig 3.1

A model of the end mill which has two orthogonal degrees of freedom in x and y directions is considered in Fig 3.2 The vibration normal to the cut surface is used to determine the instantaneous chip thickness The trajectory of the cutter is assumed to be a circular path

Figure 3.1 The model of helical end mill geometry

Figure 3.2 Dynamic cutting force model

The tool vibration in x and y directions must be projected into surface normal direction

in order to evaluate chip thickness Those displacements can be represented following Eq

(3-1), in which the positive direction for n is out of the cut



     (3-2)

The lag angle due to helix angle of cutting edge, z  

( z ) tan r

   (3-3) The position angle varies with time and depends on spindle speed shown in Eq (3-4)

6

   (deg) (3-5) The instantaneous chip thickness for ith disk element, jth flute, and kth angular position is then represented by Eq (3-5)

( , , ) tsin( i j k) ( ) ( ) ( i j k)

h i j k  f   n t  n t g  (3-5) where the tooth period is  60/ (N ) t

If the effect of runout of cutting edge for ith disk element and jth flute is considered [16], the instantaneous runout is represented by ( , , ) i j k The instantaneous chip thickness for ith disk element, jth flute, and kth angular position is now determined following Eq (3-6)

( , , ) tsin( i j k) ( ) ( ) ( , , ) ( , 1, ) ( i j k)

h i j k  f   n t  n t   i j k  i j k g  (3-6) The switching function g(i j k, , ) describes the engagement between each tooth and workpiece in one revolution Its value equals to one when the jth tooth is engaged in the cut and equals to zero otherwise:

, , , ,

1,

,

,

( ) ( )

The cutting force components with respect to workpiece coordinate frame in x and y direction are now become as Eq (3-12)

,

cos sinsin cos

can expand into a Fourier series with the Fourier coefficients calculated by Eq (3-19)

x

t y

j t

The limiting chip width or critical depth of cut is evaluated from the real and imaginary part

of the eigenvalue    Re j Im, its relationship to  is expressed by Eq (3-27)

2 Im lim

 

1

2

ph c

3.2 Dynamic Cutting Force Model

The instantaneous chip thickness including tool runout errors was presented in Eq 6) The different cutting forces in radial, axial, and tangential directions regarding effects of plowing are written as [2]:

Figure 3.3 Flow chart of dynamic cutting force

Chapter 4

Experimental Setup and Model Verification

4.1 Overview and Aim

The accuracy of the cutting force model which has been investigated in previous section was evaluated by experiments This chapter introduces the machines and equipment used to conduct the experiments Three experiments were conducted Firstly, a set of experiments was conducted to find specific cutting force coefficients and cutting force angle Secondly, impact testing was carried out in order to investigate the tool tip dynamics in the system Finally, a series of end milling experiments was conducted to examine the dynamic force model and to investigate the proposed methodology for chatter detection All of the experimental setups are also described in detail for each kind of experiment

4.2 Experimental Setup

4.2.1 Machine, Cutter and Workpiece

Machining experiments were performed in a 3-axis CNC milling machine MV-154 (with Heidenhain TNC620 controller) as shown in Fig 4.1 The machine features are provided in Table 4-1 The workpiece was a block of Al6061-T6 with the size of 30x30x40mm Figure 4.2 represents an uncoated carbide end mill cutter used in the experiments Its diameter is 12 mm, with helix angle of 26° and two flutes The cutting tool parameters in detail are shown in Table 4-2

Table 4-1 Machine features

Max Speed Max

Figure 4.1 The three- axis CNC milling machine at NTUST

Figure 4.2 A carbide end mill

4.2.2 Cutting Force Measurement Equipment

Figure 4.3 illustrates a Kistler dynamometer (type 9129AA) which used to measure the cutting force signals directly The dynamometer was mounted between the workpiece and workbench Figures 4.4 shows the experimental setup

Figure 4.3 Kistler Dynamometer

Figure 4.4 Experimental setup

Table 4-2 Parameters of Tool Tool company Model Tool diameter

(mm)

Flutes Cutting edge length

(mm) Lasting

4.2.3 Surface Topography Measurement Equipment

In order to validate the proposed chatter detection method, the surface topography images and surface roughness are considered and utilized in the following sections The topographical images of the machined surface from cutting experiments were obtained using the optical microscope (Olympus BX51) with CCD sensor as shown in Fig 4.5

Figure 4.5 Olympus BX51 at NUTST

4.2.4 Surface Roughness Measurement Equipment

Figure 4.6 shows a MITUTOYO portable surface roughness tester which used to measure the surface roughness after cutting The model is SJ-210 with the electronic probe

type The measuring range in x axis is 17.5 mm, detector range in z axis is 360 μm with the

resolution of 0.002 μm The measurement speed of the experiment was setup at 0.5 mm/s

Figure 4.6 A Mitutoyo portable surface roughness tester

4.3 Cutting Force Coefficients

A set of experiments was conducted to find specific cutting force coefficients and cutting force angle for the milling force model Slot milling tests were conducted The start cutting angle s, exist cutting angles e , cutting angle βand resultant cutting force are modeled

in Fig 4.7 According to Schmitz [2], the average cutting force in x and y directions, in this

case, can be represented as shown in Eqs (4-1) and (4-2)

Figure 4.7 The modeling of slot milling

°



Six slot milling tests corresponding to different feed rates were conducted to achieve

average cutting force in x and y directions The experimental parameters are provided in Table

4-3 Figure 4.8 shows the measured cutting forced signal at a spindle speed of 2500 rpm, 1

mm axial depth of cut and feed rate of 0.03 mm/tooth in (a) x direction and (b) y direction Table 4-4 represents the average cutting force in x and y directions obtained from those slot

2500 1.0 12 100, 150, 175, 200, 225, 250

Figure 4.8 Measured cutting force signal at feed rate of 0.03 mm/tooth in (a) x direction and

(b) y direction

Table 4-4 Average cutting forces

Feed per tooth (mm/tooth)

0.02 0.03 0.035 0.04 0.045 0.05 𝐹𝑥⁡(N)

𝐹𝑦⁡(N)

The linear regression method which was summarized by Schmitz [2] is then utilized to determine four unknown cutting force coefficients in Eqs (4-1) and (4-2) Those values can

be obtained by Eqs (4-3) and (4-4)

The results of linear regression are shown in Fig 4.9 The four unknown cutting force

coefficients were carried out in Table 4-5 The specific cutting force coefficient K s and cutting force angle  are then calculated following Eq (3-11)

Figure 4.9 Relationship between feet per tooth and mean force

Table 4-5 Specific cutting force coefficients

N/mm deg N/mm2 N/mm2 N/mm N/mm

780 65.32 326.02 709.49 12.32 24.9

4 4 Experimental Design Parameters

In order to examine the dynamic cutting force model and to investigate the proposed methodology for chatter identification, a series of end milling experiments were conducted The cutting force signals in the feed direction were measured from the cutting experiments conducted with different sets of cutting parameters shown in Table 4-6, in which the feed rate was a constant value of 150 mm/min The DOCs ranged from 0.4 to 1.6 mm, while the spindle speeds ranged from 1200 to 5250 rpm The topographical images of the machined surface from cutting experiments were obtained by using the optical microscope (Olympus BX51)

0.0150 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 5

10 15 20 25 30 35 40