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Innovative Systems Design and Engineering www.iiste.org

ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)

Vol 3, No.11, 2012

17

Chatter Stability Characterization of a Plastic End-Milling CNC

Machine

Ozoegwu C.G1, Omenyi S.N1, Achebe C.H1, Chukwuneke J.L1*.

1Department of Mechanical Engineering, Nnamdi Azikiwe University, P.M.B 5025 Awka, Nigeria

1*E-mail of the corresponding author: jl.chukwuneke@unizik.edu.ng

Abstract

The desire to carry out this work arose from an observation during a practical work on a typical plast ic end milling

CNC machine. It was noticed that under certain conditions of cutting, operation of the machine became noisy with

increasing depth of cut and eventual perforation of workpiece therefore the basic aim is to generate stability

characterization of the machine in the form of a chart on the plane of cutting parameters on which stable operation is

demarcated from the unstable operation . In modelling this machine, a slot creating mode of operation is used since

the machine is mainly used for creating logos which are basically collection of slots. The significance of the resulting

stability chart lies in the result that the cause of the aforementioned noisy operation is due to unstable parameter

combination. For example a laboratory operation at spindle speed of 1500rpm and depth of cut of 1.5mm was noisy

while that at spindle speed of 1500rpm and depth of cut of 1mm was serene. The stability chart generated for the

system thus shows close agreement with both practice and theory. A unique impact this work will have on the

reading community will be in the area of validity of the resulting stability chart on the basis of MATLAB dde23

numerical simulation. The parameters of the end milling process are; tool mass tool natural

frequency damping factor and workpiece cutting coefficient .

Keywords: chatter, time history, Fargue approximation, Floquet theory, bifurcation

1. Introduction

Components of high dimensional integrity are in ever increasing need. Machine tools such as Lathe and Milling

machines are needed for production of such components. They would not perform effectively under highly disturbed

situations thus the need for vibration control in such machines. Achieving good surface finish and high p roductivity

are two opposed demands in machining operation. This means that ascertaining safe operation range for good

product, improved tool life and design of machine tools is necessary.

A typical machining process of major importance is the end-milling in which a machined surface that is at right angle

with the cutter axis results as shown Figure1. End milling cutters equipped with shanks for mounting on the spindle

are utilized for end milling.

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Vol 3, No.11, 2012

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Figure1. End-milling

Machine tool vibration is basically called chatter. Chatter invariably results whenever there is dynamic interaction

between the tool and the workpiece (project) of a milling process. Forced, self -exited and damped natural vibrations

combine to compound the dynamics of milling process. The forced vibration component is a periodic disturbance

that stems from regular engagement and dis-engagement of tool and workpiece. Regenerative effect is

underscored as the major cause of the self-exited vibrations (mechanical chatter) in machining (Stepan et al, 2003;

Insperger, 2002). Regenerative effect is a concept used to explain the sustained vibration occurring during machining

as resulting from cutting force variation due to vibration induced surface waviness. Arnold first suggested

regenerative effects as the potential cause of chatter and is now arguably considered the cause of detrimental type of

machine tool vibration (Davies et al, 1999). The effect of delayed position on the present position of the tool cause s

modelling of regenerative vibrations to result in delay differential equations (DDEs). Major milestones have been

made in the area of milling regenerative vibrations. Some of the most popular achievements in contemporary milling

machine vibration studies are stability charts. Among the Various methods utilized in their works in tracking the

milling stability boundary are; the finite element in time (Insperger et al, 2003; Butcher and Mann, 2011), Chebyshev

Polynomials (Butcher and Mann, 2011), semi-discretization (Insperger and Stepan, 2002) and fargue-type

approximation (Insperger, 2002; Insperger and Stepan, 2000). The aim in this work is to adapt some of these

achievements in the stability characterization of a Perspex or wood end milling CNC machine es timated in

(Ozoegwu, 2011) to have the following modal parameters; mass

milling CNC ma

Natural frequency

(Ozoegwu, 2011)

to

and damping factor

owing moda

for systematic operation. As part of contribution of this work

the resulting stability chart is validated with numerical integration of the governing DDE at selected points of the

parameter space by MATLAB dde23.

2. Mathematical Model

In the dynamical model shown in figure2, the tool is given a spindle speed in revolutions per minute while the

workpiece has a prescribed feed velocity

the

imparted on it via the worktable. The model being considered is a

milling tool with three teeth creating a slot through a workpiece. The parameters of the milling p rocess as depicted

on the dynamical model are;

crea

mass of tool,

roug

the equivalent viscous damping coefficient modelling the hysteretic

damping of the tool system and

ma

the stiffness of the tool system. These modal parameters could be extracted from

plot of the tool frequency response function in a scheme of experimental modal analysis. Figure2 is a single degree

of freedom vibration model of an end milling tool. Most encountered resonance in machining involves the

fundamental natural frequency thus single degree of freedom vibration is satisfactory when it is well separated from

the higher frequencies (Stepan, 1998). The wavy regenerative machined surface that sustains chatter vibration is

shown enlarged on figure 2.

girder

table

tool

workpiece

W

f

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Vol 3, No.11, 2012

19

Figure2. Dynamical model of milling

The free-body diagram for the tool dynamics is as shown in figure3.

Figure3. Free-body diagram of tool dynamics

The differential equation governing the motion of the tool as seen from the free-body diagram is

(1)

A tool-workpiece disposition as shown in figure4 is considered for the tooth of the tool. The component of

cutting force for the tool thus becomes

(2)

is the number teeth on the milling tool indexed with the values 1, 2, 3...... . The instantaneous angular

position of a tooth

teet

is

on the

. In this work

l in

dexe

is measured clockwise relative to the negative

stan

ta

axis to give

(3)

Where is the initial angular position of the tooth indexed . Screen or switching function for the tooth

could either have the values

osition of

depending on whether the tooth is active or not. Since the tool is creating

a slot on the workpiece as shown in figure2, the start and end angles will have the values

a slot on the workpi

respectively. Under this operating condition it becomes clear from the workpiece-tool

disposition of figure4 that the screen function could be expressed thus;

( )

vttxk -)(

÷

ø

ö

ç

è

æ-v

dt

dx

c

x

F

)(tx

2

2

dt

xd

m

x

F

m

k

c

tool

W

v

x

workpiece

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Vol 3, No.11, 2012

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(4)

Figure4. Milling tooth workpiece disposition

The tangential cutting force for the tooth is given by the non-linear law (Insperger, 2002)

(5)

where is depth of cut, is the cutting coefficient associated with the workpiece which is assumed to have the

value for Perspex for reasons given in (Ozoegwu, 2011), is the actual feed and is an

exponent that is usually less than one having a value of for the three-quarter rule. It is written in (Insperger,

2002) that empirical relationship connects the milling tangential and normal cutting forces in the works of Balint,

Bali and Tlusty according to the equation

(6)

The actual feed rate is the difference between present and one period delayed position of tool, thus

(7)

Equations (5), (6) and (7) taken together give

(8)

where is a periodic function. Introducing

Equation (8) into the equation of motion of the tool system (1) gives

x

y

jg

F

,tan

jnor m

F

,

W

)(t

j

q

v

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Vol 3, No.11, 2012

21

(9)

Suppose the motion of the tool is assumed to be a linear superposition of prescribed feed motion , tool response

with period

mo

due to periodic force of tool-workpiece interaction

on of

and perturbation

motion

(Insperger, 2002)

mainly due to regenerative effects then

(10)

Substitution of equation (10) into equation (9) givess

(11)

Without perturbation (that is ), equation (11) simplifies to

(12)

Equation (12) means that equation (11) becomes

(13)

Put in Taylor series about and linearizing equation (13) becomes

(14)

Where is the time-varying specific force variation (Insperger, 2002).

Equation (14) is re-written with the following compact notations; and to give a form

similar to damped delayed Mathieu equation (15) which is the equation of regenerative vibration of the system.

(15)

With the substitutions made, equation (15) could be put in state differential equation form as

(16)

Where for .

The natural frequency and damping ratio of the tool system are given in terms of modal parameters

respectively as . These modal parameters are easily extracted from experimental

plot of the tool frequency response function for forced single degree of freedom

vibration. As shown in figure2 the number of teeth on the tool considered is three. The tool tooth indexed 1 is

assumed to have an initial angular position at the beginning of milling feed, then equation (3)

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ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)

Vol 3, No.11, 2012

22

gives: , and giving rise to specific force variation

becoming

(17)

The damped delayed Mathieu equation (16) can be solved upon substitution of equation(17). Making use of the

parameters of the system; , and

, two MATLAB dde23 sample time histories of the system based on equation (16) together with the

determining cutting parameters ( ) are as shown in figure5. It is seen that the response of the tool at an

operating condition of spindle speed and depth of cut is asymptotically stable while

that at a spindle speed for same depth of cut is unstable. The perturbation history used is

where . It is found in (Ozoegwu, 2011) through numerical simulation

of MATLAB dde23 that condition of stability and instability is determined entirely by cutting parameter combination

of spindle speed and depth of cut. This means that any arbitrary choice of perturbation history whether determinist or

stochastic will not influence stability result of MATLAB dde23.

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Vol 3, No.11, 2012

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3. Stability Analysis of the End-Milling Process

Systematic selection process for cutting parameters that will result in good surface texture and integrity can only

result from proper mathematical modelling and stability analysis of milling process. Stable milling process is needed

for surface accuracy and integrity in which case the possibility of failure by fatigue, corrosion and wear of a

machined element is reduced by avoiding adverse alteration of machined surface. Stable milling operation could then

be deemed to be a form of proactive milling machine tool maintenance since tool-breaking and machine damaging

vibrations are jettisoned.

Stability investigation of the system presently considered is based on equation (15). Via Fargue-type approximation,

periodic delay-differential equation (DDE) could be transformed into periodic ordinary differential equation(ODE).

The resulting ODE is investigated based on the Floquet theory for stability analysis of periodic ODE. As proved by

Fargue, the following holds (Insperger, 2002).

(18)

Where is the Fargue weight function. In light of equation (18), equation (15)

becomes

(19)

Fargue approximation becomes For appreciably high finite , thus good

results still results from the equation

(20)

It is then seen that is the Fargue approximation parameter. State variables of form

(21)

where , enables the transformation of equation (20) into the state vector equation

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Vol 3, No.11, 2012

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(22)

Where the periodic coefficient matrix is given thus

(23)

According to the Floquet theory the time varying periodic ODE (43) has a solution of form;

(24)

Where is the fundamental matrix of the system which has been proven by Floquet to have the form

(25)

Where is a constant matrix. has the following properties; periodicity such that if is the principal period

then

ere

is a constan

and initial condition of identity matrix such that

odicity su

ch

. These two properties imply that

(26)

is called the principal or monodromy or Floquet transition matrix. At time of one period after the initial

condition equation (24) becomes

(27)

The eigenvalues of designated and eigenvalues of B designated are called the characteristic

multipliers and characteristic exponents respectively. It can be seen that the relationship between characteristic

multipliers

and

and corresponding characteristic exponents

can

is

(28)

If a characteristic exponent is given as then . It follows that

(29)

Stability of equation (15) requires all characteristic exponents to have negative real parts, that is . From

equation (29) the stability criterion for the system can also be stated to mean that all characteristic multipliers have

modulus less than one, that is

riterion for

. It can be implied from equation (29) that at critical operating conditions when

there exist characteristic multipliers such that

implied fr

that the corresponding characteristic exponents are pure

imaginary. This is a loss of stability condition that occurs when a least one characteristic multiplier is moving out of

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Vol 3, No.11, 2012

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a unit circle centred on the origin of the complex plane by either period two bifurcation or secondary Hopf

bifurcation(Insperger, 2002).

The Floquet fundamental matrix is difficult to achieve thus numerical approximations of the monodromy matrix are

used in stability analysis of periodic ODE’s. A typical method of achieving an estimate of the principal matrix is by

piecewise constant approximation of the of the time-varying coefficient matrix

ate of t

(Insperger, 2002). This

involves dividing the principal period

th

of the system into

ma

time intervals

rger, 2002

where

involves dividing the

and approximating the coefficient matrix of the system

me i

at the midpoint of each of

the time intervals. In this work equal time intervals

coefficient matrix

are used such that

(30)

Also is the constant time interval. By coupling the piecewise constant approximations of the coefficient

matrix , an approximate Floquet transition matrix becomes

(31)

Stability of milling process described by equation (15) could be derived by investigating the nature of the

eigenvalues of

illing p

for various parameter operations of the system. Any combination of cutting parameters that

result in any of the eigenvalues of the approximate monodromy matrix becoming greater than one is a chatter

operating point.

A critical look on the last row of the coefficient matrix of equation (30) shows that maximum row sum norm

approaches infinity as both and increase. This suggests ill-conditioning if is high enough to be useful in the

Fargue approximation. This type of numerical problem could be avoided by making use of the dimensionless time

(Insperger, 2002) to obtain an approximate piecewise constant matrix for the system as

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Vol 3, No.11, 2012

26

(32)

Row sum norm of interest now stems from row 2 of (32). It is clear that as and increase that the ratio

decreases thus if the stability investigation starts from a-not-too low spindle speed the computer computations

involving the coefficient matrix will be well-conditioned while requiring a much lower memory space. The

estimation of the Floquet transition matrix utilizing equal time intervals becomes

(33)

Equations (32) and (33) are the basis of stability analysis of the milling process under consideration. Stability

boundary curve is tracked as the locus of points on the plane of the cutting parameters (

onsi

and

ti

on

) at which

maximum magnitude of the eigenvalues is one.

4. Results

Making use of Fargue approximation parameter , principal period and dividing integer for piecewise

constant approximation , the stability chart generated for the system under study with parameters;

is as shown in figure 6.

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Vol 3, No.11, 2012

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spindle speed in rpm

depth of cut in meters

0.5 1 1.5 2 2.5 3

x 104

0

0.005

0.01

0.015

First limitation of this study is that the resulting stability chart are time costly in that it takes not less than 38 hours of

computation time of a modern laptop with processing speed of 2.10Gh. Another obvious limitation of this work is

poor reliability of the stability chart at spindle speeds below 5000rpm. This stems from the method of stability

analysis used. On this chart is a delineation the stable cutting domains from the unstable ones on the plane of

cutting parameters; spindle speed

a d

and depth of cut

ble

. The region of the chart below the boundary curve is the

stable region while the one above it is the unstable region as shown labelled in figure6. Sample stable operating

conditions as validated by MATLAB dde23 solution of equation (16) are shown marked with star while the unstable

(chatter) conditions are marked with diamond on the stability chart. The close agreement between MATLAB time

histories of equation (16) with the chart is a testimony of its accuracy.

5. Conclusion

The operational spindle speed range of of the studied system falls within the poorly approximated

low spindle speed domain of the chart. It will be noticed from the chart that chatter will be very unlikely for a depth

of cut not more than 1mm. This result is in line with the specification on the actual machine. This could explain why

laboratory operation became noisy at spindle speed of 1500rpm and depth of cut of 1.5mm. If the manufacturers and

designers of the plastic end milling CNC machine under study were to provide operators and programmers of the

machine with this chart, they (the manufacturers and designers) would be free to increase the power rating of the

machine to achieve higher spindle speed range that allows for stable operation at deeper depths of cut. For example,

under full immersion condition the machine operation at spindle speed

peratio

n at deepe

and depth of cut

or example,

stable

unstable

stable

unstable

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Vol 3, No.11, 2012

28

or spindle speed and depth of cut is asymptotically stable. This chart is thus seen to

have the potential to culminate in improved productivity of the machine from the point of design to usage.

References

Butcher, E., & Mann, B. P. Stability Analysis and Control of Linear Periodic Delayed Systems using Chebyshev and

Temporal Finite Element Methods. Retrieved July 6, 2011, from

http://mae.nmsu.edu/faculty/eab/bookchapter_final.pdf

Insperger, T. Stability Analysis of Periodic Delay-Differential Equations Modelling Machine Tool Chatter (Doctoral

dissertation, Budapest University of Technology and Economics, 2002).

Insperger, T., Mann, B. P., Stepan, G., & Bayly, P. V. (2006). Stability of up-milling and down-milling, part 1:

alternative analytical methods. International Journal of Machine Tools & Manufacture 43, 25–34.

Insperger, T., & Stepan, G. (2000). Stability of Milling Process. PERIODICA POLYTECHNICA SER. MECH.

ENG. 44(1), 47–57.

Insperger, T., & Stepan, G. (2002). Semi-discretization method for delayed systems. International Journal For

Numerical Methods In Engineering, 55, 503–518. doi: 10.1002/nme.505

National Institute of Standards and Technology. (1999) High-Speed Machining Processes: Dynamics of Multiple

Scales, (Pub. No. 20899), 100 Bureau Drive, Gaithersburg USA: Davies, M.A., Burns, T. J., & Schmitz, T. L.

Ozoegwu, C. G. (2011). Chatter of Plastic Milling CNC Machine. Unpublished Master’s thesis, Nnamdi Azikiwe

University Awka, Nigeria.

Stepan, G. (1998). Delay-differential Equation Models for Machine Tool Chatter: In F. C. Moon (Ed.), Nonlinear

Dynamics of Material Processing and Manufacturing (p. 165-192). New York: John Wiley & Sons.

Stépán, G., Szalai, R. & Insperger, T. (2003). Nonlinear Dynamics of High-Speed Milling Subjected to Regenerative

Effect: In Gunther Radons (Ed.), Nonlinear Dynamics of Production Systems (pp. 1-2). New York: Wiley-VCH.