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Innovative Systems Design and Engineering Chatter Stability Characterization of a Plastic End-Milling CNC Machine

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Abstract

The desire to carry out this work arose from an observation during a practical work on a typical plastic 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 .
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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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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
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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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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(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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(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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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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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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(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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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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(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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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
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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.
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Article
Full-text available
A new technique for determining the stability conditions of delayed differential equations with time- periodic coefficient is presented. The method is based on a special kind of approximation of the delayed term. As a practical application, the stability of the milling process with respect to the technological parameters is analysed, and an unstable zone in the domain of high cutting speed is shown.
Chapter
Full-text available
This chapter provides a brief literature review together with detailed descrip-tions of the authors' work on the stability and control of systems represented by linear time-periodic delay-differential equations using the Chebyshev and temporal finite element analysis (TFEA) techniques. Here, the theory and examples assume that there is a single fixed discrete delay which is equal to the principal period. Two Chebyshev-based methods, Chebyshev polynomial expansion and collocation, are developed. After the computational techniques are explained in detail with illustrative examples, the TFEA and Chebyshev collocation techniques are both applied for comparison purposes to determine the stability boundaries of a single degree-of-freedom model of chatter vibra-tions in the milling process. Subsequently, it is shown how the Chebyshev polynomial expansion method is utilized for both optimal and delayed state feedback control of periodic delayed systems.
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The dynamic stability of the milling process is investigated through a single degree of freedom mechanical model. Two alternative analytical methods are introduced, both based on finite dimensional discrete map representations of the governing time periodic delay-differential equation. Stability charts and chatter frequencies are determined for partial immersion up-and down-milling, and for full immersion milling operations. A special duality property of stability regions for up-and down-milling is shown and explained.
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The paper presents an efficient numerical method for the stability analysis of linear delayed systems. The method is based on a special kind of discretization technique with respect to the past effect only. The resulting approximate system is delayed and also time periodic, but still, it can be transformed analytically into a high-dimensional linear discrete system. The method is applied to determine the stability charts of the Mathieu equation with continuous time delay. Copyright © 2002 John Wiley & Sons, Ltd.
Delay-differential Equation Models for Machine Tool Chatter
  • G Stepan
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.
Chatter of Plastic Milling CNC Machine
  • C G Ozoegwu
Ozoegwu, C. G. (2011). Chatter of Plastic Milling CNC Machine. Unpublished Master's thesis, Nnamdi Azikiwe University Awka, Nigeria.
Analysis of Periodic Delay-Differential Equations Modelling Machine Tool Chatter (Doctoral dissertation
  • T Insperger
  • Stability
Insperger, T. Stability Analysis of Periodic Delay-Differential Equations Modelling Machine Tool Chatter (Doctoral dissertation, Budapest University of Technology and Economics, 2002).
Semi-discretization method for delayed systems
  • T Insperger
  • G Stepan
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