Overview of stability analysis in
machining processes
M.J.J. van Ballegooijen B.Sc. (0533569)
DCT 2008.134
Traineeship report
Coach(es):
Z. Dombóvári, PhD candidate
Supervisor:
Prof. H. Nijmeijer
Prof. Y. Altintas
Eindhoven University of Technology
Department Mechanical Engineering
Dynamics and Control Group
Eindhoven, December, 2008
Summary
The main purpose of this project is to investigate the stability of a two degree-of-freedom machining process, by taking process damping into account and using semi-discretization to investigate
its equation of motion. By constructing a so-called stability chart in which the stability of the cutting process is indicated for a range of depth of cuts and spindle speeds, the stability for a certain
cutting process can be determined. The stability chart is divided into a stable area and an unstable
area, which are separated by stability lobes.
First a model with one degree of freedom (DOF) is investigated, namely the orthogonal cutting
process. The model describes the regenerative effect of a machining process. This effect, the
most fundamental effect during machining, occurs due to the returning pattern on the workpiece
induced by the tool vibration. Thus, the current tool position and the tool position in the past are
related. The lag between the two tool positions can be taken into account by one constant delay
term in the equation of motion. The resulting equation of motion thus contains a delay term and
is therefore called a delay differential equation.
After the equation of motion is formulated for orthogonal cutting, the influence of the depth
of cut and the spindle speed of the tool is investigated analytically and the stability lobes are
constructed. This analytical method is appropriate to understand the major relations between
the cutting parameters in orthogonal cutting, however it is not suitable for cutting processes with
two or more degrees of freedom. For these models, semi-discretization is used to investigate the
delay equation of motion numerically. Semi-discretization is a robust method with which linear
first order delay differential equations can be investigated. This method discretizes the infinite
dimensional phase space of the system and formulates a finite linear map. The accuracy of this
method is related to the step size chosen to discretize the infinite dimensional phase space. The
step size strongly depends on the ratio between the constant delay and the period of the occurring
periodic orbit on the stability border.
One effect that should be taken into account when considering metal cutting processes is process
damping. Process damping can result in additional damping and inertia in the cutting process
at low spindle speeds; the experimentally found stability lobes lay much higher in the stability
chart than the theoretically found lobes. Although much research has been undertaken, little is
known about the cause of the process damping. Therefore, several models are used to describe
the process damping. One of them assumes that it is described by the addition of one linear
damping term and one linear inertia term to the equation of motion of the orthogonal cutting
process. The resulting equation of motion is investigated by semi-discretization and the resulting
stability chart reveals that the stable region increases at low spindle speeds.
The one DOF theory of orthogonal cutting with process damping is used in a two DOF model
that can be used to describe a milling process. In milling, the teeth of the tool enter and exit the
workpiece, resulting in the state matrix that determines the stability of the system being timedependent. Therefore, this state matrix is calculated by numerical integration. The stability of
this cutting process depends also on the number of teeth of the tool: an increase in the tooth
number results in a decrease of the stable region. Process damping is added to this model by
assuming that this effect can be described by using the spatial derivative and the second spatial
derivative of the radial position of the tool.
ii
Samenvatting
Het doel van dit onderzoek is het bepalen van de stabiliteit van een verspaningsproces met twee
graden van vrijheid, waarbij het effect van proces demping in de modellering is opgenomen en
gebruik gemaakt wordt van semidiscretisatie voor het oplossen van de bewegingsvergelijkingen.
Met behulp van een zogeheten stabiliteitsgrafiek, waarin de stabiliteit van het proces is aangegeven voor een bereik van snijdieptes en -snelheden, kan de stabiliteit van een verspaningsproces
worden bepaald. De grafiek is in een stabiel en een instabiel gebied verdeeld, welke zijn gescheiden door stabiliteitslobben.
Allereerst is een model onderzocht met één vrijheidsgraad, namelijk het orthogonale verspaningsproces. Het model beschrijft het zogenaamde regenerative effect van een verspaningsproces. Dit is het belangrijkste effect dat optreedt tijdens het verspanen en is te wijten aan het
terugkerende patroon op het werkstuk dat veroorzaakt wordt door trillingen van de beitel. Hierdoor zijn de huidige beitel positie en de positie in het verleden aan elkaar gerelateerd. Deze
relatie wordt meegenomen in de bewegingsvergelijking door het toevoegen van een constante
vertraging. De bewegingsvergelijking heeft dus een zogeheten vertragingsterm, en wordt nu delay differentiaalvergelijking genoemd.
Na het afleiden van de bewegingsvergelijking voor het orthogonale verspaningsproces, wordt de
invloed van de snijdiepte en snijsnelheid op de stabiliteit van het systeem analytisch bepaald, en
worden de stabilteitsgrafieken getekend. Deze analytische manier is een goede manier om de grafiek te tekenen voor een eenvoudig model met slechts één vrijheidsgraad, maar kan niet gebruikt
worden als het model complexer wordt. In dergelijke gevallen kan semidiscretisatie worden gebruikt om de delay differentiaalvergelijkingen numeriek op te lossen. Semidiscretisatie is een
robuuste methode waarmee lineaire eerste orde delay differentiaalvergelijkingen kunnen worden
geanalyseerd. Deze methode discretiseert de oneindig dimensionale toestandsruimte van het
systeem en formuleert een eindige lineaire vergelijking. De nauwkeurigheid van deze methode hangt samen met de stapgrootte die is gekozen voor de discretisatie van de vertragingsterm.
Deze stapgrootte is afhankelijk van de ratio tussen de constante vertraging en de periode van de
periodieke baan op de stabiliteitsgrens.
Een effect dat mee genomen moet worden bij het onderzoek naar stabiliteit in verspaningsprocessen is proces demping. Proces demping kan bij lage snijsnelheden extra demping en inertia in
het verspaningsproces teweeg brengen: de experimentele stabiliteitslobben liggen namelijk veel
hoger in de stabiliteitsgrafiek dan de theoretisch gevonden stabiliteitslobben. Hoewel er al veel
onderzoek is gedaan, is er weinig bekend over de oorzaak van proces demping. Daarom worden
momenteel nog verschillende modellen gebruikt voor het beschrijven van proces demping. Een
van deze modellen neemt aan dat proces demping kan worden beschreven door toevoeging van
lineaire demping en lineaire inertia aan de bewegingsvergelijking van het orthogonale verspaningsproces. De hieruit volgende bewegingsvergelijking is geanalyseerd met semidiscretisatie
en de resulterende stabiliteitsgrafiek wijst uit dat het stabiele gebied is vergroot bij kleine snijsnelheden.
Het model van het orthogonale verspaningsproces met één vrijheidsgraad wordt gebruikt in een
model met twee vrijheidsgraden, waarmee een freesproces beschreven wordt. Bij frezen is de
systeemmatrix waaruit de stabiliteit van het systeem wordt bepaald tijdsafhankelijk, doordat de
tanden van de frees het werkstuk ingaan en ook weer verlaten. Daarom wordt deze matrix bepaald
iii
door numerieke integratie. De stabiliteit van het freesproces hangt af van het aantal tanden op de
frees: een vergroting van het aantal tanden resulteert in een verkleining van de stabiele gebieden
in de stabiliteitsgrafiek. Ook aan dit model wordt proces demping toegevoegd, door aan te nemen
dat proces demping in een freesproces kan worden benaderd door de ruimtelijke afgeleide en de
tweede ruimtelijke afgeleide van de radiale positie van de frees.
iv
Contents
1
Introduction
1
1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Aim of project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Outline of report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2 Introduction to Stability Charts
3
3
Basic Linear Stability Chart of the Orthogonal Cutting Process
5
3.1
The orthogonal cutting process . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3.2
Equation of motion for an orthogonal cutting process . . . . . . . . . . . . . . . .
6
3.3
Derivation of the dimensionless spindle speed and depth of cut . . . . . . . . . .
7
3.4
Analysis of the linear stability chart . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.5
Implementation of the basic lobe structure . . . . . . . . . . . . . . . . . . . . .
9
3.6
Validation of the stability lobe structure . . . . . . . . . . . . . . . . . . . . . . .
11
3.7
Analytical analysis of intersection of two lobes . . . . . . . . . . . . . . . . . . . .
12
3.8
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4 Semi-Discretization
Discretization of delay term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
4.2 Stability investigation using a linear map . . . . . . . . . . . . . . . . . . . . . . .
17
4.3
Implementation of the semi-discretization method . . . . . . . . . . . . . . . . .
18
4.3.1
Choice of step size m . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4.4 The motion of the eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
Process Damping in Orthogonal Cutting Processes
21
5.1
Addition of process damping in equation of motion . . . . . . . . . . . . . . . . .
21
5.2
Implementation of process damping in semi-discretization code . . . . . . . . . .
23
4.1
5
15
6 Stability Chart for Milling
25
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
6.2 Equations of motion for milling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
6.3
Stability lobes using semi-discretization . . . . . . . . . . . . . . . . . . . . . . .
29
6.4 Process damping for milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
6.1
v
6.5
7
Construction of stability chart for milling with process damping . . . . . . . . . .
34
Conclusions and recommendations
35
7.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
7.2
Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Bibliography
36
A Derivation of W (ω) and Ω(ω)
37
B Derivation of W (ω) and Ω(ω) with ζ = 0
39
C Proof of reordering exponential matrix
40
D Background of simplification of semi-discretization method
42
E Derivation of the dimensionless equation of motion for process damping
44
F Matrices Ipd (t), Cpd (t) and Kpd (t)
46
G Parameters used for stability charts for milling
47
Acknowledgements
48
vi
1
Introduction
For the manufacturing of metal parts of e.g. cars and planes, turning and milling are two commonly used metal cutting processes. In turning, the workpiece rotates, while the fixed tool cuts
in the workpiece; in milling the tool rotates and cuts in the fixed workpiece. During the past
decades, much research has been undertaken in the field of stability during a turning or milling
process. One laboratory that investigates this stability problem is the Manufacturing Automation
Laboratory at the Department of Mechanical Engineering of the University of British Columbia
in Vancouver, Canada. This research project has been undertaken in the MAL.
1.1
Background
During a metal cutting process, a tool cuts in the workpiece with a particular depth of cut, and
either the tool or the workpiece has a rotational speed, resulting in a cutting speed. The feed of
the cutting process is determined by a translational motion of the workpiece or the tool. In the
ideal situation, the tool cuts in the workpiece and leaves a perfectly smooth surface. However,
in every cutting process vibrations occur, and the tool cut results in a wave on the surface of the
workpiece. This wave can magnify the tool vibrations which results in such heavy tool vibrations
that the tool edge can be released from the workpiece. This, from a technical point of view,
unstable tool vibration is called chatter.
One important issue in cutting processes is the so-called process damping. Although the exact
cause of process damping is still unknown, it is recently accepted that the effect is induced by the
friction force between the flank edge of the tool and the workpiece. This force induces velocity
dependent excitation of the tool, which increases the damping of the system.
The stability of a cutting process with a certain damping ratio, natural frequency and cutting
parameters depends on the depth of cut and the spindle speed of either the tool or the workpiece.
To determine the stability for a certain cutting process, a stability chart is drawn. In this chart
so-called lobes are plotted which indicate the border between the stable region and the unstable
of the cutting process.
In order to construct this stability chart, first the equation of motion for the cutting process is
derived. This equation is not an ordinary differential equation (ODE) because the tool position
depends on the past shape of the surface. The surface of the workpiece is cut by the tool one
rotation of the tool or workpiece earlier. So, the current tool position depends on the history of
the position of the tool. This results in a delay term in the equation of motion, and the resulting
(second order) differential equation is a so-called a delay differential equation (DDE).
A delay differential equation has an infinite dimensional state space and an infinite number of
characteristic roots. Therefore it is, contrary to ODEs, not possible to directly perform analyses,
although reliable numerical solvers exist. The structural behavior of a DDE can be analyzed using
semi-discretization, which discretizes and approximates the delay terms. The approximation is
used to treat the DDE as a high, but finite, dimensional ODE system.
1
1.2
Aim of project
In the Manufacturing Automation Laboratory at the University of British Columbia, research is
done in the field of chatter in metal cutting processes in general and process damping in particular. The process damping is investigated especially in turning, but there is only few information
about process damping in milling. Therefore, the aim of this report is to construct stability lobes
for process damping in milling using one of the possible explanations of this effect. The DDEs
are solved using the semi-discretization method.
1.3
Outline of report
The report is organized as follows: first an introduction to stability charts in cutting processes
is given in Chapter 2, after which a stability chart is constructed for the one degree-of-freedom
orthogonal cutting process in Chapter 3. Chapter 4 explains how the delay differential equation that describes the orthogonal cutting process is solved by discretizing the delay term. In
Chapter 5 process damping is added to the orthogonal cutting process. Having explained semidiscretization and process damping for the one degree-of-freedom model, the theory is extended
to a two degree-of-freedom model used to describe milling in Chapter 6. Lastly, Chapter 7 discusses the conclusions that are drawn and the recommendations for future research.
2
2
Introduction to Stability Charts
The stability of a cutting process can be determined by the construction of a stability chart. In this
chart, the border between the stable and the unstable region is indicated. This chapter explains
the stability chart and its shape.
During a metal cutting process, a tool cuts in the workpiece with a depth of cut and a certain
velocity with respect to the workpiece, see Figure 2.1. The graph on the left side is a schematic
picture of a turning process in which the workpiece rotates with spindle speed Ω, the figure on
the right shows a milling operation in which the tool rotates with spindle speed Ω. The depth
of cut w is also indicated in both graphs. Furthermore, in both graphs the regenerative effect is
visible: the tool cuts on the wavy workpiece surface, induced by tool vibrations in the past.
W
work-piece
work-piece
W
w h0
2p
w
w
v
tool
tool
Figure 2.1: Two metal cutting processes: turning (left) and milling (right)
In certain positions of the parameter space of the spindle speed and depth of cut, the system
can become unstable. To determine the stability for a certain cutting process, a stability chart is
drawn. In this chart so-called lobes are plotted that indicate the border between the stable and the
unstable region of the cutting process, see Figure 2.2.
Figure 2.2: A stability chart with four lobes that indicate the stability limit of an orthogonal
cutting process
In the figure, a stable and an unstable region are indicated; the border between the two region
consists of several lobes. The stable region of a cutting process is the area underneath the lobes,
3
the unstable region is the area above the lobes. In Figure 2.2, four separate lobes are indicated by
four solid lines. Each lobe has a vertical asymptote on its left side, indicated in the figure by a thin
dashed line, and a minimum point, which is located on the thick dashed line. This thick dashed
line is the so-called minimum depth of cut Wmin . If a depth of cut below this line is chosen, the
cutting process is always stable, regardless of the spindle speed.
As mentioned before, the lobes indicate the border between the asymptotically stable and the unstable region. So, on the border the system is critically stable and the tool vibrates with vibration
frequency ω. For the critically stable orthogonal cutting process two equations can be derived
analytically: one equation for the depth of cut W and a second equation for the spindle speed Ω,
both as function of ω. The function Ω(ω) results from a trigonometric function, and, therefore,
the periodic solution contains a term that indicates the recurrence: NL π. In this term, NL is the
lobe number and NL = 1 indicates the rightmost lobe in the stability chart. Using these two
functions, the lobes can be constructed analytically by evaluating the two functions for a range of
vibration frequencies, and lobe numbers NL = 1, 2, ....
The derivation of the two functions W (ω) and Ω(ω) is discussed in Section 3.3 and the lobes are
constructed analytically in Section 3.5.
4
3
Basic Linear Stability Chart of the Orthogonal Cutting Process
This chapter deals with the explanation and construction of stability charts for the orthogonal cutting process. In these charts so-called lobes are plotted, which indicate the border between stable
stationary cutting and the unstable cutting process. The lobes can be constructed analytically by
calculating the dimensionless spindle speed and the dimensionless depth of cut for a range of
vibration frequencies. To this end, first the orthogonal cutting process is explained, after which
the equation of motion of this particular cutting process is derived, which is used to investigate
the analytical part of the construction of the lobe structure. Then, the structure of the lobes is
implemented in a Matlab code and the plotted lobes are validated. Lastly, the system’s behavior
in an intersection of two lobes is investigated analytically.
3.1
The orthogonal cutting process
This section explains the orthogonal cutting process. In Figure 3.1 the orthogonal cutting process
on the left hand side is projected schematically in the right side of the figure. This picture shows
the workpiece that rotates with spindle speed Ω and the tool that cuts in the workpiece with depth
of cut w. The tool has mass m, and is assumed to be connected to the toolholder by a spring with
spring stiffness k and a damper with damping coefficient c. The tool cut results in a chip with
desired chip thickness h0 and momentary chip thickness h(t). The momentary position of the
tool is x(t) and as the workpiece rotates, the current tool position depends also on the position of
the tool τ earlier, x(t − τ ). This constant delay τ is the time it takes the workpiece to rotate once.
The dependency of the current tool position on the previous tool position creates a delay term in
the equation of motion of the orthogonal cutting process, see Section 3.2. In this figure, the tool
work-piece
k
c
w h0
2p
x(t-t)
w
m
chip
h0
h(t)
W
tool
F
x
x(t)
W
work-piece
tool
Figure 3.1: The schematic representation (right) of an orthogonal cutting process (left)
position is considered only in x-direction, so this is a one degree-of-freedom (1 DOF) model. As
the direction of the tool vibration is perpendicular to the direction of the cutting edge of the tool,
this is called an orthogonal cutting process [2]. Section 3.2 derives the equation of motion for an
orthogonal cutting process.
5
3.2
Equation of motion for an orthogonal cutting process
Section 3.1 explained the orthogonal cutting process and this section derives the equation of motion for the one degree-of-freedom model that describes this cutting process. To this end, the
system is modeled as a 1 DOF linear mass-spring-damper system excited by a nonlinear cutting
force, [8]:
(3.1)
mẍ(t) + cẋ(t) + k(q0 + x(t)) = wfa (h(t)).
In this equation m, c and k represent the mass [kg], damping [Ns/m] and the spring stiffness
[N/m] respectively. Furthermore, q0 is the static deflection [m] of the spring caused by the static
part of the cutting force, w is the depth of cut [m] and fa (h(t)) is the resulting unit force [N/m]
as function of the momentary chip thickness h(t) [m]. This momentary chip thickness can be
derived form Figure 3.1 and depends on the current tool position x(t) [m], the position at the
previous revolution x(t − τ ) and the desired chip thickness h0 [8]:
(3.2)
h(t) = x(t − τ ) − x(t) + h0 .
In this equation, τ [s] is the delay of the vibration and is equal to the duration of one revolution
of the workpiece, i.e. τ = 2π/Ω, with Ω the spindle speed [rad/s]. The fa (h(t)) term can be
approximated by the Taylor series around the desired chip thickness h0 :
fa (h(t)) = fa0 +
∂fa (h(t) − h0 ) + H.O.T.
∂h h=h0
(3.3)
The higher order terms (H.O.T.) can be neglected and fa (h(t)) is expressed as
(3.4)
fa (h(t)) = fa0 + k1 (h(t) − h0 ),
with fa0 the initial unit force due to static deflection, so wfa0 = kq0 , and k1 =
∂fa ∂h h=h
the
0
cutting coefficient. The second term in (3.4) is the dynamic force term. Substitution of (3.2) and
(3.4) into (3.1) obtains:
mẍ(t) + cẋ(t) + (k + wk1 )x(t) = wk1 x(t − τ ).
(3.5)
This is the linear equation of motion for the orthogonal cutting process and it is now transformed
into a dimensionless parameter space. To this end, first the whole equation is divided by m. Then,
the following two expressions are used:
c
√
,
2 km
r
k
:=
,
m
(3.6a)
ζ :=
ωn
(3.6b)
with ζ the (dimensionless) damping ratio and ωn the natural angular frequency [rad/s] of the
system. For general turning operations, ζ is small; usually ∼ 0.02 [2]. (3.5) is now rewritten into:
k1 w
k1 w
2
x(t) =
x(t − τ ).
(3.7)
ẍ(t) + 2ζωn ẋ(t) + ωn +
m
m
6
This equation is dimensionless after the following substitutions:
t̃ := t ωn ,
τ̃
:= τ ωn ,
(3.8a)
→ dt̃ = dt ωn ,
(3.8b)
in which t̃ is the dimensionless time and τ̃ is the dimensionless delay. After dividing (3.7) by ωn2 ,
and introducing the dimensionless depth of cut W̃ = mk1ωw2 , a dimensionless equation of motion
n
that depends on only two system parameters results:
ẍ(t̃) + 2ζ ẋ(t̃) + (1 + W̃ )x(t̃) = W̃ x(t̃ − τ̃ ).
(3.9)
From now on, all equations in this chapter are dimensionless, therefore the tilde is dropped for
convenience.
In this section, the dimensionless second order delay differential equation was derived for an
orthogonal cutting process. This equation is used in the next section to derive the separate equations for the dimensionless depth of cut W and the dimensionless spindle speed Ω as function
of the vibration frequency ω.
3.3
Derivation of the dimensionless spindle speed and depth of cut
In Section 3.2 the dimensionless equation of motion was derived for the orthogonal cutting process and its characteristic equation is used in this section to formulate the equations of the dimensionless depth of cut W and the dimensionless spindle speed Ω. To derive the characteristic
equation of (3.9), it is assumed that the general solution of (3.9) has the following form [8]:
x(t) = beλt ,
(3.10)
with b a coefficient dependent on the initial conditions and λ a certain eigenvalue of the system.
This formulation is used to derive ẋ(t) and ẍ(t), which are substituted into (3.9). Dividing all
terms by beλt one obtains the characteristic equation D(λ) = 0:
D(λ) = λ2 + 2ζλ + 1 + W − W e−λτ .
(3.11)
This equation has infinite many solutions due to the e−λτ term.
As the borders between the stable and the unstable regions are interesting, at which the real part
of an eigenvalue is zero, iω is substituted for λ. Using Euler’s formula in complex analyses, the
following equation results:
D(ω) = −ω 2 + 2ζωi + 1 + W − W cos(ωτ ) + iW sin(ωτ ).
(3.12)
With this equation, an expression for the dimensionless depth of cut W on the border between
the stable and the unstable region can be derived. To this end, (3.12) is split into the real and the
imaginary part and as both should be zero, the following two equations are obtained [8]:
− ω2 + 1 + W
(3.13a)
= W cos(ωτ ),
(3.13b)
2ζω = −W sin(ωτ ).
When both equations are squared and then added together, the following expression for the dimensionless depth of cut W of the critically stable cutting process as function of ω can be derived:
W (ω) =
(ω 2 − 1)2 + 4ζ 2 ω 2
.
2(ω 2 − 1)
(3.14)
7
From this equation it follows that the dimensionless vibration frequency ω should be larger than
1 for this equation to make sense, as the depth of cut must be real and positive.
To obtain a derivation for the dimensionless delay τ (ω), the following expressions for sin(ωτ )
and cos(ωτ ) are used:
sin(ωτ ) =
2 tan( ωτ
2 )
,
2
1 + tan ( ωτ
2 )
(3.15a)
cos(ωτ ) =
1 − tan2 ( ωτ
2 )
ωτ .
2
1 + tan ( 2 )
(3.15b)
By substitution of (3.15a) into (3.13b) a new expression for W (ω) is obtained:
W (ω) = −
1 + tan2 ( ωτ
2 )
ζω
ωτ
tan( 2 )
(3.16)
Substituting this equation into (3.13a) obtains the following formula for the dimensionless delay,
[8]:
2
1 − ω2
τ (ω) =
arctan
+ NL π .
(3.17)
ω
2ζω
Having an expression for the dimensionless delay, the expression for the dimensionless spindle
speed Ω(ω) is obtained too:
−1
1 − ω2
2π
= πω arctan
+ NL π
.
Ω(ω) =
τ (ω)
2ζω
(3.18)
In this equation, NL is the integer number that indicates the number of the lobe, the border
between the stable and the unstable region of the stability chart in Figure 2.2. From (3.18) it
follows that NL ≥ 1 as a negative spindle speed results when NL = 0. Therefore, the numbering
of the lobes starts at 1.
3.4
Analysis of the linear stability chart
In Section 3.3, the equations for the dimensionless depth of cut and the dimensionless spindle
speed were derived. These equations can be used to plot the linear stability chart. However,
without implementing the equations in some computer code, it is in fact possible to roughly
sketch the shape of the stability lobes if the damping ratio ζ of the system is known. Namely, with
this information, the minimum value of all lobes can be approximated together with the vertical
asymptotes, see also Figure 2.2.
First, the minimum value of a lobe is calculated by differentiating the equation for the dimensionless depth of cut W (ω), (3.14), with respect to the dimensionless frequency ω. The frequency
for which the derivative equals zero is called the critical frequency ωcrit , and the corresponding
depth of cut is the minimum depth of cut Wmin as for all cutting depths below this value the
cutting process described by (3.9) is always stable. The expression for the critical frequency is
derived step by step in Appendix A and is given as:
p
ωcrit = 1 + 2ζ.
(3.19)
8
Substitution of this equation into (3.14) yields the minimum value of the dimensionless depth of
cut for all lobes:
(3.20)
Wmin = 2ζ(1 + ζ).
By substituting (3.19) into (3.18) and assuming that ζ is very small, which is true for orthogonal
cutting processes [2], obtains the value for the dimensionless spindle speed in the minimum
point for each lobe NL = 1, 2, ... is obtained:
Ωmin = Ω(ωcrit ) ≃
1
.
NL − 41
(3.21)
The derivation of this equation is also found in Appendix A.
Second, the vertical asymptote of each lobe is calculated by using the fact that the dimensionless
depth of cut reaches infinity when ω approaches 1. Substituting ω = 1 into (3.18), the location of
the vertical asymptote is determined for each lobe:
lim Ω(ω) =
ω→1
1
.
NL
(3.22)
The location of the asymptotes and minimum points of a lobe was explained in this section, with
which the rough shape of the stability chart for a cutting process with a certain damping ratio ζ
can be drawn. In the next section, the equations of Section 3.2 are used to plot the stability lobes.
3.5
Implementation of the basic lobe structure
The stability lobes can be plotted by evaluating the value for the dimensionless depth of cut W (ω)
and the dimensionless spindle speed Ω(ω) for a range of vibration frequencies ω, starting at ω
slightly larger than 1, which followed from (3.14). The intersections of the lobes can be found by
finding the vibration frequencies ω for which the values of W and Ω(ω) of the NLth lobe have the
same values as for lobe NL + 1. Finding these intersections is interesting because the behavior of
the system near the intersections is rather different from its behavior close to the minimum point
of a lobe, which is shown in Section 3.6. Furthermore, knowledge of the locations of these intersections enables plotting only the interesting parts of the lobes. The locations of the intersections
is derived analytically in Section 3.7.
For the system described in (3.9) with ζ = 0.02 the stability lobes are constructed, see Figure 3.2.
The solid lines are the stability lobes and the dashed line indicates the minimum depth of cut
Wmin . Using (3.20) and (3.21) for the location of the minimum points in the lobes and (3.22) for
the location of the vertical asymptote of a lobe, it is clear that they are located in Figure 3.2 as
expected. Furthermore, four points are indicated in the figure; in Section 3.6 time simulations
of these points are performed. In Figure 3.3 the dimensionless vibration frequency ω is plotted
against the dimensionless spindle speed Ω(ω). This figure shows that on the intersection between
two lobes, for example at Ω(ω) = 1, the vibration frequency actually has two values. The resulting
effect on the behavior of the system is treated in Section 3.6.
9
Figure 3.2: The linear stability chart for the orthogonal cutting process with ζ = 0.02. Wmin
is indicated by the dashed line
Stability chart
1.7
Dimensionless frequency ω
1.6
1.5
1.4
1.3
1.2
1.1
1
0
0.5
1
1.5
2
Dimensionless spindle speed Ω
2.5
3
Figure 3.3: The dimensionless frequency ω plotted against the dimensionless spindle speed Ω
10
3.6
Validation of the stability lobe structure
To validate the stability charts constructed in Section 3.5, time simulations are performed for the
four different locations indicated in the stability chart in Figure 3.2: point 1 lies in the asymptotically stable region, point 2 in the unstable region, point 3 on the critically stable border and point
4 near an intersection between two lobes. The resulting vibrations are shown in Figure 3.4.
W= 0.01; Ω= 0.575
−3
1
x 10
W= 0.2; Ω= 0.575
0.015
0.8
0.01
0.6
Displacement x
Displacement x
0.4
0.2
0
−0.2
−0.4
−0.6
0.005
0
−0.005
−0.01
−0.8
−1
0
20
40
60
Dimensionless time
80
−0.015
0
100
20
(a) Point 1 (Ω=0.575, W =0.01)
100
W= 0.31; Ω= 0.505
−4
10
0.8
8
0.6
6
0.4
4
Displacement x
Displacement x
x 10
80
(b) Point 2 (Ω=0.575, W =0.2)
W= 0.0435; Ω= 0.575
−3
1
40
60
Dimensionless time
0.2
0
−0.2
x 10
2
0
−2
−0.4
−4
−0.6
−6
−0.8
−1
0
20
40
60
Dimensionless time
80
−8
0
100
(c) Point 3 (Ω=0.575, W =0.04104)
20
40
60
Dimensionless time
80
100
(d) Point 4 (Ω=0.505, W =0.31)
Figure 3.4: Time simulation in the four different regions in the stability chart as indicated in
Figure 3.2
From these figures it follows that the system responds as one would expect from Figure 3.2: in the
stable region the system is indeed asymptotically stable, and in the unstable region the system is
unstable. Furthermore, the system is critically stable on the border, where the eigenvalues equal
±iω; the amplitude of the vibration is constant. Lastly, near an intersection where, as mentioned
before, two vibration frequencies are present, the system is stable as the amplitude does not
increase in time. However, the system does not vibrate with an obvious period. The resulting
11
vibration is periodic only if the ratio between the two vibration frequencies is a rational number.
If this ratio is not a rational number, the vibration is quasi-periodic.
3.7
Analytical analysis of intersection of two lobes
In Sections 3.5 and 3.6 it was shown that at the intersection of two lobes the vibration has two
frequencies. Therefore, it is interesting to find the location of the intersection between two lobes,
which is done in this section.
To investigate where the lobes intersect and how the dimensionless depth of cut W (ω) at intersections decreases for increasing lobe number, the intersections are calculated analytically in this
section. To this end, first the damping ratio ζ is assumed to be very small and so negligible. With
ζ around 0.02 for most cutting processes, this is a reasonable assumption, as is also shown later
in this section. The new characteristic equation following from this assumption is given as:
D(λ) = λ2 + 1 + W − W e−λτ .
(3.23)
Following the same method as was used in Section 3.2, two expressions for the dimensionless
depth of cut W and dimensionless spindle speed Ω can be derived:
W (ω) =
Ω(ω) =
(ω 2 − 1)
NLζ=0
1 − (−1)
2ω
.
NLζ=0
(3.24a)
,
(3.24b)
In these equations NLζ=0 refers to the lobe number, but differs from the lobe number NL used in
Section 3.2. The latter one namely refers to the shift in the tangent function, while NLζ=0 results
from a shift in the sine function, see Appendix B. From (3.24a) it follows that NLζ=0 can only be an
odd integer number for the dimensionless depth of cut to make sense. To determine the accuracy
of this approximation, the stability chart in Figure 3.2 is compared to the chart drawn using (3.24),
see Figure 3.5. From this figure it follows that ζ = 0 seems a reasonable approximation for lobes
Figure 3.5: The stability chart for ζ = 0 and ζ = 0.02
2,3,... . It is not a good approximation for lobe 1, but as lobe 1 does not intersect with a lobe with a
lower number, as lobe 0 does not exists, this is not a drawback of the approximation for this case.
12
As mentioned before, NLζ=0 can only be an odd number. Therefore, by using Figure 3.5 it follows
that the relation between the lobe number NL and NLζ=0 is: NLζ=0 = 2NL −1. As the intersection
of lobes NL and NL +1 is the point of interest of this section, lobe NL +1 is approximated by (3.24),
and lobe NL by its vertical asymptote (defined by (3.22)). The location of the intersection lies at
the vertical asymptote, but the vibration frequencies at the intersection ωint are still unknown.
As lobe NL is approximated by its vertical asymptote, the dimensionless vibration frequency is 1
in this lobe, see also the derivation of (3.22) in Section 3.5. The vibration frequency ωint in lobe
NL + 1 at the intersection can be calculated by equating (3.24b) with the equation of the vertical
asymptote, and using NLζ=0 = 2NL − 1:
ωint =
2(NL + 1) − 1
.
2NL
(3.25)
The value for the vibration frequency in (3.25) is also the ratio between ωint in lobe NL + 1 and
ωint in lobe NL in the intersection between the two lobes. To check whether the approximation
is also valid for the determination of the vibration frequencies in the intersection and the ratio
between the vibration frequencies ωint in the two lobes, the ratios for both the approximation and
vibration frequencies in the intersection, found using the analytic method, are compared. For
both cases, the ratios for the first four intersection for are shown in Table 1. The errors between
these ratios for higher lobes are plotted in Figure 6(a). Also, the error in the vibration frequency
Intersection between lobe
1&2
2&3
3&4
4&5
Ratio for ζ=0.02
1.5290
1.2757
1.1905
1.1475
Ratio for ζ=0
1.500
1.333
1.250
1.167
Table 1: Ratio of the intersections between NL +1 and NL , for ζ=0.02 and ζ=0
in lobe NL + 1 between the two cases is plotted in Figure 6(b), the error of the spindle speed
between both cases in lobe NL + 1 at an intersection is plotted in Figure 6(c), and Figure 6(d)
shows the error between the approximated and the calculated depth of cut W in lobe NL + 1 at an
intersection. From these figures it follows that the approximation is reasonable for the vibration
frequencies ωint and the spindle speed. However, it cannot be used to approximate the depth of
cut W at an intersection, therefore it is recommended to estimate the spindle speed with (3.24b)
but to use (3.14) to determine the dimensionless depth of cut at an intersection.
3.8
Summary
This chapter discussed the analytical way to construct stability lobes for a stability chart. However, it is not possible to use this method in more complicated delay differential equations (DDEs).
Therefore, the next chapter explains semi-discretization, a linear method to solve a DDE numerically.
13
−1.8
−0.5
−1.9
Error (%)
Error (%)
0
−1
−1.5
−2
−2.5
0
50
100
150
Lobe number NL
−2.3
0
200
50
100
150
Lobe number NL
200
(b) Error in vibration frequencies ω
2.5
100
2.4
0
Error (%)
Error (%)
−2.1
−2.2
(a) Error in ratios in intersections
2.3
2.2
−100
−200
2.1
2
0
−2
50
100
150
Lobe number NL
−300
0
200
(c) Error in dimensionless spindle speed Ω
50
100
150
Lobe number NL
200
(d) Error in dimensionless depth of cut W
Figure 3.6: Error in percentage in the ratios, in ω, in Ω and in W between the case in which
ω for ζ = 0.02 is used and the case in which ω for ζ = 0 is used.
14
4
Semi-Discretization
This chapter deals with semi-discretization and explains how it can be used to investigate the
structural behavior of linear delay differential equations. The chapter starts with the basic explanation of semi-discretization and then explains how it can be used in the investigation of the
stability of metal cutting processes.
4.1
Discretization of delay term
Semi-discretization is a robust and powerful tool to determine the structural behavior of linear
DDE’s in time domain, like (3.9). As a result of the constant delay, the present state cannot
identify the state of the DDE. This indicates that the past state x(t − s), s ∈ [0, τ ] is necessary
and an infinite dimensional function space is formulated. Semi-discretization can be used to
perform the stability investigation of an existing steady state solution (for an autonomous system
like turning) or an existing periodic orbit (for a non-autonomous system like milling).
In this section the second order equation of motion of the orthogonal cutting process (as defined
in (3.9)) is used to explain the semi-discretization method. This equation of motion is again given
below, and should be first converted into a first order delay differential equation.
ẍ(t) + 2ζ ẋ(t) + (1 + W )x(t) = W x(t − τ ).
For this purpose, a vector y is defined as y = [y1 y2 ]T . This results in the following first order
system:
y2 (t)
.
(4.1)
ẏ(t) =
W y1 (t − τ ) − (1 + W )y1 (t) − 2ζy2 (t)
In matrix representation this is:
(4.2)
ẏ(t) = Ly(t) + Ry(t − τ ),
with the linear matrix L [6] defined as:
0
1
L=
,
−(1 + W ) −2ζ
(4.3)
and the retarded matrix R [6] is defined as:
0 0
R=
.
W 0
(4.4)
τ
,
Having the first order ODE, the system can be discretized in m steps [6], with time step ∆t = m
see Figure 4.1. In this figure, a function with a delay is considered. The graph is divided into
three parts, namely:
• the past from time ti −τ (= ti−m ) to the current time ti , indicated by the dotted light-colored
line;
• the present at the current time ti , indicated by the black x-mark;
15
Figure 4.1: A function with a time delay
• the future from the current time increment ti until the last time increment the function is
considered, indicated by the dashed dark line.
The aim of semi-discretization for DDEs is to predict the state y between time ti and ti+1 using
the information from the discretized history of the function between time ti−m and ti−m+1 .
In semi-discretization, during each step the function is considered between the current time ti
and ti+1 , so the delay term is considered between ti−m and ti−m+1 . The delay term between ti−m
and ti+m+1 is approximated to be constant and the delay differential equation turns into an ODE.
From Figure 4.1 it follows that the solution between ti−m and ti−m+1 can be approximated by:
1
yi (t − τ ) ≃ (yi−m+1 + yi−m ).
2
(4.5)
Substitution of this in (4.2) obtains:
1
ẏi (t) = Lyi (t) + R(yi−m+1 + yi−m ),
2
(4.6)
The last two equations are only valid for t ∈ [ti , ti+1 ). This differential equation is an inhomogeneous ordinary differential equation, and is solved using the method of variation of coefficients:
1
yi (t) = eL(t−ti ) c0 − eL(t−ti ) R(yi−m+1 + yi−m ),
2
(4.7)
1
c0 = yi + R(yi−m+1 + yi−m ).
2
(4.8)
with
The complete derivation is found in Appendix C. Because the solution is considered only at time
interval t ∈ [ti , ti+1 ), t − ti is replaced by ∆t and instead of yi (t), yi (ti+1 ) (= yi+1 ) is used:
yi+1 = eL∆t yi +
1 L∆t
e
− I L−1 R(yi−m+1 + yi−m ).
2
16
(4.9)
In the next section, (4.9) is used to derive a linear map that can describe the connection between
two discrete points in the solution of this equation, with which the stability of the system can be
determined.
4.2
Stability investigation using a linear map
In Section 4.1, the inhomogeneous ODE was solved and (4.9) resulted. This equation is used here
to derive a linear map B, with which the stability behavior of the DDE in (3.9) can be investigated.
With the introduction of the multi-dimensional vector zi [6], the discretized state space of the
DDE can be considered in the following way:

yi
 yi−1

zi =  .
 ..
yi−m



,

(4.10)
and zi+1 is:



zi+1 = 

yi+1
yi
..
.
yi−m+1



.

(4.11)
As the delay only exists in position and does not exist in velocity, y2,i−1 = 0. Therefore, only the
first element each vector yi−j , j = 1, ..., m is stored in zi , and zi becomes:

yi
 y1,i−1

 y1,i−2

zi = 
..

.

 y1,i−m+1
y1,i−m





.



(4.12)
As yi = [xi ẋi ], and ẋi 6= 0 nothing changes in yi . The linear map in (4.9) can be reformulated
using the vector zi and a system matrix B = B1 + B2 with

0

0

 1 0 0


B1 =  0 1 0
 . . .
 .. .. ..

 0 0 0
eL∆t
...
...
...
...
..
.
0
0
0
0
..
.
0
0
0
0
..
.






 , and



... 0 0 
0 0 0 ... 1 0
(4.13)
17






B2 = 




O
O
...
O
O
...
..
.
..
.
..
O
O
...
.
A11 A11
A21 A21
0
0
0
0
..
..
.
.
0
0
0
0






.




(4.14)
With this linear map, zi+1 can be written as zi+1 = Bzi and the state of the system is written in
one linear time-independent map, which can be used to investigate the stability. In these matrices
O the null matrix and A is:
1
A = (eL∆t − I)L−1 R.
2
(4.15)
The exact derivation of the matrices B1 and B2 are found in Appendix D.
The stability of the system can be investigated by calculating the eigenvalues µk of the B matrix.
The system is stable if all eigenvalues lie within the unit circle, or: |µk | < 1, k = 1, ..., Nsd ,
with Nsd = (m + 1)N , the number of eigenvalues of the state matrix. m is the number of
discretization steps and N the order of the system. In the next section, this matrix is used to
evaluate the stability of the system for various dimensionless depths of cut and spindle speeds.
4.3
Implementation of the semi-discretization method
The theory of semi-discretization was explained in Section 4.1. At the end of Section 4.2, a system
matrix B was derived, with which the stability of the system can be evaluated. This matrix can be
used to check the stability of a metal cutting process for various values of the dimensionless depth
of cut and the dimensionless spindle speed. The stability lobes are constructed by determining
for each (Ω, W ) point whether it is stable or not.
The number of discretization steps m still has to be determined. To investigate the influence
of m on the results, the stability of this system is determined for a range of the dimensionless
depth of cut W ∈ [0.001, 0.5] and a range of the dimensionless spindle speed Ω ∈ [0.01, 2]. For
each point (Ω, W ) the stability at that point is determined. This is done for four different values
for m, namely 5, 10, 50 and 100. The results are shown in Figure 4.2. The black area indicates
the stable region, the gray color indicates the unstable region, and the white line indicates the
stability lobe constructed according to the theory explained in Chapter 3. From these figures it
follows that for higher values of m the borders between the unstable and stable region approach
the stability lobes better, although there is only little difference between m = 50 and m = 100.
A drawback is that more discretization steps also results in longer calculation time. For example,
if the stability of the cutting process is investigated for 160 points for W and 160 points for Ω, a
step size m = 10 takes 30 seconds, while m = 100 takes 12 minutes. Section 4.3.1 explains how
the step size m should be chosen appropriately.
18
Stablity chart using semi discretization with m=10
Dimensionless depth of cut
Dimensionless depth of cut
Stablity chart using semi discretization with m=5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.5
1
1.5
Dimensionless spindle speed Ω
0.7
0.6
0.5
0.4
0.3
0.2
0.1
2
0.5
1
1.5
Dimensionless spindle speed Ω
(a) m = 5
(b) m = 10
Stablity chart using semi discretization with m=100
Dimensionless depth of cut
Dimensionless depth of cut
Stablity chart using semi discretization with m=50
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.5
1
1.5
Dimensionless spindle speed Ω
2
2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.5
1
1.5
Dimensionless spindle speed Ω
(c) m = 50
2
(d) m = 100
Figure 4.2: The stability of each point (Ω, W ) in the stability chart for various values of m
19
4.3.1
Choice of step size m
When considering the low spindle speed lobes, the possibility exists that the time step ∆t is larger
than the period of the vibration of the tool T : ∆t > 2π
ω , with ω the vibration frequency. If this
happens, information about the vibration is lost. Therefore, the number of discretization steps m
2π
τ
should be chosen appropriately. Using T = 2π
ω , τ = Ω and ∆t = m , the following expression
Tω
ω
ω
< 1, so m > Ω
.
for ∆t can be derived: ∆t = Ωm . ∆t must be smaller than T , which is true if Ωm
4.4
The motion of the eigenvalues
In this section, the motion of the eigenvalues are discussed. For stability analysis it is interesting
to know how the eigenvalues evolve when moving from the stable region towards the unstable
region or when moving along the border between the two regions. Therefore, the eigenvalues of
the matrix B are investigated for these two cases. In the first case, the dimensionless depth of cut
W increases, while the dimensionless spindle speed Ω remains constant at Ω = 0.5855. In the
second case, both parameters are changed along the lobe. The resulting motion of eigenvalues
with respect to the unit circle are shown in Figure 4.3.
From Figure 3(a) it follows that the two critical eigenvalues move outside the unit circle in the
right half plane, which indicates Hopf bifurcation. The other eigenvalues stay inside the unit
circle. From Figure 3(b) it follows that the two critical eigenvalues move along on the border of
the unit circle, while the other eigenvalues remain inside the unit circle.
(a) The motion of the eigenvalues in the unit circle
when the system is evaluated while crossing the
lobe
(b) The motion of the eigenvalues in the unit circle
when the system is evaluated along the lobe
Figure 4.3: Motion of eigenvalues of B when W and/or Ω are changed
20
5
Process Damping in Orthogonal Cutting Processes
In Chapters 3 and 4, a simple mass spring damper system was used to describe the orthogonal
cutting process. However, an important issue has not been considered up until now, namely process damping. Process damping has a significant effect in low speed metal cutting, as in these
processes the stability lobes obtained from experiments lie much higher in the stability chart
than is predicted with the mathematical stability theorem. Although currently no physical model
is yet able to describe this effect completely, several explanations have been introduced. These
theorems assume that the additional damping is mainly caused by the contact mechanism of the
metal cutting. However, a proper description of this mechanism requires difficult mechanical
and mathematical techniques. Therefore, in this report it is proposed to use a simple assumption
for this contact mechanism to create a sufficiently simple mechanical model for dynamical investigation. This model includes a friction force in the direction tangential to the cut surface and a
contact force that depends on the surface curvature. Furthermore, it is assumed that the friction
force is constant, while it probably oscillates in time during a cutting process.
This chapter first discusses the theory of the process damping, after which the stability lobes with
process damping are compared to a stability chart of the same cutting process without process
damping.
5.1
Addition of process damping in equation of motion
Many research has been undertaken on the field of process damping, and some results are used
in this report. First, Das and Tobias [4] proposed to add a velocity dependent term with a static
cutting force coefficient, with which the damping in the system increases at low speeds; this is
the friction force. However, Altintas describes in [3] the process damping by two additional force
terms: the friction force introduced in [4], and an additional term that describes the contact force,
which depends on the curvature of the wave. This is an improvement with respect to the theory
in [4]. Therefore, the latter process damping theory for the orthogonal cutting process is used in
this section.
Figure 5.1 shows the friction force Ff (t) and the contact force Fc (t) in the orthogonal cutting
process. Furthermore, vc is the surface velocity of the workpiece.
Figure 5.1: Additional contact and friction forces that describe process damping
21
First the derivation of the friction force is discussed. In Figure 5.2 the slope of the surface is
indicated. In this figure, x(t) is the position of the tool, u(t) is the tangential distance of tool
traveled along the workpiece, φ(t) is the angle between the friction force Ff (t) and the u-axis,
and t is the time. The slope of the surface, and thus the direction of the friction force, is the
Figure 5.2: The slope of the wavy surface of the workpiece
dx
spatial derivative of x(t) with respect to u, that is du
. This slope can be resolved into the temporal
dt
dt
1
dx
derivative of x(t) and the temporal derivative of u: du
= dx
dt du . With du = − vc , and vc = Ωr, the
slope of the wave is rewritten as:
dx
ẋ(t)
=−
.
du
Ωr
(5.1)
From Figure 5.2 it follows that the slope of the wave is equal to tan φ(t). As it is assumed that
dx
this angle is very small, tan φ(t) ≃ φ(t) = du
. In this report, for the orthogonal cutting process
only forces in x-direction are considered, so the x-component of Ff (t) is derived here. With
Ffx (t) = Ff sin φ(t), and sin φ(t) ≃ φ(t), Ffx (t) = −Ff ẋ(t)
Ωr . Ff is the constant magnitude of the
friction force and it is assumed to result from a static cutting force coefficient Cf [N/m] and the
depth of cut w: Ff = wCf . The friction force in x-direction is:
Ffx (t) = −
wCf
ẋ(t).
vc
(5.2)
Having the friction force, the contact force is now derived. The contact between the tool and
the workpiece is assumed to be similar to a static Hertzian contact. Therefore, the contact force
Fc (t) depends on the curvature of the wave of the surface cut. The curvature of the wave is
ẍ(t)
d2 x
proportional to the second spatial derivative of x(t) with respect to u: du
2 = v 2 . The contact
c
force is also linearly proportional with the length of the edge in cut, that is the depth of cut w.
The proportional constant is the contact force coefficient Cc [N]:
(5.3)
Fc (t) = Fc0 + ∆Fc (t),
where
∆Fc (t) =
wCc
ẍ(t).
vc2
(5.4)
Fc0 is the static contact force, which can be neglected as this has no influence on the stability
investigation.
22
The two process damping forces are added to (3.1) in Section 3.2:
mẍ(t) + cẋ(t) + k(q0 + x(t)) = wfa (h(t)) + Ffx (t) + Fc (t).
Using Ω =
2π
τ
(5.5)
the equation of motion for an orthogonal cutting process with process damping is:
Cc wτ 2
m−
4π 2 r2
Cf wτ
ẍ(t) + c +
2πr
ẋ(t) + (k + wk1 )x(t) = wk1 x(t − τ ).
(5.6)
The coefficients Cc and Cf can be found experimentally [3]. Using (3.6) and (3.8), (5.6) is converted into a dimensionless parameter space. Two dimensionless parameters are defined for the
process damping, namely If and Ic :
If
=
Ic =
Cf
,
2πrk1
Cc
.
2
4π r2 k1
(5.7a)
(5.7b)
The dimensionless equation of motion with process damping is given as:
(1 − Ic W τ 2 )ẍ(t) + (2ζ + If W τ )ẋ(t) + (1 + W )x(t) = W x(t − τ ),
(5.8)
with W the dimensionless depth of cut as derived in Section 3.2. The complete derivations of the
dimensionless parameters in (5.7) are found in Appendix E.
In the next section, (5.8) is used to plot the stability lobes. Because the two terms that were added
to describe the process damping both depend on the delay τ , so the spindle speed Ω, and the
depth of cut W , it is not possible to derive analytically dimensionless expressions for the angular
velocity and the depth of cut as was done in Section 3.3. Therefore, it is not possible to analyze
the lobes analytically like in Section 3.4, so the lobes are constructed using semi-discretization.
5.2
Implementation of process damping in semi-discretization code
To construct the stability lobes by semi-discretization, the L and R matrices of the system described in (5.8) are derived, using the method of Section 4.1. The matrix L is:
"
#
0
1
L=
,
(5.9)
−(1+W )
−2ζ−Ic W τ
1+Ic W τ 2
1+If W τ 2
and the retarded matrix R is :
#
"
0
0
.
R=
W
0
1+If W τ 2
(5.10)
These two matrices are used to calculate the discretized system matrix B, see (4.13), (4.14) and
(4.15).
Having the system matrix B, the eigenvalues of this matrix are calculated for several (Ω, W )
points in the stability chart for three different cases, and the corresponding stability lobes are
plotted in Figure 3(a). In this figure, the blue line is the cutting process without process damping,
23
the red line is the cutting process with only the velocity term, as proposed in [4], and the black line
indicates the stability lobes for a cutting process described in (5.8). From this figure it follows that
the stable area for the orthogonal cutting process with process damping is significantly larger than
the stable area for the orthogonal cutting process without process damping. Furthermore, the
stable area of the process damping theory proposed in this report is slightly larger than the stable
area of the orthogonal cutting process damping where only a velocity term is taken into account.
These theoretical lobes are compared to a stability test in [3], indicated with the o-marks (stable)
and x-marks (unstable). From this comparison it follows that the theoretical and experimental
results do not match. This is explained by the fact that it is quite difficult to determine the contact
force coefficient Cc , [3].
The step size of the semi-discretization method has a significant influence on the reliability of the
results. To show this, in Figure 3(b) stability lobes for the same cutting process were constructed,
using a step size m = 100, instead of m = 400 in Figure 3(a).
The theory about the stability of the one degree-of-freedom-model explained in Chapters 3 till 5
can be used to determine an approximation for the stability for a 2 DOF non-autonomous cutting
process, namely the milling process, although the dynamics of an orthogonal cutting process and
a milling process are not the same. This is done in the next chapter.
−3
−3
2 x 10
2 x 10
Cc=0 [N], Cf=0 [N/m]
Cc=0 [N], Cf=0.611e6 [N/m]
1.8
1.6
Cc=332 [N], Cf=0.611e6 [N/m]
1.6
1.4
stable
unstable
Depth of cut [m]
Depth of cut [m]
1.8
1.2
1
0.8
0.6
Cc=0 [N], Cf=0 [N]
Cc=0 [N], Cf=0.611e6 [N]
Cc=332 [N], Cf=0.611e6 [N]
1.4
1.2
1
0.8
0.6
0.4
0.4
1000
1500
2000
2500
3000
Spindle speed [rpm]
3500
4000
1000
(a) The stability lobes for orthogonal cutting without process damping, with only friction force and
with both friction and contact forces. using m =
400 Experimental results are depicted by the xmarks and circles.
1500
2000
2500
3000
Spindle speed [rpm]
3500
4000
(b) The stability lobes for orthogonal cutting without process damping, with only friction force and
with both friction and contact forces using m =
100, resulting in an unreliable stability chart
Figure 5.3: Stability lobes for orthogonal cutting without and with process damping, for
different step sizes
24
6
Stability Chart for Milling
The theory in Chapters 3, 4 and 5 was applied to an orthogonal cutting process, a one degreeof-freedom autonomous model. In this chapter, it is used in a two degree-of-freedom, nonautonomous system that can describe milling. With this model, the stability of a milling process
is investigated, first for a model without process damping, after which process damping is added
to the model.
6.1
Introduction
The equation of motion of a milling process is periodic, and thus time-dependent, because of
the periodic parameter excitation due to entering and exiting teeth as a result of the tool revolving. It is clear that as a result of the time dependence of the model, the linear stability limits of the milling process cannot be investigated in the same way as for turning, which was a
time-independent system. For example, the equations of motion have more than one degree of
freedom: in milling the forces in both the x and the y-direction, and also in z-direction for helical tools, should be taken into account. In this report, it is assumed that the milling tool is
straight-fluted, so only a two degree-of-freedom model is used and the z-direction is not taken
into account. As the state vector is indicated with x further onwards, in this section the directions
are indicated with x1 and x2 , instead of x and y respectively.
The milling process is represented schematically in Figure 6.1. In this figure, the spring and
damper forces in both the x1 and the x2 direction are indicated, as are the motions due to vibration in these directions, indicated by χ and γ respectively. Furthermore, V is the speed of the tool
with respect to the workpiece resulting from the feed of the tool:
V =
N fd Ω
.
2π
(6.1)
In this equation, N is the number of teeth of the tool and fd is the feed per tooth [m].
In milling, the distinction between up- and down-milling is made, see Figure 6.2. The difference
between these two types is the direction of the motion of the workpiece with respect to the rotation
of the tool. In the figure, two important angles are indicated: the angle with which a tooth enters
the work piece, φen , and the angle with which the tooth exits the workpiece, φex . In the coordinate
system chosen the following holds: for down-milling, the exit angle is always π [rad], for upmilling, the enter angle is 0 [rad] [5]. These angles are important, as the equations of motion
for milling, which are described in Section 6.2, depend on the angle of the tool with respect to
the workpiece. Generally, in practice down-milling is used, as the teeth cut with a larger chip
thickness when they enter the workpiece, which causes a high impact, but the rubbing effect can
be avoided.
25
Figure 6.1: Schematic model of milling
Figure 6.2: Schematic representation of down milling (left) and up milling (right). The entrance angle φen and the exit angle φex are indicated
26
6.2
Equations of motion for milling
The general equation of motion for a milling proces is given as:
(6.2)
M ẍ(t) + C ẋ(t) + Kx(t) = F (t).
In this equation, the vector x(t) contains the motions in x1 and x2 -direction, and the vector F (t)
contains the forces in x1 and x2 -direction. Furthermore, the mass matrix is indicated by M [kg],
the damping matrix by C [Ns/m] and the stiffness matrix by K [N/m].
Because a milling tool has more than one tooth, the forces acting on all teeth should be considered. For each tooth, the resulting cutting forces in radial and tangential direction have been
determined in [5]:
Fj,r = g(φj (t))Kr ap h(t) [N]
and
(6.3a)
Fj,t = g(φj (t))Kt ap h(t) [N],
(6.3b)
respectively. In these equations, g(φj (t)) is a switching function, indicating whether tooth j
is cutting or not, the parameters Kr and Kt indicate the linear radial and tangential cutting
coefficients [N/m2 ] respectively, ap is the depth of cut [m], and h(t) is the momentary thickness
of the chip [m] at time t [s]. A tooth j is cutting when the angular position of the tooth at time t is
between φen and φex . The angular position φj (t) of tooth j can be calculated by (6.4), assuming
that the teeth are equally distributed over the tool:
φj (t) = Ωt +
2π(j − 1)
,
N
(6.4)
in which N is the number of teeth, j is the tooth number and Ω is the spindle speed [rad/s].
Knowing the angular position, the switching function g(φj (t)) can be determined:
g(φj (t)) =
n 1, if φ < φ (t) < φ ,
en
j
ex
0,
otherwise.
(6.5)
The chip thickness h(t) can be derived from Figure 6.3 [5]. In the left hand side of this figure,
the current tool position is shown in gray, the previous tool position is shown in black, and the
respective positions of tooth j at angle φj (t) [rad] are indicated with A and B. The difference in
time between both positions is τ . Because the tool is not stiff, it vibrates due to the periodic force
excitation and it is unable to follow the path of an ideally stiff tool. The distance of the actual
tool position with respect to the ideal tool position is indicated by the variables x1 and x2 . The
chip thickness is the distance between the previous tool movement and the current tool position,
measured along the normal of the current tool position. This region is enlarged in the right hand
side of the figure, where again the points A and B are indicated. From this figure it follows that,
assuming that the feed per tooth fd is much smaller than the tool radius R, the chip thickness h
at time t can be approximated by [5]:
(6.6)
h(t) = α sin φj (t) + β cos φj (t),
with
α = x1 (t − τ ) + fd − x1 (t)
and
(6.7a)
(6.7b)
β = x2 (t − τ ) − x2 (t).
27
Figure 6.3: The chip thickness is calculated using the previous and the current tool position
Now the forces acting on each tooth in radial and tangential direction are known, the forces acting
on each tooth in x1 and x2 -direction can be derived, see also Figure 6.1:
Fj,x1 (t) = Fj,t cos φj (t) + Fj,r sin φj (t) →
Fj,x1 (t) = g(φj (t)) Kt cos φj (t) + Kn sin φj (t) ap h(t),
Fj,x2 (t) = −Fjt sin φj (t) + Fj,r cos φj (t) →
Fj,x2 (t) = g(φj (t)) (−Kt sin φj (t) + Kn cos φj (t)) ap h(t).
(6.8)
(6.9)
(6.10)
(6.11)
To obtain the total forces on the tool the forces acting on each tooth should be added for the x1 and x2 -direction:
Fx1 (t) =
N
X
g(φj (t)) (Kt cos φj (t) + Kn sin φj (t)) ap h(t),
(6.12)
N
X
g(φj (t)) (−Kt sin φj (t) + Kn cos φj (t)) ap h(t).
(6.13)
j=1
Fx2 (t) =
j=1
Using (6.2), (6.6), (6.12) and (6.13), the following equation of motion results:
M ẍ(t) + C ẋ(t) + Kx(t) = ap H(t)(x(t − τ ) − x(t)) + G(t).
(6.14)
In this equation, H(t) is the specific cutting force variation matrix and G(t) is the stationary
cutting force vector introduced in [5]. The matrix H(t) consists of the following four elements:
28
H11 (t) =
N
X
g(φj (t)) (Kt cos(φj (t) + Kn sin φj (t)) sin φj (t),
(6.15a)
g(φj (t)) (Kt cos(φj (t) + Kn sin φj (t)) cos φj (t),
(6.15b)
g(φj (t)) (−Kt sin(φj (t) + Kn cos φj (t)) sin φj (t),
(6.15c)
g(φj (t)) (−Kt sin(φj (t) + Kn cos φj (t)) cos φj (t).
(6.15d)
j=1
H12 (t) =
N
X
j=1
H21 (t) =
N
X
j=1
H22 (t) =
N
X
j=1
The vector G(t) consists of the two elements:
Gx1 (t) = ap fd H11 (t),
(6.16a)
Gx2 (t) = ap fd H21 (t).
(6.16b)
The vector x(t) that describes the motion of the tool can be divided into two parts [5]: a
τ -periodic part xp (t), so xp (t) = xp (t + τ ), and a part ξ(t) that describes the motion due to
chatter. Substituting this for x(t) in (6.14) results in:
¨ + C ξ(t)
˙ + Kξ(t) + M ẍp (t) + C ẋp (t) + Kxp (t) = ap H(t)(ξ(t − τ ) − ξ(t)) + G(t). (6.17)
M ξ(t)
For the investigation of the stability of the solution x(t), only the terms that contain ξ(t) have to
be taken into account, and the second order delay differential equation that describes the stability
due to the regenerative effect is:
¨ + C ξ(t)
˙ + Kξ(t) = ap H(t)(ξ(t − τ ) − ξ(t)).
M ξ(t)
(6.18)
This equation is investigated in Section 6.3 using semi-discretization.
6.3
Stability lobes using semi-discretization
By using the second order delay differential equation that describes the motion of the tool due
to the regenerative effect, (6.18), the stability lobes for milling can be constructed. This is done
using the semi-discretization method explained in Section 4.1.
The second order delay differential equation in (6.18) is first transformed into a first order delay
differential equation:
(6.19)
ẏ(t) = L(t)y(t) + R(t)y(t − τ ),
with
L(t) =
O
I
−1
−M (K + ap H(t)) −M −1 C
29
,
(6.20)
and
R(t) =
O
O
ap M −1 H(t) O
(6.21)
.
Because the specific cutting force variation matrix H(t) is periodically time-dependent, it is not
possible to use exactly the same matrices as were used in Section 4.1 to check the stability. To
calculate the B-matrix, the matrices R(t) and L(t) are evaluated for the time interval t = [ti , ti+1 ).
Within this interval, the matrix L(t) can be approximated by numerical integration:
Z ti+1
Nint
1
1 X
Li =
Li,k ,
(6.22)
L(t)dt ≃
∆t ti
Nint
k
)
k N∆t
int
with Li,k = L(ti +
retarded matrix R(t).
and Nint the number of integration steps. The same holds for the
With these matrices the Bi matrix can be constructed in the same way as was done for autonomous orthogonal cutting: Bi = Bi1 + Bi2 with
 Li ∆t

e
O ... O O
 I
O ... O O 


 O
I ... O O 
Bi1 = 
(6.23)
,
 .

.
.
.
.
.
.
.
.
.
 .
. . . 
.
O

and


Bi2 = 

O ...
I
O

O O . . . Ai Ai
O O ... O O 

.. .. . .
..
..  ,
. .
. .
. 
O O ... O O
(6.24)
1
Ai = (eLi ∆t − I)L−1
(6.25)
i Ri .
2
To determine the stability of the cutting process, the Floquet theorem is used [7]. The Floquet
theorem states that a linear periodic system has a periodic solution. This theory is used to derive
the transition matrix Φ such that
(6.26)
y(t0 + T ) = Φy(t0 ).
If none of the so-called Floquet multipliers µk , which are the eigenvalues of the transistion matrix
Φ, lie outside the unit circle, the stationair periodic orbit is orbitally stable. The Floquet matrix is
calculated using semi-discretization with the step matrices Bi :
(6.27)
Φ = Bm Bm−1 ...Bi−1 Bi .
The linear stability limits can now be constructed by determining for each (Ω, w) point whether
the Floquet multipliers lie inside or outside the unit circle. This is done for two different milling
processes: half immersion down-milling with a tool with one tooth (solid line in Figure 4(a)) and
a tool with four teeth (solid line in Figure 4(b)). The process parameters used to construct these
lobes are found in Appendix G. Now the basic stability lobes are constructed for milling, the
influence of process damping is investigated in Section 6.4.
30
−4
x 10
−4
x 10
14
2.5
10
Depth of cut [m]
Depth of cut [m]
12
8
6
4
2
2
1.5
1
0.5
1
2
3
Spindle speed [rpm]
4
5
2000
4
x 10
(a) Down-milling process with a tool with one
tooth
4000
6000
Spindle speed [rpm]
8000
10000
(b) Down-milling process with a tool with four
teeth
Figure 6.4: Stability lobes for a down-milling process with two different tooth number of the
tool
6.4
Process damping for milling
In Section 6.2 the equations of motion for the two degree-of-freedom model used to describe the
milling process were derived, and in Section 6.3 the stability chart was constructed using semidiscretization As in turning, the stability of milling processes is influenced by process damping,
so this is added to the model in this section.
It is assumed that process damping can be described in the same way as in turning, see Section
5.1, so by adding a friction force and a contact force. To this end, the (x1 , x2 )-coordinate system
that is fixed to the world is converted into a (u1,j , u2,j )-coordinate system that is fixed to a tooth,
see Figure 6.5. In this figure, the friction force on a tooth is indicated by Ff,j (t), the cutting force
is indicated by Fc,j (t) and the cutting speed of a tooth is indicated by vc . The cutting force acts
only in u2,j -direction, the cutting velocity acts in −u1,j -direction. The coordinate system x can be
Figure 6.5: Additional contact and friction forces that describe process damping
31
converted into the coordinate system uj using the transformation matrix Txu :
(6.28)
uj = Txu x,
where
Txu =
− cos φj (t) sin φj (t)
− sin φj (t) − cos φj (t)
(6.29)
.
The analogy of Figure 6.5 to Figure 5.1 is used in the derivation of the friction force and the contact
force.
cos φj (t)
,
First, the friction force is derived in the same way as in Section 5.1: Ff,j (t) = Ff
sin φj (t)
with Ff = Cf ap the constant magnitude of the friction force, and Cf is the unit friction force
du2,j
coefficient [N/m]. In Section 5.1 φj (t) = du1,j
= − v1c u̇2,j (t) was derived. As the angle φj (t)
is quite small, cos φj (t) ≃ 1 and sin φj (t) ≃ φj (t). Furthermore, from (6.29) it follows that
u2,j (t) = −x1 (t) sin φj (t) − x2 (t) cos φj (t), and using this:
u̇2,j = −ẋ1 (t) sin φj (t) − x1 (t)φ̇j (t) cos φj (t) − ẋ2 (t) cos φj (t) + x2 φ̇j (t) sin φj (t).
(6.30)
With vc (t) = rΩ, the friction force on the flank face of the j th tooth is now:
Ff,j (t) = Cf ap
"
1
− rΩ
1
−ẋ1 (t) sin φj (t) − x1 (t)φ̇j (t) cos φj (t) − ẋ2 (t) cos φj (t) + x2 φ̇j (t) sin φj (t)
(6.31)
Second, the contact force is derived. As explained in Section 5.1 the contact force Fc,j (t) is assumed to be similar to a static Hertzian contact force and acting in u2,j -direction:
Fc,j (t) = Fc0 + ∆Fc,j (t), with
∆Fc,j (t) = Cc ap
"
0
d2 u2,j (t)
du21,j (t)
#
(6.32)
.
The second spatial derivative of u2,j (t) is
d2 u2,j
1 d2 u2,j (t)
= 2
,
2
vc
dt2
du1,j
(6.33)
with
d2 u2j (t)
dt2
= −ẍ1 (t) sin φj (t) − 2ẋ1 (t)φ̇j (t) cos φj (t) − x1 (t)φ̈j (t) cos φj (t) + x1 (t)φ̇j (t)2 sin φj (t)
−ẍ2 (t) cos φj (t) + 2ẋ2 (t)φ̇j (t) sin φj (t) + x2 (t)φ̈j (t) sin φj (t) + x2 (t)φ̇j (t)2 cos φj (t),
(6.34)
with φ̇j (t) = Ω and, as a constant spindle speed is assumed, φ̈j (t) = Ω̇ = 0. Both the friction
force and the contact force are now derived.
32
#
.
The derivations for the friction force and the contact force are valid in the u-coordinate system.
As the equations of motion for a 2 DOF milling process as defined in (6.14) are derived using the
x-coordinate system, the friction and contact force has to be transformed into the x-coordinate
T .
system according to Fx = Tux Fu using the transformation matrix Tux : Tux = Txu
The friction force Ff,j (t) in the x-coordinate system is:
1
sin2 φj (t)
cos φj (t) sin φj (t)
Ff,j (t) = Cf ap −
ẋ(t)+
(6.35)
cos2 φj (t)
rΩ cos φj (t) sin φj (t)
1 cos φj (t) sin φj (t)
− cos φj
− sin2 φj (t)
x(t) +
,
−
cos2 φj (t)
− cos φj (t) sin φj (t)
sin φj (t)
r
the contact force Fc,j (t) in the x-coordinate system is:
1
sin2 φj (t)
cos φj (t) sin φj (t)
− sin φj (t)
ẍ(t)+
+ Cc ap 2 2
Fc,j (t) = Fc0
cos φj (t) sin φj (t)
cos2 φj (t)
− cos φj (t)
r Ω
cos φj (t) sin φj (t)
− sin2 φj (t)
2Ωẋ(t)+
(6.36)
+
cos2 φj (t)
− cos φj (t) sin φj (t)
− sin2 φj (t)
− cos φj (t) sin φj (t)
2
Ω x(t) .
+
− cos φj (t) sin φj (t)
− cos2 φj (t)
The friction force and contact force are added to (6.14) and the new equation of motion for a 2
DOF non-autonomous cutting process is:
M ẍ(t) + C ẋ(t) + Kx(t) = ap H(t)(x(t − τ ) − x(t)) + G(t) − Ipd (t)ẍ(t) −
Cpd (t)ẋ(t) − Kpd (t)x(t).
(6.37)
To this order three new matrices, which follow from (6.35) and (6.36), are defined: Ipd (t), Cpd (t)
and Kpd (t). These matrices and their derivations are found in Appendix F. The stationary cutting
force vector G(t) is now:
Gx1 (t) = ap (fd H11 (t)) + g(φj (t))
N
X
j=1
Gx2 (t) = ap (fd H21 (t)) + g(φj (t))
N
X
j=1
(− cos φj (t)Cf ap − Fco sin φj (t)) ,
(6.38a)
(sin φj (t)Cf ap − Fco cos φj (t)) ,
(6.38b)
while the specific cutting force variation matrix H(t) does not change.
The vector x(t) in (6.37) is divided into a stationary periodic part xp (t) and a vibration motion
ξ(t), see Section 6.2. As the vibration is interesting for the stability analysis, the stability chart for
process damping in milling is constructed using the following equation:
¨
˙
(M +ap Ipd (t))ξ(t)+(C
+ap Cpd (t))ξ(t)+(K
+ap Kpd (t))ξ(t) = ap H(t)(ξ(t−τ )−ξ(t)), (6.39)
This is the equation of motion of the two degree-of-freedom model for milling with process damping and is solved using semi-discretization in the next section, after which the stability charts are
constructed.
33
6.5
Construction of stability chart for milling with process damping
The equation of motion for a non-autonomous two degree-of-freedom cutting model with process
damping was derived in Section 6.4 and is used in this section to construct the stability lobes.
The analysis of the stability of (6.39) is done using semi-discretization.
To construct the stability lobes, first (6.39) is converted into the first order delay differential equation ẏ(t) = L(t)y(t) + R(t)y(t − τ ) in same way as in Section 6.3. The linear matrix L(t) is now:
L(t) =
O
I
,
−(M + ap Ipd (t))−1 (K + ap Kpd (t) + ap H(t)) −(M + ap Ipd (t))−1 (C + ap Cpd (t))
(6.40)
and the retarded matrix is:
O
O
.
R(t) =
ap (M + ap Ipd (t))−1 H(t) O
(6.41)
(6.23) and (6.24) are used to investigate the stability of the two degree-of-freedom cutting process, and the stability chart is constructed for the same cutting processes as in Section 6.3, see
Figure 6.6. From earlier experiments it followed that process damping occurs especially after the
twentieth lobe in the stability chart. Therefore, the stability charts are plotted from the thirteenth
until the twentieth lobe. From these figures it follows that the addition of the process damping
increases the stable area of the stability charts, as was expected, although the results are less significant for the cutting process with four teeth, compared to the simulation results of the tool
with only one tooth. The parameters used to construct these charts are found in Appendix G.
−4
−5
4 x 10
5 x 10
with Process damping
no process damping
No Process damping
with process damping
3
Depth of cut [m]
Depth of cut [m]
4.5
2
4
3.5
3
2.5
1
1400
1500
1600
1700
1800
Spindle speed [rpm]
1900
2000
400
(a) Half-immersion down-milling process with one
tooth
500
600
Spindle speed [rpm]
700
800
(b) Half-immersion down-milling process with four
teeth
Figure 6.6: Comparison of stability lobes for a down-milling process with one or four teeth,
with and without process damping
34
7
Conclusions and recommendations
This section discusses the conclusions that are drawn from this report, and the recommendations
that follow from these for future research.
7.1
Conclusions
The main purpose of this project is to investigate the stability of a milling process, by taking
process damping into account and using the semi-discretization method. To this end, first a
model with one degree of freedom is investigated, namely the orthogonal cutting process turning.
After the equation of motion, which is a delay differential equation, is formulated for turning, the
influence of the depth of cut and the spindle speed of the tool are investigated first analytically and
later by the construction of the stability lobes. Then, the semi-discretization method is used to
investigate the structural behavior of the delay equation of motion, and compared to the analytical
solution. From this it followed that especially at low spindle speeds a high step size is required
for the semi-discretization to be accurate. Following, process damping is investigated for turning
and added to the semi-discretization model. It is assumed that process damping can be described
by a friction force and a contact force. From the stability lobes constructed it is concluded that
by adding these forces to the equation of motion of the orthogonal cutting processing, the stable
area in the stability chart has increased, as expected.
Having completed the one dimensional orthogonal cutting process, the theory is used in a model
with two degrees of freedom in order to investigate milling. The difficulty of this in comparison
to the 1 DOF model is the fact that one of the system matrices is time dependent. To solve this,
numerical integration is used. From the stability lobes constructed it followed that the tooth
number of the tool has a significant influence on the location and size of the stable regions in the
stability chart. Process damping is added to this model by describing it by the spatial derivative
of the radial tool position and its second spatial derivative. The stability lobes that are constructed
using this theory revealed that process damping in a 2 DOF model increases the stable region in
the stability chart, although the significance of the increase depends on the number of teeth of
the tool.
7.2
Recommendations
In this report, some simplifications are used to make things less complicated. For example, only
milling tools with straight cutting edges are considered. However, most tools are helical shaped,
as these damp the oscillatory vibration more than straight fluted tools. Therefore, it is recommended to take the effect of helical shaped tools into account in future research. Furthermore,
only the forces in x and y direction are considered. When the axis along the tool, so the z direction, is also taken into account, the model is more accurate. The last and most important
recommendation is to conduct experiments to verify the process damping model.
35
References
[1] Adams, R.A., 2003, Calculus :a complete course, Addison-Wesley Longman, Toronto.
[2] Altintas, Y., 2000, Manufacturing Automation, Cambridge University Press, Cambridge.
[3] Altintas, Y., Eynian, M., Onozuka, H., 2008, "Identification of Dynamic Cutting Force Coefficients and Chatter Stability with Process Damping", CIRP Annuals - Manufacturing Technology, 57, pp. 371-374.
[4] Das, M.K., Tobias, S.A., 1967, "The Relation Between the Static and the Dynamic Cutting
Forces of Metals", International Journal of Machine Tool Design and Research, 7, pp. 63-89.
[5] Insperger, T., Gradišek, J., Kalveram, M., Stépán, G., Weinert, K., Govekar, E., 2006, "Machine Tool Chatter and Surface Location error in Milling Processes", Journal of Manufacturing Science and Engineering, 128(4), pp. 913-920.
[6] Insperger, T., Stépán, G., 2004, "Stability Analysis of Turning with Periodic Spindle Speed
Modulation via Semi-Discretization", Journal of Vibration and Control, 10, pp. 1835-1855.
[7] Insperger, T., Stépán, G., 2004, "Updated Semi-Discretization Method for Periodic DelayDifferential Equations with Discrete Delay", International Journal for Numerical Methods
in Engineering, 61, pp.117-141.
[8] Stépán, G., 2001, "Modelling Nonlinear Regenerative Effects in Metal Cutting", Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 259(1781), pp 739-757
36
A
Derivation of W (ω) and Ω(ω)
In Section 3.4 the equations of the dimensionless depth of cut W and dimensionless spindle
speed Ω are used to explain the basic shape of the stability chart. To this end, the minimum
points and asymptotes are determined. This section derives the equations for the minimum
points and asymptotes step by step.
In Section 3.3 the following two equations for the dimensionless depth of cut and spindle speed
were derived (Equations 3.14 and 3.18 respectively in Section 3.3):
W (ω) =
(ω 2 − 1)2 + 4ζ 2 ω 2
,
2(ω 2 − 1)
πω
Ω(ω) =
arctan
1−ω 2
2ζω
(A.1)
(A.2)
.
+ NL π
The minimum point of a lobe can be calculated by first determining the dimensionless depth of
cut at this point. This is done by differentiating W (ω) with respect to ω. Using the chain rule,
this derivative is:
2(ω 2 − 1)(4(ω 2 − 1)ω + 8ζ 2 ω) − ((ω 2 − 1)2 + 4ζ 2 ω 2 )4ω
dW
=
.
dω
4(ω 2 − 1)2
(A.3)
Simplification of this equation obtains:
dW
4ζ 2 ω
=ω− 2
.
dΩ
(ω − 1)2
(A.4)
Having this equation, the vibration frequency for which (A.4) equals zero is calculated:
p
ωmin = 1 + 2ζ.
(A.5)
This frequency is substituted for ω in (A.1) and the resulting depth of cut is called the minimum
depth of cut Wmin :
(A.6)
Wmin = 2ζ(1 + ζ).
With this equation the depth of cut at the minimum point in the lobe is known.
Second, the dimensionless spindle speed at the minimum point of a lobe is determined. This is
done by substitution of the vibration frequency in the minimum point, ωmin , for ω in (A.2):
√
π 1 + 2ζ
.
(A.7)
Ωmin =
1
+ NL π
arctan − √1+2ζ
√
The damping ratio ζ is about 0.02 in metal cutting processes, and therefore the term 1 + 2ζ
is approximately 1. With arctan(−1) = − 14 , the dimensionless spindle speed at the minimum
point in the lobe is:
Ωmin =
1
.
NL − 14
(A.8)
37
The location of the minimum point in a lobe is known, and now expression of the asymptote of
a lobe is derived. The depth of cut W goes to infinity as the vibration frequency reaches 1 as a
result of the denominator being 2(ω 2 − 1). This asymptote lies at the point Ω(ω) = Ω(1), so by
using arctan(0) = 0:
Ωas =
1
.
NL
(A.9)
38
B
Derivation of W (ω) and Ω(ω) with ζ = 0
In Section 3.7 the intersections of the lobes were found analytically by assuming that ζ = 0. This
assumption results in a new characteristic equation, from which new equations for Ω and W as
function of the vibration frequency ω result. These two equations are derived in this section.
The characteristic equation D(λ) = 0 for the dimensionless orthogonal cutting was derived in
Section 3.3 and is given again in (B.1):
D(λ) = λ2 + 2ζλ + 1 + W − W e−λτ .
(B.1)
Assuming that ζ = obtains:
D(λ) = λ2 + 1 + W − W e−λτ .
(B.2)
As the border between the asymptotically stable and instable region is interesting, i.e. when the
system is critically stable, λ = iω. Substitution of this in (B.2) results in the following term in
the characteristic equation: −W e−iωτ . This term can be rewritten in a function with a sine and
cosine according to Euler’s formula in complex functions:
(B.3)
e−iωτ = cos(−ωτ ) + i sin(−ωτ ).
Using cos(−x) = cos(x) and sin(−x) = − sin(x):
D(ω) = −ω 2 + 1 + W − W cos(ωτ ) + iW sin(ωτ ).
(B.4)
As D(ω) = 0, both the real and the imaginary part of this equation are zero:
(B.5a)
iW sin(ωτ ) = 0,
2
(B.5b)
W − W cos(ωτ ) = ω − 1.
From Equation B.5a it follows that ωτ = NLζ=0 π. Using τ =
as function of the vibration frequency ω is:
Ω(ω) =
2π
Ω,
the dimensionless spindle speed
2ω
.
NLζ=0
(B.6)
NLζ=0
Substitution of ωτ = NLζ=0 π in Equation B.5b and using cos(NLζ=0 π) = (−1)
W (ω) =
ω2 − 1
NLζ=0
1 − (−1)
results in:
(B.7)
.
39
C
Proof of reordering exponential matrix
In Section 4.1 the delay term of a delay differential equation was approximated using semidiscretization and an inhomogeneous ordinary differential equation remained. This section
solves the inhomogeneous ODE by using the method of variation of coefficients [1].
The general solution yi (t) of (4.6) consists of the homogeneous or transient part yiT (t) and the
inhomogeneous or particular part yiP (t). The transient part is the solution of (C.1).
(C.1)
ẏiT (t) = LyiT (t),
with the following general solution:
yiT (t) = eL(t−ti ) c0 ,
(C.2)
with c0 a parameter, c0 = yiT (ti ). The particular solution of the inhomogeneous ODE is the
solution of the following equation:
1
ẏiP (t) = LyiP (t) + R(yi−m+1 + yi−m ),
2
(C.3)
with as general solution:
yiP (t) = eL(t−ti ) c(t).
(C.4)
Differentiating (C.4) with respect to time t yields a differential equation for the unknown function
c(t):
ẏiP (t) = LeL(t−ti ) c(t) + eL(t−ti ) ċ(t).
(C.5)
When (C.5) is substituted for ẏiP (t) in (C.3), and (C.4) is substituted for yiP (t) in (C.3), the following equation results:
1
LeL(t−ti ) c(t) + eL(t−ti ) ċ(t) = LeL(t−ti ) c(t) + R(yi−m+1 + yi−m ).
2
(C.6)
From this equation, an expression for the parameter c(t) can be derived, which is:
1
c(t) = − L−1 e−L(t−ti ) R(yi−m+1 + yi−m ).
2
(C.7)
This equation can be substituted into (C.4):
1
yiP (t) = − eL(t−ti ) L−1 e−L(t−ti ) R(yi−m+1 + yi−m ).
2
(C.8)
Due to the following property of exponential matrices, the matrices in (C.8) can be reordered
such that the eL(t−ti ) term and the e−L(t−ti ) term cancel:
eY XY
−1
= Y eX Y −1 .
(C.9)
40
The proof of this is given below:
A2 A3
+
+ ...
2!
3!
Y XY −1 , then:
(Y XY −1 )2 (Y XY −1 )3
I + Y XY −1 +
+
+ ...
2!
3!
Y X 2 Y −1 ,
X2 X3
+
Y −1 = Y eX Y −1 .
Y I +X +
2!
3!
|
{z
}
eA = I + A +
Take A =
eY XY
−1
=
and with (Y XY −1 )2 =
eY XY
−1
=
(C.10)
(C.11)
(C.12)
(C.13)
(C.14)
eX
This property is used to simplify Equation C.8 in the following way:
1
yiP (t) = − eL(t−ti ) L−1 e−L(t−ti ) R(yi−m+1 + yi−m )
2
multiply both sides with L:
1 L(t−ti ) −1 −L(t−ti )
LyiP (t) = − Le
R(yi−m+1 + yi−m ), then:
{z L } e
2|
(C.15)
(C.16)
−1
eLL(t−ti )L
1
yiP (t) = − L−1 R(yi−m+1 + yi−m ).
2
(C.17)
In this way, an expression for yi (t) is derived, in which only the parameter c0 is unknown:
1
yi (t) = eL(t−ti ) c0 − eL(t−ti ) R(yi−m+1 + yi−m ).
2
This problem is solved by investigating the solution at time t = ti :
1
c0 = yi + R(yi−m+1 + yi−m ).
2
(C.18)
(C.19)
Substituting this in (C.18), the derivation for the solution of the inhomogeneous differential equation is complete.
41
D
Background of simplification of semi-discretization method
In Section 4.2 the semi-discretization method is used to compute system matrices with which
there stability of the system can be investigated. This section derives the system matrices.
The first order discretized delay differential equation was derived in Section 4.1 and (4.9) is given
below:
yi+1 = eL∆t yi +
1 L∆t
e
− I L−1 R(yi−m+1 + yi−m ).
2
(D.1)
This equation is rewritten using one new system matrix B:

yi+1
yi
..
.





 yi−m+3

 yi−m+2
yi−m+1


 
 
 
 
=
 
 
 
eL∆t O . . .
I
O ...
..
.. . .
.
.
.
O
O ...
O
O ...
O
O ...

yi
O A A
 yi−1
O O O 


.. ..
..

. .
.



O O O   yi−m+2
I O O   yi−m+1
O I O
yi−m





.



(D.2)
The matrix is called B, in which I is the identity matrix, O the null matrix and A is
A = 21 (eL∆t − I)L−1 R. The two vectors are called zi+1 and zi respectively:

yi
 yi−1


..

.
zi = 
 yi−m+2

 yi−m+1
yi−m
with yi = [y1,i


yi+1
yi
..
.








 and zi+1 = 
 yi−m+3



 yi−m+2

yi−m+1





,



(D.3)
y2,i ]T , etc. Now, zi+1 = Bzi .
The size of matrix B is 2(m + 1). This size can be reduced for the orthogonal cutting process as
the delay only occurs in the position of the tool x(t) and not in the velocity of the tool ẋ(t). From
this it follows that yi−1 = [y1,i−1 0]T , yi−2 = [y1,i−2 0]T etc, and:

y1,i+1
y2,i+1
y1,i
yi,2
..
.









 y1,i−m+2


0

 y1,i−m+1
0


 
 
 
 
 
 
 
=
 
 
 
 
 
 
E(1, 1) E(1, 2) 0 0
E(2, 1) E(2, 2) 0 0
1
0
0 0
0
1
0 0
..
..
.. ..
.
.
. .
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0

y1,i
. . . A(1, 1) A(1, 2) A(1, 1) A(1, 2)


y2,i
. . . A(2, 1) A(2, 2) A(2, 1) A(2, 2)  
  y1,i−1
...
0
0
0
0


0
...
0
0
0
0


..
.
.
.
.
..
.
.
.
.

.
.
.
.
.
.



...
0
0
0
0
  y1,i−m+1

0
...
0
0
0
0



y1,i−m
...
1
0
0
0
0
...
0
1
0
0
(D.4)
with E = eL∆t .
42








.






From this it follows easily that after the fifth row and column, each even row and column can be
omitted, A(1, 2) and A(2, 2) are zero, and the following matrix results:


E(1, 1) E(1, 2) 0 0 . . . 0 0 A(1, 1) A(1, 1)
 E(2, 1) E(2, 2) 0 0 . . . 0 0 A(2, 1) A(2, 1) 




1
0
0
0
.
.
.
0
0
0
0




0
1
0 0 ... 0 0
0
0
B=
(D.5)
.


..
..
.. .. . . .. ..
..
..


. . .
.
.
. .
.
.




0
0
0 0 ... 0 1
0
0
0
0
0 0 ... 0 0
1
0
43
E
Derivation of the dimensionless equation of motion for process damping
In Section 5.1 the equation of motion for an orthogonal cutting process with process damping was
derived. The method for converting an equation in a parameter space with dimensions into an
equation in a dimensionless parameter space was already explained in Section 3.2. This method
is used in Section 5.1 for the derivation of (5.8), however it is only briefly explained how this
equation and the dimensionless parameters Cpd and Dpd are derived. This is explained in this
appendix.
The equation of motion in a parameter space with dimensions for orthogonal cutting with process
damping, (5.6) is given below:
Cc wτ 2
m+
4π 2 r2
Cf wτ
ẍ(t) + c +
2πr
(E.1)
ẋ(t) + (k + wk1 )x(t) = wk1 x(t − τ ),
with Cc [N] and Cf [N/m] the process damping coefficients. Dividing this by the mass m obtains:
Cc wτ 2
1+ 2 2
4π r m
ẍ(t) +
Cf wτ
c
+
m 2πrm
ẋ(t) +
k + wk1
m
x(t) =
wk1
x(t − τ ).
m
(E.2)
Using (3.6):
c
√
,
2 km
r
k
=
,
m
(E.3a)
ζ =
ωn
(E.3b)
results in:
Cf wτ
Cc wτ 2
wk1
wk1
2
1+ 2 2
x(t − τ ). (E.4)
ẍ(t) + 2ζωn +
ẋ(t) + ωn +
x(t) =
4π r m
2πrm
m
m
Substituting the following dimensionless parameters (as defined in (3.8)) for t and τ
t̃ := t ωn ,
τ̃
(E.5a)
→ dt̃ = dt ωn ,
:= τ ωn ,
(E.5b)
(E.5c)
results in the following equation:
1+
Cc wτ̃ 2
4π 2 r2 mωn2
ωn2 ẍ(t̃) +
Cf wτ̃
2ζωn +
2πrmωn
ωn ẋ(t̃) +
ωn2
wk1
+
m
x(t̃) =
wk1
x(t̃ − τ̃ ).
m
(E.6)
Dividing this equation by ωn2 results in:
1+
Cc wτ̃ 2
4π 2 r2 mωn2
Cf wτ̃
wk1
wk1
ẍ(t̃) + 2ζ +
ẋ(t̃) + 1 +
x(t̃) =
x(t̃ − τ̃ ). (E.7)
2
2
2πrmωn
mωn
mωn2
44
The coefficient
wk1
2
mωn
is called the dimensionless depth of cut W̃ . The two coefficients due to
C w
f
process damping, 4πC2crw2 m and 2πrmω
2 , can be now divided into two dimensionless parameters,
n
namely W̃ , and D˜pd and C˜pd respectively, with
Ic =
If
=
Cc
,
4π 2 r2 k1
Cf
.
2πrk1
(E.8)
(E.9)
Omitting the tilde for convenience and substitution W for
Cf w
2
2πrmωn
wk1
2 , Ic W
mωn
for
Cc w
2
4π 2 r 2 mωn
and If W for
obtains the dimensionless equation of motion:
(1 + Ic W τ )ẍ(t) + (2ζ + If W τ )ẋ(t) + (1 + W )x(t) = W x(t − τ ).
45
(E.10)
F
Matrices Ipd (t), Cpd (t) and Kpd (t)
In Section 6.4 the friction and contact force during milling were derived. These two forces were
added to the equation of motion for a milling process, Equation (6.14), and three new matrices
were defined: Ipd (t), Cpd (t) and Kpd (t). These three matrices follow from the contact and friction
force as defined in Equations (6.35) and (6.36). For each tooth j, the matrices are:
1
sin2 φj (t)
cos φj (t) sin φj (t)
Ipd,j (t) = Cc ap 2 2
(F.1a)
cos φj (t) sin φj (t)
cos2 φj (t)
r Ω
1
sin2 φj (t)
cos φj (t) sin φj (t)
+
Cpd,j (t) = −Cf ap
cos2 φj (t)
rΩ cos φj (t) sin φj (t)
2
cos φj (t) sin φj (t)
− sin2 φj (t)
(F.1b)
+Cc ap 2
cos2 φj (t)
− cos φj (t) sin φj (t)
r Ω
1 cos φj (t) sin φj (t)
− sin2 φj (t)
+
Kpd,j (t) = −Cf ap
cos2 φj (t)
− cos φj (t) sin φj (t)
r
Ω2
− sin2 φj (t)
− cos φj (t) sin φj (t)
+Cc ap 2
.
(F.1c)
− cos φj (t) sin φj (t)
− cos2 φj (t)
r
To obtain the three matrices Ipd (t), Cpd (t) and Kpd (t) for all teeth the matrices are evaluated for
each tooth and summed:
Ipd (t) = g(φj (t))
N
X
Ipd,j (t),
j=1
Cpd (t) = g(φj (t))
N
X
Cpd,j (t)and
N
X
Kpd,j (t).
j=1
Kpd (t) = g(φj (t))
j=1
46
G
Parameters used for stability charts for milling
In Section 6.3 the stability lobes for half immersion down-milling were constructed. The values
of the parameters used are found in Table 1.
Parameter
M [N]
Value
0.0199
0
0
0.0201
C[Ns/m]
1.8043
0
0
3.6445
K[N/m]
Kt [MPa]
Kr [MPa]
Cf [N/m]
Cc [N]
m
Nint
φen [rad]
φex [rad]
r [m]
fd [mm]
4.0900 ∗ 105
0
0
4.1300 ∗ 105
644
237
6.44 ∗ 106
2370
50
40
π/2
π
0.035
−1.6 ∗ 10−4
Table 1: The values of the parameters used to construct Figures 6.4 and 6.6
47
Acknowledgements
I would like to thank several people who made it possible for me to do my research project at
the University of British Columbia in Vancouver, Canada. First of all I would like to thank Prof.
Nijmeijer who introduced me to Prof. Altintas and I would like to thank Prof. Altintas for inviting me to his Manufacturing and Automation Laboratory at the UBC. Going to Canada for five
months would not have been possible without all the support my parents gave me and I owe them
many thanks. At the MAL, I would like to thank Zoltan Dombóvári for all his help during my
stay there, and also for his help with my report after my return to The Netherlands. I would like
to thank all the other guys cheerfulness in the MAL. Furthermore, I would like to thank Nicole
Kommer for her help with my report, especially when it comes to English language. Lastly, I
would like to thank Mark Nievelstein, who keeps supporting me in every possible way while he
had to deal with when I was entirely happy for going to Canada, or completely stressed while
writing my report, and every emotion in between.
48