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. NijmeijerProf. Y. Altintas

Eindhoven University of TechnologyDepartment Mechanical EngineeringDynamics and Control Group

Eindhoven, December, 2008

Summary

The main purpose of this project is to investigate the stability of a two degree-of-freedom machi-ning process, by taking process damping into account and using semi-discretization to investigateits equation of motion. By constructing a so-called stability chart in which the stability of the cut-ting process is indicated for a range of depth of cuts and spindle speeds, the stability for a certaincutting process can be determined. The stability chart is divided into a stable area and an unstablearea, which are separated by stability lobes.

First a model with one degree of freedom (DOF) is investigated, namely the orthogonal cuttingprocess. The model describes the regenerative effect of a machining process. This effect, themost fundamental effect during machining, occurs due to the returning pattern on the workpieceinduced by the tool vibration. Thus, the current tool position and the tool position in the past arerelated. The lag between the two tool positions can be taken into account by one constant delayterm in the equation of motion. The resulting equation of motion thus contains a delay term andis therefore called a delay differential equation.

After the equation of motion is formulated for orthogonal cutting, the influence of the depthof cut and the spindle speed of the tool is investigated analytically and the stability lobes areconstructed. This analytical method is appropriate to understand the major relations betweenthe cutting parameters in orthogonal cutting, however it is not suitable for cutting processes withtwo or more degrees of freedom. For these models, semi-discretization is used to investigate thedelay equation of motion numerically. Semi-discretization is a robust method with which linearfirst order delay differential equations can be investigated. This method discretizes the infinitedimensional phase space of the system and formulates a finite linear map. The accuracy of thismethod is related to the step size chosen to discretize the infinite dimensional phase space. Thestep size strongly depends on the ratio between the constant delay and the period of the occurringperiodic orbit on the stability border.

One effect that should be taken into account when considering metal cutting processes is processdamping. Process damping can result in additional damping and inertia in the cutting processat low spindle speeds; the experimentally found stability lobes lay much higher in the stabilitychart than the theoretically found lobes. Although much research has been undertaken, little isknown about the cause of the process damping. Therefore, several models are used to describethe process damping. One of them assumes that it is described by the addition of one lineardamping term and one linear inertia term to the equation of motion of the orthogonal cuttingprocess. The resulting equation of motion is investigated by semi-discretization and the resultingstability 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 modelthat can be used to describe a milling process. In milling, the teeth of the tool enter and exit theworkpiece, resulting in the state matrix that determines the stability of the system being time-dependent. Therefore, this state matrix is calculated by numerical integration. The stability ofthis cutting process depends also on the number of teeth of the tool: an increase in the toothnumber results in a decrease of the stable region. Process damping is added to this model byassuming that this effect can be described by using the spatial derivative and the second spatialderivative of the radial position of the tool.

ii

Samenvatting

Het doel van dit onderzoek is het bepalen van de stabiliteit van een verspaningsproces met tweegraden van vrijheid, waarbij het effect van proces demping in de modellering is opgenomen engebruik gemaakt wordt van semidiscretisatie voor het oplossen van de bewegingsvergelijkingen.Met behulp van een zogeheten stabiliteitsgrafiek, waarin de stabiliteit van het proces is aangege-ven voor een bereik van snijdieptes en -snelheden, kan de stabiliteit van een verspaningsprocesworden bepaald. De grafiek is in een stabiel en een instabiel gebied verdeeld, welke zijn geschei-den door stabiliteitslobben.

Allereerst is een model onderzocht met één vrijheidsgraad, namelijk het orthogonale verspa-ningsproces. Het model beschrijft het zogenaamde regenerative effect van een verspanings-proces. Dit is het belangrijkste effect dat optreedt tijdens het verspanen en is te wijten aan hetterugkerende patroon op het werkstuk dat veroorzaakt wordt door trillingen van de beitel. Hier-door zijn de huidige beitel positie en de positie in het verleden aan elkaar gerelateerd. Dezerelatie wordt meegenomen in de bewegingsvergelijking door het toevoegen van een constantevertraging. De bewegingsvergelijking heeft dus een zogeheten vertragingsterm, en wordt nu de-lay differentiaalvergelijking genoemd.

Na het afleiden van de bewegingsvergelijking voor het orthogonale verspaningsproces, wordt deinvloed van de snijdiepte en snijsnelheid op de stabiliteit van het systeem analytisch bepaald, enworden de stabilteitsgrafieken getekend. Deze analytischemanier is een goedemanier om de gra-fiek te tekenen voor een eenvoudig model met slechts één vrijheidsgraad, maar kan niet gebruiktworden als het model complexer wordt. In dergelijke gevallen kan semidiscretisatie worden ge-bruikt om de delay differentiaalvergelijkingen numeriek op te lossen. Semidiscretisatie is eenrobuuste methode waarmee lineaire eerste orde delay differentiaalvergelijkingen kunnen wordengeanalyseerd. Deze methode discretiseert de oneindig dimensionale toestandsruimte van hetsysteem en formuleert een eindige lineaire vergelijking. De nauwkeurigheid van deze metho-de 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 deperiodieke baan op de stabiliteitsgrens.

Een effect dat mee genomenmoet worden bij het onderzoek naar stabiliteit in verspaningsproces-sen is proces demping. Proces demping kan bij lage snijsnelheden extra demping en inertia inhet verspaningsproces teweeg brengen: de experimentele stabiliteitslobben liggen namelijk veelhoger in de stabiliteitsgrafiek dan de theoretisch gevonden stabiliteitslobben. Hoewel er al veelonderzoek is gedaan, is er weinig bekend over de oorzaak van proces demping. Daarom wordenmomenteel nog verschillende modellen gebruikt voor het beschrijven van proces demping. Eenvan deze modellen neemt aan dat proces demping kan worden beschreven door toevoeging vanlineaire demping en lineaire inertia aan de bewegingsvergelijking van het orthogonale verspa-ningsproces. De hieruit volgende bewegingsvergelijking is geanalyseerd met semidiscretisatieen de resulterende stabiliteitsgrafiek wijst uit dat het stabiele gebied is vergroot bij kleine snij-snelheden.

Het model van het orthogonale verspaningsproces met één vrijheidsgraad wordt gebruikt in eenmodel met twee vrijheidsgraden, waarmee een freesproces beschreven wordt. Bij frezen is desysteemmatrix waaruit de stabiliteit van het systeem wordt bepaald tijdsafhankelijk, doordat detanden van de frees het werkstuk ingaan en ook weer verlaten. Daaromwordt dezematrix bepaald

iii

door numerieke integratie. De stabiliteit van het freesproces hangt af van het aantal tanden op defrees: een vergroting van het aantal tanden resulteert in een verkleining van de stabiele gebiedenin de stabiliteitsgrafiek. Ook aan dit model wordt proces demping toegevoegd, door aan te nemendat proces demping in een freesproces kan worden benaderd door de ruimtelijke afgeleide en detweede 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 15

4.1 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

5 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

6 Stability Chart for Milling 25

6.1 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

v

6.5 Construction of stability chart for milling with process damping . . . . . . . . . . 34

7 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

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1 Introduction

For the manufacturing of metal parts of e.g. cars and planes, turning and milling are two com-monly used metal cutting processes. In turning, the workpiece rotates, while the fixed tool cutsin the workpiece; in milling the tool rotates and cuts in the fixed workpiece. During the pastdecades, much research has been undertaken in the field of stability during a turning or millingprocess. One laboratory that investigates this stability problem is the Manufacturing AutomationLaboratory at the Department of Mechanical Engineering of the University of British Columbiain 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, andeither the tool or the workpiece has a rotational speed, resulting in a cutting speed. The feed ofthe cutting process is determined by a translational motion of the workpiece or the tool. In theideal 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 theworkpiece. This wave can magnify the tool vibrations which results in such heavy tool vibrationsthat 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 exactcause of process damping is still unknown, it is recently accepted that the effect is induced by thefriction force between the flank edge of the tool and the workpiece. This force induces velocitydependent 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 cuttingparameters 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 chartso-called lobes are plotted which indicate the border between the stable region and the unstableof the cutting process.

In order to construct this stability chart, first the equation of motion for the cutting process isderived. This equation is not an ordinary differential equation (ODE) because the tool positiondepends on the past shape of the surface. The surface of the workpiece is cut by the tool onerotation of the tool or workpiece earlier. So, the current tool position depends on the history ofthe 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 ofcharacteristic 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 usingsemi-discretization, which discretizes and approximates the delay terms. The approximation isused 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 isdone in the field of chatter in metal cutting processes in general and process damping in particu-lar. The process damping is investigated especially in turning, but there is only few informationabout process damping in milling. Therefore, the aim of this report is to construct stability lobesfor process damping in milling using one of the possible explanations of this effect. The DDEsare 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 processesis given in Chapter 2, after which a stability chart is constructed for the one degree-of-freedomorthogonal cutting process in Chapter 3. Chapter 4 explains how the delay differential equa-tion that describes the orthogonal cutting process is solved by discretizing the delay term. InChapter 5 process damping is added to the orthogonal cutting process. Having explained semi-discretization and process damping for the one degree-of-freedom model, the theory is extendedto a two degree-of-freedom model used to describe milling in Chapter 6. Lastly, Chapter 7 dis-cusses 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 thischart, the border between the stable and the unstable region is indicated. This chapter explainsthe stability chart and its shape.

During a metal cutting process, a tool cuts in the workpiece with a depth of cut and a certainvelocity with respect to the workpiece, see Figure 2.1. The graph on the left side is a schematicpicture of a turning process in which the workpiece rotates with spindle speed Ω, the figure onthe right shows a milling operation in which the tool rotates with spindle speed Ω. The depthof cut w is also indicated in both graphs. Furthermore, in both graphs the regenerative effect isvisible: the tool cuts on the wavy workpiece surface, induced by tool vibrations in the past.

tool

work-pieceW

w

p

w

2

0h

tool

work-pieceW

v

w

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 systemcan become unstable. To determine the stability for a certain cutting process, a stability chart isdrawn. In this chart so-called lobes are plotted that indicate the border between the stable and theunstable 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 orthogonalcutting process

In the figure, a stable and an unstable region are indicated; the border between the two regionconsists 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 byfour solid lines. Each lobe has a vertical asymptote on its left side, indicated in the figure by a thindashed line, and a minimum point, which is located on the thick dashed line. This thick dashedline is the so-called minimum depth of cut Wmin. If a depth of cut below this line is chosen, thecutting process is always stable, regardless of the spindle speed.

As mentioned before, the lobes indicate the border between the asymptotically stable and the un-stable region. So, on the border the system is critically stable and the tool vibrates with vibrationfrequency ω. For the critically stable orthogonal cutting process two equations can be derivedanalytically: 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: NLπ. In this term, NL is thelobe number and NL = 1 indicates the rightmost lobe in the stability chart. Using these twofunctions, the lobes can be constructed analytically by evaluating the two functions for a range ofvibration 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 areconstructed analytically in Section 3.5.

4

3 Basic Linear Stability Chart of the Orthogonal Cutting Pro-

cess

This chapter deals with the explanation and construction of stability charts for the orthogonal cut-ting process. In these charts so-called lobes are plotted, which indicate the border between stablestationary cutting and the unstable cutting process. The lobes can be constructed analytically bycalculating the dimensionless spindle speed and the dimensionless depth of cut for a range ofvibration frequencies. To this end, first the orthogonal cutting process is explained, after whichthe equation of motion of this particular cutting process is derived, which is used to investigatethe analytical part of the construction of the lobe structure. Then, the structure of the lobes isimplemented in a Matlab code and the plotted lobes are validated. Lastly, the system’s behaviorin 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 processon the left hand side is projected schematically in the right side of the figure. This picture showsthe workpiece that rotates with spindle speed Ω and the tool that cuts in the workpiece with depthof cut w. The tool has mass m, and is assumed to be connected to the toolholder by a spring withspring stiffness k and a damper with damping coefficient c. The tool cut results in a chip withdesired chip thickness h0 and momentary chip thickness h(t). The momentary position of thetool is x(t) and as the workpiece rotates, the current tool position depends also on the position ofthe 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 inthe equation of motion of the orthogonal cutting process, see Section 3.2. In this figure, the tool

tool

work-pieceW

w

p

w

2

0h

h0

ht()

x t( )F

W

x

m

k c

tool

work-piece

chip

x t-( )t

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. Asthe 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 anorthogonal 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 mo-tion for the one degree-of-freedom model that describes this cutting process. To this end, thesystem is modeled as a 1 DOF linear mass-spring-damper system excited by a nonlinear cuttingforce, [8]:

mx(t) + cx(t) + k(q0 + x(t)) = wfa(h(t)). (3.1)

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 staticpart 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 bederived form Figure 3.1 and depends on the current tool position x(t) [m], the position at theprevious revolution x(t − τ) and the desired chip thickness h0 [8]:

h(t) = x(t − τ) − x(t) + h0. (3.2)

In this equation, τ [s] is the delay of the vibration and is equal to the duration of one revolutionof the workpiece, i.e. τ = 2π/Ω, with Ω the spindle speed [rad/s]. The fa(h(t)) term can beapproximated by the Taylor series around the desired chip thickness h0:

fa(h(t)) = fa0 +∂fa

∂h

∣∣∣h=h0

(h(t) − h0) + H.O.T. (3.3)

The higher order terms (H.O.T.) can be neglected and fa(h(t)) is expressed as

fa(h(t)) = fa0 + k1(h(t) − h0), (3.4)

with fa0 the initial unit force due to static deflection, so wfa0 = kq0, and k1 = ∂fa

∂h

∣∣∣h=h0

the

cutting coefficient. The second term in (3.4) is the dynamic force term. Substitution of (3.2) and(3.4) into (3.1) obtains:

mx(t) + cx(t) + (k + wk1)x(t) = wk1x(t − τ). (3.5)

This is the linear equation of motion for the orthogonal cutting process and it is now transformedinto a dimensionless parameter space. To this end, first the whole equation is divided bym. Then,the following two expressions are used:

ζ :=c

2√

km, (3.6a)

ωn :=

√

k

m, (3.6b)

with ζ the (dimensionless) damping ratio and ωn the natural angular frequency [rad/s] of thesystem. For general turning operations, ζ is small; usually ∼ 0.02 [2]. (3.5) is now rewritten into:

x(t) + 2ζωnx(t) +

(

ω2n +

k1w

m

)

x(t) =k1w

mx(t − τ). (3.7)

6

This equation is dimensionless after the following substitutions:

t := t ωn, → dt = dt ωn, (3.8a)

τ := τ ωn, (3.8b)

in which t is the dimensionless time and τ is the dimensionless delay. After dividing (3.7) by ω2n,

and introducing the dimensionless depth of cut W = k1wm ω2

n, a dimensionless equation of motion

that depends on only two system parameters results:

x(t) + 2ζx(t) + (1 + W )x(t) = Wx(t − τ). (3.9)

From now on, all equations in this chapter are dimensionless, therefore the tilde is dropped forconvenience.

In this section, the dimensionless second order delay differential equation was derived for anorthogonal cutting process. This equation is used in the next section to derive the separate equa-tions for the dimensionless depth of cut W and the dimensionless spindle speed Ω as functionof 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 pro-cess and its characteristic equation is used in this section to formulate the equations of the di-mensionless depth of cut W and the dimensionless spindle speed Ω. To derive the characteristicequation 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 x(t) and x(t), which are substituted into (3.9). Dividing allterms by beλt one obtains the characteristic equation D(λ) = 0:

D(λ) = λ2 + 2ζλ + 1 + W − We−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 partof an eigenvalue is zero, iω is substituted for λ. Using Euler’s formula in complex analyses, thefollowing equation results:

D(ω) = −ω2 + 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 betweenthe stable and the unstable region can be derived. To this end, (3.12) is split into the real and theimaginary part and as both should be zero, the following two equations are obtained [8]:

− ω2 + 1 + W = W cos(ωτ), (3.13a)

2ζω = −W sin(ωτ). (3.13b)

When both equations are squared and then added together, the following expression for the di-mensionless depth of cutW of the critically stable cutting process as function of ω can be derived:

W (ω) =(ω2 − 1)2 + 4ζ2ω2

2(ω2 − 1). (3.14)

7

From this equation it follows that the dimensionless vibration frequency ω should be larger than1 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 )

1 + tan2(ωτ2 )

, (3.15a)

cos(ωτ) =1 − tan2(ωτ

2 )

1 + tan2(ωτ2 )

. (3.15b)

By substitution of (3.15a) into (3.13b) a new expression for W (ω) is obtained:

W (ω) = −1 + tan2(ωτ2 )

tan(ωτ2 )

ζω (3.16)

Substituting this equation into (3.13a) obtains the following formula for the dimensionless delay,[8]:

τ(ω) =2

ω

[

arctan

(1 − ω2

2ζω

)

+ NLπ

]

. (3.17)

Having an expression for the dimensionless delay, the expression for the dimensionless spindlespeed Ω(ω) is obtained too:

Ω(ω) =2π

τ(ω)= πω

[

arctan

(1 − ω2

2ζω

)

+ NLπ

]−1

. (3.18)

In this equation, NL is the integer number that indicates the number of the lobe, the borderbetween the stable and the unstable region of the stability chart in Figure 2.2. From (3.18) itfollows that NL ≥ 1 as a negative spindle speed results when NL = 0. Therefore, the numberingof 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 spindlespeed 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 roughlysketch the shape of the stability lobes if the damping ratio ζ of the system is known. Namely, withthis information, the minimum value of all lobes can be approximated together with the verticalasymptotes, see also Figure 2.2.

First, the minimum value of a lobe is calculated by differentiating the equation for the dimen-sionless depth of cut W (ω), (3.14), with respect to the dimensionless frequency ω. The frequencyfor which the derivative equals zero is called the critical frequency ωcrit, and the correspondingdepth of cut is the minimum depth of cut Wmin as for all cutting depths below this value thecutting process described by (3.9) is always stable. The expression for the critical frequency isderived step by step in Appendix A and is given as:

ωcrit =√

1 + 2ζ. (3.19)

8

Substitution of this equation into (3.14) yields the minimum value of the dimensionless depth ofcut for all lobes:

Wmin = 2ζ(1 + ζ). (3.20)

By substituting (3.19) into (3.18) and assuming that ζ is very small, which is true for orthogonalcutting processes [2], obtains the value for the dimensionless spindle speed in the minimumpoint for each lobe NL = 1, 2, ... is obtained:

Ωmin = Ω(ωcrit) ≃1

NL − 14

. (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 dimensionlessdepth of cut reaches infinity when ω approaches 1. Substituting ω = 1 into (3.18), the location ofthe vertical asymptote is determined for each lobe:

limω→1

Ω(ω) =1

NL

. (3.22)

The location of the asymptotes and minimum points of a lobe was explained in this section, withwhich the rough shape of the stability chart for a cutting process with a certain damping ratio ζcan 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 ωslightly larger than 1, which followed from (3.14). The intersections of the lobes can be found byfinding the vibration frequencies ω for which the values of W and Ω(ω) of the N th

L lobe have thesame values as for lobe NL +1. Finding these intersections is interesting because the behavior ofthe system near the intersections is rather different from its behavior close to the minimum pointof a lobe, which is shown in Section 3.6. Furthermore, knowledge of the locations of these inter-sections enables plotting only the interesting parts of the lobes. The locations of the intersectionsis 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 cutWmin. Using (3.20) and (3.21) for the location of the minimum points in the lobes and (3.22) forthe location of the vertical asymptote of a lobe, it is clear that they are located in Figure 3.2 asexpected. Furthermore, four points are indicated in the figure; in Section 3.6 time simulationsof these points are performed. In Figure 3.3 the dimensionless vibration frequency ω is plottedagainst the dimensionless spindle speedΩ(ω). This figure shows that on the intersection betweentwo lobes, for example at Ω(ω) = 1, the vibration frequency actually has two values. The resultingeffect 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

0 0.5 1 1.5 2 2.5 31

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Dimensionless spindle speed Ω

Dim

ensi

onle

ss fr

eque

ncy

ω

Stability chart

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 thefour different locations indicated in the stability chart in Figure 3.2: point 1 lies in the asymptoti-cally stable region, point 2 in the unstable region, point 3 on the critically stable border and point4 near an intersection between two lobes. The resulting vibrations are shown in Figure 3.4.

0 20 40 60 80 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10

−3 W= 0.01; Ω= 0.575

Dimensionless time

Dis

plac

emen

t x

(a) Point 1 (Ω=0.575, W =0.01)

0 20 40 60 80 100−0.015

−0.01

−0.005

0

0.005

0.01

0.015W= 0.2; Ω= 0.575

Dimensionless time

Dis

plac

emen

t x

(b) Point 2 (Ω=0.575, W =0.2)

0 20 40 60 80 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10

−3 W= 0.0435; Ω= 0.575

Dimensionless time

Dis

plac

emen

t x

(c) Point 3 (Ω=0.575, W =0.04104)

0 20 40 60 80 100−8

−6

−4

−2

0

2

4

6

8

10x 10

−4 W= 0.31; Ω= 0.505

Dimensionless time

Dis

plac

emen

t x

(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 inFigure 3.2

From these figures it follows that the system responds as one would expect from Figure 3.2: in thestable region the system is indeed asymptotically stable, and in the unstable region the system isunstable. 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 mentionedbefore, two vibration frequencies are present, the system is stable as the amplitude does notincrease 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 twofrequencies. 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 inter-sections decreases for increasing lobe number, the intersections are calculated analytically in thissection. To this end, first the damping ratio ζ is assumed to be very small and so negligible. Withζ around 0.02 for most cutting processes, this is a reasonable assumption, as is also shown laterin this section. The new characteristic equation following from this assumption is given as:

D(λ) = λ2 + 1 + W − We−λτ . (3.23)

Following the same method as was used in Section 3.2, two expressions for the dimensionlessdepth of cut W and dimensionless spindle speed Ω can be derived:

W (ω) =(ω2 − 1)

1 − (−1)NLζ=0

, (3.24a)

Ω(ω) =2ω

NLζ=0

. (3.24b)

In these equations NLζ=0refers 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=0results

from a shift in the sine function, see Appendix B. From (3.24a) it follows thatNLζ=0can only be an

odd integer number for the dimensionless depth of cut to make sense. To determine the accuracyof 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 alower number, as lobe 0 does not exists, this is not a drawback of the approximation for this case.

12

As mentioned before, NLζ=0can only be an odd number. Therefore, by using Figure 3.5 it follows

that the relation between the lobe numberNL andNLζ=0is: NLζ=0

= 2NL−1. As the intersectionof lobesNL andNL+1 is the point of interest of this section, lobeNL+1 is approximated by (3.24),and lobe NL by its vertical asymptote (defined by (3.22)). The location of the intersection lies atthe 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 1in this lobe, see also the derivation of (3.22) in Section 3.5. The vibration frequency ωint in lobeNL + 1 at the intersection can be calculated by equating (3.24b) with the equation of the verticalasymptote, 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 andωint in lobe NL in the intersection between the two lobes. To check whether the approximationis also valid for the determination of the vibration frequencies in the intersection and the ratiobetween the vibration frequencies ωint in the two lobes, the ratios for both the approximation andvibration frequencies in the intersection, found using the analytic method, are compared. Forboth cases, the ratios for the first four intersection for are shown in Table 1. The errors betweenthese ratios for higher lobes are plotted in Figure 6(a). Also, the error in the vibration frequency

Intersection between lobe Ratio for ζ=0.02 Ratio for ζ=0

1 & 2 1.5290 1.5002 & 3 1.2757 1.3333 & 4 1.1905 1.2504 & 5 1.1475 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 speedbetween 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 anintersection. From these figures it follows that the approximation is reasonable for the vibrationfrequencies ωint and the spindle speed. However, it cannot be used to approximate the depth ofcut 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. How-ever, it is not possible to use this method inmore complicated delay differential equations (DDEs).Therefore, the next chapter explains semi-discretization, a linear method to solve a DDE numeri-cally.

13

0 50 100 150 200−2.5

−2

−1.5

−1

−0.5

0

Lobe number NL

Err

or (

%)

(a) Error in ratios in intersections

0 50 100 150 200−2.3

−2.2

−2.1

−2

−1.9

−1.8

Lobe number NL

Err

or (

%)

(b) Error in vibration frequencies ω

0 50 100 150 2002

2.1

2.2

2.3

2.4

2.5

Lobe number NL

Err

or (

%)

(c) Error in dimensionless spindle speed Ω

0 50 100 150 200−300

−200

−100

0

100

Lobe number NL

Err

or (

%)

(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 whichω 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 thestructural behavior of linear delay differential equations. The chapter starts with the basic ex-planation of semi-discretization and then explains how it can be used in the investigation of thestability of metal cutting processes.

4.1 Discretization of delay term

Semi-discretization is a robust and powerful tool to determine the structural behavior of linearDDE’s in time domain, like (3.9). As a result of the constant delay, the present state cannotidentify the state of the DDE. This indicates that the past state x(t − s), s ∈ [0, τ ] is necessaryand an infinite dimensional function space is formulated. Semi-discretization can be used toperform the stability investigation of an existing steady state solution (for an autonomous systemlike 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 definedin (3.9)) is used to explain the semi-discretization method. This equation of motion is again givenbelow, and should be first converted into a first order delay differential equation.

x(t) + 2ζx(t) + (1 + W )x(t) = Wx(t − τ).

For this purpose, a vector y is defined as y = [y1 y2]T . This results in the following first order

system:

y(t) =

[y2(t)

Wy1(t − τ) − (1 + W )y1(t) − 2ζy2(t)

]

. (4.1)

In matrix representation this is:

y(t) = Ly(t) + Ry(t − τ), (4.2)

with the linear matrix L [6] defined as:

L =

[0 1

−(1 + W ) −2ζ

]

, (4.3)

and the retarded matrix R [6] is defined as:

R =

[0 0W 0

]

. (4.4)

Having the first order ODE, the system can be discretized in m steps [6], with time step ∆t = τm,

see Figure 4.1. In this figure, a function with a delay is considered. The graph is divided intothree parts, namely:

• the past from time ti−τ(= ti−m) to the current time ti, indicated by the dotted light-coloredline;

• 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 isconsidered, 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 usingthe 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 tiand 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:

yi(t − τ) ≃ 1

2(yi−m+1 + yi−m). (4.5)

Substitution of this in (4.2) obtains:

yi(t) = Lyi(t) +1

2R(yi−m+1 + yi−m), (4.6)

The last two equations are only valid for t ∈ [ti, ti+1). This differential equation is an inhomoge-neous ordinary differential equation, and is solved using the method of variation of coefficients:

yi(t) = eL(t−ti)c0 −1

2eL(t−ti)R(yi−m+1 + yi−m), (4.7)

with

c0 = yi +1

2R(yi−m+1 + yi−m). (4.8)

The complete derivation is found in Appendix C. Because the solution is considered only at timeinterval 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∆tyi +1

2

(eL∆t − I

)L−1R(yi−m+1 + yi−m). (4.9)

16

In the next section, (4.9) is used to derive a linear map that can describe the connection betweentwo discrete points in the solution of this equation, with which the stability of the system can bedetermined.

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 hereto 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 theDDE can be considered in the following way:

zi =

yi

yi−1...

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 thefirst element each vector yi−j , j = 1, ..., m is stored in zi, and zi becomes:

zi =

yi

y1,i−1

y1,i−2...

y1,i−m+1

y1,i−m

. (4.12)

As yi = [xi xi], and xi 6= 0 nothing changes in yi. The linear map in (4.9) can be reformulatedusing the vector zi and a system matrix B = B1 + B2 with

B1 =

eL∆t 0 . . . 0 00 . . . 0 0

1 0 0 . . . 0 00 1 0 . . . 0 0...

......

......

...0 0 0 . . . 0 00 0 0 . . . 1 0

, and (4.13)

17

B2 =

O O . . .A11 A11

A21 A21

O O . . .0 00 0

......

. . ....

...

O O . . .0 00 0

. (4.14)

With this linear map, zi+1 can be written as zi+1 = Bzi and the state of the system is written inone linear time-independent map, which can be used to investigate the stability. In these matricesO the null matrix and A is:

A =1

2(eL∆t − I)L−1R. (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 ofdiscretization steps and N the order of the system. In the next section, this matrix is used toevaluate 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 systemmatrix B was derived, with which the stability of the system can be evaluated. This matrix can beused to check the stability of a metal cutting process for various values of the dimensionless depthof cut and the dimensionless spindle speed. The stability lobes are constructed by determiningfor each (Ω, W ) point whether it is stable or not.

The number of discretization steps m still has to be determined. To investigate the influenceof m on the results, the stability of this system is determined for a range of the dimensionlessdepth of cut W ∈ [0.001, 0.5] and a range of the dimensionless spindle speed Ω ∈ [0.01, 2]. Foreach point (Ω, W ) the stability at that point is determined. This is done for four different valuesfor m, namely 5, 10, 50 and 100. The results are shown in Figure 4.2. The black area indicatesthe stable region, the gray color indicates the unstable region, and the white line indicates thestability lobe constructed according to the theory explained in Chapter 3. From these figures itfollows that for higher values of m the borders between the unstable and stable region approachthe 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, astep size m = 10 takes 30 seconds, while m = 100 takes 12 minutes. Section 4.3.1 explains howthe step size m should be chosen appropriately.

18

0.5 1 1.5 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Stablity chart using semi discretization with m=5

Dimensionless spindle speed Ω

Dim

ensi

onle

ss d

epth

of c

ut

(a) m = 5

0.5 1 1.5 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Stablity chart using semi discretization with m=10

Dimensionless spindle speed Ω

Dim

ensi

onle

ss d

epth

of c

ut

(b) m = 10

0.5 1 1.5 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Stablity chart using semi discretization with m=50

Dimensionless spindle speed Ω

Dim

ensi

onle

ss d

epth

of c

ut

(c) m = 50

0.5 1 1.5 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Stablity chart using semi discretization with m=100

Dimensionless spindle speed Ω

Dim

ensi

onle

ss d

epth

of c

ut

(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 largerthan 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 stepsmshould be chosen appropriately. Using T = 2π

ω, τ = 2π

Ω and ∆t = τm, the following expression

for ∆t can be derived: ∆t = TωΩm

. ∆t must be smaller than T , which is true if ωΩm

< 1, so m > ωΩ .

4.4 The motion of the eigenvalues

In this section, the motion of the eigenvalues are discussed. For stability analysis it is interestingto know how the eigenvalues evolve when moving from the stable region towards the unstableregion or when moving along the border between the two regions. Therefore, the eigenvalues ofthe matrix B are investigated for these two cases. In the first case, the dimensionless depth of cutW increases, while the dimensionless spindle speed Ω remains constant at Ω = 0.5855. In thesecond case, both parameters are changed along the lobe. The resulting motion of eigenvalueswith 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 theright half plane, which indicates Hopf bifurcation. The other eigenvalues stay inside the unitcircle. From Figure 3(b) it follows that the two critical eigenvalues move along on the border ofthe unit circle, while the other eigenvalues remain inside the unit circle.

(a) The motion of the eigenvalues in the unit circlewhen the system is evaluated while crossing thelobe

(b) The motion of the eigenvalues in the unit circlewhen 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 orthogonalcutting process. However, an important issue has not been considered up until now, namely pro-cess damping. Process damping has a significant effect in low speed metal cutting, as in theseprocesses the stability lobes obtained from experiments lie much higher in the stability chartthan is predicted with the mathematical stability theorem. Although currently no physical modelis yet able to describe this effect completely, several explanations have been introduced. Thesetheorems assume that the additional damping is mainly caused by the contact mechanism of themetal cutting. However, a proper description of this mechanism requires difficult mechanicaland mathematical techniques. Therefore, in this report it is proposed to use a simple assumptionfor this contact mechanism to create a sufficiently simple mechanical model for dynamical inves-tigation. This model includes a friction force in the direction tangential to the cut surface and acontact force that depends on the surface curvature. Furthermore, it is assumed that the frictionforce 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 withprocess damping are compared to a stability chart of the same cutting process without processdamping.

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 usedin this report. First, Das and Tobias [4] proposed to add a velocity dependent term with a staticcutting force coefficient, with which the damping in the system increases at low speeds; this isthe friction force. However, Altintas describes in [3] the process damping by two additional forceterms: 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 theoryin [4]. Therefore, the latter process damping theory for the orthogonal cutting process is used inthis section.

Figure 5.1 shows the friction force Ff (t) and the contact force Fc(t) in the orthogonal cuttingprocess. 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 isindicated. In this figure, x(t) is the position of the tool, u(t) is the tangential distance of tooltraveled 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

spatial derivative of x(t) with respect to u, that is dxdu. This slope can be resolved into the temporal

derivative of x(t) and the temporal derivative of u: dxdu

= dxdt

dtdu. With dt

du= − 1

vc, and vc = Ωr, the

slope of the wave is rewritten as:

dx

du= − x(t)

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 thatthis angle is very small, tan φ(t) ≃ φ(t) = dx

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 sinφ(t), and sinφ(t) ≃ φ(t), Ffx

(t) = −Ffx(t)Ωr

. Ff is the constant magnitude of thefriction force and it is assumed to result from a static cutting force coefficient Cf [N/m] and thedepth of cut w: Ff = wCf . The friction force in x-direction is:

Ffx(t) = −wCf

vcx(t). (5.2)

Having the friction force, the contact force is now derived. The contact between the tool andthe workpiece is assumed to be similar to a static Hertzian contact. Therefore, the contact forceFc(t) depends on the curvature of the wave of the surface cut. The curvature of the wave is

proportional to the second spatial derivative of x(t) with respect to u: d2x

du2 = x(t)v2

c. The contact

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]:

Fc(t) = Fc0 + ∆Fc(t), (5.3)

where

∆Fc(t) =wCc

v2c

x(t). (5.4)

Fc0 is the static contact force, which can be neglected as this has no influence on the stabilityinvestigation.

22

The two process damping forces are added to (3.1) in Section 3.2:

mx(t) + cx(t) + k(q0 + x(t)) = wfa(h(t)) + Ffx(t) + Fc(t). (5.5)

Using Ω = 2πτthe equation of motion for an orthogonal cutting process with process damping is:

(

m − Ccwτ2

4π2r2

)

x(t) +

(

c +Cfwτ

2πr

)

x(t) + (k + wk1)x(t) = wk1x(t − τ). (5.6)

The coefficients Cc and Cf can be found experimentally [3]. Using (3.6) and (3.8), (5.6) is con-verted into a dimensionless parameter space. Two dimensionless parameters are defined for theprocess damping, namely If and Ic:

If =Cf

2πrk1, (5.7a)

Ic =Cc

4π2r2k1. (5.7b)

The dimensionless equation of motion with process damping is given as:

(1 − IcWτ2)x(t) + (2ζ + IfWτ)x(t) + (1 + W )x(t) = Wx(t − τ), (5.8)

with W the dimensionless depth of cut as derived in Section 3.2. The complete derivations of thedimensionless 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 addedto describe the process damping both depend on the delay τ , so the spindle speed Ω, and thedepth of cut W , it is not possible to derive analytically dimensionless expressions for the angularvelocity and the depth of cut as was done in Section 3.3. Therefore, it is not possible to analyzethe 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 de-scribed in (5.8) are derived, using the method of Section 4.1. The matrix L is:

L =

[

0 1−(1+W )1+IcWτ2

−2ζ−IcWτ1+If Wτ2

]

, (5.9)

and the retarded matrix R is :

R =

[

0 0W

1+If Wτ2 0

]

. (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 areplotted 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 lineindicates the stability lobes for a cutting process described in (5.8). From this figure it follows thatthe stable area for the orthogonal cutting process with process damping is significantly larger thanthe stable area for the orthogonal cutting process without process damping. Furthermore, thestable area of the process damping theory proposed in this report is slightly larger than the stablearea 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 experimentalresults do not match. This is explained by the fact that it is quite difficult to determine the contactforce coefficient Cc, [3].

The step size of the semi-discretization method has a significant influence on the reliability of theresults. 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 5can be used to determine an approximation for the stability for a 2 DOF non-autonomous cuttingprocess, namely the milling process, although the dynamics of an orthogonal cutting process anda milling process are not the same. This is done in the next chapter.

1000 1500 2000 2500 3000 3500 4000

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10−3

Spindle speed [rpm]

Dep

th o

f cut

[m]

Cc=0 [N], C

f=0 [N/m]

Cc=0 [N], C

f=0.611e6 [N/m]

Cc=332 [N], C

f=0.611e6 [N/m]

stableunstable

(a) The stability lobes for orthogonal cutting with-out process damping, with only friction force andwith both friction and contact forces. using m =

400 Experimental results are depicted by the x-marks and circles.

1000 1500 2000 2500 3000 3500 4000

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10−3

Spindle speed [rpm]

Dep

th o

f cut

[m]

Cc=0 [N], C

f=0 [N]

Cc=0 [N], C

f=0.611e6 [N]

Cc=332 [N], C

f=0.611e6 [N]

(b) The stability lobes for orthogonal cutting with-out process damping, with only friction force andwith 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, fordifferent 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 degree-of-freedom autonomous model. In this chapter, it is used in a two degree-of-freedom, non-autonomous system that can describe milling. With this model, the stability of a milling processis investigated, first for a model without process damping, after which process damping is addedto the model.

6.1 Introduction

The equation of motion of a milling process is periodic, and thus time-dependent, because ofthe periodic parameter excitation due to entering and exiting teeth as a result of the tool revolv-ing. It is clear that as a result of the time dependence of the model, the linear stability lim-its of the milling process cannot be investigated in the same way as for turning, which was atime-independent system. For example, the equations of motion have more than one degree offreedom: in milling the forces in both the x and the y-direction, and also in z-direction for he-lical tools, should be taken into account. In this report, it is assumed that the milling tool isstraight-fluted, so only a two degree-of-freedom model is used and the z-direction is not takeninto account. As the state vector is indicated with x further onwards, in this section the directionsare 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 anddamper forces in both the x1 and the x2 direction are indicated, as are the motions due to vibra-tion in these directions, indicated by χ and γ respectively. Furthermore, V is the speed of the toolwith respect to the workpiece resulting from the feed of the tool:

V =NfdΩ

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 differencebetween these two types is the direction of themotion of the workpiece with respect to the rotationof the tool. In the figure, two important angles are indicated: the angle with which a tooth entersthe work piece, φen, and the angle with which the tooth exits the workpiece, φex. In the coordinatesystem chosen the following holds: for down-milling, the exit angle is always π [rad], for up-milling, the enter angle is 0 [rad] [5]. These angles are important, as the equations of motionfor milling, which are described in Section 6.2, depend on the angle of the tool with respect tothe workpiece. Generally, in practice down-milling is used, as the teeth cut with a larger chipthickness when they enter the workpiece, which causes a high impact, but the rubbing effect canbe avoided.

25

Figure 6.1: Schematic model of milling

Figure 6.2: Schematic representation of down milling (left) and up milling (right). The en-trance 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:

Mx(t) + Cx(t) + Kx(t) = F (t). (6.2)

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 consid-ered. For each tooth, the resulting cutting forces in radial and tangential direction have beendetermined in [5]:

Fj,r = g(φj(t))Kraph(t) [N] and (6.3a)

Fj,t = g(φj(t))Ktaph(t) [N], (6.3b)

respectively. In these equations, g(φj(t)) is a switching function, indicating whether tooth jis cutting or not, the parameters Kr and Kt indicate the linear radial and tangential cuttingcoefficients [N/m2] respectively, ap is the depth of cut [m], and h(t) is the momentary thicknessof the chip [m] at time t [s]. A tooth j is cutting when the angular position of the tooth at time t isbetween φen and φex. The angular position φj(t) of tooth j can be calculated by (6.4), assumingthat 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)) = 1, if φen < φj(t) < φex,

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 therespective positions of tooth j at angle φj(t) [rad] are indicated with A and B. The difference intime between both positions is τ . Because the tool is not stiff, it vibrates due to the periodic forceexcitation and it is unable to follow the path of an ideally stiff tool. The distance of the actualtool position with respect to the ideal tool position is indicated by the variables x1 and x2. Thechip 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 handside 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 hat time t can be approximated by [5]:

h(t) = α sinφj(t) + β cos φj(t), (6.6)

with

α = x1(t − τ) + fd − x1(t) and (6.7a)

β = x2(t − τ) − x2(t). (6.7b)

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 actingon 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) → (6.8)

Fj,x1(t) = g(φj(t))(Kt cos φj(t) + Kn sinφj(t)

)aph(t), (6.9)

Fj,x2(t) = −Fjt sinφj(t) + Fj,r cos φj(t) → (6.10)

Fj,x2(t) = g(φj(t)) (−Kt sinφj(t) + Kn cos φj(t)) aph(t). (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∑

j=1

g(φj(t)) (Kt cos φj(t) + Kn sinφj(t)) aph(t), (6.12)

Fx2(t) =N∑

j=1

g(φj(t)) (−Kt sinφj(t) + Kn cos φj(t)) aph(t). (6.13)

Using (6.2), (6.6), (6.12) and (6.13), the following equation of motion results:

Mx(t) + Cx(t) + Kx(t) = apH(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 stationarycutting force vector introduced in [5]. The matrix H(t) consists of the following four elements:

28

H11(t) =N∑

j=1

g(φj(t)) (Kt cos(φj(t) + Kn sin φj(t)) sinφj(t), (6.15a)

H12(t) =N∑

j=1

g(φj(t)) (Kt cos(φj(t) + Kn sin φj(t)) cosφj(t), (6.15b)

H21(t) =

N∑

j=1

g(φj(t)) (−Kt sin(φj(t) + Kn cos φj(t)) sin φj(t), (6.15c)

H22(t) =N∑

j=1

g(φj(t)) (−Kt sin(φj(t) + Kn cos φj(t)) cos φj(t). (6.15d)

The vector G(t) consists of the two elements:

Gx1(t) = apfdH11(t), (6.16a)

Gx2(t) = apfdH21(t). (6.16b)

The vector x(t) that describes the motion of the tool can be divided into two parts [5]: aτ -periodic part xp(t), so xp(t) = xp(t + τ), and a part ξ(t) that describes the motion due tochatter. Substituting this for x(t) in (6.14) results in:

Mξ(t)+Cξ(t)+Kξ(t)+Mxp(t)+Cxp(t)+Kxp(t) = apH(t)(ξ(t−τ)−ξ(t))+G(t). (6.17)

For the investigation of the stability of the solution x(t), only the terms that contain ξ(t) have tobe taken into account, and the second order delay differential equation that describes the stabilitydue to the regenerative effect is:

Mξ(t) + Cξ(t) + Kξ(t) = apH(t)(ξ(t − τ) − 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 dueto the regenerative effect, (6.18), the stability lobes for milling can be constructed. This is doneusing 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 delaydifferential equation:

y(t) = L(t)y(t) + R(t)y(t − τ), (6.19)

with

L(t) =

[O I

−M−1(K + apH(t)) −M−1C

]

, (6.20)

29

and

R(t) =

[O O

apM−1H(t) O

]

. (6.21)

Because the specific cutting force variation matrix H(t) is periodically time-dependent, it is notpossible to use exactly the same matrices as were used in Section 4.1 to check the stability. Tocalculate theB-matrix, the matricesR(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:

Li =1

∆t

∫ ti+1

ti

L(t)dt ≃ 1

Nint

Nint∑

k

Li,k, (6.22)

with Li,k = L(ti + k ∆tNint

) and Nint the number of integration steps. The same holds for theretarded matrix R(t).

With these matrices the Bi matrix can be constructed in the same way as was done for au-tonomous orthogonal cutting: Bi = Bi1 + Bi2 with

Bi1 =

eLi∆t O . . . O O

I O . . . O O

O I . . . O O...

.... . .

......

O O . . . I O

, (6.23)

Bi2 =

O O . . . Ai Ai

O O . . . O O...

.... . .

......

O O . . . O O

, (6.24)

and

Ai =1

2(eLi∆t − I)L−1

i Ri. (6.25)

To determine the stability of the cutting process, the Floquet theorem is used [7]. The Floquettheorem states that a linear periodic system has a periodic solution. This theory is used to derivethe transition matrix Φ such that

y(t0 + T ) = Φy(t0). (6.26)

If none of the so-called Floquet multipliers µk, which are the eigenvalues of the transistion matrixΦ, lie outside the unit circle, the stationair periodic orbit is orbitally stable. The Floquet matrix iscalculated using semi-discretization with the step matrices Bi:

Φ = BmBm−1...Bi−1Bi. (6.27)

The linear stability limits can now be constructed by determining for each (Ω, w) point whetherthe Floquet multipliers lie inside or outside the unit circle. This is done for two different millingprocesses: half immersion down-milling with a tool with one tooth (solid line in Figure 4(a)) anda tool with four teeth (solid line in Figure 4(b)). The process parameters used to construct theselobes are found in Appendix G. Now the basic stability lobes are constructed for milling, theinfluence of process damping is investigated in Section 6.4.

30

Spindle speed [rpm]

Dep

th o

f cut

[m]

1 2 3 4 5x 10

4

2

4

6

8

10

12

14

x 10−4

(a) Down-milling process with a tool with onetooth

Spindle speed [rpm]

Dep

th o

f cut

[m]

2000 4000 6000 8000 10000

0.5

1

1.5

2

2.5

x 10−4

(b) Down-milling process with a tool with fourteeth

Figure 6.4: Stability lobes for a down-milling process with two different tooth number of thetool

6.4 Process damping for milling

In Section 6.2 the equations of motion for the two degree-of-freedom model used to describe themilling process were derived, and in Section 6.3 the stability chart was constructed using semi-discretization 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 Section5.1, so by adding a friction force and a contact force. To this end, the (x1, x2)-coordinate systemthat 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 forceis indicated by Fc,j(t) and the cutting speed of a tooth is indicated by vc. The cutting force actsonly 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:

uj = Txux, (6.28)

where

Txu =

[− cos φj(t) sinφj(t)− sinφj(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 contactforce.

First, the friction force is derived in the same way as in Section 5.1: Ff,j(t) = Ff

[cos φj(t)sinφj(t)

]

,

with Ff = Cfap the constant magnitude of the friction force, and Cf is the unit friction force

coefficient [N/m]. In Section 5.1 φj(t) =du2,j

du1,j= − 1

vcu2,j(t) was derived. As the angle φj(t)

is quite small, cos φj(t) ≃ 1 and sinφj(t) ≃ φj(t). Furthermore, from (6.29) it follows thatu2,j(t) = −x1(t) sinφj(t) − x2(t) cos φj(t), and using this:

u2,j = −x1(t) sin φj(t) − x1(t)φj(t) cos φj(t) − x2(t) cos φj(t) + x2φj(t) sinφj(t). (6.30)

With vc(t) = rΩ, the friction force on the flank face of the jth tooth is now:

Ff,j(t) = Cfap

[1

− 1rΩ

(

−x1(t) sin φj(t) − x1(t)φj(t) cos φj(t) − x2(t) cos φj(t) + x2φj(t) sinφj(t))

]

.

(6.31)

Second, the contact force is derived. As explained in Section 5.1 the contact force Fc,j(t) is as-sumed 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) = Ccap

[0

d2u2,j(t)

du21,j(t)

]

. (6.32)

The second spatial derivative of u2,j(t) is

d2u2,j

du21,j

=1

v2c

d2u2,j(t)

dt2, (6.33)

with

d2u2j(t)

dt2= −x1(t) sin φj(t) − 2x1(t)φj(t) cos φj(t) − x1(t)φj(t) cos φj(t) + x1(t)φj(t)

2 sinφj(t)

−x2(t) cos φj(t) + 2x2(t)φj(t) sinφj(t) + x2(t)φj(t) sinφj(t) + x2(t)φj(t)2 cos φj(t),

(6.34)

with φj(t) = Ω and, as a constant spindle speed is assumed, φj(t) = Ω = 0. Both the frictionforce 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 thex-coordinate system, the friction and contact force has to be transformed into the x-coordinatesystem according to Fx = TuxFu using the transformation matrix Tux: Tux = T T

xu.

The friction force Ff,j(t) in the x-coordinate system is:

Ff,j(t) = Cfap

(

− 1

rΩ

[sin2 φj(t) cos φj(t) sin φj(t)

cos φj(t) sinφj(t) cos2 φj(t)

]

x(t)+ (6.35)

−1

r

[cos φj(t) sin φj(t) − sin2 φj(t)

cos2 φj(t) − cos φj(t) sin φj(t)

]

x(t) +

[− cos φj

sinφj(t)

])

,

the contact force Fc,j(t) in the x-coordinate system is:

Fc,j(t) = Fc0

[− sinφj(t)− cos φj(t)

]

+ Ccap1

r2Ω2

([sin2 φj(t) cos φj(t) sin φj(t)

cos φj(t) sin φj(t) cos2 φj(t)

]

x(t)+

+

[cos φj(t) sin φj(t) − sin2 φj(t)

cos2 φj(t) − cos φj(t) sin φj(t)

]

2Ωx(t)+ (6.36)

+

[− sin2 φj(t) − cos φj(t) sin φj(t)

− cos φj(t) sin φj(t) − cos2 φj(t)

]

Ω2x(t)

)

.

The friction force and contact force are added to (6.14) and the new equation of motion for a 2DOF non-autonomous cutting process is:

Mx(t) + Cx(t) + Kx(t) = apH(t)(x(t − τ) − x(t)) + G(t) − Ipd(t)x(t) −Cpd(t)x(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 cuttingforce vector G(t) is now:

Gx1(t) = ap(fdH11(t)) + g(φj(t))N∑

j=1

(− cos φj(t)Cfap − Fco sin φj(t)) , (6.38a)

Gx2(t) = ap(fdH21(t)) + g(φj(t))

N∑

j=1

(sinφj(t)Cfap − 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 motionξ(t), see Section 6.2. As the vibration is interesting for the stability analysis, the stability chart forprocess damping in milling is constructed using the following equation:

(M +apIpd(t))ξ(t)+(C+apCpd(t))ξ(t)+(K+apKpd(t))ξ(t) = apH(t)(ξ(t−τ)−ξ(t)), (6.39)

This is the equation of motion of the two degree-of-freedommodel for milling with process damp-ing and is solved using semi-discretization in the next section, after which the stability charts areconstructed.

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 processdamping 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 equa-tion y(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 + apIpd(t))−1(K + apKpd(t) + apH(t)) −(M + apIpd(t))

−1(C + apCpd(t))

]

,

(6.40)

and the retarded matrix is:

R(t) =

[O O

ap(M + apIpd(t))−1H(t) O

]

. (6.41)

(6.23) and (6.24) are used to investigate the stability of the two degree-of-freedom cutting pro-cess, and the stability chart is constructed for the same cutting processes as in Section 6.3, seeFigure 6.6. From earlier experiments it followed that process damping occurs especially after thetwentieth lobe in the stability chart. Therefore, the stability charts are plotted from the thirteenthuntil the twentieth lobe. From these figures it follows that the addition of the process dampingincreases the stable area of the stability charts, as was expected, although the results are less sig-nificant for the cutting process with four teeth, compared to the simulation results of the toolwith only one tooth. The parameters used to construct these charts are found in Appendix G.

Spindle speed [rpm]

Dep

th o

f cut

[m]

1400 1500 1600 1700 1800 1900 2000

1

2

3

4x 10−4

with Process dampingno process damping

(a) Half-immersion down-milling process with onetooth

400 500 600 700 800

2.5

3

3.5

4

4.5

5x 10−5

Spindle speed [rpm]

Dep

th o

f cut

[m]

No Process dampingwith process damping

(b) Half-immersion down-milling process with fourteeth

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 recommendationsthat 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 takingprocess damping into account and using the semi-discretization method. To this end, first amodel 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, theinfluence of the depth of cut and the spindle speed of the tool are investigated first analytically andlater by the construction of the stability lobes. Then, the semi-discretization method is used toinvestigate the structural behavior of the delay equation of motion, and compared to the analyticalsolution. From this it followed that especially at low spindle speeds a high step size is requiredfor the semi-discretization to be accurate. Following, process damping is investigated for turningand added to the semi-discretization model. It is assumed that process damping can be describedby a friction force and a contact force. From the stability lobes constructed it is concluded thatby adding these forces to the equation of motion of the orthogonal cutting processing, the stablearea in the stability chart has increased, as expected.

Having completed the one dimensional orthogonal cutting process, the theory is used in a modelwith two degrees of freedom in order to investigate milling. The difficulty of this in comparisonto 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 toothnumber of the tool has a significant influence on the location and size of the stable regions in thestability chart. Process damping is added to this model by describing it by the spatial derivativeof the radial tool position and its second spatial derivative. The stability lobes that are constructedusing this theory revealed that process damping in a 2 DOF model increases the stable region inthe stability chart, although the significance of the increase depends on the number of teeth ofthe tool.

7.2 Recommendations

In this report, some simplifications are used to make things less complicated. For example, onlymilling 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 recom-mended 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 di-rection, is also taken into account, the model is more accurate. The last and most importantrecommendation 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 Coef-ficients and Chatter Stability with Process Damping", CIRP Annuals - Manufacturing Tech-nology, 57, pp. 371-374.

[4] Das, M.K., Tobias, S.A., 1967, "The Relation Between the Static and the Dynamic CuttingForces 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, "Ma-chine Tool Chatter and Surface Location error in Milling Processes", Journal of Manufactur-ing Science and Engineering, 128(4), pp. 913-920.

[6] Insperger, T., Stépán, G., 2004, "Stability Analysis of Turning with Periodic Spindle SpeedModulation via Semi-Discretization", Journal of Vibration and Control, 10, pp. 1835-1855.

[7] Insperger, T., Stépán, G., 2004, "Updated Semi-Discretization Method for Periodic Delay-Differential Equations with Discrete Delay", International Journal for Numerical Methodsin Engineering, 61, pp.117-141.

[8] Stépán, G., 2001, "Modelling Nonlinear Regenerative Effects in Metal Cutting", Philosoph-ical 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 spindlespeed Ω are used to explain the basic shape of the stability chart. To this end, the minimumpoints and asymptotes are determined. This section derives the equations for the minimumpoints and asymptotes step by step.

In Section 3.3 the following two equations for the dimensionless depth of cut and spindle speedwere derived (Equations 3.14 and 3.18 respectively in Section 3.3):

W (ω) =(ω2 − 1)2 + 4ζ2ω2

2(ω2 − 1), (A.1)

Ω(ω) =πω

arctan(

1−ω2

2ζω

)

+ NLπ. (A.2)

The minimum point of a lobe can be calculated by first determining the dimensionless depth ofcut at this point. This is done by differentiating W (ω) with respect to ω. Using the chain rule,this derivative is:

dW

dω=

2(ω2 − 1)(4(ω2 − 1)ω + 8ζ2ω) − ((ω2 − 1)2 + 4ζ2ω2)4ω

4(ω2 − 1)2. (A.3)

Simplification of this equation obtains:

dW

dΩ= ω − 4ζ2ω

(ω2 − 1)2. (A.4)

Having this equation, the vibration frequency for which (A.4) equals zero is calculated:

ωmin =√

1 + 2ζ. (A.5)

This frequency is substituted for ω in (A.1) and the resulting depth of cut is called the minimumdepth of cut Wmin:

Wmin = 2ζ(1 + ζ). (A.6)

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 isdone by substitution of the vibration frequency in the minimum point, ωmin, for ω in (A.2):

Ωmin =π√

1 + 2ζ

arctan− 1√1+2ζ

+ NLπ. (A.7)

The damping ratio ζ is about 0.02 in metal cutting processes, and therefore the term√

1 + 2ζis approximately 1. With arctan(−1) = −1

4 , the dimensionless spindle speed at the minimumpoint 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 ofa lobe is derived. The depth of cut W goes to infinity as the vibration frequency reaches 1 as aresult of the denominator being 2(ω2 − 1). This asymptote lies at the point Ω(ω) = Ω(1), so byusing 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. Thisassumption results in a new characteristic equation, from which new equations for Ω and W asfunction 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 inSection 3.3 and is given again in (B.1):

D(λ) = λ2 + 2ζλ + 1 + W − We−λτ . (B.1)

Assuming that ζ = obtains:

D(λ) = λ2 + 1 + W − We−λτ . (B.2)

As the border between the asymptotically stable and instable region is interesting, i.e. when thesystem is critically stable, λ = iω. Substitution of this in (B.2) results in the following term inthe characteristic equation: −We−iωτ . This term can be rewritten in a function with a sine andcosine according to Euler’s formula in complex functions:

e−iωτ = cos(−ωτ) + i sin(−ωτ). (B.3)

Using cos(−x) = cos(x) and sin(−x) = − sin(x):

D(ω) = −ω2 + 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:

iW sin(ωτ) = 0, (B.5a)

W − W cos(ωτ) = ω2 − 1. (B.5b)

From Equation B.5a it follows that ωτ = NLζ=0π. Using τ = 2π

Ω , the dimensionless spindle speedas function of the vibration frequency ω is:

Ω(ω) =2ω

NLζ=0

. (B.6)

Substitution of ωτ = NLζ=0π in Equation B.5b and using cos(NLζ=0

π) = (−1)NLζ=0 results in:

W (ω) =ω2 − 1

1 − (−1)NLζ=0

. (B.7)

39

C Proof of reordering exponential matrix

In Section 4.1 the delay term of a delay differential equation was approximated using semi-discretization and an inhomogeneous ordinary differential equation remained. This sectionsolves 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 theinhomogeneous or particular part yiP (t). The transient part is the solution of (C.1).

yiT (t) = LyiT (t), (C.1)

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 thesolution of the following equation:

yiP (t) = LyiP (t) +1

2R(yi−m+1 + yi−m), (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 functionc(t):

yiP (t) = LeL(t−ti)c(t) + eL(t−ti)c(t). (C.5)

When (C.5) is substituted for yiP (t) in (C.3), and (C.4) is substituted for yiP (t) in (C.3), the fol-lowing equation results:

LeL(t−ti)c(t) + eL(t−ti)c(t) = LeL(t−ti)c(t) +1

2R(yi−m+1 + yi−m). (C.6)

From this equation, an expression for the parameter c(t) can be derived, which is:

c(t) = −1

2L−1e−L(t−ti)R(yi−m+1 + yi−m). (C.7)

This equation can be substituted into (C.4):

yiP (t) = −1

2eL(t−ti)L−1e−L(t−ti)R(yi−m+1 + yi−m). (C.8)

Due to the following property of exponential matrices, the matrices in (C.8) can be reorderedsuch that the eL(t−ti) term and the e−L(t−ti) term cancel:

eY XY −1= Y eXY −1. (C.9)

40

The proof of this is given below:

eA = I + A +A2

2!+

A3

3!+ ... (C.10)

Take A = Y XY −1, then: (C.11)

eY XY −1= I + Y XY −1 +

(Y XY −1)2

2!+

(Y XY −1)3

3!+ ... (C.12)

and with (Y XY −1)2 = Y X2Y −1, (C.13)

eY XY −1= Y

(

I + X +X2

2!+

X3

3!

)

︸ ︷︷ ︸

eX

Y −1 = Y eXY −1. (C.14)

This property is used to simplify Equation C.8 in the following way:

yiP (t) = −1

2eL(t−ti)L−1e−L(t−ti)R(yi−m+1 + yi−m) (C.15)

multiply both sides with L:

LyiP (t) = −1

2LeL(t−ti)L−1︸ ︷︷ ︸

eLL(t−ti)L−1

e−L(t−ti)R(yi−m+1 + yi−m), then: (C.16)

yiP (t) = −1

2L−1R(yi−m+1 + yi−m). (C.17)

In this way, an expression for yi(t) is derived, in which only the parameter c0 is unknown:

yi(t) = eL(t−ti)c0 −1

2eL(t−ti)R(yi−m+1 + yi−m). (C.18)

This problem is solved by investigating the solution at time t = ti:

c0 = yi +1

2R(yi−m+1 + yi−m). (C.19)

Substituting this in (C.18), the derivation for the solution of the inhomogeneous differential equa-tion 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 whichthere 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 givenbelow:

yi+1 = eL∆tyi +1

2

(eL∆t − I

)L−1R(yi−m+1 + yi−m). (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 . . . O A A

I O . . . O O O...

.... . .

......

O O . . . O O O

O O . . . I O O

O O . . . O I O

yi

yi−1...

yi−m+2

yi−m+1

yi−m

. (D.2)

The matrix is called B, in which I is the identity matrix, O the null matrix and A isA = 1

2(eL∆t − I)L−1R. The two vectors are called zi+1 and zi respectively:

zi =

yi

yi−1...

yi−m+2

yi−m+1

yi−m

and zi+1 =

yi+1

yi

...

yi−m+3

yi−m+2

yi−m+1

, (D.3)

with yi = [y1,i 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 asthe delay only occurs in the position of the tool x(t) and not in the velocity of the tool x(t). Fromthis 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

0y1,i−m+1

0

=

E(1, 1) E(1, 2) 0 0 . . . A(1, 1) A(1, 2) A(1, 1) A(1, 2)E(2, 1) E(2, 2) 0 0 . . . A(2, 1) A(2, 2) A(2, 1) A(2, 2)

1 0 0 0 . . . 0 0 0 00 1 0 0 . . . 0 0 0 0...

......

.... . .

......

......

0 0 0 0 . . . 0 0 0 00 0 0 0 . . . 0 0 0 00 0 0 0 . . . 1 0 0 00 0 0 0 . . . 0 1 0 0

y1,i

y2,i

y1,i−1

0...

y1,i−m+1

0y1,i−m

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 beomitted, A(1, 2) and A(2, 2) are zero, and the following matrix results:

B =

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 00 1 0 0 . . . 0 0 0 0...

......

.... . .

......

......

0 0 0 0 . . . 0 1 0 00 0 0 0 . . . 0 0 1 0

. (D.5)

43

E Derivation of the dimensionless equation of motion for pro-

cess damping

In Section 5.1 the equation of motion for an orthogonal cutting process with process damping wasderived. The method for converting an equation in a parameter space with dimensions into anequation in a dimensionless parameter space was already explained in Section 3.2. This methodis used in Section 5.1 for the derivation of (5.8), however it is only briefly explained how thisequation and the dimensionless parameters Cpd and Dpd are derived. This is explained in thisappendix.

The equation of motion in a parameter space with dimensions for orthogonal cutting with processdamping, (5.6) is given below:

(

m +Ccwτ2

4π2r2

)

x(t) +

(

c +Cfwτ

2πr

)

x(t) + (k + wk1)x(t) = wk1x(t − τ), (E.1)

with Cc [N] and Cf [N/m] the process damping coefficients. Dividing this by the mass m obtains:

(

1 +Ccwτ2

4π2r2m

)

x(t) +

(c

m+

Cfwτ

2πrm

)

x(t) +

(k + wk1

m

)

x(t) =wk1

mx(t − τ). (E.2)

Using (3.6):

ζ =c

2√

km, (E.3a)

ωn =

√

k

m, (E.3b)

results in:

(

1 +Ccwτ2

4π2r2m

)

x(t) +

(

2ζωn +Cfwτ

2πrm

)

x(t) +

(

ω2n +

wk1

m

)

x(t) =wk1

mx(t − τ). (E.4)

Substituting the following dimensionless parameters (as defined in (3.8)) for t and τ

t := t ωn, → dt = dt ωn, (E.5a)

τ := τ ωn, (E.5b)

(E.5c)

results in the following equation:

(

1 +Ccwτ2

4π2r2mω2n

)

ω2nx(t)+

(

2ζωn +Cfwτ

2πrmωn

)

ωnx(t)+

(

ω2n +

wk1

m

)

x(t) =wk1

mx(t− τ).

(E.6)

Dividing this equation by ω2n results in:

(

1 +Ccwτ2

4π2r2mω2n

)

x(t)+

(

2ζ +Cfwτ

2πrmω2n

)

x(t)+

(

1 +wk1

mω2n

)

x(t) =wk1

mω2n

x(t− τ). (E.7)

44

The coefficient wk1mω2

nis called the dimensionless depth of cut W . The two coefficients due to

process damping, Ccw4π2r2m

andCf w

2πrmω2n, can be now divided into two dimensionless parameters,

namely W , and Dpd and Cpd respectively, with

Ic =Cc

4π2r2k1, (E.8)

If =Cf

2πrk1. (E.9)

Omitting the tilde for convenience and substitution W for wk1mω2

n, IcW for Ccw

4π2r2mω2nand IfW for

Cf w

2πrmω2nobtains the dimensionless equation of motion:

(1 + IcWτ)x(t) + (2ζ + IfWτ)x(t) + (1 + W )x(t) = Wx(t − τ). (E.10)

45

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 wereadded to the equation of motion for a milling process, Equation (6.14), and three new matriceswere defined: Ipd(t), Cpd(t) and Kpd(t). These three matrices follow from the contact and frictionforce as defined in Equations (6.35) and (6.36). For each tooth j, the matrices are:

Ipd,j(t) = Ccap1

r2Ω2

[sin2 φj(t) cos φj(t) sinφj(t)

cos φj(t) sin φj(t) cos2 φj(t)

]

(F.1a)

Cpd,j(t) = −Cfap1

rΩ

[sin2 φj(t) cos φj(t) sin φj(t)

cos φj(t) sinφj(t) cos2 φj(t)

]

+

+Ccap2

r2Ω

[cos φj(t) sin φj(t) − sin2 φj(t)

cos2 φj(t) − cos φj(t) sin φj(t)

]

(F.1b)

Kpd,j(t) = −Cfap1

r

[cos φj(t) sin φj(t) − sin2 φj(t)

cos2 φj(t) − cos φj(t) sin φj(t)

]

+

+CcapΩ2

r2

[− sin2 φj(t) − cos φj(t) sinφj(t)

− cos φj(t) sinφj(t) − cos2 φj(t)

]

. (F.1c)

To obtain the three matrices Ipd(t), Cpd(t) and Kpd(t) for all teeth the matrices are evaluated foreach tooth and summed:

Ipd(t) = g(φj(t))N∑

j=1

Ipd,j(t),

Cpd(t) = g(φj(t))N∑

j=1

Cpd,j(t)and

Kpd(t) = g(φj(t))N∑

j=1

Kpd,j(t).

46

G Parameters used for stability charts for milling

In Section 6.3 the stability lobes for half immersion down-milling were constructed. The valuesof the parameters used are found in Table 1.

Parameter Value

M [N]

[0.0199 0

0 0.0201

]

C[Ns/m]

[1.8043 0

0 3.6445

]

K[N/m]

[4.0900 ∗ 105 0

0 4.1300 ∗ 105

]

Kt [MPa] 644Kr[MPa] 237Cf [N/m] 6.44 ∗ 106

Cc[N] 2370m 50

Nint 40φen [rad] π/2φex [rad] π

r [m] 0.035fd [mm] −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 atthe 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 invit-ing me to his Manufacturing and Automation Laboratory at the UBC. Going to Canada for fivemonths would not have been possible without all the support my parents gave me and I owe themmany thanks. At the MAL, I would like to thank Zoltan Dombóvári for all his help during mystay there, and also for his help with my report after my return to The Netherlands. I would liketo thank all the other guys cheerfulness in the MAL. Furthermore, I would like to thank NicoleKommer for her help with my report, especially when it comes to English language. Lastly, Iwould like to thank Mark Nievelstein, who keeps supporting me in every possible way while hehad to deal with when I was entirely happy for going to Canada, or completely stressed whilewriting my report, and every emotion in between.

48