Stability of orthogonal turning processes · This report is divided into six chapters. The process...

M.A.M. Haring

DC 2010.016

Stability of orthogonal turningprocesses

Traineeship report

Coach: A. Aygun, M.A.Sc. candidate (The University of British Columbia)

Supervisors: Prof. H. NijmeijerProf. Y. Altintas

(Eindhoven University of Technology)(The University of British Columbia)

Eindhoven University of TechnologyDepartment of Mechanical EngineeringDynamics and Control Group

Eindhoven, May, 2010

Abstract

Stability is an important topic in metal cutting. Unstable vibrations in the displacement of the cuttingtool with respect to the workpiece result in a poor surface finish and cause the vibrations in the cuttingforces to grow. Large vibrations in the cutting forces may damage the cutting tool, the workpiece andthe machine. In this report, the dynamics of an orthogonal turning process are modeled. The obtainedmodel includes the influence of the geometry of a triangular cutting tool and the effect of processdamping. The cutting forces are linked to the uncut chip thickness to find an expression for thestability of the turning process. The stability of the turning process is solved using the Nyquist stabilitycriterion. An algorithm is presented to find the cutting conditions for which the turning process iscritically stable. A stability chart is made to depict the stability of the turning process for differentcutting conditions. The predicted stability is validated by cutting experiments.

3

Contents

1 Introduction 51.1 Aim of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Outline of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Modeling of orthogonal turning dynamics 72.1 Structural dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Regenerative chip model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Chip area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Chip flow angle and chord length . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Approximated chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Cutting force model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Stability of the orthogonal turning process . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Experimental setup 193.1 Identification of the frequency response functions . . . . . . . . . . . . . . . . . . . . 203.2 Identification of the cutting force coefficients . . . . . . . . . . . . . . . . . . . . . . . 223.3 Identification of the process damping coefficient . . . . . . . . . . . . . . . . . . . . . 23

4 An algorithm to construct stability charts for turning 304.1 An algorithm to determine the critically stable cutting conditions . . . . . . . . . . . . 30

4.1.1 Calculation of the cutting speed and the spindle revolution time . . . . . . . . . 314.1.2 Calculation of the chip flow angle and the chord length . . . . . . . . . . . . . 324.1.3 Generation of the Nyquist plot data . . . . . . . . . . . . . . . . . . . . . . . . 324.1.4 Calculation of the stability limit value . . . . . . . . . . . . . . . . . . . . . . . 374.1.5 Stability check and update of the depth of cut . . . . . . . . . . . . . . . . . . . 41

5 A stability chart for the experimental setup 455.1 Construction of the stability chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Experimental validation of the stability chart . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Conclusions and recommendations 496.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 Recommendations for future research . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Acknowledgements 52

Nomenclature 53

A Coefficients of the fitted frequency response functions 56

B Calculation of the partial derivative ∂∂a (1 + CR,lim) 58

References 62

4

1 Introduction

In an aim to increase the productivity and reduce the costs in metal cutting operations, metal removalrates are pushed to the bounds of instability. Vibrations in the displacement of the tool with respect tothe workpiece lead to vibrations in the cutting forces. In turn, vibrating cutting forces contribute to thetool vibrations, resulting in a process known as regenerative chatter. Unstable chatter vibrations leadto a poor surface finish and may damage the cutting tool, the workpiece and the machine. To avoidinstability, accurately predicting the stability of the performed cutting operation is key.

The stability of a metal cutting operation depends on the cutting conditions and the dynamic prop-erties of the cutting tool, the workpiece and the machine. In most cases, the dynamic properties cannot be changed. The productivity and cost savings can be increased by finding cutting conditions thatresult in larger metal removal rates while still guaranteeing a stable cutting process.

A stability chart is a convenient way to depict the stability of a metal cutting process for a largerange of cutting conditions. In a stability chart, so-called chatter stability lobes are plotted that indicatethe boundary between cutting conditions that result in stable chatter vibrations and cutting conditionsthat result in unstable chatter vibrations.

Basic regenerative chatter stability theory has been around for a few decades. Improved predictionof chatter stability lobes led to significant increases in metal removal rates in high speed operations,such as high speed milling. In low speed operations, such as turning, the chatter stability lobes aresmaller and the increase in metal removal rates is less. When the ratio of the vibration frequency overcutting speed is high, the stable depth of cut increases. This effect is known as process damping andis either attributed to the change in the direction of the cutting speed due to the cutting force or tofriction between the flank of the tool and the vibrations on the workpiece surface.

Among others such as [1, 2, 3, 4], two recently published articles related to the stability of turn-ing processes are presented by Eynian and Altintas [5] and Altintas et al. [6]. Both articles originatefrom the Manufacturing Automation Laboratory in Vancouver, Canada. The Manufacturing Automa-tion Laboratory at the Department of Mechanical Engineering of the University of British Columbiais a research facility where metal cutting operations and their stability are investigated. The researchpresented in this report was conducted at the Manufacturing Automation Laboratory.

1.1 Aim of the report

The aim of this report is to develop a method to accurately predict the stability of orthogonal turningprocesses. In turning, the workpiece is attached to a rotating spindle (figure 1.1). The rotational veloc-ity of the workpiece results in a cutting velocity when the tool touches the surface of the workpiece.The tool performs a translational motion with respect to the rotating workpiece to remove the excessmaterial.

Figure 1.1: Turning process

A cutting process is called orthogonal if the cutting edge of the tool is perpendicular to the directionof the cutting velocity (figure 1.2a). If the cutting edge is not perpendicular to the cutting velocitydirection, the cutting process is called oblique (figure 1.2b).

5

(a) Orthogonal cutting (b) Oblique cutting

Figure 1.2: Orthogonal and oblique cutting

1.2 Outline of the report

This report is divided into six chapters. The process dynamics are modeled to find conditions forthe stability of orthogonal turning processes. The model in chapter 2 is based on models in [5, 6].The structural dynamic model, the influence of the tool geometry and the stability analysis based onthe regenerative chip thickness are part of the regenerative chip model presented in [5]. The modelin [6] is used to model the process damping. The predicted process stability is validated by cuttingexperiments. In chapter 3, the experimental setup is discussed and unknown model parameters areidentified. The stability of the turning process is solved for different cutting conditions using theNyquist stability criterion. In chapter 5, an algorithm is presented to find the critically stable cuttingconditions: the boundary between stability and instability. The predicted and the measured processstability are depicted in a stability chart. The construction of the stability chart and the validation ofthe predicted stability are presented in chapter 5. In chapter 6, conclusions are drawn with respect tothe presented algorithm and the accuracy of the stability prediction, and recommendations for futureresearch are made.

6

2 Modeling of orthogonal turning dynamics

As mentioned in the introduction of chapter 1, unstable chatter vibrations can be prevented by select-ing suitable cutting conditions. For a given orthogonal turning operation, the spindle speed, the feedrate and the depth of cut are parameters of the turning process. The spindle speed n is the numberof revolutions of the spindle per period of time. The feed rate is the change in feed per spindle rev-olution. After one revolution, the workpiece comes back to its original angular position. Imposed bythe feed rate, during that revolution, the tool is desired to travel a certain distance in the feed direc-tion. It is customary to assign the same symbol to the change in feed per spindle revolution and tothe corresponding desired tool travel during one revolution. In this report, both are called feed rateand are indicated by c, although technically speaking they are two different quantities and do not havethe same unit (m/rev and m). The depth of cut a is the length of the cut perpendicular to the cuttingvelocity direction and perpendicular to the feed direction. A representation of the feed rate and thedepth of cut is shown in figure 2.1.

Figure 2.1: Feed rate c and depth of cut a

It is important to note that all cutting conditions (the spindle speed, the feed rate and the depth of cut)are assumed to be constant during the turning process. To identify which conditions result in stablevibrations, the dynamics of the turning process are modeled. The model presented in this chapter isbased on models by Eynian and Altintas [5] and Altintas et al. [6]. First, the structural dynamics of themachine and the workpiece are modeled to obtain the dynamic response evoked by the cutting forces(section 2.1). The regenerative chip model of section 2.2 and the cutting force model of section 2.3 areused to link the tool displacement back to the cutting forces. The models are combined in section 2.4to evaluate the stability of the orthogonal turning process for a given set of cutting conditions.

2.1 Structural dynamic model

This section contains the structural dynamic model presented in [5]. During a turning operation, thetool tip is pushed into the workpiece to remove the excess material. The cutting forces generated at thecontact area between the tool tip and the workpiece are transmitted through the tool and the workpieceto other machine components. A force loop is created that runs through the machine from the tool tipto the workpiece (figure 2.2). The displacements of the tool tip and the workpiece at the contact areaare dependent on the effective deformation of each component in the force loop (figure 2.3).

The measurement coordinate system (−→ex,−→ey ,−→ez ) is introduced, where−→ex,−→ey and,−→ez are unit vectorsof the coordinate system in the x-, y- and z-direction. The x-, y- and z-direction are parallel to thedirections of the feed, the depth of cut and the cutting velocity, respectively.

7

Figure 2.2: Force loop

Figure 2.3: Displacements of the tool tip dt and the workpiece dw due to cutting forces acting onthe tool tip Ft and the workpiece Fw

The displacements of the tool tip−→dt and the displacement of the workpiece

−→dw are expressed in the

measurement coordinate system as

−→dk = dx,k

−→ex + dy,k−→ey + dz,k

−→ez for k ∈ t, w, (2.1)

where the displacements of the tool tip and the workpiece in the x-, y- and z-direction are given bydx,k, dy,k and dz,k, for k ∈ t, w.

The cutting forces acting on the tool tip−→Ft and the cutting forces acting on the workpiece

−→Fw are

expressed in the measurement coordinate system as

−→Fk = Fx,k

−→ex + Fy,k−→ey + Fz,k

−→ez for k ∈ t, w, (2.2)

where the cutting forces acting on the tool tip and the workpiece in the x-, y- and z-direction are givenby Fx,k, Fy,k and Fz,k, for k ∈ t, w.

It is assumed that the structural dynamics of the machine and the workpiece are linear. In Laplace

8

domain, the relation between the displacements−→dt and

−→dw, and the cutting force

−→Ft and

−→Fw is given

by

−→dk(s) = Φk(s) ·

−→Fk(s) for k ∈ t, w. (2.3)

The transfer tensor Φk(s) is expressed in the measurement coordinate system as

Φk(s) =(−→em)T

Φmk (s)−→em for k ∈ t, w, (2.4)

with

−→em =

−→ex−→ey−→ez

and Φmk (s) =

φxx,k(s) φxy,k(s) φxz,k(s)φyx,k(s) φyy,k(s) φyz,k(s)φzx,k(s) φzy,k(s) φzz,k(s)

for k ∈ t, w,

where φij,k(s) is the transfer function from the cutting force in j-direction, Fj,k(s), to the displace-ment in i-direction, di,k(s), for i, j ∈ x, y, z and k ∈ t, w.

The cutting forces acting on the tool tip have the same magnitude but the opposite sign of the cut-ting forces acting on the workpiece. Therefore, the following relation holds:

−→F =

−→Ft = −

−→Fw. (2.5)

The amount of removed material is dependent on the relative displacement of the tool tip with respectto the workpiece. The relative displacement of the tool tip with respect to the workpiece due to thecutting forces is given by

−→d =

−→dt −

−→dw. (2.6)

By combining (2.3), (2.5) and (2.6), the following expression for the relative displacement between thetool tip and the workpiece is obtained:

−→d (s) = (Φt(s) + Φw(s)) ·

−→F = Φ(s) ·

−→F (s), (2.7)

where the combined transfer matrix of the tool tip and the workpiece dynamics is expressed in themeasurement coordinate system as Φm(s) = Φmt (s) + Φmw (s).

2.2 Regenerative chip model

The cutting forces are dependent on the distribution of the chip created by the turning process. Thechip distribution depends on the tool geometry and the cutting conditions. In [5], a chip model ispresented that includes the effect of the nose radius of the tool. Small modifications are made to thismodel to include the effect of a triangular tool with an approach angle κr and a nose radius rε (figure2.4a). The cutting conditions that influence the chip distribution are the feed rate c and the depth ofcut a (figure 2.4b). In order to simplify the complex geometry of the chip in section 2.2.3, first, the chiparea, the chip flow angle and the chord length are modeled in sections 2.2.1 and 2.2.2.

9

(a) Approach angle κr and nose radiusrε

(b) Feed rate c and depth of cut a

Figure 2.4: Chip distribution

2.2.1 Chip area

The amount of removed material is dependent on the uncut chip area. The uncut chip area is thearea of the chip projected in the plane normal to the cutting velocity direction (the xy-plane). For anorthogonal turning process with a rake angle of zero, the rake face of the tool is normal to the cuttingvelocity direction and, therefore, parallel to this plane (figure 2.5).

Figure 2.5: Rake face and rake angle

Assuming that the rake angle is zero, that there are no disturbances and that the surface of the work-piece is smooth, the area of the chip is equal to the static chip area A0 (figure 2.6). The static chip areais given by

A0 = c · a−Acusp, (2.8)

where Acusp is the cusp area (figure 2.6). The cusp area can be calculated as follows:

Acusp =c · rε

1− 12

√1−

(c

2rε

)2− r2

ε arcsin(

c

2rε

)if c ≤ cgeom and a ≥ acusp,

(2.9)

10

Figure 2.6: Static chip area A0 and cusp area Acusp

where the geometrical upper limit to the feed rate cgeom and the cusp depth acusp are given by

cgeom = 2rε cos(∣∣∣κr − π

3

∣∣∣+π

6

)(2.10)

and

acusp = rε

1−

√1−

(c

2rε

)2 . (2.11)

Eynian and Altintas [5] did not include the influence of the cusp areaAcusp. Presumably, they assumedthat the cusp area is negligibly small. This is a valid assumption if the nose radius of the tool andthe depth of cut are large compared to the feed rate. In that case, the static chip area in (2.8) can beapproximated by

A0 ≈ c · a if c · a Acusp. (2.12)

2.2.2 Chip flow angle and chord length

The chip flow direction is defined as the direction that the chip leaves the workpiece. According toColwell [7], the chip flow direction can be approximated by the direction normal to the chord thatconnects the two ends of the cutting edge engaged with the cut. The approximated chip flow directionand the chord that connects the two ends of the cutting edge are shown in figure 2.7.

Figure 2.7: Chord and corresponding chip flow direction

The chip flow coordinate system (−→ep ,−→eq ,−→er ) is introduced, where −→ep , −→eq and, −→er are unit vectors of thecoordinate system. It is assumed that the cutting velocity is normal to the rake face of the tool (see

11

section 2.2.1). The unit vector −→ep points in the chip flow direction. The unit vector −→er is parallel tothe cutting velocity. The unit vector −→eq is perpendicular to the other two unit vectors of the coordi-nate system (figure 2.8). The coordinate transformation between the measurement coordinate system(−→ex,−→ey ,−→ez ) and the chip flow coordinate system (−→ep ,−→eq ,−→er ) is given by

−→ec =

−→ep−→eq−→er

=

cos(θ) sin(θ) 0− sin(θ) cos(θ) 0

0 0 1

−→ex−→ey−→ez

= T cm(θ)−→em

⇔−→em = (T cm(θ))T

−→ec ,

(2.13)

with the coordinate transformation matrix T cm(θ) and the chip flow angle θ.

Figure 2.8: Geometrical features of the chip

The length of the chord in x-direction Lx and the length of chord in y-direction Ly (figure 2.8) aregiven by

Lx =

c2 +

√r2ε − (rε − a)2 if acusp ≤ a ≤ ageom

c2 + a cot(κr) + rε

1− cot(κr)

[1 + tan

(2κr−π

4

)]if a > ageom

if c ≤ cgeom (for cgeom and acusp see equations (2.10) and (2.11)),

(2.14)

with

ageom = rε [1− cos(κr)] , (2.15)

and

Ly = a− acusp if a ≥ acusp. (2.16)

The equations (2.14) and (2.16) are used to find expressions for the chip flow angle θ and the length ofthe chord L:

θ = arctan(LxLy

), (2.17)

12

and

L =√L2x + L2

y. (2.18)

2.2.3 Approximated chip

Instead of using the complex chip geometry created by the triangular tool (see the introduction of sec-tion 2.2), the chip geometry is approximated by a rectangle (figure 2.9). The width of the approximatedchip is equal to the chord length L. The thickness of the approximated chip is chosen such that thearea of the chip created by the triangular tool (see section 2.2.1) is equal to the area of the approximatedchip. The (static) chip thickness of the rectangular chip h0 is given by

h0 =A0

L. (2.19)

(a) Chip created by triangular tool (b) Approximated chip

Figure 2.9: Approximation of the static chip thickness h0

If the cusp areaAcusp and the cusp depth acusp are small, the static chip thickness can be approximatedby

h0 ≈c · aL≈ c · Ly

L= c cos(θ) if c · a Acusp and a acusp, (2.20)

see (2.8) and (2.16) for the approximations.

Disturbances lead to vibrations in the displacement of the tool tip with respect to the workpiece. Tool vi-brations in the plane normal to the cutting velocity (the xy-plane or the pq-plane) influence the amountof removed workpiece material and, therefore, the cutting forces. If the feed rate is small compared tothe nose radius of the tool and the depth of cut, the chip width is much larger than the chip thickness(L h0). In that case, tool vibrations in the direction of the chip thickness (the chip flow direction)have a much larger effect on the amount of removed material than similar sized tool vibrations in thedirection of the chip width. Therefore, only tool vibrations in the chip flow direction are assumed toinfluence the cutting forces. The contribution to the cutting forces of tool vibrations other than in thechip flow direction is neglected. In other words, only the chip thickness is assumed to be influencedby the tool vibrations, while the chip width is assumed to be constant.

Current vibrations of the cutting tool with respect to the workpiece in the chip flow direction areindicated by dp(t), where t is the current time. Current vibrations do not only influence the currentchip thickness, they also influence the chip thickness after one revolution because of the vibrationsleft on the surface of the workpiece. This means that the current chip thickness is both dependent on

13

the current tool vibrations and on the tool vibrations of the previous revolution. The tool-workpiecevibrations of the previous revolution are given by dp(t− τ), where τ is the time it takes to perform onerevolution. The dynamic chip thickness hc is given by the following equation:

hc(t) = h0 − [dp(t)− dp(t− τ)]. (2.21)

The equation (2.21) is depicted in figure 2.10. The delay τ is related to the spindle speed n as follows:

τ =60n, (2.22)

where τ is given in seconds and n is given in revolutions per minute.

Figure 2.10: Dynamic chip thickness hc

Similar to the static chip area, the dynamic chip area Ac is given by

Ac = L · hc. (2.23)

2.3 Cutting force model

The cutting forces are estimated using the regenerative chip model of section 2.2. The method ofestimating the cutting forces presented in this section resembles the estimation method used in [5].The differences will be pointed out further in this section.

Instead of deriving expressions for the cutting forces in the three directions of the measurementcoordinate system (−→ex,−→ey ,−→ez ) (see section 2.1), the cutting forces in the x-direction and the y-directionare combined to obtain one cutting force in the xy-plane. Assuming that the rake face of the toolis parallel to the xy-plane (see section 2.2.1), the chip leaves the tool in the same direction as thecombined cutting force in the xy-plane. Hence, the direction of the cutting forces acting on the chipdetermine the direction that the chip leaves the tool (the chip flow direction). The chip flow directionis equal to the p-direction of the chip flow coordinate system (−→ep ,−→eq ,−→er ) (see section 2.2.2). Because thecombined cutting force in the xy-plane only acts in the p-direction, the cutting force in the xy-planeperpendicular to the p-direction (the q-direction) is zero. Therefore, the cutting force vector

−→F is given

by

−→F = Fp

−→ep + Fr−→er . (2.24)

The cutting forces in the p-direction and the r-direction (Fp and Fr) are depicted in figure 2.11.Measurements presented in [5] showed that there is still a cutting force in the q-direction. However,

this force is small compared to the cutting forces in the other directions of the chip flow coordinate

14

Figure 2.11: Cutting forces Fp and Fr

system. In this report, the cutting force in the q-direction is assumed to be zero.

A traditional model for estimating the cutting forces is given in [8]. This model consists of a forceterm related to the deformation and shearing of the chip and a force term related to edge effects. Edgeeffects are related to side effects of the deforming chip, such as friction and hardening of the material.The traditional model for estimating the cutting forces is given by

Fp = Fpc + Fpe = K ′pcAc +K ′peL,

Fr = Frc + Fre = K ′rcAc +K ′reL,

(2.25)

where the force term related to the chip formation is modeled by a cutting force coefficient times thechip area and where the force term related to the edge effects is modeled by a cutting force coefficienttimes the chip width. To include the influence of the cutting speed, the cutting force coefficients weremeasured for different cutting speeds in [5]. In this report, the cutting force coefficients are assumedto be independent of the cutting speed.

In addition to the force contributions of the chip formation and the edge effects, process dampingis added to the model. This additional damping force is either attributed to the change in the directionof the cutting speed due to the cutting force or to friction between the flank of the tool and the vibra-tions on the workpiece surface (figure 2.12). Altintas et al. [6] modeled the process damping as twoforce contributions related to the velocity and acceleration of the tool in the chip flow direction. In thisreport, only the force contribution related to the velocity of the cutting tool is used to model the processdamping:

Fd(t) = −CiVL

˙dp(t), (2.26)

where Ci is the process damping coefficient. The cutting speed V is given by

V =πD

τ, (2.27)

where D is the diameter of the cylindrical workpiece.It was shown in [6] that the process damping coefficient Ci is dependent on tool wear. However,

the tool wear dependency is not modeled in this report.

In the direction parallel to the cutting velocity (the r-direction), the contribution of friction betweenthe workpiece and the tool is modeled by the friction coefficient µ times the normal force Fp (Figure2.12). The friction coefficient is modeled as a constant and is assumed to be 0.3 for steel [9]. Thefriction force Ff is given by

15

Ff = µFp. (2.28)

Figure 2.12: Process damping force Fd and friction force Ff

Although friction was already included in the traditional cutting force model of (2.25), the influenceof process damping on the friction between the workpiece and the tool was not modeled. Subtractingthe friction from the traditional cutting force model only changes its cutting force coefficients. Byincluding the process damping of (2.26) and adding the force contribution of friction in (2.28), thecutting force model of (2.25) becomes

Fp(t) = Fpc(t) + Fpe + Fd(t) = K ′pcAc(t) +K ′peL−CiVL

˙dp(t),

Fr(t) = Frc(t) + Fre + Ff (t) = K ′rcAc(t) +K ′reL+ µFp(t),

which can be rewritten to

Fp(t) = KpcAc(t) +KpeL−CiVL

˙dp(t),

Fr(t) = KrcAc(t) +KreL− µCiVL

˙dp(t),

(2.29)

with cutting force coefficients

Kpc = K ′pc Kpe = K ′pe

Krc = K ′rc + µK ′pc Kre = K ′re + µK ′pe.

2.4 Stability of the orthogonal turning process

The chip thickness is used as a measure for the stability of the turning process. The stability of the chipthickness is determined by the roots of the characteristic equation of the transfer function between thestatic chip thickness h0 and the dynamic chip thickness hc. The structural dynamic model of section2.1, the regenerative chip model of section 2.2 and the cutting force model of section 2.3 are combinedto determine the transfer function between the static and dynamic chip thickness. As time independentterms do not influence the stability of the chip thickness, they are dropped from the model. Vibrations

in the tool-workpiece displacement−→d are linked to the vibrations in the cutting forces

−→F using the

transfer tensor Φ(s) of (2.7):

−→d (s) = Φ(s) ·

−→F (s). (2.30)

16

Because only tool vibrations with respect to the workpiece in the chip flow direction are assumed toinfluence the cutting forces (see section 2.2.3), the vibrations in the tool-workpiece displacement inLaplace domain are expressed as

−→d (s) = dp(s)−→ep . (2.31)

The vibrations in the cutting forces in Laplace domain are expressed as

−→F = L (Kpc

−→ep +Krc−→er )hc(s)− s

CiVL (−→ep + µ−→er ) dp(s). (2.32)

The transfer tensor Φ(s) is expressed in the chip flow coordinate system as

Φ(s) =(−→ec)T

Φc(s)−→ec , (2.33)

with

−→ec =

−→ep−→eq−→er

and Φc(s) =

φpp(s) φpq(s) φpr(s)φqp(s) φqq(s) φqr(s)φrp(s) φrq(s) φrr(s)

.Using the coordinate transformation of (2.13), the transfer matrix Φc(s) is linked to the transfer matrixΦm(s) as follows:

Φc(s) = T cm(θ)Φm(s) (T cm(θ))T . (2.34)

The equation (2.21) gives an expression for the chip thickness hc in time domain. Converting (2.21) toLaplace domain results in

hc(s) = h0 −(1− e−sτ

)dp(s). (2.35)

The closed-loop transfer function between the static chip thickness h0 and the dynamic chip thicknesshc is obtained from (2.30) to (2.35):

hc(s)h0(s)

=1 + sCi

V L [φpp(s) + µφpr(s)]1 + L

φpp(s)

[Kpc (1− e−sτ ) + sCi

V

]+ φpr(s)

[Krc (1− e−sτ ) + sµCi

V

] . (2.36)

The stability of this closed-loop transfer function is determined by the roots (s) of its characteristicequation, that is

1 +H(s) = 0, (2.37)

with

H(s) = L

φpp(s)

[Kpc

(1− e−sτ

)+ s

CiV

]+ φpr(s)

[Krc

(1− e−sτ

)+ sµ

CiV

].

By assuming that the input is harmonic, the open-loop transfer function H(s) can be transformed toa frequency response function by replacing s with jω, where j =

√−1 and ω is the angular frequency

of the harmonic input. The corresponding frequency response function is given by

H(jω) = L

φpp(jω)

[Kpc

(1− e−jωτ

)+ jω

CiV

]+ φpr(jω)

[Krc

(1− e−jωτ

)+ jωµ

CiV

].

(2.38)

17

Transfer matrix Φm(s) is symmetric (φij(s) = φji(s) for i, j ∈ x, y, z). Therefore, transfer functionsφpp(s) and φpr(s) can be expressed as

φpp(s) = φxx(s) cos2(θ) + φxy(s) sin(2θ) + φyy(s) sin2(θ),φpr(s) = φxz(s) cos(θ) + φyz(s) sin(θ),

(2.39)

or in the frequency domain as

φpp(jω) = φxx(jω) cos2(θ) + φxy(jω) sin(2θ) + φyy(jω) sin2(θ),φpr(jω) = φxz(jω) cos(θ) + φyz(jω) sin(θ).

(2.40)

18

3 Experimental setup

The proposed stability model is experimentally validated. The cutting tests are conducted on a HardingeSuperslant CNC turning machine. A cylindrical workpiece is mounted on the spindle of the machine.The tool is placed in a toolholder that is mounted on one of the two turrets. A Kistler 9121 dynamome-ter is placed between the toolholder and the turret to measure the forces acting on the tool tip. Thedynamometer is able to measure the cutting forces in the three directions of the measurement coordi-nate system. The experimental setup is shown in figure 3.1.

Figure 3.1: Experimental setup

The workpiece is a cylindrical bar made of AISI 1045 steel. Having a length of 25 cm, the workpiece isthe most flexible component of the setup. Sandvik Coromant TNMA 16 08 04-KR 3205 inserts are usedas cutting tools. These triangular-shaped inserts have a flat rake face. This makes them suitable for themodeled orthogonal turning processes. In combination with the toolholder, the inclination angle γnof the cutting tool is not zero, which makes the turning process oblique instead of orthogonal (figure3.2). Furthermore, the rake angle λs of the cutting tool is not zero either. The model presented inchapter 2 requires that both angles are zero. Because the inclination angle and the rake angle are small(γn = λs = −6), the influence of both angles on the cutting forces is neglected.

Figure 3.2: Inclination angle γn and rake angle λs

The geometric properties of the tool and the workpiece are given in table 3.1.Other model parameters that are not dependent on geometric properties are identified using cut-

ting tests. The identification of the frequency response functions, the cutting force coefficients and theprocess damping coefficient is described in sections 3.1 to 3.3.

19

Table 3.1: Geometric properties of the tool and the workpiece

Quantity Symbol Size Unit

Tool

Nose radius rε 0.8 mmApproach angle κr 91

Inclination angle γn −6

Rake angle λs −6

WorkpieceDiameter D 41 mmLength Lw 25 cm

3.1 Identification of the frequency response functions

The frequency response functions that are required for the model presented in chapter 2 are obtainedfrom frequency response measurements of the tool and the workpiece in the directions of the measure-ment coordinate system (see section 2.1). For each measurement, an impact force is applied to the toolor the workpiece by hitting it with an impulse force hammer. Both the applied impact force and theacceleration response are measured. The measured force and acceleration data are used to calculate anacceleration-force frequency response function which is converted to a displacement-force frequencyresponse function. A Kistler 9722A500 impulse force hammer and a PCB Piezotronics 353B11 ac-celerometer are used to gather the required force and acceleration data (figure 3.3). The measuredfrequency response functions are shown in figure 3.4.

(a) Frequency response function measure-ment of the workpiece

(b) Frequency response function measure-ment of the tool tip

Figure 3.3: Frequency response measurements

By looking at the amplitude of the dominant mode at 348 Hz, it can be concluded that the workpieceis more flexible than the tool, and that the workpiece is more flexible in the radial direction than in theaxial direction. Modal analysis has shown that the mode at 348 Hz corresponds to the first bendingmode of the workpiece.

The measured frequency response functions are affected by measurement noise. The frequencyresponse functions are fitted to obtain a smooth result. Because only the relative displacement betweenthe tool tip and the workpiece is important, a fit is made of the combined frequency response data ofthe tool tip and the workpiece. A nonlinear least-squares optimization technique is used to fit thefrequency response functions to the following model:

20

200 400 600 800 1000−0.08

−0.04

0

0.04

0.08

Frequency [Hz]

Am

plitu

de [µ

m N

−1 ]

Real partImaginary part

(a) φxx,t(jω)

200 400 600 800 1000−6

−4

−2

0

2

4

Frequency [Hz]

Am

plitu

de [µ

m N

−1 ]


(b) φxx,w(jω)

200 400 600 800 1000−0.08

−0.04

0

0.04

0.08

Frequency [Hz]

Am

plitu

de [µ

m N

−1 ]


(c) φxy,t(jω)

200 400 600 800 1000−0.4

−0.2

0

0.2

Frequency [Hz]

Am

plitu

de [µ

m N

−1 ]


(d) φxy,w(jω)

200 400 600 800 1000−0.04

−0.02

0

0.02

0.04

Frequency [Hz]

Am

plitu

de [µ

m N

−1 ]


(e) φxz,t(jω)

200 400 600 800 1000−1

−0.5

0

0.5

1

Frequency [Hz]

Am

plitu

de [µ

m N

−1 ]


(f) φxz,w(jω)

200 400 600 800 1000−0.06

−0.03

0

0.03

0.06

Frequency [Hz]

Am

plitu

de [µ

m N

−1 ]


(g) φyy,t(jω)

200 400 600 800 1000−0.04

−0.02

0

0.02

0.04

Frequency [Hz]

Am

plitu

de [µ

m N

−1 ]


(h) φyy,w(jω)

200 400 600 800 1000−0.04

−0.02

0

0.02

Frequency [Hz]

Am

plitu

de [µ

m N

−1 ]


(i) φyz,t(jω)

200 400 600 800 1000−0.6

−0.3

0

0.3

0.6

0.9

Frequency [Hz]

Am

plitu

de [µ

m N

−1 ]


(j) φyz,w(jω)

Figure 3.4: Measured frequency response functions

21

φij(ω) =N∑k=1

(αk,ij + jωβk,ij

−ω2 + 2jζk,ijωn,k,ijω + ω2n,k,ij

)− 1mijω2

+ sij for i, j ∈ x, y, z, (3.1)

where

N : number of modes

αk,ij , βk,ij : modal coefficients of mode k

ζk,ij : damping coefficient of mode k

ωn,k,ij : angular natural eigenfrequency of mode k

mij : residual mass

sij : residual stiffness.

Note that all coefficients are different for each frequency response function.The fitted coefficients are given in appendix A. The frequency range of the frequency response

data is 200 Hz to 1000 Hz. The most dominant modes are within this frequency range. The Nyquistdiagrams of the measured and the fitted frequency response functions are shown in figure 3.5.

−3 −2 −1 0 1 2 3−5

−4

−3

−2

−1

0

1

Real part [µm N−1]

Imag

inar

y pa

rt [

µm N

−1 ]

MeasurmentFit

(a) φxx(jω)

−0.2 −0.1 0 0.1 0.2−0.4

−0.3

−0.2

−0.1

0

0.1


Imag

inar

y pa

rt [

µm N

−1 ]

MeasurmentFit

(b) φxy(jω)

−0.8 −0.4 0 0.4 0.8 1.2−0.8

−0.4

0

0.4

0.8


Imag

inar

y pa

rt [

µm N

−1 ]

MeasurmentFit

(c) φxz(jω)

−0.02 0 0.02 0.04 0.06 0.08−0.06

−0.04

−0.02

0

0.02

0.04


Imag

inar

y pa

rt [

µm N

−1 ]

MeasurmentFit

(d) φyy(jω)

−0.4 −0.2 0 0.2 0.4−0.2

0

0.2

0.4

0.6

0.8


Imag

inar

y pa

rt [

µm N

−1 ]

MeasurmentFit

(e) φyz(jω)

Figure 3.5: Nyquist diagrams of the measured and fitted frequency response functions

3.2 Identification of the cutting force coefficients

The cutting force coefficients are determined by cutting force measurements. The cutting force modelis given by (2.29) in section 2.4:

Fp(t) = KpcAc(t) +KpeL−CiVL

˙dp(t),

Fr(t) = KrcAc(t) +KreL− µCiVL

˙dp(t).

22

By neglecting the vibrations in the cutting forces, (2.29) can be written as

Fp = KpcA0 +KpeL = L (Kpch0 +Kpe) ,Fr = KrcA0 +KreL = L (Krch0 +Kre) .

(3.2)

The cutting force coefficients Kpc, Kpe, Krc and Kre are determined by measuring the cutting forcesfor different values of the static chip thickness h0 and fitting the model to the measurement results.Because the static chip thickness is strongly related to the feed rate (h0 ≈ c cos(θ), see (2.20) in section2.2.3), the cutting forces are measured for a constant depth of cut a = 1 mm and different feed ratesc = 0.02, 0.04, 0.06, 0.08, 0.10 mm. To reduce the vibrations in the cutting forces, the length of theworkpiece is shortened to Lw = 8 cm. The cutting conditions are given in table 3.2.

Table 3.2: Cutting conditions for the identification of the cutting force coefficients

Quantity Symbol Size UnitDepth of cut a 1 mmFeed rate c 0.02, 0.04, 0.06, 0.08, 0.10 mm/revSpindle speed n 2000 rev/minWorkpiece diameter D 41 mmWorkpiece length Lw 8 cm

The cutting forces are measured in the measurement coordinate system. The cutting forces of themeasurement coordinate system are converted to the chip flow coordinate system as follows:

Fp =√

(Fx)2 + (Fy)2,

Fq = 0,Fr = Fz.

(3.3)

Five measurements are taken for each feed rate. The converted cutting forces and the fitted curves areshown in figure 3.6. The corresponding cutting force coefficients are given in table 3.3.

0.02 0.04 0.06 0.08 0.10

100

200

300

400

c [mm/rev]

Fp [N

]

MeasurementFit

(a) Fp

0.02 0.04 0.06 0.08 0.10

100

200

300

400

c [mm/rev]

Fr [N

]

MeasurementFit

(b) Fr

Figure 3.6: Cutting forces Fp and Fr for different feed rates c

3.3 Identification of the process damping coefficient

Instead of identifying the process damping coefficient Ci with a piezo actuator driven fast tool servo,as presented in [6], orthogonal plunge turning tests are conducted. The tool is fed into the workpiecein radial direction until the rotational axis is reached (figure 3.7).

23

Table 3.3: Cutting force coefficients

Coefficient Size UnitKpc 1680 · 106 N/m2

Kpe 104.3 · 103 N/mKrc 2600 · 106 N/m2

Kre 62.8 · 103 N/m

Figure 3.7: Decrease in workpiece diameter D and cutting speed V

The decrease in the diameter results in a decrease in the cutting speed. For some cutting conditionsholds that if the cutting speed is low enough, the process damping becomes high enough such that theunstable turning process becomes stable. The process damping force was modeled in (2.26) of section2.3 as

Fd(t) = −CiVL

˙dp(t).

The critical cutting speed for which this change in stability takes place is indicated by Vlim.Orthogonal plunge turning tests are conducted for two sets of cutting conditions (table 3.4). Be-

cause the effect of process damping is largest at low cutting speeds, a relatively low spindle speed of1000 rev/min is used for all cutting tests. Chatter vibrations are not only present in displacement andcutting force measurements, they can also be recorded by sound measurements. A microphone isadded to the experimental setup as shown in figure 3.8.

Table 3.4: Cutting conditions for the process damping coefficient identification


Set 1

Depth of cut a 0.6 mmFeed rate c 0.05 mm/revSpindle speed n 1000 rev/minWorkpiece diameter D [41→ 0] mmWorkpiece length Lw 25 cm

Set 2

Depth of cut a 0.7 mmFeed rate c 0.10 mm/revSpindle speed n 1000 rev/minWorkpiece diameter D [41→ 0] mmWorkpiece length Lw 25 cm

Five cutting tests are conducted for each set of cutting conditions. Figure 3.9a shows the cutting forcein p-direction for one of the measurements for the first set of cutting conditions. The correspondingsound data is presented in figure 3.9b. Both measurements show an unstable region with large chattervibrations and a stable region with small chatter vibrations. Time is zero corresponds to the first

24

Figure 3.8: Force and sound measurement setup

contact between the tool and the workpiece. The critical cutting speed Vlim is reached when the chattervibrations start to attenuate. For the cutting force measurement, this happens after approximately 13.1seconds, which corresponds to a workpiece diameter of 19.2 mm and a critical cutting velocity of 60.3m/min. For the sound measurement, the critical cutting speed is reached after 13.6 seconds, whichcorresponds to a workpiece diameter of 18.3 mm and a cutting speed of 57.5 m/min.

0 2 4 6 8 10 13.1 16 18 20 22 24.6−100

0

100

200

300

Time [s]

Fp [N

]

D = 41.0 mm D = 19.2 mm D = 0 mm

(a) Force measurement

0 2 4 6 8 10 13.6 16 18 20 22 24.6−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

Mic

roph

one

D = 41.0 mm D = 19.2 mm

D = 18.3 mm

D = 0 mm

(b) Sound measurement

Figure 3.9: Force and sound measurement in the time domain for the cutting conditions of set 1

The frequency content of the force and sound data is calculated by taking the fast Fourier transformof the two signals. For time is 1 to 6 seconds, the (unstable) chatter vibrations are clearly visible inthe frequency domain plots of the force and the sound data. The largest peaks in the frequency do-main correspond to the chatter frequency of 381 Hz and its higher order harmonics (figures 3.10a and3.10b). For 15 to 23 seconds, the chatter vibrations are small and are barely visible in the correspondingfrequency domain plots (figures 3.10c and 3.10d).

25

0 1000 2000 3000 4000 50000

20

40

60

80

Frequency [Hz]

|Fp| [

N]

(a) Force measurement for time is 3 to12 seconds

0 1000 2000 3000 4000 50000

0.1

0.2

0.3

0.4

0.5

0.6

Frequency [Hz]

|Mic

roph

one|

(b) Sound measurement for time is 3to 12 seconds

0 1000 2000 3000 4000 50000

20

40

60

80

Frequency [Hz]

|Fp| [

N]

(c) Force measurement for time is 15to 23 seconds

0 1000 2000 3000 4000 50000

0.1

0.2

0.3

0.4

0.5

0.6

Frequency [Hz]

|Mic

roph

one|

(d) Sound measurement for time is 15to 23 seconds

Figure 3.10: Force and sound measurement in the frequency domain for the cutting conditions ofset 1

For the second set of cutting conditions, the feed rate is altered from 0.05 mm/rev to 0.10 mm/rev. Al-though chatter is still present in the measurement data, the force contribution of the chatter vibrationsis less compared to the measurements for the first set of cutting conditions (figure 3.11a). The criticalcutting speed can no longer be obtained by looking at the amplitude of the vibrations. However, it isstill possible to obtain the critical cutting speed by looking at the magnitude of the cutting forces. Afterthe critical cutting speed is reached, a sudden drop in the magnitude of the cutting forces is observed.The cutting forces start to drop at a time of 7.2 seconds. The corresponding workpiece diameter andcutting speed are 17.0 mm and 53.4 m/min.

Because the critical cutting speed is not obtained from the amplitude of the chatter vibrations, thesound data in the time domain can not be used to find the critical cutting velocity (figure 3.11b).

Contrary to the time domain, in the frequency domain, the region with unstable chatter vibrationscan clearly be distinguished from the region with stable chatter vibrations by looking at the frequencycontent (figure 3.12). Similar to the measurements for the cutting conditions of set 1, the measuredchatter frequency is 381 Hz.

Because it is not always possible to use the sound measurements to determine the critical cuttingvelocity, the force measurements are used to calculate the process damping coefficient. The given val-ues for the critical cutting velocities are representable for all measurements. In general, the calculatedcritical cutting velocities do not deviate more than 5 m/min from the given values. The calculatedvalues for critical cutting speed are collected in table 3.5.

Recall that the stability of the turning process can be determined by roots of the characteristic equa-tion of the closed-loop transfer function between the static and the dynamic chip thickness (see (2.37)section 2.4), that is

26

0 1 2 3 4 5 6 7.2 8 9 10 11 12.3−100

0

100

200

300

400

Time [s]

Fp [N

]

D = 41.0 mm D = 17.0 mm D = 0 mm

(a) Force measurement

0 1 2 3 4 5 6 7.2 8 9 10 11 12.3−0.8

−0.4

0

0.4

0.8

Time [s]

Mic

roph

one

D = 41.0 mm D = 17.0 mm D = 0 mm

(b) Sound measurement

Figure 3.11: Force and sound measurement in the time domain for the cutting conditions of set 2

0 1000 2000 3000 4000 50000

5

10

15

20

Frequency [Hz]

|Fp| [

N]

(a) Force measurement for time is 1 to6 seconds

0 1000 2000 3000 4000 50000

0.1

0.2

Frequency [Hz]

|Mic

roph

one|

(b) Sound measurement for time is 1to 6 seconds

0 1000 2000 3000 4000 50000

5

10

15

20

Frequency [Hz]

|Fp| [

N]

(c) Force measurement for time is 8 to11 seconds

0 1000 2000 3000 4000 50000

0.1

0.2

Frequency [Hz]

|Mic

roph

one|

(d) Sound measurement for time is 8to 11 seconds

Figure 3.12: Force and sound measurement in the frequency domain for the cutting conditions ofset 2

27

Table 3.5: Critical cutting speed


Set 1 Critical cutting speed Vlim60.3 m/min1.005 m/s

Set 2 Critical cutting speed Vlim53.4 m/min0.890 m/s

1 + L

φpp(s)

[Kpc

(1− e−sτ

)+ s

CiV

]+ φpr(s)

[Krc

(1− e−sτ

)+ sµ

CiV

]= 0.

The process damping coefficient Ci is calculated by transforming the characteristic equation to thefrequency domain by substituting s = jω and solving for the two unknowns ω and Ci. As the criticalcutting speed Vlim corresponds to the critically stable condition (the boundary between stability andinstability), the two unknowns ω and Ci are obtained by finding the corresponding values for whichthe characteristic equation is critically stable. After substituting s = jω and V = Vlim, the left-handside of characteristic equation has a real and an imaginary part. For critical stability, both parts ofthe characteristic equation need to be zero. This condition is necessary, but not sufficient. A stabilitycheck is necessary to ensure that the correct value of the process damping coefficient is obtained. Forevery angular frequency ω, two possible values for the process damping coefficient are calculated: onefor the real part of the characteristic equation Ci,1 and one for the imaginary part of the characteristicequation Ci,2:

Ci,1 =Vlim

[φpp,I(ω) + µφpr,I(ω)]ω

1L

+ [φpp,R(ω)Kpc + φpr,R(ω)Krc] [1− cos(ωτ)]

− [φpp,I(ω)Kpc + φpr,I(ω)Krc] sin(ωτ)

(3.4)

and

Ci,2 =−Vlim

[φpp,R(ω) + µφpr,R(ω)]ω

[φpp,I(ω)Kpc + φpr,I(ω)Krc] [1− cos(ωτ)]

+ [φpp,R(ω)Kpc + φpr,R(ω)Krc] sin(ωτ), (3.5)

where φpp,R(ω) and φpp,I(ω) are the real and imaginary part of φpp(ω), and φpr,R(ω) and φpr,I(ω) arethe real and imaginary part of φpr(ω).

The process damping coefficient Ci is obtained by looking for a value of ω close to the measuredchatter frequency of 381 Hz for which the following condition holds:

Ci = Ci,1 = Ci,2 ≥ 0. (3.6)

By using (3.6), it is possible to find more than one value for Ci. In order to find the correct value, eachCi value is substituted in the characteristic equation (in Laplace domain) and checked for critical stabil-ity using the Nyquist stability criterion. The calculated Ci values for the two sets of cutting conditions(see table 3.4) are given in table 3.6. The corresponding values of ω of 2286 rad/s and 2288 rad/s (bothapproximately 364 Hz) are close to the measured chatter frequency of 381 Hz. For further calculationsa round-off Ci value of 2.9 · 105 N/m is used.

28

Table 3.6: Process damping coefficient


Set 1Proc. damp. coeff. Ci 2.87 · 105 N/m

Ci = 2.9 · 105 N/mAngular freq. ω 2286 rad/s

Set 2Proc. damp. coeff. Ci 2.90 · 105 N/mAngular freq. ω 2288 rad/s

29

4 An algorithm to construct stability charts for turning

Stability charts are an easy way to visualize the stability of a cutting process for a large range of cuttingconditions. Usually, a stability chart portrays the stability of a cutting process for a range of spindlespeeds and depths of cut while the feed rate is kept constant. An example of a stability chart that iscreated using the algorithm in this chapter is shown in figure 4.1.

0 1000 2000 3000 40000

0.2

0.4

0.6

0.8

1

Spindle speed [rev/min]

De

pth

of

cut

[mm

]

Stable

Unstable

Figure 4.1: Stability chart example

The black curve indicates the boundary between the stable region (grey) and the unstable region(white). Unstable chatter vibrations result in a poor surface finish and can damage the tool, the work-piece and the machine. To avoid unstable chatter vibrations, the cutting conditions of the cuttingprocess should lie in the stable region of the stability chart.

The stability of the modeled orthogonal turning process is predicted by solving the stability of thechip thickness using the characteristic equation of (2.37) in section 2.4. The boundary between thestable region and the unstable region is obtained by finding the cutting conditions that result in a crit-ically stable chip thickness. In section 4.1, an algorithm is presented to identify the critically stablecutting conditions using the Nyquist stability criterion. For every spindle speed and feed rate, criticalstability is obtained by finding the corresponding depth of cut. This operation is executed for a rangeof spindle speeds to obtain a stability chart similar to the one in figure 4.1.

4.1 An algorithm to determine the critically stable cutting conditions

As mentioned, the algorithm presented in this section is designed to find the depth of cut a for whichthe modeled turning process is critically stable. Inputs to this algorithm are the feed rate c, the spindlespeed n and several model parameters, frequency response functions, limit values and initial condi-tions. The outline of the algorithm is presented in figure 4.2.

A Nyquist plot is an easy tool to determine the stability of a system using the Nyquist stabilitycriterion. A Nyquist plot of the open-loop system H(jω) is used to determine the stability of the chipthickness and, therefore, the stability of the turning process. The open-loop systemH(jω) was definedin (2.38) of section 2.4 as

H(jω) = L

φpp(jω)

[Kpc

(1− e−jωτ

)+ jω

CiV

]+ φpr(jω)

[Krc

(1− e−jωτ

)+ jωµ

CiV

].

In order to obtain the necessary Nyquist plot data, the cutting speed and the spindle revolution time arecalculated in the subroutine: calculation of the cutting speed V and the spindle revolution time τ (section4.1.1), and the chip flow angle and the chord length are calculated in the subroutine: calculation ofthe chip flow angle θ and the chord length L (section 4.1.2). After the Nyquist plot data is obtained in thesubroutine: generation of the Nyquist plot data (section 4.1.3), the stability limit valueCR,lim is calculatedin the subroutine: calculation of the stability limit value CR,lim (section 4.1.4). The stability limit value

30

n,D

Calculation of the

cutting speed V and thespindle revolution time τ

c, rε, κr

Initial inputa

BC

oo_ _ _ _ _φxx(jω), φxy(jω),φxz(jω), φyy(jω), φyz(jω) ED

@A//

Calculation ofthe chip flow angle θ

and the chord length LGF

Kpc,Krc, Ci, µ, αp // Generation of theNyquist plot data

Initial inputΩ

oo_ _ _ _ _ _ _

Initial inputCR,high, alow ED

@A

//____

Calculation of the stabilitylimit value CR,lim

αR, αa //_ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _

Stability check and updateof the depth of cut a

BC__________________

ED

oo_ _ _ _ _

a

Figure 4.2: Outline of the algorithm for finding the depth of cut for which the modeled turningprocess is critically stable

is not only a measure for the stability of the chip thickness, it also gives an indication how close thecurrent value of the depth of cut is to the value for which the chip thickness is critically stable. Thestability limit value is used to determine the stability of the chip thickness and to re-estimate the valuefor the depth of cut in the subroutine: stability check and update of the depth of cut a (section 4.1.5).

4.1.1 Calculation of the cutting speed and the spindle revolution time

The cutting speed V and spindle revolution time τ are parameters of the open loop system H(jω).The cutting speed and the spindle revolution time are calculated from the spindle speed n and theworkpiece diameter D using (2.22) and (2.27) (see sections 2.2.3 and 2.3 or figure 4.3). The subroutineto calculate V and τ is depicted in figure 4.3.

n,D // τ = 60n

Equation (2.22)

// V = πDτ

Equation (2.27)

// τ, V

Figure 4.3: Subroutine: calculation of the cutting speed V and the spindle revolution time τ

31

4.1.2 Calculation of the chip flow angle and the chord length

The subroutine: calculation of the chip flow angle θ and the chord length L consists of the subroutine:calculation of parameters acusp and ageom and the subroutine: loop calculation of the chip flow angle θ andthe chord length L (figure 4.4).

c, rε, κr

Calculation of parameters

acusp and ageom

Initial inputa

BC

oo_ _ _ _ _θ, LLoop calculation ofthe chip flow angle θ

and the chord length Loo

a

EDoo

Figure 4.4: Subroutine: calculation of the chip flow angle θ and the chord length L

The chip flow angle θ and the chord length L are dependent on the geometry of the chip. The geometryof the chip depends on the tool geometry, the feed rate c and the depth of cut a. Because the chip flowangle and the chord length are related to the depth of cut, the calculation of θ and L is placed insidethe iteration loop for a. The parameters cgeom, acusp and ageom are required for the calculation of θand L, but are not dependent on the depth of cut. These parameters are calculated in the subroutine:calculation of acusp and ageom (figure 4.5). This subroutine is placed outside the iteration loop for thedepth of cut as shown in figure 4.4.

If the feed rate is larger than cgeom, the expression for acusp can be different than the one given in(2.11) in section 2.2.1 and figure 4.5. Therefore, the algorithm is stopped. Although not given in thisreport, the correct expression for acusp for feed rates larger than cgeom can be obtained from the chipgeometry.

The part of the calculation of θ and L that is related to the depth of cut is given in the subroutine:loop calculation of θ and L (figure (4.6)). The chip flow angle θ and the chip length L are calculatedusing the equations in section 2.2.2.

The depth of cut is either the output of the subroutine: stability check and update of the depth of cuta (see section 4.1.5), or an initial estimate for the first step of the iteration process. It is necessary topick the initial estimate for the depth of cut larger than the converged value, because an initial valuefor the depth of cut that results in an unstable chip thickness provides an upper bound for the iterationprocess (see section 4.1.5 for more details).

4.1.3 Generation of the Nyquist plot data

The generation of the Nyquist plot data of the open-loop system H(jω) is an iterative process. Be-cause of the limited frequency resolution of the measured frequency response functions, there are notenough data points to get an accurate Nyquist plot for low spindle speeds, as shown in figure 4.7a. It-erative interpolation is used to generate more data points and increase the accuracy (figure 4.7b). Notethat only the frequency responses for positive frequencies are calculated and plotted in figure 4.7.

A schematic view of the subroutine: generation of the Nyquist plot data is given in figure 4.8. For agiven list of angular frequencies Ω, the corresponding frequency responses of the frequency responsefunctions φpp(jω) and φpr(jω) are calculated in the subroutine: calculation of the lists of frequency

32

c, rε, κr

cgeom = 2rε cos

(∣∣κr − π3

∣∣+ π6

)Equation (2.10)

_ _ _ _ _

_ _ _ _ _

c ≤ cgeom no //_______

yes

stop

acusp = rε

[1−

√1−

(c

2rε

)2]

Equation (2.11)

ageom = rε [1− cos(κr)]

Equation (2.15)

acusp, ageom

Figure 4.5: Subroutine: calculation of parameters acusp and ageom

responses Ψpp and Ψpr. The frequency response functions φpp(jω) and φpr(jω) were given in (2.40) ofsection 2.4 by

φpp(jω) = φxx(jω) cos2(θ) + φxy(jω) sin(2θ) + φyy(jω) sin2(θ),φpr(jω) = φxz(jω) cos(θ) + φyz(jω) sin(θ).

The frequency response data is used to calculate the frequency responses for the Nyquist plot ofH(jω)in the subroutine: calculation of the Nyquist plot data G. The relative distance between two neighboringfrequency responses in the complex plane is used as a measure for the accuracy of the Nyquist plot.The relative distances between the frequency responses are calculated in the subroutine: calculation ofthe dimensionless lengths βp. If two neighboring frequency responses are too far apart in the complexplane, angular frequencies are added to the list of angular frequencies Ω in the subroutine: update ofthe angular frequency list Ω, and a new iteration step is started.

The angular frequency list Ω is a list of positive discrete values of ω in an ascending order. Ω[k] isthe kth element of the list Ω with k ∈ 1, 2, ...,K, where K is the number of elements of the list Ω.The angular frequencies corresponding to the measured frequency response functions in section 3.1can serve as an initial input for the angular frequency list Ω. It might not be necessary to take the wholerange of measured responses. However, the angular frequency list Ω should at least contain angularfrequencies that correspond to the dominant modes of the measured frequency response functions.

The lists of frequency responses Ψpp and Ψpr corresponding to the frequency response functionsφpp(jω) and φpr(jω) are calculated in the subroutine: calculation of the lists of frequency responses Ψpp

and Ψpr. This subroutine is depicted in figure 4.9. First, the frequency responses of the frequency re-sponse functions φxx(jω), φxy(jω), φxz(jω), φyy(jω) and φyz(jω) corresponding to the list of angularfrequencies Ω are calculated. Linear interpolation between two measured frequency responses can beused to obtain the required frequency responses. The corresponding lists of frequency responses Ψxx,Ψxy , Ψxz , Ψyy and Ψyz are used to populate the lists Ψpp and Ψpr using (2.40) of section 2.4 (or seefigure 4.9).

33

a, c, rε, κr, acusp, ageom

stop

_ _ _ _ _

_ _ _ _ _a > ageom

no //_______________

yes

_ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _

acusp ≤ a ≤ ageom

noOO

yes

Lx = c2 + a cot(κr) + rε

1− cot(κr)

[1 + tan

(2κr−π

4

)]Equation (2.14)

Lx = c2 +

√r2ε − (rε − a)2

Equation (2.14)

BCGF

Ly = a− acuspEquation (2.16)

θ = arctan(Ly

Lx

)Equation (2.17)

L =√L2x + L2

y

Equation (2.18)

θ, L

Figure 4.6: Subroutine: loop calculation of the chip flow angle θ and the chord length L

34

−2 0 2 4−6

−4

−2

0

2

Real part [−]

Imag

inar

y pa

rt [−

]

(a) No interpolation between measurementpoints

−2 0 2 4−6

−4

−2

0

2

Real part [−]

Imag

inar

y pa

rt [−

]

(b) Interpolation between measurementpoints

Figure 4.7: Nyquist plots of H(jω) for an angular frequency resolution of 2π rad/s and a spindlespeed of n = 360 rev/min

L,Kpc,Krc, Ci, µ, V, τ

θ

Calculation of theNyquist plot data G

Calculation of thelists of frequency

responses Ψpp and Ψpr

oo φxx(jω), φxy(jω),φxz(jω), φyy(jω), φyz(jω)

oo

Calculation of thedimensionless lengths βp

Initial inputΩ

@A_ _ _ _ _ _ _ _ _ _

OO

αp //_ _ _ _ _ _ _

_ _ _ _ _ _ _∃r, βp[r] > αp

yes //_____

no

Update of the angularfrequency list Ω

OO

G,GR, GI r ∈ 1, 2, ...,K − 1

Figure 4.8: Subroutine: generation of the Nyquist plot data

The list of frequency responses G represents the frequency responses of the open-loop system H(jω)for the angular frequencies of the list Ω. The subroutine: calculation of the Nyquist plot dataG calculatesthe frequency responses of G using (2.38) of section 2.4. This subroutine is depicted in figure 4.10.

The dimensionless lengths βp are a measure for the accuracy of the Nyquist plot of the open-loop

system H(jω). The dimensionless length βp[r], the rth element of the list βp, is defined as the ratiobetween the distance in the complex plane between the frequency response G[r] and the frequencyresponse G[r+ 1], defined as Lp[r], and the length of the diagonal of the smallest rectangular box thatcontains all data points of the Nyquist plot, defined as Lb (figure 4.11). The dimensionless length βp[r]is given by

βp[r] =Lp[r]Lb

∀r ∈ 1, 2, ...,K − 1. (4.1)

35

Ω, θ

Calculation of the frequency responses

Ψxx[k](jΩ[k]), Ψxy[k](jΩ[k]), Ψxz[k](jΩ[k]),Ψyy[k](jΩ[k]) and Ψyz[k](jΩ[k])

φxx(ω), φxy(ω),φxz(ω), φyy(ω), φyz(ω)

oo

Ψpp[k] = Ψxx[k] cos2(θ) + Ψxy[k] sin(2θ) + Ψyy[k] sin2(θ)Equation (2.40)

Ψpr[k] = Ψxz[k] cos(θ) + Ψyz[k] sin(θ)

Equation (2.40)

Ψpp,Ψpr ∀k ∈ 1, 2, ...,K

Figure 4.9: Subroutine: calculation of the lists of frequency responses Ψpp and Ψpr

L,Kpc,Krc, Ci, µ, V, τ,Ω,Ψpp,Ψpr

G[k] = L

Ψpp[k][Kpc

(1− e−jΩ[k]τ

)+ jΩ[k]Ci

V

]+Ψpr[k]

[Krc

(1− e−jΩ[k]τ

)+ jΩ[k]µCi

V

]Equation (2.38)

G ∀k ∈ 1, 2, ...,K

Figure 4.10: Subroutine: calculation of the Nyquist plot data G

−4 −2 0 2 4−6

−4

−2

0

2

rr+1

Real part [−]

Imag

inar

y pa

rt [−

]

Lb

Lp[r]

Figure 4.11: Length of the diagonal of the smallest rectangular box that contains all data points,Lb, and the distance between point r and point r + 1, Lp[r]

36

The list of dimensionless lengths βp is calculated in the subroutine: calculation of the dimensionlesslengths βp (figure 4.12).

G,Ω

@AED

GR[k] = real(G[k])

GI [k] = imag(G[k])

Lp,R[r] = GR[r + 1]−GR[r]

Lp,I [r] = GI [r + 1]−GI [r]

Lb,R = max(GR)−min(GR)

Lb,I = max(GI)−min(GI)

BCGF

Lb =√

(Lb,R)2 + (Lb,I)2

Lp[r] =

√(Lp,R[r])2 + (Lp,I [r])2

βp[r] = Lp[r]

Lb

Equation (4.1)

GR, GI , βp

∀k ∈ 1, 2, ...,K∀r ∈ 1, 2, ...,K − 1

Figure 4.12: Subroutine: calculation of the dimensionless lengths βp

If one or more elements of the list βp are larger than the limit value αp, angular frequencies are addedto the list Ω. The number of angular frequencies that are added to the list Ω is given by

γp[r] = floor

(βp[r]αp

)∀r ∈ 1, 2, ...,K − 1, (4.2)

where the function floor rounds the number inside the brackets to the nearest integer less than or equalto that number. The subroutine: update of the angular frequency list Ω is shown in figure 4.13. The newangular frequencies corresponding to γp[r] are linearly distributed between Ω[r] and Ω[r + 1], and areadded at the right position to the angular frequency list Ξ. Note that the list Ξ contains elements of Ωand Γ, and that Ω[r] does not have to be equal to the rth element of the list Ξ.

4.1.4 Calculation of the stability limit value

The stability of the chip thickness is evaluated by applying the Nyquist stability criterion to the open-loop system H(s) of (2.37) in section 2.4:

37

Ω, βp, αp

γp[r] = floor(βp[r]αp

)∀r ∈ 1, 2, ...,K − 1

Equation (4.2)

Linear interpolation:

Γ[r][m] =(

1− mγp[r]+1

)Ω[r] + m

γp[r]+1Ω[r + 1]

∀m ∈ 1, 2, ..., γp[r]∀r ∈ 1, 2, ...,K − 1

Ξ = [Ω[1], ...,Ω[r],Γ[r],Ω[r + 1], ...,Ω[K]]

∀r ∈ 1, 2, ...,K − 1

Ω = Ξ

Ω

Figure 4.13: Subroutine: update of the angular frequency list Ω

H(s) = L

φpp(s)

[Kpc

(1− e−sτ

)+ s

CiV

]+ φpr(s)

[Krc

(1− e−sτ

)+ sµ

CiV

].

According to the Nyquist stability criterion (see [10]), the closed-loop system in (2.36) is stable if

Z = N + P = 0, (4.3)

where

Z : number of zeros of the open-loop system (poles of the closed-loop system)

in the right half-plane

N : net number of clockwise encirclements of −1P : number of poles of the open-loop system in the right half-plane.

The poles of the open-loop system H(s) are equal to the poles of transfer functions φxx(s), φxy(s),φxz(s), φyy(s) and φyz(s). Because these transfer functions represent a stable mechanical system: theturning machine, the transfer functions φxx(s), φxy(s), φxz(s), φyy(s) and φyz(s) are assumed to bestrictly proper, minimum-phase and stable. Therefore, the open-loop system H(s) has no poles in theright half-plane (P = 0). This means that the chip thickness and, therefore, the turning process isunstable if the Nyquist plot of H(jω) encircles −1 (N 6= 0). The chip thickness is stable if the Nyquistplot of H(jω) does not encircle −1 (N = 0). If the Nyquist plot of H(jω) passes through −1 withoutencircling it, the chip thickness is critically stable (on the boundary between stability and instability).

The net number of encirclements of −1 is determined from the intersections of H(jω) and thereal axis of the Nyquist plot. The list of real values corresponding to the intersections with H(jω) is

38

defined as CR. The stability limit value CR,lim is one of the elements of the list CR. The stability limitvalue is not only a measure for the stability of the chip thickness, it also gives an indication how closethe current value of the depth of cut is to the value for which the chip thickness is critically stable. Thestability limit value CR,lim is calculated in the subroutine: calculation of the stability limit value CR,lim(figure 4.14).

G,GR, GI

Calculation of the magnitude‖G‖ and the angle ∠G ofthe frequency responses G

Calculation of the listof intersections CR

Calculation of the stabilitylimit value CR,lim and the

corresponding indices rlim and wlim

‖G‖,∠G, δ, CR,lim, rlim, wlim

Figure 4.14: Subroutine: calculation of the stability limit value CR,lim

Generally, the spindle speeds for a turning process are relatively low (below 10, 000 rev/min). Forrelatively low spindle speeds, the Nyquist plot of H(jω) is dominated by the clockwise circles of thedelay difference term 1− e−jωτ (for increasing ω). Because of the clockwise circles, it is assumed thatthe Nyquist plot of H(jω) encircles −1 in a clockwise direction for unstable cutting conditions.

The list G consists of frequency responses of the open-loop system H(jω). Each intersection ofthe Nyquist plot of H(jω) and the real axis is approximated by interpolation of the magnitude andthe angle of two neighboring frequency responses of G on opposite sides of the real axis (if the twofrequency responses are on the same side of the imaginary axis). The magnitude and the angle of thefrequency response G[k] is given by

‖G[k]‖ =√

(GR[k])2 + (GI [k])2 ∀k ∈ 1, 2, ...,K, (4.4)

and

∠G[k] = arctan(GI [k]GR[k]

)∀k ∈ 1, 2, ...,K, (4.5)

where GR[k] and GI [k] are the real and imaginary part of G[k]. Note that the arctangent has onlyone argument: GI [k]/GR[k]. The lists ‖G‖ and ∠G are calculated in the subroutine: calculation of themagnitude ‖G‖ and the angle ∠G of the frequency responses G (figure 4.15).The values on the real axis corresponding to the estimated intersections withH(jω) are defined as CR.If the frequency responses G[r] and G[r + 1] are on opposite sides of the real axis and on the sameside of the imaginary axis (with real parts unequal to zero), the corresponding element of the list CR isgiven by

CR[w] = sgn (GR[r]) (1− δ[w]) ‖G[r]‖+ δ[w]‖G[r + 1]‖ , (4.6)

39

GR, GI

‖G[k]‖ =

√(GR[k])2 + (GI [k])2

Equation (4.4)


)Equation (4.5)

‖G‖,∠G ∀k ∈ 1, 2, ...,K

Figure 4.15: Subroutine: calculation of the magnitude ‖G‖ and the angle ∠G of the frequencyresponses G

where sgn is the signum function with sgn(0) = 0. The fraction δ[w] is given by

δ[w] =∠G[r]

∠G[r]− ∠G[r + 1]. (4.7)

If the frequency responses G[r] and G[r+ 1] are on opposite sides of the real axis but not on the sameside of the imaginary axis, the corresponding element of the list CR is given by

CR[w] = (1− δ[w])GR[m[w]] + δ[w]GR[m[w] + 1], (4.8)

with

δ[w] =GI [m[w]]

GI [m[w]]−GI [m[w] + 1]. (4.9)

The list of intersections CR is calculated in the subroutine: calculation of the list of intersections CR(figure 4.16).

The stability limit value CR,lim corresponds to one of the intersections of H(jω) and the real axis,and is a measure for the stability of the chip thickness. If the smallest element of CR is larger thanor equal to −1, there are no encirclements of −1 and the stability limit value CR,lim is equal to thesmallest element of CR. If the smallest element of CR is smaller than −1, encirclements of −1 arepossible. It is assumed that the centers of the clockwise circles of H(jω) that encircle −1 are in thelower half of the complex plane. Therefore, the first crossing of the real axis of each clockwise circlehas a smaller real value than the second crossing of the same circle (for increasing ω). The point −1can only be encircled if the value of the first crossing is smaller than −1 and the value of the secondcrossing of the same circle is larger than −1. If the first value is smaller than −1 and the second valueis larger than −1, the stability limit value CR,lim corresponds to the first value. If the values of bothcrossings are smaller than −1, limit value CR,lim corresponds to neither of the values, because theclockwise circle does not encircle −1.

The largest possible value of CR,lim is zero, because H(jω)|ω=0 = 0 independent of the modelparameters. The stability limit value CR,lim is calculated in the subroutine: calculation of the stabilitylimit value CR,lim and the corresponding indices rlim and wlim (figure 4.17).

40

G,GR, GI , ‖G‖,∠G

m = [u ∈ R| sgn(GI [u]GI [u+ 1]) = −1 ∨GI [u] = 0]R = 1, 2, ...,K − 1, K: number of elements of list Ω

m[v] < m[v + 1], ∀v ∈ 1, 2, ...,M − 1M : number of elements of list m

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _

sgn (GR[m[w]]GR[m[w]]) = 1∀w ∈ 1, 2, ...,M

M : number of elements of list m

yes

EDno _____________

δ[w] = ∠G[m[w]]∠G[m[w]]−∠G[m[w]+1]

Equation (4.7)

δ[w] = GI [m[w]]GI [m[w]]−GI [m[w]+1]Equation (4.9)

CR[w] = sgn (GR[m[w]])(1− δ[w]) ‖G[m[w]]‖

+δ[w]‖G[m[w] + 1]‖Equation (4.6)

CR[w] = (1− δ[w])GR[m[w]]+δ[w]GR[m[w] + 1]Equation (4.8)

BCGF

CR, δ,m

Figure 4.16: Subroutine: calculation of the list of intersections CR

4.1.5 Stability check and update of the depth of cut

In the subroutine: stability check and update of the depth of cut a (figure 4.18), the stability of the chipthickness is checked by comparing the limit value CR,lim to −1. If CR,lim is close enough to −1, inother words, if |1 + CR,lim| ≤ αR, the iteration process to find the depth of cut for which the chipthickness is critically stable is stopped.

If |1 + CR,lim| > αR, a new estimate for the depth of cut a is calculated in the subroutine: updateof the depth of cut a. There are many methods to come up with a new estimate for the depth of cut. Thebisection method and the Newton-Raphson method are two of these methods. The bisection methodbisects an interval and then selects a subinterval in which the critically stable value for the depth ofcut must lie. An initial lower bound for the interval is given by alow = 0. For this value, the chipthickness is assumed to be stable (1 + CR,lim > 0). The initial upper bound for the interval ahigh hasto correspond to a value of the depth of cut for which the turning process is unstable (1 +CR,lim < 0).A sufficiently high initial estimate for the depth of cut that results in an unstable chip thickness canserve as the upper bound for the bisection interval. For other iteration steps, a new estimate for a isgiven by

a =alow + ahigh

2. (4.10)

Before calculating a new estimate, the bounds of the interval should be updated. If 1 +CR,lim > 0, thelower bound should be updated by selecting alow = a if a > alow. If 1 + CR,lim < 0, the upper bound

41

CR,m

Initial values:

w = 1, CR,lim = 0wlim = 1

_ _ _ _ _ _ _

_ _ _ _ _ _ _CR[w] < CR,lim

no

yes //______ _ _ _ _ _

_ _ _ _ _ _CR[w] < −1

no ED______

yes

CR,lim = CR[w]wlim = wGF

w = w + 1

CR,lim = CR[w]wlim = w

_ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _CR[w + 1] > −1

yesoo_ _ _

BC

nooo_ _ _ _ _ _ _w = w + 2GF_ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _

w < MM : number of elements

of list m

@A_ _

GF

yes

//____

no

rlim = m[wlim]

CR,lim, rlim, wlim

Figure 4.17: Subroutine: calculation of the stability limit value CR,lim and the correspondingindices rlim and wlim

G,GR, GI , ‖G‖,∠G,δ, CR,lim, rlim, wlim

a

αR //_ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _|1 + CR,lim| ≤ αR no //___

yes

Update of thedepth of cut a

BC

OO

a αa

OO

Initial inputCR,high, alow

ED@A

_ _ _ _ _ _ _

OO

Figure 4.18: Subroutine: stability check and update of the depth of cut a

42

should be updated by selecting ahigh = a if a < ahigh.Another method to calculate a new estimate for the depth of cut is the Newton-Raphson method.

The Newton-Raphson method is used to find zeroes of a real-valued function. By selecting 1 + CR,limas function for the Newton-Raphson, a new estimate for the depth of cut is obtained:

a = a− 1 + CR,lim∂∂a (1 + CR,lim)

. (4.11)

The partial derivative ∂∂a (1 + CR,lim) is calculated in appendix B.

In most cases, the depth of cut a converges faster to the critically stable value if the Newton-Raphsonmethod is used. However, convergence is not guaranteed. For the bisection method, convergence isguaranteed providing that a lower bound and a upper bound on a are present. Therefore, both theNewton-Raphson method and the bisection method are used in the subroutine: update of the depth ofcut a (figure 4.19). The choice between the two methods is dependent on the reduction factor βa, whichis given by

βa =2|1 + CR,lim|

CR,high − CR,low, (4.12)

where CR,low and CR,high are calculated values of CR,lim that form the lower and upper bound of aninterval that contains−1. The lower bound CR,low corresponds to the CR,lim value of the upper boundon the depth of cut ahigh; the upper bound CR,high corresponds to the CR,lim value of the lower boundon the depth of cut alow. An initial value for CR,high is given by CR,high = 0. The lower bound isdetermined in the first iteration step if 1 + CR,lim < 0 (unstable conditions).

The Newton-Raphson method is used if reduction factor βa is smaller than or equal to limit valueαa. If the new estimate of a by the Newton-Raphson method is not within the interval of possiblevalues for the depth of cut (alow < a < ahigh) or if βa is too large, the bisection method is used. Itis assumed that there is only one value for the depth of cut for which the chip thickness is criticallystable, as the algorithm only finds one critically stable value for the depth of cut.

43

G,GR, GI , ‖G‖,∠G,δ, CR,lim, rlim, wlim

stop

_ _ _ _ _ _ _

_ _ _ _ _ _ _

First iterationstep?

no

yes //_______ _ _ _ _ _ _

_ _ _ _ _ _ _

1 + CR,lim < 0

no

OO

yes

Initial inputCR,high = 0alow = 0

//_____ βa = 2|1+CR,lim|CR,high−CR,low

Equation (4.12)

_ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _

1 + CR,lim > 0and CR,lim < CR,highGF

@A

no

ED_________________

yes

_ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _

1 + CR,lim < 0and CR,lim > CR,low

nooo_ _ _

yes

CR,high = CR,limalow = a

@AED

CR,low = CR,limahigh = a

CR,low = CR,limahigh = a

αa //_ _ _ _

_ _ _ _βa ≤ αa

no

ED

yes _________

Newton-Raphson:a = a− 1+CR,lim

∂∂a (1+CR,lim)

Equation (4.11)

Bisection:a = alow+ahigh

2Equation (4.10)

_ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _alow < a < ahigh

nooo_ _ _ _ _

BCyes

GF_ _ _ _ _ _ _ _ _ _ _

a

Figure 4.19: Subroutine: update of the depth of cut a

44

5 A stability chart for the experimental setup

Using the algorithm presented in chapter 4, a stability chart is constructed for the experimental setupdescribed in chapter 3. The feed rate is fixed at 0.05 mm/rev. A stability chart is constructed for spindlespeeds up to 4000 rev/min. The cutting conditions are summarized in table 5.1. The parameters relatedto the experimental setup are gathered in table 5.2.

Table 5.1: Cutting conditions for the stability chart

Quantity Symbol Size UnitDepth of cut a 0 ≤ a ≤ 1 mmFeed rate c 0.05 mm/revSpindle speed n 300, 301, ..., 3999, 4000 rev/min

Table 5.2: Parameters related to the experimental setup

Quantity Symbol Size UnitNose radius of the tool rε 0.8 mmApproach angle of the tool κr 91

Workpiece diameter D 41 mmCutting force coefficient Kpc 1680 · 106 N/m2

Cutting force coefficient Krc 2600 · 106 N/m2

Process damping coefficient Ci 2.9 · 105 N/mFriction coefficient µ 0.3

The initial inputs and the user defined limit values that are required for the algorithm presented insection 4.1 are given in tables 5.3 and 5.4.

Table 5.3: Initial inputs

Quantity Symbol Size UnitDepth of cut a 1 mmAngular frequency list Ω 2π · [200, 201, ..., 999, 1000] rad/sUpper bound on the stability limit value CR,high 0Lower bound on the depth of cut alow 0 mm

Table 5.4: User defined limit values (dimensionless)

Quantity Symbol SizeLimit value related to βp αp 10−2

Limit value related to CR,lim αR 10−4

Limit value related to βa αa 0.5

5.1 Construction of the stability chart

Either the measured frequency response functions or the fitted frequency response functions (see sec-tion 3.1) can be used to obtain the stability chart for the experimental setup. Using the fitted frequencyresponse functions, the depth of cut converges to the critical value within a few iteration steps, provid-ing that the initial value for the depth of cut results in an unstable chip thickness (see section 4.1.5). If

45

the measured frequency response functions are used, convergence is not always achieved. Because ofthe measurement noise in the data, the following assumptions do not necessarily hold: there is onlyone value for the depth of cut for which the chip thickness is critically stable; the point −1 is encircledin a clockwise direction; and the centers of the circles of H(jω) that encircle −1 are in the lower half-plane. If there is convergence, it takes more iteration steps before the depth of cut is converged to thecritical value.

The overall shape of the stability chart is the same using both the measured or the fitted frequencyresponse functions (figure 5.1). Using the measured frequency response functions, the effects of themeasurement noise and the interpolation between the measured frequency responses are visible in thestability chart (figure 5.2a). The curve of the stability chart is not as smooth as for the fitted frequencyresponse functions (figure 5.2b).

0 1000 2000 3000 40000

0.2

0.4

0.6

0.8

1


Dep

th o

f cut

[mm

]

Measured FRFFitted FRF

Figure 5.1: Stability charts using the measured and the fitted frequency response functions

3000 3200 3400 3600 3800 40000.4

0.5

0.6

0.7

0.8


Dep

th o

f cut

[mm

]

(a) Measured frequency response functions

3000 3200 3400 3600 3800 40000.4

0.5

0.6

0.7

0.8


Dep

th o

f cut

[mm

]

(b) Fitted frequency response functions

Figure 5.2: Stability charts for spindle speeds n from 3000 to 4000 rev/min

The similarity of the shape of both stability charts can be explained by looking at the generated Nyquistplot data. Figure 5.3 shows that the Nyquist plot data close to the negative real axis is similar for boththe Nyquist plots using the measured and the fitted frequency response functions. This part of theNyquist plot data determines the stability of the chip thickness. Hence, the shape of the stability chartsis similar.

46

−2 −1 0 1 2−4

−3

−2

−1

0

1

2

Real part [−]

Imag

inar

y pa

rt [−

]

(a) Measured frequency response functions

−2 −1 0 1 2−4

−3

−2

−1

0

1

2

Real part [−]

Imag

inar

y pa

rt [−

]

(b) Fitted frequency response functions

Figure 5.3: Nyquist plot data G(Ω) for a spindle speed of n = 4000 rev/min

5.2 Experimental validation of the stability chart

Cutting tests are conducted to validate the constructed stability chart of section 5.1. Cutting tests areconducted for a feed rate of 0.05 mm/rev, several depths of cut ranging from 0.3 mm to 0.75 mm,and several spindle speeds ranging from 250 rev/min to 4000 rev/min. The experimental results andthe constructed stability chart (using the fitted frequency response functions) are shown in figure 5.4.As the stability of the chip thickness is a measure for the stability of the turning process, cuttingconditions in the grey area below the curve are predicted to result in a stable turning process, whilecutting conditions above the curve are predicted to result in an unstable turning process. Stable cuttingtests are indicated by a green circle; unstable cuttings tests are indicated by a red cross.

0 500 1000 1500 2000 2500 3000 3500 4000

0.3

0.4

0.5

0.6

0.7

0.8


De

pth

of cu

t [m

m]

Stable

Unstable

Figure 5.4: Experimental validation of the stability chart

The stability for low depths of cut (a ≤ 0.4 mm) and the stability for low spindle speeds (n ≤ 750rev/min) is predicted correctly. The wrongly predicted points are close to lobes of the stability chart.Figure 5.5 shows that the stability prediction for about half of the points close to stability lobes is correct.

47

0 500 1000 1500 2000 2500 3000 3500 4000

0.3

0.4

0.5

0.6

0.7

0.8


De

pth

of cu

t [m

m]

Incorrect

Correct

Figure 5.5: Correctness of the stability predictions

48

6 Conclusions and recommendations

6.1 Conclusions

In this report, a model for the dynamics of an orthogonal turning process is presented. The modelis based on models in [5] and [6], and includes the influence of the geometry of a triangular cuttingtool and the effect of process damping. An algorithm is presented to find the critically stable cuttingconditions using the Nyquist stability criterion. Due to measurement noise, convergence issues canoccur if measured frequency response functions are used. Experimental results indicate that the stabil-ity prediction of the modeled turning process is accurate for low depths of cut and low spindle speeds.The stability prediction is less accurate in the region of the stability lobes.

The algorithm of chapter 4 is used to find the cutting conditions for which the turning process iscritically stable. This algorithm is based on several assumptions:

• The transfer functions that relate the cutting forces to the displacement between the cutting tooland the workpiece are obtained by frequency response measurements. These transfer functionsare assumed to be strictly proper, minimum-phase and stable.

• The stability of the turning process is predicted by determining the stability of the chip thickness.To determine the stability of the chip thickness, the closed-loop transfer function between thestatic (desired) chip thickness and the dynamic (actual) chip thickness is calculated. This closed-loop transfer function contains the measured transfer functions that relate the cutting forces tothe tool-workpiece displacement and is dependent on the cutting conditions (the depth of cut,the feed rate and the spindle speed). It is assumed there is only one value for the depth of cut forwhich the chip thickness is critically stable. Every value of the depth of cut larger than the thisvalue results in an unstable chip thickness; every value of the depth of cut lower than this valueresults in a stable chip thickness.

• A Nyquist plot is made of the open-loop transfer function that corresponds to the closed-looptransfer function between the static and the dynamic chip thickness. It is assumed that if theturning process is unstable, the point−1 in the Nyquist plot is encircled by the open-loop transferfunction in a clockwise direction for a frequency sweep from −∞ to∞.

• The open-loop transfer function contains a time delay. Because of the time delay, the Nyquistplot of the open-loop transfer function displays circles in the complex plane. It is assumed thatthe centers of the circles are in the lower half-plane.

Because of measurement noise in the frequency response measurements, it is possible that the lastthree assumptions do not hold. Therefore, convergence is not always guaranteed.

The accuracy of the stability predictions is dependent on the structure of the model and the modelparameters. The majority of the model parameters is obtained by measurements. Measurement noisecontributes to the error in the stability prediction. Although accurate measurements are important,the largest error contribution can probably be attributed to the various model assumptions. The mostimportant assumptions are summarized below:

• The depth of cut, the feed rate and the spindle speed are constant.

• The structural dynamics of the cutting tool, the workpiece and the machine are linear.

• The model parameters are independent of the cutting speed and tool wear.

• The rake face of the tool is orthogonal to the cutting velocity.

• The geometrically complex chip created by the triangular tool can be approximated by a simplerectangular chip.

49

• Only tool-workpiece vibrations in the chip flow direction affect the vibrations in the cuttingforces.

• The friction coefficient corresponding to the friction between the flank of the tool and the work-piece surface is 0.3 for steel.

The deviations in the depth of cut, the feed rate and the spindle speed are related to the turning ma-chine. As the cutting conditions are well within the operating range of the machine, the deviations inthe cutting conditions are expected to be small.

The structural dynamics of the machine and the workpiece are assumed to be linear. Nonlinearitiessuch as play and friction (beside viscous friction) are not modeled.

Although the model parameters are assumed to be independent of the cutting speed, especially theeffect of the cutting speed on the cutting force coefficients can be quite substantial. The value of thecutting force coefficients can even differ by a factor of two for different cutting speeds as shown in [5].

Altintas et al. [6] have shown that tool wear can have a large effect on the stability of the turningprocess. Because the tool wear was relatively small for all conducted cutting tests, tool wear probablydoes not have a large effect on the stability of the cutting tests in this report. Nonetheless, tool wear issomething that should be considered when predicting the stability of a turning process.

As indicated in chapter 3, the rake face of the tool is not orthogonal to the cutting velocity. Neitherthe inclination angle nor the rake angle are zero (both −6). The error contribution of both angles cannot be ruled out.

Simplifying the complex chip geometry is an essential part of the cutting force model. Althoughthe chip area remains the same, this simplification definitely affects the accuracy of the model. Howmuch it affects the accuracy is probably related to the assumption that only the tool-workpiece vibra-tions in the chip flow direction affect the vibrations in the cutting forces. The stability charts presentedin chapter 5 are made for a feed rate of 0.05 mm/rev. For this feed rate the chip width is much largerthan the chip thickness. This favors the assumption that the tool-workpiece vibrations in the directionof the chip thickness (the chip flow direction) have a much larger effect on the vibrations in the cuttingforces than tool-workpiece vibrations in other directions. Nonetheless, the assumption that vibrationsin other directions than the chip flow direction do not contribute to the cutting forces is likely over-stated.

It is assumed that the friction coefficient corresponding to the friction between the flank of thetool and the workpiece surface is 0.3. Whether this assumption holds for different tool-workpiececombinations and cutting conditions is questionable.

6.2 Recommendations for future research

The algorithm that is used to find the cutting conditions for which the turning process is critically sta-ble (see chapter 4) can be improved. As a result of the bisection method (see section 4.1.5), the depth ofcut does not converge if there is more than one value for which the turning process is critically stable.Although it might not be possible to find all critically stable values, at least one critically stable valuecould be obtained by reselecting the bounds of the bisection interval.

The stability of the chip thickness and, therefore, the cutting process is obtained from a Nyquistplot of the open-loop transfer function corresponding to the closed-loop transfer function between thestatic and the dynamic chip thickness. The stability limit value corresponds to the value of one of theintersections of the open-loop transfer function and the real axis and serves as a stability measure (seesection 4.1.4). To determine this value, first, all real values corresponding to the crossings of the realaxis by the open-loop transfer function are calculated. By also including the direction of the crossings(from the lower half-plane to the upper half-plane or vice versa), the direction of the encirclementsof −1 could be determined (clockwise or counterclockwise). Including the direction of the crossingscould help to correctly determine the stability limit value even if the centers of the circles of the open-loop transfer function are in the upper half-plane. This could greatly increase the robustness of thealgorithm.

50

To increase the accuracy of the stability prediction, the influence of tool-workpiece vibrations otherthan in the chip flow direction should be included; the model could be extended to oblique turning;the influence of the cutting speed and tool wear could be added; and the friction coefficient correspond-ing to the friction between the flank of the tool and the workpiece surface could be measured.

51

Acknowledgements

I would like to thank several people who made it possible to do my traineeship at the ManufacturingAutomation Laboratory (MAL) at the University of British Columbia. First, I would like to thank Prof.Nijmeijer for introducing me to Prof. Altintas. I would like to thank Prof. Altintas for inviting me toMAL and for his hospitality during my stay. I would also like to thank all members and visitors of MALfor their company and help. Special thanks go to Adem Aygun for his time and his assistance duringmy stay at MAL.

52

Nomenclature

A0 static chip area, 10 m2

Ac dynamic chip area, 14 m2

Acusp cusp area, 10 m2

CR list of real values corresponding to the intersection of H and the real axis, 40

CR,high, CR,low highest/lowest value of CR,lim (iteration process), 43

CR,lim stability limit value, 40

Ci process damping coefficient, 15 N/m

D workpiece diameter, 15 m

−→F cutting force vector, 9 N

Fd cutting force contribution related to process damping, 15 N

Ff cutting force contribution related to friction, 16 N

Fp, Fr cutting force in the p/r-direction, 14 N

Fpc, Frc cutting force contribution related to chip formation in the p/r-direction, 15 N

Fpe, Fre cutting force contribution related to edge effects in the p/r-direction, 15 N

−→Ft cutting force vector acting on the tool tip, 8 N

−→Fw cutting force vector acting on the workpiece, 8 N

Fx, Fy , Fz cutting force in x/y/z-direction, 9 N

Fx,t, Fy,t, Fz,t cutting force acting on the tool tip in x/y/z-direction, 8 N

Fx,w, Fy,w, Fz,w cutting force acting on the workpiece in x/y/z-direction, 8 N

G list of frequency responses of H corresponding to Ω, 35

H open-loop transf. func./freq. resp. func. related to the chip thickness, 17

K number of elements of Ω, 33

Kpc, Krc cutting force coefficient related to chip formation in the p/r-direction, 16 N/m2

Kpe, Kre cutting force coefficient related to edge effects in the p/r-direction, 16 N/m

L chord length / chip width, 13 m

Lb length of the diagonal of the smallest box in the complex plane containing G, 36

Lp list of distances between frequency responses of G in the complex plane, 36

Lx, Ly chord length in x/y-direction, 12 m

T cm coord. trans. matrix from the meas. coord. sys. to the ch. fl. coord. sys., 12

V cutting speed, 15 m/s

a depth of cut, 7 m

53

acusp cusp depth, 11 m

ageom geometrical bound on the depth of cut, 12 m

ahigh, alow upper/lower bound on a (bisection method), 41 m

c feed rate, 7 m/rev or m

cgeom geometrical bound on the feed rate, 11 m

−→d vector of the displacement of the tool tip with respect to the workpiece, 9 m

dp vibrations in the tool-workpiece displacement in p-direction, 14 m

−→dt displacement vector of the tool tip, 8 m

−→dw displacement vector of the workpiece, 8 m

dx, dy , dz displacement of the tool with respect to the workpiece in x/y/z-direction, 9 m

dx,t, dy,t, dz,t displacement of the tool tip in x/y/z-direction, 8 m

dx,w, dy,w, dz,w displacement of the workpiece in x/y/z-direction, 8 m

−→em column of unit vectors of the measurement coordinate system, 9

−→ep , −→eq , −→er unit vector of the chip flow coordinate system in the p/q/r-direction, 12

−→ex, −→ey , −→ez unit vector of the measurement coordinate system in the x/y/z-direction, 7

h0 static chip thickness, 13 m

hc dynamic chip thickness, 14 m

j imaginary number, 17

n spindle speed, 7 rev/min

rε nose radius of the tool, 9 m

s Laplace variable, 9 Hz

Φ transfer tensor related to the displacements and cutting forces, 9 m/N

Φt transfer tensor related to the displ. and cut. for. of the tool tip, 9 m/N

Φw transfer tensor related to the displ. and cut. for. of the workpiece, 9 m/N

Φc transfer matrix in the chip flow coordinate system, 17 m/N

Φm transfer matrix in the measurement coordinate system, 9 m/N

Φmt transfer matrix related to the displ. and cut. for. of the tool tip, 9 m/N

Φmw transfer matrix related to the displ. and cut. for. of the workpiece, 9 m/N

Ψpp, Ψpr list of frequency responses of φpp/φpr corresponding to Ω, 33 m/N

Ψxx, Ψxy , Ψxz list of frequency responses of φxx/φxy/φxz corresponding to Ω, 33 m/N

Ψyy, Ψyz list of frequency responses of φyy/φyz corresponding to Ω, 33 m/N

Ω list of angular frequencies, 33 rad/s

54

αR limit value related to CR,lim, 41

αa limit value related to βa, 43

αp limit value related to βp, 37

βa reduction factor, 43

βp list of dimensionless lengths, 36

γp list of number of angular frequencies to add to Ω, 37

δ list of interpolation fractions corresponding to CR, 40

θ chip flow angle, 13 rad

κr approach angle of the tool, 9 rad

µ friction coefficient, 16

τ time of one spindle revolution, 14 s

φpp, φpr transf. func. / freq. resp. func. from the p/r- to the p-direction, 18 m/N

φxx, φxy , φxz transf. func. / freq. resp. func. from the x/y/z- to the x-direction, 18 m/N

φyy , φyz transf. func. / freq. resp. func. from the y/z- to the y-direction, 18 m/N

ω angular frequency, 17 rad/s

55

A Coefficients of the fitted frequency response functions

This appendix contains the coefficients of the model described in (3.1) of section 3.1. The coefficientsof the fitted frequency response functions φxx(jω), φxy(jω), φxz(jω), φyy(jω) and φyz(jω) are givenin tables A.1 to A.5.

Table A.1: Coefficients of φxx(jω)

k αk,xx [m/N s2] βk,xx [m/N s] ζk,xx [-] ωn,k,xx [rad/s]1 4.5788 · 10−2 2.5503 · 10−5 0.0969 1698.62 4.1921 · 10−3 −4.1907 · 10−6 0.0308 1837.93 −4.0320 · 10−3 −7.9246 · 10−7 0.0134 1974.14 −3.3969 · 10−3 6.9463 · 10−7 0.0043 2050.65 5.4530 · 10−4 7.4929 · 10−7 0.0027 2107.96 4.2973 · 10−1 2.0929 · 10−5 0.0353 2145.87 6.3288 · 10−1 −2.3097 · 10−6 0.0227 2193.88 5.5681 · 10−2 −2.9324 · 10−5 0.0196 2254.99 4.8211 · 10−3 3.8824 · 10−7 0.0155 2463.110 5.3205 · 10−3 −8.4969 · 10−7 0.0207 2990.011 4.8627 · 10−1 2.8048 · 10−5 0.2537 3534.512 2.2691 · 10−2 −2.0728 · 10−5 0.0519 3678.913 −3.3965 · 10−3 3.7373 · 10−6 0.0286 4093.714 2.5610 · 10−2 −3.6606 · 10−6 0.0231 4605.4

mxx [N/ms2] 2.9465 · 101

sxx [m/N ] 1.0439 · 10−8

Table A.2: Coefficients of φxy(jω)

k αk,xy [m/N s2] βk,xy [m/N s] ζk,xy [-] ωn,k,xy [rad/s]1 7.1513 · 10−3 5.0003 · 10−6 0.0519 1665.32 4.9497 · 10−2 −2.2101 · 10−5 0.1217 1881.53 4.9889 · 10−3 5.5273 · 10−5 0.0967 2145.94 7.7681 · 10−2 −2.3467 · 10−5 0.0318 2213.25 2.0252 · 10−2 1.2429 · 10−6 0.0399 2974.56 1.0115 · 10−1 −2.0737 · 10−6 0.1018 3451.27 9.0339 · 10−3 −4.5740 · 10−6 0.0331 3718.48 7.5041 · 10−3 1.3120 · 10−6 0.0258 4202.49 6.1021 · 10−3 3.8598 · 10−7 0.0255 4635.2

mxy [N/ms2] −5.1187 · 103

sxy [m/N ] 5.5078 · 10−10

56

Table A.3: Coefficients of φxz(jω)

k αk,xz [m/N s2] βk,xz [m/N s] ζk,xz [-] ωn,k,xz [rad/s]1 2.4940 · 10−4 −2.1025 · 10−7 0.0016 2052.82 −6.0525 · 10−4 −1.4480 · 10−6 0.0050 2145.73 −2.4322 · 10−1 −5.2857 · 10−5 0.0398 2199.74 1.8124 · 10−1 5.9151 · 10−5 0.0152 2223.75 −1.4500 · 10−2 −4.3549 · 10−6 0.0485 2454.76 −1.6645 · 10−2 −1.7692 · 10−5 0.3002 2495.47 −3.3925 · 10−2 1.3873 · 10−5 0.0484 3352.38 2.3701 · 10−2 −1.4203 · 10−5 0.1232 3837.9

mxz [N/ms2] 1.2879 · 102

sxz [m/N ] −1.0796 · 10−10

Table A.4: Coefficients of φyy(jω)

k αk,yy [m/N s2] βk,yy [m/N s] ζk,yy [-] ωn,k,yy [rad/s]1 −9.3072 · 10−1 −9.3321 · 10−5 0.3552 1616.62 1.3274 · 10−3 1.7739 · 10−6 0.0333 1660.33 2.0104 · 10−1 −2.1215 · 10−4 0.2327 1798.54 2.0416 · 100 2.3967 · 10−4 0.5207 2121.35 4.3464 · 10−3 −6.6045 · 10−7 0.0158 2232.66 1.2732 · 10−1 −3.9456 · 10−6 0.0883 3023.37 −2.1893 · 10−2 5.1521 · 10−5 0.0975 3221.78 4.7464 · 10−3 −2.8798 · 10−6 0.0287 3711.59 −1.5553 · 100 4.0140 · 10−4 0.8289 3805.310 5.8912 · 10−3 7.1013 · 10−7 0.0267 4205.911 9.7431 · 10−3 −5.2389 · 10−7 0.0451 4706.1

myy [N/ms2] 5.2160 · 101

syy [m/N ] −3.5467 · 10−9

Table A.5: Coefficients of φyz(jω)

k αk,yz [m/N s2] βk,yz [m/N s] ζk,yz [-] ωn,k,yz [rad/s]1 2.7590 · 10−3 4.0742 · 10−6 0.0611 1280.92 3.4970 · 10−3 −1.6453 · 10−6 0.0382 1475.23 −1.1120 · 10−1 4.7353 · 10−5 0.0156 2227.04 −5.0883 · 10−4 −4.7546 · 10−5 0.0224 2230.85 −2.4847 · 10−2 −5.4433 · 10−6 0.0331 2840.36 3.0535 · 10−2 5.8259 · 10−6 0.0348 2845.97 1.0400 · 10−2 −6.5590 · 10−6 0.0491 3342.38 −9.4159 · 10−3 −6.3601 · 10−7 0.0283 3649.39 2.3317 · 10−3 −2.5534 · 10−7 0.0143 4243.5

myz [N/ms2] −5.6785 · 101

syz [m/N ] −6.2331 · 10−10

57

B Calculation of the partial derivative ∂∂a (1 + CR,lim)

In this appendix, the partial derivative ∂∂a (1 + CR,lim) for the Newton-Raphson method is calculated

(see section 4.1.5). The calculation of this derivative is required for estimating the value for the depthof cut for which the chip thickness is critically stable. The limit value CR,lim is given by

CR,lim = (δ[wlim]− 1) ‖G[rlim]‖ − δ[wlim]‖G[rlim + 1]‖. (B.1)

The fraction δ[wlim] in (B.1) is calculated by

δ[wlim] =∠G[rlim]

∠G[rlim]− ∠G[rlim + 1]. (B.2)

The magnitude and angle of the responses G[rlim] and G[rlim + 1] are given by

‖G[k]‖ =√

(GR[k])2 + (GI [k])2, (B.3)

and


), (B.4)

where GR[k] and GI [k] are the real and imaginary part of the frequency response G[k], with k ∈rlim, rlim + 1. The real and imaginary part of the frequency response G[k] are calculated by

GR[k] =L

Ψpp,R[k]Kpc [1− cos(Ω[k]τ)]−Ψpp,I [k][Kpc sin(Ω[k]τ) + Ω[k]

CiV

]+ Ψpr,R[k]Krc [1− cos(Ω[k]τ)]−Ψpr,I [k]

[Krc sin(Ω[k]τ) + Ω[k]µ

CiV

],

(B.5)

and

GI [k] =L

Ψpp,R[k][Kpc sin(Ω[k]τ) + Ω[k]

CiV

]+ Ψpp,I [k]Kpc [1− cos(Ω[k]τ)]

+ Ψpr,R[k][Krc sin(Ω[k]τ) + Ω[k]

CiV

]+ Ψpr,I [k]Krc [1− cos(Ω[k]τ)]

,

(B.6)

where Ψpp,R[k] and Ψpp,I [k] are the real and imaginary part of Ψpp[k] and, Ψpr,R[k] and Ψpr,I [k] arethe real and imaginary part of Ψpr[k]. The real and imaginary parts of Ψpp[k] are given by

Ψpp,R[k] = Ψxx,R[k] cos2(θ) + Ψxy,R[k] sin(2θ) + Ψyy,R[k] sin2(θ), (B.7)

and

Ψpp,I [k] = Ψxx,I [k] cos2(θ) + Ψxy,I [k] sin(2θ) + Ψyy,I [k] sin2(θ), (B.8)

where Ψxx,R[k] and Ψxx,I [k], Ψxy,R[k] and Ψxy,I [k], and, Ψyy,R[k] and Ψyy,I [k], are the real andimaginary parts of Ψxx[k], Ψxy[k] and Ψyy[k], respectively. The real and imaginary parts of Ψpr[k] aregiven by

Ψpr,R[k] = Ψxz,R[k] cos(θ) + Ψyz,R[k] sin(θ), (B.9)

and

Ψpr,I [k] = Ψxz,I [k] cos(θ) + Ψyz,I [k] sin(θ), (B.10)

58

where Ψxz,R[k] and Ψxz,I [k], and, Ψyz,R[k] and Ψyz,I [k], are the real and imaginary parts of Ψxz[k]and Ψyz[k], respectively.

The chip flow angle θ and the chip length L are given in (2.17) and (2.18) of section 2.2.2.

The partial derivative ∂∂a (1 + CR,lim) is given in (B.11). The partial derivatives that are required for

the calculation of ∂∂a (1 + CR,lim) are presented in (B.12) to (B.22).

∂

∂a(1 + CR,lim) =

∂CR,lim∂δ[wlim]

[∂δ[wlim]∂∠G[rlim]

(∂∠G[rlim]∂GR[rlim]

∂GR[rlim]∂a

+∂∠G[rlim]∂GI [rlim]

∂GI [rlim]∂a

)+

∂δ[wlim]∂∠G[rlim + 1]

(∂∠G[rlim + 1]∂GR[rlim + 1]

∂GR[rlim + 1]∂a

+∂∠G[rlim + 1]∂GI [rlim + 1]

∂GI [rlim + 1]∂a

)]+

∂CR,lim∂‖G[rlim]‖

(∂‖G[rlim]‖∂GR[rlim]

∂GR[rlim]∂a

+∂‖G[rlim]‖∂GI [rlim]

∂GI [rlim]∂a

)+

∂CR,lim∂‖G[rlim + 1]‖

(∂‖G[rlim + 1]‖∂GR[rlim + 1]

∂GR[rlim + 1]∂a

+∂‖G[rlim + 1]‖∂GI [rlim + 1]

∂GI [rlim + 1]∂a

)(B.11)

∂CR,lim∂δ[wlim]

= ‖G[rlim]‖ − ‖G[rlim + 1]‖ (B.12)

∂δ[wlim]∂∠G[rlim]

=−∠G[rlim + 1]

(∠G[rlim]− ∠G[rlim + 1])2 (B.13)

∂δ[wlim]∂∠G[rlim + 1]

=∠G[rlim]

(∠G[rlim]− ∠G[rlim + 1])2 (B.14)

∂CR,lim∂‖G[rlim]‖

= δ[wlim]− 1 (B.15)

∂CR,lim∂‖G[rlim + 1]‖

= −δ[wlim] (B.16)

∂∠G[k]∂GR[k]

=−GI [k]‖G[k]‖2

, k ∈ rlim, rlim + 1 (B.17)

∂∠G[k]∂GI [k]

=GR[k]‖G[k]‖2

, k ∈ rlim, rlim + 1 (B.18)

∂‖G[k]‖∂GR[k]

=GR[k]‖G[k]‖

, k ∈ rlim, rlim + 1 (B.19)

∂‖G[k]‖∂GI [k]

=GI [k]‖G[k]‖

, k ∈ rlim, rlim + 1 (B.20)

The partial derivatives ∂GR[k]∂a and ∂GI [k]

∂a (see (B.21) and (B.22)) are calculated by multiplying andadding the partial derivatives in (B.23) to (B.34).

∂GR[k]∂a

=∂GR[k]∂L

∂L

∂a

+(

∂GR[k]∂Ψpp,R[k]

∂Ψpp,R[k]∂θ

+∂GR[k]∂Ψpp,I [k]

∂Ψpp,I [k]∂θ

+∂GR[k]∂Ψpr,R[k]

∂Ψpr,R[k]∂θ

+∂GR[k]∂Ψpr,I [k]

∂Ψpr,I [k]∂θ

)∂θ

∂a, k ∈ rlim, rlim + 1

(B.21)

59

∂GI [k]∂a

=∂GI [k]∂L

∂L

∂a

+(

∂GI [k]∂Ψpp,R[k]

∂Ψpp,R[k]∂θ

+∂GI [k]∂Ψpp,I [k]

∂Ψpp,I [k]∂θ

+∂GI [k]∂Ψpr,R[k]

∂Ψpr,R[k]∂θ

+∂GI [k]∂Ψpr,I [k]

∂Ψpr,I [k]∂θ

)∂θ

∂a, k ∈ rlim, rlim + 1

(B.22)

∂GR[k]∂L

=GR[k]L

, k ∈ rlim, rlim + 1 (B.23)

∂GI [k]∂L

=GI [k]L

, k ∈ rlim, rlim + 1 (B.24)

∂GR[k]∂Ψpp,R[k]

=∂GI [k]∂Ψpp,I [k]

= LKpc [1− cos(Ω[k]τ)] , k ∈ rlim, rlim + 1 (B.25)

∂GR[k]∂Ψpp,I [k]

= − ∂GI [k]∂Ψpp,R[k]

= −L[Kpc sin(Ω[k]τ) + Ω[k]

CiV

], k ∈ rlim, rlim + 1 (B.26)

∂GR[k]∂Ψpr,R[k]

=∂GI [k]∂Ψpr,I [k]

= LKrc [1− cos(Ω[k]τ)] , k ∈ rlim, rlim + 1 (B.27)

∂GR[k]∂Ψpr,I [k]

= − ∂GI [k]∂Ψpr,R[k]

= −L[Krc sin(Ω[k]τ) + Ω[k]µ

CiV

], k ∈ rlim, rlim + 1 (B.28)

∂Ψpp,R[k]∂θ

= (Ψyy,R[k]−Ψxx,R[k]) sin(2θ) + 2Ψxy,R[k] cos(2θ), k ∈ rlim, rlim + 1 (B.29)

∂Ψpp,I [k]∂θ

= (Ψyy,I [k]−Ψxx,I [k]) sin(2θ) + 2Ψxy,I [k] cos(2θ), k ∈ rlim, rlim + 1 (B.30)

∂Ψpr,R[k]∂θ

= Ψyz,R[k] cos(θ)−Ψxz,R[k] sin(θ), k ∈ rlim, rlim + 1 (B.31)

∂Ψpr,I [k]∂θ

= Ψyz,I [k] cos(θ)−Ψxz,I [k] sin(θ), k ∈ rlim, rlim + 1 (B.32)

More partial derivatives are calculated (see (B.35) to (B.40)) to find the values for ∂L∂a and ∂θ

∂a in (B.33)and (B.34).

∂L

∂a=

∂L

∂Lx

∂Lx∂a

+∂L

∂Ly

∂Ly∂a

(B.33)

∂θ

∂a=

∂θ

∂Lx

∂Lx∂a

+∂θ

∂Ly

∂Ly∂a

(B.34)

∂L

∂Lx=LxL

(B.35)

∂L

∂Ly=LyL

(B.36)

60

∂θ

∂Lx=LyL2

(B.37)

∂θ

∂Ly=−LxL2

(B.38)

∂Lx∂a

=

rε−a√

r2ε−(rε−a)2if acusp ≤ a ≤ ageom

cot(κr) if a > ageom, if c ≤ cgeom (B.39)

∂Ly∂a

= 1 if a ≥ acusp (B.40)

The parameters cgeom, acusp and ageom are given in (2.10), (2.11) and (2.15) in section 2.2.1.

61

References

[1] R.Y. Chiou and S.Y. Liang. Chatter stability of a slender cutting tool in turning with wear effects.Int. J. Mach. Tools Manuf., 38(4):315–327, 1998.

[2] B. E. Clancy and Y. C. Shin. A comprehensive chatter prediction model for face turning operationincluding tool wear effect. Int. J. Mach. Tools Manuf., 42:1035–1044, 2002.

[3] E. Ozlu and E. Budak. Analytical modeling of chatter stability in turning and boring operations -part i: Model development. J. Manuf. Sci. Eng., 129(4):726–732, 2007.

[4] E. Ozlu and E. Budak. Analytical modeling of chatter stability in turning and boring operations -part ii: Experimental verification. J. Manuf. Sci. Eng., 129(4):733–739, 2007.

[5] M. Eynian and Y. Altintas. Chatter stability of general turning operations with process damping.J. Manuf. Sci. Eng., DOI:10.1115/1.3159047, 2009.

[6] Y. Altintas, M. Eynian, and H. Onozuka. Identification of dynamic cutting force coefficients andchatter stability with process damping. CIRP Ann., 57:371–374, 2008.

[7] L. V. Colwell. Predicting the angle of chip flow for single-point cutting tools. Trans. ASME, 76:199–204, 1954.

[8] Y. Altintas. Manufacturing Automation. Cambridge University Press, 2000.

[9] D. W. Wu. A new approach of formulating the transfer function of dynamic cutting processes. J.Eng. Ind., 111(1):37–47, 1989.

[10] G.F. Franklin, J.D. Powell, and A. Emami-Naeini. Feedback Control of Dynamic Systems. PearsonPrentice Hall, 2006.

62

Stability of orthogonal turning processes · This report is divided into six chapters. The process...

Documents

Transcript of Stability of orthogonal turning processes · This report is divided into six chapters. The process...