Influence of frictional mechanism on chatter vibrations in the ...The cut-ting process stability (trivial solution) is determined in order to produce stability lobe diagrams and determine

Int J Adv Manuf Technol (2017) 89:2837–2844DOI 10.1007/s00170-016-9520-5

ORIGINAL ARTICLE

Influence of frictional mechanism on chatter vibrationsin the cutting process–analytical approach

Andrzej Weremczuk1 ·Rafal Rusinek1

Received: 12 July 2016 / Accepted: 21 September 2016 / Published online: 7 October 2016© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract The paper examines a nonlinear one-degree-of-freedom model of the cutting process. The classical regener-ative mechanism of chatter is enriched by an additional fric-tion phenomenon which generates frictional chatter. Addi-tionally, the nonlinear cubic stiffness of the tool is taken intoaccount. The aim of the paper is to investigate interactionsbetween regeneration and the frictional effect. The proposedmodel is solved by the multi-time scale method. The cut-ting process stability (trivial solution) is determined in orderto produce stability lobe diagrams and determine the influ-ence of friction on the process. Finally, the maps of chatteramplitudes are presented and new frictional stability lobediagrams are proposed to analyse the influence of friction.

Keywords Frictional chatter · Regenerative chatter ·Cutting process

1 Introduction

Nowadays the cutting process is still one of the most popularmanufacturing methods. Given increased industrial compe-tition, the manufacturers must reduce costs and improvedimensional accuracy. The efficiency of a machining oper-ation is determined by the metal removal rates, cycle time,machine down time and tool wear. The primary factor

� Rafal [email protected]

Andrzej [email protected]

1 Lublin University of Technology, Nadbystrzycka 36,20-618 Lublin, Poland

that limits machining process efficiency is a phenomenoncalled chatter. Chatter is a dynamic instability that can limitmaterial removal rates, cause poor surface finish and evendamage the tool or the workpiece. From a historical pointof view, the knowledge of machine tool chatter goes back toalmost 100 years ago when Taylor first described this phe-nomenon [1]. Next, Tlusty et al. [2], Tobias [3] and Kudinov[4, 5] gave background of the so-called regenerative chatter.This effect has become the most commonly accepted expla-nation for machine tool chatter. Later, however, anotherchatter mechanism produced by friction was developed byGrabec [6]. This mechanism, called frictional chatter, cancause interesting phenomena such as deterministic chaos[6–11]. While the frictional mechanism is based on fric-tion between the tool and the workpiece, the regenerativeeffect is related to the wavy workpiece surface generatedby the previous cutting tooth pass. Wiercigroch et al. definefour chatter mechanisms [12, 13]. Besides regenerative andfrictional chatter, they also report mode coupling and termo-mechanical mechanisms. Although trace regeneration andfriction are very important practical operations, there arefew papers which consider regenerative and frictional mech-anisms together, for example [14]. Since friction alwaysexists in a real cutting process, excluding this phenomenonwould be a too big simplification.

In the literature, the most often discussed operationsare orthogonal cutting operations, e.g. turning and milling.As for turning, the governing equation is relatively sim-ple because the tool has one cutting tooth which still is incontact with the workpiece, so the depth of cut is positive[12, 13, 15, 16]. In the case of milling, the direction andvalue of the cutting force change due to rotation of the multi-blade tool, and the cutting is interrupted as each tooth entersand leaves the workpiece. Consequently, the resulting equa-tion of motion is non-smooth [17, 18]. This causes problems

2838 Int J Adv Manuf Technol (2017) 89:2837–2844

during numerical and analytical calculations. An analyticalsolution of nonlinear problems is not exact but approximateand difficult to obtain. Nonetheless, it is frequently used dueto its universality [19]. Sometimes, the impact of ploughingmechanism on chatter stability is presented as well [20].

Recently, scientists pay attention on dynamics of cuttingprocess where multifunctional tools [21] and tools for spe-cial operations e.g. thread milling [22] are used. Moreover,the problem of stability lobes in milling process with multi-ple modes is analysed [23]. In this paper, the useful methodof the lowest envelop stability lobes is developed.

In order to get knowledge about the influence of fric-tional chatter on regenerative chatter and complete field ofmathematical approach, the method of time multi-scales isapplied here. An explanation of interactions between thefrictional and regenerative mechanisms is the main aimof the paper. Therefore, the dynamics of a one-degree-of-freedom model of the cutting process is examined. Specialattention is devoted to the stability problem of trivial andnon-trivial solutions and their dependence on system param-eters. Finally, some practical conclusions regarding thecutting process are drawn from the results.

2 Mathematical model

For the purpose of analysing the regenerative and frictionalmechanisms of chatter, a one-degree-of-freedom model oforthogonal cutting is presented in Fig. 1. In order to explaininteractions between the regenerative and frictional mecha-nisms, only the feed direction (x) is considered here. Fromour point of view, the feed direction is more important, par-ticularly because the regenerative mechanism depends ontool position in the x (feed) direction and friction betweenthe tool and the chip. The tool is modelled as a lumped masswhich is suspended with a nonlinear spring and a linear (vis-cous) damper. The nonlinear spring is sometimes used inthe literature (e.g. [19, 24]) to model the nonlinear prop-erties of the tool and tool holder, although a linear spring

Fig. 1 Model of orthogonal cutting

is more popular. The differential equation of tool motion ispresented as

mx1(t) + cx1(t) + k1x13(t) + kx1(t) = Krw ·

(ho−x1(t)+x1(t−τ))+Kt(sgn(vr)−arvr +brv3r ) (1)

where, m is the tool mass, c is damping, k and k1 are thelinear and nonlinear stiffness coefficients, w is the width ofcut, and ho is the initial depth of cut. Kr is the regenera-tive component of the specific cutting force which is relatedto material shearing (regenerative effect), while Kt is thefrictional component of the specific cutting force. DividingEq. 1 by m and introducing the non-dimensional coordinate(x) and time, after some calculations the non-dimensionalspring and damping forces (Fs and Fd ) are expressed as

Fs = γ x3(t) + ω20x(t))

Fd = δx(t) (2)

The delay differential equation of motion can be presentedin a non-dimensional form as

x(t) + δx(t) + γ x3(t) + ω20x(t)

= α(ho−x(t)+x(t−τ))+β(sgn(vr)−arvr +brv3r ) (3)

Despite the fact that the regenerative effect is the main causeof chatter, one cannot neglect friction phenomena betweenthe tool and the workpiece as well as between the chip andthe tool. Therefore, the present model of the cutting forcehas two components. A regenerative force, which occurswhen the favourable phase develops between the inner andouter modulations, and a friction force between the tool andthe workpiece. Then, α denotes the cutting resistance of theregenerative force (regenerative force factor) while β is thecutting resistance of the friction force component (frictionforce factor). In other words, α and β tell us how strong theregenerative and the friction components are. The regenera-tive force depends on the depth of cut (ho), the present toolposition x(t) and the previous position x(t − τ). In turn, thetime delay x(τ) is connected with the spindle speed Ω bythe equation

Ω = 2π

τ(4)

The friction force depends on the relative velocity (vr )between the tool and the workpiece (chip) which isexpressed as

vr = vc − x(t), vc = d/τ (5)

where vc means the cutting speed which also depends onthe time delay τ and a workpiece or a tool diameter d. Thecoefficients ar and br are responsible for the friction forcecharacteristic presented in Fig. 2. The shape of this charac-teristic is consistent with the experimental results reported

Int J Adv Manuf Technol (2017) 89:2837–2844 2839

Fig. 2 Friction force characteristic

in [25–29]. The relative velocity vr can be positive and neg-ative. Therefore, the friction force characteristic has twobranches.

3 Analytical solution of chatter vibrations

The non-dimensional equation of motion of the cutting tool(3) is solved analytically by the multiple scale method [30].At the beginning, it is assumed that the relative velocity (vr )is still positive and equals 1. Next, two scales—the fast To

and the slow T1 are introduced and defined as follows:

T0 = t, T1 = εt (6)

Then, a solution in the first-order approximation has theform:

x(t) = x0(T0, T1) + εx1(T0, T1)

x(t − τ) = xτ = x0τ (T0, T1) + εx1τ (T0, T1) (7)

It is assumed that:

ω20 = ω2 + εσ, δ = εδ, γ = εγ , α = εα, β = εβ (8)

where ε is a formal small parameter. Next, in order to facil-itate notation, the tilde is omitted. Using the chain rule, thetime derivative is transformed according to the expressions:

d

dt= ∂

∂T0+ ε

∂

∂T1

d2

dt2= ∂2

∂T 20

+ ε∂2

∂T0∂T1+ ε

∂2

∂T1∂T0+ ...

= ∂2

∂T 20

+ 2ε∂2

∂T0∂T1+ ... (9)

Substituting Eqs. 6–9 into Eq. 3 we get:

∂2x(t)

∂T 20

+ 2ε∂2x(t)

∂T0∂T1+ εβbr

(∂x(t)

∂T0+ ε

∂x(t)

∂T1

)3

− 3εβbrvc

(∂x(t)

∂T0+ ε

∂x(t)

∂T1

)2

+ ε(

3βbrv2c − βar + δ

) (∂x(t)

∂T0+ ε

∂x(t)

∂T1

)

+ εα (μx(t) − xτ (t) − h0) + εγ x(t)3

+ εσx(t) + ω2x(t) + εβ(arvc − brv

3c − th − c

)= 0

(10)

For clarity, some part of the mathematical derivations is putin the appendix. Finally, we obtain the modulation equationsin the form

f1 = a′(T1) = −1

2δa(T1) − 1

2αa(T1) sin τ

+1

2βara(T1) − 3

8βbra(T1)

3 − 3

2βbrv

2c a(T1)

f2 = β ′(T1) = 1

2μα + 1

2σ + 3

8γ a(T1)

2 − 1

2α cos τ (11)

Then, for the steady-state solution, Eq. 11 take the form:

− 1

2δa − 1

2αa sin τ + 1

2βara − 3

8βbra

3 − 3

2βbrv

2c a = 0

1

2μα + 1

2(ω2

o − ω2) + 3

8γ a2 − 1

2α cos τ = 0

(12)

Solving Eq. 12 ,one trivial (a1) and two non-trivial (peri-odic) solutions (a2) are found.

a1 = 0

a2,3 = 2

√ar − δ

β

3br

∓ α sin(τ )

3βbr

− d2

τ 2(13)

The trivial solution (a1) is important from a practical pointof view because here the cutting process is stable with-out chatter vibrations. When the trivial solution is unstable,chatter appears. Therefore, the problem of solution stabilityis of great importance.

To analyse the stability of steady-state solutions, Eq. 11are linearised with respect to a(T1) and β(T1). Next, theJacobian matrix is defined as

J =(

df1da

df1dβ(T1)

df2da

df2dβ(T1)

)(14)

The eigenvalue of the Jacobian (Eq. 14) should have a neg-ative real part in order to produce a stable solution. The

2840 Int J Adv Manuf Technol (2017) 89:2837–2844

eigenvalue, which defines stability of trivial and non-trivialsolution is expressed as

1

8(4arβ − 9a2brβ − 12brβ(d/τ)2 − 4δ − 4α sin τ) (15)

Trivial solution stability For the trivial solution (a1 = 0),the eigenvalue (Eq. 15) takes the form

β(ar − 3br(d/τ)2) − δ − α sin τ < 0 (16)

The stability borders of the trivial solution determine theso-called stability lobe diagram (SLD) which is showngraphically in Fig. 3 assuming the following parameters:δ = 0.1, β = 0.8, ar = 0.5, br = 0.1 and d = 1. TheSLD shows the plane of the parameters Ω −α where cuttingprocess is stable (the trivial solution is stable). This area iswhite in the SLD while the colour lobes point to the chattervibration amplitude.

Inside the lobes, the non-trivial (periodic) solution exists.Its amplitude and the lobe width depend on the friction forcefactor (β). At β = 0.01, the chatter region is smaller, butthe amplitude is higher approaching even to 30 (Fig. 3a).At stronger friction (β = 0.1 and especially β = 0.8), thechatter region is wider and the amplitudes of chatter are sig-nificantly smaller. Thus, friction broadens the chatter regionbut limits the vibration amplitude.

Stability of non-trivial solutions The non-trivial (peri-odic) solutions (a2,3) are stable when the following equationis satisfied

β(ar − 3br(d/τ)2) − δ + α(1

2∓ 3

2) sin τ > 0 (17)

The first periodic solution a2 is stable exactly when thetrivial solutions is unstable, but the second non-trivial solu-tion a3 is stable in the regions where the trivial solutionsis stable. Thus, two solutions: trivial a1 and periodic a3,can exist in the same region of the SLD depending onthe initial conditions. The same behaviour observed forthe nonlinear regenerative model is reported in [31]. Bothperiodic solutions (a2 and a3) are presented in Fig. 4.Interestingly, that in the first-order approximation chattervibrations do not depend on cubic nonlinearity determinedby the γ coefficient. Probably the solution of the secondorder approximation reveals the influence of γ on the sys-tem’s dynamics. Similar diagrams with unstable lobes areobtained on the plane Ω −β (Fig. 5). In this case, three vari-ants of the coefficient of regenerative effect (α) are analysedα = 0.01, α = 0.1 and α = 0.4. For α = 0.01 (Fig. 5a)there is a critical value of β = 0.2. This critical β meansthat, below this value, the cutting process is free of chatterregardless of ω. Unstable lobes are hardly visible becausethe whole region β > 0.2 is unstable. In other words, the

Ω

α

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

5

10

15

20

25

30a

Ω

α

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

5

10

15b

Ω

α

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5c

Fig. 3 Stability lobe diagrams. Influence of regeneration mechanism(β) on stability of trivial solution for β = 0.01 (a), β = 0.1 (b) andβ = 0.8 (c)

Int J Adv Manuf Technol (2017) 89:2837–2844 2841

Fig. 4 Stability lobe diagrams. Influence of regeneration mechanism(β) on stability of non-trivial solution for β = 0.01 (a), β = 0.1 (b)and β = 0.8 (c)

Fig. 5 Influence of friction (β) on stability of non-trivial solution forα = 0.01 (a), α = 0.1 (b) and α = 0.4 (c)

2842 Int J Adv Manuf Technol (2017) 89:2837–2844

periodic solutions are stable. Unstable lobes are more visi-ble when α = 0.1 (Fig. 5b). In the analysed system, the mostinteresting behaviour can be observed for α = 0.4 (Fig. 5c).The highest amplitudes occur for the small β and unstablelobes seem to be inverted. Here, the regenerative mechanismdominates over the frictional one.

4 Discussion and conclusions

Chatter vibrations as a result of classical regenerativeand extra frictional mechanisms are investigated here withrespect to interactions between them. The analytical methodof multiple time scales is used successfully to solve thenonlinear problem of the cutting process. Although the non-linear properties of the tool stiffness are assumed, theirinfluence on cutting dynamics is not allowed for in the first-order approximation. Probably, the second order approxi-mation would be better to this aim; however, the influenceof the frictional mechanism on regenerative chatter is visi-ble. Classical unstable lobes generated by the regenerativeeffect are modified by the action of friction. The frictionphenomenon widens unstable cutting regions, but on theother hand, it reduces the chatter vibration amplitude. Theregenerative model of cutting with friction has trivial andtwo periodic (non-trivial) solutions. The periodic and triv-ial solutions can exists simultaneously at specific cuttingspeeds because both solutions can be stable. From prac-tical point of view it means that any disturbance causinga change of initial conditions can lead to chatter even inthe region where the classical regenerative cutting processshould be stable, this is, for a small α. Such a change of ini-tial conditions can be caused for example by chip break. Theinteresting phenomenon of reverse unstable lobes is shownon the plane of rotational speed-friction force coefficient(Ω −β). We can observed an untypical behaviour where thesmall β generates a higher vibration amplitude than largeone. The stability diagram on the plane of rotational speed(ω)-friction force component (β) is an equivalent of theclassical stability lobe diagram (SLD) and can be called africtional stability lobe diagram—FSLD.

Investigation of friction and regenerative chatter will becontinued using the numerical method in order to find ape-riodic and irregular vibrations in the nonlinear model of thecutting process. Moreover, experimental tests are planned tobe performed in order to verify the theoretical results, andmost of all, to obtain the real coefficient of frictional andregenerative force components.

Acknowledgments The work of the first author is financially sup-ported by the National Science Centre under the project no. DEC-2013/09/N/ST8/01202. The contribution of the second author is finan-cially supported by the National Science Centre under the project no.DEC-2011/01/B/ST8/07504.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricteduse, distribution, and reproduction in any medium, provided you giveappropriate credit to the original author(s) and the source, provide alink to the Creative Commons license, and indicate if changes weremade.

Appendix

Expanding derivatives of the Eq. 10, we obtain:

∂x(t)

∂T0= ∂x0

∂T0+ ε

∂x1

∂T0

∂2x(t)

∂T 20

= ∂2x0

∂T 20

+ ε∂2x1

∂T 20

∂2x(t)

∂T0∂T1

= ∂2x0

∂T0∂T1+ ε

∂2x1

∂T0∂T1(18)

ε∂2x1

∂T 20

+ ∂2x0

∂T 20

+ 2ε∂2x0

∂T0∂T1+ εβbr

(∂x0

∂T0

)3

−3εβbrvc

(∂x0

∂T0

)2

+ε(

3βbrv2c −βar +δ

) (∂x0

∂T0

)

+εα (μx0 − x0τ − h0) + εγ x30

+εσx0+ω2x0+εω2x1

+εβ(arvc − brv

3c − th − c

)= 0 (19)

Equating coefficients of powers of ε0 and ε1, we obtain:

ε0 ⇒ ∂2x0

∂T 20

+ ω2x0 = 0

ε1 ⇒ ∂2x1

∂T 20

+ 2∂2x0

∂T0∂T1+ βbr

(∂x0

∂T0

)3

−3βbrvc

(∂x0

∂T0

)2

+(

3βbrv2c −βar +δ

) (∂x0

∂T0

)

+α (μx0 − x0τ − h0) + γ x30 + ω2x1

+σx0 + β(arvc − brv

3c − th − c

)= 0 (20)

It is convenient to express the solution of first Eq. 20 in thecomplex form:

x0(T0, T1) = A(T1)eiT0 + A(T1)e

−iT0

x0τ (T0, T1) = A(T1)ei(T0−τ) + A(T1)e

−i(T0−τ) (21)

where A is the complex conjugate of A, which is an arbitrarycomplex function of T1. Substituting Eq. 21 into secondEq. 20 and expanding the derivatives, we get:

∂x0

∂T0= A(T1)ie

iT0 − A(T1)ie−iT0

∂2x0

∂T0∂T1= A′(T1)ie

iT0 − A′(T1)ie−iT0 (22)

Int J Adv Manuf Technol (2017) 89:2837–2844 2843

and then the following equation is obtained:

∂2x1

∂T 20

+ω2x1+2(A′(T1)ie

iT0 −A′(T1)ie−iT0

)

+βbr

(A(T1)ie

iT0 − A(T1)ie−iT0

)3

−3βbrvc

(A(T1)ie

iT0 − A(T1)ie−iT0

)2

+(

3βbrv2c −βar +δ

) (A(T1)ie

iT0 −A(T1)ie−iT0

)

+αμ[A(T1)e

iT0 + A(T1)e−iT0

]

−α[A(T1)e

i(T0−τ) + A(T1)e−i(T0−τ)

]− αh0

+γ(A(T1)e

iT0 + A(T1)e−iT0

)3

+σ(A(T1)e

iT0 + A(T1)e−iT0

)

+β(arvc − brv

3c − th − c

)= 0 (23)

Ordering Eq. 23, we get its final form

∂2x1

∂T 20

+ ω2x1 + (iδA(T1) + αμA(T1)

+σA(T1) − iβarA(T1) + 3iβbrv2cA(T1)

+3γA(T1)2A(T1)

+3iβbrA(T1)2A(T1) + 2iA′(T1) − αA(T1)e

−iτ )eiT0

+(−iδA(T1) + αμA(T1)

+σA(T1) + iβarA(T1) − 3iβbrv2c A(T1)

+3γ A(T1)2A(T1) − 3iβbrA(T1)

2A(T1)

−2iA′(T1) − αA(T1)e−iτ )e−iT0

+3βbrvcA(T1)2e2iT0 + 3βbrvcA(T1)

2e−2iT0

+ (γ −iβbr) A(T1)3e3iT0 +(γ +iβbr) A(T1)

3e−3iT0

−6βbrvcA(T1)A(T1)

+β(arvc−brv

3c −th−c

)−αh0 =0 (24)

The secular term of Eq. 24 vanishes if and only if:

ST1eiT0 = 0, ST2e

−iT0 = 0 (25)

where ST1 and ST2 are the secular generating terms. Thisleads to the equations:

iδA(T1) + αμA(T1) + σA(T1) − iβarA(T1)

+3iβbrv2cA(T1) + 3γA(T1)

2A(T1)

+3iβbrA(T1)2A(T1) + 2iA′(T1) − αA(T1)e

−iτ = 0

−iδA(T1) + αμA(T1) + σA(T1) + iβarA(T1)

−3iβbrv2c A(T1) + 3γ A(T1)

2A(T1)

−3iβbrA(T1)2A(T1) − 2iA′(T1) − αA(T1)e

−iτ = 0 (26)

Substituting into Eq. 26, the polar form of the complexamplitude:

A(T1) = 1

2a(T1)e

iβ(T1)

A′(T1) = 1

2a′(T1)e

iβ(T1) + 1

2ia(T1)β

′(T1)eiβ(T1)

A(T1) = 1

2a(T1)e

−iβ(T1)

A′(T1) = 1

2a′(T1)e

−iβ(T1)− 1

2ia(T1)β

′(T1)e−iβ(T1) (27)

results in:

− 1

2αa(T1)e

−iτ+iβ(T1) + 1

2iδa(T1)e

iβ(T1)

+1

2μαa(T1)e

iβ(T1) + 1

2σa(T1)e

iβ(T1)

+3

8γ a(T1)

3eiβ(T1) − 1

2iβara(T1)e

iβ(T1)

+3

8iβbra(T1)

3eiβ(T1) + 3

2iβbrv

2c a(T1)e

iβ(T1)

+2i

[1

2a′(T1)e

iβ(T1) + 1

2ia(T1)β

′(T1)eiβ(T1)

]= 0

−1

2αa(T1)e

iτ−iβ(T1) − 1

2iδa(T1)e

−iβ(T1)

+1

2μαa(T1)e

−iβ(T1) + 1

2σa(T1)e

−iβ(T1)

+3

8γ a(T1)

3e−iβ(T1)

+1

2iβara(T1)e

−iβ(T1)

−3

8iβbra(T1)

3e−iβ(T1)

−3

2iβbrv

2c a(T1)e

−iβ(T1)

−2i

[1

2a′(T1)e

−iβ(T1) − 1

2ia(T1)β

′(T1)e−iβ(T1)

]= 0 (28)

After the transformations of the first Eq. 28, we obtain:

− 1

2αa(T1)e

−iτ + 1

2iδa(T1) + 1

2μαa(T1)

+1

2σa(T1) + 3

8γ a(T1)

3 − 1

2iβara(T1)

+3

8iβbra(T1)

3 + 3

2iβbrv

2c a(T1) + ia′(T1)

−a(T1)β′(T1) = 0 (29)

Then recalling

e−iτ = cos τ − i sin τ (30)

2844 Int J Adv Manuf Technol (2017) 89:2837–2844

The normal form is obtained:

1

2iδa(T1) + 1

2μαa(T1) + 1

2σa(T1) + 3

8γ a(T1)

3

−1

2αa(T1) cos τ + 1

2iαa(T1) sin τ

−1

2iβara(T1) + 3

8iβbra(T1)

3 + 3

2iβbrv

2c a(T1)

+ia′(T1) − a(T1)β′(T1) = 0 (31)

Separating the real and imaginary parts, the two, so-called,modulation equations are found:

1

2δa(T1) + 1

2αa(T1) sin τ − 1

2βara(T1)

+3

8βbra(T1)

3 + 3

2βbrv

2c a(T1) + a′(T1) = 0

1

2μαa(T1) + 1

2σa(T1) + 3

8γ a(T1)

3

−1

2αa(T1) cos τ − a(T1)β

′(T1) = 0 (32)

References

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2. Tlusty J, Polacek M (eds) (1963) Stability of machine tools againstself-excited vibration in machining, Proc ASME Prod Eng ResConf

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