milling bifurcations from nonlinear...

Milling Bifurcations from Nonlinear Regeneration

B.P. Mann and N.K. GargNonlinear Dynamics Laboratory, Department of Mechanical and AerospaceEngineering, University of Florida-Gainesville, FL 32611, U.S.A.

T. Insperger and G. StepanDepartment of Applied Mechanics, Budapest University of Technology andEconomics, H-1521 Budapest, Hungary.

Abstract. This paper investigates various modeling choices for analyzing the richand complex dynamics of high-speed milling processes. A variational system is for-mulated that includes a nonlinear expression for regeneration in the discontinuouscutting force model. Stability is determined through the development of a dynamicmap for the resulting variational system. The general case of asymmetric structuralelements is investigated with a fixed frame and rotating frame model. Analytical andnumerical investigations are confirmed through a series of experimental cutting test.The principal results show the nonlinear cutting force model provides hysteresisin bifurcation diagrams, suggesting the presence of multiple attractors, which isconfirmed through experimentation.

Keywords: Milling Bifurcations, Parametric Excitation, Delay Differential Equa-tions, Multiple Attractors

1. Introduction

A current trend within major manufacturing industries is the rapidadoption of high-speed milling. For instance, recent improvements bymachine tool builders have enabled a paradigm shift within the aerospacecommunity. The new design methodology replaces many labor-intensivesheet metal assemblies with monolithic aluminum components. How-ever, the opportunity to capture substantial cost savings typically re-quires accurate predictive analysis tools to avoid unstable oscillations,meet dimensional requirements, and to take full advantage of a ma-chining center’s capabilities.

A primary limiting factor in material removal applications is therelative oscillations between a cutting tool and workpiece system [1, 2].More specifically, cutting force models become nonideal energy sourcesthat must capture force modulations created from relative motions be-tween the cutting tool and workpiece system [1, 3]. The most commonapproach is to prescribe a cutting force models that is a function of theuncut chip area [1, 2, 4]. This type of model connects the relative systemoscillations to cutting force and chip load variations; these motions cancause dynamic cutting forces and self-excitation of the machine-tool

c© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

Mill_nonlinear.tex; 8/07/2004; 21:52; p.1

2 B.P. Mann

structural modes that lead to instability. Unless avoided, these bifurca-tion regions may cause large dynamic loads on the machine spindle andtable structure, damage to the cutting tool, and a poor surface finish[1, 2, 5]. Therefore, a key strategy is to apply analytical and numericalmethods to avoid the parameter regions where instabilities exist.

The explanation for machine-tool chatter was first given by Tlusty[6], Tobias [7], and Merritt [8] as “regeneration of waviness.” Theirresearch provided the development of stability lobe diagrams that com-pactly represent stability information as a function of the control pa-rameters (i.e. spindle speed and depth of cut). An important resultfrom these analyses was the ability to identify stable cutting regionsin which larger metal removal rates could be obtained by cutting athigher spindle speeds.

Stability predictions from many early analysis are only approximatefor the case of milling, because they rely on the fundamental assumptionof continuous cutting. In milling, the cutting forces change directionwith tool rotation and cutting is interrupted as each tooth enters andexits the workpiece (see Figure 1). This leads to cutting force coeffi-cients which change from zero (when the tool is free) to large numbers(when the tool is cutting). While numerical simulation can be usedto capture the interrupted nature of the milling process [1, 9, 10], theexploration of parameter space by time domain simulation is clearlyinefficient.

The focus of many recent investigations has been the occurrence ofnew bifurcation phenomena in interrupted cutting processes. In addi-tion to Niemark-Sacker or secondary Hopf bifurcations, period-doublingbifurcations have been analytically predicted in references [11–17] andconfirmed experimentally in references [5, 11–13, 18–22]. Other inves-tigations have shown additional physical mechanisms may influencethe relative oscillations between the tool and the workpiece [24]. Forinstance, thermoplastic behavior in chip formation have been reportedin reference [25] and frictional effects at the tool-chip interface havebeen examined in reference [26].

The goal of this paper is to obtain a more thorough understand-ing of the appropriate modeling choices and system dynamics for ahigh-speed milling process. Specifically, the influence of asymmetricstructural elements are investigated in the presence of a nonlinearcutting force model. Experimental cutting test, performed while ob-taining tool displacement measurements, serve as a basis for evaluatingdifferent structural model choices. Additionally, independent cuttingtest are performed to estimate cutting force coefficient model parame-ters. A variational equation is formulated to determine the stabilityof the nonlinear, nonhomogeneous, delay-differential system from a


Milling Nonlinear Regeneration 3

temporal finite element analysis method. Experimental and numericalinvestigations show hysteresis in bifurcation diagrams and the presenceof coexisting periodic and quasiperiodic attractors. Analytical resultsadditionally show a better match to experimentally observed behaviorfor an uncommon modeling choice - a generalized nonlinear cuttingforce expression.

2. Model development

Several alternatives exist for modeling the dynamic behavior of a typ-ical milling process (see Figure 1). For instance, this section developstwo different lumped element models: 1) a model with rotating dampingand restoring forces; and 2) the most common model applied in millingdynamics literature that assumes non-rotating restoring and dampingforces and a negligible mass imbalance. A schematic of each modelconsidered is shown in Figure 2.

2.1. Rotating Frame Model

A top view of an end mill whirling at the spindle rotational frequencyis shown in Figure 2a. The elastic restoring forces, written as Fku andFkv, are assumed to follow the rotating coordinate frame (u, v). Therelationship between the restoring forces in the fixed coordinate frame(Fkx, Fky) and the rotating coordinate frame are[

Fku

Fkv

]= T (θ)

[Fkx

Fky

]. (1)

where θ(t), written as θ, is the cutter rotation angle and T (θ) is acoordinate transformation matrix defined by

T (θ) =[cos θ(t) − sin θ(t)sin θ(t) cos θ(t)

]. (2)

In a similar fashion, the relationship between the displacements ineach coordinate system can be written as[

uv

]= T (θ)

[xy

]. (3)

The influence of rotating asymmetric structural elements is investi-gated by assuming: 1) the u-direction spring and damping coefficientsare k − ∆k and d − ∆d; and 2) the spring and damping coefficientsin the v-direction are k + ∆k and d + ∆d. The resulting equation ofmotion is


4 B.P. Mann

[m 00 m

] [x(t)y(t)

]+[d−∆d cos 2θ(t) ∆d sin 2θ(t)

∆d sin 2θ(t) d+ ∆d cos 2θ(t)

] [x(t)y(t)

]+[k −∆k cos 2θ(t) ∆k sin 2θ(t)

∆k sin 2θ(t) k + ∆k cos 2θ(t)

] [x(t)y(t)

]=[Fx(x, y, θ(t), τ)Fy(x, y, θ(t), τ)

],

(4)

where m, d ± ∆d, and k ± ∆k are the modal mass, damping, andstiffness terms identified from impact test on a stationary tool. Thecutting forces, which are further described in Section 2.3, are writtenas Fx(x, y, θ, τ) and Fy(x, y, θ, τ) to explicitly show their dependenceupon the current tool position, the cutter rotation angle, and the toolposition in the previous tooth passage (i.e. this necessitates the use of adelayed position variable τ). A more generalized expression for multiplemodes acting along two orthogonal directions is

M ~X + D(θ) ~X + K(θ) ~X = ~Fc(x, y, θ(t), τ). (5)

where ~Fc(x, y, θ, τ) is the cutting force vector, M, D(θ), and K(θ) arethe spatial mass, damping, and stiffness matrices of the rotating system.

2.2. Fixed Frame Model

A more common approach for investigating the dynamic behavior ofthe milling process, see references [1, 2, 5, 16, 17, 20, 21, 14, 27],assumes a lumped element model in the fixed framed and negligiblemass imbalance. This results in the following equation of motion

[m 00 m

] [x(t)y(t)

]+[dx 00 dy

] [x(t)y(t)

]+[kx 00 ky

] [x(t)y(t)

]=[Fx(x, y, θ(t), τ)Fy(x, y, θ(t), τ)

],

(6)where the terms m, dx,y, and kx,y are the modal mass, damping, andstiffness in the x- and y-directions of the system. The generalizedlumped element model for this system can be written as

M ~X + D ~X + K ~X = ~Fc(x, y, θ(t), τ) , (7)

where M, D, and K are the spatial mass, damping, and stiffness ma-trices associated with the fixed x- and y-axes and ~Fc(x, y, θ(t), τ) is avector describing the cutting forces.



2.3. Cutting force model

The cutting forces in a milling operation change direction with toolrotation and cutting is interrupted as each tooth enters and leaves theworkpiece (see Figure 1). This leads to cutting force coefficients whichchange from zero (when the tool is free) to large numbers (when thetool is cutting). The total cutting force in each direction can be writtenas a summation over the total number of cutting teeth N :

[Fx(x, y, θ(t), τ)Fy(x, y, θ(t), τ)

]=

N∑p=1

gp(t)T (θ)−1[−Ftp(x, y, θ(t), τ)−Frp(x, y, θ(t), τ)

]. (8)

where gp(t) is a square wave function that provides a discontinuity,or basic nonlinearity, in the cutting force by assuming a value of onewhen the pth tooth is active and zero when this tooth is not cutting.The tangential and radial cutting force components, Ftp(t) and Frp(t)respectively, are considered to be a function of cutting pressures Kt

and Kr [1, 2], the axial depth of cut b, and the instantaneous chipthickness wp(t),

Ftp(t) = Ktbwp(t) , (9)

Frp(t) = Krbwp(t) , (10)

where wp(t) depends upon the feed per tooth, h, the cutter rotationangle θp(t), and regeneration in the compliant tool directions. Here, anapproximate nonlinear relationship is proposed between the tool feedand cutting forces

wp(t) ≈[h sin θp(t) + [x(t)− x(t− τ)] sin θp(t)

+ [y(t)− y(t− τ)] cos θp(t)]γ. (11)

where the time delay between consecutive tooth passages is taken tobe τ = 2π/NΩ [s]. Although the relationship shown in equation (11)differs from those defined in references [28, 29], similar relationshipshave been proposed by numerous researchers for turning and drillingstudies [1, 30]. Additionally, this expression provides a particularlyconvenient connection to the linear model - where the linear model(γ = 1) can be viewed as a special case of the more general nonlinearrelationship.


6 B.P. Mann

The proposed nonlinear expression was investigated after obtainingunsatisfactory results between a linear model and experimental data.For instance, Figure 3 shows a comparison between the proposed forcemodel and the measured cutting forces. Parameter estimates for alinear model only provide a “locally” good fit throughout the cutterrotation (i.e. notice a good fit at cutter entry and a poor fit nearcutter exit in Figure 3). If an iterative approach is applied to refine thecutting coefficients, the result is just a shift the location of a good andpoor fit. The application the proposed nonlinear relationship, resultsshown in Figure 3, clearly provides a better correlation to the exper-imentally observed values. Parameters for the nonlinear force model,where both cutting forces and tool displacements were measured for a19.05 [mm] diameter tool and aluminum (7050-T7451) workpiece, wereestimated to be γ = 0.94, Kt = 7.0 × 108 [N/m(1+γ)], Kn = 2.1 × 108

[N/m(1+γ)]. Cutting force measurements were obtained using a Kistler1

Model 9255B rigid dynamometer and tool displacements were measuredwith a Lion Precision capacitance probe system.

3. Stability from solution perturbation

The models derived in the previous sections illustrate the governingequations for the milling process are discontinuous and contain bothperiodic coefficients and a time delay. A variational approach is appliedhere to determine local stability by applying a perturbation, written as~ξ(t) = [ξx(t) ξy(t)]

T, about the desired τ -periodic motion

~X(t) = ~Xp(t) + ~ξ(t) (12)

where ~Xp(t) = [xp(t) yp(t)]T is the stable periodic solution. Expand-

ing the nonlinear forces from equation (8) about the desired periodicmotion provides the following after substitution into equation (5),

M ~Xp + D(θ) ~Xp + K(θ) ~Xp + M~ξ + D(θ)~ξ + K(θ)~ξ = b~fo(θ)

+ bKc(θ)[~ξ(t)− ~ξ(t− τ)

], (13)

where Kc(θ) and ~fo(θ) are compact notation for

1 Commercial equipment is identified for completeness and does not implyendorsement by the authors.



Kc(θ) =N∑

p=1

γgp(t)(h sin θp(t)

)γ−1[−Ktsc−Krs

2 −Ktc2 −Krsc

Kts2 −Krsc Ktsc−Krc

2

],

(14)

~fo(θ) =N∑

p=1

gp(t)(h sin θp(t)

)γ[−Ktc−KrsKts−Krc

], (15)

with s = sin θp(t) and c = cos θp(t).While perturbation growth will results in an unstable solution, per-

turbation decay (i.e. ~ξ(t) = 0) will yield a stable periodic solution tothe following equation

M ~Xp + D(θ) ~Xp + K(θ) ~Xp = b~fo(θ) . (16)

Equation (16) describes a linear system with a τ -periodic excitationterm and a τ -periodic solution. Subtracting equation (16) from equa-tion (13) results in an equation for the perturbed motion

M~ξ(t) + D(θ)~ξ(t) + K(θ)~ξ(t) = bKc(θ)[~ξ(t)− ~ξ(t− τ)

]. (17)

3.1. Discrete map development

This section describes the development of a discrete map for the result-ing variational system (equation (17)). The most general case of a lowradial immersion process, where the radial immersion is defined as theradial engagement divided by the tool diameter, is handled by a tem-poral finite element approach to match solutions between cutting andnon-cutting regimes. The previous work for applications to fixed-framesystems with linear force models is described in references [5, 17, 22, 23].

When the tool is out of contact with the workpiece, the system isgoverned by the equation for free vibration

M~ξ(t) + D(θ)~ξ(t) + K(θ)~ξ(t) = 0. (18)

This equation can be rearranged into state-space form~ξ(tf )~ξ(tf )

= Φ(tf , 0)

[~ξ(0)~ξ(0)

], (19)

where tf is the duration for free vibration and Φ(tf , 0) is a state transi-tion matrix that relates the states of the perturbation at the beginningof free vibration to the perturbed states at the end of free vibration.


8 B.P. Mann

During the cutting process, there is no exact solution for the per-turbed motion. However, we can divide the cutting time into elementsand approximate the perturbation as a linear combination of polyno-mials

~ξ(t) =4∑

i=1

~anjiφi(σj(t)) (20)

Here σj(t) = t−nτ−∑j−1

k=1 tk is the “local” time within the jth elementof the nth period, the time span for the kth element is tk and the trialfunctions or polynomial are written as φi(σj(t)).

Substitution of the assumed solution (equation (20)) into the vari-ational equation leads to a non-zero error. The error from the assumedsolution is “weighted” by multiplying by a set of test functions andsetting the integral of the weighted error to zero to obtain two equationsper element [31, 32],

∫ tj

0

[M

(4∑

i=1

anjiφi(σj)ψr(σj)

)+ D(σj)

(4∑

i=1


)+

(K(σj)− bKc(σj)

)( 4∑i=1


)+

bKc(σj)

(4∑

i=1

an−1ji φi(σj)ψr(σj)

)]dσj = 0,

r = 1, 2 , (21)

The test functions are chosen to be: ψ1(σj) = 1 (constant) andψ2(σj) = σj/tj − 1/2 (linear). The integrals, shown in equation (21),are taken over the time for each element, tj = tc/E, thereby dividingthe time in the cut tc into E elements. The terms D(σj), K(σj), andKc(σj) have been used in place of the previously defined D(θ), K(θ),and Kc(θ) to explicitly show their dependence on the local time withineach element. The state transition matrix and the equations for eachelement, equation (21), can be arranged into a global matrix relatingthe coefficients of the states in the current period to the coefficients ofthe states in previous period. The following expression is for the casewhen the number of elements is three



I 0 0 0

N11 N1

2 0 00 N2

1 N22 0

0 0 N31 N3

2

~a11

~a12

~a21

~a22

~a31

~a32

~a33

~a34

n

=

0 0 0 ΦP1

1 P12 0 0

0 P21 P2

2 00 0 P3

1 P32

~a11

~a12

~a21

~a22

~a31

~a32

~a33

~a34

n−1

, (22)

where the sub-matrices and elements of the sub-matrices for the jth

element are:

Nj1 =

[N j

11 N j12

N j21 N j

22

], Nj

2 =

[N j

13 N j14

N j23 N j

24

], (23)

Pj1 =

[P j

11 P j12

P j21 P j

22

], Pj

2 =

[P j

13 P j14

P j23 P j

24

], (24)

N jri =

∫ tj

0

[Mφi(σj) + D(σj)φi(σj) + (K(σj)− bKc(σj))φi(σj)

]ψr(σj) dσj ,

(25)

P jri =

∫ tj

0−bKc(σj)φi(σj)ψr(σj) dσj . (26)

Equation (22) describes a discrete dynamical system, or map, thatcan be written as:

A~an = B~an−1 , or ~an = Q~an−1 . (27)

3.2. Stability prediction

The eigenvalues of the transition matrix Q = A−1B are the dynamicmap characteristic multipliers (CMs) that contain local stability infor-mation [33]. The condition for stability requires the CM magnitudesto be in a modulus of less than one for a given spindle speed (Ω) anddepth of cut (b) for an asymptotically stable milling process. Figure 4shows the boundaries between stable and unstable cutting as a functionof spindle speed and depth of cut.

Two distinct types of instability are illustrated by CM trajectories inthe complex plane: (1) a Flip bifurcation or period doubling phenomenaoccurs when a negative real CM passes through passes through the unit


10 B.P. Mann

circle; and (2) a Niemark-Sacker or secondary Hopf bifurcation occurswhen a complex CM obtains a magnitude greater than one. Theseroutes to instability are illustrated in the bottom graphs of Figure 4with the corresponding speed and depth of cut points shown in the topstability chart. One particularly interesting result from this diagram isan abrupt jump in the largest eigenvalue just prior to the occurrence ofa secondary Hopf bifurcation. The first two modes of vibration in thex- and y-direction of Figure 1, identified from impact test on a cuttingtool, were used to construct this stability chart. The modal parameters,listed in Table I, were assumed identical (symmetric) in each directionof the tool for the presented results.

Table I. Single direction structural modes.

Diameter [mm] M [Kg] D [Ns/m] K [N/m]×106

19.050.6519 0.000

0.000 0.2242

36.87 0.000

0.000 43.67

0.8783 0.0000

0.0000 2.1987

3.3. Bifurcation and Hysteresis

Relative oscillations between the cutting tool and workpiece system willinitially grow in an unstable cutting process. As the relative oscillationsbuild, cutting forces will also grow until the tool to jumps out of thecut. As the tool jumps out of the cut, the cutting forces become zeroand free vibrations occur until the tool reenters the cut. The result isbounded post-bifurcation behavior that occurs as the tool continues toreenter and jump out of the cut [5]. This basic nonlinearity, associatedwith the tool jumping out of the cut, is not captured by the modelequations defined earlier. However, this post-bifurcation behavior canbe captured in numerical integration by simply setting the cuttingforces to zero when negative values of the radial chip thickness wp(t)arise. One alternative for capturing the post-bifurcation behavior isto remesh the time domain (i.e. truncate the temporal elements withdiscontinuities) while using the same criteria for negative wp(t) values[5].

Figure 5 shows a forward and reverse sweep bifurcation diagram forthe experimental system studied in Section 5. This graph was createdby varying a control parameter, the depth of cut b, during numericalintegration of equation (4) while capturing τ -periodic samples of the y-direction displacements. An interesting result from the nonlinear forcemodel is the presence of hysteresis in these graphs; this gives a clear



indication that multiple attractors can coexist for certain parametercombinations. The numerical settings used for the simulation were 1000time steps per revolution, 100,000 cutter revolutions, and the depth ofcut was varied linearly.

4. Equilbria solutions

Parameter selection based entirely upon stability considerations is ofteninadequate for the production of precision components. Specifically,certain stable parameter combinations can still result in an inaccuratelymachined surface. This can occur because the amplitude and phasingof cutter oscillations is dependent upon the cutting process parameters(i.e. spindle speed (Ω), axial depth of cut (b), and tool feed (h)). Theintent of this section is to describe equilibria solution prediction fromthe formulation of a dynamic map - the additional decision makinginformation necessary for a priori parameter selection.

An asymptotically stable cutting process results from perturbationdecay. The remaining dynamical system, described by equation (16),now contains an ideal energy source. Discretizing this equation, byfollowing the approach outlined in Section 3, results in

I 0 0 0

N11 N1

2 0 00 N2

1 N22 0

0 0 N31 N3

2

~a11

~a12

~a21

~a22

~a31

~a32

~a33

~a34

n

=

0 0 0 Φ0 0 0 00 0 0 00 0 0 0

~a11

~a12

~a21

~a22

~a31

~a32

~a33

~a34

n−1

+

~0~0~C1

1~C1

2~C2

1~C2

2~C3

1~C3

2

, (28)

where the sub-matrices and elements of the sub-matrices are givenin equation (23) and equation (25). The remaining term required forequilibria solution prediction is

Cjp =

∫ tj

0b~fo(σj)ψp(σj) dσj . (29)

Equation (28) can be written as

~an = Q~an−1 + ~D , (30)

after multiplying by the inverse of the matrix on the left hand side.The coefficient vector ~an identifies the velocity and displacement at


12 B.P. Mann

the beginning and end of each element. Surface location error is givenby the displacement coefficient value when the cutting tooth is normalto the surface. With the assumption of no tool helix angle, this occurs atcutter entry into the cut for up-milling and cutter exit for down-milling.

Stable milling processes have τ -periodic cutting forces and τ -periodicsolutions. The steady state coefficients are found from the fixed points(~a∗n) of the dynamic map

~an = ~an−1 = ~a∗n . (31)

Substitution of equation (31) into equation (30) gives the fixed pointmap solution or steady-state coefficient vector

~a∗n = (I−Q)−1 ~D . (32)

Since Q and ~D can be computed exactly for each spindle speed anddepth of cut, the fixed point displacement solution can be found andused to specify surface location error as a function of machining processparameters.

5. Experimental stability test

A five-axis linear motor Ingersol machining center with a Fischer, 40,000[rpm] and 40 [kW] spindle was used to perform stability cutting testswith the two flute, 19.05 [mm] diameter, 106 [mm] overhang, end mill.Modal parameters for the experimental system were obtained from im-pact tests at the cutter tip (see Table II). An aluminum (7050-T7451)workpiece was down-milled at a 5% radial immersion and a feedrateof h = 0.178 [mm/tooth]; the spindle speed (Ω) and depth of cut (b)were changed for each cutting test to determine the onset of unstablevibrations. Since multiple cuts were performed on the same workpiece,a clean-up pass was performed prior to every recorded cut to create areliable reference surface.

Table II. Modal parameters for experimental system.

Diameter [mm] m [Kg] d [Ns/m] ∆d [Ns/m] k [N/m] ∆k [N/m]

19.05 0.061 4.092 0.234 1.668×106 -1.223×103

Tool displacements, measured approximately 22 [mm] from the tooltip, were recorded at a rate of 25 [KHz] per channel using a laptopdata acquisition system and Lion Precision capacitance probes (see



Figure 6). A once per revolution timing signal was measured, using alaser tachometer, to read a barcode that was painted onto the non-fluted segment of the tool. Experimental stability results have beensuperimposed onto stability predictions for the fixed-axis model in Fig-ure 7 and the rotational dynamics model in Figure 8. Stability test,labeled as cases A–G, are further examined in Figure 9 and Figure 10.However, before diverting attention away from Figure 7 and Figure 8,it is worthy to note that there are several regions where the nonlinearforce model provides a much better match with experiment and thedifferences between fixed frame and rotating frame stability predictionsis nearly insignificant.

Raw displacement measurements were periodically sampled at thetooth passage frequency to create 1/tooth passage displacement sam-ples and Poincare sections shown in displacement vs. delayed displace-ment coordinates; these plots are also shown with the PSD (PowerSpectral Density) of the continuously sampled displacement in Figure 9.Tests were declared stable if the 1/tooth passage sampled displacment,or τ -periodic samples, approached a fixed point value (see Figure 9,case A).

Unstable behavior predicted by complex characteristic multiplierswith a magnitude greater than one, shown by cases B and D, corre-sponds to Niemark-Sacker or secondary Hopf bifurcation. These post-bifurcation test results show the 1/tooth-passage displacement sam-ples are incommensurate with the tooth passage frequency and quasi-periodic motions can be observed in the Poincare sections. Due to staticand dynamic runout in the cutter teeth, multiple circular rings appearin the Poincare sections - a separate circle for each cutting tooth. Sincethe resulting sequential mapping from one cutting tooth to the nextis predicted to be uniform in the absence of cutter runout, these testsprovide further insight about the role of runout in the dynamic toolmotions. Case D is particularly interesting since unstable behavior isonly predicted by the nonlinear cutting force model.

Unstable period-doubling behavior, commonly referred to as a flipbifurcation [5, 11, 18, 19, 21, 22], is predicted when the dominantcharacteristic multiplier of the discrete map model is negative and realwith a magnitude greater than one. Experimental evidence of this typeof post-bifurcation behavior is shown by case C of Figure 9.

An example cutting test that does not agree with either model isshown by case E of Figure 10. The experimental 1/tooth-passage dis-placement samples, Poincare section, and PSD graph clearly indicatespost-bifurcation behavior. Case F confirms the predicted unstable re-gion at the higher spindle range for both structural models. The cuttingtest for the final example, represented by case G, were repeated four


14 B.P. Mann

times with two stable and two unstable outcomes (see examples in thebottom two graphs of Figure 10). These results provide experimentalevidence of a sensitivity to initial conditions and the coexistence of aperiodic and quasiperiodic attractor.

6. Conclusions

Several alternatives exist to model the complex dynamical behaviorof a high-speed milling processes. This paper investigates the choicebetween a typical rotating structural element model and a fixed-axismodel. A variational system is formulated from a nonlinear relationshipobserved during experimental test that provides a general relationshipbetween the tool feed rate and cutting forces. Stability is determinedfor the time-varying system by expanding the perturbed periodic solu-tion about the stable periodic motion. This provides a suitable delay-differential equation for temporal finite element analysis. The temporalfinite element method forms an approximate solution by dividing thetime in the cut into a finite number of elements. The approximatesolution is then matched with the exact solution for free oscillationsto obtain a discrete map. Stability is determined directly from theeigenvalues of map characteristic multipliers. Since parameter selec-tion based entirely upon stability considerations is often inadequatein the production of precision components, an approach to determineequilibria solutions is described.

Better agreement is obtained between cutting test and predictionswhen using a nonlinear force model. Although it is believed that asym-metric structural elements can cause added unstable regions due to thecoupling of parametric excitation terms in the rotating system, onlysmall differences were observed in the particular experimental systemexamined. Also, despite several cases that appear to be in direct agree-ment with predictions, some differences were still observed betweenanalysis and experiment. The primary differences between experimentand analysis are believed to be related to: 1) relative motions fromfrictional and/or thermoplastic effects; 2) the assumption that mea-sured cutting coefficients are unaffected by changes in spindle speed;and 3) neglecting the helix angle of the cutting tool.

Acknowledgements

Support from from the following sources is gratefully acknowledged:1) National Science Foundation CAREER Award (CMS-0348288); 2) Zoltan



Magyary Postdoctoral Fellowship Foundation for Hungarian HigherEducation and Research provided by the Hungarian National ScienceFoundation (OTKA F047318 and OTKA T043368); and 3) The BoeingCompany.


16 B.P. Mann

References

1. Tlusty, J.: 2000, Manufacturing Processes and Equipment. Upper Saddle River,NJ: Prentice Hall, 1 edition.

2. Altintas, Y.: 2000, Manufacturing Automation. New York, NY: CambridgeUniversity Press, 1 edition.

3. Nayfeh, A. H. and D. T. Mook: 1979, Nonlinear Oscillations. New York: JohnWiley & Sons, Inc.

4. Davies, M. and B. Balachandran: 2001, ‘Impact Dynamics in Milling of Thin-Walled Structures’. Nonlinear Dynamics 22, 375–392.

5. Mann, B. P., P. V. Bayly, M. A. Davies, and J. E. Halley, ‘Limit cycles, bifur-cations, and accuracy of the milling proces’. Journal of Sound and Vibrationin press.

6. Tlusty, J., A. Polacek, C. Danek, and J. Spacek: 1962, ‘Selbsterregte schwingun-gen an werkzeugmaschinen’. VEB Verlag Technik. Berlin.

7. Tobias, S. A.: 1965, Machine Tool Vibration. London: Blackie.8. Merritt, H.: 1965, ‘Theory of Self-Excited Machine Tool Chatter’. Journal of

Engineering for Industry 87(4), 447–454.9. Smith, S. and J. Tlusty: 1991, ‘An overview of the modeling and simulation of

the milling process’. Journal of Engineering for Industry 113, 169–175.10. Montgomery, D. and Y. Altintas: 1991, ‘Mechanism of cutting force and surface

generation in dynamic milling’. Journal of Engineering for Industry 113, 160–168.

11. Davies, M. A., J. R. Pratt, B. Dutterer, and T. J. Burns: 2002, ‘Stabilityprediction for low radial immersion milling’. Journal of Manufacturing Scienceand Engineering 124(2), 217–225.

12. Davies, M. A., J. R. Pratt, B. Dutterer, and T. J. Burns: 2000, ‘The stabilityof low immersion milling’. Annals of the CIRP 49, 37–40.

13. Bayly, P. V., J. E. Halley, B. P. Mann, and M. A. Davies: 2001b, ‘Stability ofinterrupted cutting by temporal finite element analysis’. In: Proceedings of the18th Biennial Conference on Mechanical Vibration and Noise, Pittsburg, PA.

14. Insperger, T., G. Stepan,: 2000, ‘Stability of high-speed milling’. Proceedingsof symposium on nonlinear dynamics and stochastic mechanics AMD-241,119–123.

15. Corpus, W. T. and W. J. Endres: 2000, ‘A high order solution for the addedstability lobes in intermittent machining’. Proceeding of the Symposium onMachining Processes (MED-11), 871–878.

16. Zao, M. X. and B. Balachandran: 2001, ‘Dynamics and stability of millingprocess’. International Journal of Solids and Structures 38, 2233–2248.

17. Bayly, P. V., B. P. Mann, T. L. Schmitz, D. A. Peters, G. Stepan, and T.Insperger: 2002, ‘Effects of radial immersion and cutting direction on chatterinstability in end-milling’. In: Proceedings of ASME Engineering Congress andExposition, New Orleans, LA.

18. Mann, B. P., T. Insperger, P. V. Bayly, and G. Stepan: 2003, ‘Stability ofup-milling and down-milling, Part 2: Experimental verification’. InternationalJournal of Machine Tools and Manufacture 43, 35–40.

19. Insperger, T., B. P. Mann, G. Stepan, and P. V. Bayly: 2003a, ‘Stabil-ity of up-milling and down-milling, Part 1: Alternative analytical methods’.International Journal of Machine Tools and Manufacture 43, 25–34.

20. Insperger, T., G. Stepan, P. V. Bayly, and B. P. Mann: 2003b, ‘Multiple chatterfrequencies in milling processes’. Journal of Sound and Vibration 262, 333–345.



21. Balachandran, B.: 2001, ‘Non-Linear Dynamics of Milling Process’. Philosoph-ical Transactions of the Royal Society of London A 359, 793–819.

22. Bayly, P. V., J. E. Halley, M. A. Davies, and J. R. Pratt: 2001a, ‘Stabilityanalysis of interrupted cutting with finite time in the cut’. In: Proceed-ings of ASME Design Engineering Technical Conference, Manufacturing inEngineering Division, Orlando, Florida. Orlando, FL.

23. Bayly, P., J. Halley, B. Mann, and M. Davies: 2003, ‘Stability of interruptedcutting by temporal finite element analysis’. Journal of Manufacturing Scienceand Engineering 125, 220-225.

24. Stepan G.: 2001, ‘Modelling nonlinear regenerative effects in metal cutting’.Philosophical Transactions of the Royal Society of London A 359, 739–757.

25. Davies M., and T. Burns: 2001, ‘Thermomechanical oscillations in material flowduring high-speed machining’. Philosophical Transactions of the Royal Societyof London A 359, 821–846.

26. Grabec, I.: 1988, ‘Chaotic dynamics of the cutting process’. InternationalJournal of Machine Tools and Manufacture 28, 19–32.

27. Altintas, Y. and E. Budak: 1995, ‘Analytical prediction of stability lobes inmilling’. CIRP Annals 44(1), 357–362.

28. Hanna, N. H. and S. A. Tobias: 1974, ‘A theory of Nonlinear regenerativechatter’. Journal of Engineering for Industry 96, 247–255.

29. Nayfeh, A. H., C. M. Chin, and J. Pratt: 1997, Applications of perturbationmethods to tool chatter dynamics, in Dynamics and Chaos in ManufacturingProcesses. New York, NY: John Wiley & Sons, Inc.

30. Stephenson, D. A. and J. S. Agapio: 1997, Metal Cutting Theory and Practice.New York, NY: Marcel Dekker, Inc., 1 edition.

31. Jaluria, Y.: 1986, Computational Heat Transfer. Washington: HemispherePublishing Corporation, 1 edition.

32. Peters, D. A. and A. P. Idzapanah: 1988, ‘HP-version finite elements for thespace-time domain’. Computational Mechanics 3, 73–78.

33. Virgin, L. N.: 2000, Introduction to Experimental Nonlinear Dynamics. Cam-bridge, UK: Cambridge University Press.

34. Yamamoto, T. and Y. Ishida: 2001, Linear and Nonlinear Rotordynamics,Wiley Series in Nonlinear Science. New York, NY: John Wiley and Sons.


18 B.P. Mann

Workpiece

axial depth of cut

radial depth of cut

Machine

Spindle

X

YZ

Figure 1. Schematic diagram showing the milling process is an interrupted cuttingprocess.



X

θ

kycy

kx

cx

Y( b )

Y

Fkv

Fku

X

θ V

U

( a )

Figure 2. Schematic of two different structural dynamics models: (a) rotatingrestoring and viscous damping forces; and (b) stationary restoring and dampingforces.


20 B.P. Mann

9.65 9.66 9.67 9.68 9.69 9.7 9.71 9.72 9.73

0

50

100

150

200

250

F x [N]

Linear Force Model vs. Measured Forces

( c )

9.65 9.66 9.67 9.68 9.69 9.7 9.71 9.72 9.73

0

50

100

150

200

250Nonlinear Force Model vs. Measured Forces

Time [s]

F y [N]

( d )

Figure 3. Measured vs. estimated y-direction cutting forces for three consecutivetooth passages made at a depth of cut (b = 2 [mm]) and spindle speed (Ω =900 [rpm])for a two flute 19.05 [mm] tool. Two comparisons are shown: (c) linear force modelvs. measured cutting force data; and (d) nonlinear force model vs. measured cuttingforce data.



4500 5000 5500 6000 6500 7000 7500 8000 8500 90000

1

2

3

4

5

Ω [rpm]

b [m

m]

Fixed Frame Model Stability Chart: 26 % immersion

( e )

UnstableUnstable

Stable Stable

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

b = 0 − 4 [mm] Ω = 5505 [rpm]

Hopf Bifurcation

Real

Imag

|µ|=1

( f )

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5

b = 0 − 4 [mm] Ω = 7105 [rpm]

Flip Bifurcation

Real

Imag

|µ|=1

( g )

Figure 4. Stability is predicted from characteristic multiplier magnitudes. Unsta-ble parameter combinations penetrate the unit circle in the complex plane: (e)Down-milling stability predictions; and (f) Characteristic multiplier trajectories for adiscontinuous Hopf bifurcation (Ω=5505 [rpm], b=0–4 [mm]); and (g) Characteristicmultiplier trajectories for a Flip bifurcation (Ω=7105 [rpm], b=0–4 [mm]). Modalparameters for this system are listed in Table I.


22 B.P. Mann

Figure 5. Numerical integration bifurcation diagrams for the experimental systemdescribed in Table II. Results show τ -periodic samples of y-direction displacements,labeled as yn, for a cutting speed of Ω=35000 [rpm]. The nonlinear force model showshysteresis in the bifurcation diagrams when comparing the: (h) forward sweep; and(i) reverse sweep cases.



Tool

Spindle

Fixture

Capacitance

Probes

Machine

Spindle

Laser Tachometer

Data Acquisition

Workpiece

Figure 6. Schematic diagram of test and measurement system used for millingexperiments.


24 B.P. Mann

1.5 2 2.5 3 3.5

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

b [m

m]

Ω [rpm]

A

B

C

D E

F

G

Fixed Frame Model Stability Chart: 5 % immersion

StableBorderlineUnstable

Figure 7. Down-milling experimental results vs fixed frame model stability predic-tions for the 19.05 [mm] tool described in Table II. Linear force model stabilityboundaries are shown by a solid line and nonlinear force model stability boundariesare shown by a dotted line. Symbols in the above diagram are as follows: 1) is aclearly stable case; 2) O is an unstable cutting test; and 3) + is a borderline unstablecase (i.e. not clearly stable or unstable).



1.5 2 2.5 3 3.5

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

b [m

m]

Ω [rpm]

A

B

C

D E

F

G

Rotating Frame Model Stability Chart: 5 % immersion

StableBorderlineUnstable

Figure 8. Down-milling experimental results vs rotating frame model stability pre-dictions for the 19.05 [mm] tool described in Table II. Linear force model stabilityboundaries are shown by a solid line and nonlinear force model stability boundariesare shown by a dotted line. Symbols in the above diagram are as follows: 1) is aclearly stable case; 2) O is an unstable cutting test; and 3) + is a borderline unstablecase (i.e. not clearly stable or unstable).


26 B.P. Mann

−5

0

5x 10−5 Displacement 1/tooth

y n [m

] A

Poincare′

y n2 [

m] A

10−12

10−8

10−5

PSD

A

−5

0

5x 10−5

y n [m

] By n2

[m

] B

10−12

10−8

10−5B

−5

0

5x 10−5

y n [m

] C

y n2 [

m] C

10−12

10−8

10−5C

500 1000 1500−5

0

5x 10−5

nth tooth passage

y n [m

] D

−5 0 5

x 10−5yn [m]

y n2 [

m] D

0 500 1000 1500 2000 250010−12

10−8

10−5

freq [Hz]

D

Figure 9. Experimental down-milling measurement data for cases (A,B,C,D) shownin Figures 7 and 8. Each row contains a y-axis 1/tooth displacement plot, a Poincaresection shown in displacement (yn) vs. delayed displacement (yn2) coordinates, andthe tooth passing frequency is marked by on the power spectrum (PSD). A stableprocess is shown for Case A(Ω=12285 rpm, b=2.0 mm). Unstable cutting processesare shown in cases B(Ω=15112 rpm, b=3.6 mm), C(Ω=16868 rpm, b=2.5 mm), andD(Ω=18516 rpm, b=3.6 mm). Cases C and D result from unstable Niemark-Sackeror secondary Hopf bifurcations and case D shows a flip bifurcation.



−5

0

5x 10−5 Displacement 1/tooth

y n [m

] E

Poincare′

y n2 [

m] E

10−12

10−8

10−5

PSD

E

−5

0

5x 10−5

y n [m

] Fy n2

[m

] F

10−12

10−8

10−5F

−5

0

5x 10−5

y n [m

] G

y n2 [

m] G

10−12

10−8

10−5G

7500 8000 8500−5

0

5x 10−5

nth tooth passage

y n [m

] G

−5 0 5

x 10−5yn [m]

y n2 [

m] G

0 500 1000 1500 2000 250010−12

10−8

10−5

freq [Hz]

G

Figure 10. Experimental down-milling measurement data for cases (E,F,G) shownin Figures 7 and 8. Each row contains a y-axis 1/tooth displacement plot, aPoincare section shown in displacement (yn) vs. delayed displacement (yn2) coor-dinates, and the tooth passing frequency is marked by on the power spectrum(PSD). Quasiperiodic motions arise in the post-bifurcation behavior of casesE(Ω=21439 rpm, b=3.6 mm) and F(Ω=27285 rpm, b=3.6 mm). Repeated testfor Case G(Ω=26796 rpm, b=2.0 mm) provides experimental evidence of multipleattractors and a sensitivity to initial conditions.


milling bifurcations from nonlinear...

Documents