Period-Doubling Island Investigation in Milling with Simultaneously Engaged Helical Flutes
1Department
Brian P. Mann1(PI), Eric A. Butcher2 (PI), Firas A. Khasawneh1, Oleg A. Bobrenkov2
of Mechanical Engineering and Material Science, Duke University, 2Department of Mechanical and Aerospace Engineering, New Mexico State University;
Down-Milling Stability Results for
Even Number of Teeth
Stability of Linear Ordinary and
Delay Differential Equations
Motivation
Chatter:
• Self-excited vibrations due to the regenerative effect of cutting
a previously cut surface
Dependent on:
• Cutting speed and depth of cut
• Mass, stiffness, and damping of tool
Results in:
• Wavy workpiece surface
• Decreased life of tool and workpiece
ODEs (with initial condition) DDEs (with initial function φ(t))
Constant coefficients
(Stability obtained from
characteristic equations –
left half plane stability criteria)
x (t ) = Ax(t ), x(t ) ∈ n
Time-periodic coefficients
x (t ) = A(t )x(t )
x (t ) = Ax(t ) + Bx(t − τ )
Down-milling stability diagrams for 4-, 6-, and 8-flute cutters. The light line represents the
zero-helix case, while the thick line represents a helical tool with β=30⁰. The radial immersions
used are (a) 0.05, (b) 0.25, (c) 0.50, (d) 0.75, and (e) 1.0. “H” indicates Hopf bifurcations, while
λ I − A − Be
λI − A = 0
− λτ
=0
“P” indicates period-doubling bifurcations.
A(t ) = A(t + T )
where
(Stability obtained using
Floquet multipliers –
unit circle stability criteria)
The combinations of radial immersion and helix angle where islands occur are shown in the
diagrams below for a 4-flute cutter and both up-milling and down-milling. When the combination
of radial immersions and helix angles falls into a shaded region, at least one island is present in
x (t ) = A(t)x(t ) + B(t )x(t − τ )
the corresponding stability chart while no islands exist for the parameter combinations in the
(a)
Who cares:
• Machinists and researchers in machining dynamics
• Aerospace industry (machining of aluminum and titanium aircraft components)
Objective of this research: Perform a numerical investigation of milling dynamics and
stability
Milling machine
Stability chart divides regions of chatter
(unstable cutting) and no chatter (stable
cutting)
Island Occurrence for Different
Combinations of
Radial Immersion and Helix Angle
unshaded region.
A(t ) = A(t + T ), B(t ) = B(t + T )
Up-Milling
Finite-dimensional
monodromy matrix Φ(T)
Infinite-dimensional compact
monodromy operator U
Unit Circle Routes of Destabilization
Down-Milling
(b)
(c)
Spectral radius ρ = max(abs(µ i)) where µ i are the eigenvalues (Floquet multipliers) of U.
If ρ < 1 (all eigenvalues are in the unit circle), the DDE system is stable; otherwise unstable if ρ > 1.
(d)
chatter
no chatter
 µ = x + iy Secondary Hopf

Fold
If ρ = 1  µ = 1

Period doubling
 µ = −1
µ is the multiplier with
largest absolute value
(e)
Secondary
Fold
Period
Doubling
Hopf
1-DOF Nonzero Helix Angle Milling Model
The equation of motion for a single-mode helical mill compliant in
the y-direction
y(t) + 2ζωny(t) + ωn2 y(t) =
Kc (t , b)
[y(t)
−
y(t −τ
)]
my Nonzero helix
tool geometry
(1)
dy(t)
= A(t ) y(t ) + B(t ) y(t − τ )
dt
proportional to the chip
thickness due to cutting the
surface that was cut on the
previous tooth pass
zb (n ,t )
n =1
za (n ,t )
K c (t , b) = ∑ gn (t )
∫
y n+1 = Uy n
(3)
(Kt sinθ n (t , z) cosθ n (t , z) − K r cos θ n (t , z)) dz ,
operator for time-periodic DDEs. Equation (3) represents a discrete solution form for Eq. (2)
that maps the states of the system over one delay period τ. The eigenvalues of U determine
if θ st ≤ θ n ≤ θ st + θtip ,
0
θ n − θ st
if θ st ≤ θ n ≤ θ st + θ lag ,


za (n,t ) = θ n − (θ st + θtip )
and zb (n,t ) =  κ
if θ st + θtip < θ n ,

b
if θ st + θ lag < θ n ,
κ
where θn (t , z) = Ωt − (n − 1)θp − κ z , n = 1,2,…,N is the rotation angle of the nth flute θ tip =cos−1 (1 − 2RI)
is the angular distance over which the tool is cutting at z = 0, (RI) is the ratio of the radial step-over
distance to the tool diameter, θst is the entry angle, κ = 2tan β / D is a helix parameter, θ p = 2π / N
is the tool pitch angle (the angular distance between two consecutive teeth), p = Dπ / ( N tan β ) is
the mill helix pitch (the distance between two adjacent flutes along the axis of the tool), Nt is the
total number of simultaneously engaged flutes, and θlag = κ b is the angular distance shown in the
picture.
Milling Model (continued)
The Different Cases of
Flute Engagement
The stability criteria dictates that all the
eigenvalues, µ, of the monodromy operator U,
should lie within the unit circle in the complex
plane. Moreover, the manner in which the
eigenvalues depart the unit circle produces different
bifurcation behavior. For example, an eigenvalue
leaving the unit circle through -1 or 1 results in a
period-doubling bifurcation or a fold bifurcation,
respectively, whereas two complex conjugate
eigenvalues departing the unit circle result in
secondary Hopf bifurcation.
Stability Results
A parametrically induced period-doubling island appears when down-milling at 0.05 radial
immersion with a zero-helix 3-flute cutter. The stability of several points is also noted using a
region whereas triangles were used to denote unstable period-doubling regions.
(tool rotation opposite
to the feed direction)
θst = 0
(tool rotation in
the direction of the feed)
θst = π − θ tip
(A): only a single flute is cutting at
any instant; (B) and (C): multiple
flutes are cutting due to a high
depth of cut, and a high radial
step-over distance, respectively.
(D): combined high depth of cut
and a high radial stepover distance.
The shaded area represents the
cutting zone.
Multiple Flute
Engagement Condition
(θ tip + θlag ) / θ p  > 1
(c)
the asymptotic stability of the DDE according to the conditions below.
notation consistent with the one used in the figure above: circles were used to denote a stable
Down-Milling
(b)
where U is a finite-dimensional approximation of the infinite-dimensional monodromy
where the limits of integration are written as
Up-Milling
(a)
τ. The methods used can transform Eq. (2) into a dynamic map of the form
2
Integration Limits
as a Function of the
Flute Rotation Angle
Down-milling stability diagrams for 3-, 5-, 7-, and 9-flute cutters with β=30⁰ at the radial
immersion levels of (a) 0.05, (b) 0.25, (c) 0.35, (d) 0.50, (e) 0.75, and (f) 1.0.
(2)
where A(t+T)=A(t) and B(t+T)=B(t) are time-periodic with the period T equal to the time delay
where τ=60/(NΩ) is the time delay, Ω is the spindle
speed in r.p.m., b is the depth of cut in mm, N is the total
number of cutting flutes, my is the mass of the tool, ωn is
the natural frequency, ξ is the damping ratio, and Kc(t,b)
is the τ –periodic directional force coefficient given by
Nt
and the Chebyshev collocation (ChC) approach [1,2] were used. For implementation, Eq. (1) is
first rewritten in its state-space form
↑
Down-Milling Stability Results for
Odd Number of Teeth
Stability Analysis of the Milling Model
For stability analysis, the state-space temporal finite element analysis (TFEA) method [2,3]
(d)
(e)
(f)
Stability Results for Up-Milling
Up-milling stability charts for the 4 cutting flutes and for the radial immersion values of 0.10,
0.20, 0.75 and 0.80 (columns) and the helix angle values of 1, 13, 23, 25, 45, 60, and 63
degrees (rows).
Conclusions
1. In comparison to prior works, different stability behavior is demonstrated: perioddoubling regions are shown to appear at relatively high radial immersions when multiple
flutes with either a zero or non-zero-helix angle are simultaneously cutting.
2. Period-doubling regions appear as lobes for zero-helix tools, while closed islands
characterize the period-doubling regions for helical tools. However, as the number of
cutting edges is increased, the helical flutes smooth out the force discontinuities and
eliminate period-doubling bifurcations.
3. In agreement with prior works, period-doubling bifurcations are shown to cease to exist at
depths of cut equal to integer multiples of the mill helix pitch.
4. Whereas islands disappear at full radial immersion for helical cutters with an even parity,
cutters with odd parity produce islands even at full radial immersion (shown for cutters
with 3, 5, and 7 flutes). This effect in cutters with odd parity is a result of the abrupt
changes in the directional force coefficient as the leading flute starts exiting the cut
midway through the period. In contrast, at full radial immersion, cutters with even parity
maintain a constant number of flutes in the cut throughout the period and only produce
Hopf lobes.
5. Certain combinations of radial immersion and helix angle give rise to Hopf lobes with
pronounced waviness in the horizontal direction. This contrasts the Hopf lobes usually
reported in milling literature and it reflects the strong effect of the helical flutes on the
shown cases. The change in the mutual orientation of the islands with respect to the
secondary Hopf lobe when varying the helix angle and the radial immersion is shown
(agrees with prior works). This transition of the lobes is attributed to the sign change of
the cutting force coefficient averaged over one period.
6. The radial immersions at which the transitions in the stability characteristics occur
correspond to the two minima of the lower boundary of the shaded region. In addition, it
is shown for both up-milling and down-milling that the period-doubling islands are
sensitive to the radial immersion.
References
1. E.A. Butcher, O.A. Bobrenkov, E.L. Bueler, and P. Nindujarla, “Analysis of Milling Stability by the
Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels,” Journal of
Computational and Nonlinear Dynamics, vol.4, 031003: 1-12, 2009.
2. O.A. Bobrenkov, F.A. Khasawneh, E.A. Butcher, and B.P. Mann, “Analysis of Milling Dynamics
for Simultaneously Engaged Cutting Teeth,” Journal of Sound and Vibration, 329(5), pp.585–
606, 2010.
3. P.V. Bayly, J.E. Halley, B.P. Mann, and M.A. Davis, “Stability of Interrupted Cutting by Temporal
Finite Element Analysis,” Journal of Manufacturing Science and Engineering, 125, pp.220–225,
2003.
4. F.A. Khasawneh, B.P. Mann, O.A. Bobrenkov, and E.A. Butcher, “Self-Excited Vibrations in a
Delay Oscillator: Application to Milling with Simultaneously Engaged Helical Flutes,”
proceedings of the 22nd Biennial Conference on Mechanical Vibration and Noise, ASME IDETC
’09, San Diego, CA, Aug.30–Sep.2, 2009.
Acknowledgements
Support from US National Science Foundation Grants
No. CMMI-0900266 and CMMI-0900289
is gratefully acknowledged.
Contact Information
Brian P. Mann (email: brian.mann@duke.edu)
Firas A. Khasawneh (email: firas.khasawneh@duke.edu)
Dept. of Mechanical Engineering & Material Science
Box 90300 , Hudson Hall,
Duke University, Durham, NC 27708
Eric A. Butcher (email: eab@nmsu.edu)
Oleg A. Bobrenkov (email: chaalis@nmsu.edu)
Dept. of Mechanical & Aerospace Engineering
P.O. Box 30001 / MSC 3450,
New Mexico State University,
Las Cruces, NM 88003