The Effect of Runout on the Stability of Milling with Variable Helix Tools

Abstract

The stability of chatter vibrations in milling processes with variable helix tools is studied. Variable helix tools are characterized by different helix angles at different teeth of the cutter. With the tool rotation a distributed time delay, specifying the time between two subsequent cuts, arises in the model equations for the description of the machine-tool vibrations. The stability analysis of the distributed delay differential equation is done in the frequency domain and the effect of cutter runout on the stability is studied.

Full text

XXIII ICTAM, 19-24 August 2012, Beijing, China
THE EFFECT OF RUNOUT ON THE STABILITY OF MILLING
WITH VARIABLE HELIX TOOLS
Andreas Otto
*a)
& Günter Radons
*
*
Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
Summary T
he stability of chatter vibrations in milling processes with variable helix tools is studied. Variable helix tools are
characterized by different helix angles at different teeth of the cutter. With the tool rotation a distributed time delay, specifying the time
between two subsequent cuts, arises in the model equations for the description of the machine-tool vibrations. The stability analysis of
the distributed delay differential equation is done in the frequency domain and the effect of cutter runout on the stability is studied.
INTRODUCTION
Large vibrations in milling processes often called chatter vibrations cause noise, increased tool wear and poor surface
finish. The chatter stability analysis for milling processes can help to understand the reason for the occurrence of chatter,
to find stable cutting conditions without undesired vibrations and strategies to suppress chatter vibrations. The main
reason for chatter is an unstable self-excited mechanism. Vibrations at the contact zone between the tool and the
workpiece result in a wavy workpiece surface, which affect the chip thickness and the value of the cutting force at the
next cut. These dynamical variations of the cutting force induce new vibrations of the structure at the current cut. As a
result, the underlying model equation for chatter vibrations is a non-autonomous delay differential equation (DDE).
The stability analysis can be done by numerical simulations of the DDE and the analysis of the time series of the
vibrations. Alternatively, the semidiscretization method [1] can be used to construct a discrete version of the solution
operator of the DDE in form of a monodromy matrix. The eigenvalues of this monodromy matrix are the Floquet
multipliers and determine the stability of the chatter vibrations. In [2] a frequency domain stability analysis is proposed
for the calculation of the stability lobes for milling processes. The stability lobes separate stable cutting conditions from
unstable conditions with large chatter vibrations.
For milling processes with variable helix tools the time delay in the model equation is a distributed one, because of the
special geometrical nature of the milling cutter. For this case, the frequency domain stability analysis of regenerative
chatter is presented in [3]. In the present contribution the method of [3] is used to calculate the stability lobes for a
prototype milling process with a variable helix tool and the effect of cutter runout on chatter vibrations is shown.
MILLING MECHANICS
The process force in milling depends on the geometry of the milling cutter, the cutting conditions and the cutting force
law. The geometry of the variable helix tool is specified by the number of teeth N, the radius R
i
, the initial pitch ∆φ
i
(0) at
the tool tip and the helix angles η
i
for each tooth i (see Figure 1). For a regular tool with uniform pitch and uniform helix
angles the values of ∆φ
i
(0) and η
i
are independent of the tooth number i. The effect of runout is considered by slightly
different R
i
for different teeth. The cutting force law describes the local behavior dF
t
(h) and dF
r
(h) of the cutting force
in tangential and radial direction at differential cutting edge segments dz of the milling tool axis z. The form and the
parameters of the cutting force law are often identified empirically. In this paper a power law dependence of the cutting
force on the chip thickness h is considered:
a)
Corresponding author. Email: andreas.otto@physik.tu-chemnitz.de.
Structural model
mode number k=1 k=2 k=3
m
k
in kg 3 0.62 0.12
ζ
k
0.02 0.025 0.034
ω
k
in Hz 410 1353 2234
mode orientation
(x, y, z)
T
-2
1
2
5
-3
1
1
4
-2
Parameters of cutting force law
K
t
=950N/mm
1.7
γ
t
=0.7 K
t
=170N/mm
1.5
γ
r
=0.5
Variable helix tool with runout N=4
∆φ
i
(0) in ° 120 60 80 100
η
i
in ° 35 32 29 26
R
i
in mm 10.010 9.995 10.005 9.990
Figure 1: Circumference of the variable helix tool.
Table 1: Parameters of milling example.

XXIII ICTAM, 19-24 August 2012, Beijing, China
(1)
K
t
and K
r
are the tangential and radial cutting force coefficients. If the tangential and radial exponents, γ
t
and γ
r
, are
equal to one, eq. (1) reduces to a simple linear cutting force model, which is often studied in the literature.
For milling processes with variable helix tools the chip thickness h=h(t, r(t), r(t-τ(i,z)), i, z) depends explicitly on time t
due to the tool rotation and on the tooth number i and the axial height z due to the non-uniform geometry of the cutting
tool. Furthermore, current and time-delayed vibrations, r(t) and r(t-τ(i,z)), at the inner and outer surface of the chip
influence the chip thickness h. The time delay τ(i,z) is the time between two subsequent cuts and depends on the spindle
speed Ω and the pitch ∆φ
i
(z) at the axial height z between the present tooth i and the previous tooth (cf. Figure 1). With
the transformation matrix T=T(t, i, z) the cutting forces can be transformed from local radial and tangential coordinates
at the cutting edge segment dz into x, y and z coordinates of a fixed machine tool coordinate system. The complete
process force is the sum of the local forces over all teeth and the integral from z=0 to the axial depth of cut z=a
p
.
(2)
The cutting force F depends on the current time t, the current dynamical displacements r(t) and all W(t, ϑ)-weighted
time-delayed displacements r(t-ϑ) bounded by a minimum and a maximum value of the delay τ(i,z). The time
dependence is periodically with the period Ω of the spindle rotation.
RESULTS OF LINEAR STABILITY ANALYSIS
Regenerative chatter occurs due to dynamical process-machine interactions. The dynamical model of the machine tool is
a system of uncoupled harmonic oscillators, with the dominant modal masses m
k
, damping ratios ζ
k
and eigenfrequencies
ω
k
. The connection of the process force of eq. (2) with the structural dynamical model and a following linearization with
respect to the dynamical displacements r leads to a DDE with time-varying distributed delay:
(3)
The coefficient matrices A(t) and B(t, ϑ) are Ω-periodic. If the system is at the stability border, a periodic solution can be
assumed for the displacements r(t). Then, the coefficient matrices A, B and the state variable r can be developed into
Fourier series and transformed into the frequency domain. If a finite truncation of the Fourier series is considered, it is
possible to determine the critical cutting conditions at the stability border from the so-called Hill determinant, which can
be interpreted as the characteristic equation of the problem. Details of the frequency domain stability analysis for the
case of milling processes with variable helix tools can be found in [3].
The structural dynamical, the force and the tool geometry parameters of a milling example with a variable helix tool
with cutter runout are shown in Table 1. The cutting conditions are half immersion down milling with a feed per round
f
R
=0.4mm in x-direction. The chatter stability borders, indicating the limiting depth of cut a
p
dependent on Ω, are shown
for four milling cutters in Figure 2. The ‘Variable helix tool’ is characterized by the parameters of Table 1 with the tool
circumference, illustrated in Figure 1. ‘Regular tool’ refers to a comparable tool with uniform pitch ∆φ
i
(z)= ∆φ=90° and
uniform helix angle η
i
=η=30.5°. The exact radius of the teeth for the tools without runout is uniformly R
i
=R=10.000mm.
CONCLUSIONS
In general, variable helix angles and cutter
runout can stabilize milling processes and
suppress chatter. It can be expected, that the
stable island for the variable helix tool without
runout (black, dashed) can not be detected in
experiments, since for its calculation only three
structural modes were considered. Though, only
radial deviations due to runout in the size of a few
micrometers are assumed, the limiting depth of
cut is up to 30% larger than for the comparable
tool without runout. The effects of runout on the
stability are much larger for variable helix tools in
contrast to uniform pitch and uniform helix tools.
References
[1]
Insperger T., Stepan G.: Updated semi-discretization method for periodic delay-differential equations with discrete delay. Int. J. Numer. Meth. Engng
61:117-141, 2004.
[2]
Altintas Y., Budak E.: Analytical Prediction of Stability Lobes in Milling. CIRP Annals 44:357-362, 1995.
[3]
Otto A., Radons G.: Frequency domain stability analysis of milling processes with variable helix tools. Proc. 9
th
Int. Conf. on High Speed
Machining, March 7-8, San Sebastian, Spain, 2012.
.
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Figure 2: Stability lobes for milling processes with variable helix