TOC
Active in-process chatter control
E.J.J. Doppenberg*, R.P.H. Faassen**, N. van de Wouw**, J.A.J. Oosterling*,
H. Nijmeijer**,
*TNO Science and Industry, P.O. Box 155, 2600 AD Delft, The Netherlands1
**Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Ed.doppenberg@tno.nl
Abstract: A control stategy for high speed milling has been developed to prevent the
onset of chatter, and at the same time to increase machining efficiency and surface
finish quality. The main objective of this approach is to detect the onset and adjust the
spindle speed in such a manner, that the machining process is forced into a region of
stable operation. In the paper, this approach - active in-process chatter control - is
described and illustrated in simulations.
Keywords: Active chatter control, time-evolving regenerative-process-database,
machining simulation.
1. INTRODUCTION
To some degree, all milling tools exhibit vibrations as their teeth cut into the work
piece, leaving a wavy surface. If the waves of the succeeding cuts match, the chip
thickness is constant and the milling process is stable. When the waves are out of
synchronization, the chip thickness varies, and the start of the minute regenerative
vibrations can grow to undesired high vibration levels, which can excess to the
phenomenon ‘chatter’. This results in heavy vibrations of the entire cutting system from the cutting edge geometry to the machine tool spindle - which degrades the
machining process. Besides the degradation of for instance the surface quality, life time
of the tool, the process reliability and the life span of the machine tool spindle will be
negatively affected.
The key to prevent chatter is to maintain the synchronization of succeeding
wavy surfaces, keeping the chip thickness constant. Machine designers use passive
strategies to prevent regenerative vibrations by absorbing the vibration energy, or by
redirecting the vibration energy [Semercigil and Chen, 2002;Tarng et al, 2000]. A trend
in the design approach to control chatter behaviour, is to optimize the machine’s
dynamic behaviour at the design process, maximizing stiffness and optimizing damping
of the entire cutting system [Zhang and Sims, 2005; Kyung and Lee, 2003].
The objective of the abovementioned strategies is to minimize the energy
feedback of the unwanted regenerative vibrations to the cutting process for a vast range
of process parameters. In a schematic diagram, depicted in Figure 1, the ideal, but
unfeasible situation would be the total elimination of the feedback path to the
____________________________________
1
This research is supported in part by Jabro Tools, Philips ETG and Somatech.
http://www.tno.nl/chattercontrol.
process, in which case the dynamic chip thickness (hdyn(t)) would be zero. From a
control-engineering viewpoint, this design strategy focuses on the control (indirectly) of
the properties of the regenerative vibration process, by ‘tuning’ the machine dynamics
at the design process [Altintas and Cao, 2005].
On the other hand, high-speed milling increasingly demands a very high
response, high feed rates, which requires machines with relatively short response time,
and with constrained energy properties. Therefore the trend in high speed milling
machines is to design a relatively lightweight cutting tool system with increased cutting
power, which will constrain the machine developer in the design process even more.
Thus, the set of requirements for robust cutting without chatter and high response are
becoming more opposed, making machine designing a time consuming, critical path in
the machine development.
To ease the inevitable design trade offs in the ever-demanding machine
development, the indirect control (at the design process) of the regenerative vibration
process (to prevent chatter) will increasingly require complementary control
approaches. One of these complementary alternatives is the active control strategy of
the regenerative vibration process itself during the machining process to prevent the
onset of chatter. The main objective of this approach is to detect the onset of chatter and
to adjust the spindle speed in such a manner, that the machining process is forced into a
region of stable operation. This approach - active in-process control of chatter - is
presented in the paper. The paper is organized as follows:
In Section 2, a brief mathematical model of the regenerative vibration process
is presented. In Section 3, the alternative control approach is described. In Section 4, the
simulation set up is presented. In Section 5, the simulation results are discussed. The
conclusions are drawn up in Section 6.
Delay
hstat +
F (t )
Cutting
hdyn (t )
−
Machine
+
+
Trigonometric
functions
Figure 1; Block diagram of the milling process.
2. THE MILLING PROCESS
In Figure 1, a block diagram of the milling process is shown. The static thickness hstat is
a result of the pre-defined motion of the tool with respect to the work piece. This chip
thickness results in the force F(t) that acts on the tool via the cutting process (block
Cutting). Interaction of this force with the spindle and tool dynamics (block Machine),
results in a dynamic displacement of the tool v(t), which is superimposed on the predefined tool motion. The block Delay is due to the regenerative effect (see e.g. [Tlusty
and Polacek, 1963; Altintas and Budak, 1995; Stépán, 2001;Faassen et al., 2003]). Via
trigonometric relations, the tool motion results in a dynamic chip thickness hdyn(t),
which is added to the static chip thickness.
In the milling process, the static chip thickness is periodic. The movement of
the cutter v(t) can therefore be described by the movement vp(t), which is periodic with
1
60
=
. Here, τ is the delay as mentioned in the block Delay, z is
period time T = τ =
f t zΩ
the number of teeth on the cutter, and Ω the spindle speed in rpm. A perturbation on that
periodic movement is denoted by vu(t). If no chatter occurs, the periodic movement vp(t)
is an asymptotically stable solution of the set of delay differential equations describing
the milling process, and the perturbation motion vu(t) tends to zero asymptotically.
When the periodic solution loses its stability (e.g. with an increasing axial depth-of-cut),
in most cases a secondary Hopf bifurcation occurs, and in other cases a periodic
doubling bifurcation occurs [Insperger et al., 2003]. This means that the original
periodic solution vp(t) becomes unstable, and a new periodic motion vu(t) with a
different frequency fc is superimposed on the original periodic solution. This new
periodic motion vu(t) is strongly correlated with the dynamic chip thickness hdyn(t) and
can be used as a measure for hdyn(t). In the remainder of this paper fc is referred to as the
basic chatter frequency.
3. THE COMPLEMENTARY CONTROL STRATEGY
The main objective of the alternative control strategy is to enhance machining
performance, and at the same time to guarantee robustness of performance. This means
that chatter has to be inhibited. There are a number of secondary objectives that the
control strategy has to fulfil:
1. The control should not alter essential machining process parameters: e.g. surface
properties of the end products should maintain their quality;
2. The control is complementary in the sense that the imposed control action on the
cutting system is an enhancement, not a vital system to sustain the machining
process.
The main objective, inhibiting chatter, can be defined as to synchronize the wavy
surfaces of each succeeding cut, minimizing the variation of the dynamic chip thickness
hdyn(t). The objective of the controller can be re-defined as the optimization of cost
function J(t), which is rudimentarily defined as: J (t ) = E{hdyn (t ) 2 }.
(1)
The perturbation motion vu(t) can be used as a measure of chip thickness
hdyn(t). Then an alternative cost function becomes: J v (t ) = E vu (t ) 2 .
(2)
To meet the first secondary control objective, the machining parameters (depth-of-cut,
radial depth-of-cut, and chip load) must remain unchanged. To meet the second
secondary objective, the automated machining process (CNC software program) should
not be altered (from a practical viewpoint the altering of the CNC program during the
milling process is not an option). From the entire set of machining parameters, the
spindle speed Ω is a suitable candidate with which to manipulate the regenerative
vibration process during cutting, if the chip load is kept constant [Smith and Delio,
1992]. It is expected, that when altering the spindle speed, it will have a negligible
impact on the performance of the milling process. Therefore, the use of the spindle
system as a ‘control actuator’ will not violate the secondary objectives.
However, unlimited changes of the spindle speed do endanger machining
processing and machine safety. To limit the control effort (large speed changes), the
demanded power level of the spindle system to adjust to the new speed must be
guarded, and should not exceed the maximum power level. It is preferable to have the
spindle system power handled by the control system rather than to leave the handling to
the spindle motor amplifier system, since the latter would introduce undesirable nonlinear behaviour in the control actuator system. Augmenting the cost function (2) to
Jvp(t) will incorporate the spindle power handling by the control system. The augmented
cost function is:
J v p (t , Ω) = E{Avu (t , Ω) 2 + B∆P (t , Ω) }.
(3)
{
}
Here, ∆P(t,Ω) the spindle power is required to adjust to the new spindle speed. The
parameters A and B are tuneable weighting parameters, which will influence the
performance and robustness of the control system.
Summarizing, the control strategy will constitute four main functionalities:
1. Optimization of the cost function to realize the main control objective;
2. Sensing of the cutting process. In particular the perturbation motion vu(t);
3. The controller. A description of the controller input and output properties
will be presented;
4. Actuation of the cutting process. In particular the actuation of the
regenerative process by changing the spindle speed Ω.
3.1. Optimizing of the cost function
To maintain the objective of enhancing machining performance, and at the same time to
guarantee robustness of performance, the defined cost function (3) is optimized
according to the following rule:
min{J vp (t , Ω)}, subject to vu (t , Ω) ≤ δ 0 and Ω min < Ω < Ω max
(4)
Here δ0 is a parameter, that determines the maximum allowable value of the
perturbation vu(t). The parameters Ω min , Ω max are the restrictive bounds of spindle
speed. All parameters are determined from the model of the cutting process [Faassen et
al., 2003], from the properties of the spindle system, and by the selected machining
working point. To estimate the cost function, the perturbation motion vu(t) is estimated
from the movement of the cutter v(t), and the required spindle power of the spindle
system is estimated from the measured spindle motor torque.
3.2. Sensing of the perturbation motion
Detection of the perturbation motion vu(t,Ω) from the cutter movement v(t) is performed
by a dedicated method, which is described in [Faassen et al., 2006]. To demonstrate the
performance of this method, the measured, and detected perturbation motion vu,xy(t) in
x- and y-direction (the normal direction) of the tool is reconstructed real time, and
depicted in Figure 2. The perturbation motion distinctively grows at t > 2 s.
Figure 2; Reconstruction of the perturbation motion vu,xy(t) of the cutter.
- Using eddy current sensors;
- The spindle speed = 42000 rpm, axial Depth-Of-Cut (DOC) = 3 ->5 mm.
For the control system only one sensor signal in one direction is used to detect the
perturbation motion vu(t,Ω). Simulations and experiments [Faassen et al., 2006] have
revealed that the measured acceleration in the normal direction near the lower spindle
bearing is a suitable signal for detection purposes of the onset of chatter. In Figure 3, the
perturbation motion vu(t,Ω) is detected real time from the measured acceleration at the
spindle bearing and is presented. At time = 1.65 s the tool enters the cut. At t = 2.2 s the
onset of chatter is (visually) detectable. Full-blown chatter starts at t = 2.3 s, which can
be clearly seen.
Figure 3; Detected vu(t) from the measured acceleration at the spindle bearing.
- Axial Depth-Of-Cut (DOC) = 3 ->5 mm.
3.3 The controller
The design of the controller depends on the optimization of the cost function (4). The
properties of the cost function are estimated real time from the perturbation motion
vu(t,Ω). More precisely, the perturbation motion vu(t,Ω) is modelled with a parametric
model and is estimated at discrete time instances, discrete spindle speeds, and is denoted
by Vu(f,t,Ω). The function Vu(f,t,Ω) can be considered as a four-dimensional database,
which describes the properties of the regenerative vibration process for four domains:
frequency, time, magnitude and spindle speed. The database innovates as the machine
process proceeds, and will therefore evolve with time. From the database different types
of queries and combination of queries can be made.
To optimize the cost function (4) the spindle speed is adjusted. However, to
adjust the spindle speed while at the same time the constraint not to exceed the
maximum allowable spindle power is preserved, the adjustment is performed via an
optimal speed trajectory. This trajectory is generated using a gradient-based adaptation
scheme [Haykin, 2001]. The trajectory will be dependent of the ‘evolving’ function
Vu(f,t,Ω), of the estimated spindle power and of the selected weighting parameters A and
B. Specifically, a recursive method is used to implement the adaptation scheme to
determine the trajectory. The trajectory will therefore vary and evolve with the time due
to the nature of the adaptation scheme. To have a better understanding of the relation
between the control design and the evolving function Vu(f,t,Ω), the function is estimated
in a simulation for a spindle speed sweep. The start and end values of the spindle speed
are selected such, that the ‘left’ and ‘right’ boundaries of stable/unstable cut of the
stability lobe diagram are deliberately exceeded. The axial DOC is kept constant at 3.5
mm. Two queries are made. The first query is the function fcd(Ω), the dominant chatter
frequency with which the function fonset(Ω) = kfrpm(Ω) - fcd(Ω) is derived. The second
query is the function V(Ω) = 20log(|Vu(f,t,Ω)|) at f = fcd(Ω).
Figure 4; The frequency-domain decomposition of the perturbation motion vu(t) for a
‘spindle speed sweep’.
Here kfrpm(Ω) is the higher harmonic of the spindle speed frequency where the dominant
chatter frequency fcd(Ω) is manifest (k is the harmonic number). The basic chatter
frequency equals: f c = f onset (Ω) at full blown chatter, which occurs at approximately
Ωmin < 32 103 rpm and at Ωmax > 36 103 rpm. In the graph, depicted in Figure 4, both
functions are displayed as function of the spindle speed. In the graph a rectangle
boundary box is drawn which intersects with the function V(Ω) at the level of 60 dB.
This is a graphical representation of the optimization rule vu (t , Ω) ≤ δ 0 . The control
action can now be graphically interpreted. To optimize the cost function the spindle
speed is forced to the middle of the stability lobe close to (but outside) the red boundary
box at V(Ω) ≅ 60 dB. The direction of the approach is determined by the sign of
fonset(Ω). The speed of adjustment is determined by the time evolving trajectory. The
onset of chatter will be prevented by the controller without exceeding the maximum
allowable spindle power.
4. THE SIMULATION SETUP
The defined controller approach has been simulated and tested using the Simulink
software package. The set up is presented in Figure 5. The model consists of the
following functions:
1. The cutting system [Faassen et al., 2003];
2. Detection of the regenerative process [Faassen et al., 2006];
3. Estimation of the regenerative process;
4. The control design and action of the controller;
5. The spindle system (Mikron HSM 700)
The simulations are performed with the following settings:
1. Sampling frequency fs = 20 kHz.
2. Spindle speed is 30000 rpm.
3. The axial depth-of-cut (DOC) varies from 3 to 5 mm in 1 second
4. The feed rate is 0.15 mm/teeth, full immersion.
5. The maximum allowed relative perturbation δ0= 0.510-3 m/s2.
6. The maximum available incremental spindle power = 1 kW.
5. SIMULATION RESULTS
The results depicted in Figure 6 and 7 show that the control system minimizes the
perturbation motion Vu(f,t,Ω) in the event that the machining process tends to become
unstable. There where the spindle system has unlimited available power to adapt to the
new spindle speed, the control system reacts immediately and maintains stable
processing. The repeated experiment with limited spindle system power, depicted in
Figure 7, indicates that the response of the system is relatively slow, thereby affecting
the performance. The perturbation motion exceeds at DOC = 4.2 mm the desired
performance of δ0= 0.510-3 m/s2 (unscaled). This is also due to the inaccurate modelling
of the spindle system dynamics. Despite a relatively ‘soft’ control action the model of
the dynamics is excited unrealistically. Upcoming experiments with the Mikron HSM
700 milling machine will be used to validate the simulations and most importantly, the
relation between control performance and the quality of machining will be analyzed.
rpmdesired
Spindle system
rpmadj (t )
rpm(t )
a(t )
Onset
detection−
Cutting system
rpm(t )
Torque(t )
vu (t , Ω)
Controller
Vu ( f , t , Ω)
Onset
estimation
rpm(t )
Figure 5; Model of the active in-process chatter control.
Figure 6; Control of chatter, with unlimited available spindle power.
Figure 7; Control of chatter, with limited spindle power.
6. THE CONCLUSIONS
In this paper an alternative control approach to prevent chatter is presented. The control
approach is considered as a complementary tool or an add-on tool to give answer to the
growing demand for higher process efficiency and to ease the inevitable design tradeoffs in the ever-demanding machine development at the design process. The approach
aims to in-process control the regenerative vibration process, which is the underlying
phenomenon of onset of chatter during machining. The controller uses a sophisticated
detection and estimation method to determine this regenerative vibration process and to
construct and maintain the time-evolving regenerative-process-database. The controller
calculates real-time the optimal spindle speed trajectory to prevent the onset of chatter
from this database and from the measured spindle system power. The simulations that
are performed with Simulink reveal that a spindle system with limited power will
deteriorate the performance of the control. With the new results of the upcoming
experiments with Mikron HSM 700 milling machine, the relation between control
performance, and the quality of machining will be analyzed.
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