15. Uluslararası Makina Tasarım ve İmalat Kongresi
19- 22 Haziran 2012, Pamukkale, Denizli, Türkiye
CHATTER STABILITY OF PARALLEL MILLING OPERATIONS
Alptunç ÇOMAK, Erhan BUDAK
Manufacturing Research Laboratory, Sabanci University, Tuzla, Istanbul, Turkey
alptunc@sabanciuniv.edu
ebudak@sabanciuniv.edu
ABSTRACT
Parallel milling process is the process where more than one milling tool cut the same
workpiece simultaneously offering increased productivity. However, the dynamic interaction
between the tools may introduce additional stability problems, and thus needs to be
analyzed. In this study, a frequency domain model is developed for the chatter stability of
parallel milling processes. The stability diagrams are obtained for different parallel machining
conditions demonstrating the effects of dynamic interaction between the tools through a
flexible workpiece. The predictions are verified by experiments carried out on a multi-tasking
machining center.
1. INTRODUCTION
Simultaneous machining operations have been continuing to spread in various sectors due to
various advantages they offer. Parallel turning and parallel milling are two common examples
of simultaneous machining operations which are usually performed on multi-purpose
machine tools. Parallel milling involves more than one milling tool cutting the same or
different surfaces at the same time. Cutters can be located at the same or different spindles
or turrets. Parallel milling has the potential to increase productivity if correct machining
conditions are used. As in standard machining operations, chatter vibrations may limit the full
potential for productivity in parallel milling, too. On the other hand, if milling conditions are
selected properly, chatter-free material removal rate can be increased. In such a case,
dynamics cutting forces on both tools may cancel each other increasing stability limits.
Workpiece dynamics have great impact on the stability of parallel milling process due to its
effects on dynamic coupling of two cutting tools.
Dynamics of the simultaneous milling operation is more complex compared to conventional
single tool milling due to the existence of dynamic coupling between the tools and the
workpiece. If the workpiece is not flexible, dynamic coupling may exist only between the
cutting tools through the machine tool structure. If there is no dynamic coupling between the
tools they behave like two separate single-tool milling operations. Therefore, two sources of
15. Uluslararası Makina Tasarım ve İmalat Kongresi
19- 22 Haziran 2012, Pamukkale, Denizli, Türkiye
the dynamic coupling can be discussed in parallel milling operations. First one is the dynamic
workpiece compliance that the first cutting tool affects the dynamics of the other through a
flexible workpiece. The second type of dynamic coupling is due to the dynamic machine
compliance. This type of dynamic coupling is generally rare since the path between two tools
is usually very long and rigid.
Dynamics and stability of machining have been studied in detail in many works. The theory of
chatter in machining was first introduced by Tobias and Fishwick [1] and Tlusty and Polacek
[2]. They demonstrated the coupling between the cutting forces and dynamic displacements
and estimated the chatter stability limits. Tlusty and İsmail [3] carried out time domain
simulations and acquired more accurate results for the stability limits by including the basic
nonlinearity in cutting which is the loss of contact between the cutting tool and the material.
Budak and Altintas [4] presented an analytical method for the stability of milling which can be
used to generate stability diagrams in frequency domain very efficiently. Added lobe
phenomenon has been presented by several authors [5], [6].
Dynamics and stability of parallel machining, on the other hand, has been studied very little.
Lazoglu [7] developed a time domain model for parallel turning operation. Then, Ozdoganlar
and Endres [8] formulated the dynamics of the parallel turning process. Ozturk and Budak
[9] have simulated the dynamics of parallel milling in time domain and generated stability
diagrams for various cutting conditions. Brecher and Trofimov [10] also used time domain
simulation method and showed the effect of relative angular position offset and spindle
speed on the stable depth of cut. Shamoto [11] posed the suppression of chatter in
simultaneous milling by speed difference.
In this paper, a new analytical solution method for generation of stability diagrams in parallel
milling is proposed. Effect of the workpiece dynamics on the stability of the process is shown.
The simulation results are verified by parallel milling tests. Finally, some practical
recommendations are made based on the observations from the simulations and the tests.
2. DYNAMICS OF PARALLEL MILLING OPERATION
Unlike the conventional single tool milling operations, cross transfer functions of both cutting
tools and workpiece must be identified to be able to determine the stability lobes in
simultaneous milling. In parallel milling, the machined part is excited by the cutting forces
generated from both tools. Thus, each cutting tool is affected from the cutting forces
originated from itself and the other cutting tool. At this point, identification of cross transfer
functions is critical and significant for the dynamics and chatter analysis of the parallel cutting
process. Dynamic cutting forces on two cutting tools are also dependent on each other and
must be solved together.
15. Uluslararası Makina Tasarım ve İmalat Kongresi
19- 22 Haziran 2012, Pamukkale, Denizli, Türkiye
Dynamic Responses and Chip Thickness Definition
In order to derive the dynamic cutting force expressions, dynamic responses at the toolworkpiece contact points should be identified. As shown in Figure 1, 1 and 2 refers to the
contact points of the cutter 1 and the cutter 2 with the workpiece, respectively. The machined
stock is flexible, so there is dynamic interaction between point 1 and 2. When both cutting
tools perform the operation simultaneously, generated forces at point 1 affect also the
dynamic response at point 2 and vice versa.
Figure 1 Parallel milling operation
Dynamics responses at the first and the second points can be formulated as follows:
𝑐𝑜𝑢𝑝
𝑇
𝑤
𝑤
𝑇
𝑤
𝑤
𝑤
𝑅𝑖𝑥 = 𝐹𝑖𝑥 𝐺𝑖𝑥𝑥
+ 𝐹𝑖𝑥 𝐺𝑖𝑖𝑥𝑥
+ 𝐹𝑖𝑦 𝐺𝑖𝑖𝑦𝑥
+ 𝐹𝑖𝑦 𝐺𝑖𝑦𝑥
+ 𝐹𝑗𝑥 𝐺𝑗𝑖𝑥𝑥
+ 𝐹𝑗𝑦 𝐺𝑗𝑖𝑦𝑥
+ 𝐹𝑗𝑥 𝐺𝑗𝑖𝑥𝑥 + 𝐹𝑗𝑦 𝐺𝑗𝑖𝑦𝑥
𝑐𝑜𝑢𝑝
𝑇
𝑤
𝑤
𝑇
𝑤
𝑤
𝑤
𝑅𝑖𝑦 = 𝐹𝑖𝑦 𝐺𝑖𝑦𝑦
+ 𝐹𝑖𝑦 𝐺𝑖𝑖𝑦𝑦
+ 𝐹𝑖𝑥 𝐺𝑖𝑖𝑥𝑦
+ 𝐹𝑖𝑥 𝐺𝑖𝑥𝑦
+ 𝐹𝑗𝑦 𝐺𝑗𝑖𝑦𝑦
+ 𝐹𝑗𝑥 𝐺𝑗𝑖𝑥𝑦
+ 𝐹𝑗𝑦 𝐺𝑗𝑖𝑦𝑦 + 𝐹𝑗𝑥 𝐺𝑗𝑖𝑥𝑦
(1)
where 𝑖 𝑎𝑛𝑑 𝑗 = (1,2) indicates the first and second contact points, respectively. 𝐹𝑖𝑎 is the
𝑤
dynamic cutting force at 𝑖 𝑡ℎ contact point in direction a (𝑎 = 𝑥, 𝑦). Similarly, 𝐺𝑖𝑗𝑎𝑏
is the
workpiece transfer function measured at point “i” along the “a” direction and excited at point
“j” along the “b” direction.
Response at point 1 is affected by the first and second cutting tool’s dynamic forces, (𝐹1 ) and
(𝐹2 ). Dynamic coupling between the points 1 and 2 is represented by the cross talk transfer
𝑐𝑜𝑢𝑝
𝑤
𝑤
functions 𝐺12
and 𝐺21
. 𝐺12
is the cross transfer function due to the compliance through the
machine structure.
Dynamic responses at points 1 and 2 include tool and workpiece vibrations which are caused
by the dynamic forces from both tools as shown below.
∆𝑥1 = [𝑥1𝑇 (𝑡) − (𝑥1𝑤 (𝑡) + 𝑥2𝑤 (𝑡))] − [𝑥1𝑇 (𝑡 − 𝜏1 ) − (𝑥1𝑤 (𝑡 − 𝜏1 ) + 𝑥2𝑤 (𝑡 − 𝜏2 ))]
15. Uluslararası Makina Tasarım ve İmalat Kongresi
19- 22 Haziran 2012, Pamukkale, Denizli, Türkiye
∆𝑦1 = [𝑦1𝑇 (𝑡) − (𝑦1𝑤 (𝑡) + 𝑦2𝑤 (𝑡))] − [𝑦1𝑇 (𝑡 − 𝜏1 ) − (𝑦1𝑤 (𝑡 − 𝜏1 ) + 𝑦2𝑤 (𝑡 − 𝜏2 ))]
(2)
where 𝑥1𝑇 (𝑡) and 𝑥1𝑤 (𝑡) are the displacements of the first tool and the workpiece (due to the
first tool) in the x direction and 𝑥2𝑤 (𝑡) is the displacement in the x direction due to the second
tool’s contribution. The second part of the dynamic chip thickness represents the terms which
belong to the previous pass on the same surface. Thus, the dynamic chip thickness for the
first tool can be written as;
ℎ1 = ∆𝑥1 𝑠𝑖𝑛∅𝑗1 + ∆𝑦1 𝑐𝑜𝑠∅𝑗1
(3)
where ℎ1 is the dynamic chip thickness at point 1 and ∅𝑗1 is the immersion angle of the first
tool.
Dynamic chip thickness for the second cutting tool can be written in a similar way for ∅𝑗2 .
Another significant point is the delay terms in the equations. Unlike for the conventional
milling processes, in parallel milling there are two delays, 𝜏1 and 𝜏2 , since in general the
rotational speeds are different for both tools. Both dynamic responses are affected by these
two delay terms which make them dynamically coupled.
Milling Force Equations
Derivation of the force equations for parallel milling is similar to that of conventional milling.
Dynamic cutting forces in tangential, radial and axial directions for both cutting tools can be
written as follows:
𝐹𝑡𝑖 = 𝐾𝑡𝑖 𝑎𝑖 ℎ𝑖 , 𝐹𝑟𝑖 = 𝐾𝑟𝑖 𝑎𝑖 ℎ𝑖 , 𝐹𝑎𝑖 = 𝐾𝑎𝑖 𝑎𝑖 ℎ𝑖
(4)
where 𝐾𝑡𝑖 , 𝐾𝑟𝑖 and 𝐾𝑎𝑖 are tangential, radial and axial cutting force coefficients, respectively,
and 𝑎𝑖 is the axial depth of cut. Then, the milling forces in x and y directions can be given as
𝐹𝑥𝑖 = −𝐹𝑡𝑖 𝑐𝑜𝑠𝜃𝑗𝑖 − 𝐹𝑟1 𝑠𝑖𝑛𝜃𝑗𝑖
𝐹𝑦𝑖 = 𝐹𝑡𝑖 𝑠𝑖𝑛𝜃𝑗𝑖 − 𝐹𝑟𝑖 𝑐𝑜𝑠𝜃𝑗𝑖
(5)
Finally, force equations for both cutting tools can be written as,
𝛼𝑥𝑥1
𝐹𝑥1
1
{𝐹 } = 2 𝐾𝑡1 𝑎1 [𝛼
𝑦𝑥1
𝑦1
𝛼𝑥𝑦1 ∆𝑥1
𝛼𝑦𝑦1 ] (∆𝑦1 )
𝛼𝑥𝑥2
𝐹𝑥2
1
{𝐹 } = 2 𝐾𝑡2 𝑎2 [𝛼
𝑦𝑥2
𝑦2
𝛼𝑥𝑦2 ∆𝑥2
𝛼𝑦𝑦2 ] (∆𝑦2 )
(6)
𝛼𝑥𝑥𝑖 is the directional dynamic force coefficient for the first and second tool where 𝑖 = 1,2.
Directional dynamic cutting force coefficients are the same as in conventional milling process
[12].
15. Uluslararası Makina Tasarım ve İmalat Kongresi
19- 22 Haziran 2012, Pamukkale, Denizli, Türkiye
Having two delays in the formulation makes the solution complex compared to conventional
milling. The force equations must be solved together due to the existence of dynamic
coupling between the forces at points 1 and 2.
Delay Matrix and Transfer Functions of Workpiece and Tools
Since different delay terms affect dynamic response for each point, they can be grouped into
a single matrix called as delay matrix. Each delay term in the matrix corresponds to its
equivalent force and transfer function pair in the general force equation.
1 − 𝑒−𝑗𝑤𝑐𝜏1
−𝑗𝑤𝑐𝜏1
[1 − 𝑒−𝑗𝑤𝑐𝜏1
1−𝑒
1 − 𝑒−𝑗𝑤𝑐𝜏1
1 − 𝑒−𝑗𝑤𝑐𝜏1
1 − 𝑒−𝑗𝑤𝑐𝜏1
1 − 𝑒−𝑗𝑤𝑐𝜏1
1 − 𝑒−𝑗𝑤𝑐𝜏1
1 − 𝑒−𝑗𝑤𝑐𝜏2
1 − 𝑒−𝑗𝑤𝑐𝜏2
1 − 𝑒−𝑗𝑤𝑐𝜏2
1 − 𝑒−𝑗𝑤𝑐𝜏2
1 − 𝑒−𝑗𝑤𝑐𝜏2
1 − 𝑒−𝑗𝑤𝑐𝜏2 ]
1 − 𝑒−𝑗𝑤𝑐𝜏2
1 − 𝑒−𝑗𝑤𝑐𝜏2 4×4
(7)
Due to the dynamic coupling, cross transfer functions play an important role on the dynamics
of parallel milling processes. The transfer function matrix includes all direct and cross
transfer functions as given below:
𝐺𝑇1,𝑥𝑥 + 𝐺𝑤
11,𝑥𝑥
𝐺𝑇1,𝑥𝑦 + 𝐺𝑤
11,𝑦𝑥
𝐺𝑤
21,𝑥𝑥 + 𝐺21,𝑥𝑥
𝑐𝑜𝑢𝑝
𝐺𝑤
21,𝑦𝑥 + 𝐺21,𝑦𝑥
𝐺𝑇1,𝑦𝑥 + 𝐺𝑤
11,𝑥𝑦
𝐺𝑇1,𝑦𝑦 + 𝐺𝑤
11,𝑦𝑦
𝐺𝑤
21,𝑥𝑦 + 𝐺21,𝑥𝑦
𝑐𝑜𝑢𝑝
𝐺𝑤
21,𝑦𝑦 + 𝐺21,𝑦𝑦
𝑐𝑜𝑢𝑝
𝐺𝑇2,𝑥𝑥 + 𝐺𝑤
22,𝑥𝑥
𝐺𝑇2,𝑥𝑦 + 𝐺𝑤
22,𝑦𝑥
𝑐𝑜𝑢𝑝
𝐺𝑇2,𝑦𝑥 + 𝐺𝑤
22,𝑥𝑦
𝐺𝑇2,𝑦𝑦 + 𝐺𝑤
22,𝑦𝑦 ]
4×4
𝐺𝑤
12,𝑥𝑥 + 𝐺12,𝑥𝑥
𝑐𝑜𝑢𝑝
𝐺𝑤
12,𝑦𝑥 + 𝐺12,𝑦𝑥
𝑐𝑜𝑢𝑝
𝐺𝑤
12,𝑦𝑦 + 𝐺12,𝑦𝑦
𝑤
[𝐺12,𝑥𝑦 + 𝐺12,𝑥𝑦
𝑐𝑜𝑢𝑝
𝑐𝑜𝑢𝑝
(8)
The delay terms in the equation (7) should correspond the correct transfer function pair in the
transfer function matrix given by equation (8).
General Force Formulation of Parallel Milling Operation
If ∆𝑥𝑖 and ∆𝑦𝑖 in equation (6) are substituted with equation (2), the general form of the
characteristic force equation is obtained as follows:
𝐹𝑥1
𝐹𝑥1
𝐾𝑡1 𝑁𝑡1 𝑎1
𝑂𝑟𝑖𝑒𝑛𝑡𝑒𝑑
𝐹𝑦1 𝑖𝑤 𝑡
𝐹
𝐾
𝑁
𝑎
1
𝑦1
{ } 𝑒 𝑐 = 4𝜋 ( 𝑡1 𝑡1 1 ) .∗ [𝐷𝑒𝑙𝑎𝑦 𝑀𝑎𝑡𝑟𝑖𝑥].∗ [ 𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 ] { } 𝑒 𝑖𝑤𝑐 𝑡
𝐾𝑡2 𝑁𝑡2 𝑎2
𝐹𝑥2
𝐹𝑥2
𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐹
𝐾𝑡2 𝑁𝑡2 𝑎2
𝐹𝑥
𝑥
(9)
Oriented transfer function matrix is a four by four matrix obtained by the multiplication of the
transfer function matrix with the directional dynamic cutting force coefficients. Delay and
oriented transfer function matrices are multiplied by scalar product.
Equation (9) represents an eigenvalue problem similar to the one obtained for conventional
milling operations. On the other hand, in order to solve this eigenvalue problem, a new
solution methodology is developed to find the stable depth of cut for each cutting tool for
preset spindle speeds of the tools.
15. Uluslararası Makina Tasarım ve İmalat Kongresi
19- 22 Haziran 2012, Pamukkale, Denizli, Türkiye
3. CHATTER STABILITY OF PARALLEL MILLING OPERATION
The major difference in generating stability diagrams for parallel milling comes from the
nature of the process which has two cutting tools with different spindle speeds. Another
concern is that the relation between the chatter frequency and spindle speed cannot be
established obviously. Due to the fact that there are two separate spindle speeds, chatter
frequency cannot be linked directly to the spindle speeds. On the other hand, the chatter
frequency is still unique; so to speak there is a single chatter frequency that corresponds to
two spindle speeds.
There are three cases considered for the solution of the stability problem. First one is
that 𝑛1 and 𝑛2 are given beside the depth ratio and stable depth of cuts of both tools can be
determined. In the second case, process parameters of one of the cutting tools are fixed to
obtain the stability diagram for other cutting tool. For example, the spindle speed of the
second tool and depth of cut is preset. Then the stability diagram for the first cutting tool can
be generated by solving the characteristic force equation of the parallel milling system.
However, because the one of the spindle speed is not defined, the delay matrix cannot be
generated completely and to be able to solve the characteristic equation, iteration method
has to be applied. In the third case 𝑎1 and 𝑎2 are given, then the force equation is solved to
determine the 𝑛1 and 𝑛2 . Same method has to be applied as the second case. On the other
hand, because of some problems about the iteration method of second and third case, the
first case is the most feasible solution to implement. In this paper first case is taken into
consideration.
At first, depth ratio (𝑎𝑟 ) is defined then matrices are rearranged and the eigenvalue problem
becomes simpler to implement. Equation (9) has a non-trivial solution only if its determinant
is zero.
𝑑𝑒𝑡[𝐼 − 𝜆(𝐴)] = 0
(10)
where 𝐼 is the 4 × 4 identity matrix. A matrix is the multiplication of the oriented transfer
function, delay matrix and the ratio matrix, B, which is defined as
1 1 1 1
1
𝐵 = [1𝑟 1𝑟 1
𝑟 𝑟]
𝑟 𝑟 𝑟 𝑟
(11)
r is the ratio of the cutting parameters, that is (𝐾𝑡1 𝑁𝑡1 /𝐾𝑡2 𝑁𝑡2 )𝑎𝑟 .
λ in equation (10) is defined as:
1
𝜆 = 4𝜋 [𝐾𝑡1 𝑁𝑡1 𝑎1 ]
(12)
15. Uluslararası Makina Tasarım ve İmalat Kongresi
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Finally, the characteristic eigenvalue problem can be solved for predefined value of 𝑎𝑟 , and
then the stable depth for the first cutting tool 𝑎1 can be calculated as follows:
𝑎1 = (4𝜋𝜆/𝐾𝑡1 𝑁𝑡1 )
(13)
The eigenvalue solution of equation (10) provides four roots which, in general, are complex.
However, considering equation (12) the eigenvalues with zero imaginary part must be sought
for the solution. Calculating stable depths for different spindle speeds, the stability diagram
for the first tool is constructed. The stable depths for the second cutting tool can easily be
identified using the preset depth-of-cut ratio.
4. SIMULATION AND EXPERIMENT RESULTS
The presented analytical parallel milling chatter stability method is simulated for different
cases. First, the effect of workpiece dynamics on the stable depth of cut is analyzed and
stability diagram is constructed. In addition to the the transfer functions of the tools, the
workpiece transfer functions including the cross terms are also considered but the transfer
𝑐𝑜𝑢𝑝
functions for the machine (i.e. 𝐺21,𝑦𝑥 ) are taken as zero. The same cutting force coefficients
are used for both tools since they are identical with 4 flutes and 12 mm diameter. The first
cutting tool is in up milling mode with clockwise rotational direction, whereas the second tool
is performing a down milling cut in counterclockwise direction. Since the edge forces have no
contribution on the regenerative mechanism, they are taken as zero.
All simulations and experiments are conducted for zero offset angle between two cutting
tools. However, this offset angle may change the stable depth of cut for both tools. Some
simulations are carried for different offset angles. Effect of the offset angle on the stable
depth of cut is observed, and is still being investigated.
Modal Testing
The workpiece is designed to be more flexible than the tools using FEA. The workpiece that
used in the experimentations is shown in Figure 2 and the parts of the workpiece are marked
on figure. The material of the workpiece body structure is cold drawing steel and the
cartridges are selected aluminum. In order to be able to change the workpiece dynamics the
cartridge holder is moved on the body. The main objective here is to design a workpiece
structure such that the variation of the workpiece dynamics due to cutting is minimized. Tools
only cut the cartridges and the main structure of the workpiece is conserved so the
workpiece dynamic does not change during machining.
15. Uluslararası Makina Tasarım ve İmalat Kongresi
19- 22 Haziran 2012, Pamukkale, Denizli, Türkiye
Upper Component
Cartridge
Body Structure
Figure 2. Designed workpiece
In order to determine the tool and the workpiece dynamics, modal testing is performed as
shown in Figure 3.
1
2
Figure 3. Modal Testing Setup
In the modal test, two accelerometers are attached at points 1 and 2 to determine the
transfer functions at each point. The tools and the workpiece are excited by an instrumented
hammer conducting total of 16 modal tests. Frequency responses of the workpiece and the
tools are plotted together in Figure 4.
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Magnitude [m/N]
Most flexible
workpiece mode
First and second tool’s
modes
Hertz
Figure 4. Workpiece and tool frequency responses.
As expected the workpiece is an order of magnitude more flexible than the tools causing
dynamic coupling between them. The measured modal data for the tools and the workpiece
are given in Table 1 and Table 2, respectively. Only some important modal data for the
workpiece is showed.
Table 1. Modal data for the tools.
Modal Parameter of
Tool 1
Tool 2
Workpiece
X
Y
X
Y
Natural Frequency [Hz]
935
940
1587
1594
Modal Stiffness [m/N]
1.09e7
1.18e7
2.43e7
2.41e7
Damping Ratio [%]
4.363
4.352
3.633
3.412
Table 2. Modal data for the workpiece.
Mode Name
Frequency [Hz]
Stiffness [m/N]
Damping [%]
Gw11xx
593
1.73E+07
2.487
Gw11yy
284
1.12E+06
3.430
Gw12yy
285
2.19E+06
2.766
Gw22yy
278
1.14E+06
4.052
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Effect of Workpiece Flexibility on Process Stability
In the first example case, effect of workpiece flexibility on the chatter stability is investigated.
The flexibility of the workpiece is varied for rigid, flexible and an intermediate states.
Tangential cutting force coefficients (𝐾𝑡𝑐 ) are taken as 877 MPa for both tools. The spindle
speeds for the first and the second tool are 8000 rpm and 4000 rpm, respectively. Radial
depth of cut is 4 mm for both tools.
Stable Depth of Cut (a1) [mm]
Effect of Workpiece Dynamics
3.0
Measured Wp Dynamic Case
2.0
Wp flexibility is increased 2
times
1.0
0.0
0
0.5
1
1.5
2
Wp flexiblity is increased 10
times
ar
Figure 5. Effect of workpiece dynamics on the stable depth of cut
The simulation result shows that workpiece dynamics have great impact on the stability of
cutting process (Figure 5). If the tool is very rigid, the process becomes like two separate
single milling operations due to lost coupling between the workpiece. In the first case, the
measured workpiece modal data is used in the stability analysis which yielded stable depth
of 3.08 mm. Then, flexibility of the workpiece is increased two times which resulted in the
stable depth of cut to decrease down to 1.49 mm. In the last case, the workpiece becomes
too flexible thus the stable depth of cut is decreased to 0.33 mm. While in the first case the
workpiece is 12.5 times more flexible than the tools in the last case it becomes 145 times
more flexible. Other important point is that as the 𝑎𝑟 ratio is increased, stable depth of cut of
the first cutting tool is also decreased.
Stability Diagrams
By solving the characteristic equation (eq. 12), stability diagrams for the first tool can be
obtained. Since the depth ratio is preset as 𝑎𝑟 , the stable depth of cut for the second tool can
easily be calculated once the first tool’s stability limits are determined. In the stability
simulations, the depth ratio is taken as 0.2 whereas all the other process parameters are the
same as the first example. The predicted stability diagram is shown in Figure 7. The absolute
stability limit can be seen as 0.5 mm whereas in some lobes the stable depth exceeds 3 mm.
15. Uluslararası Makina Tasarım ve İmalat Kongresi
19- 22 Haziran 2012, Pamukkale, Denizli, Türkiye
Experimental Verification
The simulated cases are verified experimentally on a multi-purpose machine tool (Mori Seiki
NTX2000 (Figure 6).
Figure 6. Mori Seiki NTX2000 Machining Center
Experimental cuts have been performed at different spindle speeds of the first cutting tool for
different 𝑎1 and 𝑎2 combinations. Sound and acceleration data are measured for each test.
Also, the machined workpiece surface is analyzed in order to identify the process as unstable
(chatter), marginally stable and stable (no chatter). Experimental results show relatively good
agreement with the predictions.
Stable
B
Marginally
Stable
Chatter
A
Figure 7. Experimental Results on Stability Diagram
Surface photos and acceleration spectrums are shown in Figure 8 for the tests A and B
(marked on Figure 7). Chatter frequency is around 280 Hz which is the natural frequency of
15. Uluslararası Makina Tasarım ve İmalat Kongresi
19- 22 Haziran 2012, Pamukkale, Denizli, Türkiye
the most flexible workpiece mode, and can be seen in the vibration spectrum for the unstable
test.
Tooth Passing Freq.
of 1st Tool
Condition A
Condition A
Tooth Passing Freq.
of 2nd Tool
Condition B
Condition B
Chatter Frequency
Figure 8. Acceleration spectrums and surface photos for conditions A and B.
5. CONCLUSION
A frequency domain method was developed for the chatter stability of parallel milling
operations. The simulation results are verified by experimental measurements. The effect of
workpiece flexibility on the stable depth of cut is demonstrated where the importance of the
workpiece dynamics on the coupling between the tools is emphasized. As a next step, the
presented parallel milling stability formulation will be extended by including the relative
angular position offset.
ACKNOWLEDGEMENT
The authors acknowledge the support from Mori Seiki Company for the NT4250DCG/1500SZ
multi-tasking machining center, TUBITAK (Project No:110M522) and Pratt & Whitney
Canada for this research.
15. Uluslararası Makina Tasarım ve İmalat Kongresi
19- 22 Haziran 2012, Pamukkale, Denizli, Türkiye
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