Grinding
Manufacture
of the
Aircraft’s
Suppression
of Chatter
Vibrations
in Parts
Simultaneous Machining Operations
Professor Erhan Budak
Manufacturing Research Laboratory
Sabanci University
Istanbul, Turkey
Abstract
In simultaneous machining processes more than one cutting 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 paper,
analytical models are presented for the chatter stability of parallel turning and milling processes.
The stability diagrams are obtained for different parallel machining conditions demonstrating the
effects of dynamic interaction between the tools. Conditions for increased stability are also
demonstrated through examples. The predictions are verified by time-domain simulations and
experiments carried out on multi-tasking machining centers.
1
1. Introduction
Simultaneous machining operations have been continuing to spread due to various advantages
they offer. These operations involve more than one tool cutting the same or different surfaces at
the same time. Cutters can be located at the same or different spindles or turret. Parallel turning
and parallel milling are two common examples of simultaneous machining operations which are
usually performed on multi-purpose machine tools [1]. Parallel operations have 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 simultaneous
processes, as well. On the other hand, if the conditions are selected properly, chatter-free
material removal rate can be increased. In such a case, dynamic cutting forces on both tools may
cancel each other increasing stability limits.
Dynamics of simultaneous operations is more complex compared to conventional single tool
processes due to the existence of dynamic coupling between the tools and the workpiece. If the
workpiece is not flexible, the dynamic coupling may exist only between the cutting tools through
the machine tool structure. In cases where two tools cut the same surface, such as in parallel
turning, the coupling is generated due to the chatter marks left on the surface.
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 [2] and Tlusty and Polacek [3].
They demonstrated the coupling between the cutting forces and dynamic displacements and
estimated the chatter stability limits. 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.
Dynamics and stability of parallel machining, on the other hand, has been studied very little.
Lazoglu et al. [5] developed a time domain model for parallel turning operation. Then,
Ozdoganlar and Endres [6] formulated the dynamics of the parallel turning process. Budak and
Ozturk studied dynamics of parallel turning processes and demonstrated the effects of process
coupling on chatter stability [7]. The research on simultaneous milling processes is even more
limited. Ozturk and Budak [8] have simulated the dynamics of parallel milling in time domain
and generated stability diagrams for various cutting conditions. This study was extended by
Comak and Budak [9] through frequency domain solution where effects of dynamic coupling
and process conditions on chatter stability limits were demonstrated. Brecher et al. [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 et al. [11] posed the suppression of chatter in
simultaneous milling by properly the speed difference of the tools.
In this paper, dynamics and stability of simultaneous, or parallel, turning and milling processes
are investigated. Chatter stability diagrams are generated using analytical models which include
the structural and process coupling in these processes. Effect of the workpiece dynamics and
process conditions on the stability limits are demonstrated through example cases. The
simulation results are verified by experiments.
2. Chatter Stability of Simultaneous Turning Operations
2.1. Dynamic forces and chatter stability
A simultaneous turning process where two cutting tools machine the same part is illustrated in
Fig.1. The feed velocity of both tools is equal whereas their cutting depths can be different. The
cutting depths of tool 1 and tool 2 are represented by a1 and a2, respectively. The cutting tools are
dynamically dependent since vibration waves left by each tooth on the workpiece surface affect
the other tooth’s dynamic chip thickness.
2
Fig. 1 Two turning tools simultaneously cutting a work piece and its schematic representation
showing process and structural parameters.
Due to the dynamic cutting forces on each tool (F1 and F2), the dynamic displacements (z1 and
z2) develop on the tools. These displacements affect the dynamic chip thicknesses, and thus the
dynamic cutting forces. Unless the cutting depths on each tool are equal, there are two different
regions with different mechanical and dynamic characteristics in the process. The region with
depth of a1, which is assumed to be less than a2, is removed by both tools. In this region, the
dynamic chip thickness on a tool is affected by the displacement of the tool at the present time
and the displacement of the other tool at a half rotation period ( /2) before. The feed per
revolution ho is shared between the tools in this region as the static chip thickness. On the other
hand, the region with a depth of a2-a1 is only removed by the second tool. Hence, the dynamic
chip thickness depends on the dynamic displacement of the second tool at the present time and at
one rotational period () before. Moreover, the static chip thickness on the second tool is equal to
the feed per revolution in this region. As dynamic displacements can be calculated in terms of
cutting forces and transfer functions at each tool tip, G11 and G22, the cutting forces at the limit of
stability can be written as follows [7]:
 F1  ict
 F1  ict
 F e  B  F e
 2
 2
(1)
where B matrix for this case is presented below:

 a1G11
BKf 


 i c
2
a
G
e
 1 11
a 2 G 22 (e ic




 1)  a1G 22 e ic 
a1G 22 e
 i c
2
(2)
After some arrangements, the relations take the following form [7]:
I  B
F1 
0
 F2 
(3)
where I is the 2*2 identity matrix. In order to have non-trivial solutions at the stability limit, the
determinant of [I-B] matrix should be equal to zero. The determinant results in a complex
3
equation with variables a1, a2, c and τ. When the real and imaginary parts of the equation are
grouped and equated to zero, two independent equations are obtained as follows [7]:
Real (det I  B )  0
Imag (det I  B )  0
(4)
For the stability solution, first the cutting depth on the second tool, a2, should be selected and the
stability diagram for a1 can be obtained. Since a2 is selected as higher than a1 only the stability
limit values of a1 which are less than a2 are real solution. Equation (4) is numerically solved in
order to obtain the stability diagrams for a1 by sweeping the chatter frequency around the
important vibration modes of the system, i.e. G11 and G22 in equation (2).
2.2. Example cases
The stability analysis of parallel turning is demonstrated by an example case. In this example
case, both turning inserts are selected as Taegutec TPGN 110304 TT8115. The workpiece
material is a 32 mm diameter cylinder made out of 1050 steel. The edge and cutting force
coefficients in the feed direction are calibrated mechanistically using the linear-edge force model
as 86.5N/mm and 1100 MPa, respectively. The calibration was performed for 200 m/min cutting
speed and maximum feed per revolution of 0.13mm. The direct frequency response functions of
the turning tools are measured by tap testing. There was no dynamic coupling between the tools
since they are located on different turrets, and thus cross FRFs were neglected. Modal data fit to
the measured transfer functions is tabulated in Table 1.
G11
G22
Table 1 Modal data of the tools
Mode
fn(Hz)  k(N/m)
1
1688. 3.85 1.495*107
1
2
2060. 0.87 2.483*108
2
1
1922. 4.72 6.429*106
1
The absolute stability of the first and the second tools working alone are calculated as 1.1 mm
and 0.6mm, respectively. For parallel turning stability analysis, stability limits for a1 are
calculated for different values of a2. It is observed that when a2 is selected higher than the second
tool’s stability limit for the single tool operation (i.e. when a2 is selected from the unstable
region for the single tool process), two stability limits are observed in the stability diagram of a1
as presented in Fig. 2. Between these two limits, the parallel process is stable, otherwise it is
unstable. These predictions by the frequency domain model are verified by time-domain
solutions and shown in Fig. 2. This example shows that the stability of parallel turning is
complicated as there is a complex relationship between the stability limit of one tool and the
other tool’s depth of cut. The simulation and experimental results show that the stability limit of
a tool does not decrease by increasing the depth of the other tool and vice a versa. This complex
correlation can be used to determine the conditions for high stability in parallel turning which
will be demonstrated next. For this, frequency domain model is used to determine the stability
limits for the first tool for different a2 values. Predicted lower and higher stability limits for a1 at
2100 rpm are presented in Fig. 3. It should be noted that the predicted stability limits for a1 are
not real solutions if they are higher than a2 since a2 is assumed to be equal or greater than a1 in
4
the formulation. It is seen that the curve for the higher stability limit crosses a2=a1 line when a2
is 1.25mm. Hence, the higher stability limit predictions are not real when a2 is less than 1.25
mm. In order to predict the higher stability limits in this region, dynamic properties of the tools
can be swapped in the simulations and by this way, stability limits for a2 can be predicted for
several values of a1 by the frequency domain model and the calculated limits of a2 for given a1
values can be added in the figure. Alternatively, after swapping the dynamic properties of the
tools, stability limits for a1 can be predicted for several a2 values below 1.25mm using the timedomain model. The predictions of the time domain model are added in Fig. 3 in order to show
the agreement between the frequency and time domain models once more, e.g. both solutions
predict the same a1 limit of 1.25mm at a2=1.25 mm. The area that is surrounded by the calculated
lower and higher limits shows the stable process parameter combinations and it is highlighted
with gray to differentiate it from the unstable area. There are some more special regions in this
figure. When a2 is less than 0.6mm which is the predicted stability limit for the single turning
with tool 2, it is seen that the lower stability limit is zero. Hence, the stability diagrams have only
higher limit in this region. When a2 is higher than 0.6mm, lower limits for a1 become non-zero
and stability diagrams with two stability limits are observed. As a2 increases more, lower and
higher limits converge each other and become equal around 2mm when a2 is 3.1mm. For higher
values of a2, the process becomes unstable regardless of the value of a1.
a1=0mm
Frequency(Hz)
Sound
a1=0.5mm
Frequency(Hz)
Sound
a1=1.5mm
Sound
Fig 2 Stability diagram with two limits (a2=1.5mm). Time domain solution for the 2nd tool’s
displacement at 3 different a1 values (a1=1.5mm, a1=1mm, a1=0.5mm) when a2 is 1.5mm.
Frequency(Hz)
Fig 3 Lower and higher stability limits predicted by frequency domain model at 2100 rpm and
the experimental results. Surface photos and sound spectrums for different a1 when a2 is 1mm
5
Another important observation is that there are 4 stability limits of a2 when a1 is between 1.1 and
1.6 mm. It means that the stability behavior of the process shifts between unstable and stable 4
times when a2 increases from 0 to 3mm. In order to demonstrate this, time domain solution for
variation of the second tool’s displacement is presented in Table 2 for a2 values of 0, 1, 1.5, 2
and 3 mm when a1 is equal to 1.5mm.
Table 2 Variation of stability behavior for different values of a2 when a1 is 1.5 mm. (The plots
in each case show the variation of second tool’s displacement (z2) in time)
a2=0mm
Unstable
a2=1mm
Stable
2 tool // z2 displacement
a2=1.5mm
Unstable
2 tool // z2 displacement
0.4
2 tool // z2 displacement
0.03
0.3
a2=2mm
Stable
2 tool // z2 displacement
0.12
0.1
0.1
0.09
0.025
0.2
0.07
0.06
0.04
0.02
2
z2(mm)
0.015
z2(mm)
0
z2(mm)
z2(mm)
z2(mm)
4
0.06
0.1
0.05
0
0.04
0.01
0
-0.02
0.02
0.005
-0.2
-0.04
0
5
10
15
20
25
30
Revolution
35
40
45
50
0
0
5
10
15
20
25
30
Revolution
35
40
45
50
-0.06
-2
0.03
-0.1
-0.3
2 tool // z2 displacement
6
0.08
0.08
0.02
a2=3mm
Unstable
-4
0.01
0
5
10
15
20
25
30
Revolution
35
40
45
50
0
0
5
10
15
20
25
30
Revolution
35
40
45
50
-6
0
5
10
15
20
25
30
Revolution
35
40
45
50
The total depth removed from the workpiece is determined by either a1 or a2 in this parallel
turning case depending on which one has a greater value. According to Fig. 3, it is possible that
3.1 mm depth can be removed in a stable way when a1 and a2 are selected as 3.1 and 2mm,
respectively. Considering that in turning operations with single tool the stability limits were 1.1
mm and 0.6mm, the parallel turning tools offer considerable productivity advantage.
Experimental cuts have been performed at 2100 rpm for different a1 and a2 values. By analyzing
the measured sound data and resulting surface quality, the tests are classified as chatter,
marginally stable and stable. Experimental results are also given in Fig. 3. As a result of the
experiments, it is confirmed that stability diagrams with two stability limits exist for certain a2
values. When a2 is 1mm, it is shown that tests with a1=1.5mm and a1=0mm result in poor surface
quality with chatter marks, while the test with a1=0.5 mm creates a chatter free surface.
3. Dynamics and Stability of Simultaneous Milling Operations
3.1 Dynamic response, coupling and cutting forces
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. Fig 4 shows 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.
Dynamics responses at the first and the second points can be formulated as follows [9]:
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
(5)
where i and j (=1,2) indicate the first and second contact points, respectively. Fia is the dynamic
𝑤
cutting force at the ith contact point in direction a (a=x, y). Similarly, 𝐺𝑖𝑗𝑎𝑏
is the workpiece
transfer function measured at point i along a direction and excited at point j along the b direction.
6
Fig 4 Parallel milling operation
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 [8, 9]:
∆𝑥1 = [𝑥1𝑇 (𝑡) − (𝑥1𝑤 (𝑡) + 𝑥2𝑤 (𝑡))] − [𝑥1𝑇 (𝑡 − 𝜏1 ) − (𝑥1𝑤 (𝑡 − 𝜏1 ) + 𝑥2𝑤 (𝑡 − 𝜏2 ))]
∆𝑦1 = [𝑦1𝑇 (𝑡) − (𝑦1𝑤 (𝑡) + 𝑦2𝑤 (𝑡))] − [𝑦1𝑇 (𝑡 − 𝜏1 ) − (𝑦1𝑤 (𝑡 − 𝜏1 ) + 𝑦2𝑤 (𝑡 − 𝜏2 ))]
(6)
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. 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. Force equations for both cutting tools can be written as [9]:
𝛼𝑥𝑥1
𝐹𝑥1
1
{𝐹 } = 𝐾𝑡1 𝑎1 [𝛼
𝑦𝑥1
𝛼𝑥𝑦1 ∆𝑥1
𝛼𝑦𝑦1 ] (∆𝑦1 )
𝛼𝑥𝑥2
𝐹𝑥2
1
{𝐹 } = 𝐾𝑡2 𝑎2 [𝛼
𝛼𝑥𝑦2 ∆𝑥2
𝛼𝑦𝑦2 ] (∆𝑦2 )
𝑦1
𝑦2
2
2
𝑦𝑥2
(7)
𝛼𝑥𝑥𝑖 are the directional dynamic force coefficients and Kti are the cutting force coefficients for
the first and the second tool (𝑖 = 1,2). Directional dynamic cutting force coefficients are the same
as in conventional milling process [4].
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. The general form of the characteristic force equation is
obtained as follows:
7
𝐹𝑥1
𝐹𝑥1
𝐾𝑡1 𝑁𝑡1 𝑎1
𝑂𝑟𝑖𝑒𝑛𝑡𝑒𝑑
𝐹𝑦1
𝐹𝑦1
𝐾 𝑁 𝑎
1
{ } 𝑒 𝑖𝑤𝑐 𝑡 = 4𝜋 ( 𝑡1 𝑡1 1 ) .∗ [𝐷𝑒𝑙𝑎𝑦 𝑀𝑎𝑡𝑟𝑖𝑥].∗ [𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 ] { } 𝑒 𝑖𝑤𝑐 𝑡
𝐾
𝑁
𝑎
𝐹𝑥2
𝐹𝑥2
𝑡2 𝑡2 2
𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝐾𝑡2 𝑁𝑡2 𝑎2
𝐹𝑥
𝐹𝑥
(8)
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 (8) 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.
3.2. Chatter stability limits
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, and thus the chatter
frequency cannot be linked directly to the spindle speeds. On the other hand, the chatter
frequency is still unique; there is a single chatter frequency that corresponds to two spindle
speeds.
Different scenarios can be considered for the solution of the simultaneous milling stability
problem. For example, 𝑛1 and 𝑛2 can be specified for a depth ratio and the corresponding
stability limits can be determined. In another case, process parameters for one of the cutting tools
can be fixed to obtain the stability diagram for the other tool.
Equation (8) has a non-trivial solution only if its determinant is zero:
𝑑𝑒𝑡[𝐼 − 𝜆(𝐴)] = 0
(9)
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 1
𝐵 = [1𝑟 1
𝑟 𝑟 𝑟]
𝑟 𝑟 𝑟 𝑟
(10)
r is the ratio of the cutting parameters, that is (𝐾𝑡1 𝑁𝑡1 /𝐾𝑡2 𝑁𝑡2 )𝑎𝑟 . λ in equation (10) is defined as
follows [9]:
𝜆=
1
4𝜋
[𝐾𝑡1 𝑁𝑡1 𝑎1 ]
(11)
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 )
(12)
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.
8
3.3. Simulation and experimental 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 the stability
diagram is constructed. In addition to the transfer functions of the tools, the workpiece transfer
functions including the cross terms are also included. 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.
The workpiece used in the experiments is shown in Fig. 5. The material of the workpiece body
structure is cold drawing steel and the cartridges are aluminum. Tools only cut the cartridges and
the main structure of the workpiece is conserved so the workpiece dynamic does not change
during machining.
1
Upper Component
Cartridge
Body Structure
2
Fig 5 Designed workpiece and modal test set-up
In the modal test, two accelerometers are attached at points 1 and 2 to determine the transfer
functions of each point. The tools and the workpiece are excited by an instrumented hammer.
Frequency responses of the workpiece and the tools are plotted together in Fig. 6. As expected
the workpiece is an order of magnitude more flexible than the tools causing dynamic coupling
between them.
Magnitude [m/N]
Most flexible
workpiece mode
First and second tool’s
modes
Fig 6 Workpiece and tool frequency responses
In the first example case, the effect ofHertz
workpiece flexibility on the chatter stability is
investigated. The flexibility of the workpiece is varied to demonstrate its effects on stability.
9
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
Fig 7 Effect of workpiece dynamics on the stable depth of cut
The simulation result illustrated in Fig. 7 show that workpiece dynamics have great impact on
the stability of cutting process. 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
and thus the stable depth of cut decreases 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 depth ratio,𝑎𝑟 , is increased the stable depth of cut of the first cutting
tool decreases substantially.
The predictions are verified experimentally on a multi-purpose machine tool (Mori Seiki
NTX2000) shown in Fig. 8. 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 given above. The predicted
stability diagram is shown in Fig. 9. The absolute stability limit can be seen as 0.5 mm whereas
in some lobes the stable depth exceeds 3 mm. 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 reasonably good agreement with the predictions. Surface photos and
acceleration spectrums are shown in Fig. 10 for tests A and B (marked on Fig. 9). Chatter
frequency is around 280 Hz which is the most flexible workpiece mode, and can be seen in the
vibration spectrum for the unstable test.
4. Conclusions
Stability models for different simultaneous machining processes are presented in this paper.
These models are useful for understanding the dynamic interaction of parallel working tools and
predicting the effect of this interaction on stability limits. The models’ results are demonstrated
on several examples and advantages of parallel machining in terms of process stability are
shown. Moreover, experimental and time domain verifications of the models are presented. It is
demonstrated that the total cutting stability in a parallel process can be increased due to dynamic
interaction between the tools.
10
Fig 8 Mori Seiki NTX2000 Machining Center used in simultaneous milling tests.
Stable
B
Marginally
Stable
Chatter
A
Fig 9 Chatter test results and predicted stability diagram
Tooth Passing Freq.
of 1st Tool
Condition A
Condition A
Tooth Passing Freq.
of 2nd Tool
Condition B
Condition B
Chatter Frequency
Fig 10 Acceleration spectrums and surface photos for conditions A and
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Acknowledgement
The author acknowledges the support from Mori Seiki Company for the NTX2000 multi-tasking
machining center, TUBITAK (Project No: 110M522) and Pratt & Whitney Canada for this
research.
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