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Receptance Coupling for Tool Point Dynamics Prediction
Iker Mancisidor, Mikel Zatarain, Jokin Munoa, Zoltan Dombovari
To cite this version:
Iker Mancisidor, Mikel Zatarain, Jokin Munoa, Zoltan Dombovari. Receptance Coupling for Tool Point Dynamics Prediction. 17th CIRP International Conference on Modelling of Machining Operations, May 2011, Sintra, Portugal. �hal-01352695�
Receptance Coupling for Tool Point Dynamics Prediction
I. Mancisidor 1*,M. Zatarain 1, J. Muñoa 1, Z. Dombovari 2
1 Control and Dynamics, Ideko-Ik4, Elgoibar, Spain
2 Budapest University of Technology and Economics, Department of Applied Mechanics, Budapest, Hungary
Abstract
In many applications, chatter free machining is limited by the flexibility of the tool.
Estimation of that capacity requires obtaining the dynamic transfer function at the tool tip. Experimental calculation of that Frequency Response Function (FRF) is a time consuming process, because it must be done using an impact test for any combination of tool, toolholder and machine. The bibliography proposes the receptance coupling substructure analysis (RCSA) to reduce the number of experimental test. A new approach consisting of calculating the fixed boundary dynamic behavior of the tool is proposed in the paper. This way the number of modes that have to be considered is low, just one or two for each bending plane, and it supposes an important improvement in the application of the RCSA to the calculation of stability diagrams.
The predictions of this new method have been verified experimentally.
1. INTRODUCTION
In recent years, high-speed machining (HSM) has been widely applied in aerospace industry, due to the good machinability provided by aluminum alloys and the improvements in machine and spindle designs. Under these conditions, the major limitation in the material removal rate are self- excited vibrations. These vibrations, also known as chatter, cause a reduction in the surface quality and in the lifetime of mechanical elements and tools.
Since several years, many researchers have investigated methods to avoid the regenerative chatter [1-2]. One of the most popular methods is to create a stability lobe diagram in order to determine the best cutting conditions [3]. The stability lobes separate the stable and unstable regions depending on the spindle speed and depths of cuts. These diagrams are built by a stability model [4-5]
fed with four inputs: tool geometry, cutting coefficients, cutting conditions and dynamic parameters.
To obtain good dynamic parameters, accurate measurements of FRF at the tool tip are required. The problem is that the measurements have to be repeated for each combination of tool/holder/spindle, which are time consuming and disturb the production.
In order to reduce the duration of the measurements and to increase the efficiency of the process, Schmitz et al. [6-8] proposed a receptance coupling technique to predict the dynamic response at the tip of the tool. This method allows coupling of theoretical and/or experimental frequency responses of individual components to obtain the general response of the assembly.
In the simplest case, Schmitz et al. [6]
considered only translational degree of freedom at the spindle portion of the substructure and introduced a interface flexibility between the tool and the spindle- toolholder. The method was improved introducing the rotational degree of freedom with a torsional joint flexibility value [7]. The joint parameters were identified performing a correlation between theoretical and experimental values and updating the joint flexibility values to provide the best fitting.
Park et al. [9] proved that the rotational displacement of the tool at the joint cannot be neglected for accurate construction of FRF at the tool tip, and they proposed a new methodology to identify the receptance at the tool holder nose using blanks with different lengths.
The major advantage of RCSA is that it could save considerable time by means of the
calculation of the FRF in the tool tip for different tools, combining the theoretical response of the different tools and the experimental results of the toolholder/machine assembly.
There are some ways to obtain the theoretical FRFs of the tools. Schmitz et al. [6-7] used analytical Euler-Bernoulli beam theory, which is simple and provides reasonable engineering approximations for many problems. However, this model tends to overestimate slightly the natural frequencies, especially for high frequency modes [10].
Also, the prediction is better for slender beams than non-slender beams.
Another classic beam theory was proposed by Timoshenko to overcome these inaccuracies, which adds the effect of shear as well as the effect of rotation to the Euler-Bernoulli beam.
However, the Timoshenko beam model introduces a cut-off frequency that sometimes can be a low frequency. The results obtained above this frequency cannot be taken into account, and sometimes an insufficient number of modes is available.
Other way to obtain theoretical responses is by Finite Element Method (FEM). It applies a beam theory, normally Timoshenko's model, and the system is discretized in several elements. This method was used by Park et al. [9] and Liljehren et al. [11]. The tools can be me meshed using 3D elements too.
All these methods are proposed in the bibliography with 'free-free' boundary conditions. Therefore, the modes are calculated considering the tool without any kind of of restraint or connection. This way, it requires the calculation of a high number of modes (natural frequencies) to obtain an accurate representation of the dynamics of the tool, because the mathematical description of strain distribution near the joint requires high frequency modes [6].
In this paper, this drawback is overcome proposing the fixed boundary approach. This model calculates a free-free beam response based on clamped-free boundary conditions beam results. Hence, the number of modes that have to be considered is very low and the Timoshenko beams can be used.
2. RECEPTANCE COUPLING
In RCSA, experimental or analytical FRFs of the individual components can be coupled to predict the final dynamic response of the assembly at any selected spatial coordinate.
The objective of this paper is to couple an analytical model of the tool with an experimentally identified dynamic model of the toolholder-spindle assembly as illustrated in Figure 1. The RCSA will be explained briefly.
According to Park et al. [9], a rigid joint has been considered.
Figure 1: Tool and toolholder-spindle substructures Mathematically, the Gij matrix describes the frequency response function of the final assembly measured in point i and the excitation in point j (see equation (1)).
j j j i jy i
iyj iyjy i
i
M f G G
G y G
(1)
The Point FRF in the tool tip G11 is one of the main inputs to feed the stability model [3]. By means of RCSA, this main receptance is estimated using receptances measured or calculated when the two substructures are disconnected (Hij). Operating according to RCSA the next basic expressions can be obtained:
21 1 33 22 12 1111 H H H H H
G (2)
G12 H12 H12
H22 H33
1
H22 (3) where:
j i jy i
iyj iyjy ij j i jy i
iyj iyjy
ij h h
h H h
g ; g
g G g
The response in the substructure A will be obtained theoretically with free-free boundary conditions. The direct FRF in the substructure B (h3y3y), can be obtained experimentally by measuring the spindle with a short blank, but the other two receptances (h3y3θ, h3θ3θ) related
to the rotational degree of freedom, are difficult to measure directly. However, following the methodology proposed by Park et al. [9], this functions can be obtained using a long blank, measuring the direct response in the tool tip and finally subtracting the effect of the blank.
To simplify the formulation, the names of variables of equations 2 and 3 have been changed, obtaining the equations 4 and 5:
. h h
; h
h
; g h h
; f h
; e h
; d h
; c h
; b h
; a h
; v g
; u g
y y
y y y y y
y y
y y y
y y
y y
y y
y
3 3 2 2 3
3 2 2
3 3 2 2 2
2 2
2 2
2
1 2 2
1 1
1 2
1 1
1
g
b e c eg b a e
u
2 (4)
g
b e d eg b b f
v
2 (5)
Two unknowns (β and δ) are going to be considered, and they can be solved by the utilization of the symbolic non-linear analytical toolbox:
cb du da vc
fcb gfa gfu edb egv gbe
(6)
ucefd ufegb
ugevf gevfa
gebfa edbfa
cb uf a ugf vb ge
efv c efb c dvc e cbd e acb f
fc eb cefad evfcb
vdb e d ue
gu f a gf a d e d b e uedbf
b ge v vc ge
da du cb
2 2
2
2 2
2
1
2 2 2
2 2
2 2
2
2 2
2 2
2 2 2 2 2 2 2 2
2 2 2
2
(7)
After this extraction, h3y3θ and h3θ3θ
receptances can be obtained:
2 2 3
3
2 2 3
3
h h
h
h y y
(8)
Furthermore, the damping of the joint interface has been taken into account with this methodology.
3. FREE-FREE TOOL FRFS
In the receptance coupling, FRFs of substructure A (the tool) in free-free conditions are required. An analytical formulation is used to calculate this response, and predict the receptance of the complete system before the mounting of the tool. In this chapter, a new approach to calculate these responses is presented based on fixed boundary model.
Several authors ([6]-[7]) have reported that a high number of modes (more than 80) are required to obtain proper theoretical receptances oriented to RCSA. This fixed boundary model calculates the final free-free state beam response by means of clamped- free boundary conditions with an addition of the effect of fixed point movement. The advantage of this solution is that only few modes are enough to have a good receptance estimation. This approach provides the possibility to use Timoshenko beam theory solving the problems created by the cut-off frequency and introduces a new way to calculate the final response.
The first part of this section shows the development of this new model, while the second verifies theoretically the improvement introduced.
3.1. Mathematical development of the Fixed Boundaries Model
Before applying this approach, it is necessary to calculate the modes of the beam with clamped-free boundary conditions. A good strain distributions near the joint is provided, but it is not the required free-free beam response for RCSA. The next step is to add the effect of movements restrained in the clamped-free model (see Figure 2).
Figure 2: Addition of fixed point movements Mathematically, the equation (9) represents the general equation of vibration, where a
combination of rigid body and modal coordinates is going to be introduced (10):
K
x
M
x f (9)
n n n
l
l l k
k k
q q q q x
2 1 2 2
2 1 2 1
1 2
1
(10)
According with the last equation (10), the first term describes the movement of the joint and the second refers to clamped-free modes. n is the number of modes considered. This change drives to the next equation (11):
fM q
K q T
T T
T T
T
(11)
If each part of this equation is analyzed a simple form of the stiffness matrix is obtained:
2
2 2
2 2 1
diag 0
0 0
0 0 0 0
0 0
0 0
0 0
0 0
0 0
0 0 0
0 0
0 0 0
f
n T T
T T T
T
K K
K K K
(12)
However, the mass matrix presents more complexity in some of its terms:
M M
M
M TM T
T T T
T
(13)
Operating:
1 0 0
0 1 0
0 0 1
0 0
2 0
1
0 0
2 0
1
nxn T
L n L
L
L n L
L
T
tot g tot
g tot T tot
I M
dm x dm
x dm x
dm dm
dm '
M M
I x m
x m ' m
M M
where xg is the distance from the gravity center of the beam and Φ are the mode shapes along x. Finally the equation (13) can be written as:
nxn T T
T
I ' M
' M ' M M
(14)
If in the general equation (11) a sinusoidal force with a certain frequency () is considered, the next expression is obtained:
T
f' M
' M '
M q
T f
T
1 2 2 2
2
2
(15)
where
n n y y
y
n n y y
y
T L
1 2
1 1 1
1 2
1 1 1
2 2
2 1 2
2 2
2 1 2
1 0 1
1 0
0 1
Finally, the damping can be added introducing a loss factor, and obtaining the final receptance ready to introduce in the RCSA:
Tf
T T
i ' M
' M '
T M H
1 2 2
2
2 2
1
(16)
Taking into account the numeration defined in Figure 2:
1 1 1 1 2 1 2 1
1 1 1 1 2 1 2 1
1 2 1 2 2 2 2 2
1 2 1 2 2 2 2 2
h h h h
h h h h
h h h h
h h h h H
y y
y y y y y y
y y
y y y y y y
This way, the free-free state response is obtained describing the correct strain distribution in the joint with a low number of modes.
3.2. Mathematical model of the tool.
The previous solution has been combined with Timoshenko beam theory to calculate analytically the receptance of the tool.
The geometry of the tool should be modeled properly considering the complexity of the cross-section in his fluted part. Furthermore, the tool has been divided in two beams related to the fluted part and the pure cylindrical side. This way, different beam properties could be introduced along the different segments of the model.
The calculation of the inertia of the fluted part is an important issue to obtain accurate predictions. In this paper an approach proposed by Kivank et al. [13] has been used.
Figure 3: a) Cross-sections (Kivank et al. [12]); b) Region considered for each cross-section.
They presented cross-section models of 2, 3 and 4 fluted mills. Each cross-section was divided in regions which were described by three variables: the radius r of the arc, the position of the center of the arc (a) and the diameter fd of the flute.
3.3. Theoretical verification
To verify theoretically the new fixed boundary model, a simple case has been selected to compare the result with the usual free boundary model. The results of both approaches to perform the RCSA have been correlated with the real theoretical response of the assembly. The Timoshenko beam formulation has been used in all the calculations of this comparison.
The substructure A which simulates the tool, is a simple 20mm diameter and 160mm length beam. The rigid substructure (B) is a 300mm diameter and 500mm length beam. The density and the modulus of two beams are 7820 kg/m3 and 207 Gpa, respectively.
In the substructure A, the cut-off frequency is reached analytically for the 12th mode, therefore only 11 mode can be introduced using pure Timoshenko beams. This drawback can overcome using a FEM based model [5].
Figure 4: Results of RCSA with the usual free boundaries model.
Figure 5: Results of RCSA with the new fixed boundaries model.
Figure 4 shows the effect of the number of considered modes in the free-free response for A substructure calculated by free boundaries formulation. Furthermore, in this
case it is not possible to create an accurate result. However, when the new approach is used (Figure 5), two modes are enough to obtain an exact receptance.
Therefore, the great advantage introduced by the fixed boundaries model has been demonstrated.
4. EXPERIMENTAL RESULTS
In the first step, several experimental tests have been carried out to obtain the response of the assembly formed by the machine, the spindle-and the toolholder following the method proposed by Park et al. [9]. This methodology has been chosen because it presents the possibility of characterize the damping of the interface between tool and toolholder in the experimental receptances of the assembly.
This way, the receptance h3y3y is measured directly using a short blank cylinder, and a long blank cylinder is introduced to extract (equation 4-8) the other two responses of the machine-spindle-toolholder assembly (h3y3θ, h3θ3θ). The properties of these blank cylinders are showed in Table 1.
On the other hand, three carbide tools with different properties have been tested. The material and geometrical properties, including the cross-section variables, of all of them are showed in the Table 1. The machine used in the different test is Soraluce SV6000 milling machine with a Kessler 34kW spindle.
4.1. Practical application
A thermal toolholder has been selected to perform the first part of the experimental verification (LAIP 1214121420 – HSK A63) due to their simple joint.
The next figures (Figures 6-8) shows the results of the receptance coupling compared with an experimental measurement of this response. In general, the results with three tools are good even though the signals are dynamically complex. The signals obtained by RCSA reflect the reality properly in frequency and amplitude, therefore it verifies that the estimation damping is correct. However in spite of these good results, there are some peaks that introduce an error.
Figure 6: Measured and predicted FRF of the tool 1 with thermal toolholder.
Figure 7: Measured and predicted FRF of the tool 2 with thermal toolholder
Figure 8: Measured and predicted FRF of the tool 3 with thermal toolholder..
Short Blank Long Blank Tool 1 Tool 2 Tool 3 Material Carbide Carbide Carbide Carbide Carbide
E (GPa) 580 580 580 580 510
(Kg/m3) 14500 14500 14350 14000 13800
Diameter (mm) 20 20 20 20 20
Total length (mm) 50 120 145 104 100
Flute length (mm) - - 36 44 36
Number of flutes - - 3 4 3
r (mm) - - 8.52 7.30 9.07
a (mm) - - 2.50 2.70 2.30
fd (mm) - - 3.47 3.22 2.99
Table 1: Properties of the blank cylinders and tolos.
Hydraulic toolholder Collet chuck toolholder
Tool 1Tool 2Tool 3
Figure 9: Measured and predicted FRF of the tool 3 with thermal toolholder.
Thereby, the results with the more slender tool are better than with the other two tools. The reason of this is that when the tool is a slender tool, the mode shapes are clearer due to the dominant bending mode of the tool and in this way, the receptance coupling method obtains better results.
4.2. Effect of toolholder
The objective of this section is to validate this method for other types of toolholders with more complex interfaces. Therefore, two more toolholders have been tested: a hydraulic toolholder (Gühring GM 300 – HSK A63) and
a collet chuck toolholder (ROMH CMBH 753216 – HSK A63).
Analyzing the results (see Figure 9), it is clear that the proper predictions of the RCSA are maintained for these toolholders too.
Comparing the results for the same tools in different tool holders, the results are not clear, In general, the amplitudes are the same for the first and third tools, but for the second tool the dynamic stiffness is lower with the thermal toolholder. In the other hand, it is clear that the collet chuck toolholder drives to lower frequencies due to its big mass. Anyway, considering that the toolholders present some nonlinearity a wider study is required to obtain general conclusions.
4.3. Stability prediction
The accuracy of the predictions obtained using the RCSA with the new approach has been measured using a stability model proposed by Altintas and Budak [3].
In this paper, the stability lobes have been calculated for slotting operation taking into account the cutting coefficients showed in Table 3.
Cutting coefficients Value Tangential direction, Kt 700 N/mm2
Radial direction, Kr 245 N/mm2 Table 2: Cutting forcé coefficients.
In the Figures 10-12, the stability lobes diagrams obtained using the RCSA has been compared with the results provided by directly measured FRF. (see section chapter 4.1).
The diagrams show that the approximation is good enough and the RCSA is able to predict the general shape of the diagram. The minimum stable depth of cut and the spindle speed related to the maximum stability point has been compared in the Table 4.
Deviation ap lim Maximum stability spindle speed Tool 1 29.8% -1.1%
Tool 2 49.2% -1.1%
Tool 3 33.7% -1.9%
Table 3: Stability lobes results.
Figure 10: Stability lobe diagram of the tool 1 with the thermal toolholder.
Figure 11: Stability lobe diagram of the tool 2 with the thermal toolholder.
Figure 12: Stability lobe diagram of the tool 3 with the thermal toolholder.
5. CONCLUSION
A new approach to perform the receptance coupling (RCSA) has been proposed based on fixed boundary calculation. Applying this method, it is not necessary to calculate a high number of modes, and therefore they can be calculated analytically with Timoshenko beam theory. Furthermore, due to its good behavior with the receptance coupling method, this model is valid to predict dynamic parameters of different assemblies with different toolholders. Consequently, it is possible to obtain a fairly accurate stability diagram, especially reliable to determine the most stable spindle speeds.
On the other hand, this article shows the good results obtained by the methodology proposed by Park et al. [9] to characterize the
experimental data. In fact, this method provides a useful tool to consider the damping of the interface between tool and toolholder without any additional parameter.
6. ACKNOWLEDGEMENTS
This research was partially supported by the OPENAER project (CENIT program of Technical Industrial Development Center (CDTI) of the Spanish government.
7. REFERENCES
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