Stability Limits of Milling Considering the Flexibility of the Workpiece

International Journal of Machine Tools & Manufacture 45 (2005) 16691680 www.elsevier.com/locate/ijmactool

Stability limits of milling considering the exibility of the workpiece and the machine U. Bravo, O. Altuzarra, L.N. Lopez de Lacalle*, J.A. Sanchez, F.J. CampaDepartment of Mechanical Engineering. University of the Basque Country, Escuela Superior de Ingenieros Alameda de Urquijo s/n., 48013 Bilbao, Spain Received 29 November 2004; accepted 3 March 2005 Available online 18 April 2005

Abstract High speed machining of low rigidity structures is a widely used process in the aeronautical industry. Along the machining of this type of structures, the so-called monolithic components, large quantities of material are removed using high removal rate conditions, with the risk of the instability of the process. Very thin walls will also be milled, with the possibility of lateral vibration of them in some cutting conditions and at some stages of machining. Chatter is an undesirable phenomenon in all machining processes, causing a reduction in productivity, low quality of the nished workpieces, and a reduction of the machine-spindles working life. In this study, a method for obtaining the instability or stability lobes, applicable when both the machine structure and the machined workpiece have similar dynamic behaviours, is presented. Thus, a 3-dimensional lobe diagram has been developed based on the relative movement of both systems, to cover all the intermediate stages of the machining of the walls. This diagram is different and more exact than the one that arises out of the mere superposition of the machine and the workpiece lobe diagrams. A previous step of rejecting resonance modes that are not involved in the milling at the bottom zones of the thin walls must be previously performed. Finally, the proposed method has been validated, by machining a series of thin walls, applying cutting conditions contrasted with the limits previously obtained in the three-dimensional lobe diagram. q 2005 Elsevier Ltd. All rights reserved.Keywords: Chatter; High speed milling; Low-rigidity parts; Machining vibrations; Stability lobes

1. Introduction The manufacturing process of monolithic components is widely used in the aeronautical sector for the manufacturing of airframe components. The great number of these workpieces are manufactured by high speed milling, where problems can arise related to the breakage of the tools, instability in the process and dimensional errors in the workpieces. The instability of the process is a vibration phenomenon known as chatter, which causes the abovementioned effects. Chatter phenomena appear in the high removal rate roughing, as well as in the nishing of low rigidity airframe components. Taylor made rst evidence and description of chatter in 1907. However, the study of chatter goes back to the 1950s* Corresponding author. Tel.: C34 9460 14216; fax: C34 9460 14215. E-mail address: [email protected] (L.N. Lopez de Lacalle).

0890-6955/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.03.004

when Tobias [1] established the basis of the regenerative chatter theory. In the 1970s other authors concentrated on the study of the nonlinear aspects of the process. Sexton [2] established that the two main nonlinear phenomena of the process are the variation in cutting forces and temperature with time, and proposed the search for stability varying the spindle speed. At the end of the 1970s and along the 1980s, Tlusty et al. [35] carried out several studies in this eld focusing on the determination of cutting forces, machine design and chatter detection. These studies led to the proposal of a nonlinear milling model and the stability compared with the modal coupling technique was examined. They proposed a monitoring and control system [6] consisting of an active control system implemented in a computer with communication to the numerical control of the machining centre, where a algorithm for detecting the breakage of the tool is implemented along with a chatter detection system. Weck [7], performed an in depth study on the dynamic behaviour of the machines, in which constructive and analytical solutions were presented for the prevention of

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U. Bravo et al. / International Journal of Machine Tools & Manufacture 45 (2005) 16691680

regenerative vibrations. Smith et al. [8], based on simulation studies of the lobe diagram in the time domain, carried out a study in which they optimised the machining parameters at high speeds. Using a dynamometric table, the frequencies produced in the process were measured. Ioanis [9], presented a new method for creating lobe diagrams based on a theoretical method for the analysis of differential equations with periodic coefcients. Using this method, they studied the nonlinear problem, such as the loss of contact of the tools teeth. In the 1990s, Winfough [10] developed an on-Line method for the detection and stabilisation of the process with regard to the chatter phenomenon, using microphones. Altintas [11], developed an off-Line analytical utility for chatter prediction, making use of the lobe diagrams; carrying out a transformation of this diagram from time to frequency domain, based on the Fourier spectrum. Soliman and Ismail [12] advocated an on-Line method based on the detection of cutting forces by means of a Kistler table, from which they obtained a relationship between high and low frequency forces. With this method, they proposed that the on-line tunning is faster than methods proposed by the above-mentioned authors, in which adjustment time is around 9.75 s. Even so, they had the same delay problems that invalidate the method for an application in real time. Ismail and Ziaei [13,14] presented a study in which they demonstrated the effectiveness of acoustic systems for chatter detection. They later developed a combined on-line and off-line methodology for the suppression of chatter, focused on machining in ve axes. The on-line method is based on a control that varies the speed from the acoustic data and the off-line method uses the feed programming based on the lobe diagrams. Insperger [15] proposed a new method to obtain dynamic stability in milling studying the importance of machining in upmilling and downmilling, using a single degree of freedom system. Also, Schmitz et al. [16] and Park et al. [17] have studied the inuence of the tool overhang in the stable area of the lobes diagram to achieve optimum material removal rates. Recently, several solutions to predict the inuence of tool passing frequency harmonics in low immersion milling have been studied by several authors [1820]. New stability(a) (b) h(t T) y(t T)

lobes (ip bifurcation lobes) that were not covered by classical chatter theories have appeared as a result; they can be detected by the strong vibration in machining, at the integer harmonics of the half tooth passing frequencies. Up to date, efforts for the suppression of chatter have concentrated on considering it as a phenomenon which arises exclusively from the machine/spindle/tool system; the cutting forces produce a displacement of the machine/tool with respect to the workpiece being machined. In contrast, in recent studies [21], the behaviour of the workpiece is studied, but without considering the behaviour of the machine. Recent research by Schmitz et al. [22] resulted in the development of a three-dimensional stability chart that combined the traditional rotational speed versus chip width with the tool overhang length (as third axis of the chart). For its calculation, a model of the tool is coupled to an experimental measurement of the machine/spindle/toolholder subsystem. In this study, an off-line chatter detection method is proposed. Chatter is tackled as a phenomenon related to the relative movement between the two mechanical systems that are in contact with each other, that is to say, the machine/tool and the workpiece (in this case with low rigidity). In some cases, the dynamics of the machine will be predominant, for example, in hard roughing operations. In others the vibration comes from the workpiece, for example, in the nishing of thin walls (thickness lower than 1 mm). The simultaneous movement of both subsystems can also occur. Therefore, in this study we allow on the possibility of both subsystems vibrating at the same time, and the method to predict this situation. From here on, the optimum machining conditions will be deduced. Another aspect to be considered is that the dynamic response of exible parts changes along the machining process.

2. Proposed model Fig. 1a presents the case of a single degree of freedom cutting process. The tool body (or entire system tool/

Y y ho h y(t) e Ff x h(t) j(t) X Dynamic chip load hj(t)

Fig. 1. Schematic representation of a SDOF cutting and milling process.


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toolholder/machine) is exible in the perpendicular direction to the cutting speed, where the feed movement will be applied. This exibility, under the action of the cutting forces, produces a vibration, giving rise to an irregular surface (wavy). This will be the surface that the tool will machine in the next pass. For this reason, during the second pass, the width of the chip will vary constantly. The vibration will increase in the case of the gap between a pass and the previous one causes an increase in the width of the chip. Similarly, the case for milling is proposed, as shown in Fig. 1b. In this case, the deection of the tool can take place in two perpendicular axes (plane XY), making it necessary to take into account the dynamic behaviour in both these axes. For these basic models, the state of the art studies, referred to in Section 1, have been taken into account as the starting point for obtaining the lobe diagrams. However, none of these studies considers the event in which both mechanical systems (the machine assembly and the workpiece assembly) present similar dynamic behaviours. In this case, a movement of both systems could simultaneously happen. Some of the current available software (CutProe, BestSpeede, SimMille) calculate the lobe diagrams separately for the machine and the workpiece, to later carry out a superposition of the two. That is to say, although these models do take into account the exibility of both subsystems, they nally treat each mode of vibration and each system independently. In those situations when the machinetool subsystem and the workpiece have similar dynamic behaviours, the two systems have a simultaneous displacement produced by the cutting forces. These displacements are intimately related, as both systems are in contact through the tool teeth action. The cutting force appears out along this contact; this is really an internal force, with which the responses of both subsystems are related, as we will propose in the work here presented. 2.1. Model of the displacement between workpiece/tool The basic idea is the following: along the cutting process, the tool creates a cutting force on the machined workpiece, which then generates a reaction force against the tool in the same direction and magnitude but in the opposite direction to the rst. From this equality of the reacting forces, the ratio between the responses of both mechanical systems is obtained, from both static and dynamic points of view. In this study, how to proceed from this point to obtain the lobe diagrams is proposed. As a prior step, for the study of this phenomenon it must be studied the dynamic exibility (Frequency Response Function, FRF) of each subsystem. For the machinetool subsystem, and for a single degree of

freedom, the FRF is fs Z ys u2 n Z 2 Ff s ks C 2z$un $s C u2 n (1)

where k is rigidity (N/mm2), z the damping ratio, and u the frequency (Hz). And, for multiple n degree of freedom the FRF fsm Zn X kZ1 n X kZ1

a m C bm s il;k il;k s2 C 2xm un;k s C u2 k n;k s2 Rm k m C 2xk un;k s C u2 n;k (2)

Z

where the machine residue unit is (m/s2)/N. In the same way, for the subsystem workpiece the FRF matrix is: fsw Zn X kZ1 n X kZ1

aw C b w s il;k il;k s2 C 2xw un;k s C u2 k n;k Rw k s2 C 2xw un;k s C u2 k n;k (3)

Z

To study the relationship between these two subsystems, it is necessary to consider the transfer functions in the same directions. Therefore, the transfer functions are projected on the orthogonal axes, X- and Y. For the machine In the Y-axis; fyym Z Gyy:m w C Hyy:m w Z dyy:m Fyy:m dxx:m Fxx:m (4)

In the X-axis; fxxm Z Gxx:m w C Hxx:m w Z

(5)

where Gyy.m (w) and Hyy.m (w) are the real and imaginary parts of the transfer function of the machine in the Y-axis. dyy.m is the displacement of the machine in the Y-axis and Fyy.m is the force on the machine in the Y axis. In (5), the same for the machine X-axis is presented. For the workpiece In the Y-axis; fyyw Z Gyy:w w C Hyy:w w Z dyy:w Fyy:w dxx:w Fxx:w (6)

In the X-axis; fxxw Z Gxx:w w C Hxx:w w Z

(7)

where Gyy.w (w) and Hyy.w (w) are the real and imaginary parts of the transfer function of the workpiece in the Y-axis dyy.w is the displacement of the workpiece in the Y-axis and Fyy.w the force of the workpiece in the Y axis. In (7), this is presented for the X-axis of the workpiece. Once the transfer functions have been analysed, the interrelation between the responses of both mechanical

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Fo

Fo

Fo

The radial depth of the cut does not exceed the radius of the tool, so that the cutting force components denot change in direction. Therefore, the immersion angle of tool into material ranges from 0 to 908.Fo

Downmilling

Upmilling

Downmilling. With this technique, the relative movement of both systems is the addition of displacements in the Y-axis, the two bodies (tool and wall) tend to move away from each other due to the action of the cutting force. The absolute displacement, where d0 coincides with the radius of the tool, is: dyy Z d0yy C dmyy C dwyy Z d0yy C drelativeyy (10)

Fyyo

Fxxo

Fyyo

Analysing the relationship, In the Y-axis; drelativeyy Z dmyy C dwyy (11) (12)

Fyyo Fxxo Y X

Fxxo Fxxo Fyyo

In the X-axis; drelativexx Z dmxx C dwxx

Upmilling. With this other technique, due to cutting forces that are produced, the bodies tend to move towards each other in the Y axis, and therefore the relative displacement will be sum of the displacements of each component. The absolute displacement is, dyy Z d0yy K dmyy C dwyy Z d0yy K drelativeyy Analysing the relationship, In the Y-axis; drelativeyy Z dmyy C dwyy (14) (15) (13)

w w 0 m

In the X-axis; drelativexx Z dmxx C dwxxm

=0+(w+m)

=0(w+m)

Fig. 2. Representation of forces on the X- and Y-axes, in downmilling and in upmilling, respectively.

systems in the Cartesian X- and Y-axes is analysed. To do this, it is necessary to previously consider the machining technique employed, downmilling or upmilling. Both cases present totally opposed directions of the interaction force, as can be observed in Fig. 2. In both cases: Force in X; Fxx Z jFxx0 j Z j K Fxx0 j Forces in Y; Fyy Z jFyy0 j Z j K Fyy0 j (8) (9)

That is to say, in downmilling, the effect of the forces on the Y-axis causes the workpiece and the tool to move away from each other (Eq. (10)) while in upmilling the effect is just the opposite (Eq. (13)), that is to say, they move towards each other. However, the absolute value of the relative displacement always comes from the sum of both displacements (Eqs. (11), (12), (14) and (15)). 2.1.1. Transfer function of the relative movement. Analysis of the downmilling case In this case, the tool and workpiece tend to split up from each other. The transfer function is obtained from fZd/F. Therefore, the relative transfer function between the tool and the workpiece is the following, dividing displacement by the force: drelative d d d d d d Z mC w Z mC wZ mC w F Fm Fw Fo Fo Fo Fo Replacing in (11) and (12): drelativeyy dmyy dwyy Z C F F F (17) (16)

Milling has been studied without including the nonlinear phenomena due to the tools run-out, the non machined gaps that occur, the thermal effects of the process, etc. From the above relations (8) and (9), the relative displacements are proposed, for both cases of downmilling and upmilling, with the following two restrictions: The tool and workpiece bodies are always in contact with each other.


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drelativexx dmxx dwxx Z C F F F

(18)

In the domain of the frequency the sums of Eqs. (17) and (18) are analysed as: frelativeyy Z fyym C fyyw frelativexx Z fxxm C fxxw (19) (20)

That is to say, the components of the Relative FRF are the addition between the machines FRF and the FRFs of the workpiece for the Y- and X-axis. The resulting FRF will be used for the calculation of the lobe diagram. 2.1.2. Transfer function of the relative movement. Analysis of the upmilling case The case of machining in upmilling is also analysed, when the bodies tend to move towards each other. Operating in similar way to the previous case, dividing the displacement by the force: drelative d d d d d d Z mC w Z mC wZ mC w F F m Fw Fo F o Fo F o Replacing in (14) and (15): drelativeyy dmyy dwyy Z C F F F drelativexx dmxx dwxx Z C F F F Finally, it is possible to propose that: frelativexx Z fyym C fyyw frelativexx Z fxxm C fxxw (23) (24) (21) (22)

amplitude of 0.5!10K6 dB and the workpieces an amplitude of 2!10K6 dB, whilst the relative transfer function is 0.8!10K6 dB (the axis of the frequency is linear, whilst the axis of the FRF module is logarithmic). Consequently, when calculating the lobe diagrams this difference will cause a variation in the form of the diagrams. This will be shown later on, in the section corresponding to experimental validation (Section 4), in a case in which the machine and the workpiece will present very similar dynamic behaviours along the Y axis. 2.3. Calculation of the stability lobes After the development of this relative transfer function, the obtaining of the lobe diagram analytically in the frequency domain is the next step. This calculation is carried out using the theory developed by Altintas for milling in [11,23]. As usual, here the spindle speed (abscises) versus the maximum depth of milling (ordinates) are presented. However, in the proposed methodology, it is applied to only one of the transfer functions, and the calculation is carried out in all the scan of the frequency range. To do this, the transfer function matrix is calculated using the proposed method, where Fxx(iu) y Fyy(iu) are the direct transfer functions on the x and the y-axis while Fxy(iu) Fyx(iu) are the cross-transfer functions: " # Fxx iu Fxy iu Fiu Z (25) Fyx iu Fyy iu Afterwards, the behaviour of the system in the presence of the cutting forces is found, using Fourier, to nally resume the calculation process of the lobe diagram in the following steps: Select a chatter frequency with the transfer functions near to a predominant mode. Resolve the eigenvalues of the equation. Calculate the maximum cutting depth. Calculate the spindle speed for each stability lobe kZ0,1,2,. Repeat the methodology analyzing the chatter frequencies in the whole frequency scan. In the last point, the calculation used in the present study is different to that used by Altintas [11]. Although the computational time is longer, the problem of evaluating very similar and near but different behaviour modes is eliminated in this way.

That is to say, the Relative FRF is analysed as the addition of the machines FRF and the workpieces FRF, for the degrees of freedom in both the X- and Y-direction. This case is similar to the downmilling case. 2.2. Graphical representation of the FRFs In Fig. 3, the difference between the transfer function of the relative movement with respect to the treatment of the machines and workpieces transfer functions as two independent entities seems obvious. Thus, in Fig. 3 the workpieces transfer function (FRFw) is represented in stripes, the machines transfer function (FRFm) in dots and the machine and workpiece assemblys relative transfer function (FRFr) in a continuous line, the latter calculated with the proposed method. The difference in the amplitude of these functions can be observed next to the natural frequencies of both the machine and workpiece. For example, if a frequency of 800 Hz is chosen (see vertical dash line in Fig. 3), the amplitude values are very different, the machines transfer function has an

3. Development in 3D When a monolithic component is machined, the initial rough state (the stiff raw block) becomes the nal nished

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Fig. 3. Comparison of the transfer functions in the Y-axis. Machine, workpiece, and relative. They are semi-logarithmic representations.

state (the weak thin walled component) after several tool passes. Between these two states, the dynamic parameters of the workpiece vary along the milling process, as both mass and rigidity are reduced. Given this, along machining, numerous situations arise in which the dynamic behaviour of the workpiece and the machine can be very similar. Therefore, the evolution of the lobe diagrams along the operation should be studied, giving rise to a threedimensional lobe diagram. The three coordinates involved in this case are: spindle speed, N (rpm), cutting depth, blimit (mm), in end milling is the axial depth of cut ap,

the geometric state of the workpiece along machining. The last value should be parametered, given that the geometry of a workpiece along machining, even in simple cases as a vertical and plane thin wall, is a complex entity. In Table 1, representation of the machining of an aluminium wall with a nal measurement of 150!50! 0.8 mm is shown. This has been obtained from an initial block measuring 150!50!4.8 mm. In this table, four geometric situations at intermediate stages are detailed (gures of the last row of the table), and the values of the rst natural frequencies (mode shapes right aside the table). Thus, step 0 is the starting raw block, step 3 is

U. Bravo et al. / International Journal of Machine Tools & Manufacture 45 (2005) 16691680 Table 1 Representation of modes of four geometric states of the workpiece (last row) (right, modes shape)

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Fig. 4. (Up) 2D Diagram for the successive stages. (Down) 3D lobe diagram, composed of spindle speed, machining stages and maximum depth.

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Table 2 Shear specic coefcients, tangential and radial Cutting coefcients Ktc-tor (N/mm) Ktc-end (N/mm) Krc-tor (N/mm) Krc-end (N/mm) 2668.50 1182.60 1087.40 274.18

Table 3 Dynamic parameters of the machine and workpiece Hz Machine subsystem Mode X 1 650 2 1037.5 3 2025 4 2550 Mode Y 1 675 2 1062.5 Workpiece subsystem Stage 0 1 1225 2 1650 Stage 14 1 425 2 675 3 800 4 1025 N/mm2 x

1.73622!10C8 2.06620!10C8 1.23255!10C9 4.24559!10C9 6.56267!10C7 5.16145!10C8

0.00923 0.00675 0.00568 0.00196 0.01185 0.00800

The FRF of the milled thin walled components has been obtained from impact tests, using a micro-accelerometer (0.8 gr) placed on the base. After the presentation of this table, the lobe diagram of each geometric situation is calculated at these steps 0, 3, 6 and 14. Fig. 4 (up) presents the two-dimensional diagrams for each step; and it can be seen that is difcult to make use of it. For this reason in Fig. 4 (down) the four lobe diagrams are presented in a three-dimensional graph. Here, the geometric state of the wall is introduced as the third dimension. From the analysis of this case, it is demonstrated how the machining parameters indicated in the 3D lobe diagram are more restrictive for the rst geometric stages of the workpiece, since it is here when the workpiece (the wall being studied) has a weaker behaviour in the presence of the dynamic excitation coming from the contact of the tool and workpiece at the top of the wall. This aspect is studied in Section 3.1. 3.1. Geometric inuence of the wall machining stage. Mode discrimination When the dynamic behaviour of a thin wall is studied, it varies at the same time as its geometric state along milling also changes, as has been seen in the previous point. For this reason, the following problem arises: after all the modes of the workpiece are experimentally obtained, not all of these will inuence in the elaboration of the lobe diagram. Thus, if all the modes obtained experimentally are introduced in the analysis, erroneous lobes may result. That is because the experimental modes can be classied in three types:

5.18369!10C6 5.09148!10C6 5.3322!10C5 4.3593!10C5 1.6332!10C7 1.5634!10C7

0.01124 0.01216 0.04824 0.02815 0.01045 0.01552

the wall after three longitudinal milling passes (milling the second z-level), step 6 after six passes (milling the third zlevel), and nally step 14 after 14 tool passes (milling the seventh z-level), in this case part is near to the nal thin wall shape.

Fig. 5. Lobe diagram at stage 0: machine, workpiece, and relative displacement.


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Fig. 6. Detail of the lobe diagram at stage 0, next to 20,000 rpm.

global modes of the entire wall, those of the upper band, that is, the thin wall just machined, and those of the wall not yet milled, that is, the bottom thick wall still to be machined. The low frequency modes are those from the upper band of the thin wall, but these modes do not produce displacements in the place where the tool is just machining (the tool is milling at the bottom of the thin wall). This is the case detailed in the picture of the fourth step shown in the last row of Table 1. In consequence, once the global modes are calculated, they will have to be discriminated, selecting only those modes that produce displacement where the tool is machining. In other case, if all the modes are evaluated in the diagram, lobes will be obtained according to which, the machining of a wall in the last steps (when the wall has less mass and rigidity) would be the most restrictive situation. However, this is not the case, and it has been proven in the experimental analyses shown in the experimental validation (Section 4). Discrimination of modes can be done analysing their shape by means of a nite element analysis; those modes without movement in the milled zone must be rejected.

Machining a thin 150!50!0.8 mm wall, Aluminium 7075 T6, starting with a wall thickness of 4.8 mm. The specic shear cutting coefcients used for the couple tool/material for the end and bull nose (toroidal) mills are in Table 2. The dynamic parameters of the machine and the workpiece in different geometric situations, classied by the steps (stages), are presented in Table 3. From the results obtained, it can be seen in Fig. 5 how a big difference exists between the lobes obtained either considering independent both subsystems or by means of the relative analysis. In this gure, the behaviour of the machine is represented with crosses as an independent entity, the workpiece as an independent entity with triangles and the method analysed here, considering the relative movement, with circles. At this stage 0 of the milling process, at a speed of 20,000 rpm the lobe diagram indicates very different maximum chatter depths, blimit, for each methodology: for the independent workpiece the blimit is 14 mm, and the limit that the machine imposes is 56 mm. In such a way, the result of the superposition would be the less of both, 14 mm.

4. Experimental validation The example case is dened by the following cutting parameters: Radial depth of cut ar 2 mm, upmilling, with a 16 mm diameter carbide tool (K10) with a tip corner radius of 1.5 mm.

Fig. 7. Analysis of sound frequency, at 20,000 rpm and apZ4 mm.

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Fig. 8. Lobe diagram at stage 14, without discriminating modes.

However, with the proposed method, the maximum cutting depth, blimit, is 2.5 mm, which indicates that 14 mm depth will be really a chatter situation. (Fig. 6). After this theoretical study, machining was carried out in these two situations. Thus, in Fig. 7, with 4 mm depth, the Fast Fourier transform (FFT) of the sound collected along the machining of this wall can be observed. All peaks are multiples of the tooth passing frequency, except that one at 1559 Hz. In the lobe diagram for these 20,000 rpm, the maximum machining depth is 2.5 mm (see Fig. 6), and so when machining at this speed at a depth of 4 mm, chatter arises, can be extracted from the analysis at a frequency of 1559 Hz, Fig. 7. This is a vibration frequency close to the second mode of the wall, 1650 Hz. For concluding this experiment, machining was carried out under this depth (2.5 mm), resulting in a stable milling. On the other hand, validation was then carried out in pass 14 (stage 14), where the problem (already mentioned in

Section 3.1) of considering the global workpiece modes can affect the phenomenon. That is to say, when we have a workpiece at the nal stage of nishing, modes exist (the rst ones) that do not produce displacements at the cutting area. If discrimination of modes is not previous performed, all the modes obtained experimentally would be introduced in the theoretical procedure, concluding that chatter would arise at depths greater than 0.4 mm at 15,000 rpm, see Fig. 8.

Fig. 9. Analysis of sound frequency, at 15,000 rpm and apZ7 mm.

Fig. 10. Lobe diagram, step 14, without the non-inuencing modes.


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(a) 1 2 chatter 3 4 5 6 7 (b) 6 no chatter 7

This zone is the base, not machined

Fig. 11. Photographs at several z-level (numbered): (a) chatter at stage 0. (b) No chatter at stage 14 (seventh z-level).

Machining was carried out at this speed and at a depth of cut of 7 mm; from the analysis of frequency it was determined that chatter did not happen, frequency plot is in Fig. 9. That is to say, for the calculation of the lobe diagram we had introduced some modes which did not affect the stability of the system, as they did not product local displacement where machining was being carried out. Finally, the supposed non-inuencing modes, mode 1 at 425 Hz and mode 2 at 675 Hz after the analysis of displacements by Finite Element analysis, were eliminated from the calculation. Therefore, the only ones that cause displacement were modes 3 at 800 Hz and 4 at 1025 Hz, producing the lobe diagram at Fig. 10. It was determined through calculation that chatter did not exist at this depth of 7 mm, since the maximum depth of cut here was 12 mm. Therefore, it explains the experimental evidence. Moreover, with a 12, 5 mm depth of cut milling was unstable, as predicted. To end, in Fig. 11 photographs of the experimental results above commented are presented; in Fig. 11a a machined pass can be observed in which chatter arises at depth of 4 mm. Fig. 11b presents a photograph of pass 14 without chatter, in a pass of ap 7 mm. 5. Conclusions At date, studies have taken into account either the rigid workpiece or the very rigid machine. However, the real case is that along machining the rigidity of the two can be similar. Chatter comes from both the machine and the workpiece. However, at the present state of the art, the phenomenon of both mechanical systems presenting similar dynamic behaviour has not been taken into account. In those

situations, the relative displacement between the two bodies (tool and workpiece) is due to the sum of the tool and part. In the proposed method, a scientic methodology was developed in which the chatter phenomenon is treated from its origin, according to the relationship of the wall/tool interaction, in upmilling and downmilling. Finally, from the experiments carried out, the calculation of the lobe diagram for the case of similar dynamic behaviours has been tested and validated. The lobe diagrams obtained considering only the machine or only the tool are not in touch with reality, only through the consideration of the relative FRF the real region of stability can be obtained. In the proposed method, not only this problem was pointed out, but also the incidence of the workpiece modes that do not affect the process. The FRFs to be introduced in the model were not only those obtained through modal hammer tests, but also the discrimination of the modes that do not inuence signicantly in the cutting area is necessary. The development of a threedimensional diagram is recommend, to show the variation experimented in the lobe diagram along the machining process. This methodology has been applied to a thin wall example, allowing the prediction of the ap (N) conditions that lead to stability/instability states.

Acknowledgements This project has been supported by the Spanish Ministry of Science, from Project number DPI DPI2004-07569-C0201 (Sky-Skin). The support of Interreg EU Aerosn project is also acknowledged. Thanks are also addressed to

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U. Bravo et al. / International Journal of Machine Tools & Manufacture 45 (2005) 16691680 [13] F. Ismail, R. Ziaei, Chatter suppression in ve-axis machining of exible parts, International Journal of Machine Tools and Manufacture 42 (2002) 115122. [14] F. Ismail, R. Ziaei, Monitoring machining chatter using acoustic intensity, World Automation Congress Conference, Maui, Hawaii, USA, 2000. [15] T. Insperger, B.P. Mann, G. Stepan, P.V. Bayly, Stability of upmilling, Part 1: Alternative analytical methods, International Journal of Machine Tools and Manufacture 43 (2003) 2534. [16] T.L. Schmitz, M. Davies, M.D. Kennedy, Tool point frequency response prediction for high-speed machining by RCSA, Journal of Manufacturing Science and Engineering 123 (2002) 700707. [17] S.S. Park, Y. Altintas, M. Movahhedy, Receptance coupling for end mills, International Journal of Machine Tools and Manufacture 43 (2003) 889896. [18] M.A. Davies, J.R. Pratt, B. Dutterer, T.J. Burns, Stability Prediction for Low Radial Immersion Milling, Journal of Manufacturing Science and Engineering 124 (2002) 217225. ` [19] T. Insperger, G. Stepan, P.V. Bayly, B.P. Mann, Multiple chatter frequencies in milling processes, Journal of Sound and Vibration 262 (2003) 333345. [20] S.D. Merdol, Y. Altintas, Multi frequency solution of chatter stability for low immersion milling, Journal of Manufacturing Science and Engineering 126 (2004) 459466. [21] V. Thevenot, L. Arnaud, G. Dessein, G. Cazenave-Larroche, Inuence of material removal on dynamic behaviour of thin walled structure in peripheral milling, Seventh CIRP International Workshop on Modelling of Machining Operations, Cluny, France, 2004. [22] T.L. Schmitz, T.J. Burns, J.C. Zieger, B. Dutterer, W.R. Winfoug, Tool length-dependent stability surfaces, Machining Science and Technology 8 (3) (2004) 377397. [23] Y. Altintas, E. Budak, Analytical prediction of stability lobes in milling, Annals of the CIRP 44 (1995) 357362.

technological centre Fatronik for help in projects O/D Freemach and Prot Perfomill. Also to Dr. Zatarain for his valuable suggestions.

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Documents

Stability Limits of Milling Considering the Flexibility of the Workpiece