G Aguirre

A Iglesias
email: aiglesias@ideko.es

J Muñoa

A Astarloa

J Ciurana

Real milling force based dynamic parameter extraction method

Although chatter prediction stability lobes have been successfully used in many industrial machining process optimizations, currently they lack for enough reliability to apply them intensively in the manufacturing industry. Machine dynamics accurate determination is one of the main sources of errors in current cutting stability models. For this reason, traditional dynamic characterization procedures have been called into question.

In this paper, the milling force excitation modal analysis is proposed, which consists of exciting the structure through the milling force and measuring its response. After working on the measured data, a standard FRF is obtained and the dynamic parameters are extracted in order to feed the stability model.

The obtained dynamic parameters are compared to those of the standard methods, analyzing the deviations and, therefore, quantifying the errors of the traditional dynamic parameter extraction methods.

Introduction

Vibrations are probably the most harmful and complex problem existing in machining processes. Machine tools are mechanical structures that vibrate according to their dynamic characteristics as a result of the forces generated during the machining process. Chatter vibrations are the most critical type of vibrations, since they affect machined workpiece quality and tool life and jeopardize machine integrity. Many efforts have been put into vibration origin and mechanism study in machine tools [START_REF] Tobias | Theory of Regenerative Machine Tool Chatter[END_REF]- [START_REF] Altintas | Manufacturing automation: metal cutting mechanics, machine tool vibrations, and CNC design[END_REF].

Different solutions have been proposed in the literature to increase the stability of the process and suppress chatter vibrations [START_REF] Munoa | Chatter suppression in ram type travelling column milling machines using a biaxial inertial actuator[END_REF]- [START_REF] Bediaga | Reducción de la inestabilidad en cortes interrumpidos en fresado a alta velocidad mediante variación de la velocidad del husillo, XVI Congreso de Máquinas-Herramienta y Tecnologías de Fabricación[END_REF]. Chatter stability lobes are the main tool to predict unstable machining and optimize the process accordingly, however, currently they lack for enough reliability to apply them intensively in the manufacturing industry [START_REF] Stepan | Cylindrical milling tools: comparative real case study for process stability[END_REF]- [START_REF] Brecher | Stability prediction: advances and current restrictions[END_REF].

Stability lobes have been used mainly in aluminum roughing process planning to maximize the Material Removal Rate (MRR). In this case, the stability is associated with the flexibility of the tool, tool holder, or spindle (chatter frequencies between 300 and 4000 Hz). Therefore, the main dynamic properties remain constant at different machine positions and they change with variations in the tool and the tool holder. However, this work is focused on steel heavy duty face milling operations where the critical modes are related to the whole machine tool structure (chatter frequency is usually between 15 and 200 Hz) [START_REF] Munoa | Optimization of hard material roughing by means of a stability model[END_REF]. Considering the shape of the critical modes, the stability lobes cannot be used as effectively as in aluminum case due to inaccuracies [START_REF] Iglesias | Optimisation of face milling operations with structural chatter using a stability model based process planning methodology[END_REF], [START_REF] Brecher | Stability prediction: advances and current restrictions[END_REF]. The source of these problems is usually attributed to nonlinearities of the process or dynamic parameter identification errors. Several authors have emphasized this fact in different research works. Geometry and cutting parameters are considered as reliable information but a high level of uncertainty in the compliance of the machine tool and the stiffness of the process is claimed [START_REF] Brecher | Stability prediction: advances and current restrictions[END_REF].

Nowadays, impact hammer testing is the usual method for dynamic parameter identification in industrial environments. This FRF traditional concept is doubtful since the shape of the response curves vary significantly with the level of force or displacement. When a non-linear system is analyzed, sometimes a shaker is used for excitation, although the time consuming set-up makes it usually not worthwhile. Big differences have been observed in the dynamics of a machine measured with an impact hammer compared to the results obtained by means of a shaker excitation [START_REF] Rasper | Analysis of uncertainties in the stability prediction for milling processes[END_REF]. Certain variability in the mobility when changing the angular position of the spindle was also observed.

On the other hand, hammer or shaker testing is always performed with the idle spindle, which can result in dynamic parameters significantly different from those obtained when the spindle is rotating, or going one step further, when the actual cutting process is taking place. Effects such as the spindle bearing preload change with centrifugal or cutting forces [START_REF] Ozturk | Investigation of spindle bearing preload on dynamics and stability limit in milling[END_REF], the gyroscopic effect [START_REF] Movahhedy | Prediction of chatter in high speed milling including gyroscopic effects[END_REF] or tool and workpiece contact effect [START_REF] Jensen | Stability analysis in face milling operations. Part 2: Experimental validation and influencing factors[END_REF] may have a considerable effect in system's dynamic parameter identification.

In order to overcome these problems, alternative experimental methods can provide more accurate input data. Experimental inverse methods rely on experimental data which are later used as inputs in the stability model. They provide in-process information and can handle process non-linearities if they are conducted under conditions close to the real case.

The alternative experimental methods could improve significantly model input data quality and therefore enhance chatter prediction model reliability. This is of paramount importance, since chatter continues to be nowadays one of the most harmful and most difficult problems to avoid in any kind of machining process.

Some authors [START_REF] Zaghbani | Estimation of machine-tool dynamic parameters during machining operation through operational modal analysis[END_REF] have tried to increase the dynamic knowledge of the critical modes combining impact tests with data obtained from operational modal analysis (OMA). However, the data obtained from OMA is not enough to feed a stability model.

Several authors have used the real cutting force in order to obtain the system's dynamic parameters. Liu, Cheng and Webb [START_REF] Liu | Prediction and simulation on the machining dynamics and instability in peripheral milling[END_REF] measured the force by means of a 3-component dynamometric plate and the response by means of an accelerometer in real-time. The ARMAX module of Simulink (Matlab) was used to perform the experimental modal analysis.

Özsahin, Budak and Özgüven [START_REF] Özsahin | Investigating dynamics of machine tool spindles under operational conditions[END_REF] used real cutting forces in order to obtain the FRF of a spindle. They used two different approaches. First, they obtained the FRF for discrete frequency points varying the cutting speed and measuring the response at the corresponding speed. Secondly, in order to avoid the harmonic content problem of the cutting forces, specially designed workpiece with randomly distributed channels was cut. This way a random excitation of the system was achieved. Both approaches showed significant changes in system dynamic parameters in comparison with the standard hammer test.

The randomly distributed channels for random excitation approach had been previously tested by Opitz and Weck [START_REF] Opitz | Determination of the transfer function by means of spectral density measurements and its application to the dynamic investigation of machine tools under machining conditions[END_REF] and Minis, Magrab and Pandelidis [START_REF] Minis | Improved Methods for the Prediction of Chatter in Turning. Part 1: Determination of Structural Response Parameters[END_REF]. They measured system compliance through random cutting force excitation, they built theoretical stability lobes out of these measurements and finally they correlated them with actual cutting tests. Deviation with respect to the compliance measurements through impact hammer excitation was observed and parameters like preload, direction of preload, feed rate or damping were addressed as the main causes of this deviation.

In this paper the actual milling cutting force will be used as the input excitation to estimate the FRF function. The difference with previous authors is that in this case the cutting force is steadily changed in a frequency sweep excitation. This way the frequency range of interest is excited in a continuous way. In previous works, this technique was used to disturb the regenerative effect and increase the stability of the cutting process [START_REF] Bediaga | Reducción de la inestabilidad en cortes interrumpidos en fresado a alta velocidad mediante variación de la velocidad del husillo, XVI Congreso de Máquinas-Herramienta y Tecnologías de Fabricación[END_REF], [START_REF] Alvarez | Effectiveness of continuous workpiece speed variation (CWSV) for chatter avoidance in throughfeed centerless grinding[END_REF].

Moreover, a rotary dynamometric tool holder is used to measure cutting forces and accelerometers are used to measure the response in different directions. The advantage of this technique is that the cutting is not limited by a dynamometric plate (as it was the case with previous authors), the cutting test can be directly performed over the actual workpiece, exactly in the same conditions as in the real machining. This speeds up the process and makes the procedure feasible for dynamic parameter estimation in an industrial workshop.

Stability model

In this chapter the time domain milling stability model is presented. This model will be used in order to simulate the cutting forces and the theoretical FRFs that would be obtained by means of the milling force sweep technique.

        t t t t p c, F r K r C r M          (1)
The tangential F t,j , radial F r,j , and F a,j cutting forces are calculated according to the linear model [START_REF] Movahhedy | Prediction of chatter in high speed milling including gyroscopic effects[END_REF].

) ( ) , ( ) ( ) ( ) ( j ae re te j j ac j rc j tc a,j r,j t,j g S K K K b h h K h K h K F F F                                                   (2)

K

tc , K rc and K ac are the cutting coefficients, which are proportional to the chip section, whereas the function g(ϕ s ) is defined as:

        s j e j e j s j and g         0 1 ) ( , (3)
where ϕ s and ϕ e are the cutter entry and exit angles, respectively. K te , K re and K ae are the "ploughing" coefficients, which are proportional to the length of the flute (or insert) . S

The tangential F t,j , radial F r,j and axial F a,j cutting forces are projected onto the xyz Cartesian axes and the summation of the contribution of each insert Z of the tool is computed:

                                 1 0 1 0 , , , c , Z j a,j r,j t,j j Z j j z j y j x F F F F F F t   β F (4) where                                sin cos 0 ) cos( cos ) cos( sin ) sin( ) sin( cos ) sin( sin ) cos( , j j j j j j j β (5)
The M, C and K matrices are the mass, damping and stiffness matrices respectively, which are obtained experimentally.

Equation 1 has to be solved by a numerical integration method, due to the impossibility of solving it analytically. The fourth-order Runge-Kutta method is the most used in dynamic system simulation. As the equation is solved numerically, the obtained result is an approximation of the exact solution.

The spindle speed is varied in a linear way, from an angular speed of  1 to  2 . The integration is performed along the time frame of the spindle speed sweep, keeping a constant ϕ angle step. As the speed is variable, the time integration step is also variable:

    t , ( 6
) i total n             1 2 1 , ( 7
)
where ϕ total is the total angle rotated during the sweep and n i is the integration step.

FRF estimation procedure

This work shows a new method to obtain the dynamic parameters in a fast and easy way using the cutting force itself as the input excitation for the FRF calculation.

First of all, the frequency range of interest is defined. The main modes involved in the process stability must be within this range, in order to do a complete dynamic characterization. The cutting speed range is selected accordingly, in such way that the tooth passing frequency is swept over the desired frequency range. The milling process is carried out with a constant slope rotating speed along time (see Figure 2). As the excitation of the main modes is achieved by sweeping the first harmonics of the cutting force over their frequency range, it is important that these harmonics have high energy content in order to have a strong excitation. An interrupted cutting process, with a small width of cut, is therefore used for this purpose.

The cutting force is divided into three components, according to the model derived in chapter 2. The system is regarded as a MIMO system, where multiple inputs and multiple outputs are produced with each cutting speed sweep. The frequency response function H(f) relates these inputs and outputs:

      ) ( ) ( ) ( f F f H f X   (8)
There are several methods to calculate H(f). In this case the H 1 estimate will be used:

FF rF G G H  1 (9)
If a 90º lead angle tool is used, the Z force can be neglected and a 2-input 2-output system is obtained (1 input and 1 response in each direction x and y).

1 1 1 1 1 1 fx fy xy fx fx xx fx x G H G H G     1 1 1 1 1 1 fx fy yy fx fx yx fx y G H G H G     (10)
In the case of 2-input 2-output case (where F z and its response are neglected), the [H(f)] matrix has four unknown terms H xx , H xy , H yx and H yy . Therefore at least an extra cutting test is needed in order to obtain as many equations as unknown terms. It has to be considered that the measurement must be independent from the previous test, modifying the relation of amplitudes or phases between F

x and F y . An easy way to accomplish this condition is changing the cutting strategy from down-milling to up-milling direction. When the cutting direction is changed from down-milling to up-milling direction, F y remains in the same sign whereas F

x changes sign (see Figure 3). With this change, a not linearly dependent system of equations is achieved. Moreover, the amplitude relation also changes slightly from down-milling to upmilling mode. x is opposite sign.

The down-milling and up-milling strategies are conducted according to the sketch depicted in Figure 4.

        a a N i a rr N i a rF f r f r N G f F f r N G 1 1 ) ( ) ( 1 ) ( ) ( 1 (11)
where N a is the number of averages. The coherence function will be also calculated as:

  FF rr rF rF rF G G G G     2  (12)

FRF Simulations

Based on the theoretical time domain model described in chapter 2 and the FRF estimation procedure explained in chapter 3, theoretical FRFs have been estimated from a real milling force sweep.

The system to analyze is a flexible fixture with a dominant mode in each axis x and y (see Figure 8). A hammer test is conducted over the setup in order to extract the modal parameters in the traditional way. The milling sweep is simulated and the FRF estimation procedure is tested and compared with the "static" FRF. The cutting conditions of the milling sweep are described in The milling sweep borders, that is, the rotating speeds between which the simulation is done, are selected according to the frequency range of interest. From the hammer test it is known that the flexible fixture presents two modes at around 71Hz (y axis) and 159Hz (x axis). If the 71Hz mode must be excited, a frequency sweep from 50Hz to 100Hz should be considered. This implies the use of a rotation speed from 1500rpm to 3000rpm, since the excitation is achieved by means of the insert passing frequency (2X). This means that the second harmonic of the insert passing frequency will excite the range between 100-200Hz and therefore, the x axis mode.

In order to have enough equations to solve the system, two independent tests are simulated. The conditions are exactly the same for both tests but the cutting strategy is changed from down-milling to up-milling (see Figure 4).

In order to guarantee the coherence of the system, each test is repeated four times and a frequency averaging of the results is done. Uncorrelated noise is added to the theoretical system with the purpose of simulating a real system in which noise is also recorded. The coherence of the direct FRFs is calculated from the four repetitions (see Figure 5) and, in this way, the useful frequency range is detected. As the harmonic content is basically limited to the insert passing frequency 2X (50-100Hz) and its second harmonic 4X(100-200Hz), the useful frequency range is about 50-200Hz . Out of this range, the harmonic content is very weak and the signal to noise ratio is too small to consider the frequency response function as a reliable value. According to Figures 6 and7, the milling sweep test introduces a certain delay in the frequency peaks related to the main modes. The faster the sweep the bigger this delay is, although the difference is not that significant. The amplitude is also well simulated, so it is concluded that the procedure to obtain the FRF is theoretically valid.

Experimental tests

In this chapter the real cutting force excited system FRF is calculated. The tests described in chapter 4 have been carried out. The simulated slowest milling sweep (T=30s) has been selected in order to obtain the most accurate results. F x and F y have been measured by means of a dynamometric tool holder and two accelerometers have been used to measure the response (see Figure 8). The signal is clearer for the x axis, since its main mode is excited by the fundamental insert passing frequency.

Theoretical versus experimental result comparison

In the theoretical analysis performed in chapter 4, the useful frequency span, as defined by the coherence function, ranged from 50Hz to about 200Hz. In case of the experimental case, however, the useful frequency range was larger due to the tool runout. In an ideal tool case, as defined in the theoretical simulation, the tool presents harmonics only as multiples of the insert passing frequency (2X), whereas in a real case the tool will also show harmonics of the rotating frequency (1X). Ideally the frequency range of interest should be excited through the cutting harmonics, preferably with the first cutting harmonic (insert passing frequency) but the harmonics of the rotation speed might be also used in case of need. The bigger the runout of the tool the higher the energy content of these harmonics will be.

Next From the experimental tests it can be concluded that the FRF obtained from the milling sweep, which is the most accurate according to what it has been exposed in previous chapters, is smaller than the amplitude obtained with the impact hammer. This would imply that the cutting stability achieved through the stability models is underestimated. Nevertheless, the dynamic parameter change is not dramatic, what means that a stability simulation with the dynamic parameters obtained from the hammer test would provide a good approximation for this particular case.

Analyzing the dynamic parameters separately, the overall result for the two modes studied is that the frequency decreases slightly and the stiffness increases slightly. Damping behavior is uneven; a damping increase has been measured for the main mode in y direction, whereas a damping decrease has been observed in the x direction mode. It has to be noticed that mode in y direction is better defined, due to higher harmonic amplitude excitation at this frequency range. For this reason the parameter estimation is more accurate and conclusions about y mode variation are more reliable.

The system under test is a clear dynamic structure with two well defined main modes. It might happen that in a real machine tool mode, where many joints and interfaces are involved in the mode definition, bigger differences in the parameters obtained by these two types of FRF estimators is observed.

Conclusions and final remarks

A new method for dynamic testing in machine tools has been proposed. In some cases, the usual dynamic excitation techniques do not offer enough accuracy for dynamic parameter identification and more realistic system excitation has been sought in order to account for process or machine non-linearities. The objective has been to assume an excitation method in which the system undergoes excitation conditions closer to the real working conditions. With this purpose, the real milling force has been used as input to the system for dynamic characterization. By means of a cutting speed run-up the spectrum of interest is excited by the harmonics of the cutting force, in a milling sweep process.

The method is easy to put in practice, as just several standard cutting tests over a test workpiece are needed. The main problem could be the availability of a cutting force measurement system.

Simulations have been carried out considering the milling force model and the results show a good correlation with the standard FRF. There is however, a small delay in the milling sweep FRF that can be improved but not completely removed decreasing the sweep rate.

It is very important that two independent measurements are carried out in order to obtain two linearly independent relations and solve the four unknown system H xx , H xy , H yx and H yy . Down-milling and upmilling tests are proposed, keeping the rest of cutting conditions unchanged.

The experimental results are coherent and show a good agreement with hammer tests. Slight changes are observed, which are probably related to the cutting process effect on dynamic parameters. In this way, a slight decrease in the main mode amplitude peak has been observed. According to the extracted dynamic parameters a very slight frequency decrease, a very slight damping drop and a more noticeable stiffness increase are observed. The higher accuracy of these new dynamic parameters should be verified by means of a stability check of the process over the analyzed fixture.

The influence of the cutting parameters has to be deeply analyzed as well. The presence of higher or lower milling excitation forces that those used in this study could give raise to variations in the estimated FRF due to non-linear behavior.

Even if the observed changes in the FRFs are not very significant, higher differences are expected in other type of systems, for instance machine tool dynamics, where the higher amount of joints and mechanical elements involved in the modal shape could result in bigger differences between the classical hammer test and the milling sweep test. Machine tool own dynamics study through the newly developed FRF estimation procedure will be carried out in the near future. In this case a three input MIMO system will be considered, since Fz force will be also involved in the main mode description.

Figure 1 :

 1 Figure 1: Milling force calculation layout

Figure 2 :

 2 Figure 2: Cutting speed variation in a T=20s milling sweep test

Figure 3 :

 3 Figure 3: Simulated cutting force comparison of down-milling cut versus up-milling cut. F y keeps the same sign whereas Fx is opposite sign.

Figure 4 :

 4 Figure 4: Milling sweep test layout

Figure 5 :Figure 6 :

 56 Figure 5: Simulated coherence functions of the sweep milling FRFs

Figure 7 :

 7 Figure 7: Hammer test and milling sweep test comparison in y axis

Figure 8 :Figure 9 :

 89 Figure 8: Experimental setup: workpiece on flexible fixture with one dominant mode in each direction x and y

Figure 10 :

 10 Figure 10: Hammer test FRF comparison with milling sweep test FRF in y axis

Table 1 :

 1

	Direction	Down-milling / Up-milling
	Workpiece material	C45k
	Tool	490-032A25-08M
	Inserts	490R-08T308M-PH
	Diameter	32 [mm]
	Number of inserts	2
	Cutting speed range	150-300 [m/min]
	Rotation speed range	1500-3000 [rpm]
	Depth of cut	1 [mm]
	Width of cut	6.4 [mm]

Table 1 :

 1 Cutting conditions of the milling sweep tests

Table 2 :

 2 table compares the dynamic parameters obtained with hammer and with milling sweep tests: Dynamic parameter comparison

		Dynamic parameter	Hammer test	Milling sweep test
		Natural frequency	159.3 [Hz]	157.6 [Hz]
	x axis	Damping	0.66 [%]	0.60 [%]
		Modal stiffness	100.875 [N/µm] 120.738 [N/µm]
		Natural frequency	71.1 [Hz]	71 [Hz]
	y axis	Damping	1.77 [%]	1.94 [%]
		Modal stiffness	32.546 [N/µm]	34.555 [N/µm]


T He

G J Zhang

X Zhao
email: zhaox@mail.neu.edu.cn

Effects of Magnetic Field Strength on Microstructure and Texture Evolution in Cold-Rolled Pure Copper by Magnetic Field Annealing

Keywords: Magnetic field strength, Texture, Recrystallization, Pure copper

Effects of magnetic field strength on the evolution of recrystallization microstructure and texture in cold-rolled pure copper are investigated after annealing at 300℃ in a magnetic field up to 12 T. The average intensity of Cube texture of the annealing specimens with the field is higher than that without the field (0T). As the strength of field increases, the intensity of Cube texture strengthened to a maximum value at 8T, and then slightly decreased. The results may be attributed to boundary migration induced by the magnetic field during grain growth process.

Introduction

There is a nice bit of experimental evidence that the microstructure and texture evolution during heat treatment subsequence to plastic deformation in metallic materials can be affected by an external high magnetic field [1,2]. Most researches in this field are related to magnetically affected recrystallization in ferromagnetic materials [3,4]. By the development of high magnetic field technologies, the effect of a high magnetic field on metallurgical phenomena in non-ferromagnetic materials has been hot researching issues. The results as shown in the past by Molodov [5][6][7] for Zn, Ti and Al polycrystalline can reveal that the recrystallized microstructure and texture can be affected by the external high magnetic field. Their experiments were carried out at various annealing temperatures for various holding time at each temperature with fixed strength of magnetic field when the field was applied. However, some other investigations [8][9][10] suggested that there may be some interesting effect at certain field strengths. On the basis of this argument, the present work studied the effect of magnetic field strength on the evolution of recrystallization microstructure and texture in a cold-rolled pure copper during annealing and reported some new experimental findings.

Experimental

The material used is 98% cold-rolled pure copper of 0.1mm in thickness, with the following chemical composition (wt pct): Fe 0.01; Pb 0.08; Cu 99.9. The dimension of the specimens is 20mm×10mm×0.1mm. The specimens are annealed at 300℃ for 30min with a heating rate of 5℃/min and then cooled in water. For magnetic field annealing, a magnetic field from 4 T up to 12 T (with 2T intervals from 4 T to 12 T) is applied during the isothermal holding process. The specimens are kept in the zero magnetic area with their rolling direction (RD) parallel to the magnetic field direction (MD). For comparison, the non-field annealing is carried out in the same furnace under the same heat treatment conditions. The global textures are obtained from the 1/4 layer of specimen thickness by measuring three incomplete {111}, {200} and {220} pole figures, using the Schulz back-reflection method, with Cu K α radiation. The corresponding ODFs are calculated with LabTex 3.0 software. Microstructures of longitudinal sections of the specimens are observed by orientation imaging microscopy (OIM). The scan is carried out over the area of 613×459 measuring points with a 0.5 μm step size, and the corresponding ODFs are computed using the Channel 5 software.

Results and Discussion

Fig. 1 shows the constant φ 2 ODF sections of the as-rolled specimen which is characterized by three major ODF peaks, spread around the orientations {112}<111> (Cu-), {123}<634> (S-), and {011}<211> (Brass-component), together making up so called β-fibre. Fig. 2 compares the constant φ 2 =45° ODF sections of specimens annealed at 300℃ with different strength of magnetic field. The Cube texture component occurred in all annealing specimens with and without the field (0T). The orientation distributions of the fully recrystallized samples are very similar for both the specimens without the field (0T) and with the field despite the different strength of field. The observed variation in the orientation intensity of Cube texture for all the annealed specimens are shown in Fig. 3. It is obvious that the orientation intensity of Cube texture for the annealing specimens with the field is higher than that without field (0T). The orientation intensity of Cube texture increases with the increasing of magnetic field strength. Their intensities are up to a maximum at 8T and then slightly decrease. This indicates that magnetic field annealing promote the growth of the recrystallized grains with Cube orientation. This promotion effect increases with the field strength increase, especially when the field is 8 T, where the result is more pronounced. Fig. 4 shows the variation of volume fraction for Cube texture component with magnetic field strength. With the increasing of the magnetic field strength, the volume fraction of the Cube texture component increases firstly and then slightly decreases. The maximum value occurred when the field is 8 T. with magnetic field strength of annealed specimens.

Fig. 5 shows the microstructures of longitudinal sections of the specimens annealed at 300℃ for 30min with different magnetic field strength. In these OIM maps, both the rolling direction and the magnetic field direction are horizontal.

There are some grains with different size and shape in the figures instead of deformed structures. This indicated that the recrystallization is essentially complete and grain growth is starting for all the annealed specimens. Most of the big grains in the figures with the same or similar color (red color) are Cube orientation. This indicated that magnetic field promoted the selective growth of the grains with Cube orientation. The quantitative comparison of their recrystallized grain size with magnetic field strength is shown in Fig. 6. The recrystallized grain size in the Fig. 6 counts by their diameter. From the figure one can see that the size of recrystallized grains increases firstly and then decreases slightly with the increase of the magnetic field strength. The peak value still corresponds to the 8 T field. This result once again confirms that the magnetic field promotes the selective growth of recrystallized grains with Cube orientation during annealing. So far, the effect of high magnetic field and their influence on the recrystallization process and texture evolution is not well understood. As pointed elsewhere [11][12][START_REF] Sheikh-Ali | Materials Processing in Magnetic fields[END_REF], one of the mechanisms which can be responsible for the magnetically related effects during grain growth and recrystallization in dia-and paramagnetic materials is boundary migration induced by the differences in magnetic free energy. The magnetic force acting on the boundary of two crystals that have different magnetic free energy can be expressed as follows [START_REF] Molotskii | [END_REF]15]:

  2 1 2 0 2 1 2 P          H (1)
where 1

 and 2  are the susceptibilities of the two adjacent grains, respectively. 1  , 2  are the corresponding magnetic free energy, μ 0 is permeability of vacuum, and H is the field strength. It is known that the difference between 1

 and 2  for pure copper is very small during grain growth process. So, the driving force acting on the boundary of two crystals increases with the increasing of magnetic field strength.

In the present experiment, the results show high magnetic field can promote the growth of the recrystallized grains with Cube orientation. This promotion effect increases with the increasing of magnetic field strength, especially when the field is 8 T, where the result is more pronounced. According to equation ( 1), these results may be attributed to boundary migration induced by magnetic field which effects increase with the field strength increasing. However, it is a very complicated process that includes magnetic field effects as well as the evolution of recrystallization textures under thermodynamics. Further detailed research on this process will be done in the next step. Magnetic Field (T) Fig. 6: Grain diameter of recrystallized grains in annealed specimens with different magnetic field strength.

Conclusion

A cold-rolled pure copper is annealed under a magnetic field with different magnetic field strength. It is obvious that magnetic field annealing promote the growth of the recrystallized grains with Cube orientation. The average intensity of Cube texture of the annealing specimens with the field is higher than that without the field (0T). As the strength of field increases, the intensity of Cube texture strengthened to a maximum value at 8T, and then slightly decreased. The results may be attributed to boundary migration induced by the magnetic field during grain growth process.

Fig. 1 :Fig. 3 : 4 :

 134 Fig. 1: Constant φ 2 ODF sections of Fig. 2: Constant φ 2 =45° ODF sections of annealing specimens cold-rolled pure copper. with different strength of magnetic field. (a) 0T, (b) 4T, (c)6T, (d) 8T, (e) 10T, (f) 12T.

Fig. 5 :

 5 OIM (IPF + Grain Boundary) maps of specimens annealed with different magnetic field strength. (a) 0T, (b) 4T, (c) 6T, (d) 8T, (e) 10T, (f) 12T.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant No. 51201028), the "111" Project (Grant No. B07015), the Fundamental Research Funds for the Central Universities (N130510001) and the Program for New Century Excellent Talents in University (NCET-13-0104).


Bernard Anselmetti

Laurent Pierre
email: laurent.pierre@ens-cachan.fr

Complementary writing of maximum and least material requirements, with an extension to complex surfaces

Keywords: ISO Standard, maximum/least material condition, functional tolerancing, composite specification

Maximum and least material requirements describe the strictly necessary fittability and accuracy functional requirements for an assembly involving connections with clearances. In the ISO 2692:2014 [1] dimensioning and tolerancing standard, the writing of this requirement violates the principle of independency and is limited to features of size. This paper proposes two complementary writings and several explanations for the application of the concepts. In order to make the definitions consistent with those of ISO 1101:2012 [2] standard, the requirements are defined by means of unilateral tolerance zones. For features of size, the dimension of the tolerance zone for the specified surface and for the reference is written directly between brackets in the specification. For all complex surfaces, the tolerance zone is defined by an offset surface of the nominal surface. The offset value is written between braces. The definitions of form, location and orientation specifications with these modifiers are given for simple elements and for a pattern of holes. Composite specifications, which associate orientation and location tolerance zones with respect to the same nominal, are defined. An example with flutter on a primary reference shows that it is no longer possible to use all the degrees of freedom to associate the subsequent references. The use of an orientation plane to deal with unidirectional chains of dimensions is defined. In terms of metrology, the characteristic to evaluate is the margin between the actual surface and the limit surface of the tolerance zone when the tolerance zone on the references is respected. This margin enables one, for example, to determine a capability.

Three applications present an assembly of a mechanism with clearances, a connection with a complex surface and a 3D chain of dimensions at least material which requires a composite specification.

Introduction

The context

In functional dimensioning, maximum or least material specifications should be used systematically for all connections with clearances because they define the strictly necessary requirements.

In practice, many designers are not familiar with the use of these specifications and many metrology computer programs are still unable to verify these specifications in perfect agreement with the definitions. One reason is probably the complexity of ISO 2692:2014 standard [START_REF]Geometrical product specifications (GPS) -Geometrical tolerancing -Maximum material requirement (MMR), least material requirement (LMR) and reciprocity requirement (RPR)[END_REF], which presents the concepts through the notion of virtual boundary along with very complex rules due to the violation of the principle of independency. Under very strict criteria, the dimension of the virtual boundary depends on the nominal diameter, the dimensional tolerance, the type of surface, the modifier and, possibly, the tolerance on the element of reference. Such information is very difficult to obtain in the numerical continuity context, especially for the calculation of 3D chains of dimensions and for metrology.

Figure 1, borrowed from figure A.13 of 2692:2014 standard, gives a schematic view of the rules for calculating the diameters of the virtual states on the toleranced element and the datum. Six values must be collected in four specifications.

The definition is limited to features of size, whereas many connections with clearances are created from more complex features. Very often it is unnecessary to verify the local dimensions when the maximum or least material specifications are met. This paper develops a more direct and complementary type of writing which satisfies the principle of independency and sets some rules for the application of the specifications.

The scientific context

The principle of maximum or least material specifications by virtual boundary are defined in ISO standards 2692 (1988) and in ASME standard (1994). This virtual boundaries are used as virtual gauges since the early 1990's by [START_REF] Jayaraman | Geometric tolerancing: 1. virtual boundary requirements[END_REF][START_REF] Etesami | Position tolerance verification using simulated gaging[END_REF]. Robinson uses maximum material parts among assembly specifications, tolerance specifications and assembly tolerance analysis [START_REF] Robinson | Geometric tolerancing for assembly with maximum material parts[END_REF].

Some mathematical models can use the condition of the virtual boundary as T-maps [START_REF] Davidson | A New Mathematical Model for Geometric Tolerances as Applied to Round Faces[END_REF], domains [START_REF] Giordano | Mathematical representation of tolerance zones[END_REF], analysis lines [START_REF] Anselmetti | Generation of functional tolerancing based on positioning features[END_REF] and polytopes [START_REF] Pierre | Variational tolerancing analysis taking thermomechanical strains into account: application to a high pressure turbine[END_REF]. These different methods are applied to the tolerance analysis. Some papers confront these different methods [START_REF] Mansuy | Comparison of two similar mathematical models for tolerance analysis: T-Map and deviation domain[END_REF][START_REF] Pierre | Comparison of analysis line and polytopes methods to determine the result of a tolerance chain[END_REF]. Muthy uses simplex method for metrology [START_REF] Murthy | Minimum zone of evaluation surfaces[END_REF] Pairel and al. have exposed a conceptual model of "virtual fitting gauges" [START_REF] Pairel | The gauge model: a new approach for coordinate measurement[END_REF][START_REF] Pairel | Virtual Gauges Representation for Geometrical Tolerances in CAD-CAM Systems[END_REF]. In this model, they can exploit maximum and least material requirement. They developed some algorithms implemented in a conventional software package for metrology. In [START_REF] Pairel | Three-dimensional verification of geometric tolerances with the "fitting gauge" model[END_REF][START_REF] Pairel | Virtual gauge representation for geometric tolerances in CAD-CAM systems[END_REF], a presentation of the usability of this software is done in a pattern of holes.

Dantan and al. define the gauge with internal mobilities to limit the geometrical variations of the part [START_REF] Dantan | Functional and product specification by Gauge with Internal Mobilities[END_REF]. The permissible geometrical variations are compared to the worst geometry of its environment. [START_REF] Ballu | Virtual gauge with internal mobilities for verification of functional specifications[END_REF] use virtual gauges with internal mobilities to verify the maximum material and least material requirements.

Anselmetti uses these principles for functional tolerancing in assembling and for 3D tolerance stack-up [START_REF] Anselmetti | Synthesis of tolerances starting from a fuzzy expression of the functional requirements, Geometric Product Specification and Verification: Integration of functionality[END_REF].

Definition of the Ⓜ and Ⓛ tolerance zones

The tolerance zone for a cylinder

The new writing is simply to place in the specification the diameter of the virtual state of the specified surface and of datum. The meaning is exactly the same as classical writing Figure 1. This writing is new for ISO standard but is described in ASME Y14.5 2009 standard (section 4.1)

For a dimension, the diameter D of the tolerance zone is followed by Ⓜ or Ⓛ. For a specification, the diameter is placed between brackets after a Ø symbol. The unilateral tolerance zone is bounded by a cylinder of diameter D.

The maximum material tolerance zones are shown in figure 2:

 For a shaft, the tolerance zone is inside the cylinder of diameter D.  For a hole, the tolerance zone is outside the cylinder of diameter D. The least material tolerance zones are shown in figure 3:

 For a shaft, the tolerance zone is outside the cylinder of diameter D.  For a hole, the tolerance zone is inside the cylinder of diameter D.

Example

The maximum material is a means to guarantee the fittability of two parts by specifying a boundary between the elements (figure 4). The values of D and d must be chosen, depending on the desired minimum clearance, such that Min. clearance = Dd Using modifier Ⓜ, the writing on the drawing is immediate (figure 5).

The tolerance zone for two parallel planes facing each other

Modifiers Ⓜ or Ⓛ indicate that the specified element is the set of the two lateral sides which are identified by the arrows. For a dimension, the size D of the tolerance zone is followed by the modifier. For a specification, the dimension is placed between brackets without a Ø symbol. The tolerance zone is bounded by two planes at a distance of D.

The maximum material tolerance zones are shown in figure 6:

 For a tenon (a solid part), the tolerance zone is inside the two planes at a distance of D.  For a slot (a hollow part), the tolerance zone is outside of the two planes at a distance of D. The least material tolerance zones are shown in figure 7:

 For a tenon (a solid part), the tolerance zone is outside of the two planes at a distance of D.  For a slot (a hollow part), the tolerance zone is inside the two planes. The modifier Ⓜ or Ⓛ indicates that the toleranced element is the surface. The normal n to the surface points outside the material. The value of the offset deviation d is placed between braces without a Ø symbol. The tolerance zone is bounded by an offset surface with a deviation d (figure 8):

 At maximum material condition, the direction of the offset d follows the normal n to the surface. The surface at maximum material condition must be outside of the material.  At least material condition, the direction of the offset d is the opposite of the normal n to the surface. The surface at least material condition must be inside the material.

In figure 8, the offsets d are positive. A negative offset is designated, for example, as {-0,02} Ⓜ. The superposition of two specifications with different offsets at Ⓜ and at Ⓛ defines a tolerance zone which is not centered on the nominal surface.

Complementary definitions

Specification of a pattern

A pattern is identified by the indication Nx above the tolerance frame. The pattern is a set of N surfaces with N tolerance zones. For a pattern made up of identical surfaces, a single arrow connects the tolerance frame to one of the surfaces. If the surfaces in the pattern are different, one arrow is needed per type of surface. Each element of the pattern has its own tolerance zone (figure 9): There are N tolerance zones. The modifier CZ is not implicit.  (a) Location: each zone is centred on the nominal surface.  (b) Orientation: each zone is defined centred on the nominal surface, then translated independently of the other zones.  (c) Orientation in common zone: each zone is defined centred on the nominal surface, then translated as one common zone.  (d) Form: each zone is free with respect to its nominal surface and independent of the other zones.

Composite specification

A composite specification imposes a double condition on the specified surfaces by defining the location and orientation tolerance zones with respect to the same nominal surfaces. Where applicable, the nominal model is positioned with respect to the datum system in such a way that, if possible, the actual surfaces lie within all the tolerance zones simultaneously. One composite specification is represented on two lines, with no separator between the specification symbols and with a single datum system where applicable.

A composite specification restrains the orientation defects of a surface to within the location tolerance zone, even if the datum system leaves residual freedoms. The objective is to control the influence of the overhangs in calculating the resultant of a threedimensional chain of dimensions.

Figure 10 contains a composite specification and a form specification. The composite specification requires that the three actual cylinders be simultaneously in the Ø20.2Ⓛ location tolerance zones and in the Ø20.1Ⓛ orientation tolerance zones with respect to the same nominal model. The straightness specification is independent of the composite specification and imposes three independent tolerance zones Ø20.02Ⓜ.

Datum at maximum material condition

A datum at maximum material condition is identified by Ⓜ at the right of the datum. The real toleranced surface and the real datum surface must be simultaneously inside their tolerance zone. The specification is not consistent if this condition cannot be met, included on datum surface. The corresponding association method is obtained in two steps: (ei is negative if the point is inside the boundary)

Step 1: For each point Ai A, eAi≤0, max(|eAi|) minimal.

Step 2: For each point Bi B, eBi≤0 and for each point Ci C, eCi≤0, max(eCi) minimal.

Datum at least material condition

Figure 12, corresponds to inspection of specification (4) in figure 17 with a functional approach. A part holder simulates the maximum material boundary corresponding to the surface B. A gauge simulates the part at least material sizes.

A probe is set up to "0" on the gauge in contact with part holder. This probe demonstrates that both least material conditions on B and on D are met with real for all orientations of part into the part holder. This method is equivalent to an extended tolerance zone Ø20.19 for the hole. This method is equivalent to a tolerance zone extended in each point with the floating of the gauge in this point. In this simple case, the tolerance zone of the hole becomes Ø20. [START_REF] Anselmetti | Synthesis of tolerances starting from a fuzzy expression of the functional requirements, Geometric Product Specification and Verification: Integration of functionality[END_REF]. The association method is obtained in two steps:

Step 1: For each point Ai A, eAi≤0, max(|eAi|) minimal

Step 2: For each point Bi B, eBi≤0. For each point Di D, eDi is minimized and he condition is eDi  0.

Specification with an orientation plane

For a unidirectional chain of dimensions going through a cylindrical connection, the specification must contain an orientation plane which is perpendicular to the direction of the chain of dimensions. Specification (2) of figure 13 imposes a Ø18Ⓛ cylindrical tolerance zone at least material condition. Orientation plane P is defined in the nominal model. It indicates that it is sufficient for the tolerance zone to be respected on the two generatrices of the cylinder in the plane perpendicular to the orientation plane.

The margin on the maximum material specification

At maximum or least material condition, the specification is met if the tolerance zone is respected. The margin is obtained by maximizing the smallest deviation between the actual specified surface and the limit of the tolerance zone. The specification is met if the margin is positive.

For a specification with a datum at maximum or least material condition, the margin on the datum must also be positive or zero. If the condition on the datum fails, the defect can be characterized by the margin on the datum and the margin on the toleranced element, ignoring the modifier on the datum.

Figure 14a contains two specifications. The margin on location (1) is m1 between the actual holes and the limit of the Ø10.1Ⓜ tolerance zones (figure 14b). The margin on location (2) is defined when the holes B respect the tolerance zone on the datum. Margin m2 is defined between the actual hole and the limit of the Ø29.9Ⓛ tolerance zone (figure 14c). Figure 14 -The margin at maximum/least material condition When measuring a set of parts produced on one machine tool, if m and  are respectivelythe average and the standard deviation of the margins the capability of the process is:

3 m Cpk   (1)

Applications

Dimensioning of a mechanism

The mechanism shown in figure 15 imposes a height requirement X at the end of the shaft with respect to the datum A of the body. The cylindrical connections have some clearance. Figure 16 shows the dimensioning of the shaft. Specification (1) concerns the composed surface A. In order to ensure fittability, the two actual cylinders must be positioned, if possible, within two coaxial cylinders Ø30Ⓜ and Ø20Ⓜ. To meet requirement X, coaxiality (2) imposes on the axis of the actual cylinder a tolerance zone Ø0.2, centered on two coaxial cylinders Ø29.9Ⓛ and Ø19.9Ⓛ which must lie within the material of the actual cylinders A.

The general tolerancing is prescribed by specification (3), which must contain no modifier.

Calculation of a 3D tolerance stack-up

The mechanism shown in figure 19a includes a slide which is supposed to move along the two shafts. One must determine the minimum clearance between the shafts and the slide. The dimensioning of the body is shown in figure 19b. The deviation e is maximum when the holes respect the orientation tolerance zone Ø20.1Ⓛ while reaching the limit of the position tolerance zone Ø20.2Ⓛ. Without composite specification, orientation tolerance zones are not parallel to position tolerance zone.

The worst-case deviation is: e = (DLds)/2 + (DO-ds).L/E

Conclusion

The specifications illustrated in figure 16 and figure 17 correspond to exactly the same definitions as in the ISO 2692:2014 standard, except that the dimensions of the tolerance zones are written directly without requiring the local dimensions to be controlled, which is unnecessary because these are already verified through the specifications at Ⓜ and Ⓛ.

Thus, this new writing is complementary to the current writing and does not generate any conflict. The additional information concerning the management of the primary flutter and the use of orientation planes can be applied to classical specifications. This

Figure 1 -

 1 Figure 1 -Dimensions of the maximum material virtual states

Figure 2 -

 2 Figure 2 -The maximum material tolerance zone for a cylinder

Figure 3 -

 3 Figure 3 -The least material tolerance zone for a cylinder

Figure 4 -

 4 Figure 4 -Fittability with two pins

Figure 5 -

 5 Figure 5 -Maximum material dimensioning

Figure 6 -

 6 Figure 6 -The maximum material tolerance zone for two planes

Figure 7 -

 7 Figure 7 -The least material tolerance zone for two planes

Figure 8 -

 8 Figure 8 -The maximum/least material tolerance zone for a surface

Figure 9 -

 9 Figure 9 -Specification of a pattern of holes

Figure 10 -

 10 Figure 10 -A composite specification for a pattern of holes

Figure 11

 11 corresponds to inspection of specification (1) figure 17. The sizes of the part holder are directly defined by tolerancing. Real part must be placed inside the part holder.

Figure 11 -

 11 Figure 11 -Inspection with maximum material condition

Figure 12 -

 12 Figure 12 -Inspection with least material condition

Figure 13 -

 13 Figure 13 -A tolerance zone with an orientation plane

Figure 15 -

 15 Figure 15 -The mechanism considered

Figure 16 -

 16 Figure 16 -Tolerancing of the shaft

Figure 17 -

 17 Figure 17 -Tolerancing of the flange

Figure 18 -

 18 Figure 18 -Assembly with a complex surface

Figure 19 -

 19 Figure 19 -3D tolerance stack-up

writing simplifies the dimensioning process considerably. It improves its readability and facilitates its decoding by analysis, tolerancing and metrology software.

The complete dimensioning requires a maximum material specification to guarantee the fittability of each interface and a least material specification for each link between two interfaces. The extension to complex surfaces opens multiple possibilities.

This proposal could lead to an amendment to ISO 2692 standard which would not alter the body of the current norm.