Claudiu F Bisu
email: cfbisu@gmail.com

Jean-Yves K'nevez²

Philippe Darnis

Raynald Laheurte²

Alain Gérard²
email: alain.gerard@u-bordeaux1.fr

International Journal of Material Forming New method to characterize a machining system: application in turning

Keywords: experimental model, displacement plan, self-excited vibrations, turning} WTM Workpiece-

Many studies simulates the machining process by using a single degree of freedom springmass sytem to model the tool stiffness, or the workpiece stiffness, or the unit toolworkpiece stiffness in modelings 2D. Others impose the tool action, or use more or less complex modelings of the efforts applied by the tool taking account the tool geometry. Thus, all these models remain two-dimensional or sometimes partially three-dimensional. This paper aims at developing an experimental method allowing to determine accurately the real three-dimensional behaviour of a machining system (machine tool, cutting tool, toolholder and associated system of force metrology six-component dynamometer).

In the work-space model of machining, a new experimental procedure is implemented to determine the machining system elastic behaviour. An experimental study of machining system is presented. We propose a machining system static characterization. A decomposition in two distinct blocks of the system "Workpiece-Tool-Machine" is realized. The block Tool and the block Workpiece are studied and characterized separately by matrix stiffness and displacement (three translations and three rotations). The Castigliano's theory allows us to calculate the total stiffness matrix and the total displacement matrix.

A stiffness center point and a plan of tool tip static displacement are presented in agreement with the turning machining dynamic model and especially during the self induced vibration. These results are necessary to have a good three-dimensional machining system dynamic characterization (presented in a next paper).

Introduction

Metal cutting is one of the most important manufacturing process. The most common cutting processes are turning, milling, drilling and grinding.

During the cutting process of different materials, a whole of physic-chemical and dynamic phenomena are involved. Elasto-plastic strains, friction and thermal phenomena are generated in the contact zone between workpiece, tool and chip. These phenomena are influenced by: physical properties of the material to be machined, tool geometry, cutting and lubrication conditions, and also the machining system dynamic parameters (stiffness, damping). The machine tool vibrations are generated by the interaction between the elastic machining system and the cutting process. The elastic system is composed of: the different parts of the machine tool in movement, the workpiece and the tool. Actions of the machining process are usually forces and moments. These actions also generate relative displacements of elements composing the elastic system. They occur for example between the tool and workpiece, the tool device and bed, etc. These displacements modify the cutting conditions and in the same way the forces. Thus, the knowledge of the machining system elastic behaviour is essential to understand the cutting process [START_REF] Cano | A new approach to identifying the elastic behaviour of a manufacturing machine[END_REF].

Certain scientists developed a finite element beam model of spinning stepped-shaft workpiece to perform stability analysis using Nyquist criterion [START_REF] Wang | Stability analysis of spinning stepped-shaft workpieces in a turning process[END_REF]} or the traditional stability lobe diagram [START_REF] Ganguli | Regenerative chatter reduction by active damping control[END_REF][START_REF] Lapujoulade | Cutting forces evaluation during high speed milling[END_REF]. This traditional stability analysis technique shows that the chatter instability depends on the structural damping in the system and the spindle speed.

Chen and Tsao presented a dynamic model of cutting tool with [START_REF] Chen | A stability analysis of turning tailstock supported flexible work-piece[END_REF] and without tailstock supported workpiece using beam theory [START_REF] Chen | A stability analysis of regenerative chatter in turning process without using tailstock[END_REF]. Here, the effects of workpiece parameters are studied on the dynamic stability of turning process by treating the workpiece as a continuous system. Carrino et al., [START_REF] Carrino | Dimensional errors in longitudinal turning based on the unified generalized mechanics of cutting approach. Part I: Three-dimensional theory[END_REF] present a model that takes into account both the workpiece deflection and the cutting force between tool and workpiece. The three components of the cutting force are function of the cutting geometry. The effect of the workpiece-tool-machine deflections is a shift of the workpiece cross-section and a moving back of the tool holder in the radial and the tangential direction (2D model).

In these processes, the cutting forces measurement has important and tremendous applications within industry and research alike. The cutting forces estimation allows to supervise tool wear evolution [START_REF] Toh | Static and dynamic cutting force analysis when high speed rough milling hardened steel[END_REF], establishes material machinabilities, optimizes cutting parameters, predicts machined workpiece surface quality and study phenomena such as chip formation or vibrations appearance. Sekar and Yang propose a compliant two degree of freedom dynamic cutting force model by considering the relative motion of workpiece with cutting tool. Tool and workpiece are modelled as two separate single degree of freedom spring-mass-damper systems [START_REF] Seka | Stability analysis of turning process with tailstock-supported workpiece[END_REF].

In the literature, there are many studies concerning the cutting force measurement. Many dynamometers for this purpose have been developed [START_REF] Buyuksagis | Analysis of circular marble sawing using a block-cutter[END_REF][START_REF] Castro | Correction of dynamic effects on force measurements made with piezoelectric dynamometers[END_REF][START_REF] Couétard | Caractérisation et étalonnage des dynamomètres à six composantes pour torseur associé à un système de forces[END_REF][START_REF] Korkut | Design and manufacturing of a dynamometer connected to computer which can do measuring with strain gages on the lathe[END_REF].

Independently of the machining operation type, methods for cutting force measurements can be divided into two general categories. The first corresponds to the category that uses the current or the voltage signals emitted by the tool machine drive motor or control systems [START_REF] Mehdi | Dynamic behavior of thin wall cylindrical workpiece during the turning process, Part 1: Cutting process simulation[END_REF]. The second uses transducers mounted on the tool or the workpiece assemblies [START_REF] Karabay | Design criteria for electro-mechanical transducers and arrangement for measurement cutting forces acting on dynamometers[END_REF][START_REF] Toulouse | Contribution à la modélisation et à la métrologie de la coupe dans le cas d'un usinage tridimensionnel[END_REF][START_REF] Yaldiz | A dynamometer design for measurement the cutting forces on turning[END_REF].

The cutting forces developed in machining operations may be estimated indirectly by obtaining the power consumed or directly from metal cutting dynamometers; mechanical, hydraulic, pneumatic or several types of electro-mechanical dynamometers.

Knowing the cutting forces is essential to machine tool builders in calculating power requirements and frame rigidity. Cutting forces acting on the tool must be measured at the design tool that are strong enough to remove chip at the desired quantity from the workpiece and to calculate power of tool driver system. The dynamometer is able to measure three force components: cutting force (F c ), feed force (F a ) and radial force (F p ) but not the torque at the tool tip. Axinte et al., [START_REF] Axinte | Evaluation of cutting force uncertainty components in turning[END_REF] propose a procedure to account for both calibration and process errors in the uncertainty estimation for the specific situation of single cutting force measurements. The influence parameters considered in their work, contribution to the measurement uncertainty, workpiece, tool and machine were not considered. Perez et al., [START_REF] Pérez | Estimation of cutting forces in micromilling through the determination of specific cutting pressures[END_REF] give a mechanistic model for the estimation of cutting forces in micromilling based on specific cutting pressure. The model includes three parameters which allow to control the entry of the cutter in the workpiece. The errors in the radial and the angular position of the cutting edges of the tool is considered. The cutting forces are calculated on the basis of the engaged cut geometry, the undeformed chip thickness distribution along the cutting edges, and the empirical relationships that relate the cutting forces to the undeformed chip geometry. This relation do not take into account the elasticity of the machining system. In the measurement of the cutting forces [START_REF] Seker | Design and constrution of a dynamometer for measurement of cutting forces during machining with linear motion[END_REF], only elastic deflections of the cutting tool due to the cutting forces were measured by means of the load cells located at suitable position on the cutting tool.

However the dynamometer can measure three perpendicular cutting force components and three torque components simultaneously during turning, and the measured numerical values can be stored in computer by data acquisition system [START_REF] Couétard | Caractérisation et étalonnage des dynamomètres à six composantes pour torseur associé à un système de forces[END_REF]. This dynamometer was designed to measure up to 5,000 N maximum force and 350 N/m torque. The system sensitivity is pm 4 % in force and pm 8 % in torque.

During the cutting process, the cutting tool penetrates into the workpiece due to the relative motion between tool and workpiece and the cutting forces and torques are measured on a measuring plane in the Cartesian coordinate system. The cutting forces have been measured by the dynamometers designed for different working principles as strain gauge based [START_REF] Couétard | Caractérisation et étalonnage des dynamomètres à six composantes pour torseur associé à un système de forces[END_REF]. Thus it is necessary to have a good methodology to measure the workpiece-tool-machine rigidity before measuring forces and torques. This new methodolgy is precisely the purpose of this paper. In section 2 we present the experimental device. After (see section 3) we conceive the workpiece. The workpiece geometry and dimensions retained for these testtubes were selected using the finite element method coupled to an optimization method, by SAMCEF  software. In the following section 4 a methodology based on the virtual work (three translations and three rotations) is exhibited to study the static aims and to characterize the static equivalent stiffness values in order to identify the three-dimensional elastic behaviour of the machining system. The applied efforts are quantified with a force sensor. The torsor of small displacements, 3 linears and 3 rotations displacement, is measured by six displacement transducer. A stiffness global matrix is deduced with its various results. The sum of the two stiffness matrix displacements block tool and block workpiece determines the stiffness matrix of machining system displacement. By the Castigliano's theorem we determine the angle that characterizes the principal direction of deformation. Before concluding, in section 5 the stiffness center is obtained using the least squares method in the coordinate system based on the tool in O point that is the origin of the coordinate system.

General points

Today, machines tool are very rigid there are less and less geometrically faulty. The vibratory problems are strongly related to the cutting. Ideally, cutting conditions are chosen such that material removal is performed in a stable manner. However, sometimes chatter is unavoidable because of the geometry of the cutting tool and workpiece. In [START_REF] Cardi | Workpiece dynamic analysis and prediction during chatter of turning process[END_REF] the bulk of the motion during chatter comes from the workpiece since it has a static stiffness that is up to 3.2 times less than the cutting tool. Since it is highly impractical to instrument the workpiece during cutting the end goal is to develop an observer that can transform measurements made at the cutting tool into a prediction about the motion of the workpiece. Dassanayake [START_REF] Dassanayake | On nonlinear cutting response and tool chatter in turning operation[END_REF] approaches in the 1D case the dynamic response of the tool holder to the request of the tool which follows a regenerative surface. They consider only tool motions and disregards workpiece vibration. Insperger continues in the 2D case keeping workpiece rigid but he takes into account the flexibility of the tool [START_REF] Insperger | Criticality of Hopf bifucation in state-dependent delay model turning processes[END_REF]}. For an operation of milling [START_REF] Salgado | Evaluation of the stiffness chain on the deflection of end-mills under cutting forces[END_REF], the deflections of the machine-tool, the toolholder and the toolholder clamping in the spindle, the tool clamping in the toolholder, and the tool itself, were measured experimentally under the effects of known forces. The results of this study show that the stiffness of both the machine and the clamping in the machine-spindle-toolholder-tool system have a similar importance in the displacement of the tool tip (subjected to a cutting force) to the deflection of the tool itself. Thus, it is necessary to identify the elastic behaviour of machine parts [START_REF] Cano | A new approach to identifying the elastic behaviour of a manufacturing machine[END_REF]. These vibrations are generated and self induced by the cutting process. A conventional lathe with high rigidity is used to study these dynamic phenomena. The Workpiece-Tool-Machine (WTM) system is presented on the figure 1 for a turning operation. Our experimental approach is based on the matrix development that is presented in [START_REF] Pestel | Matrix methods in elastomechanics[END_REF]. To identify the WTM system static behaviour, the machining system is divided into two blocks, the block Tool (BT) and the block Workpiece (BW) figure 2. These two blocks are related to the turn bed supposed to be infinitely rigid. The workpiece geometry and his holding fixture are selected with D 1 = 60 mm, D 2 = 120 mm and L 2 = 30 mm (cf. figure 4). These dimensions retained for these test-tubes were selected using the finite element method coupled to an optimization method by SAMCEF  software. It is necessary to determine the holding fixture length L 1 to obtain a significant stiffness in flexion. The objective is to move away the first BW vibration mode of the lathe fundamental natural vibration mode (see [START_REF] Bisu | Etude des vibrations auto-entretenues en coupe tridimensionnelle: nouvelle modélisation appliquée au tournage[END_REF]).

Fig. 4 Geometry of holding fixture / workpiece.} 8

As [START_REF] Salgado | Evaluation of the stiffness chain on the deflection of end-mills under cutting forces[END_REF] and others the stiffness is calculated on the basis of the displacement  for a given force P :

= I E L P   3 3 (1)
with inertial moment :

I = 64 4 1 D   (2)
The figure 5 represents the displacements and stiffness values relating to the length of holding fixture / workpiece, for a force P = 1,000 N, a Young modulus E = 21.10 -5 N/mm 2 and a holding fixture diameter D 1 = 60 mm. A holding fixture length : L 1 = 180 mm, for a stiffness in flexion of 7.10 -7 N/m, is reminded. This value is including in the higher part of the interval of the acceptable rigidity values for conventional lathe (cf. figure 6), [START_REF] Ispas | Vibrations des systèmes technologiques[END_REF][START_REF] Koenigsberger | Machine Tools Structures[END_REF][START_REF] Konig | Zweikomponenten schnittkraftmesser[END_REF].

Block Tool: BT

In this case, the BT part includes the tool, the tool-holder, the dynamometer, the fixing plate on the cross slide (figure 7a). The six-component dynamometer [START_REF] Couétard | Caractérisation et étalonnage des dynamomètres à six composantes pour torseur associé à un système de forces[END_REF] is fixed between the cross slide and the tool-holder. This is necessary thereafter to measure the cutting mechanical actions. The stiffness of BT is evaluated into the next section.

Static characterization of machining system

The static study aims at characterizing the static equivalent stiffness values in order to identify the three dimensional elastic behaviour of the machining system. Generally, the static tests consist in charging by known efforts the two blocks and measuring only the associated displacements components [START_REF] Gorodetskii | A system for experimental studies of the dynamics of the process of cutting metal[END_REF][START_REF] Yaldiz | Design, development and testing of a turning dynamometer for cutting force measurement[END_REF]. Here, also the static tests consist in loading by known efforts the two blocks but measuring the small displacements torsor (i.e. three linear displacements, and three rotations). The applied efforts are quantified with a force sensor. The small displacements torsor is measured by six displacement transducers. A stiffness global matrix is deduced with its various results. It is a real 3D pattern. For instance, Carrino et al., present a model that takes into account both the workpiece deflection and the cutting force between tool and workpiece. The three components of the cutting force are a depend on the cutting geometry. The effect of the workpiece-toolmachine deflections is a shift of the workpiece cross-section and a moving back of the tool holder in the radial and the tangential direction (2D model) [START_REF] Carrino | Dimensional errors in longitudinal turning based on the unified generalized mechanics of cutting approach. Part I: Three-dimensional theory[END_REF].

Stiffness matrix

The experimental approach is based on the matrix development presented in [START_REF] Robinson | Analyse matricielle des structures à l'usage des ingénieurs[END_REF]. The deformation of a structure element is represented by displacements of nodes determining this element. The "associated forces" correspond to displacements which act as these nodes.

The transformation matrix which connects generalized displacements of an element to the "associated forces" is the rigidity matrix or the stiffness matrix of the element. In the same way the matrix which connects generalized displacements of the structure to the applied generalized discrete forces is the stiffness matrix of the structure simply named as "stiffness matrix" : [K].

The relation between forces and displacements is given by [START_REF] Pestel | Matrix methods in elastomechanics[END_REF]:

{T} = [K] x {D} (3)
where {T} represents the mechanical action torsor, [K] the stiffness matrix and {D} the small displacements torsor.

The general form of the square (6 x 6) stiffness matrix [K] is:

[K] A, xyz = xyz A CF C F FC K K K K ,       (4)
where [K F ], [K C ], [K CF ] and [K FC ] are respectively square (3 x 3) displacement matrix, rotation and rotations / displacement and displacement / rotation expressed at the point A in x, y, z machine axes.

Experimental determination of the stiffness matrix

The matrix elements of the small displacement torsor are identified thanks to the experimental device presented in the figure 8. The considered system is a cube. Displacements are measured by six displacement transducers. Two displacement transducers are positioned symmetrically on each of the 3 directions. The force is applied to each x, y, z direction in two different levels by a screw-swivel system controlled by a force sensor. Each loading point coordinates are known starting from the cube center O c . This allows for each applied force to determine the moment and thus the complete torsor of mechanical actions {T}. Fig. 8 Experimental device for the static characterization.

Induced displacements are solid body displacements and it is noted that rotations are low (> 10 -5 rad) but exist. The existence of these rotations is important and in agreement with the torque via the virtual work theory. The measure principle is presented in the figure 9 and is used to determine the components of the small displacements torsor {D} which is composed by the three rotations  x ,  y ,  z and the three displacements  x ,  y ,  z . (5)

From these relations and considering the six measurements points it results: The tests are carried out with specific assemblies which are designed for each direction of measurement. The loading (respectively unloading) is carried out by step of 30 daN (resp. -30 daN) until (resp. from) level of 200 daN, and this procedure is used for each test following known directions. To check the repeatability and accuracy of identifications all tests and measurements are carried out five times and the average is selected for each point at figure 10.

          z y x    =                                  2 1 2 1 0 0 0 0 0 0 2 1 2 1 0 0 0 0 0 0 2 1 2 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 0 0 a a a a a a                     
To exploit measurements as well as possible, the displacements curves are plotted depending on the applied force for each loading direction. A line of least squares is adjusted to determine the displacements components values for a given force. Thus, six torsors of small displacements are identified for six loadings cases. The linear behaviour observed in loading is different from the linear behaviour noted in unloading. This different linearity between loading and unloading is due to the existence of deviations and friction forces in each point surfaces of the assembly. These deviations due to the installation of the parts of the associated assembly and friction force are different in charge and discharge.

When this difference in linear behaviour appears (hysteresis), we use the line (figure 10) which passes by the middle (C point) of segment AB (charge-discharge). OC is the line which the slope, by assumption, corresponds to the "real" stiffness. Segment AB represents the double of the friction forces and deviations for the deformation , [START_REF] Kudinov | Dinamica Masinilor Unelten[END_REF].

[C O ] = {T} -1  {D} . ( 7
)
The inversion of experimental flexibility matrix gives the global stiffness matrix, [K].

Stiffness matrix of BT

The figure 11 presents a loading example in x direction of BT (see [START_REF] Bisu | Etude des vibrations auto-entretenues en coupe tridimensionnelle: nouvelle modélisation appliquée au tournage[END_REF]). A similar loading experimentation in y and z direction of BT is carried out. For each x, y or z direction measurements are taken at three points and captured by data acquisition card installed in PC. The stored data can be retrieved and used for analysis when required.

                                                                          (8)
In addition, on figure 10 we made a simplifying assumption while retaining for each level of deformation given (for example AB), the average charge (C) correspondent between the value in charge (A) and the value discharges (B) from them. Consequently, it is advisable to make sure the validity of this assumption. This must be done while estimating, on each level of loading, the error made by using the median value between the charge and the discharge. However, the use of the least squares method allows the evaluation of the error made for each level of loading, i.e. each elements of the matrix [K BT ]. Thus, a matrix of error can be built. This matrix noted [K errors% ] thus allows to know the error attached to each element of the matrix [K BT ]. This operation is performed by an errors matrix [K errors% ] given in [START_REF] Castro | Correction of dynamic effects on force measurements made with piezoelectric dynamometers[END_REF]. [START_REF] Castro | Correction of dynamic effects on force measurements made with piezoelectric dynamometers[END_REF] It is noted that the error does not exceed 6 % what is largely acceptable.

[K errors% ] =                     2 .
In addition, for a "perfect" decoupled system [K] is diagonal, and the elements are stiffness values in N/m. The matrix [K] obtained here is a matrix block. Comparing this matrix to the matrix [K] A, xyz [START_REF] Bisu | Displacements analysis of self-excited vibrations in turning[END_REF] we establish between elements the following correspondences:

 the elements of the matrix 3  3 in the right higher corner are the elements corresponding to stiffness values of displacements (N/m) noted [K F ] in (4),  the elements of the matrix 3  3 in the left lower corner are the elements corresponding to stiffness values of rotations (N/rad) noted [K C ] in (4),  \item the elements of the matrix 3  3in the left higher corner are the elements corresponding to the couplings of "displacements / rotations" noted [K FC ] in (4),  \item the elements of the matrix 3  3 in the right lower corner are the elements corresponding to the couplings of "rotations / displacements" noted [K CF ] in (4).

                      (10)

Stiffness matrix of BW

As the BW geometry was simpler, we limited ourselves to measure the stiffness values of displacements by using three displacement transducer following the three main directions. The loading was carried out with the dynamometer (cf. figure 12). Using this geometry, the BW has a very high rigidity according to z axis and this value is very small compared with the principal stiffness value. The BW behaviour is linear, with a nearly null hysteresis. At the loading and unloading points, the part is not influenced by the friction phenomenon or other various fits generated by the assembled elements, like the spindle or the ball bearings. The stiffness matrix [K F, BW ] obtained according to the three directions is:

[K F, BW ] = xyz O, 8 7 7 10 85 . 2 0 0 10 2 0 0 0 10 4 . 1               . ( 11
)
We may notice that the spindle and its ball bearings decrease the global BW rigidity compared with the calculated rigidity in the section 3.1.

Experimental determination of the machining system stiffness matrix

In order to know stiffness values of the machining system in the three directions, the elastic interaction BT BW is modelled by static stiffness which are assembled in parallel (figure 13). The chip is the common point between deformation and force. It connects the two stiffnesses (BT / BW). The sum of the two stiffness matrix of displacements BT [K F,BT ] and BW [K F,BW ] determines the stiffness matrix of machining system displacement [K F,WAM ]. [START_REF] Couétard | Caractérisation et étalonnage des dynamomètres à six composantes pour torseur associé à un système de forces[END_REF] In this form the matrix of the machining system shows that the main diagonal has elements of a higher order than the others. Moreover the diagonalization of the matrix is possible and it comes: By the Castigliano's theorem we determine the angle  K . This angle characterizes the principal direction of deformation. It takes place in the case of two blocks BT and BW interaction (figure 14). On the figure 14,  K-BT et  K-BW correspond respectively to the principal direction of deformation of the Block Tool and Block Workpiece in connexion with the machine axes. In the plane O,x,y, with  K-BT = 52° and  K-BW = 0° we obtain an angle  K = 76°. In the plane O,y,z with  K-BT = 32° and  K-BW = 0° we have an angle  K = 65°. On this direction, the maximum deformation of the system is obtained.

[K F, WAM-d ] = xyz O,

Rotation center

Deacu [START_REF] Deacu | Vibrations des Machines-Outils[END_REF] and Kudinov [START_REF] Kudinov | Dinamica Masinilor Unelten[END_REF] show that any machine tool, is characterized by the deformation principal directions. These deformation principal directions are function of the machine structure, its geometrical configurations and cutting parameters used. We can observe either a very stiff behaviour, or very rubber band according to the direction. Here, the tool is regarded as forming integral part of block BT. The aim is to determine the stiffness center CR BT of elastic system BT [START_REF] Marinescu | Handbook of Machine Tool Analysis[END_REF]. This stiffness center corresponds to the rotation center of the block BT compared to the bed. Obtaining CR BT consists in finding the points of intersection of different perpendicular to displacements.

Experimental step

The stiffness center is obtained in the coordinate system based on the tool in O point that is the origin of the (x, y, z) coordinate system.

The procedure to obtain the stiffness center is detailed in the figure 15; the imposed charge follows all three directions, F i being known; (i = x, y, z). The displacement vector measurement under the load d i,j is obtained in two points P i,j in each direction (i = x, y, z) and (j = 1, 2). The direction (D ij ) is released being straight line containing d i,j vector that include the point P i,j .

According to the figure 16, these lines are not exactly coplanar and no secant. It is possible to find for each direction the intersection point, named M i , of the lines D ij using the following relations, for x direction:

x OM = 1 , 1 , x x x Od e OP         (15)
Using the method of least squares, we can minimize the distance d x between the lines (D x,1 and D x,2 , figure 16); d x is calculated using the expression:

d x = 2 , 2 , 2 , x x x x d d OP OM         (16)
and the angle  x corresponds of angular deviation of "coplanarity":

 x = arccos               1 , 1 , 1 , 1 , x x x x x x d M P d M P (17)
The two lines (D ij ) are separated by a minimum distance  i = 1.8 mm and maximum deviation  i = 2° (see the table 1).

In the next step, three mean plans P i are defined by approximation of the points M i and containing the lines D ij . We can draw the normal i n of each plan P i that includes the point M i (figure 17). Writing the loop of geometrical closure under the three directions: [START_REF] Insperger | Criticality of Hopf bifucation in state-dependent delay model turning processes[END_REF] where: i = x, y, z.

BT OCR = i i i n f OJ   (
We obtain a linear system with three vectors' equations and three unknown factors, after that is seeking to obtain the intersection point CR BT using least squares minimization method. In reality, the directions of these three normal lines cross (nearly at r = 1.2 mm) in only one point noted CR BT (see figure 17) which corresponds with stiffness center. The obtained results using this experimental approach are coherent with other finding in the literature [START_REF] Kudinov | Dinamica Masinilor Unelten[END_REF].

Thereafter is made a verification measuring the tool point displacement O in work-space under the charge. We considerate our experimental results viewing the tool point that moves under one segment of a sphere with the center CR BT : assimilated with a normal plan (O, CR BT ) for small displacements (accordingly figure 18).

The eigenvectors associated are: (

[V] = xyz O v v v
) 20
We note that the maximum stiffness obtained is situated on the direction of third eigenvector 3 v , being at 4° nearly the direction (CR BT -O}) (see figure 19); and respectively the minimum and stiffness average on the directions of eigenvectors 1 v and 2 v .

Conclusion

In the literature many authors use only tool rigidity [START_REF] Sr | The impact of cutting conditions on cutting forces and vibration signals in turning with plane face geometry inserts[END_REF][START_REF] Saglam | Investigation of the effect of rake angle and approaching angle on main cutting force and tool tip temperature[END_REF][START_REF] Saglam | The effect of tool geometry and cutting speed on main cutting force and tool tip temperature[END_REF][START_REF] Yaldiz | A dynamometer design for measurement the cutting forces on turning[END_REF][START_REF] Yaldiz | Comparaison of experimental results obtained by designed dynamometer to fuzzy model for predicting cutting forces in turning[END_REF] and recently the elastic behaviour of the machine [START_REF] Cano | A new approach to identifying the elastic behaviour of a manufacturing machine[END_REF]; furthermore, this research considers the rigidity of the tool device system (BT). Anothers takes into account that both the workpiece deflection and the cutting force between tool and workpiece [START_REF] Carrino | Dimensional errors in longitudinal turning based on the unified generalized mechanics of cutting approach. Part I: Three-dimensional theory[END_REF]. In [START_REF] Seka | Stability analysis of turning process with tailstock-supported workpiece[END_REF] tool and workpiece are modelled as two separate single degree of freedom spring-mass-damper systems. All these models are 2D, or more or less partially 3D, and thus does not allow a real 3D modeling of the cut.

The approach presented in this paper is fully different from previous studies in the sense that we use the six-dimension of the torsors (3 force components, 3 torque components). Consequently, this way allows us to be directly in the most suitable framework for a real 3D cutting modeling. Thanks to the principle of virtual work, we determined the complete torsor of small displacements (3 linear and 3 rotations displacement) associated with the mechanical actions torsor (3 force components, 3 torque components) via the matrix of rigidities. The stiffness center and the rotation center were obtained experimentally. The minimal displacement direction was defined on the basis of experimental model. Using Castigliano's theorem, we determined the angle  K which characterizes the principal deformation direction. Then, we characterized block workpiece (BW) was characterized by its stiffness values and the diagonal matrix of displacements. This diagonal matrix of displacement validate the hypothesis and confirm the good rigidity of workpiece. Whole static behaviour analysis of the system (BT-BW) is thus validated. An application of this methodology within a framework of machining really 3D is available in [START_REF] Bisu | Displacements analysis of self-excited vibrations in turning[END_REF]. These results are required to have a good 3D modeling of the cut, as being presented in a forthcoming paper.

Fig. 1

 1 Fig. 1 Workpiece-Tool-Machine considering dynamic cutting process.

Fig. 2

 2 Fig. 2 Presentation of the experimental device.}

Fig. 3

 3 Fig.3 BW representation.}

Fig. 5

 5 Fig. 5 Displacements according to the holding fixture length.

Fig. 6

 6 Fig. 6 Acceptable aria representation of the workpiece deformation.}

Fig. 7

 7 Fig. 7 Block tool BT representation.

Fig. 9

 9 Fig. 9 Position of Displacement transducer.

Fig. 10

 10 Fig. 10 Diagrammatic representation of linear behaviour in charge and discharge.}

Fig. 12

 12 Fig. 12 Experimental device for the BW static characterization.

Fig. 13

 13 Fig. 13 Static stiffness assembled in parallel.

Fig. 14

 14 Fig. 14 Diagram of  K angle determination.

Fig. 15

 15 Fig. 15 Experimental procedure to obtain stiffness center CR BT

Fig. 16

 16 Fig. 16 Identifying the intersection point M x

Fig. 17

 17 Fig. 17 Determination of the block tool rotation center: CR BT

Fig. 18

 18 Fig. 18 Determination of the block tool center rotation: CR BT

Fig 19

 19 Fig 19 Eigenvectors.

Table . 1

 . Experimental values to obtain stiffness center CR BT 22

						0.366
		-20	1.710 -5	1.810 -3	0.28	0.042	0.56
		52	3.410 -5			0.178
		35	0.810 -5			0.366
	P x,2	-20	1.510 -5	1.810 -3	0.28	0.042	0.56
		117	1.510 -5			0.178
		116	210 -5			0.086
	P y,1	15	-210 -5	1.710 -3	1.97	0.045	-0.58
		56	-210 -5			0.081
		116	0.210 -5			0.086
	P y,2	15	-1.310 -5	1.710 -3	1.97	0.045	-0.58
		103	9.810 -6			0.081
		45	5.510 -6			0.033
	P z,1	-20	6.510 -5	8.810 -4	0.18	-0.052	-0.08
		17	5.510 -5			-0.227
		130	310 -5			0.033
	P z,2	-20	110 -5	8.810 -4	0.18	-0.052	-0.08
		6	8.710 -5			-0.227

Acknowledgements

The authors acknowledge Jean Pierre Larivière, Ingineer CNRS (Centre National de la Recherche Scintifique -France) for the numerical simulation with SAMCEF software and Professor Miron Zapciu for the helpful discussions on this subject.


Miron Zapciu

Olivier Cahuc

Claudiu F Bisu

Alain Gérard
email: alain.gerard@u-bordeaux1.fr

Jean-Yves K'nevez

Experimental study of machining system: dynamic characterization

Keywords: Dynamic characterization, Machining system, Natural frequencies analysis Nomenclature

In the workspace model of machining an experimental procedure is implemented to determine the elastic behaviour of the machining system. In an other work we have proposed a static characterization of the machining system. In this paper a dynamic characterization and vibration analysis have long been used for the detection and identification of machine tool condition. The natural frequencies of the lathe machining system (Ernault HN400 -France) according with three different situations with no cutting process were acquired. The system modal analysis is used to identify the natural frequencies. These ones and these obtained on the spindle numerical model by finite element method are compared. This research is validated by experimental tests being based to measures of the lathe machine tool frequencies domain. Main focus is to identify a procedure to obtain natural frequencies values for machine tool components in order to establish better conditions in the cutting process on the machine tool.

Introduction

Among the various manufacturing processes to remove excess material, milling, drilling, grinding, turning occupies a choice place. This process is usually used in many circumstances as the outline of the parts for example, but also as soon as appear requirements of dimensional tolerance, precision or the surfaces quality of the produced part. However these dimensional accuracies or the final shape of the part (controlled roughness) is often dependent on the vibrations appearance during the process.

The vibration in the cutting process is a phenomenon dynamically unstable. These instabilities, often regenerative, are generated by many factors such as the work piece flexibility and the properties materials of the tools, the machine rigidity, tool geometry (approaching angle, rake angle, etc.) and the cutting tool edge radius, the nominal cutting conditions such as tool wear, feed rate and depth of cut, etc.

The cutting process stability is very studied for many reason of which, particularly, its the influence on the final surface quality [START_REF] Chiou | A nalysis of tool wear effect on chatter stability in turning[END_REF][START_REF] Karabay | Design criteria for electro-mechanical transducers and arrangement for measurement cutting forces acting on dynamometers[END_REF]. The ordering of the process under various work conditions is an important problem for machining. Many modelings are implemented to optimise the cutting conditions, [START_REF] Benardos | Prediction of workpiece elastic deflections under cutting forces in turning[END_REF], [START_REF] Karube | The non-linear phenomena in vibration cutting system. The establishment of dynamic model[END_REF][START_REF] Toh | Static and dynamic cutting force analysis when high speed rough milling hardened steel[END_REF]. The case of the orthogonal cut is the process that holds more the attention because of its simplicity to implement. For example by taking into account the regenerative aspect induced by the surface previously machined which has a sinusoidal form (waviness on the surface). The introduction of the non-linear interaction between the tool and carries it part always gives interesting results which go from regular (periodic or quasiperiodic) vibrations to possibilities of chaotic ones [START_REF] Litak | Chaotic vibrations in a regenerative cutting process[END_REF] or, always within the orthogonal framework of the cut, while utilizing in more the dry friction [START_REF] Litak | Effect of high-frequency excitation in regenerative turning of metals and brittle materials[END_REF].

The case 2D is also examined while considering carries it rigid part but by taking into account the tool flexibility [START_REF] Insperger | Criticality of Hopf bifucation in statedependent delay model turning processes[END_REF], the tool holder flexibility [START_REF] Chen | A stability analysis of regenerative chatter in turning process without using tailstock[END_REF] or the rotor system (Qi,et al.,200). [START_REF] Dassanayake | On nonlinear cutting response and tool chatter in turning operation[END_REF] approach the case of the work piece dynamic response to the tool request that follows a regenerative surface. All these modelings give interesting results, but, in general, do not take account the whole of the kinematic chain of the cut process. We want to describe the whole of the kinematic chain of the cut process thereafter in a digital model richer and close to the experimental data.

Analysis of machines dynamic behaviour and equipments it's an important method to redesign the product or the manufacturing process and to assure the proper quality, maintenance and service. When we study machines or only part of them, dynamic behaviour is analysed for the following situations:

-Constant operating speed (e.g. speed rate of a rotor); -Variable speed into a limited operating domain; -Imposed speed inside the operational domain (e.g. rotational speed of the spindle 100 to 60,000 rpm);

In all of these mentioned cases the system behaviour under the external excitation effect is evaluated (Figure 1). The measurement of vibratory severity makes it possible to know if the vibratory behaviour of a machine exceeds the acceptable limits [START_REF] Chen | A stability analysis of regenerative chatter in turning process without using tailstock[END_REF]. But within results sight, it is not possible to make an assumption on the vibration causes. This information could not be obtained that by using a frequency spectrum analysis [START_REF] Moraru | Vibratiile si Stabilitatea Masinilor Unelte[END_REF].

We wish to be able to establish a model of three-dimensional turning processes nearest possible to the physical reality. Thus, the aim of this paper is to test a methodology that makes it possible to characterize the band with frequencies associated with the unit with each element of the kinematic chain apart from any operation of machining. Also, the tasks for performing the frequency analysis are given in Section 2. In Section 3 we present the experimental device with the frequency spectrum schema acquisition and the experimental results. A numerical model in finite element is used in order to find the first twelve natural frequencies. These ones confirm the first natural frequencies of the system spindle and the system spindle with work piece. The machine tool dynamic characterization is the object of Section 4 before to conclude.

Tasks for performing the frequency analysis

Technical determination of the frequency spectrum

The individual tasks for performing the frequency analysis are shown in the Figure 2.

Technical determination of the frequency spectrum is done nowadays with vibration sensors and FFT analysers. These modern analysers (LMS, Bruel & Kjaer, Schenck etc.) record the time signal of the vibration mixture in digital form, calculate the individual components and display the results in a frequency spectrum form.

All analysers and measurements can be triggered for start/stop, e.g., by RPM signal or vibration signal. Using a trigger, unwanted non-periodic components in the time signal and related spectrum, can be suppressed, e.g., for separation of other vibration signal. The analogue signal from the vibration sensor is sent to the digital-to-digital converter (ADC) via an input amplifier, an anti-alias filter and a sample-and-hold circuit. This samples the signal and forms a time-sequential individual (discrete) amplitude value. This way each discrete value is quantified and converted to a binary data value (coding).

The Shannon theorem describes the important requirement for the sampling of analogue signals (Figure 3), namely regarding the relationship between the maximum signal frequency and the minimum sampling frequency. To guarantee the digitised signal reproducibility, the sampling frequency (f sampling ) must be at least double that of the highest frequency component (f max ) contained in the measured signal.

f sampling  2 f max (1)

Figure 3 Example of signal vibration sampling

To counteract this undesirable effect (alias), all signal components with a frequency higher than half the sampling frequency must be suppressed. This is done in practice with a low-pass filter called an "anti-aliasing filter"

Experimental vibration analysis

Experimental device

All the machines vibrate and, as the state of the machines worsens (imbalance of the spindle or other important shaft, defect of bearing or spindle) the vibration level increases. While measuring and by supervising the vibration level produced by a machine, is obtained an ideal indicator on his state, especially dynamic behaviour [START_REF] Karabay | Design criteria for electro-mechanical transducers and arrangement for measurement cutting forces acting on dynamometers[END_REF][START_REF] Cardi | Workpiece dynamic analysis and prediction during chatter of turning process[END_REF].

While the increase in machine vibration makes it possible to detect a defect, the analysis of the machine vibration characteristics makes it possible to identify the cause of it. In this way we can deduce with sufficient precision the time domain before vibrations does not become critical.

Each component of spectrum FFT corresponds to a characteristic frequency well defined (imbalance, resonance, misalignment etc.) [START_REF] Lalwani | Experimental investigations of cutting parameters influence on cutting forces and surface roughness in finish hard turning of MND250 steel[END_REF]. The analysis in frequency is carried out in general when the machine vibratory level is considered to be higher than the acceptable threshold [START_REF] Rigal | Etude du comportement dynamique du tour universel Gallic 20[END_REF].

In the Figure 4 is illustrated the schema acquisition of the frequency spectrum FFT in the case of the Ernault HN400 lathe -France, LMP Laboratory from Bordeaux 1 University. Accelerometers were fixed on the horizontal and vertical plans on the bearing fix front part of the lathe spindle. The vibration speed level in mm/s (rms) is situated on the range 0,1-0,3. For this type of machine tool, in accordance with IEC 34-14 standard the upper limit of the vibration level is 1,8 mm/s (see Table 1 As many workers [START_REF] Benardos | Prediction of workpiece elastic deflections under cutting forces in turning[END_REF], [START_REF] Yaldiz | Comparaison of experimental results obtained by designed dynamometer to fuzzy model for predicting cutting forces in turning[END_REF], (Mehdi, et al., 2002a) a cylindrical geometry of the work piece is chosen. The Block Work piece (BW) represents the revolving part of the Machine-Tool-Machine system (WTM); it includes the holding fixture, the work piece and the spindle Figure 6. To make the whole frame rigid, a very rigid unit (work piece, holding fixture) is conceived in front of the WTM elements (Figure 7).

Figure 6 Representation of the BW

The work piece geometry and his holding fixture are selected with D 1 = 60 mm, D 2 = 120 mm and L 2 = 30 mm (cf. Figure 7). These dimensions retained for these test tubes were selected using the finite elements method coupled to an optimisation method by SAMCEF software. It is necessary to determine the holding fixture length L 1 to obtain a significant stiffness in flexion. The objective is to move away the first vibration mode of the Block work piece of the lathe vibration fundamental natural mode (see [START_REF] Bisu | Etude des vibrations auto-entretenues en coupe tridimensionnelle~: nouvelle modélisation appliquée au tournage[END_REF]. A holding fixture length: L 1 = 180 mm, for a stiffness in flexion of 7.10 7 N/m, is reminded. This value is in the higher limit of the acceptable zone of rigidity for conventional lathe (cf. Figure 9), [START_REF] Ispas | Vibrations des systèmes technologiques. Agir[END_REF][START_REF] Koenigsberger | Machine Tools Structures[END_REF][START_REF] Konig | Zweikomponenten schnittkraftmesser[END_REF].

Figure 9 Representation of the acceptable aria of the work piece deformation

In this case, the BT part includes the tool, the tool-holder, the dynamometer, the fixing plate on the cross slide (Figure10). The six-components dynamometer (Couetard-00) is fixed between the cross slide and the tool-holder. It is necessary thereafter for the measurement of the mechanical actions of cutting.

Experimental results

The natural frequencies experimental values obtained following the impact tests are detailed in the Figure 11 and Figure 12, considering blocks BT and respectively BW. Using an impact hammer, the natural frequencies of each block, for each element, are given accordingly the machine axes directions x (cross direction), y (cutting direction), and z (feed direction). A three directions accelerometer is positioned on each element, in each direction and the elements tested were the subject of the hammer impact. An example of result is given into the Figure 11, where the natural frequencies of whole block BT are presented, with all the components. For the BW unit the results are presented in the Figure 12. The natural frequencies domain is presented for each component of the system by carrying out a modal superposition in the Figure 13. These results are coherent with those met in the literature [START_REF] Moraru | Vibratiile si Stabilitatea Masinilor Unelte[END_REF][START_REF] Ispas | Vibrations des systèmes technologiques. Agir[END_REF][START_REF] Marinescu | Handbook of Machine Tool Analysis[END_REF][START_REF] Benmohammed | Identification des lois d'effort de coupe de tournage en régime dynamique[END_REF]Kudinov, 1970).

Figure 13 Natural frequencies domains superposition of the machine tool components at the time of the impact for each element.

Using the layout of the tool free oscillations, (presented in the Figure 14), the  i damping percentage is directly given by the logarithmic decrement curve, starting from N consecutive maximum; the measured period T of the damped oscillations give the damped natural frequency  d of the system.

Figure 14 Test of free displacement during of one impulse using an impact hammer.

With the obtained values  d and  i , we can calculate no damped natural frequency. The values of these parameters allow calculating: the stiffness K, the equivalent mass M and the equivalent damping coefficient C for each part BT and BW and for all three directions. Like values example it comes: (5) and the stiffness matrix k is given in the first part of this work (Bisu, et al., 2007).

M =           5 . 2 0 0 0 3 . 5 0 0 0 2 . 2 (4)
3.5 Numerical model of the assembly of the lathe spindle Ernault HN400.

In order to find the spindle natural frequencies domain of the machine tool and of the assembly Block Work piece BW (spindle with the work-piece attached) was used Finite Element Method. The Figure 15 shows the mesh (4,290 nods, 16,310 nets) and spindle numerical model. The Figure 16 shows the model by adding the work-piece. Accordingly numerical models were obtained the results for the first twelve natural frequencies of these two models (represented in the Figure 15 and Figure 16). Vibration mode accordingly first natural frequency is represented in the Figure 17 and the numerical values of first twelve natural frequencies are represented in the Figure 18. We can observe that the first spindle natural frequency of the system with work piece is about 675 Hz and it is lower considering only the spindle system (1,600 Hz). These results are coherent with those experimentally obtained (Figure 19).

Figure 19 Experimentally FFT spectrum; spindle turning at 1,000 rpm On the acquired signal FFT spectrum (spindle system turning, without cutting process) we can see the first frequency 675 Hz and, on the right side, the frequency with the value 1,599.1 Hz (corresponding to the spindle alone).

Machine tool dynamic characterization -turning process recommendations

An analysis according to three configurations

The dynamic characterization of the machining system is supplemented by an analysis according to three configurations: electric motor drive, electric motor turning the spindle, electric motor turning the spindle and advance movement coupled [START_REF] Bisu | Etude des vibrations auto-entretenues en coupe tridimensionnelle~: nouvelle modélisation appliquée au tournage[END_REF][START_REF] Bisu | Un nouveau modèle expérimental d'analyse des phénomènes vibratoires lors d'une opération de tournage[END_REF]. The three axes accelerometer is placed on the tool body and the one-way accelerometer is located on the bed near the spindle. In the Figure 20 are presented the frequencies values in the case when the machine tool is no charge with the cutting process, for the three configurations mentioned. The frequencies below 100 Hz, belong to the engine behaviour, the amplitudes are very weak, very low and appear on each tests configuration. The frequencies corresponding with no cutting process are given for the three directions in the case of the kinematics chain "electric motor turning the spindle and advance movement coupled" [START_REF] Bisu | Etude des vibrations auto-entretenues en coupe tridimensionnelle~: nouvelle modélisation appliquée au tournage[END_REF]. The measured significant frequencies are into the domain of 230 Hz up to 1,000 Hz (see Figure 21), and more, on the three x, y and z directions. Increasing the work piece rigidity. In order to reduce the vibrations problems at the time of the machining of the thin walls it is always well adapted, like known as previously, to rigidify the part by an adapted fixture, when that is possible (Mehdi, 2002a;[START_REF] Sutter | Analysis of the Cutting Force Components and Friction in High Speed Machining[END_REF]Mehdi, et al., 2002b). The process simulations makes possible to specify before to have begun machining the most adapted solution, and even it is possible to redesign the shape of the work-piece in order to bring more rigidity to him.

Control of the cutting stiffness by reducing axial tool engagement. The theory of the stability lobes highlighted very early the importance of the relationship between the workpiece stiffness or the tool, and the cutting stiffness (the coefficient which links the tool displacement into the material to the cutting pressure, in the vibration direction). The most significant parameter to decrease this ratio is roughly speaking the projected cutting tool edge length in the material, which always impose to the specialists in machining process to naturally reducing the tool axial engagement [START_REF] Calderon-Najera | Caractérisation dynamique du système Pièce/Outil/Machine : usinage des pièces minces[END_REF].

Modifying the cutting tool angles. Another way of decreasing the cutting stiffness is to exploit the tool angles so that the cutting pressure is parallel with the wall, thus the cutting stiffness would be theoretically null.

These two possibilities of stiffness control are major in the vibratory phenomenon during machining, in tools selection (angles, coatings etc.) and the strategies of machining in generally have the tendency to exploit these two effects [START_REF] Belhadi | Experimental and numerical study of chip formation during straight turning of hardened AISI 4340 steel[END_REF], [START_REF] Mabrouki | Material modeling and experimental study of serrated chip morphology[END_REF] Limit the possibility to generate vibrations, avoid the resonance.

-Avoiding the frequencies with problems: by choosing the proper speed of the spindle (considering the stability lobes theory); -Adding damping: by lubrication, controlling the pressure on the assembly surfaces, etc; -Obtaining the excitation spectrum: by tools with variable step, by the spindle rotation with variable speed; -Controlling actively: by the use of actuators (axes machine, turret etc.) controlled dynamically according to measurements, in order to eliminate the vibrations. For the moment, in practice, the achievements in this field are mainly regulations of average effort or power by the reduction of the tool advance [START_REF] Karabay | Design criteria for electro-mechanical transducers and arrangement for measurement cutting forces acting on dynamometers[END_REF][START_REF] Bakkal | Chip formation, cutting forces, and tool wear in turning of Zr-based bulk metallic glass[END_REF] Within the framework of this study, the objective was to find (starting from the experimental tests in turning) a correlation between the frequency founded on the acquired FFT spectrum and the dynamic process based on the elastic system Machine tool, Work piece and Tool device.

Machine tools and particularly, in this work paper, the lathe, evolved to the limits of certain parameters of cutting that consist in a better control of the process. In this context, this subject "dynamic characterization" is very important and could helping on the development of a simulation model for the cutting process on the machine tool.

This research was validated by experimental dynamic tests being based to measures of the frequencies domain of the machining system Ernault HN400 lathe, in the LMP laboratory of the University Bordeaux 1, France. Nevertheless the method used here applies to other machine tools (milling, drilling etc.).

Taking into account the recommendations summarized in the Section 4 the cutting process could be better managed.

The following step in our research will be to make a prediction for the cutting conditions and to supplement necessary knowledge toward a global and industrial model of cutting process considering a 3D model in turning.

Figure 1

 1 Figure 1 Transfer function

Figure 2

 2 Figure 2 Frequency analysis tasks for a machine tool

Figure 4

 4 Figure 4 Position of transducers in acquisition schema to acquire vibrations of the spindle ERNAULT. 1-Accelerometer AS020 -vertical plan (V); 2-Accelerometer AS020 -horizontal plan (H), 3-Gearbox of Ernault HN400 lathe; 4-Tachometer; 5-Spindle bearing front side; 6-Work piece.

Figure 5

 5 Figure 5 Vibration level of the Ernault lathe using Tracking versus time procedure

Figure 7

 7 Figure 7 Geometry of holding fixture / work pieceThe stiffness is calculated on the basis of the displacement  for a given force P :

Figure 8

 8 Figure 8 Displacements according to the holding fixture length

Figure 10

 10 Figure 10 Representation of the block tool BT

Figure 11

 11 Figure 11 Natural frequencies representation for the block-tool BT (direction: x red colour, y blue colour, z green colour)

Figure. 15

 15 Figure. 15 Spindle numerical model.

Figure 16

 16 Figure 16 Spindle with work piece BW numerical model and boundary conditions.

Figure 17

 17 Figure 17 Vibration bending mode for the first natural frequency

Figure 20

 20 Figure 20 Frequencies domains representation in the case with no cutting process

Figure 21

 21 Figure 21 Natural frequencies superposition in the case of no cutting process

Table 1

 1 ). Limit values in accordance with IEC 34-14.

	Related operating	Limit value of the effective vibration velocity for shaft heights H in mm
	speed (rpm)
	Normal	Machine is mounted on specifically design elastic	Specifically
		foundations (e.g. vibration dampers)	designed rigid
	(mm/s)	56 < H < 132	132 < H < 225	H > 225	H > 40
	> 600 < 1.800	1,8	1,8	2,8	2,8
	> 1.800 < 3.600	1,8	2,8	4,5	2,8

Acknowledgements

The authors acknowledge Jean Pierre Larivière Engineer Centre National de la Recherche Scientifique -France for the numerical simulation with SAMCEF software. The authors would like also to thank the CNRS (UMR 5469) for the financial support to accomplish the project.


P. -V Koseleff

D Pecker

On Fibonacci knots

Keywords: Fibonacci polynomials, Fibonacci knots, continued fractions

We show that the Conway polynomials of Fibonacci links are Fibonacci polynomials modulo 2. We deduce that, when n ≡ 0 (mod 4) and (n, j) = (3, 3), the Fibonacci knot F (n) j is not a Lissajous knot.

Introduction

Fibonacci knots (or links) were defined by J. C. Turner ([11]) as rational knots with Conway notation C(1, 1, . . . , 1). He also considered the generalized Fibonacci knots F (n) j = C(n, n, . . . , n), where n is a fixed integer and the sequence (n, . . . , n) has length j.

In this paper we determine the Conway and Alexander polynomials modulo 2 of Fibonacci knots. We show that the Conway polynomial of a generalized Fibonacci knot is a Fibonacci polynomial modulo 2.

As an application, we show that if n ≡ 0 (mod 4) and (n, j) = [START_REF] Conway | An enumeration of knots and links, and some of their algebraic properties[END_REF][START_REF] Conway | An enumeration of knots and links, and some of their algebraic properties[END_REF] the Fibonacci knot F

(n) j is not a Lissajous knot.

Our results are obtained by continued fraction expansions.

Conway notation and Fibonacci knots

The Conway notation (J. H. Conway, [START_REF] Conway | An enumeration of knots and links, and some of their algebraic properties[END_REF]) is particularly convenient for the important class of rational (or two-bridge) knots. The Conway normal form C(a 1 , a 2 , ..., a m ) of a rational knot (or link), is best explained by the following figure. The number of twists is denoted by the integer |a i |, and the sign of a i is defined as follows: if i is odd, then the right twist is positive, if i is even, then the right twist is negative. On Fig. 1 the a i are positive (the a 1 first twists are right twists).

The rational links are classified by their Schubert fractions β ′ are equivalent if and only if α = α ′ and β ′ ≡ β ±1 (mod α). The integer α is the determinant of the link, it is odd for a knot, and even for a two-component link.

α β = a 1 + 1 a 2 + 1 a 3 + 1 • • • + 1 a m = [a 1 , . . . , a m ], α > 0. (1)
The following result is a useful consequence of the continued fraction description of rational links (see [START_REF] Cromwell | Knots and links[END_REF] p. 207).

Theorem 1. Any rational link has a Conway normal form C(2a 1 , 2a 2 , . . . , 2a m ).

The Fibonacci knots (or links) are defined by their Conway notation F j = C(1, 1, . . . , 1), where j is the number of crossings. The Schubert fraction of F j is F j+1 F j , and its determinant is the Fibonacci number F j+1 . It is the reason why J. C. Turner named these knots Fibonacci knots. He also introduced the generalized Fibonacci knots

F (n) j = C(n, n, . . . n),
where n is a fixed integer.

We first observe

Proposition 2. F (n) j
is a knot if and only if n ≡ 0 (mod 2) and j ≡ 0 (mod 2) or n ≡ 0 (mod 2) and j ≡ 2 (mod 3).

Proof. Let us consider the Möbius transformation

P (z) = [n, z] = n + 1 z . It is convenient to consider its matrix notation P = n 1 1 0 . Let α β = [n, . . . , n] = P j (∞), it is also α β = P j 1 0 .
If n ≡ 1 (mod 2) then P ≡ 1 1 1 0 (mod 2), P 2 ≡ 0 1 1 1 (mod 2), P 3 ≡ 1l (mod 2). We deduce that α ≡ β ≡ 1 (mod 2) when j ≡ 1 (mod 3), α ≡ 0, β ≡ 1 (mod 2) when j ≡ 2 (mod 3) and α ≡ 1, β ≡ 0 (mod 2) when j ≡ 0 (mod 3). The case n ≡ 0 (mod 2) is similar.

F (1) 3
F (1) 4
F (2) 1 F (2) 2 F (2) 3 F (3) 1 F (3) 2 F
(3) 3 The Alexander polynomial, discovered in 1928, is one of the most famous invariant of knots. J. H. Conway discovered an easy way to calculate it. He introduced the "Skein relations" which relate the polynomial of a link K to the polynomials of links obtained by changing one crossing of K.

The following result is a beautiful application of his algorithm.

Theorem 3 ([4]

). Let K = C(2a 1 , 2a 2 , . . . , 2a m ) be a rational knot (or link). The Conway polynomial of K is

∇ K (z) = ( 1 0 ) -a 1 z 1 1 0 a 2 z 1 1 0 • • • (-1) m a m z 1 1 0 1 0
The Alexander polynomial of K is

∆ K (t) = ∇ K t 1/2 -t -1/2 .
Let us consider a simple example.

m 1 -m = [-2, 2, . . . , (-1) m-1 • 2].
Hence, the Conway polynomial is

∇(z) = ( 1 0 ) z 1 1 0 m-1 1 0 . It is well known that z 1 1 0 m = f m+1 (z) f m (z) f m (z) f m-1 (z)
where f m (z) are the Fibonacci polynomials defined by f 0 (z) = 0,

f 1 (z) = 1, f m+1 (z) = zf m (z) + f m-1 (z) ([12]
). We conclude that the Conway polynomial of T(2, m) is the Fibonacci polynomial f m (z) (see also [START_REF] Kauffman | On knots[END_REF]). If m = 2k + 1 (i.e. T(2, m) is a knot) we obtain the Alexander polynomial

∆(t) = f 2k+1 t 1/2 -t -1/2 = (t k + t -k ) -(t k-1 + t k-1 ) + • • • + (-1) k .
The recently introduced Lissajous knots ([2, 5, 10, 4]) are non singular Lissajous space curves. We will show that in many cases, Fibonacci knots are not Lissajous knots. Let us first recall the following Theorem 5 ([5, 10]). If K is a rational Lissajous knot then ∆ K (t) ≡ 1 (mod 2).

Consequently, we see that a non trivial torus knot is never a Lissajous knot.

Moreover, Theorem 3 provides many examples of knots which are not Lissajous knots.

Corollary 6. Let b i ≡ 2 (mod 4), m > 1. The Conway polynomial of C(b 1 , . . . , b m ) is equivalent to f m+1 (z) (mod 2). Corollary 7. If n ≡ 2 (mod 4), the modulo 2 Conway polynomial of F (n) j is f j+1 (z).
Hence these knots are not Lissajous knots by Theorem 5.

The following result is an immediate consequence of Theorem 3.

Corollary 8. If n ≡ 0 (mod 4), the modulo 2 Conway polynomial of

F (n) j
is 0 if j is odd, and 1 if j is even.

It is not known whether the knot F (4) 2 = C(4, 4) is Lissajous or not (see [START_REF] Boocher | Sampling Lissajous and Fourier knots[END_REF]).

The modulo 2 Conway polynomial of Fibonacci knots

We shall now study the knots F (n) j , where n = 2k + 1 is an odd integer. Lemma 9. Let n = 2k + 1. We have the identities

[n, n, x] = [n + 1, -2, 2, . . . , -2, 2 2k , -(1 + x)] (2)
[n, n, n, z] = [n + 1, -2, 2, . . . , -2, 2 2k , -(n + 1), -z]. (3)
Proof. Let us prove the first formula. We shall use matrix notations for Möbius transformations. Let

G(u) = [-2, 2, u] = 3u + 2 -2u -1 . Its matrix is G = 3 2 -2 -1
, and consequently we get by induction

G k = 1 + 2k 2k -2k 1 -2k = n n -1 1 -n 2 -n . (4)
Let us define the Möbius transformation

H = G k • Q 2 • R.
We obtain, using Form. ( 4)

H = n 3 + n 2 + 2n + 1 n 2 + 1 -n 3 -n n -n 2 -1 . ( 10
)
Let S be a matrix corresponding to the Möbius map s. We have

S -1 HS = 1 0 1 1 H 1 0 -1 1 = n 3 + 2n n 2 + 1 n 2 + 1 n = n 1
F 3m+2 F 3m = [2, 2, -2, -2, . . . , (-1) m-1 • 2, (-1) m-1 • 2]
Using the corollary 10 we obtain similarly

F 3m+1 F 3m = [2, -2, -2, 2, 2, . . . , (-1) m-1 • 2, (-1) m-1 • 2, (-1) m • 2]
of length 2m and

F 3m+3 F 3m+2 = [2, - F 3m+2 F 3m ] of length 2m + 1.
Of course, these fractions correspond to Fibonacci knots (or links). They are not Lissajous knots because of corollary 6.

It is straightforward to calculate the Conway polynomials of our Fibonacci knots, using Prop. 11.

Theorem 13. Let us denote by ∇ is not a Lissajous knot.

It is not known whether the knot F If n = 1 we obtain all the rational knots ( [START_REF] Koseleff | Chebyshev knots[END_REF][START_REF] Koseleff | Chebyshev diagrams for rational knots[END_REF]). If n = 2 we obtain the important class of rational fibered knots (see [START_REF] Kawauchi | A survey of knot theory[END_REF]).

In general, we obtain knots with fractions α β such that (α, β) ≡ (0, ±1) or (±1, 0) (mod n).

Figure 1 :

 1 Figure 1: Conway's normal forms, m odd, m even

Figure 2 :

 2 Figure 2: Some Fibonacci knots and links

Example 4 (

 4 The torus links). The torus link T(2, m) has Conway normal form C(m) = F We have the continued fraction (of length m -1)

2 Remark 12 .

 212 This means that s [n, n, n, x] = h • s(x), which proves our formula. By considering the case n = 1 where h(x) = [-2, -2, -x], we obtain the following interesting continued fractions of length 2m:

j 11 )

 11 (z) the modulo 2 Conway polynomial of the Fibonacci linkF (n) j . We have ∇ (n) j (z) = f N (z) where If n ≡ 1 (mod 4), N = ⌊ j+2 3 ⌋(n -2) + j + 1, If n ≡ 3 (mod 4), N = ⌊ j+2 3 ⌋(n + 2) -(j + 1). (Corollary14. If n ≡ 0 (mod 4) and (n, j) = (3, 3), the Fibonacci link F (n) j

( 3 ) 3 =

 33 C(3, 3, 3) is a Lissajous knot ([START_REF] Boocher | Sampling Lissajous and Fourier knots[END_REF]). Question 15. It would be interesting to study the wider classes of knots defined by their Conway notation C(±n, ±n, . . . ± n).

Acknowledgements: We would like to thank Pr. C. Lamm for having suggested this problem to us.

Let

M (x) = [n + 1, -2, 2, . . . , -2, 2, -(1 + x)], L(u) = [n + 1, u], T (x) = -x -1.

(

The corresponding matrices are

Consequently

that is M (x) = [n, n, x] which proves the first identity. If we substitute x = [n, z] in Formula (2), we obtain the second identity (3). 2

Corollary 10. Let n = 2k + 1. We have the continued fractions When j ≡ 1 (mod 3), there is no continued fraction expansion of [n] j with even quotients, by Prop. 2. In this case, we shall get a continued fraction expansion for α βα , which is another fraction of the same knot. Let s be the Möbius transformation defined by s(

Proposition 11. Let n = 2k + 1. We have the continued fractions

.