Thiago De

thiagof.pinto Freitas Pinto

Anna Carla Araujo
email: annaaraujo@cefet-rj.br

Carlos Frederico

Trotta Matt

ANALYTICAL, NUMERICAL AND EXPERIMENTAL INVESTIGATION OF THE DYNAMIC BEHAVIOR OF A PROTOTYPE FOR A STOCKBRIDGE DAMPER

Keywords: Stockbridge damper, FEM simulation, Dynamic absorber, Mechanical modeling

Stockbridge damper is a dynamic absorber attached to transmission line conductors in order to suppress their amplitudes of vibration induced by the wind. In order to understand the complex dynamic behavior of a typical Stockbridge damper and to further develop an optimized shape and material to increase the effectiveness of the absorber, a series of numerical and experimental simulations must be done. In the current work, a cantilever beam carried at the free end a tip mass slender in the axial direction was manufactured, experimented and simulated. Such cantilever beam prototype resembles a typical Stockbridge damper though the geometrical complexity of the wire cable has not been considered in this preliminary investigation. The governing equations for the dynamic behavior of the prototype, under some set of simplifying assumptions, are set forth and analytically solved. Numerical simulations using the commercial finite-element software ANSYS are also performed with the full three-dimensional prototype model, relaxing some simplifying hypotheses adopted in the analytical treatment. The analytical and numerical results are finally compared with measurements obtained during laboratory tests performed with the prototype.

INTRODUCTION

The reliability of a high-voltage transmission line system depends largely on the integrity of its overhead conductors. Overhead conductors are subjected to many different mechanical and electrical phenomena in the field. The damages associated with the mechanical vibrations induced by the wind are one of the main causes for forced interruptions in the supply of electric energy. These wind-induced vibrations are caused by the periodic vortex-shedding phenomenon behind the wake of the conductor and are commonly referred to as aeolian vibrations. The vortex-shedding phenomenon produces a pressure difference between the top and the bottom sides of the conductor which induces an alternating lift force with the same frequency as that of the vortex shedding in the wake. These vibrations arise for wind speeds in the range from 1 m/s to 8 m/s, the frequencies expected lie within the range from 5 Hz to 50Hz and the vibration amplitudes are of the order of the conductors' diameter or less [START_REF] Wagner | Dynamics of Stockbridge Damper[END_REF]. If not properly damped, when the predominant vortex-shedding frequency coincide with one of the resonance or natural frequencies of the conductor, the bending stresses and strains may become too high and may lead to severe fatigue damages at the conductor clamps, leading to the complete rupture of the conductor.

Overhead conductors possess very low internal damping in the frequency range expected for aeolian vibrations in the field. Modifications in the vortex-shedding frequency by introducing perturbations in the wind flow have been proved to be not a good strategy because the conductors possess a large number of resonance frequencies closely spaced. Hence, the best strategy verified has been to attach dynamic absorbers to the conductors in order to remove part of the energy input by the wind, as indicated in Fig. 1. Stockbridge dampers are the most widely used dynamic absorbers attached to transmission line conductors in order to suppress the wind-induced vibrations and, consequently, to minimize the susceptibility of conductors to fatigue damages.

Stockbridge dampers, shown in Fig. 1, generally comprise two inertial masses or counterweights resiliently suspended at the tips of a flexible steel wire cable, commonly referred to as messenger cable, and attached to the conductor at its midpoint through a rigid bolted clamp. In the majority of Stockbridge dampers available at the market, the messenger cable comprises six or eighteen wires helically wrapped around a central straight wire; the wires are normally made of galvanized steel. The purpose of these devices is to absorb part of the energy input by the wind so as to drop down the vibration amplitudes to a security level. Mechanical energy is also dissipated in these devices due to the friction among the wires of the messenger cable during its bending vibrations. When a transmission line conductor equipped with Stockbridge dampers is subjected to forced vibrations (as in the field due to wind), these dampers interact with the conductor by exerting a transverse force and a bending moment at the attachment point. The concentrated transverse force and bending moment may increase too much the local curvature of the conductor leading to large stresses and strains in the neighborhood of these attachment points [START_REF] Hagedorn | On the computation of damped wind-excited vibrations of overhead transmission lines[END_REF]. Hence, accurate predictions of the dynamic behavior of such damping devices become an important task. Although Stockbridge dampers are easily manufactured, experimental tests performed on a laboratory indicate that these devices possess a quite complex dynamic behavior (Varella et al., 2007). As a first step towards the understanding of the dynamic behavior of a typical Stockbridge damper, in the current work, a simpler prototype resembling a Stockbridge damper is to be analyzed. The prototype comprises a uniform steel cantilever beam with circular crosssection carrying a tip mass slender in the axial direction, schematically illustrated in Fig. 2. The tip mass is made of the same steel as the beam and its gravity center does not coincide with its attachment point to the uniform beam. The major difference between the prototype investigated here and a typical Stockbridge damper relies on the cantilever uniform beam, whose stiffness and damping properties are distinct from the corresponding ones of an actual messenger cable. Table 1. Prototype dimensions (the symbols are indicated on Fig. 2).

Dimension

Value (mm)

φ A 9 φ B 29 C 49 L 149

BRIEF LITERATURE REVIEW

Previous researchers have also investigated the dynamic behavior of a cantilever beam carrying a tip mass at the free end. For instance, the exact frequency equation for a uniform cantilever beam carrying a concentrated tip mass without moment of inertia was considered by Pipes (1958), Prescot (1946), Temple and Bickley (1956) and Parnell and Cobble (1976). [START_REF] Durvasula | Vibrations of a Uniform Cantilever Beam Carrying a Concentrated Mass and Moment of Inertia and a the Tip[END_REF] derived the exact frequency equation for free vibrations of a uniform cantilever beam carrying a tip mass with concentrated moment of inertia. However, the concept of a concentrated mass or moment of inertia is often violated when the tip mass is slender in the axial direction and the attachment point does not coincide with the gravity center of the tip mass, as is the case for the prototype investigated here and for a typical Stockbridge damper. In these situations, the boundary conditions at the free end of beam must be modified accordingly which increases the complexity of the problem. By using perturbation expansion techniques, Rhama Bhat and Wagner (1976) derived approximate equations to compute the undamped natural frequencies of a cantilever beam carrying a tip mass slender in the axial direction. The numerical results obtained by those authors for the natural frequencies were tabulated for selected values of non-dimensional parameters, which represent ratios between the inertia and stiffness values of the tip mass and of the beam. It is worthwhile to mention that the numerical results obtained by Rhama Bhat and Wagner (1976) lack accuracy when the hypothesis ε ≡ l/L << 1 (see Fig. 2) is not fulfilled.

Two major improvements are proposed in the current work for the theoretical modeling of the prototype shown in Fig. 2. First, an exact frequency equation is derived which allows one to compute the natural frequencies for the cantilever beam-tip mass system; such frequency equation remains accurate even when the hypothesis ε ≡ l/L << 1 is violated. Second, one derives an analytical expression for the transmissibility when the clamped end of the beam is subjected to harmonic displacements. The theoretical results for the transmissibility are then compared with the measured ones obtained during tests performed on a laboratory in order to assess the validity of the theoretical model proposed.

THEORETICAL MODELING

The simplifying hypotheses adopted in the theoretical model proposed here are: (i) the tip mass behaves as a rigid body; (ii) the rotary inertia and shear deformation of the beam may be neglected and (iii) both the beam and tip mass displacements and rotations are small. With the above assumptions, the transverse vibrations of the prototype shown in Fig. 2, in the absence of external forces, are described mathematically by the partial differential Eq. ( 1) subjected to the boundary conditions described by Eqs. ( 2), ( 3) and ( 4

). 0 ) , , ( ) , ( ) , ( 2 2 4 4
= + ∂ ∂ + ∂ ∂ w w t f t t x w x t x w EI d & µ (1) t W t w Ω = i 0 e ) , 0 ( , 0 ) , 0 ( ' = t w at x = 0 (2) ( ) ) , ( ) , ( ' ) , ( ' ' 2 t L w l m t L w l m J t L w EI - = at x = L (3) ) , ( ' ) , ( ) , ( ' ' ' t L w l m t L w m t L w EI & & & & + = at x = L. ( 4
)
In Eqs. ( 1)-( 4), x denotes the axial coordinate along the beam length; t denotes the time; w(x,t) denotes the transverse displacement field; EI and µ denote, respectively, the bending stiffness and the mass per unit length of the beam; m and J G denote, respectively, the tip mass and the tip moment of inertia about the gravity center C-G; l denotes the axial distance between the attachment point of the tip mass to the beam and the center of gravity of the tip mass; W 0 and Ω denote, respectively, the amplitude and the circular frequency for the prescribed displacement at the clamped end x = 0; and, finally, ) , , (

w w t f d
& denotes the damping forces. Here and in what follows, the dot and prime symbols over a variable denote, respectively, temporal and spatial derivatives. The bending stiffness of the beam may be computed through the expression:

64 4 A E EI φ π = (5)
where φ A denotes the nominal diameter of the beam whereas E is the Young modulus for the steel (E = 209 GPa).

Before solving the boundary-value problem given by Eqs. ( 1)-( 4), one must adopt a model for the damping forces. Several models for structural damping are available in the literature; here, as a first investigation, one adopts a quite simple and widely used hysteretic damping model for which one may write

t t x w h w w t f d ∂ ∂ Ω = ) , ( ) , , ( & (6)
where the parameter h is commonly referred to as the hysteretic damping parameter. Previous researchers correlate the hysteretic damping parameter with the logarithmic decrement [START_REF] Wagner | Dynamics of Stockbridge Damper[END_REF], a quantity measured on a laboratory with relatively ease. With the hysteretic damping model given by Eq. ( 6), the boundary-value problem defined by Eqs. ( 1)-(4) becomes linear; hence, one might expect that the steady-state vibrations are also harmonic in time; in other words, the transverse displacement field may be written as

t x W t x w Ω = i e ) ( ) , (
where W(x) denotes the (complex) amplitude of the displacement and i denotes the imaginary unity (i 2 = -1) . Substituting the above expression for w(x,t) into Eqs. ( 1)-( 4) one derives the following boundary-value problem for the complex amplitude W(x):

0 ) ( i ) ( 2 4 4 =         - Ω - x W EI h dx x W d µ (7) 0 ) 0 ( W W = , 0 ) 0 ( ' = W at x = 0 (8) (
)

) ( ) ( ' ) ( ' ' 2 2 2 L W l m L W l m J L W EI G Ω + Ω + = at x = L (9) ) ( ' ) ( ) ( ' ' ' 2 2 L W l m L W m L W EI Ω - Ω - = at x = L (10)
The analytical solution of the boundary-value problem defined by Eqs. ( 7)-( 10) may be cast in the following form:

x x x x U U U U x W ) ( -i 4 ) ( i 3 ) ( 2 ) ( 1 e e e e ) ( Ω Ω Ω - Ω + + + = β β β β (11)
where the four components of vector U are the solution of a linear system of equations of the kind KU = F, for which the components of the coefficient matrix K and load vector F are given by 1 14 13 12 11

= = = = K K K K ; i ; i ; 1 ; 1 24 23 22 21 - = = - = = K K K K (12) L L G L l m l m J EI K ) ( 2 ) ( 2 2 ) ( 2 31 e e ) ( ) ( e ) ( Ω Ω Ω Ω - + Ω Ω - Ω = β β β β β (13) L L G L l m l m J EI K ) ( 2 ) ( 2 2 ) ( 2 32 e e ) ( ) ( e ) ( Ω - Ω - Ω - Ω - + Ω Ω + Ω = β β β β β (14) L L G L l m l m J EI K ) ( i 2 ) ( i 2 2 ) ( i 2 33 e e ) ( ) ( i e ) ( Ω Ω Ω Ω - + Ω Ω - Ω - = β β β β β (15) L L G L l m l m J EI K ) ( i 2 ) ( i 2 2 ) ( -i 2 34 e e ) ( ) ( i e ) ( Ω - Ω - Ω Ω - + Ω Ω + Ω - = β β β β β (16) L L L l m m EI K ) ( 2 ) ( 2 ) ( 3 41 e ) ( e e ) ( Ω Ω Ω Ω Ω + Ω + Ω = β β β β β (17) L L L l m m EI K ) ( 2 ) ( 2 ) ( 3 42 e ) ( e e ) ( Ω - Ω - Ω - Ω Ω - Ω + Ω - = β β β β β (18) L L L l m m EI K ) ( i 2 ) ( i 2 ) ( i 3 43 e ) ( i e e ) ( i Ω Ω Ω Ω Ω + Ω + Ω - = β β β β β (19) L L L l m m EI K ) ( i 2 ) ( i 2 ) ( i 3 44 e ) ( i e e ) ( i Ω - Ω - Ω - Ω Ω - Ω + Ω = β β β β β (20) 0 ; 4 3 2 0 1 = = = = F F F W F (21) and 4 2 i ) ( EI h - Ω = Ω µ β . ( 22
)
It should be emphasized that when W 0 = 0 the boundary-value problem ( 7)-( 10) reduces to a non-self-adjoint eigenvalue problem (Mennicken and Möller, 2003) of the kind KU = 0, for which nontrivial solutions exist if and only if the determinant of the coefficient matrix K is zero. The numerical values of the circular frequency Ω which satisfy the null determinant equation for the coefficient matrix K correspond to the natural frequencies of the prototype shown in Fig. 2. Finally, the transmissibility of the prototype, T(x,Ω), is defined as the ratio between the acceleration at a given location along the beam length and the acceleration imposed at the clamped end x = 0; therefore, one may write

x x x x t t W U W U W U W U W x W W x W x T ) ( i - 0 4 ) ( i 0 3 ) ( 0 2 ) ( 0 1 0 i 0 2 i 2 e e e e
) ( e e ) ( ) , (

Ω Ω Ω - Ω Ω Ω + + + = = Ω - Ω - = Ω β β β β . ( 23
)
A mathematical software was used to compute analytical solutions and Figures ( 3) and ( 4) show the transmissibility results derived from Eq. ( 23) at the two measurement points selected during the experimental tests with the prototype, namely, point 1 at x = L + C/2 and point 2 at x = L/2 (details about the experimental tests will be described in section 4).

The transmissibility results shown in Figs.

(3) and ( 4) are computed for excitation frequencies ranging from 5 Hz to 5.605 kHz with a frequency resolution of 6.25 Hz, which is the frequency resolution adopted in the experimental tests. Moreover, in the computations performed with Eq. ( 23), the hysteretic damping coefficient is set equal to zero. The first, second, third and fourth natural frequencies calculated analytically are, respectively, 61 Hz, 741 Hz, 2.35 kHz and 5.51 kHz and correspond to the frequencies associated with the four peaks shown in Figs.

(3) and ( 4).

NUMERICAL SIMULATION

The prototype was modeled and simulated to understand its behavior when exposed to harmonic functions. The finite element software ANSYS was used with this purpose as it is commercially sated to find mechanical vibrations, frequency response function and frequency modes.

The dimensions shown on table 1 were used to manufacture the prototype and also as input for the tridimensional numerical simulation.

The material for the prototype was assumed as an elastic isotropic material with elastic module E= 209GPa and specific weight ρ= 7860kg/m³. Solid Brick 8 node 45 was used for the 3-D mesh modeling. This element type is defined by eight nodes and three degrees of freedom for each node. The element has plasticity, creep, swelling, stress stiffening, large deflection, and large strain capabilities.

The mathematic method used by the software on this simulation was the Full method due to its simplicity and its completeness compared to the other two methods available on ANSYS (reduced method and modal superposition method). This method makes use of the full stiffness and mass matrices and thus is the slower and costlier option. The tetragonal mesh was used.

The boundary restrain conditions used to the simulation had the objective to approximate to the experimental situation. The beam was fixed by the smaller diameter and this basis had a prescribed harmonic displacement. Several different forced frequencies were used between the range of frequency from 5 to 6000Hz with a frequency resolution (sub-step) of 22 Hz. For each frequency the simulation was performed and the transmissibity for the range was analyzed.

On Figure 5 can be seen the forth first modes. The first and second models dissipate bending energy. The third mode compress the beam and the forth mode dissipates a torsion elastic energy.

On Figures 6 and7 the relative displacement of two points located on the beam in respect to the prescribed displacement boundary. Analyzing those Figures, it can be considered that 60, 750, 2300 and 5300 Hz were taken as natural frequencies related to the four modes.

On Figure 6, the transmissibity taken on the mass (x=L+C/2) is presented and it can be noted that the higher amplitude was on the forth natural frequency. On Figure 7, the transmissibility related to a point located on the middle of the messenger cable and it can be noted that only two frequencies had a higher response.

EXPERIMENTAL TESTS

Experimental tests were performed with the prototype on a laboratory in order to compare the analytical and numerical results obtained for its transmissibility with the corresponding measured ones. The prototype was mounted on an electrodynamics shaker and three piezoelectric accelerometers were fixed on different positions for measurements of the excitation acceleration and response accelerations. Figure 8 illustrates a schematic view of the experimental apparatus for the laboratory tests performed. As indicated on Fig. 8, the accelerometers labeled as 1, 2 and 3 and attached to the prototype were electrically connected to an amplifier, a signal conditioner, an analogical-digital converter. The digital signals measured by the accelerometers were collected by the computer. The prototype was fixed at the shaker which transformed the amplified signal from the computer and a Digital-analogical converter to mechanical oscillations.

The selected excitation signal generated by the computer was a swept sine, that is, a harmonic waveform with variable frequency. The swept sine had 200mV RMS amplitude and its frequency changes from 5 Hz to 8 kHz with a linear sweep rate equal to XX Hz/s. The measured transmissibility at a given position was computed from the crosscorrelation function between the acceleration imposed at the clamped end and the acceleration measured at that position. Thirty independent samples of the acceleration signals in the time domain were used to compute the average cross-correlation function.

In Figs. 9 and 10, one illustrates graphically the ratio between the acceleration amplitudes at the measurement points 1 and 2 and the acceleration amplitude at the clamped end (point 3) as a function of the excitation frequency. Three natural frequencies noted on experimental results were 62,5, 75* Hz, 675* Hz, 900Hz, 2800 Hz and 3370 Hz. (* only in Figure 9)

CONCLUSIONS

In this work a prototype comprising a cantilever beam carrying a tip mass at the free end and resembling a typical Stockbridge damper was manufactured and tested on a laboratory. The prototype tested had not the exact geometrical shape of a typical Stockbridge damper and the uniform cantilever beam had geometric, stiffness and damping properties different from the corresponding ones of a typical messenger cable. The experimental tests were designed in order to identify the natural frequencies of the prototype and to measure its transmissibility when the clamped end of the cantilever beam was subjected to forced harmonic displacements in a wide frequency range. With an analytical onedimensional continuous Euler-Bernoulli beam model and a full three-dimensional numerical model one computed the natural frequencies and the transmissibility for further comparison with the corresponding measured data. The analytical and numerical results obtained for the natural frequencies and transmissibility did not take into account the damping effects in the prototype. Table 2 presents the measured, analytical and numerical results for the first four natural frequencies of the prototype.

Figure 1 .

 1 Figure 1. A typical Stockbridge damper attached to an overhead conductor.

Figure 2 .

 2 Figure 2. The prototype for a Stockbridge damper investigated in the current work. The dimensions of the prototype are indicated on Tab. 1.

Figure 3 .Figure 4 .

 34 Figure 3. Transmissibility at the mass position (position 1, x = L+C/2) obtained with the theoretical model.

Figure 5 .Figure 6 -Figure 7 .

 567 Figure 5. The first four natural modes of vibration for the prototype investigated.

Figure 8 -

 8 Figure 8 -Experimental set-up.

Figure 9 -

 9 Figure 9 -Transmissibility at point 1 obtained experimentally.

Figure 10 -

 10 Figure 10 -Transmissibility at point 2 obtained experimentally.

Table 2 .

 2 Measured, analytical and numerical results for the first four natural frequencies of the prototype. Natural Frequencies

& & & & -+

Analytical Results

Numerical Results

Experimental From the analysis of Tab. 2 one may draw the following conclusions. First, the analytical and numerical results for the first four natural frequencies of the prototype are in very good agreement; the maximum relative error between them is less than 4%. For the first natural frequency the three models were almost coincident. However, large discrepancies are verified when both the analytical and numerical results for the second, third and fourth natural frequencies when compared with the corresponding measured ones. The second, the third and the fourth natural frequencies of the prototype by, respectively, at least 20%, 17% and 61% were both the analytical and numerical models underestimate.

The first mode is the one that the accelerometer was positioned to measure, as it can be seen on Figure 5(a) and 8. It should be investigate the possibility of taking more data with other sensors and analyze the complexity of all the modes and displacements.

Concerning the transmissibility results, from the analysis of Figs. 3, 4, 6, 7, 9 and 10, one may conclude that both the analytical and numerical results reproduce qualitatively the dynamic behavior of the prototype verified experimentally on a laboratory (one may note the remarkable agreement between the plots shown in Figs. 4 and 10 in the frequency range from 0 to 1 kHz); nevertheless, quantitatively, both the analytical and numerical models are unable to reproduce the magnitudes of the transmissibility measured at the positions 1 and 2. The reasons for the large discrepancies might be attributed to the neglect of damping effects in the preliminary theoretical investigations performed in the current work. Because both the analytical and numerical transmissibility are computed for no damping, their magnitudes change appreciably depending on the frequency resolution. The authors recognize that much more research efforts must be done both theoretically and experimentally for more conclusive remarks about the validity of the theoretical models proposed and will be the subject of future investigations.

AKNOWLEDGEMENTS

The authors acknowledge the support of CEFET/RJ, CEPEL, Brazilian Research Councils CNPq and FAPERJ.

RESPONSIBILITY NOTICE

The authors are the only responsible for the printed material included in this paper.


Anna Carla Araujo

Marcelo Amorim Savi
email: savi@mecanica.ufrj.br

Pedro Manuel Lopes

Calas Pacheco

Experimental

EXPERIMENTAL AND NUMERICAL DYNAMICAL ANALYSIS OF AN END MILLING PROCESS

Keywords: End milling, force prediction, nonlinear dynamics, nonsmooth systems, chatter, machining

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Machining process is associated with a complex dynamics that involves the coupling of different phenomena. During cutting processes, for example, it is common to have temperature variations that induce dramatic changes in expected behaviors. Although all these complexity, some simple models may be used in order to obtain useful information concerning tool and workpiece behaviors as dynamic tool prediction.

The machine tool vibration during the cutting process plays an important role concerning the workpiece surface quality and also the tool durability. Chatter and squeal are some undesirable phenomena related to an improper functioning. During certain cutting conditions, motions of the workpiece-tool system are characterized by large amplitudes, which are not desirable for obtaining a good surface finish. Undesirable motions can result in wavy surfaces on the workpiece, inaccurate dimensions, and excessive tool wear.

The analysis of machining using a dynamical approach has been done in different research efforts. The first works in this way is due to [START_REF] Tobias | Machine tool vibration[END_REF], [START_REF] Merritt | Theory of self-excited machine tool chatter[END_REF] and Tlusty [START_REF] Altintas | Manufacturing automation[END_REF] that investigates the chatter on the process. Their research provided instability diagrams informing safe parameters to avoid chatter. After them, a new approach has been developed using nonlinear dynamic tools as bifurcation analysis, providing better comprehension of the process (Moon, 1978) .

Another references treated machining operations by dynamical approach. For turning operations, [START_REF] Chandiramani | Dynamics of 2 dof regenerative chatter during turning[END_REF] analyzed the dynamics of cutting considering a two-degree of freedom system (2-dof) and orthogonal cutting modeling to predict chatter. [START_REF] Pratt | Design and modelling for chatter control[END_REF] studied the boring bars for turning, which commonly presents chatter problems and experimentally determined modal properties of the tool. Also dedicated to turning, [START_REF] Kalmar-Nagy | Subcritical Hoft bifurcation in the delay equation model for machine tool vibrations[END_REF] showed the existence of a subcritical Hopf bifurcation in the delay-differential equation used to describe the machining equations of motion. The stability of the milling process was investigated by [START_REF] Zhao | Dynamics and stability of milling process[END_REF], [START_REF] Insperger | Stability of up-milling and down-milling, part 2: experimental verification[END_REF] and [START_REF] Gradisek | On stability prediction for milling[END_REF] by considering single and two degree of freedom systems for different experimental conditions. The mathematical model is represented by delay-differential equations.

Milling process has the peculiarity to have a contact/non-contact behavior due to its geometry and due to tool vibration. This kind of behavior is related to nonsmooth systems that are usually associated with either the friction phenomenon or the discontinuous characteristics as intermittent contacts [START_REF] Savi | Numerical and experimental investigations of the nonlinear dynamics and chaos in non-smooth systems[END_REF] .

This article investigates the milling process either by experimental or by numerical approaches. The mathematical model assumes that the end milling tool is represented by a nonsmooth mass-spring-dashpot system [START_REF] Araujo | Analysis of an End Milling Process-Encontro de Modelagem Computacional -Nova Friburgo[END_REF] . The process of cutting is related to a stick-slip behavior that defines whether the chip is being removed from the workpiece. The tool holder displacement is prescribed and the local force that allows the slip motion is related to the workpiece shear stress. Under these assumptions, the equation of motion is represented by a differential equation that is solved employing the Runge-Kutta method. Numerical simulations are carried out showing some situations related to proper and improper functioning during the cutting process.

DYNAMICAL MODEL

The milling process is of concern by assuming that it is a full immersed milling in the x direction [START_REF] Araujo | Dynamical Analysis of an End Milling Process[END_REF]. Figure 1(a) shows the general view of the tool and the workpiece, where it is identified the prescribed displacement, X h and the tool tip displacement, X t . This process is modeled by assuming a nonsmooth system that is composed by a primary system that represents the tool and a secondary system, representing the workpiece. The primary system consists of a linear spring-dashpot-mass oscillator with parameters m, k, c and displacement X t , and presents a gap that separates itself from the secondary system that represents the work-piece.

The workpiece system has weightless slider connected to a linear spring-dashpot system with parameters k s , c s that can present progressive motion when the acting force exceeds F h , here represented by a dry friction, and displacement X p , the tool-work piece contact position. Similarly to the stick-slip phenomena reported by Marian ( 2001), the progressive motion X c occurs when the force acting on the slider exceeds the threshold of the dry friction force. Figure 1(b) presents a schematic picture of the cutting process system related to the milling. As the system may operate in contact/non-contact situations, its dynamical behavior is nonlinear and its equations should consider each case. The system dynamics may be understood as a three-stage motion as represented in Figure 2: non-contact, contact in stick and contact in slip. The non-contact stage may be understood as two independent systems representing the situation where the tool is not in contact with the workpiece. Figure 2(a) shows that the displacements X t and X h are not connected with X c and X p by the mass.

The gap g is used to locate the work-piece position related to the tool position. This variable remains equal to X t (as in Figure 2a) in non-contact case and in stick phase, as shown in Figure 2. The displacement g changes only when the slider slips simulating the chip removal. As long as g is greater than X t , the system remains separate. When there is no more gap between X t and X p , both systems are in contact and the dynamic changes, as shown in Figure 2

Non-contact Stage

In non-contact stage, when X t < g, the secondary system has no movement. Its displacement does not change Ẋp = Ẋc and the primary system is described by the dynamic equation of a single-degree of freedom system, as follows:

m( Ẍt -Ẍh ) + c( Ẋt -Ẋh ) + k(X t -X h ) = 0 (1)
The holder force is calculated by:

F h = -k(X t -X h ) -c( Ẋt -Ẋh ) (2)
In order to rewrite the equation 2 for either contact and non-contact situations, the Heaviside function H(.) is included. Note that the damping coefficient when the tool retracts from the workpiece ( Ẋt < 0) is not considered.

F h = -k(X t -X h ) -c( Ẋt -Ẋh ) -H[X t -g] k s (X t -X c ) + H[ Ẋt ]c s Ẋt (3)
The difference between chip displacement X c and work-piece displacement X p when the secondary system is free is constant and it is defined as δ, used to calculate the chip movement X p on slip phase.

Contact -Stick Stage

The chip, represented by the slider, is not removed in stick stage. This case occurs when the friction force F f is less than a maximum friction force F f supported by the workpiece material, which can be considered as a function of the shear area and the maximum shear stress.

F f = A s τ s (4)
The friction force is calculated, in stick phase, by:

F cont = H[X t -g] k s (X t -X c ) + H[ Ẋt ]c s Ẋt ) (5)

Contact -Slip Stage

During slip stage, when the chip is being removed, the maximum friction force is achieved. The chip position X c follows the workpiece displacement, X c = X p + δ, and g assumes the work-piece position g = X p . The friction force is equal to the maximum friction force F h = F f .

The force applied in the tool tip cannot be calculated by equation (3) due to the difference between k s (X t -X c ) + H[ Ẋt ]c s Ẋt ) and F f . Therefore, during slip phase the force F h is:

F h = -k(X t -X h ) -c( Ẋt -Ẋh ) -F f (6)

Equations of motion

Under these assumptions, the system dynamics may be represented by a nonsmooth system as follows, using F cont described on Eq (5) and F f on Eq (4):

m( Ẍt -Ẍh ) + k(X t -X h ) + c( Ẋt -Ẋh ) = F cont if F cont < F f F f elseif (7)

System parameters

The machine tool is represented by the tool holder and it has a prescribed displacement X h defined by the feed velocity v f , the spindle speed w and the run-out distance ρ between the the tool axis and the spindle speed axis. All these aspects are considered by the following equation:

X h (t) = v f t + ρ sin(wt) + X h0 (8)
where X h0 is constant and the feed velocity is writen as:

v f = f t wz (9)
The run-out, ρ, is calculated based on the tool tip displacement. Therefore, the tool is only twisting without any tool deflection nor cutting force. The milling tool is considered as a flexible body with linear elastic behavior, therefore, the tool parameters m, k and c are calculated by considering solid mechanics principles from the tool geometry and material properties.

The stiffness of the primary system k is measured in laboratory and an the workpiece stiffness is evaluated by assuming that the workpiece is a compression beam, therefore:

k s = E c dt 50L ( 10
)
where workpiece material Young modulus E c is considered as 200 GPaand t is the depth of cut.

The uncut chip thickness is not considered as in Martellotti equation (Araujo et al, 2006). The dry fiction is evaluated from the shear strength calculated by the shear area as a function of the depth of cut t and approximated by:

A s = π 0 tf t sinφdφ (11)
The chip thickness is given by the simulation as the displacement steps that the work-piece material let the tool cut. In order to improve the surface quality, the chip steps must be as smaller as possible. Also, the tool cannot touch and leave the work-piece with large displacements, because it causes impacts on the tool. The spindle speed was defined as 2500 rpm and the feed per tooth 0.007 mm. The tool has diameter as 5 mm and the tool was fixes in 39.9mm from the tool tip. The elastic modulus of the tool and those parameters are presented in Table 1.

Table 1 -Tool, workpiece and machine tool properties. Diameter d Height L Elastic Modulus E t Spindle Speed (w) Feed per tooth f t 5 mm 39.9 mm 200 GPa 2500 rpm 0.007 mm

Experimental Results

Initially, it is considered the non-cutting dynamics of the system. By imposing the spindle speed as 2500 rpm as in milling, the system response is shown in Fig. 3. The runout calculated is assumed to be as half signal amplitude of free tool tip Displacement and the result has amplitude of approximately 20 microns.

After this first approach, the stiffness of the primary system k is measured in laboratory by imposition of a measured force and measuring the tip tool displacement. The measured value is 640 N/m. At this point, the influence of depth of cut on the stability and dynamic behavior of the cutting process is of concern. Basically, different situations are of concern varying the depth of cut from 0.25mm to 2.25mm. Fig. 4 shows the cutting force for all experiments while Fig. 5 presents the tool tip displacement. The phase space for all experiments is presented on Fig. 6. Results have a periodic aspect being related to sub-harmonic oscillations. For numerical simulation it was assumed that X h0 is 25 microns. The distance between the tool holder and the workpiece in t = 0 is 0.001 mm. Moreover, it is assumed that the workpiece material presents plastic behavior. Under this assumption, the damping coefficient for the workpiece is defined as a function of the damping tool coefficient (c s = 105c). Tool damping is considered to be 0.002 N s/m [START_REF] Mann | Milling bifurcation from structural asymmetry and nonlinear regeneration[END_REF]. Note that for Ẋt < 0, when the tool move back from the workpiece, c s vanishes. The material considered presenting yield shear stress as τ s = 1000MPa.

Numerical simulations are carried out in order to evaluate the model capability to describe milling process. The governing equations are solved by considering the fourth order Runge-Kutta method with time steps less than 5.10 -4 s. Basically, we want to establish a qualitative comparison between the proposed model with experimental data presented in the preceeding section. Fig. 7 presents numerical results for 1.5 mm of depth of cut. Basically, it presents the time-history of force and displacement, together with the phase space. Concerning the cutting force analysis, it presents values that are two times lower than experimental one but the force profile is close to the chip tool area profile, showing that the model needs improvements but is not far from modeling machining cutting process. The analysis of displacements (X h , X t , X p and X c ) and the tool tip displacement (X t -X h ) shows that the model is capturing the process behavior. They presents the same periodic aspects, being related to sub-hamonic response. The amplitude of the motion has the similar values, showing a qualitative agreement with experimental data.

CONCLUSIONS

This article presents a dynamical investigation of the milling process. Experimental and numerical results of cutting force and tool tip displacement for end milling processes are of concern. The cutting mechanics is modeled as a 2dof nonsmooth system. The experimental data is collected in different depth of cuts maintaining the spindle speed and cutting velocities. Numerical results are developed in order to establish a qualitative comparison with experimental data. In general, the proposed model is capturing the general behavior of experimental data. Nevertheless, it is important to

Figure 1 -

 1 Figure 1 -Tool and system modeling in contact and non-contact situations

  (b), for stick stage, and Figure 2(c) for slip stage. In stick stage the slider displacement does not change and the system is restricted to a fixed boundary. In slip stage this boundary is free and X c displacement changes. The equations of motion are formulated by considering each kind of response separately.

  Figure 2 -Experimental Set-up

  Figure 3 -Free Tool Tip Displacement

  (a) p f = 0.25mm (b) p f = 0.5mm(c) p f = 0.75mm (d) p f = 1.0mm (e) p f = 1.5mm (f) p f = 1.75mm (g) p f = 2.0mm (h) p f = 2.25mmFigure 4 -Experimental Cutting Force -Depth of cut from 0.25 to 2.25 mm (a) p f = 1.0mm (a) p f = 1.5mm (c) p f = 1.75mm Figure 5 -Experimental Tool Tip Displacement (a) p f = 0.25mm (b) p f = 0.5mm(c) p f = 0.75mm (d) p f = 1.0mm (e) p f = 1.5mm (f) p f = 1.75mm (g) p f = 2.0mm(h) p f = 2.25mm Figure 6 -Experimental Phase Space -Depth of cut from 0.25 to 2.25 mm 4. NUMERICAL SIMULATION

  (a) Cutting Force F cont and Holder Force F h (b) Secundary System Displacement -X p (c) Phase Space Figure 7 -Numerical Results

highlight that the model needs to be improved, especially concerning the cutting force behavior, in order to be used to predict the dynamic behavior of machining processes. The authors agree that this kind of model can be useful to choose important parameters to improve tool behavior.

ACKNOWLEDGEMENTS

The authors acknowledge the support of the Brazilian Research Councils CNPq and FAPERJ.

RESPONSIBILITY NOTICE

The author(s) is (are) the only responsible for the printed material included in this paper


Fabio Vinícius Castilho
email: fabioviniciuscastilho@yahoo.com.br

Mariana Pimenta

Braga Braga

Anna Carla Araujo

Pedro Manuel Lopes

Calas Pacheco

Experimental

Vinícius Fábio

Castilho

Mariana Pimenta Braga
email: marianapimentabraga@yahoo.com.br

Pedro Manuel Calas
email: calas@cefet-rj.br

Lopes Pacheco

EXPERIMENTAL AND NUMERICAL ANALYSIS OF END MILLING TOOLS DYNAMIC PARAMETERS

Keywords: End Milling, Machining Models, FEM, Chatter

published or not. The documents may come

INTRODUCTION

Several machining operations are executed by a large variety of machine-tools in order to produce all needed shapes and tolerances. Being under the desired tolerances is a target for the machine tools to produce non rejected pieces. Machined pieces presents dimensional or geometry error as a function of the inherent process and also due to vibrations and misplacing of tool and workpiece. Controlling the workpiece is a problem when chatter appears due to low stiffness of tool-machine-tool system or workpiece specially when using high cutting velocities values, as in HSM [START_REF] Patel | Uncharted islands of chatter instability in milling[END_REF]. Stability affects directly the precision and quality of workpiece and wear and breakage of the tool. Large tool displacement is one of the causes of damage on the mechanical system, due to lower stiffness. One very important factor to avoid the appearance of chatter is controlling the stiffness of the machine-tool system [START_REF] Zhao | Dynamics and stability of milling process[END_REF].

Several papers has been published to model and predict the dynamic behavior of the system, for example the articles on milling process from [START_REF] Mann | Chatter Vibration and surface location error prediction for helical end mills[END_REF], Inspegerger et al (2003) and [START_REF] Albrecht | High frequency bandwidth cutting force measurement in milling using capacitance displacement sensors[END_REF], nevertheless it is necessary to run impact tests to estimate stiffness and damping coefficients and model the machine and the tool holder to avoid unexpected instability. [START_REF] Altintas | Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design[END_REF] built a model for chatter prediction on milling that uses the impact test procedure. For each tool-holder condition, it is necessary to run a set of experiments to input on the model the dynamic parameters and the lobes diagram can be used to predict chatter. [START_REF] Araujo | Experimental and Numerical Dynamical Analysis of an End Milling Process[END_REF] and Araujo and Savi (2008) described the dynamic of milling using a mathematical model that assumes the end milling tool as a non smooth mass-spring-dashpot system. The process of cutting in that is related to a stick-slip behavior that defines whether the chip is being removed from the workpiece. The tool holder displacement is prescribed and the local force that allows the slip motion is related to the workpiece shear stress and the equation of motion is represented by a differential equation that is solved employing the Runge-Kutta method. The dynamic parameters used on this article were estimated based on mechanical assumptions. It is needed an experimental ou numerical procedure in order to approximate the assumptions to the real parameters.

This article presents a comparison between a numerical study and experimental procedure of the dynamic parameters of an end milling tool and its tool holder. The results were used as input parameter on a known chatter model proposed by [START_REF] Altintas | Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design[END_REF] as a first step of the research (Castilho e Braga, 2010). The methodology implemented on this article is applied for that specific chatter model but can be used as input data in other dynamic modeling for end milling [START_REF] Araujo | Experimental and Numerical Dynamical Analysis of an End Milling Process[END_REF].

END MILLING TOOL DYNAMICS

Fundamentals of forced vibrations

The end milling tool is similar to a cylindrical beam fixed in one extremity by the tool-holder and forced on the other side by the contact with the workpiece under cutting process. The dynamic of the system composed by the end milling tool, tool-holder, machine-tool and the workpiece can present complex nonlinear behavior associated to extreme conditions, as chatter, which can severely affect the quality of the machined workpiece. Modeling can be very useful for establish optimal operational conditions.

Several authors proposed simple models to study tool dynamics during machining using one (Inspegerger et al, 2003;Patel, Mann e Young, 2008) or two (Inspegerger et al, 2008;[START_REF] Kim | Prediction of chatter in NC machining based on a dynamic cutting force model for ball end milling[END_REF][START_REF] Mann | Chatter Vibration and surface location error prediction for helical end mills[END_REF] degrees of freedom mass-spring-dashpot mechanical system models. For one degree of freedom, the equation that represents the behavior of the tool tip displacement x and its relation to the force applied at this position is:

(1) Where m is the mass, k is the stiffness and c is the damping coefficient of the tool system. The external force F(t) is promoted by the workpiece to the tool during the machining process and t is the time. Although the simplicity, these models can capture important complex behaviors of the tool and furnish important information about the tool dynamics.

The characterization of the main parameters can be achieved through simple tests as from a condition of free vibration, where a tool free end initial displacement, x 0 , is prescribed at t = 0. It is well know that for this condition the equation of motion for the tool free end with F(t)=0 can be expressed as [START_REF] Altintas | Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design[END_REF]:

(2) Where ω n = (K/m) 1/2 is the undamped natural frequency,

ω d = ω n [(1-ξ) 2 ] 1/2 is the damped natural frequency, ξ = c/ [2 (K m) 1/2 ]
is the damping ratio and xo is static displacement.

Parameters k and ξ can be obtained from a simple free vibration experimental data using the equations above and the logarithmic decrement technique. The data signal peaks x 1 , x 2 … x n , observed at time instants t 1 , t 2 … t n , can be used to predict the stiffness and the damping ratio:

(3) The determination of accurate values for these parameters is essential to permit a precise assessment of the tool dynamics. This can be achieved through analytic, numerical or experimental approaches. Although analytic approaches generally results in lower costs and are less time consuming, the available models are very simple, and large errors can be obtained. Numerical models offer more precise predicted values, as more complex geometries and phenomena can be incorporated in the analysis. Finally, experimental approach can lead to very precise parameters evaluation, but at the expense of higher costs and larger time consuming. Although damping parameter evaluation requires an experimental analysis, mass and stiffness parameters can be obtained through analytic and numerical methods. Therefore numerical analysis offers an interesting approach to study the dynamical parameters of these simple models.

Regenerative Chatter Vibrations

Chatter vibrations are a result of auto-generative instability on chip formation. One of the natural frequencies of the machine-tool or the tool system can be achieved but not only on this critical situation. That is why it is needed a complex analysis that considered the time delay inherent to the milling process, related to the spindle speed and the number of teeth, as a potential source of instability. One tooth cuts more or less material and the next one will have more or less material to cut, consequently the forces will vary and the tool displacement will answer to the system variation closing the loop. Figure 1 shows the geometry that causes the chatter on end milling.

The main parameter that excites the chatter is the depth of cut combined to the spindle speed values. That is the reason it is important to draw a stability graph with defined regions those two parameters that causes stable or unstable situations.

To understand the generation of the phenomenology it is important to understand that there are two sources of chatter: the superficial wave generated on the piece, when the external force is an input of the system and it modulates the system modifying displacements and, as a consequence, the force; and the second source it is the big change on forces associated with the entry of cutting edge on the workpiece that create different impact forces on each teeth because of run out. Altintas (2000) presented on his book a procedure for predicting the occurrence of chatter based on block diagram theory (Fig. 2). The transfer function matrix Φ(s) identified at the cutter-workpiece contact zone and it is a square matrix with direct transfer functions related to each direction (x and y) of the tool displacement that defined the modified chip thickness h(s). It is an analytic method that is obtained from a Fourier series expansion of force coefficients based on the Eigen values (Λ) and the chatter frequency (ω c ).

The procedure proposed by Altintas stablishes the following steps:

I. Identify the chatter frequency from transfer functions around a dominant mode II. Solve the Eigen value equation III. Calculate the critical depth of cut (a lim ) IV. Calculate the spindle speed for each stability lobe (N=0, 1, 2, ecc) and construct the graphic

The final expression for chatter-free axial depth of cut is found as [START_REF] Altintas | Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design[END_REF]:

(4) Where Λ R is the real part of the eigen value and and Kt is the cutting constant for specific cutting. The parameter τ is the tooth passing period (s), the delay for the external modulation (Fig. 2).

Figure 2. Block Diagram on Chatter Dynamics

NUMERICAL MODEL

A nonlinear tridimensional model based on the finite element method is developed to study the global stiffness of the assembly composed by the end milling tool and the tool holder using the commercial software Ansys/Workbench (Ansys, 2010). The model considers the nonlinearities associated to the contact phenomenon between the end milling tool and the tool holder. As the tool fixation is obtained through the application of a radial pressure by the tool holder on the end milling tool, the magnitude of the pressure can affect the global stiffness of the assembly. First, a solid tridimensional model is developed using the soling modeling software package Solidworks (2007) considering three geometry elements of the tool: end milling tool, mandrill and support. Figure 3 shows the three elements. With this solid model is possible to predict the total mass of the assembly, considering the density of the materials. Following, the solid model is imported to Ansys/Benchmark finite element package. Figure 3 shows the final mesh obtained after a convergence analysis.

To study the assembly stiffness, a prescribed load was applied at the tip of the end milling tool. The ratio between prescribed load and the end milling tool tip displacement can be associated to the assembly stiffness. The numerical analysis considers two steps: (1) fixture and (2) loading. The first step considers the fixture between the end milling tool and the mandrill, whereas the second step considers the application of a load at the end tool. The finite element model considers the contact between the three elements. Figure 4 shows the contact surfaces between the three elements. Some simplifications are adopted in the proposed model. First, the fixation disk used to compress the mandrill onto the support in order to furnish a fixture pressure on the end milling tool is not represented in the model. Instead, an upward prescribed displacement is applied to the lower mandrill surface in order to represent the effect of the fixation disk as it is tightened into the support threads. The prescribed displacement is associated to the thread pitch. For the presented simulations a 0.25 mm upward displacement was applied, corresponding to a 45° thread rotation. Also, null displacements and rotations are applied to the upper surface of the support. Therefore in this model, except for the three elements represented, the tool machine is considered as rigid. Figure 5a presents the imposed boundary conditions. Figure 5c shows the prescribed load applied at the end tool. To reduce concentration effects, the load is applied as a pressure at a small are at the end tool which load resultant is equal to the magnitude of the desired load. To study the effect of the applied load on the assembly stiffness, several loads are applied to the end milling tool end. Figure 6 presents the obtained assembly stiffness for several load magnitudes. A near constant value of 2.03 MN/m is obtained. The proposed finite element model can be used to assess the critical regions of the three elements in terms of stress in order to study the assembly geometry optimization. Figure 7 shows von Mises equivalent stress distribution. As the prescribed displacement is associated to the thread pitch and the rotation angle of the disk that clamps the tool, it was simulated different situations of clamping angles to analyze the stiffness variation. The same force was used on the tool tip, with a value of 128 N, and the displacement was imposed by the rotation angle of the clamping disk up to 0,32 mm on vertical direction that correspond to 61,2 degrees.

The results of these situations can be seen on Fig. 8. It is noted that on the range of the experiments, around 45 degrees, the stiffness does not depend anymore on the disk displacement.

EXPERIMENTAL PROCEDURE

Two different approaches were taken to experimentally measure stiffness and natural frequency to use as input data in chatter model and also to compare with numerical results. The first one is the conventional one, when the discussion is chatter prediction: impact hammer test. The second one is static procedure that relates applied force and displacement. For both experiments, it was used a capacitive sensor with 0.6 mm range.

The calibration of the sensor was checked in order to verify if the acquisition system correspond to the calibration given by the manufacturer. To compare calibration it was used a micrometer system shown on Figure 9 and it was computed from the nearest point of displacement with a sequence of known discrete displacements. The system dynamic response needs an experimental set-up, shown on Figure 10a, to be identified. The set presented on Fig. 9 is fixed to the machine tool and a hammer is used to introduce the impact force and its range of mechanical signals in the system. The capacitive sensor is used to acquire the displacement of the tool as a response to the impact. Natural frequency and damping coefficient are the parameters that are the aim of this experiment.

Before running experiments using the capacitive sensor, a calibration set-up is used to relate the electrical signal and displacement comparing with the calibration chart given by the supplier (Figure 11).

The direction of the force should be in the same direction the displacement sensor (Figure 12a). Nevertheless, the position on the vertical direction is not the same. The distance of measuring and impact is an important parameter and it is compatible to the numerical simulation. The results of the impact test with a steel head hammer shown on Fig. 12b are used to calculate the logarithmic decrement. From this graphic it is possible to identify the damping frequency of 118.7 Hz, using the first and the fourth pics. The damping coefficient is then 0.027.

Static Stiffness Experiment

A simplified experimental procedure was developed to review the parameters obtained from dynamic identification (Figure 10b). In this activity, sequences of several displacements were imposed to the workpiece in contact with the tool, and measure reaction force for each displacement. The holding system and the tool were the same used one dynamic response procedure. A dynamometer, an amplifier and an A/D converter was used to acquire the force data.

The dynamometer was fixed in the machine tool table. the results shown when there is a contact with the tool and the displacement The displacements were taken from the display of the machine tool table in steps of 0.1mm.

Experimental data from static stiffness test is presented on Figure 13. The results shown the non linearity of the system. The average value for the range experimented is calculated as 1.85.10 6 N / m.

COMPARISON BETWEEN NUMERICAL AND EXPERIMENTAL RESULTS

Collecting data from Fig. 6 and 13, it can be claimed that the numerical results presented higher stiffness then the experimental data. It can be explained by the absence of more elements in the machine-tool in the numerical simulation. The machine-tool presents elements that can be modified with the machining time and wear of components. Experimental stiffness considers the mechanical and geometrical models of each solid.

Both curves are presented in Fig. 14 to show the difference between the two procedures. The maximum error between the curves is 15%. Figura 14. Stiffness coefficient versus force applied to experimental and numerical models.

In order to have another reference, an analytical evaluation was analyzed. A simple analytic model based on Mechanics of Solids can be achieved considering the simplified geometry associated to a cantilever beam with two regions of different transversal area. Using the end milling tool parameters, a value of 2.16 MN/m is obtained, near 6% higher than the numerical model result. It can be obtained as function of Inertia of different transversal areas I 1 and I 2 [START_REF] Craig | Mecânica dos Materiais[END_REF]:

(5) 6. CHATTER SIMULATION Altintas model predicts stable and unstable regions of space defined by axial depth of cut and spindle speed parameters ar it was presented previously. It is based on the experimental impact test that identity the stiffness, damping and its natural frequency. As shown, after successively expansions of Fourrier series to calculate the force coefficients, it is possible to generate the lobus diagram.

Using the dynamic parameters simulated in this article and the Altintas model equations, the Figure 15 shows the first four lobes for stability on end milling in a full machining operation using the tool parameters described on the experiments. In full machining the radial depth of cut is defined by the diameter and the entry and exit angles on the working material are respectively 0 and 180 degrees.

The algorithm presents the dynamic behavior of the milling tool as two degrees of freedom system. The parameters in both directions (x and y) were considered with same modules are per the symmetric shape of the four flute milling tool. Matlab software was used for this prediction.

CONCLUSION

The numerical methodology for tool dynamic parameters is used based on a nonlinear model based on the Finite Element Method (FEM). A comparison from the ones obtained through FEM and experimental data was done to evaluate if the procedure can be used.

The model allows an approach that is flexible and easy rebuilt for different cases of tool-holder and end milling tool positions and set-up. Different clampings can be simulated to find a better result of the set-up without executing experiments that can expend more time and resources.

Using the Altintas model for predicting chatter, the two sets of dynamic parameters (taken from experiments and from simulation) are computed separately and lobe diagrams for each set are constructed and compared. Results indicate that dynamic parameters obtained from FEM model can be used with literature models, as the Altintas model, to predict chatter in end milling.

The influence of the disk rotation, contact pressure and the distance between the tool tip and the clamping on the chatter dynamics can be studied with this procedure to propose to the operator changes on the machining parameters when there is a chatter occurrence during the process.

Figure 1 .

 1 Figure 1. 2 DOF Dynamics of chatter on End Milling[START_REF] Altintas | Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design[END_REF]

Figure 3 .

 3 Figure 3. Tool Holder Assembly: (a) solid model and (b) finite element mesh. Numerical model.

Figure 4 .

 4 Figure 4. Contact surfaces between elements. Numerical model.

Figure 5 .

 5 Numerical analysis: (a) Step 1 -Fixture and (b) Step 2 -Loading. Numerical model.

Figure 6 .

 6 Figure 6. Assembly stiffness as a function of the applied load. Numerical Model.

Figure 7 .

 7 Figure 7. Assembly von Mises equivalent stress distribution. Numerical Model.

Figure 8 .

 8 Figure 8. Stiffness obtained for different disk rotation angle to clamp the tool

Figure 9 .Figure 10 .Figure 11 .

 91011 Figure 9. Tool holder -End Milling Tool System[START_REF] Castilho | Modelagem Dinâmica e Experimental da Rigidez da Fresa de Topo[END_REF]

  Figure 12. Impact Test

Figure 13 .

 13 Figure 13. Stiffness as a function of the Applied Force

Figure 15 .

 15 Figure 15. Lobes Diagram using Dynamic Parameters from Model It can be seen that the lobus diagram defines the range of axial depth of cut and spindle speed under the continuous line where machining is safe from chatter.

ACKNOWLEDGEMENTS

The authors thanks CEFET/RJ and UFRJ in names of the laboratories LABUS/CEFET/RJ and CEFCON/COPPE/UFRJ, where experiments were developed, and LAVI/UFRJ for the impact hammer used on the tests.


S Crombez

P Leclerc

C Ray

N Ducros

A computational hyperspectral structured light sheet microscope

We describe here a hyperspectral light sheet microscope that is based on beam shaping and image reconstruction. Our approach has no moving parts and can be easily combined with a standard microscope.

Introduction

As in Jahr et al. [START_REF] Jahr | Hyperspectral light sheet microscopy[END_REF], we consider the acquisition of a four-dimensional (4D) hypercube Ω = (x, y, z, λ ), where (x, y, z) is the voxel position, and λ is the wavelength. A single plane illumination microscope (SPIM) images the volume f (x, y, z) by acquiring 2D optical sections (x, y) sequentially at multiples z ∈ {z } 1≤ ≤L . Let f (x, y, λ ) denote the optical section at z .

Proposed methodology

1D modulation

In standard SPIM, the illumination sheet p is designed to be uniform in the 2D plane (x, y). Here, we propose to modulate the illumination along the x-axis and use a cylinder lens to focus the light on the slit of a spectrometer, which is oriented along the y-axis. Measurements are repeated for a set of modulation patterns {p k } 1≤k≤K , which leads to the set of measurements {m k (y, λ )} 1≤k≤K that can be modelled as m k (y, λ ) = f (x, y, λ )p k (x)dx. Each of the L optical sections are acquired independently by traversing the sample. In a discrete setting, we denote m y λ ∈ R N k as the measurements obtained for all of the modulation patterns at a given λ and vertical position y, where N k is the number of modulation patterns. The discrete forward model leads to m y λ = Pf y λ , ∀(λ , y) where P ∈ R N k ×N x is the modulation matrix, and f y λ ∈ R N x is the optical profile at vertical position y and wavelength λ . Note that the number of pixels along the (compressed) x-axis is denoted by N x . When N k = N x , and assuming that P is an orthogonal matrix (e.g., Hadamard, Fourier, wavelets patterns), the optical profile can be reconstructed by f y λ = P m y λ , ∀(λ , y). As the experimental patterns differ from the theoretical ones, we reconstruct the hypercube using the Tikhonov regularized least square solution

fλ = (A A + αI) -1 A m λ (1)
where A contains the experimental patterns.

Experimental set-up

Our optical system is shown in Fig. 1. It is fed by two continuous wave lasers that illuminate a digital micro-mirror device (DMD) that can shape the beam by reflecting only some part of it. The modulated beam is fed to a modified OpenSPIM (Pitrone et al. [START_REF] Pitrone | Open-SPIM: an open access light sheet microscopy platform[END_REF])) set-up to generate the structured light sheet that illuminates the sample with a given modulation pattern. The light emitted by the sample can be sent to two detection arms. A standard arm provides classical SPIM imaging, while the hyperspectral arm focuses the light on the slit of the spectrometer, to perform hyperspectral SPIM.

Results

To assess our set-up for the unmixing of two fluorophores, we measured a hydra sample labelled with two fluorophores (Superfloder GFP for the skin, DsRed2 for the inner tissues), as shown in Fig. 2a. We measured the same slice of the hydra tail (see Fig. 2b) using both arms. In Fig. 2d, we superimpose the reconstruction that corresponds to the emission of the GFP (green channel) and that corresponds to the DsRed (red channel). Fig. 2c shows the plot of the emission spectrum of a pixel taken in the skin area, and another in the inner tissue.

The reconstruction of the compressed dimension (i.e., x-axis) remains the main challenge, as can be seen by comparing the SPIM-arm and the chSPIM-arm images. Here, we only consider 64 modulation patterns to encode the x-axis spatial dimension, while the SPIM arm has 2560 pixels along this dimension. Other limiting factors include beam shaping while maintaining high spatial frequency and increasing the imaging rate.

Conclusion

We propose a computational hyperspectral SPIM set-up based on a DMD that generates a structured light sheet. We demonstrate the acquisition of a full hypercube of a two-colour sample with high spectral resolution. In future studies, we will work on improving the spatial resolution while maintaining the spectral resolution.

Fig. 1 .

 1 Fig. 1. (a) Schematic of the structured light sheet illumination. (b) Scheme of the computational hyperspectral SPIM demonstrator (chSPIM) with a control arm that allows classical SPIM imaging.

Fig. 2 .

 2 Fig. 2. Comparison of conventional and hyperspectral imaging of a two-colour specimen of a hydra, using Superfolder GFP (λ em = 510 nm) and DsRed2 (λ em = 587 nm). (a) Picture of the sample. The cyan box indicate the SPIM field of view. Scale bar, 1 mm. (b) SPIM acquisition. Scale bar, 100 µm. (c) Spectrum of zones 1 and 2. (d) chSPIM acquisition; green: reconstruction for λ ∈ [493 -527] nm; red: reconstruction for λ ∈ [576-601] nm. Scale bar, 100 µm.