Anna Carla Araujo

José Luís Silveira

6th BRAZILIAN CONGRESS OF MECHANICAL ENGINEERING THE INFLUENCE OF SPECIFIC CUTTING FORCE ON END MILLING MODELS

Keywords: specific cutting force, end milling, machining modelling

In order to predict machining forces it is necessary to know the chip volume and calculate the specific cutting force of the process, the prediction force problem is concerned on the right formulation of this last parameter that most models considers constant. This paper analyzes the behavior of the specific cutting pressures on end milling by calculating, from experimental data force, the pressure on each moment of cut. This procedure involves an approach which do not considers the force on each cutting edge piece but a function of area and height of cutting. After the cutting pressure analysis, some comparisons with previously published experiments are done and it can be seen that for the prediction of end milling cutting forces, the variation in time of the specific cutting force should be considered.

Introduction

The end milling process is one of the most widely used and efficient means of machining materials. In this process there is a periodically varying chip section during the material removal and the cutting force also varies during this process. Accurate modelling of the cutting forces is required to predict the cutting forces, vibration, surface quality, and stability of machining processes.

A number of different methods to predict cutting forces have been developed over the last years. These models can be classified into three major categories: empirical, analytical, and mechanistic methods. In the empirical methods, a number of machining experiments are performed and performance measures such as cutting forces, tool life, and tool wear are measured [START_REF] Armarego | The Machining of Metals[END_REF]. These responses are then correlated to the cutting conditions using empirical functions and require a lot of experimentation. Analytical approaches, [START_REF] Armarego | The Machining of Metals[END_REF] and [START_REF] Altintas | Manufacturing Automation[END_REF] model the physical mechanisms that occur during cutting. This include complex mechanisms such as high strain rates, high temperature gradients and combined elastic and plastic deformations and it's not yet completely solved. Mechanistic models [START_REF] Tlusty | Dynamics of Cutting in End Milling[END_REF]Kline et all, 1982;[START_REF] Altintas | End Milling force algorithm for CAD Systems[END_REF][START_REF] Yun | Accurate 3-D cutting force prediction using cutting condition independent coefficients in end milling[END_REF][START_REF] Yun | Accurate 3-D cutting force prediction using cutting condition independent coefficients in end milling[END_REF]Kapoor et all, 2001) predict the cutting forces based on a method that assumed cutting force to be proportional to the chip cross-sectional area.

The constants of proportionality are called the specific cutting pressures and depend on the cutter geometry, cutting conditions, insert grade and workpiece material properties. This paper questions the behavior in time of specific cutting pressures for cutting force models for end milling process, estimating the specific cutting pressures directly from cutting force data previously published.

Cutting force modelling of end-milling operations

Instantaneous differential force modelling for one single flute is normally written by Martelotti formula [START_REF] Araujo | Models for the Prediction of Instantaneous Cutting Forces in End Milling[END_REF]

d F i = K dA = K t db (1)
where K is a vector called specific cutting force and the instantaneous force is the sum of differential parts calculated on small cutting tool pieces (db) multiplied by the specific force and the uncut chip thickness (t) that for end milling can be written as:

t = s t sin φ (2)
where φ is the angle of the cutting piece measured in relation of the normal direction of the feed per tooth s t ,

s t = v ω N f (3)
from known v, ω and N f as the feed velocity, rotation velocity and the number of flutes.

The cutting tool pieces can be calculated by The angle δ is calculated by

δ = 2 b tan λ d ( 5
)
where b is the the depth of cut.

The angle δ is used to classify the cutting geometry as Type I or Type II (Tab. ( 1)), where ϕ 1 and ϕ 2 are the entry and exit angles, respectively (Fig. ( 1)), which were enlightened on [START_REF] Tlusty | Dynamics of Cutting in End Milling[END_REF] and [START_REF] Araujo | Models for the Prediction of Instantaneous Cutting Forces in End Milling[END_REF].

Table 1. Classification as Type I or Type II.

Type I Type II δ ≤ ϕ 2 -ϕ 1 δ > ϕ 2 -ϕ 1
The force become:

F i = KdA = Ktdb = K d s t sin φ 2 tan λ dφ (6)
So, the total force of cut, considering the N f flutes of the mill, are calculated by the sum:

F = N f i=1 F i (7)
In this approach, all force contributions are calculated at the same time because all differential parts of the force are calculated for each cutting piece.

Force as a function of time

The milling force vector ( F(t)) is written as the multiplication of two parcels: one relative to the specific force, written as a vector function ( K(t)), and other relative to the cutting area, an scalar function (A(t)). This form is called from now on by Function of area variation approach.

F(t) = K(t)A(t) (8)
The time variable t can be substituted by the angle of rotation θ of a fixed point P in a peripheral tool and the tool velocity rotation ω: (Fig. (2))

F(t) = F θ ω = F (θ) (9a) F (θ) = K(θ) A(θ) (9b)
The function A(θ) will be calculated separately in the following section.

The Cross-sectional Area Variation

Let take the area parcel separately to construct the area function A(θ).

In Fig.

(3) can be observed the chip cross-sectional area (A 1 ) of the first flute at milling angle θ, for a cutting geometry having ϕ 1 = 0 and ϕ 2 = π.

y x z θ A 1 s sin t θ (θ) b Figure 3. Chip Cross-sectional Area.
The chip area can be calculated by:

A 1 (θ) = L2(θ) L1(θ) s t d 2 sin λ sin φ dφ (10)
where the limits L 1 and L 2 are functions of θ and are calculated differently for each cutting phase of θ as can be observed on Tab. (2).

Table 2. Integration limits for each phase.

Type I Type II Phase L 1 (θ) L 2 (θ) L 1 (θ) L 2 (θ) For e 1 < θ ≤ e 2 -Phase A e 1 θ e 1 θ For e 2 < θ ≤ e 3 -Phase B θ -δ θ e 1 e 2 For e 3 < θ ≤ e 4 -Phase C θ -δ e 3 θ -δ e 2
The values of e 1 , e 2 , e 3 and e 4 can be extracted from Tab. (3). In order to add the contributions of all flutes, the chip cross-sectional area function for any flute (n) is written by:

A n (θ) = L2(θ+ξ(n-1)) L1(θ+ξ(n-1)) s t d 2 sin λ sin ε dε (11)
Note that the functions L 1 and L 2 are now not only a function of θ but also a function of n, where ξ is the angle between the flutes.

The total area is calculated by:

A(θ) = N f n=1 A n (θ) (12

Rotated Area Function

In order to compare the present model with experimental data, the force components should be decomposed in x, y and z directions as they are usually recorded in machining tests.

F (θ) =   F x (θ) F y (θ) F z (θ)   = A(θ)   K x (θ) K y (θ) K z (θ)   (13)
But it's not convenient to write the specific cutting force in the fixed referential x, y, z. To rewrite it on the more appropriate tool referential t, r, z (tangential, radial and axial directions), another function A R (θ) has to be introduced:

  F x (θ) F y (θ) F z (θ)   = A R (θ)   K t (θ) K r (θ) K z (θ)   (14)
In fact, the function A R (θ) is the rotation matrix R(θ) multiplied by the area.

R(θ) =   cos(θ) sin(θ) 0 sin(θ) -cos(θ) 0 0 0 1   (15)
The rotation matrix R n (θ) have to be written for each flute:

R n (θ) =   cos(θ + ξ (n -1)) sin(θ + ξ (n -1)) 0 sin(θ + ξ (n -1)) -cos(θ + ξ (n -1)) 0 0 0 1   (16)
Then, for all flutes, the rotated area function becomes:

A R (θ) = N f n=1 R n (θ) A n (θ) (17)
To simplify the following calculations, C 1 (θ) and C 2 (θ) are defined:

C 1 (θ) = N f n=1 A n (θ) cos(θ + ξ (n -1)) (18a) C 2 (θ) = N f n=1 A n (θ) sin(θ + ξ (n -1)) (18b) A(θ) = N f n=1 A n (θ) (18c)
Rewriting the Eq. ( 14), the force can be expressed by:

  F x (θ) F y (θ) F z (θ)   = A R (θ)   K t (θ) K r (θ) K z (θ)   =   C 1 (θ) C 2 (θ) 0 C 2 (θ) -C 1 (θ) 0 0 0 A(θ)     K t (θ) K r (θ) K z (θ)   (19)
Multiplying the matrix A R (θ) by a fictitious specific cutting force, e.g. K t (θ) = K r (θ) = K z (θ) = 1N/mm 2 , can be analyzed the contribution only from the area, with constant specific force. (Fig. ( 6))

  f x (θ) f y (θ) f z (θ)   = A R (θ)   1 1 1   (20) 2π π x y z f f f N Figure 6. Cutting force for fictitious constant specific force (K t (θ) = K r (θ) = K z (θ) = 1 N/mm 2 ).
Note that the matrix A R (θ) is the Jacobian of the F components in relation to the specific cutting force coefficients:

A R (θ) =           ∂F x ∂K t ∂F x ∂K r ∂F x ∂K z ∂F y ∂K t ∂F y ∂K r ∂F y ∂K z ∂F z ∂K t ∂F z ∂K r ∂F z ∂K z           (21)

Specific Force Analysis

The problem is concentrated on the specific cutting force analysis. The Eq. ( 19) can be easily calculated if the specific pressure is constant in time (θ).

Empirical works, [START_REF] Tlusty | Dynamics of Cutting in End Milling[END_REF] and (Ber et all, 1988), considered the specific force as one single value for each pair of tool-workpiece and relate the components by coefficients of proportionality; these researchers considers that the specific force is time constant.

Analytical approaches [START_REF] Altintas | Manufacturing Automation[END_REF] calculate the specific cutting forces as a function of cutting parameters.

Orthogonal and Oblique are the most important models and the specific cutting forces are presented on Tab. ( 4), where φ is the shear angle, α the rake angle, β the friction angle, η the chip flow angle, and the "lower index" n means the normal component of the angles. Table 4. Analytical Specific Force.

Orthogonal Model

Oblique Model

K t τ s cos(β-α) sin φ cos(φ+β-α) τ s sin φ n (cos(β n -α n )+tan λ tan η sin β n ) √ cos 2 (φ n +β n -α n )+tan 2 η sin 2 β n K r τ s sin(β-α) sin φ cos(φ+β-α) τ s sin φ n (cos(β n -α n ) tan λ-tan η sin β n ) √ cos 2 (φ n +β n -α n )+tan 2 η sin 2 β n K z 0 τ s sin φ n cos λ (sin(β n -α n )) √ cos 2 (φ n +β n -α n )+tan 2 η sin 2 β n
If these parameters are time constant, the specific cutting force will not change in time. To analyze this behavior, for each point of θ, the Eq. ( 22)

K(θ) = A R -1 (θ)F(θ) (22)
can be applied to any experimental data from milling process, then if the specific cutting force is constant, the result must has the same value.

The graphic from Fig. ( 7) was taken from [START_REF] Altintas | A General Mechanics and Dynamics Model for Helical End Mills[END_REF]. This experiment used the following parameters: Analyzing these points, with the aim to get the best proposal for the specific cutting force, the Eq. ( 19) is recalculated and compared with the experimental data.

d = 18.1mm, λ = 30.0, N f = 4, b = 5.08, ϕ 1 = 0, ϕ 2 = π 2 , v = 30 m min , s t = 0.05.
The first test is concerned on a constant value: the mean of the function presented in Fig. (8):

K tmedio = 1 p p i=1 K t (θ) (23a) K r medio = 1 p p i=1 K r (θ) ( 23b
)
where p is the number of points.

Those values, applied on Eq. ( 19) produces the forces presented and compared with the experimental data in Fig. (10). To improve the value of the constant specific force, those who minimize the difference between the theoretical and experimental forces are the asymptotic values: (Fig. ( 8))

K tconstant = 1700 N/mm 2 K r constant = 1350 N/mm 2 (24)
Applied on Eq. ( 19) results the comparison presented in So, constant values for the specific force do not produces good agreement with the experiment. In a mechanistic approach, one proposal is modelling the radial specific force as a function of the crosssectional area, because it varies with θ.

Here, the proposed function for the radial specific cutting force is:

K r variable = k r 1 A(θ) -1 + k r 2 (25)
where Using this proposal for the radial specific force and the tangential asymptotic value (K t = 1700 N/mm 2 ) and applying on Eq. ( 19):

  F x (θ) F y (θ) F z (θ)   =   C 1 (θ) C 2 (θ) 0 C 2 (θ) -C 1 (θ) 0 0 0 A(θ)     K t k r 1 A(θ) -1 + k r 2 K z   (26)
The Fig. ( 13) presents the comparison of the result of the Eq. ( 26) with the experiment. For this case, can be observed that the cutting force model with time variable cutting pressure is more precise then the constant specific force. Applying on Eq. ( 19) and rewriting, the Eq. ( 28) becomes:

  F x (θ) F y (θ) F z (θ)   = A R (θ)   K t k r 2 K z   + k r 1   C 2 (θ) A(θ) -1 -C 1 (θ) A(θ) -1 0   (27)
and shows that, for this case, the problem is engaged.

Conclusions

This paper analyzes the behavior of the specific cutting pressures from experimental cutting force data. A procedure to estimate the specific cutting forces from data is developed using the function of area variation approach. The developed method is validated by recalculation of the cutting force and compared with the experiment published by [START_REF] Altintas | A General Mechanics and Dynamics Model for Helical End Mills[END_REF]. From the results, it can be seen that the prediction of end milling forces should consider the variation on time of the specific cutting force.

Figure 1 .

 1 Figure 1. Milling geometry.

  Figure 2. Milling Tool Referential.

Figure 4 .

 4 Figure 4. Area Function for one Single Flute.

Figure 5 .

 5 Figure 5. End Milling Area Function.

Figure 7 .

 7 Figure7. Experimental data from[START_REF] Altintas | A General Mechanics and Dynamics Model for Helical End Mills[END_REF].Taking each point of the curve (Fig. (7)) and calculating the proposed Eq. (22), results in behavior shown in the Fig. (8), that is K(θ) is not constant:

Figure 10 .

 10 Figure 10. Comparison between experimental force and the force calculated from the mean specific force.

FigFigure 11 .

 11 Figure 11. Comparison between experimental force and the force calculated from a constant specific force.

Figure 12 .

 12 Figure 12. Proposed function for radial specific cutting force.

Figure 13 .

 13 Figure 13. Comparison between experimental and calculated force from proposed specific force function.

Table 3 .

 3 Variables values.

		Type I Type II
	e 1	ϕ 1	ϕ 1
	e 2 ϕ 1 + δ	ϕ 2
	e 3	ϕ 2	ϕ 1 + δ
	e 4 ϕ 2 + δ	ϕ 2 + δ

Proceedings ofCOBEM 2001, Manufacturing Process, Vol. 14, 488

Acknowledgements

This research was partially supported by FUJB (Fundação Universitária José Bonifácio) under project 8451-4.


Anna Carla Araujo

José Luís Silveira

ANALYSIS OF THE SPECIFIC FORCE ON END MILLING

Keywords: specific cutting force, end milling, machining modelling 22 nd IBERIAN

An analysis of specific cutting pressure in end milling process is presented. To model analytically milling forces it is necessary to compute the chip volume and the specific cutting pressure. The problem is concerned on the exact geometrical formulation of the process and especially on the last parameter. Considering only the influence of shearing and friction on the rake face and not taking into consideration the effects on cutting of the flank face i.e. the edge effects, the results obtained shown an error proportional to the contribution of the tool contact with the finished surface. This work aims taking the value, or the variation, of specific pressure from experimental data considering both effects, cutting force and edge force. This procedure analyzes the influence of the specific force modelling on the cutting process.

INTRODUCTION

In the end milling process there is a periodically varying chip cross section during the material removal and due to this the cutting force also varies. Accurate modelling of the cutting forces is required to predict the cutting forces, vibration, surface quality, and stability of machining processes.

A number of different methods to predict cutting forces have been developed over the last years. These models can be classified into three major categories: analytical, empirical, and mechanistic methods.

Analytical approaches, [START_REF] Armarego | The Machining of Metals[END_REF], [START_REF] Altintas | Manufacturing Automation[END_REF], model the physical mechanisms that occur during cutting. This includes complex mechanisms such as high strain rates, high temperature gradients and combined elastic and plastic deformations and it's not yet completely solved. [START_REF] Moufki | Thermoviscoplastic modelling of oblique cutting: forces and chip flow predictions[END_REF] In the empirical methods, a number of machining experiments are performed and performance measures such as cutting forces, tool life, and tool wear are measured [START_REF] Armarego | The Machining of Metals[END_REF].

Mechanistic models, [START_REF] Tlusty | Dynamics of Cutting in End Milling[END_REF], [START_REF] Kline | The Prediction of Cutting Forces in End Milling[END_REF], [START_REF] Altintas | End Milling force algorithm for CAD Systems[END_REF], [START_REF] Yun | Accurate 3-D cutting force prediction using cutting condition independent coefficients in end milling[END_REF] and [START_REF] Jayaram | Estimation of the Specific Cutting Pressures for Mechanistic Cutting Force Models[END_REF], predict the cutting forces based on a method that assumed cutting force to be proportional to the chip crosssectional area.

The constants of proportionality are called the specific cutting pressures and depend on the cutter geometry, cutting conditions, insert grade and work piece material properties. This paper questions the behavior in time of specific cutting pressures for cutting force models for end milling process, estimating the specific cutting pressures directly from experimental cutting force data.

CUTTING FORCE MODELLING OF END MILLING OPERATIONS

Instantaneous differential cutting force for one single flute was written by Martelotti [START_REF] Tlusty | Dynamics of Cutting in End Milling[END_REF]:

d F cutting = K cutting t db (1)
where t is the uncut chip thickness and db is a differential piece of the depth of cut. And was rewritten by (Armarego, 1989) adding the edge parcel:

d F = d F edge + d F cutting d F = K edge db + K cutting t db (2)
where K edge and K cutting are vectors called specific edge force and specific cutting force, respectively.

The uncut chip thickness t for end milling is written as:

t = s t sin φ (3)
where φ is the angle of the cutting piece measured in relation of the normal direction of the feed per tooth s t ,

s t = v ω N f (4)
from known v, ω and N f as the feed velocity, rotation velocity and the number of flutes. The cutting tool pieces can be calculated by

db = d 2 tan λ dφ ( 5
)
where d is the tool diameter and λ is the helix angle, as shown in Fig. 1. The angle δ is calculated by

δ = 2 b tan λ d ( 6
)
where b is the the depth of cut.

The angle δ is used to classify the cutting geometry as Type I or Type II,

Type I -→ δ ≤ ϕ 2 -ϕ 1 Type II -→ δ > ϕ 2 -ϕ 1 (7)
where ϕ 1 and ϕ 2 are the entry and exit angles, respectively (Fig. 1), enlightened on [START_REF] Tlusty | Dynamics of Cutting in End Milling[END_REF] and [START_REF] Araujo | Models for the Prediction of Instantaneous Cutting Forces in End Milling[END_REF]. The force become:

F = K edge db + K cutting t db = K edge + K cutting s t sin φ d 2 tan λ dφ (8)
So, the total force of cut, considering the N f flutes of the mill, are calculated by the sum:

F = N f i=1 F i (9)
In this approach, all force contributions are calculated at the same time because all differential parts of the force are calculated for each cutting piece.

The time variation of the milling force vector ( F(t)) is written as the multiplication of functions in time.

F(t) = K edge (t)h(t) + K cutting (t)A(t) (10)
The specific forces, written as a vector function, ( K edge (t) and K cutting (t)), are multiplied by a scalar function relative to the height cutting (h(t)) and other relative to the cutting area, the scalar function (A(t)). This form is called from now on by Functional approach. The time variable t can be substituted by the angle of rotation θ of a fixed point P in a peripheral tool and the tool velocity rotation ω as shown in Fig. 2.

F(t) = F θ ω = F (θ) (11a) F (θ) = K edge (θ)h(θ) + K cutting (θ)A(θ) (11b)
The functions h(θ) and A(θ) will be calculated separately in the following sections.

(θ) L 2 (θ) L 1 (θ) L 2 (θ) For e 1 < θ ≤ e 2 -Phase A ϕ 1 θ ϕ 1 θ For e 2 < θ ≤ e 3 -Phase B θ -δ θ ϕ 1 ϕ 2 For e 3 < θ ≤ e 4 -Phase C θ -δ ϕ 2 θ -δ ϕ 2

THE HEIGHT FUNCTION

In Fig. 3. can be observed the height of cutting of the first flute (h 1 ) at the milling angle θ, for a cutting geometry.

The height function of cutting of the first flute h 1 (θ) can be calculated by:

h 1 (θ) = L 2 (θ) L 1 (θ) d 2 sin λ dε ( 12
)
where the limits L 1 and L 2 are functions of θ and are calculated differently for each cutting phase of θ as can be observed on Table 1 and the values of e 1 , e 2 , e 3 and e 4 can be extracted from Table 2. In order to add the contributions of all flutes, the height of cutting for any flute (n) is written by:

h n (θ) = L 2 (θ+ξ(n-1)) L 1 (θ+ξ(n-1)) d 2 sin λ dε (13)
Note that the functions L 1 and L 2 are now not only a function of θ but also a function of n, where ξ is the angle between the flutes.

e 2 ϕ 1 + δ ϕ 2 e 3 ϕ 2 ϕ 1 + δ e 4 ϕ 2 + δ ϕ 2 + δ
The total length is calculated by:

h(θ) = N f n=1 h n (θ) (14)

THE CROSS-SECTIONAL AREA VARIATION

In Fig. 5 can be observed the chip cross-sectional area (A 1 ) of the first flute at milling angle θ, for a cutting geometry having ϕ 1 = 30 o and ϕ 2 = π/2.

The chip area can be calculated by:

A 1 (θ) = L 2 (θ) L 1 (θ) s t d 2 sin λ sin φ dφ ( 15
)
where the limits L 1 and L 2 are functions of θ and are calculated differently for each cutting phase of θ as can be observed on Table 1.

y x z θ A 1 s sin t θ (θ) b Figure 5: Chip Cross-sectional Area
For a single flute and Type I cutting geometry, the chip cross-sectional area function is presented on Fig. 6. In order to add the contributions of all flutes, the chip cross-sectional area function for any flute (n) is written by:

A n (θ) = L 2 (θ+ξ(n-1)) L 1 (θ+ξ(n-1)) s t d 2 sin λ sin ε dε (16)
Note that the functions L 1 and L 2 are now not only a function of θ but also a function of n, where ξ is the angle between the flutes.

The total area is calculated by:

A(θ) = N f n=1 A n (θ) (17)
and the figure 7 shows the chip cross-sectional area function A(θ) for a milling with four flutes (ξ = 90 o ) and in bold the function of the first flute A 1 (θ), of a cut having Type I geometry.

FORCE COMPUTATION

In order to compare the present model with experimental data, the force components should be decomposed in x, y and z directions as they are usually recorded in machining tests.

F (θ) =   F x (θ) F y (θ) F z (θ)   = A(θ)   K cx (θ) K cy (θ) K cz (θ)   + h(θ)   K ex (θ) K ey (θ) K ez (θ)   (18)
But it's not convenient to write the specific cutting force in the fixed referential x, y, z. To rewrite it on the more appropriate tool referential t, r, z (tangential, radial and axial directions), another functions A R (θ) and h R (θ) has to be introduced:

  F x (θ) F y (θ) F z (θ)   = A R (θ)   K ct (θ) K cr (θ) K cz (θ)   + h R (θ)   K et (θ) K er (θ) K ez (θ)   (19)
In fact, the function A R (θ) and h R (θ) are the rotation matrix R(θ) multiplied by the area and the height, respectively.

R(θ) =   cos(θ) sin(θ) 0 sin(θ) -cos(θ) 0 0 0 1   (20)
The rotation matrix R n (θ) have to be written for each flute:

R n (θ) =   cos(θ + ξ (n -1)) sin(θ + ξ (n -1)) 0 sin(θ + ξ (n -1)) -cos(θ + ξ (n -1)) 0 0 0 1   (21)
Then, for all flutes, the rotated area function becomes:

A R (θ) = N f n=1 R n (θ) A n (θ) (22)
and the rotated length function is: To simplify the following calculations, S 1 (θ), S 2 (θ), S 3 (θ) and S 4 (θ)are defined:

h R (θ) = N f n=1 R n (θ) h n (θ) (23)
S 1 (θ) = N f n=1 A n (θ) cos(θ + ξ (n -1)) (24a) S 2 (θ) = N f n=1 A n (θ) sin(θ + ξ (n -1)) (24b) S 3 (θ) = N f n=1 h n (θ) cos(θ + ξ (n -1)) (24c) S 4 (θ) = N f n=1 h n (θ) sin(θ + ξ (n -1)) (24d)
Rewriting the Eq. ( 19), the force can be expressed by:

  F x (θ) F y (θ) F z (θ)   = A R (θ)   K ct (θ) K cr (θ) K cz (θ)   + h R (θ)   K et (θ) K er (θ) K ez (θ)   (25a)   F x (θ) F y (θ) F z (θ)   =   S 1 (θ) S 2 (θ) 0 S 3 (θ) S 4 (θ) 0 S 2 (θ) -S 1 (θ) 0 S 4 (θ) -S 3 (θ) 0 0 0 A(θ) 0 0 h(θ)           K ct (θ) K cr (θ) K cz (θ) K et (θ) K er (θ) K ez (θ)         (25b)
Note that the matrix J(θ) is the Jacobian of the F components in relation to the specific cutting force coefficients.

J(θ) =   S 1 (θ) S 2 (θ) 0 S 3 (θ) S 4 (θ) 0 S 2 (θ) -S 1 (θ) 0 S 4 (θ) -S 4 (θ) 0 0 0 A(θ) 0 0 h(θ)   =      

SPECIFIC FORCE ANALYSIS

The problem is concentrated on the specific force analysis. The Eq. ( 25) can be easily calculated if the specific pressures are constant in θ.

Empirical works, [START_REF] Tlusty | Dynamics of Cutting in End Milling[END_REF] and [START_REF] Ber | A Method for Cutting Forces in End Milling[END_REF], considered the specific force as one single value for each pair of tool-workpiece and relate the components by coefficients of proportionality; these researchers considers that the specific force is time constant.

Analytical approaches [START_REF] Altintas | Manufacturing Automation[END_REF] calculate the specific cutting forces as a function of cutting parameters.

Orthogonal and Oblique are the most well known models and these specific cutting forces are shown on Table 3, where φ is the shear angle, α the rake angle, β the friction angle, η the chip flow angle, and the lower index n means the normal component of the angles.

If these parameters are time constant, the specific cutting force do not change in time. To analyze this behavior, for each point of θ, the equation:

K(θ) = J -1 (θ)F(θ) (27)
can be applied to any experimental data from milling process, then if the specific cutting force is constant, the result must has the same value.

The graphic from Fig. 8 was taken from [START_REF] Altintas | A General Mechanics and Dynamics Model for Helical End Mills[END_REF]. This experiment used the following parameters:

d = 18.1mm, λ = 30.0, N f = 4, b = 5.08, ϕ 1 = 0, ϕ 2 = π 2 , v = 30 m min , s t = 0.05.
But, when calculating the Eq. 27, appears a singularity because there are three parameters [START_REF] Altintas | A General Mechanics and Dynamics Model for Helical End Mills[END_REF] and six unknown variables.

  F x (θ) F y (θ) F z (θ)   =   S 1 (θ) S 2 (θ) 0 S 3 (θ) S 4 (θ) 0 S 2 (θ) -S 1 (θ) 0 S 4 (θ) -S 3 (θ) 0 0 0 A(θ) 0 0 h(θ)           K ct (θ) K cr (θ) K cz (θ) K et (θ) K er (θ) K ez (θ)         (28)
To solve this problem, we suppose that two consecutive points have the same specific force, calculating the specific force for each interval:

        F x (θ) F y (θ) F z (θ) F x (θ + dθ) F y (θ + dθ) F z (θ + dθ)         = J A (θ)         K ct (θ) K cr (θ) K cz (θ) K et (θ) K er (θ) K ez (θ)         (29)
The matrix J amplified became a square matrix J A :

J A (θ) =         S 1 (θ) S 2 (θ) 0 S 3 (θ) S 4 (θ) 0 S 2 (θ) -S 1 (θ) 0 S 4 (θ) -S 3 (θ) 0 0 0 A(θ) 0 0 h(θ) S 1 (θ + dθ) S 2 (θ + dθ) 0 S 3 (θ + dθ) S 4 (θ + dθ) 0 S 2 (θ + dθ) -S 1 (θ + dθ) 0 S 4 (θ + dθ) -S 3 (θ + dθ) 0 0 0 A(θ + dθ) 0 0 h(θ + dθ)         (30)
and the problem has only one solution:

        K ct (θ) K cr (θ) K cz (θ) K et (θ) K er (θ) K ez (θ)         = [J A (θ)] -1         F x (θ) F y (θ) F z (θ) F x (θ + dθ) F y (θ + dθ) F z (θ + dθ)         (31)
The Fig. 9 shows the result for specific pressures using the data from Fig. 8. Figures 9a, 9b and9c, respectively tangential, radial and vertical specific cutting forces and 9g, 9h and 9i, are tangential, radial and vertical specific edge forces). Figures 9d,9e, 9f, 9j, 9k and 9l are them distributions.

Calculating the average value of the specific values obtained for the hole rotation K, and applying on Eq. 25 results:

  F x (θ) F y (θ) F z (θ)   = J(θ)         K ct K cr K cz K et K er K ez         (32)
as shown in Table 4. Using these values to recalculate the forces, Fig. 10 shows the comparison between experimental force and semi-empirical force. This procedure aims to confirm if the behavior of the specific pressure considered is correct.

A general code of an end milling process is given on Table 5. The input variables set by the user are cutter radius, helix angle, entry and exit angles, axial depth of cut, the number of teeth, feed per tooth and the experimental cutting force.

(θ) = N f n=1 A(θ + ξ(n -1)) cos(θ + ξ(n -1)) S 2 (θ) = N f n=1 A(θ + ξ(n -1)) sin(θ + ξ(n -1)) A t (θ) = N f n=1 A(θ + ξ(n -1)) S 3 (θ) = N f n=1 h(θ + ξ(n -1)) cos(θ + ξ(n -1)) S 4 (θ) = N f n=1 h(θ + ξ(n -1)) sin(θ + ξ(n -1)) h t (θ) = N f n=1 h(θ + ξ(n -1)) 9
Discretize all functions Sd 1 (j) = S 1 (θ = 2π (j-1) Np-1 ) 10 Compute J A (j)

J A (j) =         Sd 1 (j) Sd 2 (j) 0 Sd 3 (j) Sd 4 (j) 0 Sd 2 (j) -Sd 1 (j) 0 Sd 4 (j) -Sd 3 (j) 0 0 0 Ad t (j) 0 0 hd t (j) Sd 1 (j + 1) Sd 2 (j + 1) 0 Sd 3 (j + 1) Sd 4 (j + 1) 0 Sd 2 (j + 1) -Sd 1 (j + 1) 0 Sd 4 (j + 1) -Sd 3 (j + 1) 0 0 0 Ad t (j + 1) 0 0 hd t (j + 1)        
11 Compute the matrix KG(j) with all specific pressures

KG(j) = J A (j) -1 .         F x exp (j) F x exp (j) F x exp (j) F x exp (j + 1) F x exp (j + 1) F exp (j + 1)         12 Compute each specific pressure K ct (j) = KG(j) 1 K cr (j) = KG(j) 2 K cz (j) = KG(j) 3 K et (j) = KG(j) 4 K er (j) = KG(j) 5 K ez (j) = KG(j) 6
13 Compute average specific pressures

K ct = Np j=1 K ct (j) K cr = Np j=1 K cr (j) K cz = Np j=1 K cz (j) K et = Np j=1 K et (j) K er = Np j=1 K er (j) K ez = Np j=1 K ez (j)
14 Use this values to calculate semi-empirical force

F x (j) = K ct Sd 1 (j) + K cr Sd 2 (j) + K et Sd 3 (j) + K er Sd 4 (j) F y (j) = K ct Sd 2 (j) -K cr Sd 1 (j) + K et Sd 4 (j) -K er Sd 3 (j)
F z (j) = K ct Sd 1 (j) + K cr Sd 2 (j) + K cz Ad t (j) + K ez hd t (j)

CONCLUSIONS

This paper analyzes the behavior of the specific cutting pressures from experimental cutting force data. A procedure to estimate the specific cutting forces from data is developed using the function approach including the edge parcel in the model. The method is validated by recalculation of the cutting force and compared with the experiment published by [START_REF] Altintas | A General Mechanics and Dynamics Model for Helical End Mills[END_REF]. The contribution of the edge force is important to the model but there is no visual variation on time for them. The difference from results with and without the edge forces can be observed comparing with [START_REF] Araujo | The influence of the specific cutting force on end milling models[END_REF].

  Figure 1: Milling geometry

Figure 2 :

 2 Figure 2: Milling Tool Referential

Figure 3 :

 3 Figure 3: Height of cutting

Figure 4 :

 4 Figure 4: Function of height of cutting

Figure 6 :

 6 Figure 6: Area Function for one Single Flute

Figure 7 :

 7 Figure 7: End Milling Area Function

  βn-αn) tan λ-tan η sin βn) √ cos 2 (φn+βn-αn)+tan 2 η sin 2 βn K z 0 τs sin φn cos λ (sin(βn-αn)) √ cos 2 (φn+βn-αn)+tan 2 η sin 2 βn

Figure 8 :

 8 Figure 8: Experimental data from[START_REF] Altintas | A General Mechanics and Dynamics Model for Helical End Mills[END_REF]

Figure 9 :

 9 Figure 9: Experimental Specific Pressures and distribution

Table 1 :

 1 Integration limits for each phase.

		Type I	Type II
	Phase	L 1

Table 2 :

 2 Variables values

		Type I Type II
	e 1	ϕ 1	ϕ 1

Table 3 :

 3 Analytical Specific Force

		Orthogonal Model		Oblique Model
	K t	τs cos(β-α) sin φ cos(φ+β-α)	τs sin φn	(cos(βn-αn)+tan λ tan η sin βn) √ cos 2 (φn+βn-αn)+tan 2 η sin 2 βn

Table 4 :

 4 Average Specific Pressure Values -N/mm 2 K ct K cr K cz K et K er K ez 1150.5 341.0 436.7 24.1 42.2 0.4

Table 5 :

 5 Algorithm to specific force analysis . 1 Input experimental cutting force (Np points) . F x exp (j), F y exp (j), j ∈ [1, Np] F z exp (j). 2 Inputs: r, λ, N n , b, ϕ 1 , ϕ 2 , v, s t 3 Compute δ and ζ 4 Check for Type I or Type II 5 Compute e 1 , e 2 , e 3 and e 4

	6 Compute A(θ) =		0	0≤ θ < e 1
		θ e 1 θ θ-δ e 3 θ-δ		r sin λ s t sin φ dφ r sin λ s t sin φ dφ e 2 ≤ θ < e 3 e 1 ≤ θ < e 2 r sin λ s t sin φ dφ e 3 ≤ θ < e 4 if Type I
				0	e 4 ≤ θ < 2π
	or	A(θ) =		0	0≤ θ < e 1
		θ e 1 e 2 e 1 e 2 θ-δ	r sin λ s t sin φ dφ r sin λ s t sin φ dφ e 2 ≤ θ < e 3 if Type II e 1 ≤ θ < e 2 r sin λ s t sin φ dφ e 3 ≤ θ < e 4
				0	e 4 ≤ θ < 2π
	7 Compute h(θ) =		0	0≤ θ < e 1
		θ e 1 θ θ-δ e 3 θ-δ		r sin λ dφ r sin λ dφ r sin λ dφ 0	e 1 ≤ θ < e 2 e 2 ≤ θ < e 3 if Type I e 3 ≤ θ < e 4 e 4 ≤ θ < 2π
	or	h(θ) =		0	0≤ θ < e 1
		θ e 1 e 2 e 1 e 2 θ-δ	r sin λ dφ r sin λ dφ r sin λ dφ 0	e 1 ≤ θ < e 2 e 2 ≤ θ < e 3 if Type II e 3 ≤ θ < e 4 e 4 ≤ θ < 2π
	8 Compute
	S 1

Acknowledgements

This research was partially supported by FUJB (Fundac ¸ão Universitária José Bonifácio) under project 8451-4.


Anna Carla Araujo

Jose Luis Silveira

Shiv Gopal Kapoor

Shiv Gopal

Luis Silveira

Shiv Kapoor
email: s-kapoor@uiuc.edu

FORCE PREDICTION IN THREAD MILLING

Keywords: Thread Milling, Mechanistic Models, Force Prediction

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

INTRODUCTION

Threading a work piece is a fundamental metalworking process. Threads can be produced in a variety of ways, involving two basic methods: plastic working or metal cutting. The dominant method used in industry is plastic working. Conventional bolts and screws, for example, are mostly made by this method. Threads produced by plastic deformation are stronger because of the grain structure than those produced by cutting, although forming cannot achieve the high accuracy and precision required in many applications. Threads made of brittle materials also cannot be produced by plastic working. In such cases, thread cutting is necessary [START_REF] Grahan | Advanced Machining: The handbook of cutting technology[END_REF]. The common cutting processes for producing internal threads are tapping and thread milling [START_REF] Stephenson | Metal Cutting Theory and Practice[END_REF]. Tapping is used to make internal threads with the same diameter of the tool. It is done by feeding the cutting tool into the hole until the desired thread depth is achieved, then reversing the tap to back it out of the hole and remove it from the workpiece. Thread milling tools can produce internal threads with any diameter bigger than tool diameter as well as external threads. In thread milling, the machine tool executes the thread in one single pass. The tool goes down to the hole and begins the cutting from the deepest part to the top in a helical path, or it begins at the top and goes until the end of the hole. Some geometries of thread milling tools can be observed in Figure ( Many authors developed models for prediction of forces in machining. These include analytical, experimental, mechanistic and numerical methods [START_REF] Ehmann | Machining Process Modeling: A Review[END_REF]. In thread cutting by tapping, a mechanistic method for the prediction of forces was presented by [START_REF] Dogra | Mechanistic Model for Tapping Process with Emphasis on Process Faults and hole geometry[END_REF]. A number of papers describing the thread milling operation have been published [START_REF] Grahan | Advanced Machining: The handbook of cutting technology[END_REF][START_REF] Koelsch | Thread Milling Takes on Tapping[END_REF][START_REF] Stephenson | Metal Cutting Theory and Practice[END_REF] but there is no model to predict the forces involved in the process.

The objective of this article is to present a mechanistic model for thread milling. The tool geometry analyzed involves triangular and metric threads. The thread milling tool in this article has helical flutes and its geometry is analyzed as a modified end milling tool. This follows with the description of cutting geometry. Tool run out is added to the model and some examples of thread milling processes are presented. The relation between the diameters can be written as: The thread milling tool is very similar to an end milling tool. The tool geometry of a helical thread flute is presented in figure (2b). The helix angle λ, the rake angle α, the internal and external diameters d i and d e and the numbers of flutes N f define the tool geometry. The angle between flutes is:

d I = d E - p tan( ξ 2 ) (1)
ζ = 2π N f (2)
and the number of each flute is n, 1 ≤ n ≤ N f . The local diameter d(z) is written as a function of the height z, d i ≤ d(z) ≤ d e , calculated as follows:

d(z) =    d i + 2(z-nt(z) p 2 ) tan( ξ 2 )
, if nt(z) is odd;

d e - 2(z-nt(z) p 2 ) tan( ξ 2 )
, if nt(z) is even;

(3)

where nt(z) is:

nt(z) = IntegerP art 2z p (4)
The tool motion is circular in the plane normal to the tool-axis and linear in the direction of the tool-axis with a stationary workpiece to generate the thread. The workpiece is predrilled and the diameter of the hole is called d h . The width of cut (or radial depth of cut) is e(z):

e(z) = d(z) -d h 2 (5)

Cutting geometry

The cutting geometry of the thread milling process is different than the common end milling because the cutting edge is not a straight line and the tool follows a circular trajectory. Following the approach used by [START_REF] Tlusty | Dynamics of cutting in end milling[END_REF] the contact interface in thread milling process can be described as shown in Figure (3), where can be observed the depth of cut, b.

b λ A B C ξ b A B C Ψ Ψ Ψ Ψ 1 4 3 2 φ i φ f Flute δ Figure 3 -Contact Plane
The contact surface is divided in three phases: A, where the length of active cutting edge increases in time, B, where it is constant, and C where it decreases. There are two different types of contact surface geometry according to the relation between the angle δ, defined in Equation 6, and the contact angle: Type I and Type II. It is a Type I geometry if δ ≤ ϕ 2 , and

Type II occurs if δ ≥ ϕ 2 -ϕ 1 . δ = 2b tan λ d e (6)
The contact angle is defined as the difference between the initial angle ϕ 1 and the final angle ϕ 2 . Two auxiliary angles were defined by [START_REF] Tlusty | Dynamics of cutting in end milling[END_REF] to analyze the flute movement through the three phases A, B and C. The first one is angle Ψ, which indicates the angular position of the leading point of the cutting edge. The other one is angle φ, which indicates the position of the other points of the same flute. For a known Ψ, the range of values for φ is between φ i and φ f , and it changes for each phase and each position θ of the tool as shown in Table 1.

Table 1 -Limits of φ i (θ) and φ f (θ) in A, B and C.

Type I Type II Phase

φ i (θ) φ f (θ) φ i (θ) φ f (θ) For Ψ 1 < θ ≤ Ψ 2 -Phase A ϕ 1 θ ϕ 1 θ For Ψ 2 < θ ≤ Ψ 3 -Phase B θ -δ θ ϕ 1 ϕ 2 For Ψ 3 < θ ≤ Ψ 4 -Phase C θ -δ ϕ 2 θ -δ ϕ 2
The range of values for Ψ in each phase is shown on Table 2 and the limits of each phase are called Ψ 1 , Ψ 2 , Ψ 3 and Ψ 4 .

ϕ 1 ϕ 1 Ψ 2 ϕ 1 + δ ϕ 2 Ψ 3 ϕ 2 ϕ 1 + δ Ψ 4 ϕ 2 + δ ϕ 2 + δ
For a helical tool, the height z can be expressed as a function of the position of the tool θ and the point angle φ:

z(θ, φ) = d i (θ -φ) 2 tan λ (7)
In the case I, shown in figure (4a), the tool feed velocity does not change direction, as occurs in end milling. In case II the tool cuts in a the circular path, figure (4b). Case I: For up milling the initial angle is zero. In case I the final angle is written as:

ϕ 2 = b tan λ re (8)
Case II: In case II, the final angle is written as: (Figure 5)

ϕ 2 = arcsin d 2 h -d 2 i -4r o 4d h r o (9)
where r o = d E -de 2 . The uncut chip thickness for any point of the cutting edge, located in the height z and by the angle φ (Figure 5), can be written as: [START_REF] Sabberwall | Chip section and cutting force during the milling operation[END_REF] t(φ, z) = f t (z) sin φ (10)

FORCE PREDICTION

Elemental normal and frictional forces are required to the determination of cutting forces for a given geometry. The mechanistic modelling approach is a combination of analytical and empirical methods in which the forces are proportional to the chip load [START_REF] Kline | The effects of run-out on cutting geometry and forces in end milling[END_REF]. The specific cutting energies, K n , K f and K z , have been shown as a function of chip thickness t c and cutting velocity V c . [START_REF] Dogra | Mechanistic Model for Tapping Process with Emphasis on Process Faults and hole geometry[END_REF])

F n (θ) = K n .A(θ) F f (θ) = K f .A(θ) (11) ln(K c ) = a 0 + a 1 ln(t c ) + a 2 ln(V c ) + a 3 ln(t c ) ln(V c ) ln(K f ) = b 0 + b 1 ln(t c ) + b 2 ln(V c ) + b 3 ln(t c ) ln(V c ) (12)
The coefficients a 0 , a 1 , a 2 , a 3 , b 0 , b 1 , b 2 and b 3 are called specific cutting energy coefficients. They are dependent on the tool and workpiece materials and also on the cutting speed and the chip thickness. They are determined from calibration tests for a given tool workpiece combination and for given a range of cutting conditions.

Chip Area

The function of chip area for the first flute A 1 (θ) is:

A 1 (θ) = φ f (θ) φ i (θ) t(φ, z)db ( 13
)
where db is (Sabberwal, 1960):

db = d(z)) 2 tan λ dφ (14)
Using the Equations 3, 7 and 10, the area A 1 (θ) can be calculated.

A 1 (θ) = φ f (θ) φ i (θ) t(φ, z(θ, φ)) d(z(θ, φ)) 2 tan λ dφ (15)
The limits φ i (θ) and φ f (θ) are functions of the θ and the cutting phase of θ, as shown in Table 1. In order to add the contributions of all flutes, the chip cross-sectional area function for each flute (n) is written as:

A n (θ) = φ f (θ+ζ(n-1)) φ i (θ+ζ(n-1)) t(φ, z(θ, φ)) d(z(θ, φ)) 2 tan λ dφ (16)
In the equation ( 15) φ i and φ f are also written as a function of n. The total area A(θ) is calculated as:

A(θ) = N f n=1 A n (θ) (17)

Tool Run Out

Cutter run out exists in all kinds of milling operations and results in variations in undeformed chip thickness, local forces and machined surface characteristics. The run out can be due to cutter axis offset, eccentricity ( ) or cutting points positioning offset (ρ), shown in Figure 6, and it depends principally on the characteristics of the spindle and tool holder. The chip thickness in presence of run out is rewritten as: [START_REF] Kline | The effects of run-out on cutting geometry and forces in end milling[END_REF])

t c (θ, φ, n) = f t sin φ + ρ cos(θ --φ) -cos(θ --φ -nζ) (18)

Force Computation

To calculate the components of the cutting forces using the specific cutting forces K f , K n and K z , a function A R (θ) has to be introduced. [START_REF] Araujo | The influence of the specific cutting force on end milling models[END_REF])

F (θ) =   F x (θ) F y (θ) F z (θ)   =   F x (θ) F y (θ) F z (θ)   = A(θ)   K x K y K z   = A R (θ)   K f K n K z   (19)
In fact, the function A R (θ) is the rotation matrix R(θ) multiplied by the area.

R(θ) =   cos(θ) sin(θ) 0 sin(θ) -cos(θ) 0 0 0 1   (20)
For each flute the rotation matrix R n (θ) is:

R n (θ) =   cos(θ + ζ (n -1)) sin(θ + ζ (n -1)) 0 sin(θ + ζ (n -1)) -cos(θ + ζ (n -1)) 0 0 0 1   (21)
The area for all flutes is written as:

A R (θ) = N f n=1 R n (θ) A n (θ) (22)

EXAMPLES

In order to analyze the forces profile in thread milling, four examples are presented in this article. Specific pressure in these examples will be same and are given by K f = 900 N/mm 2 , K n = 500 N/mm 2 and K z = 100 N/mm 2 .

The tool used in the simulation has the following parameters: p=1.5 mm, λ = 30 o , N f = 4 and ξ = 60 o . The geometry of cut and the velocities are: d E = 10 mm, ω = 1400 rpm, f t = 0.06 mm. The values for eccentricity and off set for the run out case are: ρ = 10 o and = 0.04 mm.

Example I: In this example the depth of cut is b = 1.5 mm, just one pitch. To illustrate the example I, Figures (7) presents the chip cross area and the cutting force with and without tool run out. [N] (θ)

[N] (θ) [N] Fz Fy Fx (θ) [N] (θ) [N] (θ)

CONCLUSIONS

A mechanistic model have been developed for thread milling. The model takes into account the thready cutting edge of the tool and linear movement of the tool. The forces were predicted for four depths of cuts. The results show the effects of the increasing the depth of cut on the cutting forces. The thready flute contribute for another frequency oscillation in chip area and consequently in cutting force as compared to the straight cutting edge. In order to improve the model, the contact stresses between the threads and the tool need to be added.

  1). Figure (1a) shows a single cutting edge which produces one pitch per feed rotation, Figure (1b) presents a single straight tool with only one cutting edge and Figure (1c) shows a helical thread tool with some cutting flutes. In thread milling high tool pressure are generated which can result into an excessive tool deflection and tool breakage when milling at full thread.

  Figure 1 -Thread Milling Tools (Emuge, 2002)

  Figure 2 -Tool and Thread Geometries

  Figure 4 -Tool Path Cases

Figure 5 -

 5 Figure 5 -Chip Thickness for Case II

Figure 6 -

 6 Figure 6 -Tool Run Out

  Figure 7 -Example I Example II: The depth of cut in this example is b = 5 mm. Figure (8) presents the chip cross area and the cutting force with and without tool run out for this case.

  Figure 8 -Example II Example III: In this example the depth of cut is b = 10 mm and Figure (9) presents the chip cross area and the cutting force with and without tool run out for this example.

  Figure 9 -Example III

  Figure 10 -Example IV

Table 2 -

 2 Values of Ψ 1 , Ψ 2 , Ψ 3 and Ψ 4 for Type I and II Type I Type II Ψ 1

ACKNOWLEDGMENTS

This project is fulfilled with the fellowship received from CNPq, a Brazilian council for scientific and technological development. The author is grateful for the council and the University of Illinois at Urbana Champaign for the support.


Ricardo Rodrigues Alvarez

Anna Carla Araujo

José Luís Silveira

MODELING END MILLING WITH RUNOUT

Keywords: end milling, runout, uncut chip thickness, specific cutting force, machining modelling

), the force model of end milling with cutter runout is established. This paper analyzes the behavior of the specific cutting pressures on end milling by calculating, from experimental data force, the pressure on each moment of cut considering the cutting force as much the edge force effect. Runout has been a limiting factor for achievable productivity in machining due to its adverse effects on tool life and part quality. The presence of cutter runout generates a cutting force component at the spindle rotational frequency and increases the average chip thickness for those teeth actually engaged in the cut. After the cutting pressure analysis some comparisons with previously published experiments by Yun & Cho (2001) will be done.

Introduction

Cutter runout is a common phenomenon in multi-tooth machining operations, particularly in milling operations [START_REF] Kline | The Effect of Runout on Cutting Geometry and Forces in End Milling[END_REF]. In end milling, runout commonly occurs due to cutter offset with respect to the spindle rotation when set screw-type holders are employed. Many other causes of cutter runout are possible, but in this paper, cutter offset with respect to the center of rotation is the type of runout considered.

The presence of runout causes chip load to vary over the rotation of a multi-tooth cutter. Depending on the degree and nature of runout, we can observe problems as cutter breakage, cutter wear, the surface error generation mechanism and the dynamic behavior of the machine tool and cutting process.

The cutting forces in end milling have often been considered as the product of a specific cutting force and uncut chip thickness [START_REF] Araujo | Models for the Prediction of Instantaneous Cutting Forces in End Milling[END_REF][START_REF] Yun | Accurate 3-D Cutting Force Prediction using Cutting Condition Independent Coefficients in End Milling[END_REF]. The uncut chip thickness can be separated into purely geometrical and physical aspects. The physical aspects of uncut chip thickness include the effects of cutter deflection, runout, vibration, and so forth.

Cutting force modelling of end milling operations

Instantaneous differential cutting force for one single flute was written by Martelotti [START_REF] Tlusty | Dynamics of Cutting in End Milling[END_REF]:

db t K F d cutting cutting ⋅ ⋅ = → → (1)
where t is the uncut chip thickness and db is a differential piece of the depth of cut.

And was rewritten by [START_REF] Armarego | The Machining of Metals[END_REF] adding the edge parcel:

cutting edge F d F d F d → → → + = (2) db t K db K F d cutting edge ⋅ ⋅ + ⋅ = → → →
where edge K → and cutting K → are vectors called specific edge pressure and specific cutting pressure, respectively. The uncut chip thickness t for end milling is written as:

) (φ sin s t t ⋅ = ( 3
)
where φ is the angle of the cutting piece measured in relation of the normal direction of the feed per tooth t s :

f t N v s ⋅ = ω (4)
from known v , ω and f N as the feed velocity, rotation velocity and the number of flutes.

Figure 1. Milling geometry.

The cutting tool pieces can be calculated by

φ λ d d db ⋅ ⋅ = ) tan( 2 ( 5
)
where d is the tool diameter and λ is the helix angle, as shown in Fig.

(1).

The angle δ is calculated by

d b ) tan( 2 λ δ ⋅ ⋅ = ( 6
)
where b is the depth of cut and δ is the contact angle.

The angle δ is used to classify the cutting geometry as Type I or Type II, Type I

1 2 ϕ ϕ δ - ≤ → Type II 1 2 ϕ ϕ δ - > → (7)
where 1 ϕ and 2 ϕ are the entry and exit angles, respectively (Fig.

(1)), enlightened on [START_REF] Tlusty | Dynamics of Cutting in End Milling[END_REF] and [START_REF] Araujo | Models for the Prediction of Instantaneous Cutting Forces in End Milling[END_REF].

The force become:

( )

φ λ φ d d sin s K K db t K db K t cutting edge cutting edge ⋅ ⋅ ⋅ ⋅ ⋅ + = ⋅ ⋅ + ⋅ ∫ ∫ ∫ ) tan( 2 ) ( r r r r (8)
So, the total force of cut, considering the f N flutes of the mill, are calculated by the sum:

∑ = = f N i i F F 1 r r (9)
In this approach, all force contributions are calculated at the same time because all differential parts of the force are calculated for each cutting piece.

The time variation of the milling force vector ( )

) (t F r
is written as the multiplication of functions in time.

) ( ) ( ) ( ) ( ) ( t A t K t h t K t F cutting edge ⋅ + ⋅ = r r r (10)
The specific forces, written as a vector function, (

) (t K edge r and ) (t K cutting r
), are multiplied by a scalar function relative to the height cutting ( ) (t h

) and other relative to the cutting area, the scalar function (

) (t A
). This form is called from now on by Functional approach.

( ) ( θ ω θ F F t F r r =       = (11a) ) ( ) ( ) ( ) ( ) ( θ θ θ θ θ A K h K F cutting edge ⋅ + ⋅ = r r r (11b)
The functions ) (θ h and ) (θ A will be calculated separately in the following sections. Table 1. Integration limits for each phase.

The height function

In Fig.

(3) can be observed the height of cutting of the first flute ( 1 h ) at the milling angle θ , for a cutting geometry.

The height function of cutting of the first flute ) ( 1 θ h can be calculated by:

∫ ⋅ ⋅ = ) ( ) ( 1 2 1 ) ( 2 ) ( θ θ φ λ θ L L d sin d h ( 12
)
where the limits 1 L and 2 L are functions of θ and are calculated differently for each cutting phase of θ as can be observed on Table 1 and the values of 1 e , 2 e , 3 e and 4 e can be extracted from Tab. (2).

In order to add the contributions of all flutes, the height of cutting for any flute ) (n is written by:

φ λ θ ξ θ ξ θ d sin d h n L n L n ⋅ ⋅ = ∫ - ⋅ + - ⋅ + )) 1 ( ( )) 1 ( ( 2 1 ) ( 2 ) ( ( 13
)
Note that the functions 1 L and 2 L are now not only a function of θ but also a function of n , where ξ is the angle between the flutes.

∑ = = f N n n h h 1 ) ( ) ( θ θ ( 14
φ λ φ θ θ θ d sin d sin s A L L t ⋅ ⋅ ⋅ ⋅ = ∫ ) ( ) ( 1 2 1 ) ( 2 ) ( ) ( ( 15
)
where the limits 1 L and 2 L are functions of θ and are calculated differently for each cutting phase of θ as can be observed on Tab. (1). (n is written by:

φ λ φ ξ θ ξ θ d sin d sin s A n L n L t n ⋅ ⋅ ⋅ ⋅ = ∫ - ⋅ + ⋅ - ⋅ + ⋅ )) 1 ( ( )) 1 ( ( 2 1 ) ( 2 ) ( (16)
Note that the functions 1 L and 2 L are now not only a function of θ but also a function of n , where ξ is the angle between the flutes.

The total area is calculated by:

∑ = = f N n n A A 1 ) ( ) ( θ θ (17)
and the Fig.

Cutter runout

When runout is present, the uncut chip thickness ( Eq. ( 3a) ) must be modified to include the effect of the changing radius of the cutter teeth with respect to the axis of rotation.

Set screw-type tool holders commonly employed in industry to eliminate the problems of torsional and axial slippage of the cutter may induce cutter runout as the tightening action of the set screw tends to offset the cutter in the holder Fig. ( 8). As a result, the teeth on the offset side of the holder have a larger effective radius than do those on the other side of the cutter.

The position of the center of the cutter deviates from its nominal positions due to numerous factors, such as cutter deflection, runout, servo error, volumetric error, thermal error, wear, etc. Of all the factors, runout and cutter deflection account for the most deviation.

Cutter runout exists in all kinds of milling operations and results in variations in the uncut chip thickness, local forces and machined surface characteristics. The effect of runout is to redistribute the chip load among the flutes: the average chip thickness averaged over only the flutes engaged with the workpiece increases. The runout offset, ρ , can be easily measured by a dial gauge or other suitable device, whereas its location angle, ε is difficult to measure. This angle determines the initial state of the predicted cutting forces. The uncut chip thickness in presence of runout is rewritten as: [START_REF] Kline | The Effect of Runout on Cutting Geometry and Forces in End Milling[END_REF].

        ⋅ ⋅ - - - - - - ⋅ + ⋅ = ) 2 cos( ) cos( ) ( f t N n sin s t π φ ε θ φ ε θ ρ φ (3b)
Figure 8. Cutter runout.

Force computation

In order to compare the present model with experimental data, the force components should be decomposed in x, y and z directions as they are usually recorded in machining tests.

          ⋅ +           ⋅ =           = ) ( ) ( ) ( θ θ θ ez ey ex cz cy cx z y x K K K h K K K A F F F F r (18)
But it's not convenient to write the specific cutting force in the fixed referential x, y, z. To rewrite it on the more appropriate tool referential t, r, z ( tangential, radial and axial directions ), another functions

) (θ R A and ) (θ R h
has to be introduced: The rotation matrix

          ⋅ +           ⋅ =           ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
) (θ n R
have to be written for each flute:

          - ⋅ + - - ⋅ + - ⋅ + - ⋅ + = 1 0 0 0 )) 1 ( cos( )) 1 ( ( 0 )) 1 ( ( )) 1 ( cos( ) ( n n sin n sin n R n ξ θ ξ θ ξ θ ξ θ θ (21)
Then, for all flutes, the rotated area function becomes:

∑ = ⋅ = f N n n n R A R A and ) ( 4 θ S are defined: ∑ = - ⋅ + ⋅ = f N n n n A S 1 1 )) 1 ( cos( ) ( ) ( ξ θ θ θ (24a) ∑ = - ⋅ + ⋅ = f N n n n sin A S 1 2 )) 1 ( ( ) ( ) ( ξ θ θ θ (24b) ∑ = - ⋅ + ⋅ = f N n n n h S 1 3 )) 1 ( cos( ) ( ) ( ξ θ θ θ (24c) ∑ = - ⋅ + ⋅ = f N n n n sin h S 1 4 )) 1 ( ( ) ( ) ( ξ θ θ θ (24d)
Rewriting the Eq. ( 19), the force can be expressed by: Note that the matrix ) (θ J is the Jacobian of the F components in relation to the specific cutting pressure coefficients.

          ⋅ +           ⋅ =         θ θ θ ez er et R cz cr ct R z y x K K K h K K K A F F F (25a)                     ⋅           - - =           ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 ) ( 0 0 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( ) ( ) ( ) (
          - - = ) ( 0 0 ) ( 0 0 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( ) ( θ θ θ h A S S S S S S S S J                   ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =
K F K F K F K F K F K F K F K F K F K F K F K F K F K F K F K F K F K F (26)

Specific force analysis

The problem is concentrated on the specific force analysis. The Eq. ( 25) can be easily calculated if the specific pressures are constant in θ . [START_REF] Kline | The Effect of Runout on Cutting Geometry and Forces in End Milling[END_REF] assumed that the specific cutting forces are functions of cutting conditions, and the values were calculated using the average cutting forces during one revolution of a cutter.

If parameters of orthogonal and oblique models are time constant, the specific cutting force do not change in time. To analyze this behavior, for each point of θ , the equation:

θ θ θ d F d J d F d J K + ⋅ + + - ⋅ - ⋅ = - - (27)
The shown experimental cutting force data in the Fig. ( 9) had been extracted of the article (Yun & Cho, 2001, p.476, Fig. (8c)) and previously had been synchronized and adequately adjusted for the execution of the analysis. [START_REF] Yun | Accurate 3-D Cutting Force Prediction using Cutting Condition Independent Coefficients in End Milling[END_REF] test cuts were performed with high-speed steel (HSS) end mills with four flutes, 30° helix angle, 11° rake angle and 10 mm diameter in a vertical type machining center. The workpiece material was aluminum 2014-T6. A tool dynamometer (Kistler, type 9257B) was used to measure the three cutting force components. The experiment performed by Yun and Cho and used in this work has the following parameters: But, when calculating the Eq. ( 27), appears a singularity because there are three parameters and six unknown variables. To solve this problem, we suppose that two consecutive points have the same specific force, calculating the specific force for each interval:

                    ⋅ =                     + + + ) ( ) ( ) ( ) ( ) ( ) ( θ θ ez er et cz cr ct A z y x z y x K K K K K K J d F d F d F F F F (28)
The matrix J amplified became a square matrix A J : Calculating the best "average" value of the specific values obtained for hole rotation K , and applying on Eq. ( 25) results: 12) for Fx, Fy and Fz respectively, shows the comparison between experimental force and semi-empirical force. This procedure aims to confirm if the behavior of the specific pressure considered is correct.

                    + + + - + + - + + + + + - - = ) ( 0 0 ) ( 0 0 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( ) ( 0 0 ) ( 0 0 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( θ θ θ d h d A d S d
[ ] [ ]                                           ⋅ + ⋅ + ⋅ + + + + ⋅ + +                     - - - ⋅ - ⋅ =                     - - θ θ θ d F d F d F d F d F d F d J F F F d F d F d F d J K K K K K K z
                      ⋅ =          

Conclusions

This paper analyzes the behavior of the specific cutting pressures from experimental cutting force data. A procedure to estimate the specific cutting forces from data is developed using the function approach including the edge parcel and the runout parameters in the model. The method is validated by recalculation of the cutting force and compared with the experiment published by [START_REF] Yun | Accurate 3-D Cutting Force Prediction using Cutting Condition Independent Coefficients in End Milling[END_REF]. After the analysis of the results, it's evident that the proposed model is effective and accurate. In order to improve the model, the cutter deflection need to be added.

Figure 2 .

 2 Figure 2. Milling tool referential. The time variable t can be substituted by the angle of rotation θ of a fixed point P in a peripheral tool and the tool velocity rotation ω as shown in Fig. (2).

)

Figure 3 .

 3 Figure 3. Height of cutting.

  ) A height function of each flute, for a four flute tool, is present in Fig. (4).

Figure 4 .A

 4 Figure 4. Function of height of cutting. 4. The cross-sectional area variation In Fig. (5) can be observed the chip cross-sectional area ( ) 1 A of the first flute at milling angle θ , for a cutting

Figure 5 .

 5 Figure 5. Chip cross-sectional area. For a single flute and Type I cutting geometry, the chip cross-sectional area function is presented on Fig. (6). In order to add the contributions of all flutes, the chip cross-sectional area function for any flute )(n is written by:

  (7) shows the chip cross-sectional area function ) (θ A for a milling with four flutes ( ) °= 90 ξ and in bold the function of the first flute ) ( 1 θ A , of a cut having Type I geometry.

Figure 6 .

 6 Figure 6. Area function for one single flute, without runout.

Figure 7 .

 7 Figure 7. End milling area function, without runout.

  Estimated values by the authors of this paper, after the analysis of the experimental cutting force data.

Figure 9 .

 9 Figure 9. Experimental data from (Yun & Cho, 2001, p.476, Fig. (8c)).

  to recalculate the forces, Fig. (10), Fig. (11) and Fig. (

Figure 10 .

 10 Figure 10. Comparison between the experimental F x and the semi-empirical F x .

Figure 11 .

 11 Figure 11. Comparison between the experimental F y and the semi-empirical F y .

Figure 12 .

 12 Figure 12. Comparison between the experimental F z and the semi-empirical F z .

Table 2 .

 2 Variables values.

The total height is calculated by:

Table 3 .

 3 Average specific pressures values.

Acknowledgements

This project was stimulated through the fellowship received from the CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) of Brazil. The authors are grateful to this entity and also the Federal University of Rio de Janeiro for the support.