Selection of cutting conditions for a stable milling of flexible parts with bull-nose end mills

Journal of Materials Processing Technology 191 (2007) 279–282

Selection of cutting conditions for a stable milling of flexible parts with bull-nose end mills F.J. Campa ∗ , L.N. L´opez de Lacalle, A. Lamikiz, J.A. S´anchez Department of Mechanical Engineering, University of the Basque Country, ETSII, C/Alameda de Urquijo s/n, 48013 Bilbao, Spain

Abstract In this paper, a three-dimensional dynamic model for the calculation of the stability lobes of low rigidity parts is applied for the prediction of chatter vibrations during the finish milling of aeronautical parts that include thin walls and thin floors. Hence, an accurate selection of both axial depth of cut and spindle speed can be done. The following methodology is applied: first, a modal analysis of the test device is performed; second, stability lobes are calculated; finally milling tests validate the approach. In the example included in this work, the lobes diagrams obtained for the thin floors are accurate, defining an approximate borderline of stability. © 2007 Elsevier B.V. All rights reserved. Keywords: Chatter; Milling; Flexible structures; Thin floors

1. Introduction Airframes are nowadays mainly composed of monolithic components, made of thin walls and thin floors [1]. At high removal rate conditions, the main dynamic problem is the selfexcited vibration called regenerative chatter. For milling, chatter studies were explained in Refs. [2–8]. The study of the highly interrupted milling [9] has lead to the apparition of new added stability lobes, due to period-doubling vibrations. Chatter due to the excitation of the part has been also studied [1,10]. Smith et al. [10] proposed the selection of tools with zero corner radius in thin floor machining. But bull-nose end mills had to be used in complex parts. In this paper, the stability of the milling of thin floors is going to be studied by means of a frequency domain model. 2. Model of the milling stability problem The model here used is the 3D model described in Altintas [7,8]. The evidence of chatter appearance due to Z-compliant modes brings up the necessity of a variable component of the chip thickness in Z direction. This happens when ball-end mills, bull-nose end mills or tools with inserts (with an edge lead angle κ lower than 90◦ ) are ∗

Corresponding author. E-mail address: [email protected] (F.J. Campa).

0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.03.023

used. Bearing in mind the classical chatter theory, the helix angle is not taken into account in the forces model, nor the process damping, although its effect may be noticeable approximately below the fifth lobe. Also, it is supposed that there are not any contact losses between the workpiece and the tool. 2.1. Dynamic chip thickness calculation The dynamic chip thickness hj that the cutting edge j cuts is calculated as a function of the difference of position (x, y, z), between the actual edge and the previous edge (see Fig. 1): h(φj (t)) = [x sin(φj (t)) + y cos(φj (t))]sin(γ) + z cos(γ) (1) where x = x(t) − x(t − T ); z = z(t) − z(t − T ).

y = y(t) − y(t − T );

2.2. Calculation of the cutting forces The force model relates the cutting forces with the dynamic displacements through Eq. (2). The characterization of the specific cutting coefficients considers the shearing and edge force coefficients, so the tangential, radial and axial components of the cutting force along the edge are obtained. Edge forces due to friction are not taken into account for chatter, as they are not affected by the regeneration, so cutting forces (tangential Ft (j),

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F.J. Campa et al. / Journal of Materials Processing Technology 191 (2007) 279–282

Fig. 1. Left, cutting edges out of cut may cut due to Z motions of the floor. Right, projections of the dynamic displacements over the chip thickness direction.

radial Fr (j) and axial Fa (j)) are: ⎧ ⎫ ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ 1 ⎪ ⎬ ⎨ Ft (j) ⎪ Fr (j) = Kt a Kr hj (φj ) ⎪ ⎪ ⎪ ⎭ ⎩ ⎪ ⎭ ⎩ Fa (j) Ka

Introducing Eq. (6) in Eq. (4), the following equation is obtained: (2)

where Kr and Ka are the specific radial and axial shearing coefficients ‘normalized’ respect to Kt . Those forces are projected over axes x, y, z as follows: ⎧ ⎫ ⎡ ⎫ ⎤⎧ −cosφj −sinγ sinφj cosγ sinφj ⎪ ⎪ ⎨ Fx (j) ⎪ ⎬ ⎬ ⎨ Ft (j) ⎪ ⎢ ⎥ Fy (j) = ⎣ sinφj −sinγ cosφj cosγ cosφj ⎦ Fr (j) ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ ⎩ Fz (j) Fa (j) 0 −cosγ −sinγ (3) Substituting Eq. (2) in Eq. (3), summing the forces for all the cutting edges Eq. (4) is obtained, where forces and displacements are related by the matrix of directional coefficients: [A(t)] : {F (t)} =

1 aKt [A(t)]{(t)} 2

(4)

The dynamic displacement between the tool and the workpiece in the moment t and previous (t − T) is the following: t − T : ({rt } − {rw })e−iωc T

(5)

The incremental displacement for a given period is: {Δ} = (1 − e−iωc T )[G(iωc )]{F }eiωc t

(6)

where [G(iωc )] is obtained adding the matrixes Frequency Response Functions (FRF) of the tool and the workpiece, [Gt (iωc )] and [Gw (iωc )], that is, the relative FRF is: ⎡ ⎤ Gxxt (iωc ) Gxyt (iωc ) Gxzt (iωc ) ⎢ ⎥ [G(iωc )] = ⎣ Gyxt (iωc ) Gyyt (iωc ) Gyzt (iωc ) ⎦ Gzxt (iωc ) Gzyt (iωc ) Gzzt (iωc ) ⎡ ⎤ Gxxw (iωc ) Gxyw (iωc ) Gxzw (iωc ) ⎢ ⎥ + ⎣ Gyxw (iωc ) Gyyw (iωc ) Gyzw (iωc ) ⎦ Gzxw (iωc )

Gzyw (iωc )

1 aKt (1 − e−iωc t )[A(t)][G(iωc )]{F }eiωc t 2

(8)

To find an analytical solution for Eq. (8) a mono-frequency solution has been adopted, which consists on making the periodic matrix [A(t)] constant taking its average Fourier term [A0 ] of the Fourier series expansion. Other alternative is the multi-frequency solution, which considers several terms [11]. Introducing the average term, Eq. (8) becomes an eigenvalue problem:   1 −iωc T det [I] − aKt (1 − e )[A0 ][G(iωc )] = 0 (9) 2 Hence, Eq. (10) is the characteristic equation that defines the stability of the system, where the eigenvalue is Eq. (11): det[[I] + Λ[G0 (iωc )]] = 0

(10)

N aKt (1 − e−iωc T ) 2π

(11)

Λ=−

2.4. Calculation of the critical axial depth of cut and the spindle speed

2.3. Calculation of the stability limit

t : ({rt } − {rw }),

{F }eiωc t =

Gzzw (iωc )

(7)

The eigenvalues are complex values, Λ = ΛR+i ΛI . Introducing them in Eq. (11), the critical depth of cut alim for a given ωc is obtained.     ΛI 2 2π (12) alim = − ΛR 1 + NKt ΛR The phase angle of the eigenvalue of ψ is calculated from Eq. (11):   ΛI ψ = tan−1 (13) ΛR The phase between two impacts, ωc T, can be divided into an integer number of waves and the phase shift ε between present and previous vibration waves, hence ε < 2π. ε = π − 2ψ

(14)

ωc T = ε + 2πk

(15)

F.J. Campa et al. / Journal of Materials Processing Technology 191 (2007) 279–282

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From Eqs. (14) and (15), ε and T can be obtained, and so the value of the spindle speed for each lobe k. n=

60 60ωc = NT N(ε + 2πk)

(16)

2.5. Simplified 1D model for thin floor milling For thin floor machining, whenever the compliance in Z is dominant, the stability model presented here can be reduced to a 1D model in Z, considering only the FRF of the floor. Hence, the dynamic chip thickness in (1) can be simplified to: h(φj (t)) = z cos(γ)

(17)

The calculation of the tangential, radial and axial forces remains the same as (2), whilst their projection over the Z axis is as follows: ⎧ ⎫ Ft (j) ⎪ ⎪ ⎨ ⎬   Fr (j) {Fz (j)} = 0 −cosγ −sinγ (18) ⎪ ⎪ ⎩ ⎭ Fa (j) As the size of FRF matrix [Gzzw (iωc )] is 1 × 1, the eigenvalue problem of (9) is reduced to a linear equation where only one eigenvalue is obtained. 2.6. Considerations about highly interrupted milling When the milling is highly interrupted, the harmonic content of the cutting forces cannot be neglected. The multi-frequency solution leads to the truncation of the Fourier series expansion of the matrix of directional factors [A(t)]: ∞ 

[A(t)] =

[Ar ]eirωt ,

r=−∞

[Ar ] =

1 T



T

[A(t)]eirωt dt

(r = 0, ±1, ±2, . . .)

(19)

0

So Eq. (8) becomes Eq. (20): ∞ 

{Fk }eikωt

k=−∞

∞  1 = aKt (1 − e−iωc T ) [A(t)][G(iωc + ikω)] 2 k=−∞

× {Fk }eikωt

(20)

Eq. (20) leads to the determinant (21):   1 −iωc T det δrk [I] − aKt (1 − e )[Wr−k (iωc + ikω)] = 0 2 (r, k = 0, ±1, ±2, . . .)

Fig. 2. Left, variables involved in the averaging. Right, cutting coefficients for aluminium 7075T6 and the bull-nose end tool used in the experimental validation.

account and dof is the number of degrees of freedom. To solve this determinant, iteration is needed. Added lobes due to perioddoubling or flip bifurcation chatter are obtained apart from the Hopf lobes calculated applying the mono-frequency solution. 3. Averaging In the case of a bull-nose end mills or ball-end mills, the angle κ (or its complementary, the axial immersion angle γ) is variable along the tool axis direction, it is necessary to find an average value to solve the stability problem analytically. On the other hand, the cutting coefficients have to be constant to solve the stability equation, but for bull-nose and ball end mills they vary along the tool axis due to the helix angle (Fig. 2). For thin floor machining, an averaging of the axial immersion angle and the cutting coefficients has to be done though. The method here presented averages the cutting edge lead angle at the middle of the cutting arc, κm , over the volume of the chip for a depth of cut ap , so an average cutting edge lead angle is obtained, κ¯ . Then the average cutting coefficients are obtained at the corresponding z¯ . The stability lobes obtained will be accurate at the depth of cut ap only. When the ap is lower than the corner radius:   rad − z κm1 = arccos (22) rad rm1 = (R − rad) + rad sinκm1 Thus,  ap  φ1 κ¯ =

0

dκ = dz (21)

where δrk [I] is the Kronecker delta. The size of the determinant is (2r + 1)dof, where r is the number of harmonics taken into

(23)

(κm1 /2)fz sin(φ)rm1 (dκ/dz) dφ dz  φap0  φ1 0 φ0 fz sin(φ)rm1 (dκ/dz) dφ dz 

rad

1

(24)

(25)

1 − ((rad − z) /rad ) 2

2

When the depth of cut is higher than the corner radius results in (26).

rad(π/2)(π/4) + (z − rad)(π/2) rad(π/2) + (z − rad)  ap  φ1  rad  φ1 0 φ0 (κm1 /2)fz sin(φ)rm1 (dκ/dz)dφ dz + rad φ0 (κm2 /2)fz sin (φ)R dφ dz κ¯ =  ap  φ1  rad  φ1 φ0 fz sin (φ)rm1 (dκ/dz)dφ dz + rad φ0 fz sin(φ)R dφ dz 0 κm2 =

(26)

(27)

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5. Conclusions The milling of thin floors with a corner radius tool has been studied in the present work with the aim of test the viability of the use of these tools to machine floors with only one tool and without back-plate support. A 3D frequency domain model has been considered to take into account the compliances of the system tool-part in the three Cartesian directions. However, the model can be simplified to a 1D model when the modes of the floor are dominant. To solve the nonlinearities introduced in the model by the variation of the cutting coefficients along the cutting edge and the variation of the axial immersion angle, a new method of averaging has been proposed. The model has been tested in an experimental setup with an 87% of effectiveness. Fig. 3. Results:  clearly stable, x clearly unstable,  slightly unstable.

Therefore, the average axial immersion angle γ¯ and the average z¯ are: γ¯ = 180◦ − κ¯

and

z¯ = rad − rad cos(¯κ)

(28)

4. Experimental validation An aluminium block was bolt on a cantilever plate of 250 mm × 200 mm × 10 mm with overhang of 120 mm. Workpiece dynamics was frequency: 111.26 Hz; damping, 0.0153 (%); stiffness, 825,305 N/m. The tool was a bull-nose end mill Ø 16 mm, 2 th, corner radius 2, 5 mm, 30◦ helix. The step was 2, 5 mm. Six axial immersions were tested from 0, 5 to 7 mm. Workpiece vibration has been measured with an accelerometer. Results are in Fig. 3. There are two main lobes, one due to Hopf bifurcation chatter between 3500 and 6500 rpm and other due to period-doubling chatter between 7250 and 9750 rpm approximately. The frequency of chatter is expected to be near the modal frequency, and, in the case of flip bifurcation chatter, it is also approximately half the tooth passing frequency in this case. Three groups have been identified: clearly stable, clearly unstable and slightly unstable. The results show an 87% of tests successfully predicted. The main discrepancies are located next to the flip bifurcation lobe. It seems that the stable area between the Hopf bifurcation lobe and the flip bifurcation lobe should be wider. Also, the flip lobe fails to predict four cases of instability at 7000 rpm. The averaging method is not an exact method, actually, it is an approximation that pretends to be sufficiently accurate.

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