Frequency-domain stability lobe prediction for high-speed face milling process under tool–workpiece dynamic interaction

Face milling cutting forces at different coordinate systems

Three different coordinate systems are used for model description. CS0 is the global coordinate system of the cutting tool system. It is fixed and coincident with the main axes of the cutting tool system. As shown in Figure 2, Z-axis is set to be the direction of the spindle axis of the machine tool. The direction opposite to the feed direction of the workpiece is set to be X. Y is set in a direction perpendicular to X- and Z-axes, pointing to the side where the cutter exits the workpiece during an up-milling operation. The cutting force of the jth tooth can be divided into the orthogonal F_x(j), F_y(j) and F_z(j) components. CS1 is the local coordinate system located at the cutting point of each insert. It is oriented according to the radial, tangential and axial directions, and it rotates in conjunction with the tool. The cutting force of the jth tooth can be divided into tangential cutting force F_t(j), radial cutting force F_r(j) and axial cutting force F_a(j). CS2 is the cylindrical coordinate system coincident with the tool position in the machine workspace. The cutting force can be split into friction component F_f(j) and normal cutting force component F_n(j). θ_j is the angular rotation of the jth tooth rotating from the entry point to X measured in the X–Y plane. These coordinate systems are depicted in Figure 2.

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Figure 2.

Three different coordinate systems are used for model description: (a) CS0, (b) CS1 and (c) CS2.

In CS2 system, at the rake face of the jth cutting tooth, γ_r is the radial rake angle and γ_q is the axial rake angle. The angle between friction and radial cutting force is called cutting layer angle η_c as shown in Figure 2.

The transformation matrix from CS2 to CS1 can be expressed as

[ F t ( j ) F r ( j ) F a ( j ) ] = [ cos γ r sin γ r 0 − sin γ r cos γ r 0 0 0 1 ] · [ cos γ a 0 sin γ a 0 1 0 − sin γ a 0 cos γ a ] · [ F n ( i ) F f ( i ) cos η c F f ( i ) sin η c ]

(1)

The cutting force coefficient in the normal direction and friction direction is K_n and K_f, respectively. K_t, K_r and K_a are the tangential, radial and axial cutting force coefficients, respectively. K_t, K_r and K_a can be expressed as

{ K t = K n cos γ r cos γ a + K f sin γ r cos η c + K f cos γ r sin γ a sin η c K r = − K n sin γ r cos γ a + K f cos γ r cos η c − K f sin γ r sin γ a sin η c K a = − K n sin γ a + K f cos γ a sin η c

(2)

a_p is the axial depth of cut, and C(j) is the actual cut chip thickness. F_t(j), F_r(j) and F_a(j) can be expressed as

{ F t ( j ) = K t · a p · C ( j ) F r ( j ) = K r · a p · C ( j ) F a ( j ) = K a · a p · C ( j )

(3)

The cutting force in CS1 system transformed into CS0 system can be shown as follows

[ F x ( j ) F y ( j ) F z ( j ) ] = [ − cos θ j − sin θ j 0 sin θ j − cos θ j 0 0 0 1 ] · [ F t ( j ) F r ( j ) F a ( j ) ]

(4)

Instantaneous radial cutting chip thickness

Figure 3 shows the dynamic representation of the cutting insert. From Figures 2(b) and 3, it can be seen that when the jth cutting tooth rotates θ_j, Δr is the displacement in the r direction comprising r_jc (displacement for cutting tooth) and r_jw (displacement for workpiece). Δz is the displacement in the Z-direction comprising z_jc (displacement for cutting tooth) and z_jw (displacement for workpiece) and can also induce radial displacement. γ_L is the main cutting edge angle. Suppose Δr₋₁ comprising r_jcl and r_jwl are the displacements in the r direction between the cutting tooth and workpiece for the j − 1th cutting tooth. f_jt is the displacement of tangential cutting force F_t.

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Figure 3.

Dynamic representation of the cutting insert.

The radial cutting force in the r direction can be recognized as the sum of static cutting force and dynamic cutting force. The radial displacement r(θ_j) in the r direction for the jth cutting tooth can be expressed as

r ( θ j ) = ( r j c l l − r j w l l ) − ( r j c − r j w ) + f j t · sin θ j − ( z j c − z j w ) · tan γ L

(5)

When the jth cutting tooth is at rotation angle of θ_j, x_jc, y_jc, x_jw and y_jw are the displacements in the X- and Y-directions in CS0 system for cutting tooth and workpiece cutting layer. x_jcl, y_jcl, x_jwl and y_jwl are the residual displacements of the j − 1th cutting tooth. Equation (5) can be transformed into CS0 system

r ( θ j ) = [ ( x j c − x j c l ) − ( x j w − x j w l ) ] · sin θ j + [ ( y j c − y j c l ) − ( y j w − y j w l ) ] · cos θ + f j t · sin θ j − ( z j c − z j w ) · tan γ L

(6)

Equation (6) shows the instantaneous milling chip thickness in the radial direction.

Dynamic cutting force model

In order to simplify equations (5) and (6), the dynamic relative displacement between cutting tooth and workpiece cutting layer in the X-, Y- and Z-directions can be regarded as

{ Δ } = { Δ x j = ( x j c − x j c l ) − ( x j w − x j w l ) Δ y j = ( y j c − y j c l ) − ( y j w − y j w l ) Δ z j = z j c − z j w }

(7)

The angular position of the cutting tool at entry point is θ_s and at exit point is θ_e. The step function g(θ_j) is

{ g ( θ j ) = 1 , θ s < θ j < θ e g ( θ j ) = 0 , θ j < θ s , θ j > θ e

(8)

According to equation (6), the cutting force for an individual cutting tooth shown in equation (4) can be changed into

{ F x ( j ) = − ( K t cos θ j + K r sin θ j ) ( Δ x · sin θ j + Δ y · cos θ j − Δ z · cos θ j + f t sin θ j ) · g ( θ j ) F y ( j ) = ( K t sin θ j − K r cos θ j ) ( Δ x · sin θ j + Δ y · cos θ j − Δ z · cos θ j + f t sin θ j ) · g ( θ j ) F x ( j ) = K a ( Δ x · sin θ j + Δ y · cos θ j − Δ z · tan γ L + f t sin θ j ) · g ( θ j )

(9)

Equation (9) can also be expressed in matrix form as follows

[ F x ( j ) F y ( j ) F z ( j ) ] = [ F xs ( j ) F ys ( j ) F zs ( j ) ] + g ( θ j ) [ A ( t ) ] [ Δ x Δ y Δ z ]

(10)

where F_xs(j), F_ys(j) and F_zs(j) are the static cutting forces. [A(t)] is the matrix of dynamic cutting force coefficient as shown in equation (11)

[ A ( t ) ] = 1 2 · [ − K t sin 2 θ j − K r ( 1 − cos 2 θ j ) − K t ( 1 + cos 2 θ j ) − K r sin 2 θ 2 tan γ L ( K t cos θ j + K r sin θ j ) K t ( 1 − cos 2 θ j ) − K r sin 2 θ K t sin 2 θ j − K r ( 1 + cos 2 θ j ) 2 tan γ L ( − K t sin θ j + K r cos θ j ) 2 K a sin θ j 2 K a cos θ j − 2 K a tan γ L ]

(11)

where all the variables in equation (11) have been defined before. Now n_c is the number of cutting tool inserts, with ω representing the rotation frequency of face milling cutter, T representing face milling cutter rotation period and ω_t representing face milling cutter insert’s rotation frequency which gives

ω t = n c ω

(12)

T = 2 π n c ω

(13)

The total cutting force at an arbitrary given rotation angle θ is the sum of individual cutting tooth as shown in equation (14)

{ F x = ∑ j = 0 n c − 1 F x ( j ) F y = ∑ j = 0 n c − 1 F y ( j ) F z = ∑ j = 0 n c − 1 F z ( j )

(14)

Suppose the step function g(θ_j) is considered in [A(t)], and each matrix element can be stackable. Equation (10) can be simplified as shown in equation (15)

{ F } = { F s } + a p · [ A ( t ) ] · { Δ }

(15)

where F_s is the total static cutting force.

In a face milling cutting process, cutting force is induced by dynamic displacement between the cutting tooth and workpiece cutting layer. The dynamic cutting force also induces regenerative chatter. Hence, for stability analysis of face milling processes, the effect of static cutting force can be neglected.

The dynamic cutting force F_d can be expressed as

{ F d } = a p · [ A ( t ) ] · { Δ }

(16)

where [A(t)] reflects the features of face milling process: intermittent, time-varying and periodic.

Prediction of face milling stability

For e − i ω t t = cos ω t t − i sin ω t t , the Fourier series expansion of the dynamic milling force coefficient matrix [A(t)] is

{ [ A ( t ) ] = ∑ r = − ∞ ∞ [ A r ] e ir ω t t [ A r ] = 1 T ∫ 0 T [ A ( t ) ] e ir ω t t dt

(17)

The average of the Fourier series expansion is recognized as [A_o]^12,13

[ A o ] = 1 T ∫ 0 T [ A ( t ) ] dt = g ( θ j ) 2 T ∫ 0 T [ α xx α xy α xz α yx α yy α yz α zx α zy α zz ] dt

(18)

By changing time-domain integration into frequency-domain integration, where θ = ω_t, equation (18) can be replaced as

[ A o ] = 1 ω T ∫ 0 2 π g ( θ ) [ A ( θ ) ] d θ

(19)

when θ_s < θ < θ_e, g(θ) = 1. [A_o] can be expressed as

[ A o ] = n c 4 π ∫ θ s θ e [ 1 2 ( K t cos 2 θ − 2 K r θ + K r sin 2 θ ) 1 2 ( − K t sin 2 θ − 2 K t θ + K r cos 2 θ ) 2 tan γ L ( K t sin θ − K r cos θ ) 1 2 ( − K t sin 2 θ + 2 K t θ + K r cos 2 θ ) 1 2 ( − K t cos 2 θ − 2 K r θ − K r sin 2 θ ) 2 tan γ L ( K t cos θ + K r sin θ ) − 2 K a cos θ 2 K a sin θ 2 K a tan γ L · θ ] d θ

(20)

When only considering the vibration at flutter frequency ω_c, the displacement for face milling cutter and workpiece can be calculated as follows

{ d c } = [ x c y c z c ] T = [ G c ( i ω c ) ] · { F } e i ω c t

(21)

{ d w } = [ x w y w z w ] T = − [ G w ( i ω c ) ] · { F } e i ω c t

(22)

The dynamic relative displacement shown in equation (7) can also be written as

{ Δ } = ( { d c } − { d w } ) · ( 1 − e − i ω c T ) = [ G c ( i ω c ) + G w ( i ω c ) ] { F } e i ω c t · ( 1 − e − i ω c T ) = ( 1 − e − i ω c T ) [ G ( i ω c ) ] { F } e i ω c t

(23)

From equations (21)–(23), equation (16) can be expressed as

{ F d } e i ω c t = a p · ( 1 − e − i ω c T ) [ A o ] [ G ( i ω c ) ] { F } e i ω c t

(24)

where [G(iω_c)] is the 3 × 3 transfer function matrix.

In X–Y plane, there is always regenerative chatter. The face milling cutter not only rotates but also does linear feed movement at a certain speed.

The impact factor for regenerative chatter µ is

μ = 1 − f z D

(25)

where D is the cutting tool diameter, and f_z is the feed per tooth.

If µ < 0, then let µ = 0. µ only affects the factor of G_3,3 in matrix [A_o] as shown in equation (26)

G 3 , 3 ( i ω c ) = n c 4 π ( 1 − μ e − i ω c T ) [ G c z z ( i ω c ) + G w z z ( i ω c ) ] ∫ θ s θ e ( − 2 K a tan γ L ) θ d θ

(26)

The face milling stability domain can be solved according to equation (24). If the determinant has nonsingular solution, the characteristic equation of the closed-loop dynamic face milling system can be expressed as

det [ [ I ] − a p ( 1 − e − i ω c T ) ] [ A o ] [ G ( i ω c ) ] = 0

(27)

where [I] is the unitary matrix. The remaining factors including flutter frequency ω_c, axial depth of cut a_p, cutting tooth number n_c, feed per tooth f_z and cutting force coefficients K_t, K_r and K_a need to be determined.

At first, an initial value should be given to a_p as cutting force coefficient depends on a_p. The calculation process is iterative. When the calculation results reach to a certain range, the iterative process ends. So far, the critical axial depth of cut a_pl can be obtained which can be used to represent the stability of face milling process.

The characteristic equation (27) has real part and imaginary part. Hence, there are two unknown items: ω_c and a_pl

Λ = − a p l n c 4 π ( 1 − e − i ω c T ) = Λ R + i Λ I

(28)

p = Λ R Λ I = 1 − cos ω c T sin ω c T

(29)

where Λ is the eigenvalue, and p is the ratio of real part and imaginary part.

The transfer function matrix G o ( i ω c ) can be shown to be

G o ( i ω c ) = ∫ θ e θ s [ α x x α x y α x z α y x α y y α y z α z x α z y α z z ] d θ · [ G x x ( i ω c ) G x y ( i ω c ) G x z ( i ω c ) G y x ( i ω c ) G y y ( i ω c ) G y z ( i ω c ) G z x ( i ω c ) G z y ( i ω c ) G z z ( i ω c ) ]

(30)

The characteristic equation can be shown

det [ [ I ] + Λ [ G o ( i ω c ) ] ] = 0

(31)

The nonsingular solution for equation (31) is a_pl

a pl = − 2 π Λ I n c ( p + 1 p )

(32)

In order to calculate the characteristic value, ω_c have to be known.

Under normal circumstances, ω_c depends on spindle rotation speed n and the modal parameters of face milling system.

The vibration cycle number for individual cutting tooth is

ω c T = ε + 2 k π

(33)

where k is the largest possible integer, and such that ε < 2π.

From equation (33), the phase difference of the stability limit can be expressed

ε = cos − 1 ( 1 − p 2 ) ( 1 + p 2 )

(34)

The phase angle φ determined by eigenvalue Λ is defined as

φ = tan − 1 ( Λ I Λ R ) = tan − 1 ( 1 p )

(35)

The solution for equation (34) is

ε = π − 2 φ or ω c T = ( π − 2 φ ) + 2 k π

(36)

where in face milling stability limit diagram, each k corresponds to different stability phase.

Hence, for certain flutter frequency, the stability limit of the face milling system can be determined by critical axial depth of cut according to equation (32). According to the calculated critical axial depth of cut, the face milling cutter rotation period can be calculated according to equation (36).

The number of rotation period of the face milling cutter can also be expressed as

n = 60 n c T

(37)

It can be seen that when ω_c and k are determined, the spindle speed n can be calculated as shown in equation (38)

n = 60 ω c n c [ ( π − 2 φ ) + 2 k π ]

(38)

According to the above analysis, in a face milling system, when the machine tool, the face milling cutting tool and the workpiece are determined, under certain chatter frequency, the real part and imaginary part of the characteristic value can be obtained from equation (11). Then, according to equations (32) and (38), the critical axial depth of cut a_pl and the corresponding rotation speed n can be obtained. Based on the selected chatter frequency and lobe number to repeat the above step, the stability domain for face milling process can be determined.

Identifying modal parameters for face milling system

The main modal parameters that affect the stability limit lobes include modal mass, modal stiffness, modal damping and natural frequency. Hence, the identification of these parameters is necessary to model the dynamic characteristics. Figure 4 shows the principle of the modal test. The modal tests were conducted on VMC0540d vertical machine center which has an installed HSK63A holder and face milling cutter as shown in Figure 4.

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Figure 4.

Modal test system for face milling system.

Table 1 shows the components of the modal experimental setup.

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Table 1.

The testing equipment and analysis software.

Equipment	Type
Machine center	VMC0540d
Face milling cutter	Sandvik 392.41005-63 32 060B
Impact hammer	In-house design
Force sensor	CL-YD-303 70192
Acceleration sensor	CA-YD-185C 72357
Signal conditioning instrument	AZ802
Data acquisition box	AZ208R
Signal acquisition and analysis system	SsCras

The real frequency diagram and imaginary frequency diagram of the workbench-workpiece can be obtained and are shown in Figure 5.

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Figure 5.

Diagram of real frequency and imaginary frequency for workbench-workpiece.

Figure 6 shows the real frequency and imaginary frequency diagram for “spindle-shank-face milling cutter.”

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Figure 6.

Diagram of real frequency and imaginary frequency for spindle-shank-face milling cutter.

By using component analysis method, the natural frequency, modal mass, modal stiffness and modal damping can be determined and are shown in Table 2.

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Table 2.

Modal parameters for face milling system.

	Nature frequency (Hz)	Modal mass (kg)	Modal stiffness (N/m)	Modal damping (%)
Face milling cutter	1765	0.08912	1.98762	12.45
Workpiece	3890	0.06742	1.27632	13.68

Stability limit lobes for face milling system

According to the critical axial depth of cut and the peak point decided by modal parameters, the stability limit lobes can be built as shown in Figure 7. Since the face milling system is a multi-degree freedom system, reflected in FRF, there are several peak points. Taking every peak point as single degree of freedom, the corresponding “stability lobes” can be constructed. The critical axial depth of cut under every spindle speed is reserved, with the left part trimmed off (the dotted line as shown in Figure 7). Then the stability limit diagram can be obtained.

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Figure 7.

The stability limit lobe for face milling system.

Design of the experimental system

Milling experiments with proper cutting conditions were conducted based on appropriate cutting angles and other cutting parameters. In this article, cutting force and machined workpiece surface roughness are selected as the characterization to check whether the milling process is stable or not. When the milling process is unstable, the cutting force will increase dramatically, and so does the surface roughness. Workpiece material used for the experiments is an aluminum 7050 alloy.

The cutting force acquisition equipment is Kistler dynamometer (9219AA). The cutting force acquisition system is shown in Figure 8. The sensitivity of the dynamometer in the X-, Y- and Z-directions is 35.59 pC/lbf. The surface roughness and surface morphology are determined by using NT9300 optical surface profiler.

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Figure 8.

Cutting force measurement system.

Due to the stability limit, lobes reflect the relationship between axial depth of cut and the spindle speed. Hence, in this article, the feed per tooth is set to be constant with the value of 0.12 mm/tooth.

According to the stability limit lobes, the spindle speeds were selected to be 10,000, 12,000 and 14,000 r/min. Seven different axial cutting depths were adopted as shown in Table 3.

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Table 3.

Cutting parameters for experiments.

Test conditions	A	B	C	D	E	F	G
Axial cutting depth (mm)	2	3	4	5	6	7	8

Experiment results and analysis

Figure 9 shows the cutting force changes with axial depth of cut under different spindle speeds.

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Figure 9.

The relationship between cutting force and axial depth of cut under different spindle speeds: (a) spindle speed is 10,000 r/min, (b) spindle speed is 12,000 r/min and (c) spindle speed is 14,000 r/min.

From Figure 9, it can be seen that the cutting force decreases with the increase in spindle speed. This can be explained by increased heat generated at higher spindle speed, which softens the workpiece and makes it easy to cut. The reflection in cutting force is a monotonic decrease. The value of tangential milling force F_y is the largest, followed by the radial milling force F_x and axial milling force F_z which obtained the same trends as Mativenga and Hon. ¹⁴

In Figure 9(a), the milling force begins to increase sharply and face milling stability declines when the axial depth of cut was 5 mm at the spindle speed of 10,000 r/min; for Figure 9(b) with spindle speed of 12,000 r/min, the milling force begins to increase sharply when the axial depth of cut reached 4 mm and for spindle speed of 14,000 r/min, the critical axial depth of cut is 7 mm as shown in Figure 9(c).

Figure 10 shows the surface roughness change with axial depth of cut under different spindle speeds.

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Figure 10.

Surface roughness change with axial depth of cut under different spindle speeds.

From Figure 10, it can be seen that surface roughness generally increases with the increase in depth of cut and decreases with spindle speed. For spindle speed of 10,000 r/min in stable cutting region (axial depth of cut smaller than 5 mm), the surface roughness changes gently. Once unstable region (axial depth of cut greater than 5 mm) is reached, the surface roughness changes rapidly, and the surface morphology gets worse as shown in Figure 11. From Figure 10, for spindle speeds of 12,000 and 14,000 r/min, the critical axial depths of cut for dividing the stable region and unstable region of 4 and 7 mm, respectively, can also be seen.

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Figure 11.

Surface morphology in stable and unstable cutting operations: (a) surface morphology with axial depths of cut 5 and 6 mm under spindle speed 10,000 r/min, (b) surface morphology with axial depths of cut 4 and 5 mm under spindle speed 15,000 r/min and (c) surface morphology with axial depths of cut 7 and 8 mm under spindle speed 14,000 r/min.

Figure 11 shows the surface morphology in stable and unstable cutting operations. As shown in Figure 11, under different spindle speeds, only two cases of the surface morphology are selected to exhibit. One is under the critical axial depth of cut and the other one is under higher axial depth of cut. It can be seen in Figure 11, for the stable cutting region, the surface quality is good; when in the unstable cutting region, most area of the surface has been deteriorated and presents typical chatter surfaces. The surface morphology is intuitive to distinguish the stable and unstable regions.

Figure 12 shows the stability limit lobe for face milling system. The seven points A, B, C, D, E, F and G represent seven different axial depths of cut 2, 3, 4, 5, 6, 7 and 8 mm when the spindle speeds are 10,000, 12,000 and 14,000 r/min. It can be seen that for spindle speed of 10,000 r/min, A, B, C and D points are in the stable region, and there is no occurrence of chatter phenomena. It begins to have signs of chatter phenomenon when the axial cutting depth reaches 6 mm. While there was occurrence of chatter phenomena when the axial cutting depth reached 6 mm and points E, F and G are in the unstable region. For spindle speed of 12,000 r/min, points A, B and C are in the stable region, and the last four points D, E, F and G are in the unstable region. For spindle speed of 14,000 r/min, the stable region is quite broad and only point G is in the unstable region.

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Figure 12.

Verification of the stability limit lobes.

The region under the dividing line drawn at the bottom of the boundary of the stable region and unstable region can be treated as the absolute stable region as shown in Figure 12. Under the dividing line, it is a stable cutting area for different spindle speeds. In order to guarantee the stability and efficiency of milling process, the axial depth of cut should be chosen to be under the curve, while the spindle speed should be chosen to be highest possible for the given setup.