Mechanism of process damping in milling of thin-walled workpiece

International Journal of Machine Tools and Manufacture 134 (2018) 1–19

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International Journal of Machine Tools and Manufacture journal homepage: www.elsevier.com/locate/ijmactool

Mechanism of process damping in milling of thin-walled workpiece Jia Feng a b

a,b

, Min Wan

a,b,∗

a,b

, Ting-Qi Gao

, Wei-Hong Zhang

T

a,b,∗

School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi, 710072, China State IJR Center of Aerospace Design and Additive Manufacturing, Northwestern Polytechnical University, Xi'an, Shaanxi, 710072, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Velocity-dependent process damping Linearization of cutting force Milling of thin-walled workpiece Stability lobe diagram

Existing ploughing mechanism and velocity-dependent mechanism, which were used for explaining the process damping in milling, did not consider the feedback influence of vibrations in normal direction, and thus, they cannot be used to reveal the underlying cause of process damping in milling process of thin-walled workpiece. This paper systematically studies the generation mechanism of process damping in thin-wall milling, and turns out that the relative vibration of cutter-workpiece system rather than the ploughing indentation is the main source of process damping in this kind of process. Theoretically, the actual cutting velocity and the relative vibrations in both feed and normal directions are combined to formulate the expression of instantaneous uncut chip thickness, which is then used to derive the equations of dynamic cutting forces. Taylor series expansions are carried out around harmonic response to linearize the equations of dynamic cutting forces. Derivations show that the dynamic cutting forces are composed of two items, i.e. the vibration displacements-induced force and the vibration velocities-induced force. The latter, which is actually an additional dissipative force being inversely proportional to the cutting velocity, constitutes a source of process damping. It is subsequently integrated into the process's governing equation to estimate the stability lobe diagrams (SLDs). Verification shows that the SLDs predicted by the proposed velocity-dependent mechanism reasonably agree with the experimentally observed results (among 672 sets of detailed comparisons, agreement rate is more than 95%.). Especially, thin-wall milling experiments with various values of spindle speeds show that the SLDs predicted using the proposed velocitydependent mechanism are much closer to the experimental observations than those obtained by the ploughing mechanism (among 36 sets of detailed comparisons, agreement rate between the predictions and measurements is more than 94% for the proposed velocity-dependent mechanism, while the agreement rate is only 47% for the ploughing mechanism.).

1. Introduction Chatter is one of the biggest barriers for improving the productivity and the accuracy of metal cutting operations, and thus, it is of great importance to find means to avoid the occurrence of chatter vibrations [1,2]. Prediction of stability lobe diagram provides an excellent way to select chatter-free cutting process parameters [1]. Classical stability theories can give accurate prediction results if the spindle speed is at a relatively high level [3]. However, experimental observations show that the stability boundaries will be greatly deviated from the predicted values, and shift toward higher axial depths of cut once relatively low spindle speeds are adopted [4,5]. These phenomena are attributed to the process damping arising from the machining process itself. Many research efforts have been made to study the generation mechanism of process damping. Literature review shows that process damping force mainly originates from an additional dissipative force, which is inversely ∗

proportional to the spindle speed. It is explained by two mechanisms. One is the ploughing mechanism, which is the indentation effect between the clearance face of the cutter and the wavy surface of the workpiece. The other is the velocity-dependent mechanism, which modifies the dynamic cutting forces by introducing the influence of the direction change of cutting velocity. Ploughing mechanism is attributed to the indentation and friction between the clearance face of the cutter and the undulations left on the machined surface of the workpiece. Wu [6,7] assumed that the ploughing force is proportional to the total volume of the displaced material, and summarized two crucial points for predicting ploughing force, i.e. representation of the indented volume and calibration of the ploughing force coefficient.

• Many researchers have carried out studies on calculating the in-

dented volume. Lee et al. [8] presented an iterative algorithm to calculate the volume of the displaced workpiece material by solving

Corresponding authors. School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi, 710072, China. E-mail addresses: [email protected] (M. Wan), [email protected] (W.-H. Zhang).

https://doi.org/10.1016/j.ijmachtools.2018.06.001 Received 31 January 2018; Received in revised form 1 June 2018; Accepted 2 June 2018

Available online 15 June 2018 0890-6955/ © 2018 Elsevier Ltd. All rights reserved.

International Journal of Machine Tools and Manufacture 134 (2018) 1–19

J. Feng et al.

f xdy , f ydy R Ω v , v′

Nomenclature

Ft , Fr Fn, Fs FN , Ff i j pσ , pτ

σs ft , fr K sp S, S st, S dy dz μ lw r0 βs γ hp v x˙ c , y˙c x˙ w , y˙w

x˙ , y˙ φij (t ), φ′ij (t )

φij , φ′ij f tdy , f rdy

•

tangential and radial cutting forces, as shown in Fig. 1 normal and shearing forces acting on the shear plane, as shown in Fig. 1 normal and friction forces acting on the cutter-chip interface, as shown in Fig. 1 index number related to the ith axial disk element index number related to the jth flute normal and shearing stresses exerted on the contact surface, as shown in Fig. 2 material yield strength, as shown in Fig. 2 tangential and radial ploughing forces specific contact force, i.e. the ploughing cutting force coefficient overall indented, statically indented, and dynamically indented areas, as shown in Fig. 3 axial length of the axial disk element mean friction coefficient equivalent wear land length, as shown in Fig. 3 honed radius, as shown in Fig. 3 separation angle, as shown in Fig. 3 clearance angle, as shown in Fig. 3 depth of cutter penetration, as shown in Fig. 3 cutting speed vibration velocities of the cutter in x- and y-directions vibration velocities of the workpiece in x- and y-directions relative vibration velocities of the cutter-workpiece system in x- and y-directions, as shown in Fig. 5 nominal and actual cutting angles related to the jth flute of the ith axial disk element at instant t, as shown in Fig. 5 abbreviated form of φij (t ) and φ′ij (t )

v , v′ u, u′ q˙ c, q˙ w q˙ h′ c t T Ft ′, Fr ′ gij (φij (t ))

gij K t , Kr φst , φex a p , ae N Fx , Fy M c , Cc , K c Mw , C w , Kw q mc , m w Lc , L w Γc , Γw ωc , ω w ζ c, ζ w α n, β , η ψn , βn, τs

dynamic ploughing forces in tangential and radial directions

dynamic ploughing forces in x- and y-directions nominal radius of the cutter angular speed of cutter rotating nominal and actual vectors of the tooth tip's cutting velocity nominal and actual cutting directions, as shown in Fig. 5 nominal and actual normal directions, as shown in Fig. 5 vectors of vibration velocities of cutter and workpiece vector of relative vibration velocity of the cutterworkpiece system instantaneous uncut chip thickness feed per tooth cutting instant time tooth passing period cutting forces in v′ - and u′ -directions screen function related to the jth flute of the ith axial disk element at φij (t ) abbreviated form of gij (φij (t )) tangential and radial cutting force coefficients enter and exit immersion angles axial and radial depths of cut total number of flutes cutting forces in x- and y-directions mass, damping and stiffness matrices of the cutter mass, damping and stiffness matrices of the workpiece total number of discrete nodes along the axial depth of cut number of dominant modes for cutter and workpiece mode shape matrices of cutter and workpiece modal displacement vectors of cutter and workpiece diagonal natural frequency matrices of cutter and workpiece diagonal damping ratio matrices of cutter and workpiece normal rake angle, helix angle and chip flow angle shear angle, friction angle and shear stress

decomposition of vibration signals measured from stable turning tests.

the relative vibrations between the cutter and the workpiece. Endres et al. [9,10] proposed numerical summation formulations to calculate the interference volume for cutting with variable chip thickness. Chiou and Liang [11] used small amplitude assumption to approximate the cut surface as a straight line, and formulated the compressed area as an explicit function of cutter vibrations. Ahmadi and Ismail [12] replaced the nonlinear damper proposed by Chiou and Liang [11] with an equivalent linear viscous damper, and integrated it into the milling stability solution [13]. Cao et al. [14] fitted the sinusoidal expression of workpiece surface undulation with piecewise polynomial to achieve efficiently calculating the indentation area. The other focus is the calibration of the ploughing force coefficient. Shaw and DeSalvo [15] formulated an explicit function to describe the relationship between the normal load and the indented volume. Wu [6] applied Shaw and DeSalvo's contact theory [15] to estimate the ploughing force coefficient. Shawky and Elbestawi [16] decomposed the total indented volume and ploughing forces into static and dynamic components, and then calibrated the ploughing force coefficient based on the statically indented volume and the measured normal cutting forces. Eynian and Altintas [17] determined the ploughing force coefficient through forcing the tool flank to penetrate into the workpiece material. Budak and Tunc [18,19] identified the turning and milling process damping coefficients by combining the predicted analytical stability limits with the experimentally observed results. Ahmadi and Altintas [20] extracted the equivalent process damping coefficient from the frequency domain

Velocity-dependent mechanism includes the influence of the direction change of actual cutting velocity on the dynamic cutting force. Das and Tobias [21] expressed the regenerative force with the cutting velocity effect, and then realized capturing the improvement of low-speed stability. Later, Altintas et al. [22] extended Das and Tobias's model [21] by introducing both slope and curvature of the waves to reveal the influences of regenerative chip thickness, velocity and acceleration on the dynamic cutting force coefficients. Bachrathy and Stepan [23] investigated the velocity-dependent process damping in milling through calculating chip thickness according to the actual cutting velocity direction instead of the nominal one. Molnar et al. [24] analytically studied the influences of radial immersion on velocity-dependent process damping in milling, and developed a model by combining the chip thickness calculation principle proposed by Bachrathy and Stepan [23] with the force projection method developed by Das and Tobias [21]. The model can well capture the improvement of low-speed stability in milling with relatively large radial immersion, while it will result in a negative process damping for milling with low radial immersion. Based on this fact, Molnar et al. [24] concluded that a process damping model suitable for milling with low radial immersion is still needed to be studied further. Besides, it is worth pointing out that Molnar et al. [24] did not carry out experimental verification, and only included the influence of vibrations in feed direction. The vibrations in normal direction were ignored. Actually, if chatter appears in milling process of 2

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defined as the radial contact force per unit indented volume. The value of K sp can be determined by experimental means, such as static indentation tests [17], chatter tests [18,19] or stable cutting tests [20,26]. The ploughing force ft, ij , which is in the direction parallel to the instantaneous cutting velocity, can be calculated by

thin-walled workpiece, the vibrations in normal direction are obvious and will influence the process damping. The above researches studied the two mechanisms separately, and very limited works focused on studying the combined effects of both the ploughing and the velocity-dependent mechanisms. Huang and Wang [25] are the pioneers in exploring the comprehensive effects of process damping mechanisms. They firstly considered the influence of the velocity changes induced by the process's vibrations in formulating the shearing and ploughing force coefficients, and then established a dynamic cutting force model by including both the direction and magnitude variations of the cutter/workpiece relative vibrations. Huang and Wang [25] finally concluded that ploughing mechanism has by far the greatest influence on the total process damping in machining of rigid workpiece. It is also worth mentioning that the generation mechanism of process damping in thin-wall milling was not involved in the study [25]. Hence, this paper presents a velocity-dependent model to predict the process damping in thin-wall milling, which is a typical machining process with relatively low radial immersion. The major innovation lies in that the cutting velocity variation induced by the vibrations in both feed and normal directions is considered for the first time to comprehensively study the generation mechanism of process damping in milling of thin-walled workpiece. For the purpose of comparison study, the ploughing mechanism suitable for milling of rigid workpiece is also briefly introduced. The structure of this paper is summarized as follows. In section 2, process damping model together with the dynamic governing equation is established. Especially, the process damping forces, which are induced by the changes of cutting velocity, are formulated by considering the process's vibrations in feed and normal directions. Section 3 experimentally validates the proposed model by conducting thin-wall milling tests, and conclusions are drawn in Section 4.

ft, ij = μfr, ij

where μ is the average friction coefficient on the contact surface, and it is usually assumed to be a constant. The indented area Sij is separated into two components, i.e. static area S st and dynamic area Sijdy , whose geometrical definitions are shown in Fig. 3. The semicircle-like region surrounded by the horizontal line and the outline of the cutter is the statically indented area S st , while the triangle region between the trajectory of separation point and the horizontal line is the dynamically indented area Sijdy . Actually, Fig. 3 (a) and (b) show two cases. One is that the cutter vibrates toward the workpiece, while the other is that the cutter deflects from the workpiece. For both cases, Sijdy can be expressed as a unified formula if the amplitude of the cutter's vibration is relatively small [27].

Sijdy =

l w2 r˙ij 2v

(3)

where r˙ij is the vibration of the cutter, which is coupled by the vibration velocities x˙ c (t ) and y˙c (t ) in x- and y-directions.

r˙ij = x˙ c (t )sin φij + y˙c (t )cos φij

(4)

The equivalent wear land length l w can be calculated according to Ahmadi and Altintas's method [20].

l w = l w1 + l w2 + l w3, l w3 =

l w1 = r0sin βs,

l w2 = r0sin γ ,

r 0 (cos γ − cos βs)

(5)

tan γ

By substituting Eqs. (3) and (4) into Eqs. (1) and (2), the dynamic ploughing forces in tangential and radial directions have the following form.

2. Formulation of milling process damping 2.1. Process damping model in milling of rigid workpiece

dy ⎡ f t, ij (t ) ⎤ l w2 ⎡ μ sin φij μ cos φij ⎤ ⎡ x˙ c (t ) ⎤ dy f tr, dz ⎥ = K sp ⎢ ij (t ) = ⎢ dy cos φij ⎥ ⎢ y˙ (t ) ⎥ 2v sin φij f (t ) ⎢ ⎣ ⎦⎣ c ⎦ ⎣ r, ij ⎥ ⎦

There are two mechanisms controlling the cutting process, i.e. (1) the chip-formation mechanism rising in front of the cutter rake face and (2) the ploughing mechanism occurring in the vicinity of the cutter nose region [7]. In this paper, the forces induced by the chip-formation and ploughing mechanisms are called cutting force and ploughing force, respectively. Relationships of cutting forces are shown in Fig. 1. It can be seen that the shearing force Fs and the normal force Fn acting on the shear plane are in balance with the friction force Ff and the corresponding normal force FN acting on the cutter-chip interface. Note that in Fig. 1, Ft and Fr are force components in the directions parallel and vertical to the instantaneous cutting velocity. If milling process is concerned, Ft and Fr correspond to the tangential and radial forces, respectively. Under the effect of ploughing mechanism, the material below the cutter's clearance surface will be indented and restrained by its surrounding material, and thus, a very complex elastic-plastic stress field is produced inside the workpiece. Fig. 2 (a) illustratively shows the produced stress field, which must be in equilibrium with the distributed contact pressures around the cutter nose region, as indicated in Fig. 2 (b). Since the volumetric strain produced by ft is really smaller than that induced by fr , the force component fr accommodates the majority of the indented material. Under this fact, the ploughing force fr, ij , which is vertical to the instantaneous cutting velocity, is assumed to be proportional to the total indented volume of material [6,7].

fr, ij = K sp Sij dz

(2)

(6)

dy Eq. (6) indicates that f tr, ij (t ) is proportional to vibration velocities, and constitutes the source of process damping. Total dynamic ploughing forces in x- and y-directions are obtained by dy ⎡ f x (t ) ⎤ t f dy ( ) = ⎥= ⎢ xy f dy (t ) ⎥ ⎢ ⎦ ⎣ y

∑

dy Tij (t ) f tr, ij (t )

i, j

(1)

where the subscript ij means the index number related to the jth flute of the ith axial disk element. dz is the axial length of each disk element. Sij is the indented area and K sp is the specific contact force, which is

Fig. 1. Cutting force diagram. 3

(7)

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J. Feng et al.

Fig. 2. Explanatory diagrams of the ploughing mechanism. (a) Equivalent stress field distribution beneath the cutter nose region; (b) Normal and shearing stresses exerted on the contact surface.

the system's vibration is at a relatively small level. Under this fact, the tangential loads, i.e. Ft, ij and ft, ij , act approximately in the nominal cutting direction, and the radial loads, i.e. Fr, ij and fr, ij , act along the nominal normal direction. As a result, the direction changes of tangential and radial loads are weak and can be neglected. In such a machining process, ploughing mechanism has more significant effect on process damping than that from velocity effect [21], as proven by

where Tij (t ) is the coordinate transformation matrix defined as follows.

⎡− cosφij − sinφij ⎤ Tij (t ) = ⎢ sinφij − cosφij ⎥ ⎣ ⎦

(8)

It should be mentioned that the above procedure is derived based on the assumption of rigid workpiece. That is, only the vibration of cutter is taken into account. Since the workpiece's vibration is not involved,

Fig. 3. Static and dynamic components of the indented area. (a) In case of that the cutter vibrates toward the workpiece; (b) In case of that the cutter deflects from the workpiece. Note that fr and ft are in radial and tangential directions in milling. 4

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J. Feng et al.

the principal basis of ploughing-based process damping mechanism lies in that the cutter could indent the machined surface of workpiece. In milling process of rigid workpiece, the stiffness of workpiece is large enough to prevent it from deflecting, and thus, this characteristic provides the opportunity for the cutter's clearance surface to penetrate into the machined surface. Under this fact, the ploughing effect evidently appears and substantially contributes to the generation of process damping. However, in the flexible milling system shown in Fig. 4, thin-walled workpiece has relatively weak rigidity in normal direction. This will thus lead to obvious normal deflections under cutting forces [30]. This phenomenon means that once the cutter has the tendency to penetrate into the machined surface, the workpiece will deflect to resist this penetration. That is, actual penetration depth in thin-walled workpiece is much smaller than that in rigid workpiece, hence, the contribution of ploughing to the generation of process damping in milling of thin-walled workpiece is limited and can be ignored. (2) Dynamic deflections enormously enhance the contribution of velocity effect to process damping. If chatter occurs in flexible milling, there will exist obvious vibrations in normal direction (i.e. y-direction) of the machined surface [31], and correspondingly, the vibrations will influence the system's dynamic behaviors including process damping [21,24]. The following contents will focus on explaining the contribution of the cutter-workpiece system's vibrations to process damping. Self-excited vibrations in milling process of thin-walled workpiece are shown in Fig. 5, where a cutter with the diameter of R rotates at an angular speed of Ω (rad/s). The nominal vector of the tooth tip's cutting velocity, i.e. vij (t ) , is in the instantaneous tangential direction, and involves a component in x-direction and the one in y-direction. Mathematically, vij (t ) can be calculated by

Fig. 4. Schematic illustration of milling process of thin-walled workpiece with flexible cutter.

Huang and Wang [25]. Just by substituting f dy xy (t ) into the dynamic model of milling system, stability lobes including process damping can be solved by using either semi-discretization [28] or full-discretization [29] methods.

⎡ RΩ cos φij ⎤ vij (t ) = ⎢ − RΩ sin φij ⎥ ⎣ ⎦

2.2. Process damping model in milling of thin-walled workpiece

(9)

For the convenience of derivation, φij (t ) is abbreviated as φij in the following procedure. Actually, the vibrations x˙ i (t ) and y˙i (t ) of the cutter-workpiece system will cause the change of the nominal cutting velocity vector vij (t ) . That is, the tooth tip's actual cutting velocity v′ij (t ) composes of the nominal cutting velocity vij (t ) , the cutter vibration q˙ c, i (t ) and workpiece vibration q˙ w, i (t ) .

Fig. 4 schematically shows a milling process of thin-walled workpiece with flexible cutter. Since the workpiece usually has the characteristics of low rigidity, large size and complex shape in such a milling process [2], both the static and dynamic deflections will change the ploughing-based process damping mechanism from the following two points.

v′ij (t ) = vij (t ) + q˙ i (t ) with

(1) Static deflections greatly weaken the ploughing effect. As is known,

Fig. 5. Self-excited vibrations in milling process of thin-walled workpiece. (a) Geometries in peripheral milling; (b) Dynamic chip thickness. 5

(10)

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J. Feng et al.

differential cutting forces are obtained.

x˙ i (t ) ⎤ = q˙ c, i (t ) − q˙ w, i (t ) q˙ i (t ) = ⎡ ⎢ y˙i (t ) ⎥ ⎣ ⎦ x˙ c, i (t ) ⎤ q˙ c, i (t ) = ⎡ ⎢ y˙c, i (t ) ⎥ ⎣ ⎦ x˙ w, i (t ) ⎤ ⎡ q˙ w, i (t ) = ⎢ y˙ (t ) ⎥ ⎣ w, i ⎦

N

Fx, i (t ) = ∑ j = 1 − gij Kt [(c + x i (t ) − x i (t − T ))sin φ′ij cos φ′ij + (yi (t ) − yi (t − T ))cos φ′ij cos φ′ij ]dz N

+ ∑ j = 1 − gij Kr [(c + x i (t ) − x i (t − T ))sin φ′ij sin φ′ij + (yi (t ) − yi (t − T ))cos φ′ij sin φ′ij ]dz

(11)

(20)

Substituting Eqs. (9) and (11) into Eq. (10) gives N

Fy, i (t ) = ∑ j = 1 gij Kt [(c + x i (t ) − x i (t − T ))sin φ′ij sin φ′ij

v′ij (t )cos φ′ij ⎤ v′ij (t ) = ⎡ ⎢− v′ij (t )sin φ′ij ⎥ ⎣ ⎦

+ (yi (t ) − yi (t − T ))cos φ′ij sin φ′ij ]dz

(12)

N

+ ∑ j = 1 − gij Kr [(c + x i (t ) − x i (t − T ))sin φ′ij cos φ′ij

with

+ (yi (t ) − yi (t − T ))cos φ′ij cos φ′ij ]dz

(RΩ cos φij + x˙ i (t ))2 + (RΩ sin φij − y˙i (t ))2

v′ij (t ) =

(21)

RΩ cos φij + x˙ i (t )

cosφ′ij =

2

Since the actual cutting angle φ′ij is related to the vibrations x˙ i (t ) and y˙i (t ) of the cutter-workpiece system, Eqs. (20) and (21) are nonlinear time-delayed formula. The harmonic response is symbolized as pi (t ) , which is associated with the T-periodic chatter-free motion.

2

⎛RΩ cos φ + x˙ (t ) ⎞ + ⎛RΩ sin φ − y˙ (t ) ⎞ ⎜ i ⎟ i ⎟ ij ij ⎝ ⎠ ⎝ ⎠

⎜

RΩ sin φij − y˙i (t )

sinφ′ij =

2

2

⎛RΩ cos φ + x˙ (t ) ⎞ + ⎛RΩ sin φ − y˙ (t ) ⎞ ⎜ i ⎟ i ⎟ ij ij ⎝ ⎠ ⎝ ⎠

⎜

(13)

⎡ x p, i (t ) ⎤ ⎡ x p, i (t − T ) ⎤ pi (t ) = ⎢ = pi (t − T ) = ⎢ yp, i (t ) ⎥ y (t − T ) ⎥ ⎦ ⎣ ⎦ ⎣ p, i

Comparing Eq. (12) with Eq. (9) indicates that φ′ij is the actual cutting angle, as shown in Fig. 5. Correspondingly, the chip thickness should be measured in the actual radial direction u′ij rather than in the nominal radial direction uij . The total chip load can be expressed by

In order to linearize the expressions of Eqs. (20) and (21), the first order Taylor series expansions are carried out around the critical condition of harmonic response [33].

h′ij (t ) = c sin φ′ij + [x i (t ) − x i (t − T )] sin φ′ij + [yi (t ) − yi (t − T )] cos φ′ij

(14) Fx, i (t ) = Fx, i (t )

where [ x i (t ) , yi (t ) ] and [ x i (t − T ) , yi (t − T ) ] represent the dynamic displacements of the cutter-workpiece system at the present and previous tooth passing periods, respectively. The tangential F ′t, ij (t ) and radial F ′r, ij (t ) cutting forces, which are proportional to the axial disk length dz and the chip thickness h'ij (t ) , are expressed as follows [32].

F ′t, ij (t ) ⎤ F′tr, ij (t ) = ⎡ ⎢ F ′r, ij (t ) ⎥ ⎦ ⎣ K t = gij (φij (t )) ⎡ ⎤ h′ij (t )dz ⎢ Kr ⎥ ⎣ ⎦

+

(15) +

∂Fy, i (t ) ∂x˙ i (t )

∂Fx, i (t ) ∂xi (t )

(xi (t ) − xp, i (t )) + pi

∂Fx, i (t ) (xi (t − T ) − xp, i (t − T )) ∂xi (t − T ) pi

∂Fx, i (t )

pi

(yi (t ) − yp, i (t )) + (yi (t − T ) − yp, i (t − T )) ∂yi (t − T ) pi (x˙ i (t ) − x˙ p, i (t )) +

pi

+

∂Fy, i (t ) ∂xi (t )

∂Fx, i (t ) ∂y˙i (t )

(y˙i (t ) − y˙ p, i (t )) pi

(xi (t ) − xp, i (t )) + pi

(23)

∂Fy, i (t ) (xi (t − T ) − xp, i (t − T )) ∂xi (t − T ) pi

∂Fy, i (t )

pi

(yi (t ) − yp, i (t )) + (yi (t − T ) − yp, i (t − T )) ∂yi (t − T ) pi (x˙ i (t ) − x˙ p, i (t )) +

pi

∂Fy, i (t ) ∂y˙i (t )

(y˙i (t ) − y˙ p, i (t )) pi

(24) The following contents will focus on the simplification of the partial derivatives in Eqs. (23) and (24). Since harmonic response is small perturbation from the equilibrium limit cycle, the forced vibration velocities x˙ p, i (t ) and y˙p, i (t ) have small values, which are far less than the nominal cutting speed component RΩ. Thus, x˙ p, i (t ) and y˙p, i (t ) can be approximated as zeros when they are used for addition or subtraction operation with RΩ [24,33]. That is, they can be neglected in Eq. (13). Based on this approximation, a series of equations can be obtained from Eq. (13) as follows.

(16)

(17)

cosφ′ij sinφ′ij

pi

≈ cos φij

pi

≈ sin φij

∂cosφ ′ij ∂xi

N

∑ T′ij (t ) F′tr,ij (t )

∂xi

=0 =0 pi

∂cosφ ′ij ∂yi

(19)

=0 pi

∂sinφ ′ij ∂yi

By substituting Eqs. (14), (15) and (19) into Eq. (18), the following 6

(25)

pi

∂sinφ ′ij

(18)

where T'ij (t ) is the coordinate transformation matrix defined as follows.

− cosφ′ij − sinφ′ij ⎤ T′ij (t ) = ⎡ ⎢ sinφ′ij − cosφ′ij ⎥ ⎦ ⎣

pi ∂Fy, i (t ) ∂yi (t )

+

For the convenience of derivation, gij (φij (t )) is abbreviated as gij in the following procedure. Projecting the cutting forces in Eq. (15) into Cartesian coordinate system, and then summing the cutting forces related to all flutes will give

j=1

∂Fx, i (t ) ∂x˙ i (t )

Fy, i (t ) = Fy, i (t )

φst = 0 ⎧ , up milling arccos(1 φ = − ae / R) ⎨ ⎩ ex

Fx, i (t ) ⎤ Fxy, i (t ) = ⎡ = ⎢ Fy, i (t ) ⎥ ⎣ ⎦

∂Fx, i (t ) ∂yi (t )

+

with

⎧ φst = arccos(ae / R − 1) , down milling φex = π ⎨ ⎩

+ pi

where gij (φij (t )) is the screen function determining whether the disk element is in or out of cut.

1, φst ≤ φij ≤ φex gij (φij (t )) = ⎧ ⎨ otherwise ⎩ 0,

(22)

=0 pi

(26)

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∂ cos φ ′ij ∂x˙ i

≈

∂x˙ i

≈ pi

∂ cos φ ′ij ∂y˙i

≈

−sin φij cos φij RΩ

≈

dy pr stac ⎡ F xy,1 (t ) + F xy,1 (t ) + Fxy,1 (t ) ⎤ ⎥ ⎢ ⋮ ⎥ ⎢ dy pr Fxy (t ) = ⎢ F stac xy, i (t ) + F xy, i (t ) + F xy, i (t ) ⎥ ⎥ ⎢ ⋮ ⎥ ⎢ stac dy pr ⎢ F xy, q (t ) + F xy, q (t ) + Fxy, q (t ) ⎥ ⎦[2q × 1] ⎣ ⌢stac ⌢ ⌢ = F xy (t ) + D (t )[q (t ) − q (t − T )] + P (t )[q˙ (t ) − p˙ (t )]

sin φij cos φij RΩ

pi

∂ sin φ ′ij ∂y˙i

RΩ

pi

∂ sin φ ′ij

the ith axial element of the cutter. If the cutter is divided into q number of axial elements along the axial depth of cut, the total cutting force Fxy (t ) associated with the whole depth of cut can be expressed as follows.

sin2φij

−cos2φij RΩ

pi

(27)

Detailed derivations of Eq. (27) and the partial derivatives in Eqs. (23) and (24) are given in Appendix A. It should be pointed out that when harmonic response is considered, x˙ i (t ) and y˙i (t ) in Eq. (13) become x˙ p, i (t ) and y˙p, i (t ) . By substituting Eqs. (25)–(27) into Eqs. (23) and (24), Fx, i (t ) and Fy, i (t ) can be easily linearized as follows.

(35)

with

N

Fx, i (t ) = − ∑ j = 1 gij c (Kt sin φij cos φij + Kr sin φij sin φij )dz N

− ∑ j = 1 gij (Kt sin φij cos φij + Kr sin φij sin φij )((x i (t ) − x p, i (t )) − (x i (t − T ) − x p, i (t − T )))dz N

− ∑ j = 1 gij (Kt cos φij cos φij + Kr sin φij cos φij )((yi (t ) − yp, i (t )) − (yi (t − T ) − yp, i (t − T )))dz + +

N c ∑ j = 1 gij RΩ [Kt sin φij (cos2φij N c ∑ j = 1 gij RΩ [Kt cos φij (cos2φij

− sin2φij ) + Kr sinφij (2sin φij cos φij )](x˙ i (t ) − x˙ p, i (t ))dz − sin2φij ) + Kr cos φij (2sin φij cos φij )](y˙i (t ) − y˙p, i (t ))dz

(28)

N

Fy, i (t ) = + ∑ j = 1 gij c (Kt sin φij sin φij − Kr sin φij cos φij )dz N

+ ∑ j = 1 gij (Kt sin φij sin φij − Kr sin φij cos φij )((x i (t ) − x p, i (t )) − (x i (t − T ) − x p, i (t − T )))dz N

+ ∑ j = 1 gij (Kt sin φij cos φij − Kr cos φij cos φij )((yi (t ) − yp, i (t )) − (yi (t − T ) − yp, i (t − T )))dz − −

N c ∑ j = 1 gij RΩ [Kt sin φij (2sin φij cos φij ) − Kr sin φij (cos2φij − sin2φij )](x˙ i (t ) − x˙ p, i (t ))dz N c ∑ j = 1 gij RΩ [Kt cos φij (2sin φij cos φij ) − Kr cos φij (cos2φij − sin2φij )](y˙i (t ) − y˙p, i (t ))dz

Eqs. (28) and (29) indicate that with the aid of the first order Taylor series expansions, the cutting forces in Eqs. (20) and (21), which are represented as the functions of the actual cutting angle φ′ij , are converted to the expressions represented by the nominal cutting angle φij . Eqs. (28) and (29) can be rewritten as the following matrix form. dy pr Fxy, i (t ) = F stac xy, i (t ) + F xy, i (t ) + F xy, i (t )

(30)

⎡ x i (t ) − x i (t − T ) ⎤ F dy xy, i (t ) = Dij (t ) ⎢ ⎥ ⎣ yi (t ) − yi (t − T ) ⎦

(31)

⎡ x˙ i (t ) − x˙ p, i (t ) ⎤ Fpr xy, i (t ) = Pij (t ) ⎢ y˙ (t ) − y˙p, i (t ) ⎥ ⎦ ⎣ i

(32)

(29)

⌢stac ⌢ ⌢⌢ N F xy (t ) = ∑ j = 1 (⌢ g (t ) c T (t ) Ktr V2 (t )dz ) ⌢ ⌢ ⌢⌢ N D (t ) = ∑ j = 1 (⌢ g (t ) T (t ) Ktr V (t )dz ) ⌢ N P (t ) = ∑ j = 1

(

⌢ ⌢⌢ c ⌢ g (t ) T2 (t ) Ktr V (t )dz RΩ

q (t ) = q c (t ) − qw (t ),

⎡ qw,1 (t ) ⎤ ⎡ q c,1 (t ) ⎤ ⎢ ⋮ ⎥ ⎢ ⋮ ⎥ ⎥ ⎢ ⎥ ⎢ q c (t ) = ⎢ q c, i (t ) ⎥ , qw (t ) = ⎢ qw, i (t ) ⎥ ⎢ ⋮ ⎥ ⎢ ⋮ ⎥ ⎥ ⎢ ⎥ ⎢ q c, q (t ) q (t ) ⎣ w, q ⎦[2q × 1] ⎦[2q × 1] ⎣

⏜

g (t ) = diag[g1j g1j ⋯ gij gij ⋯ gqj gqj ]2q × 2q

N

N

Dij (t ) = ∑ j = 1 (gij Tij (t ) Ktr Vij (t )dz )

(

c

)

⏜ V2 (t )

= [sin φ1j ⋯ sin φij ⋯ sin φqj ]qT× 1

⎡ T1j (t ) ⎤ ⎢ ⎥ ⋱ ⏜ ⎥ Tij (t ) T (t ) = ⎢ ⎢ ⎥ ⋱ ⎢ ⎥ ⎢ Tqj (t ) ⎥ ⎣ ⎦2q × 2q

F stac xy, i (t ) = ∑ j = 1 (gij c Tij (t ) Ktrsinφij dz )

N

(36)

(37)

with

Pij (t ) = ∑ j = 1 gij RΩ T2, ij (t ) Ktr Vij (t )dz

)

(33)

⎡ Ktr ⎤ ⋱ ⎢ ⎥ ⎥ Ktr Ktr = ⎢ ⎢ ⎥ ⋱ ⎢ ⎥ Ktr ⎦2q × q ⎣ ⏜

K ⎡ cos 2φij sin 2φij ⎤ T2, ij (t ) = ⎢ , Ktr = ⎡ t ⎤, Vij (t ) = [ sin φij cos φij ] ⎢ Kr ⎥ − sin 2φij cos 2φij ⎥ ⎣ ⎦ ⎣ ⎦ (34)

⎡ V1j (t ) ⎤ ⎥ ⎢ ⋱ ⎥ Vij (t ) V (t ) = ⎢ ⎢ ⎥ ⋱ ⎢ ⎥ ⎢ Vqj (t ) ⎥ ⎣ ⎦q × 2q

⏜

dy where F stac xy, i (t ) is the static cutting force component. F xy, i (t ) is the dynamic cutting force component induced by the vibration displacements, i.e. x i (t ) , x i (t − T ) , yi (t ) and yi (t − T ) , while Fpr xy, i (t ) is the dynamic cutting force component induced by vibration velocities, i.e. x˙ i (t ) and y˙i (t ) . Obviously, Fpr xy, i (t ) actually constitutes the process damping force since it is the function of vibration velocities. It should be pointed out that Eq. (30) is the cutting force related to

⏜ T2 (t )

7

⎡ T2,1j (t ) ⎤ ⎥ ⎢ ⋱ ⎥ T2, ij (t ) =⎢ ⎢ ⎥ ⋱ ⎢ ⎥ ⎢ T2, qj (t ) ⎥ ⎣ ⎦2q × 2q

(38)

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Fig. 6. Experimental setup in milling tests. ⌢

2.3. Dynamic model of milling process of thin-walled workpiece including the effect of process damping

⌢

(42) For the convenience of analyzing stability, u c and uw are transformed into modal coordinates.

⌢stac ⌢ ⌢ Mc q¨ c (t ) + Cc q˙ c (t ) + K cq c (t ) = Fxy = F xy + D (t )[q (t ) − q (t − T )] + P [q˙ (t ) − p˙ (t )] ⌢stac ⌢ ⌢ ˙ ¨ Mw qw (t ) + C w qw (t ) + Kw qw (t ) = −Fxy = − F xy − D (t )[q (t ) − q (t − T )] − P [q˙ (t ) − p˙ (t )]

u c[2q × 1] (t ) = Lc[2q × mc] Γc[mc × 1] (t )

(39)

uw[2q × 1] (t ) = L w[2q × mw] Γw[mw × 1] (t )

Notice that the dynamic cutting force acting on the cutter and the dynamic cutting force acting on the workpiece are a pair of force and reaction force, and thus, they are in opposite directions. Replacement q(t ) in Eq. (39) with the T-periodic solution p(t ) leads to the cancellation of the delay and the velocity terms of the right-hand expressions [33]. As a result, a modified governing equation is provided for the periodic solution.

(43)

where Lc[2q × mc] and L w[2q × mw] are the mode shape matrices of cutter and workpiece, respectively. Γc[mc × 1] (t ) and Γw[mw × 1] (t ) are the modal displacement vectors of cutter and workpiece, respectively. Eq. (42) is formulated in modal coordinates as follows. + pr ¨c (t ) + 2ζ cωn,c Γ˙ c (t ) + ω2n,c Γc (t ) = LcT F dy Γ (t ) xy + pr ¨w (t ) + 2ζ w ωn,w Γ˙ w (t ) + ω2n,w Γw (t ) = −L Tw F dy Γ (t ) xy

⌢stac Mc p¨c (t ) + Cc p˙ c (t ) + K cpc (t ) = F xy

(44)

with (40)

⌢ ⎛ Γc (t ) ⎤ ⎡ Γc (t − T ) ⎤ ⎞ + pr F dy (t ) = D (t )[ Lc L w ] ⎜ ⎡ − xy ⎢− Γw (t ) ⎥ ⎢− Γw (t − T ) ⎥ ⎟ ⎦ ⎣ ⎦⎠ ⎝⎣ ˙ Γc (t ) ⎤ ⌢ + P (t )[ Lc L w ] ⎡ ⎢− Γ˙ (t ) ⎥ w ⎣ ⎦

To investigate the stability of the periodic solution p(t ) , Eq. (39) is linearized around p(t ) with respect to new “small” perturbation variable u(t ) .

u (t ) = q (t ) − p (t )

⌢

Mw u ¨ w (t ) + C w u˙ w (t ) + Kw uw (t ) = −D (t )[u (t ) − u (t − T )] − Pu˙ (t )

Based on Eq. (35), the governing equation of the machine cutterworkpiece system is given as follows.

⌢stac Mw p¨ w (t ) + C w p˙ w (t ) + Kw pw (t ) = − F xy

⌢

Mc u ¨ c (t ) + Cc u˙ c (t ) + K cu c (t ) = D (t )[u (t ) − u (t − T )] + Pu˙ (t )

(41)

(45)

Combining Eq. (44) with Eq. (45), the governing dynamic equation can be expressed as follows.

Substituting Eqs. (40) and (41) into Eq. (39) leads to

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Fig. 7. Measured FRFs along the cutter axis. (a) Locations of measurement points; (b) Measured FRFs of milling cutter in x-direction; (c) Measured FRFs of milling cutter in y-direction.

Table 2 Identified natural frequencies and damping ratios of the workpiece.

Table 1 Identified modal parameters of the milling cutter in x- and y-directions. Modal direction

Mode No.

x

1 2 3 4 1 2 3 4

y

Natural frequency (Hz)

Damping ratio (%)

Mass normalized mode

888 1807 2011 2292 972 1595 1902 2290

2.98 3.66 1.94 2.36 3.05 5.58 3.25 2.05

1.4538, 1.2438, 1.0515 3.2130, 2.1457, 1.3174 3.2175, 2.0492, 1.2033 4.2826, 2.5710, 1.2775 1.6474, 1.3825, 1.1348 2.4515, 1.7673, 1.2180 4.1486, 2.7443, 1.6788 4.4567, 2.6420, 1.3006

shape (1/

Mode No.

Natural frequency by finite element method (Hz)

Natural frequency by standard impact test (Hz)

Damping ratio (%)

1 2

1970 2333

1927 2285

4.5 3.5

kg )

process of thin-walled workpiece. It should be mentioned that if the milling process of thin-walled workpiece has large material removal rate, the workpiece dynamics will continuously change during process. Under this circumstance, the in-process workpiece dynamics induced by material removal can be estimated by using the method proposed by Budak et al. [36]. 3. Experimental verification In this section, a series of down milling tests with low radial immersion are carried out to verify the accuracy of the proposed process damping mechanism. A four-fluted carbide end mill with diameter of 12 mm, helix angle of 30∘ , rake angle of 10∘ and clearance angle of 9∘ is used in actual thin-wall milling. The hone radius of the cutter is 14 μm . Thin-walled plates with the size of 120 × 35 × 3 mm are used, as shown in Fig. 6. The workpiece material is Aluminum alloy 7050 with Young's modulus of 71.7 GPa, density of 2830 kg/m3 and Poisson's ratio of 0.33. 3.1. Impact tests Standard impact tests are carried out to measure the frequency response functions (FRFs) of both cutter and workpiece. In order to obtain the mode shapes at different axial height of the cutter, three miniature accelerometers are attached to the axial altitudes, which are located at 0, 17 and 34 mm away from the cutter tip. That is, a1 = a2 = 17 mm , as shown in Fig. 7 (a). The impact hammer exerts impulse excitation at the cutter tip. The measured FRFs of the cutter are shown in Fig. 7 (b) and (c), which correspond to the results in x- and y-directions, respectively. The modal parameters, which are identified by using software system CUTPROTM , are listed in Table 1. Based on the measurements, linear interpolation is used to evaluate the modal displacements corresponding to the medial points along axial depth of cut. The natural frequencies and mode shapes of the workpiece are obtained by using finite element method. Impact tests are performed at

Fig. 8. Workpiece's FRF corresponding to the top of the plate at the end of tool path. 2 ¨c (t ) ⎤ ⎡ 2ζ cωn,c 0 ⎤ Γc (t ) 0 ⎤ ⎡ Γ˙ c (t ) ⎤ + ⎡ ωn,c ⎡ Γ ⎡ ⎤ + ⎥ ⎢−Γ˙ (t ) ⎥ ⎢ 0 ω2 ⎥ ⎢−Γw (t ) ⎥= ⎢− Γ ¨w (t ) ⎥ ⎢ 0 2 ζ ω n,w w w ⎦ n,w ⎦ ⎣ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ T T ⌢ ⎡ Lc ⎤ ⌢ ⎡ Γ˙ c (t ) ⎤ ⎛ ⎡ Γc (t ) ⎤ ⎡ Γc (t − T ) ⎤⎞ ⎡ Lc ⎤ ⎢ T ⎥ D (t ) [ Lc L w ]⎜ ⎢−Γ (t ) ⎥ − ⎢−Γ (t − T ) ⎥⎟ + ⎢ T ⎥ P (t ) [ Lc L w ] ⎢−Γ˙ (t ) ⎥ w w L L ⎣ ⎦ ⎣ ⎦ w w ⎝ ⎠ ⎣ w ⎦ ⎣ ⎦ ⎣ ⎦

(46)

By using the discrete-time simulation methods reported in Refs. [34,35], Eq. (46) can be solved to predict the chatter stability of milling

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Table 3 : Simulated mass normalized mode shapes of the workpiece. Cutter position

Modal direction

Mode No.

Mass normalized mode shape (1/

0

x

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

0.0360, 0.0394, 0.0483, 0.0598, 0.0701, 0.0729, 0.0620 −0.1622, −0.1466, −0.1072, −0.0543, 0.0029, 0.0508, 0.0753 10.230, 8.1556, 6.1066, 4.1623, 2.4398, 1.0796, 0.2272 15.609, 12.703, 9.7603, 6.8657, 4.1895, 1.9659, 0.4711 0.0127, 0.0118, 0.0099, 0.0072, 0.0042, 0.0015, 0.0001 −0.2993, −0.2590, −0.2008, −0.1443, −0.0918, −0.0471, −0.0150 10.580, 8.5077, 6.4673, 4.5250, 2.7788, 1.3508, 0.3758 12.610, 10.192, 7.8181, 5.5349, 3.4457, 1.7003, 0.4811 0.0002, 0.0001, −0.0002, −0.0004, −0.0005, −0.0004, −0.0002 −0.4614, −0.3964, −0.3039, −0.2151, −0.1342, −0.0669, −0.0204 10.645, 8.5638, 6.5095, 4.5504, 2.7884, 1.3502, 0.3733 7.1319, 5.7546, 4.4052, 3.1108, 1.9305, 0.9490, 0.2674 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000 −0.5232, −0.4491, −0.3439, −0.2432, −0.1515, −0.0755, −0.0231 10.640, 8.5592, 6.5043, 4.5449, 2.7835, 1.3469, 0.3721 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000

y 20

x y

40

x y

60

x y

kg )

Fig. 9. Predicted 3D stability lobes. (a) Without process damping; (b) With process damping.

different axial locations to extract the damping ratios of the workpiece system. The method proposed by Cakar and Sanliturk [37] is used to eliminate the effect of transducer mass on measurements. The measured workpiece's FRF, which corresponds to the top of the plate at the end of tool path, is shown in Fig. 8. Without the loss of generality, the FRF in y-direction is plotted. To demonstrate the relative flexibility of the workpiece compared to the cutter, the FRF of the tool tip is also given in Fig. 8. It can be seen that the workpiece is more flexible than the cutter. Table 2 shows the simulated natural frequencies and identified natural frequencies and damping ratios. It is observed that the natural frequencies obtained by finite element simulation and standard impact test are in good agreement. Table 3 illustratively lists the simulated mass normalized mode shapes corresponding to the selected axial altitudes, which are located at 0, 5, 10, 15, 20, 25 and 30 mm away from the top of the thin-walled plate.

cos(βn − αn) + tan β tan η sin βn

Kt =

τs sin ψn

Kr =

τs

sin(βn − αn )

sinψn cosβ

cos2 (ψn + βn − αn) + tan2η sin2βn

cos2 (ψn + βn − αn) + tan2η sin2βn

(47)

where α n and β are normal rake angle and helix angle of the cutter. ψn , βn and τs are shear angle, friction angle and shear stress, which are determined from the orthogonal cutting tests. In this work, Kt and Kr are estimated by using Eq. (47), in which the values of ψn , βn and τs are taken from Ref. [38]. The estimated values of Kt and Kr are 828 N/mm2 and 200 N/mm2 , respectively. To check whether the obtained Kt and Kr could be effective in this work, a series of milling tests are conducted with the feed rate varying from 0.02 mm/tooth to 0.1 mm/tooth. The measured cutting forces match well with the predicted results, and this confirms that the obtained values of Kt and Kr can ensure the prediction accuracy of the following study. Actually, as described below, the good consistency between the measured and predicted stability results also validates the effectiveness of the used cutting force coefficients. Here, it should be pointed out that the cutting forces can be measured either by KISTLER™ dynamometer or by other additional devices mounted on tool-spindle system [39,40]. In this work, KISTLER™ 9255 B table dynamometer is used for measurement. Since the workpiece's FRFs vary along the whole tool path, the tool path is divided into 24 segments to obtain 25 cutter positions. Corresponding to each cutter position, stability lobe diagram is predicted in terms of spindle speed and axial depth of cut. Combining the

3.2. Cutting tests A series of down milling tests with 5% radial immersion ratio and 0.05 mm/tooth feed rate are conducted to validate the proposed process damping mechanism. The experimental setup consisted of a threeaxis table dynamometer, a displacement sensor and a microphone for the collections of cutting forces, displacements and sounds is shown in Fig. 6. The cutting force coefficients, i.e. Kt and Kr , can either be mechanistically calibrated or be calculated by the orthogonal-to-oblique transformation given in Eq. (47) [32,38].

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Fig. 10. Predicted stability lobes v. s. the cutter position at spindle speeds of (a) 3200 rpm, (b) 3500 rpm, (c) 3800 rpm and (d) 5500 rpm.

validates the effectiveness of the model, and also clearly indicates that the process damping effect is improved with the decrease of spindle speeds. To understand the contribution of different modes to instability, the predicted stability lobes related to the beginning and the middle of the workpiece are plotted in Fig. 11 (c) for comparison. It is observed that obvious difference exists in the predicted results. This phenomenon can be attributed to the following reason. Fig. 12 visually shows the first and second mode shapes of the workpiece. Obviously, the two modes have different shapes in the middle and the end. Especially, with respect to the middle of the workpiece, the second mode has higher stiffness than the first mode. This leads to that the critical axial depth of cut associated with the middle of the tool path is relatively higher than that related to the end, as can be observed from Fig. 11 (c). Figs. 13–17 illustrate the sampled cutting forces and sound signals together with their spectra of Fourier transformation. It can be observed that for the three selected spindle speeds, chatter occurs at the two ends of the workpiece. Spectrum diagrams show that for the test conducted at spindle speed of 3800 rpm and axial depth of cut of 2.4 mm, chatter mainly concentrates on the frequencies around the first mode; while for the test at spindle speed of 5500 rpm and axial depth of cut of 1.2 mm, the chatter frequency is mainly in the vicinity of the second mode. Besides, it is also noticed that in Figs. 14 and 15, there are multiple chatter frequencies, i.e. chatter frequency (1989 Hz ) and integer multiples of tooth passing frequencies spreading on either side of the chatter frequency (1735 Hz, 2242 Hz, 2495 Hz, 2749 Hz and 3002 Hz ). This phenomenon was explained by the time domain simulation results

stability lobe diagrams at all cutter positions gives a three dimensional (3D) stability lobe diagram [41,42], which is related to spindle speed, axial depth of cut and cutter position. Fig. 9 (a) and (b) show the 3D stability lobe diagrams without and with velocity-dependent process damping mechanism, respectively. It can be seen that the stability boundaries are obviously improved by considering the proposed process damping mechanism. Cutting forces and sound signals are collected to detect whether the process is stable or not. By means of Fourier transformation, the cutting process can be considered as stable cut if the spectra of the force and sound signals are distributed in the multiples of spindle and tooth passing frequencies. Chatter occurrence should have peaks around the natural frequencies of the cutter or workpiece, and the spectra are different from the harmonics of spindle and tooth passing frequencies. Fig. 10 shows the experimentally observed stability results at the spindle speeds of 3200 rpm, 3500 rpm, 3800 rpm and 5500 rpm. Fig. 10 definitely indicates that if process damping is not considered, there exist clear discrepancies between the predictions and measurements. However, once the proposed process damping mechanism is taken into account, the experimental results are in good agreement with the predicted stability boundaries. To show the growing stable parameter region at low spindle speeds, a series of experiments corresponding to the beginning and the middle of the tool path have been carried out to verify the accuracy of the proposed model. Measurements collected at different spindle speeds are compared to the predicted stability lobes, as shown in Fig. 11 (a) and (b). Good agreement between the measured and predicted results

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transformation shows that during the whole milling process, there are no strong spectra. This means that the cutter is stable during process. That is, the chatter frequencies belong to the workpiece. At the same time, from Figs. 13–17, it is observed that as the cutter travels along the tool path, the process experiences chatter, stable cut and chatter. It should be commented that as the cutter reaches close to the middle of the tool path, the process does not suddenly become stable. There exists a transition stage from chatter to stable cut. For example, the tool path corresponding to Fig. 14 can be divided into three segments, i.e. the start (0–40 mm), the middle (40–80 mm) and the end (80–120 mm). From the start to the end, the process experiences chatter, stable cut and chatter. In the beginning of the middle segment, i.e. the part of 40–45 mm, it does not show an absolute stable state, as illustrated in Fig. 19. It can be seen that the amplitudes of sound spectra gradually decay and tend to be stable. This is due to that chatter-existing part of the cycle affects the cutter-workpiece vibrations in the beginning of the stable part of the cycle. Finally, to show the influence of ploughing mechanism on chatter stability, the stability lobes predicted by both velocity-dependent mechanism and ploughing mechanism are plotted together for comparison, as shown in Fig. 20. Here, the ploughing cutting force coefficient reported in Ref. [26] is adopted to predict the SLDs corresponding to the existing ploughing mechanism. It can be clearly seen that the stability predicted with velocity-dependent mechanism is closer to measured results than the one related to the ploughing mechanism. This is attributed to the following reason. As show in Fig. 8, the workpiece is more flexible than the cutter. This fact means that the workpiece is so flexible that it resists against penetration of cutter into the machined surface. That is, the penetration effect is greatly weakened, and thus, ploughing has a very limited influence on the process damping. It confirms that the velocity-dependent mechanism rather than the ploughing mechanism dominates the process damping effect in thinwall milling. 4. Conclusions Due to the small stiffness of thin-walled workpiece, the velocitydependent mechanism rather than the ploughing mechanism dominates the process damping effect in thin-wall milling. In this paper, a velocitydependent process damping model, which considers the relative vibrations of cutter-workpiece system in both feed and normal directions, is established. To well reflect the real cutting states, instantaneous uncut chip thickness and dynamic cutting force are calculated with the actual cutting velocity, which is composed of nominal cutting velocity and the vibrations of the cutter and workpiece. By means of Taylor series expansions, the dynamic cutting forces represented by actual cutting angle are converted to a linearized form with nominal cutting angle. The governing dynamic equation of the cutter-workpiece system is given in modal space for the convenience of analyzing stability, and 3D SLDs are evaluated by including the proposed process damping mechanism. The SLDs calculated by the proposed velocity-dependent mechanism show reasonable agreements with the experimental results (among 672 sets of detailed comparisons shown in Figs. 10 and 11, only 33 sets show disagreement and the agreement rate is more than 95% if the proposed velocity-dependent mechanism is considered; while among these 672 sets of comparisons, 216 sets show disagreement and the agreement rate is only 68% if the proposed velocity-dependent mechanism is not considered.), proving the reliability and the accuracy of the proposed mechanism; while, the SLDs obtained by the existing

Fig. 11. Predicted stability lobes v. s. the spindle speed at cutter positions of (a) 5 mm and (b) 60 mm. (c) Comparison of the predicted results corresponding to the cutter positions of 5 and 60 mm.

for low immersion milling, as stated in Refs. [43,44]. It should be pointed out that as can be seen from Tables 1 and 2, the first and the second modes of the workpiece have very close frequencies to those of the third and the fourth modes of the cutter. To decide whether the chatter frequencies belong to the workpiece or cutter, a non-contact capacitive displacement sensor (see Fig. 6) is adopted to sample the displacement signals of the cutter. Fig. 18 shows the measured displacements of the cutter corresponding to Figs. 14 and 15. Fourier

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Fig. 12. Mode shapes of the workpiece. (a) First bending mode shape; (b) Second torsional mode shape.

Fig. 13. Measured sounds together with their frequency spectra at spindle speed of 3200 rpm and axial depth of cut of 3.2 mm (SF: spindle frequency, CF: chatter frequency). 13

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Fig. 14. Measured sounds together with their frequency spectra at spindle speed of 3800 rpm and axial depth of cut of 2.4 mm (SF: spindle frequency, CF: chatter frequency).

Fig. 15. Measured cutting forces together with their frequency spectra at spindle speed of 3800 rpm and axial depth of cut of 2.4 mm (SF: spindle frequency, CF: chatter frequency). 14

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Fig. 16. Measured sounds together with their frequency spectra at spindle speed of 5500 rpm and axial depth of cut of 1.2 mm (SF: spindle frequency, CF: chatter frequency).

Fig. 17. Measured cutting forces together with their frequency spectra at spindle speed of 5500 rpm and axial depth of cut of 1.2 mm (SF: spindle frequency, CF: chatter frequency). 15

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Fig. 18. Measured displacements of milling cutter together with their frequency spectra at spindle speed of 3800 rpm and axial depth of cut of 2.4 mm.

Fig. 19. Frequency spectra corresponding to (a) 40 mm, (b) 42 mm, (c) 44 mm and (d) 46 mm away from the start of the tool path in Fig. 14.

ploughing mechanism greatly deviate from the measured results (among 36 sets of detailed comparisons shown in Fig. 20, about 19 sets show disagreement between the predictions and measurements, and the agreement rate is only 47% for the ploughing mechanism. Especially, about 17 sets show relative error of more than 100%.), confirming that ploughing has little influence on the process damping of thin-wall milling. Besides, verification also shows that in thin-wall milling of the workpiece with one end being fixed, the critical axial depth of cut associated with the middle of the tool path is relatively higher than that related to the tool path's ends since the middle of the workpiece has relatively high dynamic stiffness. As a result, chatter is easy to occur at the beginning and the end of the tool path. At the same time, chatterexisting part of the cycle corresponding to the previous tool positions affects the cutter-workpiece vibrations in the beginning of the stable part of the cycle related to the subsequent tool positions. Fig. 20. Comparison of the stability lobes predicted by velocity-dependent and ploughing mechanisms at cutter position of 5 mm. 16

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Acknowledgements

National Key Research and Development Program of China under Grant no. 2017YFB1102800, and the Fundamental Research Funds for the Central Universities under Grant no. 3102018gxc025.

This research has been supported by the National Natural Science Foundation of China under Grant nos. 51675440 and 51705427,

Appendix A. Detailed derivations of Eq. (27) and the partial derivatives in Eqs. (23) and (24) In this study, harmonic response is treated as small perturbation from the equilibrium limit cycle. That is, the forced vibration velocities x˙ p, i (t ) and y˙p, i (t ) have small values, which are far less than the nominal cutting speed component RΩ. As a result, x˙ p, i (t ) and y˙p, i (t ) can be approximated as zeros when they are used for addition or subtraction operation with RΩ [24,33]. That is, they are neglected in Eq. (13) for the derivation of Eq. (27). The derivation procedure is as follows.

∂ cos φ ′ij ∂x˙ i (t )

= pi

≈

∂ sin φ ′ij ∂x˙ i (t )

RΩ − RΩ cos φij

RΩ cos φij RΩ

=

∂ cos φ ′ij ∂y˙i (t )

⎝

⎠ X2

≈

∂ sinφ ′ij

RΩ cos φij RΩ

=

R2Ω2

=

= pi

≈

RΩ

2 ⎛⎜RΩ cos φij + x˙ p, i (t ) ⎞⎟ 1 ⎠ −⎛⎜RΩ sin φij − y˙ p, i (t ) ⎟⎞ ⎝ X 2

−RΩ sin φij

pi

sin2φij

=

R2Ω2

pi

≈

∂y˙i (t )

2 ⎛⎜RΩ cos φij + x˙ p, i (t ) ⎞⎟ ⎠ X

1 X − ⎛⎜RΩ cos φij + x˙ p, i (t ) ⎞⎟ ⎝ 2 ⎝ ⎠ 2 X

−sin φij cos φij RΩ

1 −⎛⎜RΩ cos φij + x˙ p, i (t ) ⎞⎟ 2 ⎝ ⎠ X2 RΩsinφij

RΩ cos φij

RΩ

=

R2Ω2

−2 ⎛⎜RΩsinφij − y˙ p, i (t ) ⎞⎟ ⎝ ⎠ X

sinφij cosφij RΩ

1 −X − ⎛⎜RΩ sin φij − y˙ p, i (t ) ⎟⎞ 2 ⎝ ⎠ X2

−RΩ + RΩ sin φij

−2 ⎛⎜RΩ sin φij − y˙ p, i (t ) ⎞⎟ ⎠ ⎝ X

RΩ sin φij RΩ

R2Ω2

=

−cos2φij

(27)

RΩ

with

X=

(RΩ cos φij + x˙ p, i (t ))2 + (RΩ sin φij − y˙p, i (t ))2

(A.1)

With the aid of the above results, the partial derivatives in Eqs. (23) and (24), which are required to formulate Eqs. (28) and (29), can be derived as follows.

∂ (sin φ′ij cos φ′ij ) ∂x˙ i (t )

≈

≈

≈

∂y˙i (t )

≈

≈

∂y˙i (t )

RΩ

−cos2φij cos

≈ pi

φij +

(A.4)

sin2φij cos

φij

RΩ

(A.5)

sin φij cos2φij + sin φij cos2φij RΩ

pi

∂ (sin φ′ij sin φ′ij )

(A.3)

−sin2φij cos φij − sin2φij cos φij

pi

∂ (cos φ′ij cos φ′ij ) ∂y˙i (t )

RΩ

pi

∂ (sin φ′ij cos φ′ij )

(A.2)

sin2φij cos φij + sin2φij cos φij

pi

∂ (sin φ′ij sin φ′ij ) ∂x˙ i (t )

RΩ

pi

∂ (cos φ′ij cos φ′ij ) ∂x˙ i (t )

−sin φij cos2φij + sin φij sin2φij

−sin

φij cos2φij

− sin

(A.6)

φij cos2φij

RΩ

(A.7)

17

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J. Feng et al.

Then, the partial derivatives of Fx, i (t ) and Fy, i (t ) are represented as the functions of the nominal cutting angle φij . N

= − ∑ gij c (Kt sin φij cos φij + Kr sin φij sin φij )dz

Fx, i (t )

j=1

pi

∂Fx, i (t ) ∂x i (t )

= − ∑ gij (Kt sin φij cos φij + Kr sin φij sin φij )dz pi

N

= pi

∂Fx, i (t ) ∂y˙i (t )

∑ gij (Ktsin φij cos φij + Krsin φijsin φij)dz

(A.10)

j=1 N

= − ∑ gij (Kt cos φij cos φij + Kr sin φij cos φij )dz (A.11)

j=1

pi

N

∂Fx, i (t ) ∂yi (t − T ) ∂Fx, i (t ) ∂x˙ i (t )

(A.9)

j=1

∂Fx, i (t ) ∂x i (t − T ) ∂Fx, i (t ) ∂yi (t )

(A.8)

N

∑ gij (Kt cos φijcos φij + Krsin φijcos φij)dz

=

(A.12)

j=1

pi N

c

∑ gij RΩ [Ktsin φij (cos2φij − sin2φij) + Krsin φij (2sin φijcos φij)]dz

= pi

(A.13)

j=1 N

c

∑ gij RΩ [Kt cos φij (cos2φij − sin2φij) + Kr cosφij (2sin φij cos φij )]dz

=

(A.14)

j=1

pi

and N

=

Fy, i (t )

j=1

pi

= pi

= − ∑ gij (Kt sin φij sin φij − Kr sin φij cos φij )dz pi

=

(A.17)

j=1

∑ gij (Ktsin φij cos φij − Kr cos φijcos φij)dz (A.18)

j=1

pi

N

∂Fy, i (t ) ∂yi (t − T )

= − ∑ gij (Kt sin φij cos φij − Kr cos φij cos φij )dz

= − ∑ gij pi

j=1 N

∂Fy, i (t )

= − ∑ gij pi

(A.19)

j=1

pi N

∂Fy, i (t )

∂y˙i (t )

(A.16)

N

N

∂Fy, i (t )

∂x˙ i (t )

∑ gij (Ktsin φijsin φij − Krsin φij cos φij)dz j=1

∂Fy, i (t ) ∂x i (t − T )

∂yi (t )

(A.15)

N

∂Fy, i (t ) ∂x i (t )

∑ gij c (Ktsin φijsin φij − Krsin φij cos φij)dz

j=1

c [Kt sin φij (2sin φij cos φij ) − Kr sin φij (cos2φij − sin2φij )]dz RΩ

(A.20)

c [Kt cos φij (2sin φij cos φij ) − Kr cos φij (cos2φij − sin2φij )]dz RΩ

(A.21)

Finally, it should be mentioned that x˙ p, i (t ) and y˙p, i (t ) are treated as zeros in the above procedure. Directly considering nonlinear effect induced by the actual forced vibration velocities x˙ p, i (t ) and y˙p, i (t ) into the formulation of velocity-dependent process damping model remains as an open problem to be theoretically carried out in the future.

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