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Chatter free tool orientations in 5-axis ball-end milling
International Journal of Machine Tools & Manufacture 106 (2016) 89–97
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International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Chatter free tool orientations in 5-axis ball-end milling Sun Chaob, Yusuf Altintas a,n,1 a The University of British Columbia, Department of Mechanical Engineering, Manufacturing Automation Laboratory, 2054-6250 Applied Science Lane, Vancouver, BC V6T 1Z4, Canada b State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 22 December 2015 Received in revised form 14 April 2016 Accepted 18 April 2016 Available online 20 April 2016
Dies, molds and parts with complex free form surfaces are usually machined with ball end mills on 5-axis CNC machining centers. This paper presents automatic adjustment of tool axis orientations to avoid chatter along the tool path. The process mechanics and dynamics of ball end milling are modeled in cutter-workpiece engagement coordinate system. The structural dynamics of tool and workpiece are transformed to cutter-workpiece engagement coordinates by considering the tool path and the kinematics of the machine tool. The stability of the 5-axis ball end milling is modeled at each tool path location, and the chatter free tool axis orientations are searched iteratively using Nyquist criterion while avoiding gouging limits. The tool path, i.e. cutter location (CL) file, is updated to generate chatter free, 5-axis ball end milling of the parts. The proposed algorithm has been experimentally proven in 5-axis ball end milling tests. & 2016 Elsevier Ltd. All rights reserved.
Keywords: 5 axis Ball end milling Chatter stability Tool orientation Optimization
1. Introduction Ball end milling is commonly used in finish milling of dies, molds, impellers, turbine blades and other parts with complex sculpture surfaces. The tool axis direction is always constant in 3 axis machines, but can be oriented to avoid collision and zero speeds at the tool tip on 5-axis machine tools. This paper presents a method to predict tool axis orientations to avoid chatter in 5-axis ball end milling of parts with free form surfaces. There has been significant research reported in 5-axis milling, but mainly on tool path generation and collision avoidance as summarized by Lasemi et al. [1]. The tool path generation and collision avoidance methods are based on wokpiece geometry and machine configuration, hence the physics of machining are not considered. Recent research efforts considered the servo drives, contouring errors, and the kinematics of 5-axis machine tools. Sencer et al. [2] modeled the contouring errors as a function of tool tip, tool orientation, path curvature and servo dynamics. Beudaert et al. [3] and Yuen et al. [4] optimized the tool orientation to generate smooth tool paths while avoiding to violate the torque limits of all drives in 5-axis machines. The process related research in tool axis orientation considers the static and dynamic flexibilities of the workpiece and tool. n
Corresponding author. E-mail address: [email protected] (Y. Altintas). 1 www.mal.mech.ubc.ca
http://dx.doi.org/10.1016/j.ijmachtools.2016.04.007 0890-6955/& 2016 Elsevier Ltd. All rights reserved.
Lazoglu et al. [5] optimized the tool orientation to constrain the static deflections of tools and parts perpendicular to the finish surface. Although significant research has been reported in the stability of machining operations [6], there has been a very limited effort reported in optimizing the tool orientation to avoid chatter in 5-axis ball-end milling of free form surfaces. Ozturk et al. [7] modeled the influence of tool's tilt and lead angles of the tool on the stability of 5-axis ball end milling operations. They used both zero order and multi-frequency stability solutions [6] to generate stability lobes as a function of tool's lead and tilt angles. However, the engagement and the FRF directions vary with the change of tool orientation along the tool path. The chatter free tool path planning requires the identification of most optimal tool orientation and depth along the curved, 5-axis tool path, which is presented in this paper. The frequency response functions (FRF) of the machine and workpiece are assumed to be measured in their stationary coordinate systems. The relative FRFs between the tool and part are transformed to moving tool-part engagement coordinates where the ball end milling process stability is modeled. The tool-part engagement conditions, feed direction and kinematics of the machine tool are considered in constructing the stability equation in frequency domain. The rotary drive positions of the machine which determine the tool axis orientation are searched to achieve chatter free, stable cutting conditions along the tool path. The proposed method is experimentally validated with 5-axis ball end milling tests. Henceforth, the paper is organized as follows. The dynamics of ball end milling system is modeled in engagement coordinate
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Nomenclature
dynamic displacement of workpiece in WCS relative dynamic displacement between the tool and workpiece in WCS n (ϕj (z , t )) surface normal vector at cutting edge A (ϕj (z , t )) directional matrix average of directional matrix A0 FE , FM , FW dynamic cutting forces in ECS, MCS and WCS, respectively. relative FRF between the tool and workpiece in ECS ΦE FRF of the tool measured in MCS ΦM FRF of the workpiece measured in WCS ΦW ⇀ ⇀ { PW , OW } original tool tip coordinate and orientation vector, respectively. ⇀ ⇀ { PWm, OWm } modified tool tip coordinate and orientation vector, respectively. angular positions of A and C axis, respectively θA , θC rotational transformation matrix from ECS to WCS RE − W rotational transformation matrix from modified enR Em− W gagement coordinate system ECSm to WCS rotational transformation matrix from MCS to WCS RMW rotational transformation matrix from modified enR Em− E gagement coordinate system ECSm to ECS [ xm ym zm ]T Cartesian coordinate of a vertex in ECSm [ R0 ϕm κm ]T Spherical coordinate of a vertex in ECSm
ΔqW ΔqW t
WCS ECS MCS ECSm l ϕstl , ϕex
workpiece coordinate system engagement coordinate system machine coordinate system modified engagement coordinate system entry and exit angle of the engagement map of axial disk element l l l , ztop bottom and top z value of the engagement map of zbottom axial disk element l ϕj (z, t ) radial immersion angle at time t and elevation z from the tool tip axial immersion angle at elevation z from the tool tip κ (z ) unit step function that determines whether the tooth g (ϕj ) is in or out of cut Krc , Ktc , Kac oblique cutting force coefficients in radial, tangential and axial directions, respectively height of differential axial disk elements Δz dynamic chip thickness h d (t ) ΔqE , Δq Em relative dynamic displacement between the tool and workpiece in ECS and modified ECS, respectively. ΔqM , ΔqMW dynamic displacement of tool in MCS and WCS, respectively.
system in Section 2. The tool and part dynamics, which change as a function of tool path and machine drive positions, are transformed to engagement coordinate system in Section 3. Chatter free tool orientations are predicted in Section 4. The proposed, chatter free tool axis orientation along the paths are experimentally proven in Section 5, and the paper is concluded in Section 6.
2. Coordinate transformations of vectors for process dynamics A five axis ball end milling system is shown in Fig. 1. The tool path is generated in a Computer Aided Manufacturing (CAM) software environment using the Workpiece Coordinate System (WCS). The machining process is modeled in Engagement Coordinate System (ECS) where the Z is the tool axis; X-axis is in the tool axis-feed plane, and Y axis is normal to XZ plane. The chip thickness, hence the force distribution along the cutting edges, require the tool-workpiece engagement which is modeled in ECS (shown in Fig. 2). On each cutter location, the engagement of the cutter with workpiece is discretized at each tool location by series of discrete elements along the Z-axis of ECS. Each element has a thickness of Δz with an entry angle ( ϕst ) and exit angle ( ϕex ) at each elevation ( z ). The FRFs of the machine tool and workpiece are needed to predict the relative vibrations between the tool and workpiece along the tool path. The FRF of the workpiece is measured in WCS. The spindle-holder-tool assembly has the most flexibility in ball end milling. Although the low frequency structural modes may change, the FRF of the spindle assembly does not change as the machine configuration varies in 5–axis motions. The FRF of the tool attached to the spindle is measured when the machine is at its home position at the Machine Coordinate System (MCS) as shown in Fig. 1. As the tool orientation changes along the 5-axis tool path, the vibrations must be projected to the tool tip at the ECS in order to predict the dynamic chip loads. The objective is not only to predict the chatter stability of 5-axis ball end milling, but also predict the most stable tool orientations of the ball end mill along the tool path to achieve highest material removal rates.
The geometry of a helical ball-end mill is given in Fig. 3. Due to ball end and helical flutes, the axial depth of cut, radial depth of cut, and the entry ( ϕst ) and exit ( ϕex ) angles may change along the tool axis. The tool axis is divided into m number of differential axial disk elements with height Δz along the axial depth of cut. The engagement angles ( ϕst , ϕex ) are identified by an in-house developed system as shown in Fig. 2 (MACHPRO) for each axial element. The process is defined in ECS where the tool tip position is at the center, and tool axis corresponds to ZE axis. The stability of 3 axis ball end milling was previously modeled by Altintas at al. [8], and the dynamics of ball end milling is briefly summarized here for an engagement and speed. The cutting forces contributed by the differential axial disk element l at time t are summarized as:
⎡ dF l [ϕ (z, t )]⎤ ⎡ Krc ⎤ ⎥ ⎢ x j ⎢ dF l [ϕ (z, t )]⎥ = T ⎢ K ⎥ g ϕj Δz h (t ) ⎢ tc ⎥ sin κ (z ) d ⎥ ⎢ y j ⎢⎣ Kac ⎥⎦ ⎢ dF l [ϕ (z, t )]⎥ ⎦ ⎣ z j
( )
E
Kc
(1)
where ϕj (z, t ) is the radial immersion angle, κ (z ) is the axial immersion angle at elevation z from the tool tip. g (ϕj ) = 1 when the tooth is in cut, and g (ϕj ) = 0 otherwise. Matrix T transforms the forces from radial, tangential and axial directions to the Cartesian (x, y, z) directions:
⎤ ⎡ ⎢ − sin κ⋅ sin ϕj (z, t ) − cos ϕj (z, t ) − cos κ (z )⋅ sin ϕj ⎥ ⎥ ⎢ (z, t ) ⎥ ⎢ T = ⎢ − sin κ⋅ cos ϕ (z, t ) sin ϕ (z, t ) − cos κ (z )⋅ cos ϕ ⎥ j j j ⎥ ⎢ ⎥ ⎢ (z, t ) ⎥ ⎢ cos κ (z ) 0 − sin κ (z ) ⎦ ⎣
(2)
The vector K c contains oblique cutting force coefficients in radial, tangential and axial directions, respectively. The dynamic chip thickness ( hd (t )) contributed by the relative vibrations between the tool and workpiece in feed ( XE ) and normal ( YE )
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91
Fig. 1. Workpiece Coordinate System (WCS), Machine Coordinate System (MCS) and Engagement Coordinate System (ECS) in modeling 5-axis machining process with C–A rotary table configuration.
of cut as the cuter rotates at discrete time intervals, and applied as lumped dynamic load on the tool-workpiece structure.
⎡ Fx [ϕ (t )]⎤ ⎡ ΔxE (t ) ⎤ ⎛ m N ⎞ ⎡ ΔxE (t ) ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ Fy [ϕ (t )]⎥ = ⎜⎜ ∑ ∑ A (ϕj (z, t )) ⎟⎟ ⎢ ΔyE (t )⎥ = A (ϕ (t )) ⎢ ΔyE (t )⎥ ⎝ l= 1 j = 1 ⎠ ⎣⎢ Δz (t ) ⎥⎦ ⎢⎣ F [ϕ (t )]⎥⎦ ⎢⎣ ΔzE (t ) ⎥⎦ E z E
(5)
The average of directional matrix A (ϕ (t )), which is periodic at tooth passing or pitch angle (ϕp = 2π /N ) intervals, can be evaluated to make it time invariant as suggested in [8]:
A0 =
1 ϕp
∫0
ϕp
A (ϕ (t )) dϕ
(6)
A 0 is now only dependent on tool geometry, cutting coefficients, radial engagement conditions ( ϕst , ϕex ), and axial depth of cut ( a = mΔz ) at a given tool path position. The dynamics of the system can be expressed in frequency domain when it is at its critical stability state with chatter frequency ( ωc ):
⎡ F ( iω ) ⎤ ⎡ ΔxE (iωc ) ⎤ c ⎥ ⎢ x ⎥ ⎢ ⎢ Fy ( iωc )⎥ = A 0 ⎢ ΔyE (iωc )⎥ orFE ( iωc ) = A 0Δq E ( iωc ) ⎥ ⎢ ⎢⎣ Δz (iω ) ⎥⎦ ⎢⎣ Fz ( iωc ) ⎥⎦ E c E
Fig. 2. Ball end mill – workpiece engagement map along the tool axis.
directions defined in ECS: ⎡ Δx E (t ) ⎤ ⎢ ⎥ h d (z, t ) = nΔq E = ⎡⎣ sin ϕj (z, t ) sin κ (z ) cos ϕj (z, t ) sin κ (z ) − cos κ (z ) ⎤⎦ ⎢ ΔyE (t ) ⎥ ⎢ ⎣ ΔzE (t ) ⎥⎦ n Δq E
The vibrations can be expressed as:
(
(
)
( ) (
⎤ ⋅K c(3 × 1) ⎥ n ϕj (z , t ) ⎦ (3 × 3 )
)
(
Δz (1 × 3) sin κ (z )
)
)
Δq E ( iωc ) = 1−e−iwc T ΦE ( iωc ) FE ( iωc )
(3)
where ΔqE (t ) = qE (t ) − qE (t − T ) ∀ q = (x, y, z ) and T is the tooth period. The geometric terms can be collected in a directional matrix A (ϕj (z, t )) as: ⎡ A ϕj (z , t ) = g ϕj ⎢ T ϕj (z , t ) ⎣
(7)
(4)
The dynamic cutting forces are summed along the axial depth
(8)
where ΦE (iωc ) is the relative frequency response function (FRF) between the tool and workpiece in ECS:
ΦE ( iωc )
⎡ Φ ( iω ) Φ ( iω ) Φ ( iω ) ⎤ c xy c xz c ⎥ ⎢ xx = ⎢ Φyx ( iωc ) Φyy ( iωc ) Φyz ( iωc )⎥ ⎥ ⎢ ⎢⎣ Φzx ( iωc ) Φzy ( iωc ) Φzz ( iωc ) ⎥⎦
E
(9)
The stability of the ball-end milling system is determined from
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Fig. 3. Helical ball end mill.
→ → perpendicular to yE − W × zE − W plane in ECS. ECS axis directions can be expressed in WCS frame as:
the following characteristic equation in frequency domain:
(
)
I − 1 − e−iωc T A 0ΦE ( iωc ) = 0
(10)
The feed and spindle speed are kept as they were set in the NC program, but only the tool axis orientation is manipulated which influence A 0ΦE (iωc ) in Eq. (10). The FRFs of the structure ΦE (iωc ) are defined in engagement coordinate system (ECS), which needs to be transformed from the machine's coordinate (MCS) and workpiece coordinate (WCS) systems where they are measured. The transformations, which include the new orientation of the tool axis, are presented in the following sections. The chatter stability of the system is evaluated using Nyquist stability criteria at each cutter location.
3. Transformation of Machine tool and workpiece FRFs The FRFs of the machine tool and workpiece are measured in their own coordinate systems, namely in MCS and WCS, respectively. The transformations from MCS and WCS to ECS system, where the stability is analyzed, are constructed along the tool path as follows. 3.1. Transformation from WCS to ECS Tool path is digitized into small differential segments from the cutter location (CL) data generated by the CAM system in WCS. The tool tip coordinate and orientation vector are defined by CL data at the beginning of path segment (n) in WCS as:
{ ⇀P , ⇀O } = { (x, y, z) W
W
W,
( ox , oy , oz )W }
(11)
The origin of the ECS is at the center of the ball part of the cutter, which is evaluated as:
⇀ ⇀ ⇀ PE = P W + OW ⋅R 0
(12)
where R0 is the radius of the ball-end mill. Engagement coordinate system (ECS) directions can be expressed from the tool orientation and feed directions that are ⎯⎯⇀ evaluated from the CL data defined in WCS. The feed direction fW in WCS is obtained from two consecutive cutter locations along the differential path segments as:
⇀ ⇀ ⎯⎯⇀ P (n + 1) − P W (n) fW = ⇀W ⇀ P W (n + 1) − P W (n)
(13)
The z-axis of ECS, i.e. tool axis, is defined by the CL data in WCS → → → → → as zE = ox iW + oy jW + oz kW . The normal axis ( yE ) is perpendicular → to the tool axis- feed plane, and the horizontal axis ( xE ) is
→ → → → zE = ox iW + oy jW + oz kW → ⎯⎯⇀ → → → zE × fW → yE = = vx iW + vy jW + vz kW → ⎯⎯⇀ zE × fW → → → → → → xE = yE × zE = ux iW + uy jW + uz kW
(14)
Hence, the rotational transformation from ECS to WCS is represented by the following matrix:
⎡ ux vx ox ⎤ R E − W = ⎢⎢ uy vy oy ⎥⎥ ⎣ uz vz oz ⎦
(15)
The tool orientation (ox , oy , oz )W set in the original CL file may not lead to optimal material removal rate with chatter stability constraint. Tool orientation is manipulated to achieve chatter free, higher material removal rates. Optimal tool orientation (oxm , oym , ozm )W are searched by solving the stability equation iteratively in the modified engagement coordinate system ECSm . For a trial set of A axis ( θA ) and C axis ( θC ) positions, the tool orientation in WCS deviates the following amount:
⎡ cos θC − sin θC ⎢ → z Em (θA, θC ) = ⎢ sin θC cos θC ⎢⎣ 0 0
0⎤ ⎡ 1 0 0 ⎤ ⎥ ⎥⎢ T 0⎥ ⎢ 0 cos θA − sin θA ⎥ ⎡⎣ 0 0 1⎤⎦ 1⎥⎦ ⎢⎣ 0 sin θA cos θA ⎥⎦
T =⎡⎣ sin θC sin θA − cos θC sin θA cos θA ⎤⎦ → → → =oxm iW + oym jW + ozm kW
(16)
⎯⎯⇀ The feed direction fW is also used to construct the rotation matrix, and remains as in Eq. (13). However, the normal and horizontal directions need to be updated by substituting the modified tool axis direction (Eq. (16)) into (14), which will lead to new rotational transformation from the modified engagement coordinate system ECSm to WCS as:
⎡ uxm vxm oxm ⎤ R Em− W = ⎢⎢ uym vym oym ⎥⎥ ⎣ uzm vzm ozm ⎦
(17)
3.2. Transformation from MCS to WCS The CL motion is transformed as machine drive commands through the inverse kinematic model of the machine as described in [9]. The translational motions do not change the directions in WCS and MCS, but rotary motions lead to the orientation of machine's FRF along the curved path in WCS. The rotary motion passes through the machine's kinematic chain in the order of the
C. Sun, Y. Altintas / International Journal of Machine Tools & Manufacture 106 (2016) 89–97
rotary drive A first and followed by the rotary drive C with the corresponding angular positions θA and θC , respectively. The rotational motions of A and C are around x and z axes, respectively, with the following rotation matrixes: ⎡1 0 0 ⎤ ⎢ ⎥ R x (θA ) = ⎢ 0 cos θA − sin θA ⎥ ⎢⎣ 0 sin θA cos θA ⎥⎦
⎡ cos θC − sin θC 0 ⎤ ⎢ ⎥ R z (θC ) = ⎢ sin θC cos θC 0 ⎥ ⎢⎣ 0 0 1⎥⎦
;
(18)
The rotational transformation from MCS to WCS is represented by the following matrix as:
⎡ cos θC − sin θC cos θA sin θC sin θA ⎤ ⎢ ⎥ R MW = R z (θC ) R x (θA ) = ⎢ sin θC cos θC cos θA − cos θC sin θA ⎥ ⎢⎣ 0 ⎥⎦ sin θA cos θA
(19)
3.3. Transformations of FRFs from measurement coordinates to tool – workpiece engagement coordinates Cutting forces and the effects of vibrations on the chip thickness are modeled in the engagement coordinate system where the machining process is modeled. The relative vibrations between the tool and workpiece can be expressed in frequency domain as:
Δq Em (iω) = Φ Em (iω) FEm (iω)
(20)
Henceforth, the frequency terms (iω) are dropped from the notations as a convenience. The FRFs of the machine and workpiece are measured in MCS and WCS, respectively, as:
Δq M = ΦM FM
;
ΔqW = ΦW FW
(21)
They are transformed to ECSm system with the following sequence. The machine vibrations qM (ω)are transformed to vibrations on the workpiece (WCS) as:
Δq MW = R MW Δq M
(22)
The forces applied on the machine FM are transformed from the cutting forces FW represented in WCS: 1 FM = R−MW FW
(
93
)
1 1 1 Φ Em (iω) = ( RWEmR MW ) ΦM R−MW R −WE + RWEmΦW R −WE m m
where ΦM (iω)and ΦW (iω)are the FRFs of the machine and workpiece measured (or modeled) in machine and workpiece coordinate systems, respectively. 3.4. Transformations of engagement maps from original engagement coordinates to modified engagement coordinates The ball-part of the cutter in ECSm is discretized into small elements along the zm axis where the element's radial and axial immersions are ϕm and κm , respectively (Fig. 4). Each rectangular engagement element has four vertexes p1, p2, p3, p4 . The Cartesian coordinate [ xm ym zm ]T of each vertex in ECSm can be evaluated from its Spherical coordinate [ R0 ϕm κm ]T , where R0 is the radius of the ball-end mill. The size of the vertex can be selected smaller to improve the accuracy. The coordinates of each vertex ( [ xm ym zm ]T ) are transformed from modified engagement coordinate ECSm to original engagement coordinate ECS via rotation matrix R Em− E to check whether the vertex lies in the tool part engagement zone. The rotation matrix R Em− E can be evaluated from Eqs. (15) and (17) as:
R Em− E = RW − E R Em− W = R−E −1 W R Em− W
(29) T
Each vertex with a coordinate [ xm ym zm ] in ECSm can be transformed to ECS as:
⎡ xm ⎤ ⎡ xm ⎤ ⎡ x⎤ ⎢ y ⎥ = R E − E ⎢ ym ⎥ = R−1 R E − W ⎢ ym ⎥ E− W m m ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦ z ⎣ zm⎦ ⎣ zm⎦
1 FW = R −WE F m Em
(24)
The workpiece vibrations can be added to machine vibrations to obtain total relative vibrations between the tool and workpiece:
ΔqWt = Δq MW + ΔqW = R MW Δq M + ΔqW (25)
= R MW ΦM FM + ΦW FW
If all the vertexes of a rectangular differential element are inl l l side the cutting zone (i.e. ϕstl < ϕ < ϕex and zbottom ), the < z < ztop corresponding material is in cut. By checking each piece on the ball-part, the engagement map in ECS’ can be established as shown in Fig. 4.
4. Optimization of tool orientation Chatter free tool orientations are optimized at each cutter location along the tool path. The rotational positions ( θA , θC ) are constrained within upper and lower limits to avoid gauging into the part, and divided into m × n segments as shown in Fig. 5. The feasible rotational positions are searched from the center towards the limits. For each group (θA, θC )k , the stability is checked using Eq. (10). Finally, groups of feasible angular positions θA and θC are acquired in MCS as:(Fig. 6)
⎡ ( θ , θ ) , ( θ , θ ) , ( θ , θ ) , ...⎤ ⎣ A C 1 A C 2 A C k ⎦i
They are further transformed to ECSm where the machining process stability is analyzed:
Δq Em = RWEmΔqWt = RWEm ( Δq MW + ΔqW ) = RWEmR MW ΦM FM + RWEmΦW FW
(26)
By substituting FM and FW in Eq. (26) with (Eqs. (23) and 24), q Em can be expressed as:
(
)
1 Δq Em (ω) = ( RWEmR MW ) ΦM R−MW FW + RWEmΦW FW
(
)
1 1 1 R −WE FEm + RWEmΦW R −WE F = ( RWEmR MW ) ΦM R−MW m m Em
{(
(30)
(23)
Force FW Represented in WCS is transformed from the cutting forces FE , and calculated in ECSm as:
=
(28)
(
1 1 RWEmR MW ΦM R−MW R −WE m
)
)+R
−1 WEmΦ W R WEm
}F
Em
Hence, the equivalent FRF in Eq. (20) can be replaced by:
(27) Fig. 4. Transformation of the engagement maps.
(31)
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calculated by Eq. (16). The cutter location coordinates are updated in CL data as:
→ → → → P Wm = P W + R 0⋅ OW − R 0⋅ OWm
(32)
→ where PW is the original cutter location, R0 is the radius of the ball→ part, OW is the original tool orientation. The new CL file is generated from the updated tool orientation and position data.
5. Simulations and experimental results
Fig. 5. Searching scope for A axis ( θA ) and C axis ( θC ) positions.
The proposed chatter free tool orientation algorithm has been experimentally validated on QUASER U600 5-axis CNC Machining Center. The original tool path is generated in Siemens NX CAM system, and the cutter-part engagement maps are obtained from MACHPRO [11] virtual machining system powered by Moduleworks [12] graphics kernel. The workpiece is machined from a rectangular block of AL-7050 material, and directly clamped on the C-rotary table. The spindle speed and feed are set to 15500 rev/ min and 3100 mm/min, respectively in the NC program. The frequency response function (FRF) of the tool is measured by impact tests (Fig. 8), and identified modal parameters are given in Table 1 for X-direction and Y-directions. The workpiece block is highly rigid in comparison to the tool. The sound is collected by a microphone, and the surface finish is analyzed at each cutting tests to evaluate the presence of chatter. The presence of chatter is identified by checking the spectrum of the sound at frequencies close to the modes but not
Fig. 6. Searching for feasible tool orientations.
Fig. 8. FRF results in X and Y direction.
Table 1 Modal data for the 12 mm ball-end mill with 2 flutes having 30 degree helix and 5 degree rake angles. Mode No.
Modal fit in X-direction
ωn (Hz)
ζ
m (kg)
1142.481 1317.533 1750.721 2123.095 2601.653 2798.097 3651.942 3766.414 4447.001
0.013336 0.033535 0.015992 0.040358 0.018809 0.017911 0.016413 0.00875 0.015494
3.038142 0.464746 4.238352 0.227459 0.483618 0.107584 0.160221 0.18702 0.713531
Modal fit in Y-direction
ωn (Hz)
ζ
m (kg)
1116.031 1340.842 2023.32 2617.665 2795.421 3662.316 3767.235 4449.955
0.02669 0.024135 0.046681 0.016307 0.018108 0.016879 0.00924 0.015182
2.642671 0.478966 0.273383 0.634201 0.107237 0.154746 0.200178 0.753022
Fig. 7. The shortest path problem for smoothing tool orientation.
The tool orientation is selected to minimize the global movement of angular positions θA and θC , see Fig. 7. This problem is regarded as a typical shortest path searching and solved by dynamic programming method as shown in [10]. After the selection of angular positions θA and θC at cutter location → i, the corresponding modified tool orientation OWm (oxm , oym , ozm )W is
1 2 3 4 5 6 7 8 9
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coinciding with the harmonics of tooth passing/spindle frequencies, as well as the presence of rough surface finish. 5.1. Experiment #1 – tool path with varying cutting depth The ball end mill slots the block along a straight path with 10° lead and zero degree tilt angles. The axial depth of cut varies from 2.5 mm to 2 mm as shown in Fig. 9. The original tool path has been observed to be unstable along the whole tool path. A feasible tool orientation is searched along the whole tool path which has 42 cutter location points as shown in Fig. 10. The search space had m × n = 10 × 10 combinations of A and C axis positions at each cutter location. A sample search at cutter location 22 is shown in Fig. 10. Since the tool path was straight, one set of A and C axis positions, which gave the smallest incremental deviation while satisfying chatter stability at the
95
entire tool path, is selected and kept constant in the updated CL file. The most feasible A and C axis positions have been found to be 27 and 12.34 degrees, respectively. They correspond to the tool orientation vector of [ 0.097 − 0.443 0.891]. The cutting test results with the original and optimized tool orientations are shown in Fig. 11. The original tool orientation led to severe chatter at 3894 Hz which is caused by the tool mode (Mode 8 in X-direction and Mode 7 in Y-direction, Table 1), and produced a rough surface finish. When the tool orientation is adjusted by the proposed algorithm, the chatter diminished and the surface finish became smooth. The sound spectrum is dominated by the tooth passing frequency (255 Hz), which is caused by forced vibrations. 5.2. Experiment #2–5-axis tool path A curved, 5-axis tool path (Fig. 12) with A constant axial depth of cut (2.5 mm) has been used to validate the proposed tool orientation Optimization algorithm. The tool orientation varies along the tool path. The cutting test showed that the original tool path led to severe chatter along the entire path. The algorithm searched feasible tool orientations at the 40 cutter locations of the entire path as shown in Fig. 13. The cutting tests for the original and optimized tool paths are shown in Fig. 14. The original path led to severe chatter at 3901 Hz which is caused by the tool mode (mode 8 in X-direction and mode 7 in Y-direction, Table 1). When the tool orientation is adjusted automatically at each cutter location, the chatter has been totally eliminated and smooth surface finish has been obtained along the entire 5-axis tool path.
6. Conclusion
Fig. 9. Tool path for Experiment #1. Original lead angle is 10°, and the tilt angle is zero degree.
The flexibilities of the tool and part may cause severe chatter in 5-axis ball end milling operations. This paper presents the prediction of an automatic tool axis orientation algorithm that avoids chatter while improving the productivity. The tool axis re-orientation diverts the cutting forces towards the more rigid z-axis of the tool, and avoids chatter. The tool orientation is searched
Fig. 10. Chatter free tool orientation search leads to optimal but constant set of A-axis (27°) and C-axis (12.34°) angles along the straight path. Cutting conditions: Experiment #1.
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Fig. 11. Experiment #1: Surface finish and sound spectrum before and after optimizing the tool orientation for tool path given in Fig. 9. Cutting conditions: Spindle speed: 15500 rev/min, feed: 3100 mm/min. Work material: Al-7050 cut with 12 mm diameter ball end mill with 2 flutes.
Fig. 12. 5-axis tool path used in Experiment #2. The axial depth of cut is kept constant at 2.5 mm.
Fig. 13. Optimization of tool orientation along the 5-axis, curved tool path. Cutting conditions: Experiment #2.
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Fig. 14. Experiment #2: Surface finish and sound spectrum before and after optimizing the tool orientation for tool path given in Fig. 12. Cutting conditions: Spindle speed: 15500 rev/min, feed: 3100 mm/min. Work material: Al-7050 cut with 12 mm diameter ball end mill with 2 flutes.
within a small scope to avoid gouging and collision on each cutter location. The kinematics of the machine and tool-part engagement geometry is considered in evaluating the stable cutting conditions of the process. For each tool orientation within the searching scope, the characteristic equation is established and stability is checked by Nyquist criterion. The integration of the proposed algorithm to 5-axis tool path generations of CAM systems can improve both the surface quality and productivity of machining dies, molds, and parts with free-form surfaces.
Acknowledgment This research is supported by National research and Engineering Science Council of Canada (NSERC) discovery grant number 11R86164 given to Prof. Altintas, the National Natural Science Foundation of China (No. 51475302) and the Shanghai Municipal Commission of Economy and Informatization Project (No. RX-RJJC02-15-9242). We thank the help of Phd candidate Oguzhan Tuysuz from MAL in our study and experiments.
References [1] A. Lasemi, D. Xue, P. Gu, Recent development in CNC machining of freeform surfaces: a state-of-the-art review, Comput.-Aided Des. 42 (7) (2010) 641–654. [2] B. Sencer, Y. Altintas, E. Croft, Modeling and control of contouring errors for five-axis machine tools.Part I: modeling, ASME J. Manuf. Sci. Eng. 131 (3) (2009) 031006. [3] X. Beudaert, P.Y. Pechard, C. Tournier, 5-axis tool path smoothing based on drive constraints, Int. J. Mach. Tools Manuf. 51 (12) (2011) 958–965. [4] A. Yuen, K. Zhang, Y. Altintas, Smooth trajectory generation for five-axis machine tools, Int. J. Mach. Tools Manuf. 71 (2013) 11–19. [5] S.E. Layegh K, I.E. Yigit, I. Lazoglu, Analysis of tool orientation for 5-axis ballend milling of flexible parts, CIRP Ann. – Manuf. Technol. 64 (1) (2015) 97–100. [6] Y. Altintas, M. Weck, Chatter stability of metal cutting and grinding, CIRP Ann. – Manuf. Technol. 53 (2) (2004) 619–642. [7] E. Ozturk, E. Budak, Dynamics and stability of five-axis ball-end milling, ASME J. Manuf. Sci. Eng. 132 (2) (2010) 021003. [8] Y. Altintas, E. Shamoto, P. Lee, E. Budak, Analytical prediction of stability lobes in ball end milling, ASME J. Manuf. Sci. Eng. 121 (4) (1999) 586–592. [9] J. Yang, Y. Altintas, Generalized kinematics of five-axis serial machines with non-singular tool path generation, Int. J. Mach. Tools Manuf. 75 (2013) 119–132. [10] H. Thomas, Introduction to algorithms, MIT press, 2009. [11] Manufacturing Automation Laboratories Inc. British Columbia (Canada). Available from 〈www.malinc.com/products/machpro〉. [12] ModuleWorks GmbH. German. Available from 〈www.moduleworks.com〉.
Contents lists available at ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Chatter free tool orientations in 5-axis ball-end milling Sun Chaob, Yusuf Altintas a,n,1 a The University of British Columbia, Department of Mechanical Engineering, Manufacturing Automation Laboratory, 2054-6250 Applied Science Lane, Vancouver, BC V6T 1Z4, Canada b State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 22 December 2015 Received in revised form 14 April 2016 Accepted 18 April 2016 Available online 20 April 2016
Dies, molds and parts with complex free form surfaces are usually machined with ball end mills on 5-axis CNC machining centers. This paper presents automatic adjustment of tool axis orientations to avoid chatter along the tool path. The process mechanics and dynamics of ball end milling are modeled in cutter-workpiece engagement coordinate system. The structural dynamics of tool and workpiece are transformed to cutter-workpiece engagement coordinates by considering the tool path and the kinematics of the machine tool. The stability of the 5-axis ball end milling is modeled at each tool path location, and the chatter free tool axis orientations are searched iteratively using Nyquist criterion while avoiding gouging limits. The tool path, i.e. cutter location (CL) file, is updated to generate chatter free, 5-axis ball end milling of the parts. The proposed algorithm has been experimentally proven in 5-axis ball end milling tests. & 2016 Elsevier Ltd. All rights reserved.
Keywords: 5 axis Ball end milling Chatter stability Tool orientation Optimization
1. Introduction Ball end milling is commonly used in finish milling of dies, molds, impellers, turbine blades and other parts with complex sculpture surfaces. The tool axis direction is always constant in 3 axis machines, but can be oriented to avoid collision and zero speeds at the tool tip on 5-axis machine tools. This paper presents a method to predict tool axis orientations to avoid chatter in 5-axis ball end milling of parts with free form surfaces. There has been significant research reported in 5-axis milling, but mainly on tool path generation and collision avoidance as summarized by Lasemi et al. [1]. The tool path generation and collision avoidance methods are based on wokpiece geometry and machine configuration, hence the physics of machining are not considered. Recent research efforts considered the servo drives, contouring errors, and the kinematics of 5-axis machine tools. Sencer et al. [2] modeled the contouring errors as a function of tool tip, tool orientation, path curvature and servo dynamics. Beudaert et al. [3] and Yuen et al. [4] optimized the tool orientation to generate smooth tool paths while avoiding to violate the torque limits of all drives in 5-axis machines. The process related research in tool axis orientation considers the static and dynamic flexibilities of the workpiece and tool. n
Corresponding author. E-mail address: [email protected] (Y. Altintas). 1 www.mal.mech.ubc.ca
http://dx.doi.org/10.1016/j.ijmachtools.2016.04.007 0890-6955/& 2016 Elsevier Ltd. All rights reserved.
Lazoglu et al. [5] optimized the tool orientation to constrain the static deflections of tools and parts perpendicular to the finish surface. Although significant research has been reported in the stability of machining operations [6], there has been a very limited effort reported in optimizing the tool orientation to avoid chatter in 5-axis ball-end milling of free form surfaces. Ozturk et al. [7] modeled the influence of tool's tilt and lead angles of the tool on the stability of 5-axis ball end milling operations. They used both zero order and multi-frequency stability solutions [6] to generate stability lobes as a function of tool's lead and tilt angles. However, the engagement and the FRF directions vary with the change of tool orientation along the tool path. The chatter free tool path planning requires the identification of most optimal tool orientation and depth along the curved, 5-axis tool path, which is presented in this paper. The frequency response functions (FRF) of the machine and workpiece are assumed to be measured in their stationary coordinate systems. The relative FRFs between the tool and part are transformed to moving tool-part engagement coordinates where the ball end milling process stability is modeled. The tool-part engagement conditions, feed direction and kinematics of the machine tool are considered in constructing the stability equation in frequency domain. The rotary drive positions of the machine which determine the tool axis orientation are searched to achieve chatter free, stable cutting conditions along the tool path. The proposed method is experimentally validated with 5-axis ball end milling tests. Henceforth, the paper is organized as follows. The dynamics of ball end milling system is modeled in engagement coordinate
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Nomenclature
dynamic displacement of workpiece in WCS relative dynamic displacement between the tool and workpiece in WCS n (ϕj (z , t )) surface normal vector at cutting edge A (ϕj (z , t )) directional matrix average of directional matrix A0 FE , FM , FW dynamic cutting forces in ECS, MCS and WCS, respectively. relative FRF between the tool and workpiece in ECS ΦE FRF of the tool measured in MCS ΦM FRF of the workpiece measured in WCS ΦW ⇀ ⇀ { PW , OW } original tool tip coordinate and orientation vector, respectively. ⇀ ⇀ { PWm, OWm } modified tool tip coordinate and orientation vector, respectively. angular positions of A and C axis, respectively θA , θC rotational transformation matrix from ECS to WCS RE − W rotational transformation matrix from modified enR Em− W gagement coordinate system ECSm to WCS rotational transformation matrix from MCS to WCS RMW rotational transformation matrix from modified enR Em− E gagement coordinate system ECSm to ECS [ xm ym zm ]T Cartesian coordinate of a vertex in ECSm [ R0 ϕm κm ]T Spherical coordinate of a vertex in ECSm
ΔqW ΔqW t
WCS ECS MCS ECSm l ϕstl , ϕex
workpiece coordinate system engagement coordinate system machine coordinate system modified engagement coordinate system entry and exit angle of the engagement map of axial disk element l l l , ztop bottom and top z value of the engagement map of zbottom axial disk element l ϕj (z, t ) radial immersion angle at time t and elevation z from the tool tip axial immersion angle at elevation z from the tool tip κ (z ) unit step function that determines whether the tooth g (ϕj ) is in or out of cut Krc , Ktc , Kac oblique cutting force coefficients in radial, tangential and axial directions, respectively height of differential axial disk elements Δz dynamic chip thickness h d (t ) ΔqE , Δq Em relative dynamic displacement between the tool and workpiece in ECS and modified ECS, respectively. ΔqM , ΔqMW dynamic displacement of tool in MCS and WCS, respectively.
system in Section 2. The tool and part dynamics, which change as a function of tool path and machine drive positions, are transformed to engagement coordinate system in Section 3. Chatter free tool orientations are predicted in Section 4. The proposed, chatter free tool axis orientation along the paths are experimentally proven in Section 5, and the paper is concluded in Section 6.
2. Coordinate transformations of vectors for process dynamics A five axis ball end milling system is shown in Fig. 1. The tool path is generated in a Computer Aided Manufacturing (CAM) software environment using the Workpiece Coordinate System (WCS). The machining process is modeled in Engagement Coordinate System (ECS) where the Z is the tool axis; X-axis is in the tool axis-feed plane, and Y axis is normal to XZ plane. The chip thickness, hence the force distribution along the cutting edges, require the tool-workpiece engagement which is modeled in ECS (shown in Fig. 2). On each cutter location, the engagement of the cutter with workpiece is discretized at each tool location by series of discrete elements along the Z-axis of ECS. Each element has a thickness of Δz with an entry angle ( ϕst ) and exit angle ( ϕex ) at each elevation ( z ). The FRFs of the machine tool and workpiece are needed to predict the relative vibrations between the tool and workpiece along the tool path. The FRF of the workpiece is measured in WCS. The spindle-holder-tool assembly has the most flexibility in ball end milling. Although the low frequency structural modes may change, the FRF of the spindle assembly does not change as the machine configuration varies in 5–axis motions. The FRF of the tool attached to the spindle is measured when the machine is at its home position at the Machine Coordinate System (MCS) as shown in Fig. 1. As the tool orientation changes along the 5-axis tool path, the vibrations must be projected to the tool tip at the ECS in order to predict the dynamic chip loads. The objective is not only to predict the chatter stability of 5-axis ball end milling, but also predict the most stable tool orientations of the ball end mill along the tool path to achieve highest material removal rates.
The geometry of a helical ball-end mill is given in Fig. 3. Due to ball end and helical flutes, the axial depth of cut, radial depth of cut, and the entry ( ϕst ) and exit ( ϕex ) angles may change along the tool axis. The tool axis is divided into m number of differential axial disk elements with height Δz along the axial depth of cut. The engagement angles ( ϕst , ϕex ) are identified by an in-house developed system as shown in Fig. 2 (MACHPRO) for each axial element. The process is defined in ECS where the tool tip position is at the center, and tool axis corresponds to ZE axis. The stability of 3 axis ball end milling was previously modeled by Altintas at al. [8], and the dynamics of ball end milling is briefly summarized here for an engagement and speed. The cutting forces contributed by the differential axial disk element l at time t are summarized as:
⎡ dF l [ϕ (z, t )]⎤ ⎡ Krc ⎤ ⎥ ⎢ x j ⎢ dF l [ϕ (z, t )]⎥ = T ⎢ K ⎥ g ϕj Δz h (t ) ⎢ tc ⎥ sin κ (z ) d ⎥ ⎢ y j ⎢⎣ Kac ⎥⎦ ⎢ dF l [ϕ (z, t )]⎥ ⎦ ⎣ z j
( )
E
Kc
(1)
where ϕj (z, t ) is the radial immersion angle, κ (z ) is the axial immersion angle at elevation z from the tool tip. g (ϕj ) = 1 when the tooth is in cut, and g (ϕj ) = 0 otherwise. Matrix T transforms the forces from radial, tangential and axial directions to the Cartesian (x, y, z) directions:
⎤ ⎡ ⎢ − sin κ⋅ sin ϕj (z, t ) − cos ϕj (z, t ) − cos κ (z )⋅ sin ϕj ⎥ ⎥ ⎢ (z, t ) ⎥ ⎢ T = ⎢ − sin κ⋅ cos ϕ (z, t ) sin ϕ (z, t ) − cos κ (z )⋅ cos ϕ ⎥ j j j ⎥ ⎢ ⎥ ⎢ (z, t ) ⎥ ⎢ cos κ (z ) 0 − sin κ (z ) ⎦ ⎣
(2)
The vector K c contains oblique cutting force coefficients in radial, tangential and axial directions, respectively. The dynamic chip thickness ( hd (t )) contributed by the relative vibrations between the tool and workpiece in feed ( XE ) and normal ( YE )
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91
Fig. 1. Workpiece Coordinate System (WCS), Machine Coordinate System (MCS) and Engagement Coordinate System (ECS) in modeling 5-axis machining process with C–A rotary table configuration.
of cut as the cuter rotates at discrete time intervals, and applied as lumped dynamic load on the tool-workpiece structure.
⎡ Fx [ϕ (t )]⎤ ⎡ ΔxE (t ) ⎤ ⎛ m N ⎞ ⎡ ΔxE (t ) ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ Fy [ϕ (t )]⎥ = ⎜⎜ ∑ ∑ A (ϕj (z, t )) ⎟⎟ ⎢ ΔyE (t )⎥ = A (ϕ (t )) ⎢ ΔyE (t )⎥ ⎝ l= 1 j = 1 ⎠ ⎣⎢ Δz (t ) ⎥⎦ ⎢⎣ F [ϕ (t )]⎥⎦ ⎢⎣ ΔzE (t ) ⎥⎦ E z E
(5)
The average of directional matrix A (ϕ (t )), which is periodic at tooth passing or pitch angle (ϕp = 2π /N ) intervals, can be evaluated to make it time invariant as suggested in [8]:
A0 =
1 ϕp
∫0
ϕp
A (ϕ (t )) dϕ
(6)
A 0 is now only dependent on tool geometry, cutting coefficients, radial engagement conditions ( ϕst , ϕex ), and axial depth of cut ( a = mΔz ) at a given tool path position. The dynamics of the system can be expressed in frequency domain when it is at its critical stability state with chatter frequency ( ωc ):
⎡ F ( iω ) ⎤ ⎡ ΔxE (iωc ) ⎤ c ⎥ ⎢ x ⎥ ⎢ ⎢ Fy ( iωc )⎥ = A 0 ⎢ ΔyE (iωc )⎥ orFE ( iωc ) = A 0Δq E ( iωc ) ⎥ ⎢ ⎢⎣ Δz (iω ) ⎥⎦ ⎢⎣ Fz ( iωc ) ⎥⎦ E c E
Fig. 2. Ball end mill – workpiece engagement map along the tool axis.
directions defined in ECS: ⎡ Δx E (t ) ⎤ ⎢ ⎥ h d (z, t ) = nΔq E = ⎡⎣ sin ϕj (z, t ) sin κ (z ) cos ϕj (z, t ) sin κ (z ) − cos κ (z ) ⎤⎦ ⎢ ΔyE (t ) ⎥ ⎢ ⎣ ΔzE (t ) ⎥⎦ n Δq E
The vibrations can be expressed as:
(
(
)
( ) (
⎤ ⋅K c(3 × 1) ⎥ n ϕj (z , t ) ⎦ (3 × 3 )
)
(
Δz (1 × 3) sin κ (z )
)
)
Δq E ( iωc ) = 1−e−iwc T ΦE ( iωc ) FE ( iωc )
(3)
where ΔqE (t ) = qE (t ) − qE (t − T ) ∀ q = (x, y, z ) and T is the tooth period. The geometric terms can be collected in a directional matrix A (ϕj (z, t )) as: ⎡ A ϕj (z , t ) = g ϕj ⎢ T ϕj (z , t ) ⎣
(7)
(4)
The dynamic cutting forces are summed along the axial depth
(8)
where ΦE (iωc ) is the relative frequency response function (FRF) between the tool and workpiece in ECS:
ΦE ( iωc )
⎡ Φ ( iω ) Φ ( iω ) Φ ( iω ) ⎤ c xy c xz c ⎥ ⎢ xx = ⎢ Φyx ( iωc ) Φyy ( iωc ) Φyz ( iωc )⎥ ⎥ ⎢ ⎢⎣ Φzx ( iωc ) Φzy ( iωc ) Φzz ( iωc ) ⎥⎦
E
(9)
The stability of the ball-end milling system is determined from
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Fig. 3. Helical ball end mill.
→ → perpendicular to yE − W × zE − W plane in ECS. ECS axis directions can be expressed in WCS frame as:
the following characteristic equation in frequency domain:
(
)
I − 1 − e−iωc T A 0ΦE ( iωc ) = 0
(10)
The feed and spindle speed are kept as they were set in the NC program, but only the tool axis orientation is manipulated which influence A 0ΦE (iωc ) in Eq. (10). The FRFs of the structure ΦE (iωc ) are defined in engagement coordinate system (ECS), which needs to be transformed from the machine's coordinate (MCS) and workpiece coordinate (WCS) systems where they are measured. The transformations, which include the new orientation of the tool axis, are presented in the following sections. The chatter stability of the system is evaluated using Nyquist stability criteria at each cutter location.
3. Transformation of Machine tool and workpiece FRFs The FRFs of the machine tool and workpiece are measured in their own coordinate systems, namely in MCS and WCS, respectively. The transformations from MCS and WCS to ECS system, where the stability is analyzed, are constructed along the tool path as follows. 3.1. Transformation from WCS to ECS Tool path is digitized into small differential segments from the cutter location (CL) data generated by the CAM system in WCS. The tool tip coordinate and orientation vector are defined by CL data at the beginning of path segment (n) in WCS as:
{ ⇀P , ⇀O } = { (x, y, z) W
W
W,
( ox , oy , oz )W }
(11)
The origin of the ECS is at the center of the ball part of the cutter, which is evaluated as:
⇀ ⇀ ⇀ PE = P W + OW ⋅R 0
(12)
where R0 is the radius of the ball-end mill. Engagement coordinate system (ECS) directions can be expressed from the tool orientation and feed directions that are ⎯⎯⇀ evaluated from the CL data defined in WCS. The feed direction fW in WCS is obtained from two consecutive cutter locations along the differential path segments as:
⇀ ⇀ ⎯⎯⇀ P (n + 1) − P W (n) fW = ⇀W ⇀ P W (n + 1) − P W (n)
(13)
The z-axis of ECS, i.e. tool axis, is defined by the CL data in WCS → → → → → as zE = ox iW + oy jW + oz kW . The normal axis ( yE ) is perpendicular → to the tool axis- feed plane, and the horizontal axis ( xE ) is
→ → → → zE = ox iW + oy jW + oz kW → ⎯⎯⇀ → → → zE × fW → yE = = vx iW + vy jW + vz kW → ⎯⎯⇀ zE × fW → → → → → → xE = yE × zE = ux iW + uy jW + uz kW
(14)
Hence, the rotational transformation from ECS to WCS is represented by the following matrix:
⎡ ux vx ox ⎤ R E − W = ⎢⎢ uy vy oy ⎥⎥ ⎣ uz vz oz ⎦
(15)
The tool orientation (ox , oy , oz )W set in the original CL file may not lead to optimal material removal rate with chatter stability constraint. Tool orientation is manipulated to achieve chatter free, higher material removal rates. Optimal tool orientation (oxm , oym , ozm )W are searched by solving the stability equation iteratively in the modified engagement coordinate system ECSm . For a trial set of A axis ( θA ) and C axis ( θC ) positions, the tool orientation in WCS deviates the following amount:
⎡ cos θC − sin θC ⎢ → z Em (θA, θC ) = ⎢ sin θC cos θC ⎢⎣ 0 0
0⎤ ⎡ 1 0 0 ⎤ ⎥ ⎥⎢ T 0⎥ ⎢ 0 cos θA − sin θA ⎥ ⎡⎣ 0 0 1⎤⎦ 1⎥⎦ ⎢⎣ 0 sin θA cos θA ⎥⎦
T =⎡⎣ sin θC sin θA − cos θC sin θA cos θA ⎤⎦ → → → =oxm iW + oym jW + ozm kW
(16)
⎯⎯⇀ The feed direction fW is also used to construct the rotation matrix, and remains as in Eq. (13). However, the normal and horizontal directions need to be updated by substituting the modified tool axis direction (Eq. (16)) into (14), which will lead to new rotational transformation from the modified engagement coordinate system ECSm to WCS as:
⎡ uxm vxm oxm ⎤ R Em− W = ⎢⎢ uym vym oym ⎥⎥ ⎣ uzm vzm ozm ⎦
(17)
3.2. Transformation from MCS to WCS The CL motion is transformed as machine drive commands through the inverse kinematic model of the machine as described in [9]. The translational motions do not change the directions in WCS and MCS, but rotary motions lead to the orientation of machine's FRF along the curved path in WCS. The rotary motion passes through the machine's kinematic chain in the order of the
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rotary drive A first and followed by the rotary drive C with the corresponding angular positions θA and θC , respectively. The rotational motions of A and C are around x and z axes, respectively, with the following rotation matrixes: ⎡1 0 0 ⎤ ⎢ ⎥ R x (θA ) = ⎢ 0 cos θA − sin θA ⎥ ⎢⎣ 0 sin θA cos θA ⎥⎦
⎡ cos θC − sin θC 0 ⎤ ⎢ ⎥ R z (θC ) = ⎢ sin θC cos θC 0 ⎥ ⎢⎣ 0 0 1⎥⎦
;
(18)
The rotational transformation from MCS to WCS is represented by the following matrix as:
⎡ cos θC − sin θC cos θA sin θC sin θA ⎤ ⎢ ⎥ R MW = R z (θC ) R x (θA ) = ⎢ sin θC cos θC cos θA − cos θC sin θA ⎥ ⎢⎣ 0 ⎥⎦ sin θA cos θA
(19)
3.3. Transformations of FRFs from measurement coordinates to tool – workpiece engagement coordinates Cutting forces and the effects of vibrations on the chip thickness are modeled in the engagement coordinate system where the machining process is modeled. The relative vibrations between the tool and workpiece can be expressed in frequency domain as:
Δq Em (iω) = Φ Em (iω) FEm (iω)
(20)
Henceforth, the frequency terms (iω) are dropped from the notations as a convenience. The FRFs of the machine and workpiece are measured in MCS and WCS, respectively, as:
Δq M = ΦM FM
;
ΔqW = ΦW FW
(21)
They are transformed to ECSm system with the following sequence. The machine vibrations qM (ω)are transformed to vibrations on the workpiece (WCS) as:
Δq MW = R MW Δq M
(22)
The forces applied on the machine FM are transformed from the cutting forces FW represented in WCS: 1 FM = R−MW FW
(
93
)
1 1 1 Φ Em (iω) = ( RWEmR MW ) ΦM R−MW R −WE + RWEmΦW R −WE m m
where ΦM (iω)and ΦW (iω)are the FRFs of the machine and workpiece measured (or modeled) in machine and workpiece coordinate systems, respectively. 3.4. Transformations of engagement maps from original engagement coordinates to modified engagement coordinates The ball-part of the cutter in ECSm is discretized into small elements along the zm axis where the element's radial and axial immersions are ϕm and κm , respectively (Fig. 4). Each rectangular engagement element has four vertexes p1, p2, p3, p4 . The Cartesian coordinate [ xm ym zm ]T of each vertex in ECSm can be evaluated from its Spherical coordinate [ R0 ϕm κm ]T , where R0 is the radius of the ball-end mill. The size of the vertex can be selected smaller to improve the accuracy. The coordinates of each vertex ( [ xm ym zm ]T ) are transformed from modified engagement coordinate ECSm to original engagement coordinate ECS via rotation matrix R Em− E to check whether the vertex lies in the tool part engagement zone. The rotation matrix R Em− E can be evaluated from Eqs. (15) and (17) as:
R Em− E = RW − E R Em− W = R−E −1 W R Em− W
(29) T
Each vertex with a coordinate [ xm ym zm ] in ECSm can be transformed to ECS as:
⎡ xm ⎤ ⎡ xm ⎤ ⎡ x⎤ ⎢ y ⎥ = R E − E ⎢ ym ⎥ = R−1 R E − W ⎢ ym ⎥ E− W m m ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦ z ⎣ zm⎦ ⎣ zm⎦
1 FW = R −WE F m Em
(24)
The workpiece vibrations can be added to machine vibrations to obtain total relative vibrations between the tool and workpiece:
ΔqWt = Δq MW + ΔqW = R MW Δq M + ΔqW (25)
= R MW ΦM FM + ΦW FW
If all the vertexes of a rectangular differential element are inl l l side the cutting zone (i.e. ϕstl < ϕ < ϕex and zbottom ), the < z < ztop corresponding material is in cut. By checking each piece on the ball-part, the engagement map in ECS’ can be established as shown in Fig. 4.
4. Optimization of tool orientation Chatter free tool orientations are optimized at each cutter location along the tool path. The rotational positions ( θA , θC ) are constrained within upper and lower limits to avoid gauging into the part, and divided into m × n segments as shown in Fig. 5. The feasible rotational positions are searched from the center towards the limits. For each group (θA, θC )k , the stability is checked using Eq. (10). Finally, groups of feasible angular positions θA and θC are acquired in MCS as:(Fig. 6)
⎡ ( θ , θ ) , ( θ , θ ) , ( θ , θ ) , ...⎤ ⎣ A C 1 A C 2 A C k ⎦i
They are further transformed to ECSm where the machining process stability is analyzed:
Δq Em = RWEmΔqWt = RWEm ( Δq MW + ΔqW ) = RWEmR MW ΦM FM + RWEmΦW FW
(26)
By substituting FM and FW in Eq. (26) with (Eqs. (23) and 24), q Em can be expressed as:
(
)
1 Δq Em (ω) = ( RWEmR MW ) ΦM R−MW FW + RWEmΦW FW
(
)
1 1 1 R −WE FEm + RWEmΦW R −WE F = ( RWEmR MW ) ΦM R−MW m m Em
{(
(30)
(23)
Force FW Represented in WCS is transformed from the cutting forces FE , and calculated in ECSm as:
=
(28)
(
1 1 RWEmR MW ΦM R−MW R −WE m
)
)+R
−1 WEmΦ W R WEm
}F
Em
Hence, the equivalent FRF in Eq. (20) can be replaced by:
(27) Fig. 4. Transformation of the engagement maps.
(31)
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calculated by Eq. (16). The cutter location coordinates are updated in CL data as:
→ → → → P Wm = P W + R 0⋅ OW − R 0⋅ OWm
(32)
→ where PW is the original cutter location, R0 is the radius of the ball→ part, OW is the original tool orientation. The new CL file is generated from the updated tool orientation and position data.
5. Simulations and experimental results
Fig. 5. Searching scope for A axis ( θA ) and C axis ( θC ) positions.
The proposed chatter free tool orientation algorithm has been experimentally validated on QUASER U600 5-axis CNC Machining Center. The original tool path is generated in Siemens NX CAM system, and the cutter-part engagement maps are obtained from MACHPRO [11] virtual machining system powered by Moduleworks [12] graphics kernel. The workpiece is machined from a rectangular block of AL-7050 material, and directly clamped on the C-rotary table. The spindle speed and feed are set to 15500 rev/ min and 3100 mm/min, respectively in the NC program. The frequency response function (FRF) of the tool is measured by impact tests (Fig. 8), and identified modal parameters are given in Table 1 for X-direction and Y-directions. The workpiece block is highly rigid in comparison to the tool. The sound is collected by a microphone, and the surface finish is analyzed at each cutting tests to evaluate the presence of chatter. The presence of chatter is identified by checking the spectrum of the sound at frequencies close to the modes but not
Fig. 6. Searching for feasible tool orientations.
Fig. 8. FRF results in X and Y direction.
Table 1 Modal data for the 12 mm ball-end mill with 2 flutes having 30 degree helix and 5 degree rake angles. Mode No.
Modal fit in X-direction
ωn (Hz)
ζ
m (kg)
1142.481 1317.533 1750.721 2123.095 2601.653 2798.097 3651.942 3766.414 4447.001
0.013336 0.033535 0.015992 0.040358 0.018809 0.017911 0.016413 0.00875 0.015494
3.038142 0.464746 4.238352 0.227459 0.483618 0.107584 0.160221 0.18702 0.713531
Modal fit in Y-direction
ωn (Hz)
ζ
m (kg)
1116.031 1340.842 2023.32 2617.665 2795.421 3662.316 3767.235 4449.955
0.02669 0.024135 0.046681 0.016307 0.018108 0.016879 0.00924 0.015182
2.642671 0.478966 0.273383 0.634201 0.107237 0.154746 0.200178 0.753022
Fig. 7. The shortest path problem for smoothing tool orientation.
The tool orientation is selected to minimize the global movement of angular positions θA and θC , see Fig. 7. This problem is regarded as a typical shortest path searching and solved by dynamic programming method as shown in [10]. After the selection of angular positions θA and θC at cutter location → i, the corresponding modified tool orientation OWm (oxm , oym , ozm )W is
1 2 3 4 5 6 7 8 9
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coinciding with the harmonics of tooth passing/spindle frequencies, as well as the presence of rough surface finish. 5.1. Experiment #1 – tool path with varying cutting depth The ball end mill slots the block along a straight path with 10° lead and zero degree tilt angles. The axial depth of cut varies from 2.5 mm to 2 mm as shown in Fig. 9. The original tool path has been observed to be unstable along the whole tool path. A feasible tool orientation is searched along the whole tool path which has 42 cutter location points as shown in Fig. 10. The search space had m × n = 10 × 10 combinations of A and C axis positions at each cutter location. A sample search at cutter location 22 is shown in Fig. 10. Since the tool path was straight, one set of A and C axis positions, which gave the smallest incremental deviation while satisfying chatter stability at the
95
entire tool path, is selected and kept constant in the updated CL file. The most feasible A and C axis positions have been found to be 27 and 12.34 degrees, respectively. They correspond to the tool orientation vector of [ 0.097 − 0.443 0.891]. The cutting test results with the original and optimized tool orientations are shown in Fig. 11. The original tool orientation led to severe chatter at 3894 Hz which is caused by the tool mode (Mode 8 in X-direction and Mode 7 in Y-direction, Table 1), and produced a rough surface finish. When the tool orientation is adjusted by the proposed algorithm, the chatter diminished and the surface finish became smooth. The sound spectrum is dominated by the tooth passing frequency (255 Hz), which is caused by forced vibrations. 5.2. Experiment #2–5-axis tool path A curved, 5-axis tool path (Fig. 12) with A constant axial depth of cut (2.5 mm) has been used to validate the proposed tool orientation Optimization algorithm. The tool orientation varies along the tool path. The cutting test showed that the original tool path led to severe chatter along the entire path. The algorithm searched feasible tool orientations at the 40 cutter locations of the entire path as shown in Fig. 13. The cutting tests for the original and optimized tool paths are shown in Fig. 14. The original path led to severe chatter at 3901 Hz which is caused by the tool mode (mode 8 in X-direction and mode 7 in Y-direction, Table 1). When the tool orientation is adjusted automatically at each cutter location, the chatter has been totally eliminated and smooth surface finish has been obtained along the entire 5-axis tool path.
6. Conclusion
Fig. 9. Tool path for Experiment #1. Original lead angle is 10°, and the tilt angle is zero degree.
The flexibilities of the tool and part may cause severe chatter in 5-axis ball end milling operations. This paper presents the prediction of an automatic tool axis orientation algorithm that avoids chatter while improving the productivity. The tool axis re-orientation diverts the cutting forces towards the more rigid z-axis of the tool, and avoids chatter. The tool orientation is searched
Fig. 10. Chatter free tool orientation search leads to optimal but constant set of A-axis (27°) and C-axis (12.34°) angles along the straight path. Cutting conditions: Experiment #1.
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Fig. 11. Experiment #1: Surface finish and sound spectrum before and after optimizing the tool orientation for tool path given in Fig. 9. Cutting conditions: Spindle speed: 15500 rev/min, feed: 3100 mm/min. Work material: Al-7050 cut with 12 mm diameter ball end mill with 2 flutes.
Fig. 12. 5-axis tool path used in Experiment #2. The axial depth of cut is kept constant at 2.5 mm.
Fig. 13. Optimization of tool orientation along the 5-axis, curved tool path. Cutting conditions: Experiment #2.
C. Sun, Y. Altintas / International Journal of Machine Tools & Manufacture 106 (2016) 89–97
97
Fig. 14. Experiment #2: Surface finish and sound spectrum before and after optimizing the tool orientation for tool path given in Fig. 12. Cutting conditions: Spindle speed: 15500 rev/min, feed: 3100 mm/min. Work material: Al-7050 cut with 12 mm diameter ball end mill with 2 flutes.
within a small scope to avoid gouging and collision on each cutter location. The kinematics of the machine and tool-part engagement geometry is considered in evaluating the stable cutting conditions of the process. For each tool orientation within the searching scope, the characteristic equation is established and stability is checked by Nyquist criterion. The integration of the proposed algorithm to 5-axis tool path generations of CAM systems can improve both the surface quality and productivity of machining dies, molds, and parts with free-form surfaces.
Acknowledgment This research is supported by National research and Engineering Science Council of Canada (NSERC) discovery grant number 11R86164 given to Prof. Altintas, the National Natural Science Foundation of China (No. 51475302) and the Shanghai Municipal Commission of Economy and Informatization Project (No. RX-RJJC02-15-9242). We thank the help of Phd candidate Oguzhan Tuysuz from MAL in our study and experiments.
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