Stability of orthogonal turning processes

Transcription

1 Stability of orthogonal turning processes M.A.M. Haring DC Traineeship report Coach: A. Aygun, M.A.Sc. candidate (The University of British Columbia) Supervisors: Prof. H. Nijmeijer Prof. Y. Altintas (Eindhoven University of Technology) (The University of British Columbia) Eindhoven University of Technology Department of Mechanical Engineering Dynamics and Control Group Eindhoven, May, 21

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3 Abstract Stability is an important topic in metal cutting. Unstable vibrations in the displacement of the cutting tool with respect to the workpiece result in a poor surface finish and cause the vibrations in the cutting forces to grow. Large vibrations in the cutting forces may damage the cutting tool, the workpiece and the machine. In this report, the dynamics of an orthogonal turning process are modeled. The obtained model includes the influence of the geometry of a triangular cutting tool and the effect of process damping. The cutting forces are linked to the uncut chip thickness to find an expression for the stability of the turning process. The stability of the turning process is solved using the Nyquist stability criterion. An algorithm is presented to find the cutting conditions for which the turning process is critically stable. A stability chart is made to depict the stability of the turning process for different cutting conditions. The predicted stability is validated by cutting experiments. 3

4 Contents 1 Introduction Aim of the report Outline of the report Modeling of orthogonal turning dynamics Structural dynamic model Regenerative chip model Chip area Chip flow angle and chord length Approximated chip Cutting force model Stability of the orthogonal turning process Experimental setup Identification of the frequency response functions Identification of the cutting force coefficients Identification of the process damping coefficient An algorithm to construct stability charts for turning An algorithm to determine the critically stable cutting conditions Calculation of the cutting speed and the spindle revolution time Calculation of the chip flow angle and the chord length Generation of the Nyquist plot data Calculation of the stability limit value Stability check and update of the depth of cut A stability chart for the experimental setup Construction of the stability chart Experimental validation of the stability chart Conclusions and recommendations Conclusions Recommendations for future research Acknowledgements 52 Nomenclature 53 A Coefficients of the fitted frequency response functions 56 B Calculation of the partial derivative a (1 + C R,lim) 58 References 62 4

5 1 Introduction In an aim to increase the productivity and reduce the costs in metal cutting operations, metal removal rates are pushed to the bounds of instability. Vibrations in the displacement of the tool with respect to the workpiece lead to vibrations in the cutting forces. In turn, vibrating cutting forces contribute to the tool vibrations, resulting in a process known as regenerative chatter. Unstable chatter vibrations lead to a poor surface finish and may damage the cutting tool, the workpiece and the machine. To avoid instability, accurately predicting the stability of the performed cutting operation is key. The stability of a metal cutting operation depends on the cutting conditions and the dynamic properties of the cutting tool, the workpiece and the machine. In most cases, the dynamic properties can not be changed. The productivity and cost savings can be increased by finding cutting conditions that result in larger metal removal rates while still guaranteeing a stable cutting process. A stability chart is a convenient way to depict the stability of a metal cutting process for a large range of cutting conditions. In a stability chart, so-called chatter stability lobes are plotted that indicate the boundary between cutting conditions that result in stable chatter vibrations and cutting conditions that result in unstable chatter vibrations. Basic regenerative chatter stability theory has been around for a few decades. Improved prediction of chatter stability lobes led to significant increases in metal removal rates in high speed operations, such as high speed milling. In low speed operations, such as turning, the chatter stability lobes are smaller and the increase in metal removal rates is less. When the ratio of the vibration frequency over cutting speed is high, the stable depth of cut increases. This effect is known as process damping and is either attributed to the change in the direction of the cutting speed due to the cutting force or to friction between the flank of the tool and the vibrations on the workpiece surface. Among others such as [1, 2, 3, 4], two recently published articles related to the stability of turning processes are presented by Eynian and Altintas [5] and Altintas et al. [6]. Both articles originate from the Manufacturing Automation Laboratory in Vancouver, Canada. The Manufacturing Automation Laboratory at the Department of Mechanical Engineering of the University of British Columbia is a research facility where metal cutting operations and their stability are investigated. The research presented in this report was conducted at the Manufacturing Automation Laboratory. 1.1 Aim of the report The aim of this report is to develop a method to accurately predict the stability of orthogonal turning processes. In turning, the workpiece is attached to a rotating spindle (figure 1.1). The rotational velocity of the workpiece results in a cutting velocity when the tool touches the surface of the workpiece. The tool performs a translational motion with respect to the rotating workpiece to remove the excess material. Figure 1.1: Turning process A cutting process is called orthogonal if the cutting edge of the tool is perpendicular to the direction of the cutting velocity (figure 1.2a). If the cutting edge is not perpendicular to the cutting velocity direction, the cutting process is called oblique (figure 1.2b). 5

6 (a) Orthogonal cutting (b) Oblique cutting Figure 1.2: Orthogonal and oblique cutting 1.2 Outline of the report This report is divided into six chapters. The process dynamics are modeled to find conditions for the stability of orthogonal turning processes. The model in chapter 2 is based on models in [5, 6]. The structural dynamic model, the influence of the tool geometry and the stability analysis based on the regenerative chip thickness are part of the regenerative chip model presented in [5]. The model in [6] is used to model the process damping. The predicted process stability is validated by cutting experiments. In chapter 3, the experimental setup is discussed and unknown model parameters are identified. The stability of the turning process is solved for different cutting conditions using the Nyquist stability criterion. In chapter 5, an algorithm is presented to find the critically stable cutting conditions: the boundary between stability and instability. The predicted and the measured process stability are depicted in a stability chart. The construction of the stability chart and the validation of the predicted stability are presented in chapter 5. In chapter 6, conclusions are drawn with respect to the presented algorithm and the accuracy of the stability prediction, and recommendations for future research are made. 6

7 2 Modeling of orthogonal turning dynamics As mentioned in the introduction of chapter 1, unstable chatter vibrations can be prevented by selecting suitable cutting conditions. For a given orthogonal turning operation, the spindle speed, the feed rate and the depth of cut are parameters of the turning process. The spindle speed n is the number of revolutions of the spindle per period of time. The feed rate is the change in feed per spindle revolution. After one revolution, the workpiece comes back to its original angular position. Imposed by the feed rate, during that revolution, the tool is desired to travel a certain distance in the feed direction. It is customary to assign the same symbol to the change in feed per spindle revolution and to the corresponding desired tool travel during one revolution. In this report, both are called feed rate and are indicated by c, although technically speaking they are two different quantities and do not have the same unit (m/rev and m). The depth of cut a is the length of the cut perpendicular to the cutting velocity direction and perpendicular to the feed direction. A representation of the feed rate and the depth of cut is shown in figure 2.1. Figure 2.1: Feed rate c and depth of cut a It is important to note that all cutting conditions (the spindle speed, the feed rate and the depth of cut) are assumed to be constant during the turning process. To identify which conditions result in stable vibrations, the dynamics of the turning process are modeled. The model presented in this chapter is based on models by Eynian and Altintas [5] and Altintas et al. [6]. First, the structural dynamics of the machine and the workpiece are modeled to obtain the dynamic response evoked by the cutting forces (section 2.1). The regenerative chip model of section 2.2 and the cutting force model of section 2.3 are used to link the tool displacement back to the cutting forces. The models are combined in section 2.4 to evaluate the stability of the orthogonal turning process for a given set of cutting conditions. 2.1 Structural dynamic model This section contains the structural dynamic model presented in [5]. During a turning operation, the tool tip is pushed into the workpiece to remove the excess material. The cutting forces generated at the contact area between the tool tip and the workpiece are transmitted through the tool and the workpiece to other machine components. A force loop is created that runs through the machine from the tool tip to the workpiece (figure 2.2). The displacements of the tool tip and the workpiece at the contact area are dependent on the effective deformation of each component in the force loop (figure 2.3). The measurement coordinate system ( e x, e y, e z ) is introduced, where e x, e y and, e z are unit vectors of the coordinate system in the x-, y- and z-direction. The x-, y- and z-direction are parallel to the directions of the feed, the depth of cut and the cutting velocity, respectively. 7

8 Figure 2.2: Force loop Figure 2.3: Displacements of the tool tip d t and the workpiece d w due to cutting forces acting on the tool tip F t and the workpiece F w The displacements of the tool tip d t and the displacement of the workpiece d w are expressed in the measurement coordinate system as d k = d x,k ex + d y,k ey + d z,k ez for k {t, w}, (2.1) where the displacements of the tool tip and the workpiece in the x-, y- and z-direction are given by d x,k, d y,k and d z,k, for k {t, w}. The cutting forces acting on the tool tip F t and the cutting forces acting on the workpiece F w are expressed in the measurement coordinate system as F k = F x,k ex + F y,k ey + F z,k ez for k {t, w}, (2.2) where the cutting forces acting on the tool tip and the workpiece in the x-, y- and z-direction are given by F x,k, F y,k and F z,k, for k {t, w}. It is assumed that the structural dynamics of the machine and the workpiece are linear. In Laplace 8

9 domain, the relation between the displacements d t and d w, and the cutting force F t and F w is given by d k (s) = Φ k (s) F k (s) for k {t, w}. (2.3) The transfer tensor Φ k (s) is expressed in the measurement coordinate system as Φ k (s) = ( e m ) T Φ m k (s) e m for k {t, w}, (2.4) with e m = ex ey ez φ xx,k (s) φ xy,k (s) φ xz,k (s) and Φ m k (s) = φ yx,k (s) φ yy,k (s) φ yz,k (s) φ zx,k (s) φ zy,k (s) φ zz,k (s) for k {t, w}, where φ ij,k (s) is the transfer function from the cutting force in j-direction, F j,k (s), to the displacement in i-direction, d i,k (s), for i, j {x, y, z} and k {t, w}. The cutting forces acting on the tool tip have the same magnitude but the opposite sign of the cutting forces acting on the workpiece. Therefore, the following relation holds: F = Ft = F w. (2.5) The amount of removed material is dependent on the relative displacement of the tool tip with respect to the workpiece. The relative displacement of the tool tip with respect to the workpiece due to the cutting forces is given by d = dt d w. (2.6) By combining (2.3), (2.5) and (2.6), the following expression for the relative displacement between the tool tip and the workpiece is obtained: d (s) = (Φt (s) + Φ w (s)) F = Φ(s) F (s), (2.7) where the combined transfer matrix of the tool tip and the workpiece dynamics is expressed in the measurement coordinate system as Φ m (s) = Φ m t (s) + Φ m w (s). 2.2 Regenerative chip model The cutting forces are dependent on the distribution of the chip created by the turning process. The chip distribution depends on the tool geometry and the cutting conditions. In [5], a chip model is presented that includes the effect of the nose radius of the tool. Small modifications are made to this model to include the effect of a triangular tool with an approach angle κ r and a nose radius r ε (figure 2.4a). The cutting conditions that influence the chip distribution are the feed rate c and the depth of cut a (figure 2.4b). In order to simplify the complex geometry of the chip in section 2.2.3, first, the chip area, the chip flow angle and the chord length are modeled in sections and

10 (a) Approach angle κ r and nose radius r ε (b) Feed rate c and depth of cut a Figure 2.4: Chip distribution Chip area The amount of removed material is dependent on the uncut chip area. The uncut chip area is the area of the chip projected in the plane normal to the cutting velocity direction (the xy-plane). For an orthogonal turning process with a rake angle of zero, the rake face of the tool is normal to the cutting velocity direction and, therefore, parallel to this plane (figure 2.5). Figure 2.5: Rake face and rake angle Assuming that the rake angle is zero, that there are no disturbances and that the surface of the workpiece is smooth, the area of the chip is equal to the static chip area A (figure 2.6). The static chip area is given by A = c a A cusp, (2.8) where A cusp is the cusp area (figure 2.6). The cusp area can be calculated as follows: A cusp =c r ε 1 1 ( ) 2 ( ) c 1 c rε 2 arcsin 2 2r ε 2r ε (2.9) if c c geom and a a cusp, 1

11 Figure 2.6: Static chip area A and cusp area A cusp where the geometrical upper limit to the feed rate c geom and the cusp depth a cusp are given by ( κr c geom = 2r ε cos π + π ) 3 6 (2.1) and ( ) 2 c a cusp = r ε 1 1. (2.11) 2r ε Eynian and Altintas [5] did not include the influence of the cusp area A cusp. Presumably, they assumed that the cusp area is negligibly small. This is a valid assumption if the nose radius of the tool and the depth of cut are large compared to the feed rate. In that case, the static chip area in (2.8) can be approximated by Chip flow angle and chord length A c a if c a A cusp. (2.12) The chip flow direction is defined as the direction that the chip leaves the workpiece. According to Colwell [7], the chip flow direction can be approximated by the direction normal to the chord that connects the two ends of the cutting edge engaged with the cut. The approximated chip flow direction and the chord that connects the two ends of the cutting edge are shown in figure 2.7. Figure 2.7: Chord and corresponding chip flow direction The chip flow coordinate system ( e p, e q, e r ) is introduced, where e p, e q and, e r are unit vectors of the coordinate system. It is assumed that the cutting velocity is normal to the rake face of the tool (see 11

12 section 2.2.1). The unit vector e p points in the chip flow direction. The unit vector e r is parallel to the cutting velocity. The unit vector e q is perpendicular to the other two unit vectors of the coordinate system (figure 2.8). The coordinate transformation between the measurement coordinate system ( e x, e y, e z ) and the chip flow coordinate system ( e p, e q, e r ) is given by e c = ep eq er cos(θ) sin(θ) = sin(θ) cos(θ) 1 e m = (T cm (θ)) T e c, ex ey ez = T cm (θ) e m (2.13) with the coordinate transformation matrix T cm (θ) and the chip flow angle θ. Figure 2.8: Geometrical features of the chip The length of the chord in x-direction L x and the length of chord in y-direction L y (figure 2.8) are given by L x = { c 2 r + ε 2 (r ε a) 2 if a cusp a a geom c 2 + a cot(κ { r) + r ε 1 cot(κr ) [ 1 + tan ( )]} 2κ r π 4 if a > ageom if c c geom (for c geom and a cusp see equations (2.1) and (2.11)), (2.14) with a geom = r ε [1 cos(κ r )], (2.15) and L y = a a cusp if a a cusp. (2.16) The equations (2.14) and (2.16) are used to find expressions for the chip flow angle θ and the length of the chord L: ( ) Lx θ = arctan, (2.17) L y 12

13 and L = L 2 x + L 2 y. (2.18) Approximated chip Instead of using the complex chip geometry created by the triangular tool (see the introduction of section 2.2), the chip geometry is approximated by a rectangle (figure 2.9). The width of the approximated chip is equal to the chord length L. The thickness of the approximated chip is chosen such that the area of the chip created by the triangular tool (see section 2.2.1) is equal to the area of the approximated chip. The (static) chip thickness of the rectangular chip h is given by h = A L. (2.19) (a) Chip created by triangular tool (b) Approximated chip Figure 2.9: Approximation of the static chip thickness h If the cusp area A cusp and the cusp depth a cusp are small, the static chip thickness can be approximated by h c a L see (2.8) and (2.16) for the approximations. c L y L = c cos(θ) if c a A cusp and a a cusp, (2.2) Disturbances lead to vibrations in the displacement of the tool tip with respect to the workpiece. Tool vibrations in the plane normal to the cutting velocity (the xy-plane or the pq-plane) influence the amount of removed workpiece material and, therefore, the cutting forces. If the feed rate is small compared to the nose radius of the tool and the depth of cut, the chip width is much larger than the chip thickness (L h ). In that case, tool vibrations in the direction of the chip thickness (the chip flow direction) have a much larger effect on the amount of removed material than similar sized tool vibrations in the direction of the chip width. Therefore, only tool vibrations in the chip flow direction are assumed to influence the cutting forces. The contribution to the cutting forces of tool vibrations other than in the chip flow direction is neglected. In other words, only the chip thickness is assumed to be influenced by the tool vibrations, while the chip width is assumed to be constant. Current vibrations of the cutting tool with respect to the workpiece in the chip flow direction are indicated by d p (t), where t is the current time. Current vibrations do not only influence the current chip thickness, they also influence the chip thickness after one revolution because of the vibrations left on the surface of the workpiece. This means that the current chip thickness is both dependent on 13

14 the current tool vibrations and on the tool vibrations of the previous revolution. The tool-workpiece vibrations of the previous revolution are given by d p (t τ), where τ is the time it takes to perform one revolution. The dynamic chip thickness h c is given by the following equation: h c (t) = h [ d p (t) d p (t τ)]. (2.21) The equation (2.21) is depicted in figure 2.1. The delay τ is related to the spindle speed n as follows: where τ is given in seconds and n is given in revolutions per minute. τ = 6 n, (2.22) Figure 2.1: Dynamic chip thickness h c Similar to the static chip area, the dynamic chip area A c is given by A c = L h c. (2.23) 2.3 Cutting force model The cutting forces are estimated using the regenerative chip model of section 2.2. The method of estimating the cutting forces presented in this section resembles the estimation method used in [5]. The differences will be pointed out further in this section. Instead of deriving expressions for the cutting forces in the three directions of the measurement coordinate system ( e x, e y, e z ) (see section 2.1), the cutting forces in the x-direction and the y-direction are combined to obtain one cutting force in the xy-plane. Assuming that the rake face of the tool is parallel to the xy-plane (see section 2.2.1), the chip leaves the tool in the same direction as the combined cutting force in the xy-plane. Hence, the direction of the cutting forces acting on the chip determine the direction that the chip leaves the tool (the chip flow direction). The chip flow direction is equal to the p-direction of the chip flow coordinate system ( e p, e q, e r ) (see section 2.2.2). Because the combined cutting force in the xy-plane only acts in the p-direction, the cutting force in the xy-plane perpendicular to the p-direction (the q-direction) is zero. Therefore, the cutting force vector F is given by F = Fp ep + F r er. (2.24) The cutting forces in the p-direction and the r-direction (F p and F r ) are depicted in figure Measurements presented in [5] showed that there is still a cutting force in the q-direction. However, this force is small compared to the cutting forces in the other directions of the chip flow coordinate 14

15 Figure 2.11: Cutting forces F p and F r system. In this report, the cutting force in the q-direction is assumed to be zero. A traditional model for estimating the cutting forces is given in [8]. This model consists of a force term related to the deformation and shearing of the chip and a force term related to edge effects. Edge effects are related to side effects of the deforming chip, such as friction and hardening of the material. The traditional model for estimating the cutting forces is given by } F p = F pc + F pe = K pca c + K pel, F r = F rc + F re = K rca c + K rel, (2.25) where the force term related to the chip formation is modeled by a cutting force coefficient times the chip area and where the force term related to the edge effects is modeled by a cutting force coefficient times the chip width. To include the influence of the cutting speed, the cutting force coefficients were measured for different cutting speeds in [5]. In this report, the cutting force coefficients are assumed to be independent of the cutting speed. In addition to the force contributions of the chip formation and the edge effects, process damping is added to the model. This additional damping force is either attributed to the change in the direction of the cutting speed due to the cutting force or to friction between the flank of the tool and the vibrations on the workpiece surface (figure 2.12). Altintas et al. [6] modeled the process damping as two force contributions related to the velocity and acceleration of the tool in the chip flow direction. In this report, only the force contribution related to the velocity of the cutting tool is used to model the process damping: F d (t) = C i V L dp (t), (2.26) where C i is the process damping coefficient. The cutting speed V is given by V = πd τ, (2.27) where D is the diameter of the cylindrical workpiece. It was shown in [6] that the process damping coefficient C i is dependent on tool wear. However, the tool wear dependency is not modeled in this report. In the direction parallel to the cutting velocity (the r-direction), the contribution of friction between the workpiece and the tool is modeled by the friction coefficient µ times the normal force F p (Figure 2.12). The friction coefficient is modeled as a constant and is assumed to be.3 for steel [9]. The friction force F f is given by 15

16 F f = µf p. (2.28) Figure 2.12: Process damping force F d and friction force F f Although friction was already included in the traditional cutting force model of (2.25), the influence of process damping on the friction between the workpiece and the tool was not modeled. Subtracting the friction from the traditional cutting force model only changes its cutting force coefficients. By including the process damping of (2.26) and adding the force contribution of friction in (2.28), the cutting force model of (2.25) becomes F p (t) = F pc (t) + F pe + F d (t) = K pca c (t) + K pel C i V L dp (t), F r (t) = F rc (t) + F re + F f (t) = K rca c (t) + K rel + µf p (t), which can be rewritten to F p (t) = K pc A c (t) + K pe L C i V L dp (t), F r (t) = K rc A c (t) + K re L µ C i V L dp (t), with cutting force coefficients (2.29) K pc = K pc K rc = K rc + µk pc K pe = K pe K re = K re + µk pe. 2.4 Stability of the orthogonal turning process The chip thickness is used as a measure for the stability of the turning process. The stability of the chip thickness is determined by the roots of the characteristic equation of the transfer function between the static chip thickness h and the dynamic chip thickness h c. The structural dynamic model of section 2.1, the regenerative chip model of section 2.2 and the cutting force model of section 2.3 are combined to determine the transfer function between the static and dynamic chip thickness. As time independent terms do not influence the stability of the chip thickness, they are dropped from the model. Vibrations in the tool-workpiece displacement d are linked to the vibrations in the cutting forces F using the transfer tensor Φ(s) of (2.7): d (s) = Φ(s) F (s). (2.3) 16

17 Because only tool vibrations with respect to the workpiece in the chip flow direction are assumed to influence the cutting forces (see section 2.2.3), the vibrations in the tool-workpiece displacement in Laplace domain are expressed as The vibrations in the cutting forces in Laplace domain are expressed as d (s) = dp (s) e p. (2.31) F = L (Kpc ep + K rc er ) h c (s) s C i V L ( e p + µ e r ) d p (s). (2.32) The transfer tensor Φ(s) is expressed in the chip flow coordinate system as Φ(s) = ( e c ) T Φ c (s) e c, (2.33) with e c = ep eq er φ pp (s) φ pq (s) φ pr (s) and Φ c (s) = φ qp (s) φ qq (s) φ qr (s). φ rp (s) φ rq (s) φ rr (s) Using the coordinate transformation of (2.13), the transfer matrix Φ c (s) is linked to the transfer matrix Φ m (s) as follows: Φ c (s) = T cm (θ)φ m (s) (T cm (θ)) T. (2.34) The equation (2.21) gives an expression for the chip thickness h c in time domain. Converting (2.21) to Laplace domain results in h c (s) = h ( 1 e sτ ) dp (s). (2.35) The closed-loop transfer function between the static chip thickness h and the dynamic chip thickness h c is obtained from (2.3) to (2.35): h c (s) h (s) = 1 + L { φ pp (s) [ K pc (1 e sτ ) + s Ci V 1 + s Ci V L [φ pp(s) + µφ pr (s)] ] + φpr (s) [ K rc (1 e sτ ) + sµ Ci V ]}. (2.36) The stability of this closed-loop transfer function is determined by the roots (s) of its characteristic equation, that is 1 + H(s) =, (2.37) with { [ ( H(s) = L φ pp (s) K pc 1 e sτ ) + s C ] [ i ( + φ pr (s) K rc 1 e sτ ) + sµ C ]} i. V V By assuming that the input is harmonic, the open-loop transfer function H(s) can be transformed to a frequency response function by replacing s with jω, where j = 1 and ω is the angular frequency of the harmonic input. The corresponding frequency response function is given by { [ ( H(jω) = L φ pp (jω) K pc 1 e jωτ ) + jω C ] [ i ( + φ pr (jω) K rc 1 e jωτ ) + jωµ C ]} i. V V (2.38) 17

18 Transfer matrix Φ m (s) is symmetric (φ ij (s) = φ ji (s) for i, j {x, y, z}). Therefore, transfer functions φ pp (s) and φ pr (s) can be expressed as } φ pp (s) = φ xx (s) cos 2 (θ) + φ xy (s) sin(2θ) + φ yy (s) sin 2 (θ), (2.39) φ pr (s) = φ xz (s) cos(θ) + φ yz (s) sin(θ), or in the frequency domain as } φ pp (jω) = φ xx (jω) cos 2 (θ) + φ xy (jω) sin(2θ) + φ yy (jω) sin 2 (θ), φ pr (jω) = φ xz (jω) cos(θ) + φ yz (jω) sin(θ). (2.4) 18

19 3 Experimental setup The proposed stability model is experimentally validated. The cutting tests are conducted on a Hardinge Superslant CNC turning machine. A cylindrical workpiece is mounted on the spindle of the machine. The tool is placed in a toolholder that is mounted on one of the two turrets. A Kistler 9121 dynamometer is placed between the toolholder and the turret to measure the forces acting on the tool tip. The dynamometer is able to measure the cutting forces in the three directions of the measurement coordinate system. The experimental setup is shown in figure 3.1. Figure 3.1: Experimental setup The workpiece is a cylindrical bar made of AISI 145 steel. Having a length of 25 cm, the workpiece is the most flexible component of the setup. Sandvik Coromant TNMA KR 325 inserts are used as cutting tools. These triangular-shaped inserts have a flat rake face. This makes them suitable for the modeled orthogonal turning processes. In combination with the toolholder, the inclination angle γ n of the cutting tool is not zero, which makes the turning process oblique instead of orthogonal (figure 3.2). Furthermore, the rake angle λ s of the cutting tool is not zero either. The model presented in chapter 2 requires that both angles are zero. Because the inclination angle and the rake angle are small (γ n = λs = 6 ), the influence of both angles on the cutting forces is neglected. Figure 3.2: Inclination angle γ n and rake angle λ s The geometric properties of the tool and the workpiece are given in table 3.1. Other model parameters that are not dependent on geometric properties are identified using cutting tests. The identification of the frequency response functions, the cutting force coefficients and the process damping coefficient is described in sections 3.1 to

20 Table 3.1: Geometric properties of the tool and the workpiece Tool Workpiece 3.1 Quantity Nose radius Approach angle Inclination angle Rake angle Diameter Length Symbol rε κr γn λs D Lw Size Unit mm mm cm Identification of the frequency response functions The frequency response functions that are required for the model presented in chapter 2 are obtained from frequency response measurements of the tool and the workpiece in the directions of the measurement coordinate system (see section 2.1). For each measurement, an impact force is applied to the tool or the workpiece by hitting it with an impulse force hammer. Both the applied impact force and the acceleration response are measured. The measured force and acceleration data are used to calculate an acceleration-force frequency response function which is converted to a displacement-force frequency response function. A Kistler 9722A5 impulse force hammer and a PCB Piezotronics 353B11 accelerometer are used to gather the required force and acceleration data (figure 3.3). The measured frequency response functions are shown in figure 3.4. (a) Frequency response function measurement of the workpiece (b) Frequency response function measurement of the tool tip Figure 3.3: Frequency response measurements By looking at the amplitude of the dominant mode at 348 Hz, it can be concluded that the workpiece is more flexible than the tool, and that the workpiece is more flexible in the radial direction than in the axial direction. Modal analysis has shown that the mode at 348 Hz corresponds to the first bending mode of the workpiece. The measured frequency response functions are affected by measurement noise. The frequency response functions are fitted to obtain a smooth result. Because only the relative displacement between the tool tip and the workpiece is important, a fit is made of the combined frequency response data of the tool tip and the workpiece. A nonlinear least-squares optimization technique is used to fit the frequency response functions to the following model: 2

21 Amplitude [µm N 1 ] Real part Imaginary part Amplitude [µm N 1 ] Real part Imaginary part Amplitude [µm N 1 ] Real part Imaginary part Frequency [Hz] (a) φ xx,t(jω) Frequency [Hz] (b) φ xx,w(jω) Frequency [Hz] (c) φ xy,t(jω) Amplitude [µm N 1 ].2.2 Real part Imaginary part Frequency [Hz] (d) φ xy,w(jω) Amplitude [µm N 1 ] Real part Imaginary part Frequency [Hz] (e) φ xz,t(jω) Amplitude [µm N 1 ] Real part Imaginary part Frequency [Hz] (f) φ xz,w(jω) Amplitude [µm N 1 ] Real part Imaginary part Frequency [Hz] (g) φ yy,t(jω) Amplitude [µm N 1 ] Real part Imaginary part Frequency [Hz] (h) φ yy,w(jω) Amplitude [µm N 1 ].2.2 Real part Imaginary part Frequency [Hz] (i) φ yz,t(jω) Amplitude [µm N 1 ] Real part Imaginary part Frequency [Hz] (j) φ yz,w(jω) Figure 3.4: Measured frequency response functions 21

22 ( ) N α k,ij + jωβ k,ij φ ij (ω) = ω 2 + 2jζ k,ij ω n,k,ij ω + ωn,k,ij 2 where k=1 1 m ij ω 2 + s ij for i, j {x, y, z}, (3.1) N : number of modes α k,ij, β k,ij : modal coefficients of mode k ζ k,ij : damping coefficient of mode k ω n,k,ij : angular natural eigenfrequency of mode k m ij : residual mass s ij : residual stiffness. Note that all coefficients are different for each frequency response function. The fitted coefficients are given in appendix A. The frequency range of the frequency response data is 2 Hz to 1 Hz. The most dominant modes are within this frequency range. The Nyquist diagrams of the measured and the fitted frequency response functions are shown in figure 3.5. Imaginary part [µm N 1 ] Measurment Fit Imaginary part [µm N 1 ] Measurment Fit Imaginary part [µm N 1 ] Measurment Fit Real part [µm N 1 ] (a) φ xx(jω) Real part [µm N 1 ] (b) φ xy(jω) Real part [µm N 1 ] (c) φ xz(jω) Imaginary part [µm N 1 ] Measurment Fit Imaginary part [µm N 1 ] Measurment Fit Real part [µm N 1 ] (d) φ yy(jω) Real part [µm N 1 ] (e) φ yz(jω) Figure 3.5: Nyquist diagrams of the measured and fitted frequency response functions 3.2 Identification of the cutting force coefficients The cutting force coefficients are determined by cutting force measurements. The cutting force model is given by (2.29) in section 2.4: F p (t) = K pc A c (t) + K pe L C i V L dp (t), F r (t) = K rc A c (t) + K re L µ C i V L dp (t). 22

23 By neglecting the vibrations in the cutting forces, (2.29) can be written as } F p = K pc A + K pe L = L (K pc h + K pe ), F r = K rc A + K re L = L (K rc h + K re ). (3.2) The cutting force coefficients K pc, K pe, K rc and K re are determined by measuring the cutting forces for different values of the static chip thickness h and fitting the model to the measurement results. Because the static chip thickness is strongly related to the feed rate (h c cos(θ), see (2.2) in section 2.2.3), the cutting forces are measured for a constant depth of cut a = 1 mm and different feed rates c = {.2,.4,.6,.8,.1} mm. To reduce the vibrations in the cutting forces, the length of the workpiece is shortened to L w = 8 cm. The cutting conditions are given in table 3.2. Table 3.2: Cutting conditions for the identification of the cutting force coefficients Quantity Symbol Size Unit Depth of cut a 1 mm Feed rate c {.2,.4,.6,.8,.1} mm/rev Spindle speed n 2 rev/min Workpiece diameter D 41 mm Workpiece length L w 8 cm The cutting forces are measured in the measurement coordinate system. The cutting forces of the measurement coordinate system are converted to the chip flow coordinate system as follows: F p = (F x ) 2 + (F y ) 2, F q =, (3.3) F r = F z. Five measurements are taken for each feed rate. The converted cutting forces and the fitted curves are shown in figure 3.6. The corresponding cutting force coefficients are given in table F p [N] 2 F r [N] 2 1 Measurement Fit c [mm/rev] 1 Measurement Fit c [mm/rev] (a) F p (b) F r Figure 3.6: Cutting forces F p and F r for different feed rates c 3.3 Identification of the process damping coefficient Instead of identifying the process damping coefficient C i with a piezo actuator driven fast tool servo, as presented in [6], orthogonal plunge turning tests are conducted. The tool is fed into the workpiece in radial direction until the rotational axis is reached (figure 3.7). 23

24 Table 3.3: Cutting force coefficients Coefficient Size Unit K pc N/m 2 K pe N/m K rc N/m 2 K re N/m Figure 3.7: Decrease in workpiece diameter D and cutting speed V The decrease in the diameter results in a decrease in the cutting speed. For some cutting conditions holds that if the cutting speed is low enough, the process damping becomes high enough such that the unstable turning process becomes stable. The process damping force was modeled in (2.26) of section 2.3 as F d (t) = C i V L dp (t). The critical cutting speed for which this change in stability takes place is indicated by V lim. Orthogonal plunge turning tests are conducted for two sets of cutting conditions (table 3.4). Because the effect of process damping is largest at low cutting speeds, a relatively low spindle speed of 1 rev/min is used for all cutting tests. Chatter vibrations are not only present in displacement and cutting force measurements, they can also be recorded by sound measurements. A microphone is added to the experimental setup as shown in figure 3.8. Table 3.4: Cutting conditions for the process damping coefficient identification Set 1 Set 2 Quantity Symbol Size Unit Depth of cut a.6 mm Feed rate c.5 mm/rev Spindle speed n 1 rev/min Workpiece diameter D [41 ] mm Workpiece length L w 25 cm Depth of cut a.7 mm Feed rate c.1 mm/rev Spindle speed n 1 rev/min Workpiece diameter D [41 ] mm Workpiece length L w 25 cm Five cutting tests are conducted for each set of cutting conditions. Figure 3.9a shows the cutting force in p-direction for one of the measurements for the first set of cutting conditions. The corresponding sound data is presented in figure 3.9b. Both measurements show an unstable region with large chatter vibrations and a stable region with small chatter vibrations. Time is zero corresponds to the first 24

25 Figure 3.8: Force and sound measurement setup contact between the tool and the workpiece. The critical cutting speed V lim is reached when the chatter vibrations start to attenuate. For the cutting force measurement, this happens after approximately 13.1 seconds, which corresponds to a workpiece diameter of 19.2 mm and a critical cutting velocity of 6.3 m/min. For the sound measurement, the critical cutting speed is reached after 13.6 seconds, which corresponds to a workpiece diameter of 18.3 mm and a cutting speed of 57.5 m/min. 3 2 F p [N] 1 1 D = 41. mm D = 19.2 mm D = mm Time [s] (a) Force measurement D = 18.3 mm Microphone D = 41. mm D = 19.2 mm D = mm Time [s] (b) Sound measurement Figure 3.9: Force and sound measurement in the time domain for the cutting conditions of set 1 The frequency content of the force and sound data is calculated by taking the fast Fourier transform of the two signals. For time is 1 to 6 seconds, the (unstable) chatter vibrations are clearly visible in the frequency domain plots of the force and the sound data. The largest peaks in the frequency domain correspond to the chatter frequency of 381 Hz and its higher order harmonics (figures 3.1a and 3.1b). For 15 to 23 seconds, the chatter vibrations are small and are barely visible in the corresponding frequency domain plots (figures 3.1c and 3.1d). 25

26 F p [N] Frequency [Hz] (a) Force measurement for time is 3 to 12 seconds Microphone Frequency [Hz] (b) Sound measurement for time is 3 to 12 seconds F p [N] Frequency [Hz] (c) Force measurement for time is 15 to 23 seconds Microphone Frequency [Hz] (d) Sound measurement for time is 15 to 23 seconds Figure 3.1: Force and sound measurement in the frequency domain for the cutting conditions of set 1 For the second set of cutting conditions, the feed rate is altered from.5 mm/rev to.1 mm/rev. Although chatter is still present in the measurement data, the force contribution of the chatter vibrations is less compared to the measurements for the first set of cutting conditions (figure 3.11a). The critical cutting speed can no longer be obtained by looking at the amplitude of the vibrations. However, it is still possible to obtain the critical cutting speed by looking at the magnitude of the cutting forces. After the critical cutting speed is reached, a sudden drop in the magnitude of the cutting forces is observed. The cutting forces start to drop at a time of 7.2 seconds. The corresponding workpiece diameter and cutting speed are 17. mm and 53.4 m/min. Because the critical cutting speed is not obtained from the amplitude of the chatter vibrations, the sound data in the time domain can not be used to find the critical cutting velocity (figure 3.11b). Contrary to the time domain, in the frequency domain, the region with unstable chatter vibrations can clearly be distinguished from the region with stable chatter vibrations by looking at the frequency content (figure 3.12). Similar to the measurements for the cutting conditions of set 1, the measured chatter frequency is 381 Hz. Because it is not always possible to use the sound measurements to determine the critical cutting velocity, the force measurements are used to calculate the process damping coefficient. The given values for the critical cutting velocities are representable for all measurements. In general, the calculated critical cutting velocities do not deviate more than 5 m/min from the given values. The calculated values for critical cutting speed are collected in table 3.5. Recall that the stability of the turning process can be determined by roots of the characteristic equation of the closed-loop transfer function between the static and the dynamic chip thickness (see (2.37) section 2.4), that is 26

27 4 3 F p [N] D = 41. mm D = 17. mm D = mm Time [s] (a) Force measurement.8 Microphone D = 41. mm D = 17. mm D = mm Time [s] (b) Sound measurement Figure 3.11: Force and sound measurement in the time domain for the cutting conditions of set F p [N] Microphone Frequency [Hz] (a) Force measurement for time is 1 to 6 seconds Frequency [Hz] (b) Sound measurement for time is 1 to 6 seconds 2.2 F p [N] Microphone Frequency [Hz] (c) Force measurement for time is 8 to 11 seconds Frequency [Hz] (d) Sound measurement for time is 8 to 11 seconds Figure 3.12: Force and sound measurement in the frequency domain for the cutting conditions of set 2 27

28 Table 3.5: Critical cutting speed Quantity Symbol Size Unit Set 1 Critical cutting speed V lim 6.3 m/min 1.5 m/s Set 2 Critical cutting speed V lim 53.4 m/min.89 m/s { [ ( 1 + L φ pp (s) K pc 1 e sτ ) + s C ] [ i ( + φ pr (s) K rc 1 e sτ ) + sµ C ]} i =. V V The process damping coefficient C i is calculated by transforming the characteristic equation to the frequency domain by substituting s = jω and solving for the two unknowns ω and C i. As the critical cutting speed V lim corresponds to the critically stable condition (the boundary between stability and instability), the two unknowns ω and C i are obtained by finding the corresponding values for which the characteristic equation is critically stable. After substituting s = jω and V = V lim, the left-hand side of characteristic equation has a real and an imaginary part. For critical stability, both parts of the characteristic equation need to be zero. This condition is necessary, but not sufficient. A stability check is necessary to ensure that the correct value of the process damping coefficient is obtained. For every angular frequency ω, two possible values for the process damping coefficient are calculated: one for the real part of the characteristic equation C i,1 and one for the imaginary part of the characteristic equation C i,2 : C i,1 = V lim { 1 [φ pp,i (ω) + µφ pr,i (ω)] ω L + [φ pp,r (ω)k pc + φ pr,r (ω)k rc ] [1 cos(ωτ)] } [φ pp,i (ω)k pc + φ pr,i (ω)k rc ] sin(ωτ) (3.4) and C i,2 = { V lim [φ pp,i (ω)k pc + φ pr,i (ω)k rc ] [1 cos(ωτ)] [φ pp,r (ω) + µφ pr,r (ω)] ω } + [φ pp,r (ω)k pc + φ pr,r (ω)k rc ] sin(ωτ), (3.5) where φ pp,r (ω) and φ pp,i (ω) are the real and imaginary part of φ pp (ω), and φ pr,r (ω) and φ pr,i (ω) are the real and imaginary part of φ pr (ω). The process damping coefficient C i is obtained by looking for a value of ω close to the measured chatter frequency of 381 Hz for which the following condition holds: C i = C i,1 = C i,2. (3.6) By using (3.6), it is possible to find more than one value for C i. In order to find the correct value, each C i value is substituted in the characteristic equation (in Laplace domain) and checked for critical stability using the Nyquist stability criterion. The calculated C i values for the two sets of cutting conditions (see table 3.4) are given in table 3.6. The corresponding values of ω of 2286 rad/s and 2288 rad/s (both approximately 364 Hz) are close to the measured chatter frequency of 381 Hz. For further calculations a round-off C i value of N/m is used. 28

29 Table 3.6: Process damping coefficient Set 1 Set 2 Quantity Symbol Size Unit Proc. damp. coeff. C i N/m } Angular freq. ω 2286 rad/s Ci = N/m Proc. damp. coeff. C i N/m Angular freq. ω 2288 rad/s 29

30 4 An algorithm to construct stability charts for turning Stability charts are an easy way to visualize the stability of a cutting process for a large range of cutting conditions. Usually, a stability chart portrays the stability of a cutting process for a range of spindle speeds and depths of cut while the feed rate is kept constant. An example of a stability chart that is created using the algorithm in this chapter is shown in figure Depth of cut [mm] Unstable Stable Spindle speed [rev/min] Figure 4.1: Stability chart example The black curve indicates the boundary between the stable region (grey) and the unstable region (white). Unstable chatter vibrations result in a poor surface finish and can damage the tool, the workpiece and the machine. To avoid unstable chatter vibrations, the cutting conditions of the cutting process should lie in the stable region of the stability chart. The stability of the modeled orthogonal turning process is predicted by solving the stability of the chip thickness using the characteristic equation of (2.37) in section 2.4. The boundary between the stable region and the unstable region is obtained by finding the cutting conditions that result in a critically stable chip thickness. In section 4.1, an algorithm is presented to identify the critically stable cutting conditions using the Nyquist stability criterion. For every spindle speed and feed rate, critical stability is obtained by finding the corresponding depth of cut. This operation is executed for a range of spindle speeds to obtain a stability chart similar to the one in figure An algorithm to determine the critically stable cutting conditions As mentioned, the algorithm presented in this section is designed to find the depth of cut a for which the modeled turning process is critically stable. Inputs to this algorithm are the feed rate c, the spindle speed n and several model parameters, frequency response functions, limit values and initial conditions. The outline of the algorithm is presented in figure 4.2. A Nyquist plot is an easy tool to determine the stability of a system using the Nyquist stability criterion. A Nyquist plot of the open-loop system H(jω) is used to determine the stability of the chip thickness and, therefore, the stability of the turning process. The open-loop system H(jω) was defined in (2.38) of section 2.4 as { [ ( H(jω) = L φ pp (jω) K pc 1 e jωτ ) + jω C ] [ i ( + φ pr (jω) K rc 1 e jωτ ) + jωµ C ]} i. V V In order to obtain the necessary Nyquist plot data, the cutting speed and the spindle revolution time are calculated in the subroutine: calculation of the cutting speed V and the spindle revolution time τ (section 4.1.1), and the chip flow angle and the chord length are calculated in the subroutine: calculation of the chip flow angle θ and the chord length L (section 4.1.2). After the Nyquist plot data is obtained in the subroutine: generation of the Nyquist plot data (section 4.1.3), the stability limit value C R,lim is calculated in the subroutine: calculation of the stability limit value C R,lim (section 4.1.4). The stability limit value 3

31 n, D φ xx (jω), φ xy (jω), φ xz (jω), φ yy (jω), φ yz (jω) K pc, K rc, C i, µ, α p Initial input C R,high, a low α R, α a Calculation of the cutting speed V and the spindle revolution time τ Generation of the Nyquist plot data Calculation of the stability limit value C R,lim Stability check and update of the depth of cut a a c, r ε, κ r Calculation of the chip flow angle θ and the chord length L Initial input Ω Initial input a Figure 4.2: Outline of the algorithm for finding the depth of cut for which the modeled turning process is critically stable is not only a measure for the stability of the chip thickness, it also gives an indication how close the current value of the depth of cut is to the value for which the chip thickness is critically stable. The stability limit value is used to determine the stability of the chip thickness and to re-estimate the value for the depth of cut in the subroutine: stability check and update of the depth of cut a (section 4.1.5) Calculation of the cutting speed and the spindle revolution time The cutting speed V and spindle revolution time τ are parameters of the open loop system H(jω). The cutting speed and the spindle revolution time are calculated from the spindle speed n and the workpiece diameter D using (2.22) and (2.27) (see sections and 2.3 or figure 4.3). The subroutine to calculate V and τ is depicted in figure 4.3. n, D τ = 6 n Equation (2.22) V = πd τ Equation (2.27) τ, V Figure 4.3: Subroutine: calculation of the cutting speed V and the spindle revolution time τ 31