High-speed machining frequency response prediction for process optimization Tony Schmitz, Matthew Davies, Michael Kennedy National Institute of Standards and Technology Automated Production Technology Division MS 8220, Gaithersburg, MD, 20899, USA E-mail: [email protected], [email protected]

ABSTRACT. As interest in high-speed machining techniques grows among world manufacturers, efficient methods for obtaining the tool point frequency response (used as input to analytic or numerical chatter prediction/avoidance models) become more important. We describe the application of receptance coupling substructure analysis to the analytic prediction of the tool point dynamic response using frequency response measurements of individual components coupled through appropriate connections. This paper demonstrates experimental verification of the receptance coupling method for various tool geometries (e.g., diameter and length) and holders (HSK 63A collet and shrink fit). System changes (and the corresponding critical stability limit deviations) are predicted for tool length variation in a tool tuning example. Mode shape prediction is also shown. KEY WORDS: High speed machining, Sub-structure analysis, Frequency response.

1. Introduction High-speed machining (HSM) is an active area of research in the manufacturing community, particularly in the aerospace industry. A major limitation on the productivity of HSM systems in practice is regenerative chatter. Recent studies have explored practical HSM implementation techniques (e.g., accurate measurement of machine dynamics, self-excited vibration theory for the calculation of stability lobes, machine/spindle design requirements, and sensors capable of chatter recognition), as well as analytic and numerical time-domain simulation of the cutting forces and tool deflections during HSM [SMI 92,TLU 96,SMI 90,DAV 96,AGA 95,SCH 92,HEI 96,DAV 98,WIN 95,WEC 94,WEC 99,DEL 92,SMI 91,TLU 83,ALT 95]. HSM simulation, which is crucial for chatter prediction and avoidance, requires knowledge of the system dynamics reflected at the tool point. In general, a separate set of tool point frequency response function (FRF) measurements must be performed for each tool/holder/spindle combination on a particular machining center. These measurements can prove time-consuming, require a trained technician, and lead to costly machine downtime. In this work, we seek to reduce measurement time and increase process efficiency by predicting the tool point dynamic response using receptance coupling substructure analysis. This method couples component frequency responses (e.g., tool, holder/spindle) through appropriate connections using simple vector manipulations and will be described in the following sections. Experimental verification of the method will be provided for: 1) a tool tuning example (i.e., adjusting tool length to make use of local increases in machining stability [TLU 96,DAV 98]), 2) a 19.1 mm diameter endmill coupled to a shrink-fit HSK 63A holder, and 3) measured and predicted mode shapes for the 19.1 mm diameter endmill. 1.1 High-speed machining1 HSM may be defined in various ways. First, with regard to attainable cutting speeds, it is suggested that operating at cutting speeds significantly higher than those typically utilized for a particular material may be termed HSM (see Figure 1 [SCH 92]). Second, theoretical and experimental analyses have shown that increased local stability occurs when the tooth passing frequency of the cutter is equal to the natural frequency (or any of its fractional harmonics) of the most flexible system mode [TLU 96,SMI 90]. Selection of the maximum available spindle speed that corresponds to one of these stable tooth passing frequencies is also referred to as HSM. The latter definition of HSM relies on the concept of regeneration of waviness as a primary cause of instability (i.e., self-excited vibrations) in machining [SMI 90]. This waviness regeneration occurs when a cutter tooth encounters an undulating surface left by the previous tooth. The prediction of system stability depends on the phase 1

Before proceeding to a description of the receptance coupling algorithm, a brief overview of high-speed machining is provided for completeness.

relationship between the displacement of the current cutter tooth and the waviness it encounters. For certain phase relationships, the succeeding tool vibrations diminish, while for others they increase until either failure or a system non-linearity (e.g., the tool leaves the cut) limits the motion [TLU 81]. A typical stability lobe diagram, which predicts system stability as a function of spindle speed and machining parameters, is shown in Figure 2. Both stable and unstable regions are seen depending on the selected spindle speed and axial depth of cut, b. This diagram may be calculated using analytic (see, for example, [ALT 95,KOE 67,TOB 65,MER 65,MIN 90]) or numerical time-domain techniques [SMI 90,WIN 95,SMI 91,TLU 83]. In either instance, knowledge of the tool/workpiece/machine dynamics is required. In many cases, the system dynamics are obtained using impact testing and modal analysis. The direct FRF is measured at the tool point and multiple modes fit to the results. Modal parameters (i.e., mass, m, stiffness, k, and damping ratio, ξ) for each of the selected modes are extracted and used as input to the stability lobe analysis [SMI 90,DAV 98,SMI 91,TLU 83,TLU 91,EWI 95,TLU 90]. Other possibilities include computational methods for modal parameter determination and milling experiments for direct stable speed selection. For the latter, modal parameters are not determined or required. Using stability lobe theory, machining tests are completed to locate chatter frequencies and select stable spindle speeds [SMI 92]. In all cases, however, the measurements are specific to the selected components (e.g., tool and tool length, holder, workpiece, spindle, and machine) and boundary conditions (e.g., holder force and drawbar force [SMI 99]). Nickel-Base Alloys Titanium Steel HSM Range Transition Range Normal Range

Cast Iron Bronze, Brass Aluminium Fiber Reinforced Plastics 10

100 1000 Cutting Speed [m/min]

10000

Figure 1. Attainable speeds in the machining of various materials (after [SCH 92]) 2. Receptance Coupling In receptance coupling substructure analysis, experimental or analytic direct and cross FRFs for the individual components are used to predict the final assembly’s dynamic

response at any spatial coordinate selected for component measurements [FER 95,FER 96,REN 93,KLO 77]. In this method, unlike modal coupling and finite element analyses, experimental or analytic FRFs are required only at the coordinate of interest (tool point) and any connection coordinates; the number of modeled structural modes in each component does not define the number of required measurement locations (to obtain square modal matrices); and no matrix inversions are necessary (only vector manipulations are required). In the following section, a mathematical description of receptance coupling will be given, as well as comparisons between this method, complex matrix inversion, and modal analysis for a simple (contrived) system. 0.13 0.12

Unstable region

b lim (mm)

0.11 0.1 0.09 0.08 0.07 Stable region

0.06 6500

7500 8500 Spindle Speed (rpm)

9500

Figure 2. Stability lobe diagram (limiting axial depth of cut versus spindle speed)

2.1 Mathematical Example As shown in Figure 3, two single degree-of-freedom (SDOF) spring/mass/damper systems, A and B, are to be connected through the linear spring element, kc, to form the new two degree-of-freedom (2DOF) assembly C. The assembled system’s equations of motion (for this simple example) can be determined from Newton’s 2nd law of motion. The matrix form of these equations, after substituting the assumed harmonic solution, x(t) = Xeiωt for harmonic vibration, is shown in Equation 1.

[A(ω )]

X1  = X 2 

[1]

− m1ω 2 + iω c1 + (k1 + k c )  X 1   F1  − kc    =   2 − kc − m2ω + iω c 2 + (k 2 + k c ) X 2  F2  

Figure 3. Mathematical example 2.1.1 Complex Matrix Inversion Because an analytical form for the system frequency response can be written directly in this case, the complex vibrations of the two coordinates, X1 and X2, can be determined by complex matrix inversion. Inversion of the equations of motion matrix, A(ω), yields G(ω), the receptance matrix relating complex displacements to applied forces for this system (see Equation 2). The direct FRF at coordinate 2 of the assembly in Figure 3, G22(ω) = [A(ω)]22-1, is given in Equation 3.

[A(ω )]− 1 = [G (ω )]=

[2]

− m 2ω 2 + iω c2 + (k 2 + kc )  kc 1   2 det ( A(ω )) kc − m1ω + iω c1 + (k1 + k c ) G22 (ω )=

(− m ω

− m1ω 2 + iω c1 + (k1 + kc )

2

1

)(

)

+ iω c1 + (k1 + kc ) − m2ω 2 + iω c2 + (k2 + kc ) − kc 2

[3]

2.1.2 Modal Solution The equations of motion shown in Equation 1 may also be used to find the modal solution for the assembled system. If proportional damping is assumed (i.e., [c] = α[m] + β[k], where α and β are real numbers), damping can be neglected in the modal

solution. The characteristic equation for this system may then be written as shown in Equation 4. The quadratic roots of this 4th order equation give the two eigenvalues (and natural frequencies) for the 2DOF system. Substitution of these eigenvalues, normalized to the coordinate of interest (coordinate 2 in this case), into the original equations of motion (again neglecting damping) yields the eigenvectors (mode shapes). The assembly equations of motion (coupled through the connection spring, kc) can then be uncoupled using the modal matrix (composed of columns of the eigenvectors), P, as shown in Equation 5. The modal solution for the direct FRF at coordinate 2 of the assembled system can then be expressed as shown in Equation 6. Please note that the q subscripts refer to the (uncoupled) modal parameters, not the original spatial model values.

(− m ω 1

2

)(

)

+ (k1 + k c ) − m2ω 2 + (k 2 + k c ) − k c = 2

[4]

m1m 2ω + (− m1 (k 2 + k c )− m 2 (k1 + k c ))ω + (k1k 2 + k1 k c + k 2 k c ) = 0 4

2

0  m q11 mP =  m q 22   0  k 0   q11 k q = P T kP =  0 k q 22    0  cq11 c q = P T cP =    0 cq 22 

[m ]= P

T

q

[]

[5]

[]

G22 (ω )=

ω n21 1 X2 = + 2 F2 kq11 − ω + i 2ζ q1ω n1ω + ω n21

(

)

1 k q 22

ω n22 − ω 2 + i 2ζ q 2ω n2ω + ω n22

(

[6]

)

2.1.3 Receptance Coupling The receptance matrix, G(ω), for the assembled system in Figure 3 will now be derived using the receptance coupling method [FER 96]. The 2x2 receptance matrix (for the 2DOF system) will be calculated by columns. We will determine the first column by applying a virtual force, F1, to coordinate X1. Figure 4 displays the assembled and component systems with F1 applied to the assembled system. For simplicity, the spring/mass/damper systems are represented by continuous bodies (which may, in general, contain several degrees of freedom). Considering the (unassembled) substructures in Figure 4, the displacements at the two coordinates can be written as shown in Equation 7. The notation H refers to the spatial receptance matrices of the individual components before assembly. The equilibrium

condition for the components is given in Equation 8. The compatibility conditions are shown in Equation 9.

Figure 4. Assembled/component systems

x1 = H 11 f 1

[7]

x2 = H 22 f 2

f1 + f 2 = F1 , and f1 = F1 − f 2 X 1 = x1 , X 2 = x2 , and x2 − x1 =

[8]

− f2 kc

[9]

Substitution of Equations 7 and 8 into 9 yields Equation 10. Substituting the result shown in 10 into the equilibrium condition gives 11. These are expressions for the forces acting on the individual components. −1

 1 f2 =  H 11 + H 22 + k   H 11 F1  c 

[10]

 f 1 = 1 −  

[11]

−1   1    + + H H H 22  11  11 F1 k  c  

The first column of the receptance matrix is now determined by substitution into the appropriate displacement/force relationships. Equations 12 and 13 give expressions for the G11(ω) and G21(ω) receptance terms.

G11 (ω )=

−1

 X1 1 = H 11 − H 11  H 11 + H 22 + k   H 11 F1  c 

[12]

 X 1 G21 (ω )= 2 = H 22  H11 + H 22 +  F1 kc 

−1

  H11  

[13]

The second column of the receptance matrix is found by applying a virtual force, F2, to coordinate X2 of the assembly. The displacements for this case are the same as those given in Equation 7. The new equilibrium and compatibility equations are shown in Equations 14 and 15, respectively. Substitution and combining operations similar to those for the first column give expressions for G12(ω) and G22(ω).

f1 + f 2 = F2 ,and f 2 = F2 − f1

[14]

− f1 X 1 = x1 , X 2 = x2 , and x1 − x2 = kc

[15]

−1

G12 (ω )=

 1  X1  = H11  H11 + H 22 + k  H 22 F2 c 

G 22 (ω )=

 X2 1   = H 22 − H 22  H 11 + H 22 + k  H 22 F2 c  

[16] −1

[17]

2.1.4 Method Comparison A graphical comparison between complex matrix inversion, modal analysis, and receptance coupling results of the direct FRF at coordinate X2 for the simple system in Figure 3 will now be completed. The three expressions for G22(ω) derived in Equations 3, 6, and 17, as well as the original component direct FRFs (H11 and H22), are plotted in Figure 5. The parameters for the spring/mass/damper systems and coupling spring are listed in Table 1. The reader may note that proportional damping exists (α = 0, β = 1x10-4) for the selected system. Figure 5 serves to verify the receptance coupling approach mathematically. It is seen that all three methods predict essentially the same dynamic response for the assembled system. Further investigation, however, yields an interesting result. The point-by-point differences between the complex matrix inversion result and the modal and receptance coupling method results, respectively, are shown in Figure 6. It is seen that the errors introduced by the modal method, although small (less than 2x10-7), are 4x1013 times greater than the errors associated with the receptance technique. Additionally, the modal errors vary substantially with changes in the system parameters. The differences between the three techniques are, of course, introduced by numerical round-off errors (15 significant digits were carried) in the mathematical manipulations. However, the

improved numerical accuracy obtained with receptance coupling (vector manipulations) over modal coupling (matrix manipulations) is not easily dismissed. Table 1. Mathematical example system parameters Parameter Value k1 2x106 N/m m1 3 kg c1 200 N-s/m k2 1x106 m2 2 c2 100 kc 5x105 -6

Real (m/N)

x 10

G22(ω )

0 -5 0 -6 x 10

Imag (m/N)

H11(ω )

5 H22(ω ) 50

100

150

200

250

300

50

100 150 200 Frequency (Hz)

250

300

0 -5 -10 -15 0

Figure 5. G22(ω) comparison 2.2 Experimental Verification To verify the receptance coupling technique, tests were carried out on two tool geometries and holder types. First, three long, slender tools (length to diameter ratios of 8:1, 9:1, and 10:1) were selected for measurement in a 12.7 mm diameter collet holder (HSK 63A spindle connection). The tungsten carbide (cobalt binder) tools had two flutes and a relieved diameter of 11.8 mm for the portion of the tool outside the holder (12.7 mm inside). This tooling had previously been examined for machining of an aluminum part with 102 mm deep, 38 mm wide hexagonal pockets with 0.5 mm wall thickness [DAV 98]. Empirically determined, improved stability was reported for a 10:1 length/diameter ratio. Second, a 19.1 mm diameter tungsten carbide endmill with two flutes and a 98.5 mm overhang (5.2:1 length to diameter ratio) was mounted

in a shrink fit holder in the same spindle and the tool point response measured and predicted. Comparisons between predicted and measured mode shapes for this tool were also completed.

Receptance Errors (m/N)

Modal Errors (m/N)

-7

2

x 10

Real part of error

Imaginary part of error 0

-2 0 50 -21 x 10 5

100

150

200

250

300

250

300

Imaginary 0 -5 -10 0

Real 50

100

150 200 Frequency (Hz)

Figure 6. Error comparison 2.2.1 11.8 mm diameter tooling Three primary steps are involved in the prediction of the tool point response for an assembled tool/holder/spindle system: 1) define an appropriate model that includes all necessary degrees of freedom and connection terms, 2) determine the component FRFs by measurement or analytic prediction, and 3) couple the component FRFs through the modeled connections using receptance coupling. The model representing the connection of a tool (component A) to a particular holder/spindle combination (component B) is shown in Figure 7. This model is similar to the example in the previous section, except there are now three spatial coordinates of interest (with translation and rotation considered at each) and damping is included.

Figure 7. Tool/holder/spindle assembly receptance coupling model

The component FRFs were determined in two parts. First, the direct and cross freefree state (or unsupported) FRFs for coordinates x1 and x2 of component A (the tool) were calculated analytically. An analytical formulation, rather than experimental measurement, was selected due to the difficulties associated with obtaining these tool FRFs using impact testing (e.g., the free-free state is difficult to realize in practice; low mass, wide bandwidth accelerometers typically do not perform well at low frequencies and zero frequency, rigid body mode behavior may not be well represented in the measurements; and the response for low mass tools is easily corrupted by the accelerometer mass). Second, the direct FRFs at coordinate x3 of component B (the holder/spindle) were obtained experimentally.

Figure 8. Tool loading conditions A brief description of the analytic derivation of the tool FRFs will now be given. Figure 8 shows the loads applied to the tool when connected to the holder/spindle. A force is applied to the free end (x1) while the end to be inserted in the holder (x2) opposes displacement and rotation (i.e., a cantilever beam). The equation of motion for this model is shown in Equation 18. Here, u(x,t) is the temporal displacement at any point along the beam and q(x,t) represents the externally applied loads. Additionally, ρ is the beam mass per unit length, c′is the viscous damping coefficient per unit length, E is Young’s modulus (homogeneity assumed), and I is the second area moment of inertia (uniform cylindrical cross-section assumed). ?

∂2u (x,t ) ∂u (x,t ) ∂4u (x,t ) + c' + EI = q(x,t ) ∂t ∂t 2 ∂x4

[18]

If harmonic motion is assumed, a solution for the lateral vibration of the beam at any coordinate along its length (L) may be obtained from Equation 18 for each beam mode. These results are then summed to determine the total vibration at that location. In this analysis, two rigid body modes (inertial translation and rotation about the center of mass) were combined with analytic expressions for the beam free-free modes [BLE 79] to determine the tool vibration. Expressions for the direct (complex displacement over force) FRFs at coordinates x1 and x2 (H11 and H22, respectively) and a cross FRF H12 (H21 is equivalent by reciprocity) are shown in Equation 19, where λi is a dimensionless frequency parameter [BLE 79].

The receptance term H11, for example, contains three components. The first two represent the contributions by the translational and rotational rigid body modes, respectively, while the third gives the response due to the free-free modes (φi(x), expressed as shown in Equation 20) which have been evaluated at coordinate x1 (a distance L from the model origin). Response functions that relate displacement to applied moment (Lij), rotation to applied force (Nij), and rotation to applied moment (Pij) were also derived. H 11 (? ) =

H 22 (? ) =

1 − ?L? 2

1

+

+

3 − ?L? 2

3

+

+

∞ ∑ i =1 ∞ ∑ i =1

f i (L )2 − ?L? 2 + ic' L? +

EI?i4 L3

f i (0 )2

EI?i4 − ?L? 2 + ic' L? + L3 ∞ f i (L )f i (0 ) 1 3 − + ∑ H 12 (? )= 2 2 − ?L? − ?L? EI?i4 i =1 − ?L? 2 + ic' L? + L3 − ?L? 2

− ?L? 2

?x ?x ?i x ?i x    f i (x )= cosh i + cos i − s i  sinh L + sin L  L L   cosh?i − cos?i si = sinh?i − sin?i

[19]

[20]

For component B (the holder/spindle combination), only the direct FRFs at the connection coordinate (H33, L33, N33, and P33) are required. In this case, the direct FRF H33 was obtained by impact testing. A low mass, wide bandwidth accelerometer was placed at the free end of the collet HSK 63A holder and this location excited by a modally tuned impact hammer for both the X and Y coordinate directions. The Y direction experimental data is shown in Figure 9. The remaining FRFs (pertaining to moment and rotation) were assumed zero. The receptance coupling derivation for the tool point direct FRF, G11(ω), is similar to the example in section 2.1.3, except that damping terms and additional degrees of freedom have been added. The displacements, equilibrium conditions, and compatibility conditions are determined from the system model and combined to calculate the assembly FRFs. The result for G11 is shown in Equation 21 where the H′ ij, L′ ij, N′ ij, and P′ ij terms represent mobility FRFs [EWI 95], or the ratio of linear or rotational velocity to force or moment. The linear and rotational stiffness and damping terms are labeled kx, kθ, cx, and cθ, respectively.

-8

Real (m/N)

x 10 6 4 2 0 -2

Imag (m/N)

0 1000 -8 x 10 0

2000

3000

4000

5000

6000

2000 3000 4000 Frequency (Hz)

5000

6000

-5

-10 0

1000

Figure 9. Holder/spindle Y direction direct FRF (H33)  − 1  '  G11 = H 11 − H 12 E1− 1E2 − L12 E3 − 1  k N + c N  − E E E   ? 21 ? 21  4 1 2 

where,

[21]

−1 ' '  E1 =  k x H 33 + k x H 22 + cx H 33 + cx H 22 + [1]  − E3 E4   k L + k L + c L' + c L'   x 33 x 22 x 33 x 22    − 1 '    E2 = k x H 21 + cx H 21 − E3 k? N 21 + c? N'21       k L + k L + c L' + c L'   x 33 x 22 x 33 x 22    ' + [1]  '  E3 = k? P33 + k? P22 + c? P33 + c? P22    ' '   E4 = k? N33 + k? N 22 + c? N33 + c? N 22   

The analytic direct and cross FRFs for the tool and experimental holder/spindle direct FRF (note that no modal fit is necessary, experimental data is sufficient for this method) were then inserted in Equation 21 and the tool point dynamic response predicted. Experimental impact tests were also performed for each of the three selected tools. A comparison of the results follows. Figures 10-12 show the experimental and predicted, Y direction tool point FRFs (G11) for length to diameter ratios of 8:1, 9:1, and 10:1, respectively (please note changes in vertical scale). The overall agreement between the predicted and measured results is

good. However, small deviations are also seen. This is attributed to variations in collet torque between the three tool setups, imperfect knowledge of tool length/geometry, deviations in contact conditions between the collet and tool, and finite repeatability of the FRF measurement process. -6

Real (m/N)

x 10 4 0

Predicted

-2 -4 0 -6 x 10 0

Imag (m/N)

Measured

2

500

1000

1500

2000

500

1000 Frequency (Hz)

1500

2000

-2 -4 -6 -8 0

Figure 10. Experimental/predicted tool point FRF (8:1 tool) The transition from the 8:1 to 9:1 case results in the anticipated decrease in both natural frequency and stiffness. For the 10:1 tool, however, the single tool mode seen in the two previous results has been effectively split into two dynamically stiffer modes, providing an increase in stability. This is due to interaction of the cantilever tool mode with the approximately 727 Hz Y direction holder/spindle mode (shown in Figure 9). The analog to this situation is the dynamic absorber, where a small spring/mass is added to a larger vibrating system. The spring constant and mass of the added system are selected such that the natural frequency is equal to the excitation frequency of the larger structure and the vibration of the support structure is reduced, theoretically, to zero at the driving frequency. The selected linear and rotational spring and damping coefficients for the Y direction collet connection between the tool and holder/spindle for the three cases shown in Figures 10-12 are given in Table 2. It is anticipated that the system measured here (HSK 63A collet-type holder) represents a typical physical arrangement and, in general, a single set of values (with some associated uncertainty) may be used to represent the connection stiffness and damping for a particular holder. These values could be obtained in a single experimental setup (or determined from accurate modeling) and used for any tool/holder combination. In this case, a direct FRF

measurement was performed on a representative tool and the connection coefficients selected to provide a match between the predicted and measured responses. These parameters were then used to predict the responses shown in Figures 10-12. -5

Real (m/N)

x 10 1

Predicted

0 -1

Measured

Imag (m/N)

-2 0 -5 x 10 0

500

1000

1500

2000

500

1000 Frequency (Hz)

1500

2000

-1 -2 -3 0

Figure 11. Experimental/predicted tool point FRF (9:1 tool) -6

Real (m/N)

x 10

Predicted

5 0 -5

Measured

Imag (m/N)

-10 0 -6 x 10 0

500

1000

1500

2000

500

1000 Frequency (Hz)

1500

2000

-5 -10 -15 0

Figure 12. Experimental/predicted tool point FRF (10:1 tool)

Table 2. Stiffness/damping coefficients (11.8 mm diameter tooling) kx (N/m) 7

2.1x10

kθ (N m/rad) 6

1.4x10

cx(N s/m)

cθ (N m s/rad)

130

35

Figures 10-12 show that interaction between tool and holder/spindle modes can dramatically affect the tool point FRF. To illustrate the effect of this interaction on machining performance, stability lobe diagrams for slotting cuts in the machine Y direction have been developed for the 9:1 and 10:1 tools. These are shown in Figures 13 and 14. It is seen that the 10:1 cutter offers a significantly higher critical (or asymptotic) stability limit (maximum stable axial depth of cut at all spindle speeds) than the 9:1 cutter, as well as a shift in the location of the stability lobes. This implies two possible optimization parameters: 1) tool length selection for interaction with holder/spindle modes to increase the local stability limit, and 2) tool length selection to move a highly stable lobe to the top spindle speed of the machine (e.g., for a top spindle speed of 20,000 rpm, the 10:1 tool would be selected) [TLU 96,SMI 98].

Figure 13. Stability lobe diagram (9:1 tool) To further investigate the interaction of tool modes with holder/spindle modes, receptance coupling simulations were completed for 7:1 to 12:1 tools. The minimum real part of the FRF was recorded for each case and used to estimate the critical stability limit (bcritical) for Y direction slot milling in aluminum, according to Equation 22 [TLU 91]. In this equation, Ks is the specific cutting force for the work piece material (700x106 N/m2 [TLU 85]), µy is the ‘average’ direction orientation factor of the cutting force (cos 70° [TLU 85]), and z is the number of cutter teeth (2). The predicted slotting critical stability limits are plotted versus tool length in Figure 15.

Increased stability is recognized at tool frequencies which correspond to holder/spindle modes at approximately (1344, 1114, 727, and 540.5) Hz. Although the expected general trend of decreasing stability for increasing tool length is seen, significant local increases in stability due to the ‘dynamic absorber effect’ are also recognized. The improved stability offered by the 727 Hz mode interaction (10:1 tool) agrees with the empirical machining results shown in [DAV 98].

Figure 14. Stability lobe diagram (10:1 tool) −1 bcritical = K s µ y min(Re[G11( ω )])z

[22]

2.2.2 19.1 mm diameter tooling A receptance coupling analysis was also completed for a 19.1 mm diameter, 98.5 mm overhang tungsten carbide endmill secured in a shrink-fit HSK 63A holder. The analytic tool FRFs were derived and coupled to an experimental measurement of the holder/spindle combination in the X and Y coordinate directions. Similar to the previous case, a single experimental measurement of the assembly was used to determine the connection parameters (shown in Table 3), then these parameters were used to predict the system response in other configurations. Table 3. Stiffness/damping coefficients (19.1 mm diameter tooling) kx (N/m) 7

4.8x10

kθ (N m/rad) 6

3.2x10

cx(N s/m)

cθ (N m s/rad)

80

21

1

1344 Hz Mode

0.9

b

critical

(mm)

0.8 1114 Hz Mode

0.7 0.6 0.5 0.4

727 Hz Mode 540.4 Hz Mode

0.3 0.2 0.1 0

8:1 90

9:1 10:1 11:1 100 110 120 130 Tool Overhang Length (mm)

140

Figure 15. Critical stability limits vs. tool length Figures 16 and 17 display the predicted and measured tool point FRFs (G11 as shown in Figure 7) for the X and Y directions, respectively. Again, good qualitative agreement is seen with slight deviations due to uncertainties in the measurement procedure and system geometry. It is also recognized that the stiffer tool allows greater participation of the spindle modes in the tool point response, as opposed to the example in section 2.2.1 where the final response was dominated by the rather flexible cantilever tool mode. In both cases, however, it is the combined relationship of the tool and spindle modes that defines the system response. To further verify the receptance coupling method performance, direct and cross FRFs were measured on the tool/holder/spindle assembly at six axial locations (equally spaced points from the tool free end to holder face) in the Y direction and two dominant mode shapes determined (with approximate natural frequencies of 1208 and 1480 Hz, respectively). These experimental results are compared to mode shapes derived from receptance coupling predictions (i.e., G11, G12, G13, G14, G15, and G16) at the same spatial locations in Figure 18.

-7

Real (m/N)

6

x 10

Measured

4 2

Predicted

0 -2

Imag (m/N)

-4 0 -7 x 10 0

500

1000

1500

2000

1000 1500 Frequency (Hz)

2000

-2 -4 -6 -8 0

500

Figure 16. X direction tool point direct FRF

-7

Real (m/N)

6

x 10

4 2 0 -2 -4 0 -7 x 10 0

Imag (m/N)

Measured

Predicted 500

1000

1500

2000

1000 1500 Frequency (Hz)

2000

-2 -4 -6 -8 0

500

Figure 17. Y direction tool point direct FRF

Magnitude 1480 Hz (m/N) Magnitude 1208 Hz (m/N)

0

x 10

-7

-2 Predicted -4 -6 -8 0

Measured 0 x 10

20

40

60

80

100

80

100

-7

-2 -4 -6

0

20

40 60 Coordinate Location (mm)

Figure 18. Y direction mode shape comparison 3. Conclusions The application of receptance coupling substructure analysis to tool point FRF prediction was demonstrated. It was shown that analytic expressions for tool FRFs could be coupled with holder/spindle experimental FRFs to predict the assembly’s dynamic response. A tool tuning example, in which long, slender tools were coupled to a collet-type HSK 63A holder, verified the method. Experimental and predicted results showed dramatic variations in the tool point response as tool modes interacted with holder/spindle modes, similar to the effect seen in dynamic absorbers. Predicted local increases in the critical stability limit matched previous empirical results. Stability lobe diagrams were also generated and it was shown that two optimization criteria for proper tool length may be selected: 1) maximize the critical stability limit, and 2) maximize the chatter free axial depth of cut at the top available spindle speed. Further experimental verification was provided by comparisons between measured and predicted FRFs and mode shapes for shrink-fit tooling with higher dynamic stiffness. 4. Acknowledgements The authors gratefully acknowledge support from an NRC/NIST Postdoctoral Research Fellowship (for T. Schmitz). They would also like to thank Dr. Jon Pratt of NIST for helpful suggestions.

5. References [SMI 92] SMITH, S., TLUSTY, J., Stabilizing Chatter by Automatic Spindle Speed Regulation, Annals of the CIRP, Vol. 41/1, p. 433-436, 1992. [TLU 96] TLUSTY, J., SMITH, S., WINFOUGH, W., Techniques for the Use of Long Slender End Mills in High-Speed Machining, Annals of the CIRP, Vol. 45/1, p. 393-396, 1996. [SMI 90] SMITH, S., TLUSTY, J., Update on High-Speed Milling Dynamics, Transactions of the ASME Journal of Engineering for Industry, Vol. 112, p. 142-149, 1990. [DAV 96] DAVIES, M., BALACHANDRAN, B., Impact Dynamics in the Milling of ThinWalled Structures, ASME Nonlinear Dynamics and Controls, DE-Vol. 91, p. 67-72, 1996. [AGA 95] AGAPIOU, J., RIVIN, E., XIE, C., Toolholder/Spindle Interfaces for CNC Machine Tools, Annals of the CIRP, Vol. 44/1, p. 383-387, 1995. [SCH 92] SCHULTZ, H., MORIWAKI, T., High-Speed Machining, Annals of the CIRP, Vol. 41/2, p. 637-643, 1992. [HEI 96] HEISEL, U., GRINGEL, M., Machine Tool Design Requirements for High-Speed Machining, Annals of the CIRP, Vol. 45/1, p. 389-392, 1996. [DAV 98] DAVIES, M., DUTTERER, B., PRATT, J., SCHAUT, A., On the Dynamics of High-Speed Milling with Long, Slender Endmills, Annals of the CIRP, Vol. 47/1, p. 55-60, 1998. [WIN 95]WINFOUGH, W., SMITH, S., 1995, Automatic Selection of the Optimum Metal Removal Conditions for High-Speed Milling, Transactions of the NAMRI/SME, Vol. 23, p. 163-168. [WEC 94] WECK, M., SCHUBERT, I., New Interface Machine/Tool: Hollow Shank, Annals of the CIRP, Vol. 43/1, p. 345-348, 1994. [WEC 99] WECK, M., HENNES, N., KRELL, M., Spindle and Toolsystems with High Damping, Annals of the CIRP, Vol. 48/1, p. 297-302, 1999. [DEL 92] DELIO, T., TLUSTY, J., SMITH, S., Use of Audio Signals for Chatter Detection and Control, Transactions of the ASME Journal of Engineering for Industry, Vol. 114, p. 146157, 1992. [SMI 91] SMITH, S., TLUSTY, J., An Overview of Modeling and Simulation of the Milling Process, Transactions of the ASME Journal of Engineering for Industry, Vol. 113, p. 169-175, 1991. [TLU 83] TLUSTY, J., ZATON, W., ISMAIL, F., Stability Lobes in Milling, Annals of the CIRP, Vol. 32/1, p. 309-313, 1983. [ALT 95] ALTINTAS, Y., BUDAK, E., Analytical Prediction of Stability Lobes in Milling, Annals of the CIRP, Vol. 44/1, p. 357-362, 1995. [TLU 81] TLUSTY, J., ISMAIL, F., Basic Non-linearity in Machining Chatter, Annals of the CIRP, Vol. 30/1, p. 299-304, 1981. [KOE 67] KOENISBERGER, F., TLUSTY, J., Machine Tool Structures-Vol. I: Stability Against Chatter, Pergamon Press, 1967. [TOB 65] TOBIAS, S., Machine Tool Vibration, Blackie and Sons, Ltd., 1965. [MER 65] MERRIT, H., Theory of Self-Excited Machine Tool Chatter, Transaction of the ASME Journal of Engineering for Industry, Vol. 87, p. 447-454, 1965. [MIN 90] MINIS, I., YANUSHEVSKY, T., TEMBO, R., HOCKEN, R., Analysis of Linear and Nonlinear Chatter in Milling, Annals of the CIRP, Vol. 39, p. 459-462 1990. [TLU 91] TLUSTY, J., SMITH, S., ZAMUDIO, C., Evaluation of Cutting Performance of Machining Centers, Annals of the CIRP, Vol. 40/1, p. 405-410, 1991.

[EWI 95] EWINS, D., Modal Testing: Theory and Practice, Research Studies Press, Ltd., Somerset, England, 1995. [SMI 99] SMITH, S., JACOBS, T., HALLEY, J., The Effect of Drawbar Force on Metal Removal Rate in Milling, Annals of the CIRP, Vol. 48/1, p. 293-296, 1999. [FER 95] FERREIRA, J., EWINS, D., Nonlinear Receptance Coupling Approach Based on Describing Functions, Proceedings of the 14th International Modal Analysis Conference, Dearborn, MI, p. 1034-1040, 1995. [FER 96] FERREIRA, J., Transfer Report on Research Dynamic Response Analysis of Structures with Nonlinear Components, Internal Report- Dynamics Section, Imperial College, London, UK, 1996. [REN 93] REN, Y., BEARDS, C., A Generalized Receptance Coupling Technique, Proceedings of the 11th International Modal Analysis Conference, Kissimmee, FL, p. 868-871, 1993. [KLO 77] KLOSTERMAN, A., MCCLELLAND, W., SHERLOCK, I., Dynamic Simulation of Complex Systems Utilizing Experimental and Analytical Techniques, ASME, 75-WA/Aero9, 1977. [THO 98] THOMSON, W., DAHLEH, M., Theory of Vibration with Applications 5th Ed., Prentice-Hall, Upper Saddle River, NJ, 1998. [BLE 79] BLEVINS, R., Formulas for Natural Frequency and Mode Shape, Van Nostrand Reinhold Co., New York, NY, 1979. [TLU 90] TLUSTY, J., ISMAIL, F., Dynamic Structural Identification Tasks and Methods, Annals of the CIRP, Vol. 29/1, p. 251-255, 1990. [TLU 85] TLUSTY, J., Dynamics of High-Speed Milling, Handbook of High-Speed Machining Technology, R. I. King, ed., Chapman and Hall, New York, p. 48-153, 1985. [SMI 98] SMITH, S., WINFOUGH, W., HALLEY, J., The Effect of Tool Length on Stable Metal Removal Rate in High-Speed Milling, Annals of the CIRP, Vol. 47/1, p. 307-310, 1998.