Tony L. Schmitz1 e-mail: tschmitz@ufl.edu G. Scott Duncan Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611 Three-Component Receptance Coupling Substructure Analysis for Tool Point Dynamics Prediction In this paper we present the second generation receptance coupling substructure analysis (RCSA) method, which is used to predict the tool point response for high-speed machining applications. This method divides the spindle-holder-tool assembly into three substructures: the spindle-holder base; the extended holder; and the tool. The tool and extended holder receptances are modeled, while the spindle-holder base subassembly receptances are measured using a “standard” test holder and finite difference calculations. To predict the tool point dynamics, RCSA is used to couple the three substructures. Experimental validation is provided. 关DOI: 10.1115/1.2039102兴 Keywords: high-speed machining, milling, stability, beam, finite element 1 Introduction One area of manufacturing research that has made significant technological advancements in recent years is high-speed machining. Machine improvements include new spindle designs for higher rotational speed, torque, and power; increased slide speeds and accelerations; direct drive linear motor technology; and new machine designs for lower moving mass. The combination of new machine technology and tool material/coating developments often makes high-speed machining a viable alternative to other manufacturing processes. A key application example is the aerospace industry, where dramatic increases in material removal rates made possible using high-speed machining techniques have allowed designers to replace assembly-intensive sheet metal build-ups with monolithic aluminum components resulting in substantial cost savings 关1兴. A primary obstacle to the successful implementation of highspeed machining and full use of the available technology is chatter, or unstable machining. Many research efforts geared toward the understanding and avoidance of chatter have been carried out 共e.g., see early studies in 关2–11兴兲. This work has led to the development of stability lobe diagrams that identify stable and unstable cutting zones as a function of the chip width and spindle speed. However, the methods used to produce these diagrams, whether analytic or time-domain, require knowledge of the tool point dynamics. The required dynamic model is typically obtained using impact testing, where an instrumented hammer is used to excite the tool at its free end 共i.e., the tool point兲 and the resulting vibration is measured using an appropriate transducer, typically a low mass accelerometer. However, due to the large number of spindle, holder, and tool combinations, the required testing time can be significant. Therefore, a model which is able to predict the tool point response based on minimum input data is the preferred alternative. The purpose of this paper is to build on the previous work of Schmitz et al. 关12–15兴, which describes the tool point frequency response function, or receptance, prediction using the receptance coupling substructure analysis 共RCSA兲 method. In these previous 1 Author to whom correspondence should be addressed. Contributed by the Manufacturing Engineering Division for publication in the ASME JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received April 6, 2004; final revision received February 4, 2005. Review conducted by: Y. C. Shin. studies, a two component model of the machine-spindle-holdertool assembly was defined. The machine-spindle-holder displacement-to-force receptance was recorded using impact testing, while the tool was modeled analytically. The tool and machine-spindle-holder substructure receptances were then coupled through translational and rotational springs and dampers; see the model in Fig. 1, where kx and k␪ are the translational and rotational springs, cx and c␪ are the translational and rotational viscous dampers, component A represents the tool, and component B the machine-spindle-holder. While the purpose of the springs and dampers between the tool and holder was to capture the effects of a potentially nonrigid, damped connection, it is likely that these connections also served to compensate for the fact that the displacement-to-moment, rotation-to-force, and rotation-tomoment receptances at the free end of the holder were assumed zero 共i.e., perfectly rigid兲. Although it was shown in Ref. 关15兴 that this two component model provides a valid approximation for a flexible tool clamped in a stiff spindle-holder, it does not offer the most generalized solution. In order to enable RCSA predictions for a wider variety of machine-spindle-holder-tool combinations, an improved threecomponent model is presented here. In this model, the machinespindle-holder substructure is separated into two parts: 共1兲 the machine, spindle, holder taper, and portion of the holder nearest the spindle with standard geometry from one holder to another 共hereafter referred to as the spindle-holder base subassembly兲; and 共2兲 the remaining portion of the holder from the base to the free end 共hereafter referred to as the extended holder subassembly兲. A technique for determining the rotation-to-force/moment and displacement-to-moment receptances for the free end of the spindle-holder base subassembly using only displacement-to-force measurements is also described. The experimental procedure involves direct and cross displacement-to-force measurements of a simple geometry “standard” holder clamped in the spindle to be modeled. The portion of the standard holder beyond the section with consistent geometry from holder-to-holder is then removed in simulation using an inverse receptance coupling approach 共i.e., decomposition兲 to identify the four spindle-holder base subassembly receptances. These receptances are then coupled to models of the actual holder and tool. In the following sections, the method is described and experimental validation is presented. Journal of Manufacturing Science and Engineering Copyright © 2005 by ASME NOVEMBER 2005, Vol. 127 / 781 Fig. 1 Previous two-component RCSA model. An external force, Fa„t…, is applied to the free end of the tool „A… to determine the assembly Xa / Fa receptance. The tool is coupled to the machine-spindle-holder „B… through springs and dampers 2 Background and Notation Substructure analysis, or component mode synthesis, methods have been used for several decades to predict the dynamic response of complicated assemblies using measurements and/or models of the individual components, or substructures. These components can be represented by spatial mass, stiffness, and damping data, modal data, or receptances 共e.g., 关16–30兴兲. The latter representation is preferred in situations where the assembly receptances are the desired analysis output, as is the case in this research. For an assembly consisting of two rigidly connected substructures, as shown in Fig. 2, the assembly receptance, G jk共␻兲, can be expressed as shown in Eq. 共1兲, where ␻ is the frequency, X j and ⍜ j are the assembly displacement and rotation at coordinate j, and Fk and M k are the force and moment applied to the assembly at coordinate k. If coordinate j is coincident with coordinate k, the receptance is referred to as a direct receptance; otherwise, it is a cross receptance. For the purposes of this paper, the nomenclature G jk共␻兲 is used to describe the receptances that are produced when two substructures 共or subassemblies兲 are coupled to produce the final assembly. The nomenclature GS jk共␻兲 will replace G jk共␻兲 in all relevant equations when two substructures 共or subassemblies兲 are coupled that do not form the final assembly. G jk共␻兲 = 冋 册 Xj Fk Xj Mk ⍜j Fk ⍜j Mk = 冋 H jk L jk N jk P jk 册 R jk共␻兲 = 冋 册冋 ␪j fk xj mk ␪j mk = h jk l jk n jk p jk 册 共2兲 Based on the coordinates defined in Fig. 2, the equations to determine the assembly direct receptances, Gaa共␻兲 and Gdd共␻兲, and the assembly cross receptances, Gad共␻兲 and Gda共␻兲, can be written as a function of the substructure receptances as shown in Eqs. 共3兲–共6兲, where rigid connections have been applied 关32兴. Fig. 2 Two-component assembly. The component responses are coupled through a rigid connection to give the assembly receptance„s… 782 / Vol. 127, NOVEMBER 2005 Gaa共␻兲 = Gdd共␻兲 = 共1兲 The substructure receptances, R jk共␻兲, are defined in Eq. 共2兲, where x j and ␪ j are the substructure displacement and rotation at coordinate j, and f k and mk are the force and moment applied to the substructure at coordinate k 关15,31兴. xj fk Fig. 3 Example standard holder for spindle-holder base subassembly receptance identification „dimensions provided in Table 1…. Hammer impacts are completed at locations 3, 3b, and 3c to identify the required direct and cross receptances 冋 册 冋 册 冋 册 冋 册 Xa Fa ⍜a Fa Xd Fd ⍜d Fd Xa Ma ⍜a Ma Xd Md ⍜d Md = Raa共␻兲 − Rab共␻兲关Rbb共␻兲 + Rcc共␻兲兴−1Rba共␻兲 共3兲 = Rdd共␻兲 − Rdc共␻兲关Rbb共␻兲 + Rcc共␻兲兴−1Rcd共␻兲 共4兲 Gad共␻兲 = Xa Fd ⍜a Fd Xa Md ⍜a Md = Rab共␻兲关Rbb共␻兲 + Rcc共␻兲兴−1Rcd共␻兲 共5兲 Gda共␻兲 = Xd Fa ⍜d Fa Xd Ma ⍜d Ma = Rdc共␻兲关Rbb共␻兲 + Rcc共␻兲兴−1Rba共␻兲 共6兲 As noted, in order to populate the substructure receptance matrices, we apply measurement and modeling. Common modeling options include closed-form expressions for uniform EulerBernoulli beams 关33兴 and finite element solutions 共which can incorporate the more accurate Timoshenko beam model 关34兴兲. We consider both approaches in this study. As a convenience to the reader, the relevant analytical formulas and finite element Timoshenko stiffness and mass matrices are included in the Appendix. 3 Spindle-Holder Base Subassembly Identification The experimental procedure used to determine the receptances at the free end of the spindle-holder base subassembly, GS jk共␻兲, is described in this section. It is composed of three primary steps. First, the standard holder displacement-to-force direct and cross receptances are determined by impact testing. The standard holder geometry, which was selected to approximate a broad range of potential holders, is provided in Fig. 3. Second, these results are used to determine the three other direct receptances at the free end of the standard holder. Third, the section of the standard holder which is not common to other holders 共see Fig. 4兲 is removed using inverse receptance coupling to determine all four spindleholder base subassembly receptances. Each step of the procedure is described in the following sections. Example results are included. 3.1 Standard Test Holder Receptances Once the standard holder is mounted in a spindle 共see Fig. 3兲, the four subassembly Transactions of the ASME Table 1 Standard holder substructure parameters Coordinate j Coordinate k do 共mm兲 L 共mm兲 ␳ 共kg/ m3兲 E 共N / m2兲 GS34共␻兲 = receptances are determined by measuring the direct, H33, and cross, H33b, and H33c, displacement-to-force receptances on the standard holder, applying a second-order backward finite difference method to find L33 共and, equivalently, N33兲 关35兴, and then synthesizing P33. For the cross displacement-to-force measurements, the distance S should be selected to increase the difference in relative amplitudes between H33, H33b, and H33c without leading to a poor signal-to-noise ratio for the H33c measurement 共i.e., many of the lower frequency spindle-holder modes resemble a fixed-free fundamental mode shape and have very small amplitudes near the spindle face for the bandwidth of interest兲. Practically, we have observed that the finite difference results improve as S is increased; however, care must be taken to ensure that the location of the H33c measurement provides sufficient signal-tonoise. The receptance L33 is determined from the measured displacement-to-force receptances using Eq. 共7兲. By reciprocity, N33 can be set equal to L33. The remaining receptance, P33, is synthesized from H33, L33, and N33, as shown in Eq. 共8兲 关27兴. L33 = 3H33 − 4H33b + H33c 2S L332 1 ⍜3 F3 X3 ⍜3 P33 = = = L33N33 = M 3 X3 M 3 F3 H33 H33 共7兲 3.2 Extended Holder Subassembly Model. The extended holder subassembly for the steel standard holder consisted of solid, cylindrical substructures I and II as shown in Fig. 4. Equations 共9兲–共12兲 provide the direct and cross extended holder subassembly receptance matrices, where rigid coupling has been applied. These equations were determined from Eqs. 共3兲–共6兲 by appropriate substitutions. GS33共␻兲 = GS44共␻兲 = 冋 册 X3 M3 ⍜3 M3 冋 册 X4 F4 ⍜4 F4 X4 M4 ⍜4 M4 = R33共␻兲 − R33a共␻兲关R3a3a共␻兲 + R3b3b共␻兲兴−1R3a3共␻兲 共9兲 = R44共␻兲 − R43b共␻兲关R3b3b共␻兲 + R3a3a共␻兲兴−1R3b4共␻兲 Journal of Manufacturing Science and Engineering 冋 册 冋 册 共10兲 X3 M4 ⍜3 M4 X3 F4 ⍜3 F4 X4 M3 ⍜4 M3 X4 F3 ⍜4 F3 3 3a 63.3 62.8 3b 4 52.7 16.3 = R33a共␻兲关R3a3a共␻兲 + R3b3b共␻兲兴−1R3b4共␻兲 共11兲 = R43b共␻兲关R3a3a共␻兲 + R3b3b共␻兲兴−1R3a3共␻兲 共12兲 3.3 Spindle-Holder Base Subassembly Receptance. The spindle-holder base subassembly receptance matrix, G33共␻兲, can be expressed as shown in Eq. 共13兲 by rewriting Eq. 共3兲. The lefthand side of this equation is known once the steps described in Sec. 3.1 are completed. Also, the extended holder subassembly receptances, GS33, GS44, GS34, and GS43, are determined using the equations provided in Sec. 3.2. Therefore, Eq. 共13兲 can be rewritten to solve for the spindle-holder base subassembly receptances, GS55共␻兲. See Eq. 共14兲. G33共␻兲 = 冋 共8兲 Due to the subtraction of the similarly scaled H33, H33b, and H33c receptances, noise in the measurement data can detrimentally affect the quality of L33 and N33 共produced by the finite-difference method兲 and, therefore, P33. To reduce the noise effect, the measured receptance data were smoothed using a Savitzky-Golay filter, which performs a local polynomial regression to determine the smoothed value for each data point 关36兴, prior to the application of Eq. 共7兲. For this study, filters with polynomial orders of two or three were applied over windows of 31 to 81 data points. X3 F3 ⍜3 F3 GS43共␻兲 = II 7800 2 ⫻ 1011 0.0015 ␩ Fig. 4 Standard holder substructures for inverse receptance coupling I GS55共␻兲 = H33 L33 N33 P33 册 冋 册 x5 f5 ␪5 f5 x5 m5 ␪5 m5 = GS33共␻兲 − GS34共␻兲关GS44共␻兲 + GS55共␻兲兴−1GS43共␻兲 共13兲 = GS34共␻兲关GS33共␻兲 − G33共␻兲兴−1 ⫻GS43共␻兲 − GS44共␻兲 共14兲 Tests were completed to determine GS55共␻兲 for a 24,000 rpm/ 40 kW direct drive spindle 共HSK 63A interface兲 using a steel standard holder. The dimensions and material properties for the standard holder substructures are provided in Table 1, where do is the diameter, L is the length, ␳ is the density, and ␩ is the frequency-independent damping coefficient. The ␩ values used in this study were determined experimentally from free-free testing of representative cylindrical rods. During the measurement of the direct and cross receptances for the mounted standard holder, the distance S was selected as 25.40 mm. The resulting spindle receptances, h55, l55, and p55, are shown in Fig. 5. These results are based on the average of 15 complete measurement sets 共H33, H33b, and H33c—each the average of ten impacts兲. Our experience has shown that averaging is the most effective technique for reducing the inherent noise amplification during the finite difference computations. As shown in Eq. 共14兲, the Fig. 5 result was determined by removing the extended holder subassembly for the standard holder from the complete assembly in simulation. Because the measurement bandwidth for high-speed/high-power spindle testing is typically 5 kHz or less, we have found that it makes no practical difference whether the Euler-Bernoulli or Timoshenko beam model is used to describe the standard holder substructure共s兲. The standard holder behaves basically as an inertial mass since its NOVEMBER 2005, Vol. 127 / 783 Fig. 7 Measured „two nominally identical holders… and predicted H33 results for tapered thermal shrink fit holder „25.3 mm bore… Fig. 5 Spindle receptances G55„␻… determined from standard holder direct and cross receptance measurements clamped-free bending mode fundamental natural frequency, for the geometry used in this study, is outside the bandwidth of interest. 3.4 Holder Experimental Verification. Once the 24,000 rpm/ 40 kW spindle-holder base subassembly receptances, GS55共␻兲, were determined, it was possible to couple this result to arbitrary holder geometries to predict the receptance at any coordinate on the machine-spindle-holder assembly. To validate the procedure, a tapered thermal shrink fit holder 共25.3 mm bore兲 with an HSK 63A spindle interface was divided into 12 substructures beyond the spindle-holder base subassembly as shown in Fig. 6. Each substructure was assumed to be a hollow or solid cylindrical steel beam, as appropriate. Table 2 provides the holder geometry and assumed material properties. The first step in predicting the assembly response, as described in Sec. 3.2, was to couple substructures I–XII to produce the direct and cross extended holder subassembly receptances at coordinates 3 and 4. With the increase in substructures from 2 to 12, the Sec. 3.2 procedure remained the same; however, substructure I was first coupled to substructure II, then the resulting subassembly was coupled to substructure III, and so on to produce the required extended holder subassembly receptances. The next step was to rigidly couple the spindle-holder base subassembly 共determined in the previous section and shown in Fig. 5兲 to the extended holder subassembly using Eq. 共13兲 to determine the receptances at the free end of the holder, G33共␻兲. Figure 7 shows the predicted H33 result as well as measurements for two nominally identical holders. The Euler-Bernoulli beam model was applied to develop the extended holder receptances in this case. 4 Tool Point Response Prediction To predict the tool point dynamics, the modeling procedure was again applied to the 24,000 rpm/ 40 kW spindle 共HSK 63A interface兲 using a tapered thermal shrink holder with a 19.1 mm carbide tool blank inserted as shown in Fig. 8. The assembly was divided into the spindle-holder base subassembly and 13 cylindrical substructures of differing diameters; see Table 3. To model the receptances, a composite modulus and mass were substituted for substructures II–VIII to account for the material differences between the steel holder and the carbide tool blank. Also, the mass expression for these substructures 共provided in the Appendix兲 was replaced with the composite mass shown in Eq. 共15兲, where ␳h and ␳t are the density of the holder and tool, respectively. Additionally, the product of the elastic modulus and second area moment of inertia, EI, was replaced by the product shown in Eq. 共16兲, where Eh is the holder modulus, Et is the tool material modulus, and Ih and It are the second area moments of inertia for the holder and tool, respectively. The substructure parameters are shown in Table 3. Fig. 6 Tapered thermal shrink fit holder „25.3 mm bore… substructure model Table 2 Tapered thermal shrink fit holder „25.3 mm bore… substructure parameters Substructure I II III IV V VI VII VIII IX X XI XII di 共mm兲 do 共mm兲 L 共mm兲 ␳ 共kg/ m3兲 E 共N / m2兲 25.3 44.2 5.5 25.3 45.1 5.5 25.3 46.1 5.5 25.3 47.0 5.5 25.3 47.9 5.5 25.3 48.9 5.5 25.3 49.8 5.5 26.0 50.7 5.5 26.0 51.7 5.5 26.0 52.6 5.5 26.0 52.6 15.7 52.6 30.3 ␩ 784 / Vol. 127, NOVEMBER 2005 7800 2 ⫻ 1011 0.0015 Transactions of the ASME Fig. 8 Tapered thermal shrink fit holder with 19.1-mm-diam tool blank substructure model m= ␲共␳h共d2o − di2兲 + ␳tdi2兲L 4 EI = EhIh + EtIt = 共15兲 Eh␲共d4o − di4兲 + Et␲di4 64 共16兲 The next step was to rigidly couple substructures I through XIII to produce the direct and cross extended holder-tool subassembly receptances at coordinates 1 and 4. The final step in the procedure was to predict the tool point dynamics by rigidly coupling the extended holder-tool subassembly to the spindle-holder base subassembly. With the appropriate coordinate substitution in Eq. 共13兲, the tool point receptance, G11共␻兲, was determined according to Eq. 共17兲, where the receptances associated with coordinates 1 and 4 are the extended holder-tool subassembly direct and cross receptances. The predicted and measured assembly tool point displacement-to-force receptances, H11, are displayed in Fig. 9. In this figure, results for both Euler-Bernoulli and Timoshenko 共finite element兲 beam models are provided. It is seen that the finite element model 共100 elements were used for each substructure兲 dominant natural frequency is closer to the measured result, as expected. However, the predicted natural frequency is still approximately 50 Hz higher. This disagreement is explored in Sec. 5.3. G11共␻兲 = 冋 H11 L11 N11 P11 册 = GS11共␻兲 − GS14共␻兲关GS44共␻兲 + GS55共␻兲兴−1GS41共␻兲 5 共17兲 Case Studies 5.1 Geared Quill-Type Spindle. In this section, prediction and measurement results are provided for two cutters coupled to a geared, quill-type spindle with a CAT-50 spindle-holder interface 共Big-Plus tool holders were used which include both taper and face contact兲. The spindle-holder base subassembly receptances were determined using a steel cylindrical standard holder 共63.4 mm diameter and 89.0 mm length兲; the cross FRF measurements were again recorded at distances of 25.4 mm and 50.8 mm from the free end of the standard holder. The substructure receptances for the solid body tools 共i.e., both cutting tools were composed of solid steel modular bodies with carbide inserts attached兲 were then computed and the tool point FRF predicted by rigidly coupling the tool models to the spindle measurements. Figure 10 displays the H11 results for an inserted endmill with 4 “flutes” 共20 total inserts兲. The tool body geometry is defined in Table 4 共as before substructure I is nearest the free end of the clamped cutter兲. Figure 11 shows the H11 measurement and prediction for a 28-insert facemill 共see Table 5兲. In both cases, EulerBernoulli beam models were employed to describe the standard holder and cutter bodies. 5.2 Geared Spindle Comparison. In this section, the spindle-holder base subassembly receptances were measured on two nominally identical, geared spindles 共CAT-50 holder-spindle interface兲. The steel cylindrical standard holder was 63.4 mm in diameter and 89.0 mm long. The cross FRF measurement locations were the same as specified previously. Figure 12 provides standard holder direct FRF measurement results for both spindles. Three curves are shown: the solid line 共line 1a兲 represents the average of five measurement sets 共10 impacts each兲 completed without removing the holder from the first spindle 共i.e., spindle 1兲; the dotted line gives the average of three more spindle 1 measurements after removing and replacing the holder 共line 1b兲; and the dashed line shows the average of five spindle 2 measurements 共line 2兲. These curves show that, although the spindles are similar, the difference between the spindle dynamics is larger than the measurement divergence. Next, a 16-insert solid body facemill was inserted in spindle 1 and the tool point FRF recorded. Predictions were finally completed using both the spindle 1 and 2 receptances. This result is provided in Fig. 13; the facemill geometry and material properties are given in Table 6. It is seen that the prediction completed using the spindle 1 receptances 共dashed line兲 more accurately identifies the spindle 1 measured frequency content 共solid line兲. Therefore, it would be necessary to measure both spindles to make accurate predictions, rather than relying on manufacturing repeatability. It has been our experience that the dynamic consistency between spindles is manufacturer-dependent. 5.3 Shrink Fit Holder With Varying Tool Length. In this study 30 carbide tool blanks were sequentially inserted in a tapered thermal shrink fit holder and the tool point response recorded. The insertion length was maintained at 22.9 mm while the overhang length varied from 66.0 to 142.2 mm in increments of 2.5 mm 共the 139.7 overhang length test was not completed兲 for the 19.1-mm-diam tool blanks. These measurements were completed on a 16,000 rpm direct drive spindle with an HSK 63A spindle-holder interface. The substructure information is provided in Table 7. The 30 measurement results are shown in the top panel of Fig. 14, while the bottom panel shows the h55 spindle response Table 3 Tapered thermal shrink fit holder and 19.1 mm diameter tool blank substructure parameters Substructure I II III IV di 共mm兲 do 共mm兲 L 共mm兲 ␳ 共kg/ m3兲 E 共N / m2兲 19.1 111.9 19.1 33.4 5.8 19.1 34.4 5.8 19.1 35.4 5.8 ␩ V VI VII VIII IX 19.1 19.1 19.1 19.1 19.1 36.4 37.5 38.5 39.5 39.5 5.8 5.8 5.8 5.8 4.1 7800 共steel holder兲 14,500 共carbide tool blank兲 2 ⫻ 1011 共steel holder兲 5.85⫻ 1011 共carbide tool blank兲 0.0015 Journal of Manufacturing Science and Engineering X XI XII XIII 19.1 40.4 4.1 19.1 41.4 4.1 19.1 41.4 10.6 41.4 37.4 NOVEMBER 2005, Vol. 127 / 785 Fig. 9 Measured and predicted H11 results for tapered thermal shrink fit holder with 19.1-mm-diam tool blank „111.9 mm overhang length… Fig. 11 Measured and predicted H11 results for 28-insert facemill 共i.e., after removing the extended portion of the standard holder in simulation兲. It is seen in the top panel that, although the general trend is increased amplitude and reduced frequency with increasing overhang length, the tool point magnitudes are attenuated near 800 and 1200 Hz. This is due to dynamic interaction between the tool clamped-free mode and the spindle modes 关37兴. The fact that the spindle natural frequencies agree with the locations of the dynamic interactions 共see bottom panel of Fig. 14兲 suggests that the spindle response has been properly identified. Predictions of the tool point responses using the spindle receptances and Timoshenko beam elements 共100 for each substructure兲 to model the tool and holder showed similar disagreement in natural frequency to the results provided in Fig. 9. Reasonable perturbations to the model parameters were unable to close the approximately 50 Hz gap. Therefore, translational and rotational springs and viscous dampers 共as shown in Fig. 1兲 were inserted between the holder and tool to account for what was presumed to be a nonrigid connection 共even for the shrink fit test case studied here兲. The spring and damper values were then determined using a nonlinear least-squares best fit 关15兴. The least-squares algorithm was initiated using connection parameters obtained from a visual fit and continued until the frequency-dependent residual between the predicted and measured H11 results was less than 1 ⫻ 10−15 m / N. The four parameter values were constrained to be zero or greater, but no other restrictions were applied. The average values for the connection parameters 共see Table 8兲 were then used to make predictions for various overhang lengths. The predictions were carried out using Eq. 共18兲, where K= Table 5 Solid parameters 冋 kx + i␻cx 0 0 k␪ + i␻c␪ body facemill „28 册 . inserts… substructure Substructure I II III IV do 共mm兲 L 共mm兲 ␳ 共kg/ m3兲 E 共N / m2兲 126.2 55.0 130.3 18.3 80.0 62.7 69.9 18.3 ␩ 7800 2 ⫻ 1011 0.0015 Fig. 10 Measured and predicted H11 results for 20-insert endmill Table 4 Solid parameters body endmill „20 inserts… substructure Substructure I II III do 共mm兲 L 共mm兲 ␳ 共kg/ m3兲 E 共N / m2兲 99.8 85.6 80.1 94.9 7800 2 ⫻ 1011 0.0015 69.9 16.8 ␩ 786 / Vol. 127, NOVEMBER 2005 Fig. 12 Standard holder direct receptances for two nominally identical, geared spindles „CAT-50 holder-spindle interface…. Line 1a „solid… shows the average of five measurement sets completed without removing the holder from spindle 1; line 1b „dotted… gives the average of three more spindle 1 measurements after removing and replacing the holder; line 2 „dashed… shows the average of five spindle 2 measurements Transactions of the ASME Fig. 13 Measured and predicted H11 results for 16-insert facemill. Results are shown for predictions from spindle 1 „dashed… and spindle 2 „dotted… standard holder measurements. Measurement recorded using spindle 1 G11共␻兲 = 冋 H11 L11 N11 P11 册 = GS11共␻兲 − GS14共␻兲关GS44共␻兲 + GS55共␻兲 + K−1兴GS41共␻兲 共18兲 Figure 15 shows the measured and predicted results for four different overhang lengths. These lengths were selected to provide results: 共1兲 near the 1200 Hz interaction frequency shown in Fig. 14 共76.2 mm兲; 共2兲 between the interactions at 800 and 1200 Hz 共94.0 mm兲; 共3兲 near the 800 Hz interaction 共106.7 mm兲; and 共4兲 to the left of the 800 Hz interaction 共132.1 mm兲. Reasonable agreement is observed in all cases. To determine the impact of the residual disagreement, however, stability lobes were constructed using the 94.0 mm overhang case for both the measured and predicted tool point receptances 关38兴. A 50% radial immersion upmilling cut using a four-flute cutter with cutting force coefficients of 800 N / mm2 and 0.3 was assumed for demonstration purposes. This result is provided in Fig. 16. Although there is a shift toward lower speeds for the lobes computed using the predicted receptance 共due to the underprediction of the natural frequency兲, the diagram does not exhibit extreme sensitivity to this frequency error. Based on this result, while the use of finite connection stiffness values, i.e., a nonzero K−1 matrix in Eq. 共18兲, may improve the receptance prediction accuracy, a rigid connection appears to be adequate to guide the selection of stable cutting conditions provided points near the stability boundaries are not chosen. Table 6 Solid parameters body Fig. 14 Measurement results for thermal shrink fit tool holdertool blank case study. „Top panel… 30 different carbide blanks were sequentially inserted and the tool point receptance recorded. „Bottom panel… The spindle displacement-to-force receptance identified using the standard holder facemill „16 inserts… substructure Substructure I II III do 共mm兲 L 共mm兲 ␳ 共kg/ m3兲 E 共N / m2兲 279.4 27.2 63.5 88.9 7800 2 ⫻ 1011 0.0015 69.9 15.9 ␩ 6 Conclusions Tool point dynamics prediction using the second generation RCSA method was demonstrated. The improved method includes the following features: 共1兲 separation of the spindle-holder-tool assembly into three substructures—the spindle-holder base, extended holder, and tool; 共2兲 experimental identification of the spindle-holder base subassembly translational and rotational receptances using a finite difference approach; 共3兲 analytical and finite element modeling of the holder and tool substructure receptances; and 共4兲 rigid coupling of the spindle-holder base subassembly to the extended holder and rigid or flexible/damped coupling of the tool to this result to determine the tool point response. Experimental validation of the method was provided for multiple spindle-holder-tool setups. Acknowledgments This work was partially supported by the National Science Foundation 共Grant No. DMI-0238019兲, the Office of Naval Research 共2003 Young Investigator Program兲, the Naval Surface Warfare Center—Carderock Division, and BWXT Y-12. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of these agencies. The authors also wish to acknowledge contributions to the development of the RCSA method by Dr. M. Davies, University of North Carolina-Charlotte, Charlotte, NC, and Dr. T. Burns, National Institute of Standards and Technology, Gaithersburg, MD. They also acknowledge Mr. R. Ketron, Caterpillar, Inc., Aurora, IL, Ms. J. Dyer, Eastside High School, Gainesville, FL, Mr. Duke Hughes, BWXT Y-12, Oak Ridge, TN, and Dr. P. Jacobs, BWXT Y-12, for their assistance in collecting portions of the data used in this study. Table 7 Shrink fit holder case study substructure parameters Substructure I di 共mm兲 do 共mm兲 L 共mm兲 ␳ 共kg/ m3兲 E 共N / m2兲 19.1 Varied ␩ II III IV V 19.1 19.1 19.1 21 35.0 36.1 37.3 38.2 11.4 11.4 9.1 25.0 7800 共steel holder兲 14,500 共carbide tool blank兲 2 ⫻ 1011 共steel holder兲 5.85⫻ 1011 共carbide tool blank兲 0.0015 Journal of Manufacturing Science and Engineering VI 6 38.5 17.0 NOVEMBER 2005, Vol. 127 / 787 Table 8 Connection parameters for shrink fit holder case study kx共N / m兲 k␪共N / rad兲 cx共N s / m兲 c␪共N s / rad兲 6.5⫻ 107 3.4⫻ 106 520 3540 Fig. 16 Example stability lobes „50% radial immersion upmilling cut using a four-flute cutter with cutting force coefficients of 800 N / mm2 and 0.3… developed using measured „solid line… and predicted „dotted line… H11 results for 94.0 mm overhang length Fig. 15 Measured and predicted H11 results for four different overhang lengths „132.1, 106.7, 94.0, and 76.2 mm…. The overhang length for each of the four results is identified. Predictions were completed using the flexible/damped connection „connection parameters are provided in Table 8… l jj = − lkk = − F1 EI共1 + i␩兲␭2F3 l jk = − lkj = 共A2兲 n jj = − nkk = − F1 EI共1 + i␩兲␭2F3 n jk = − nkj = Bishop and Johnson 关33兴 showed that the displacement and rotation-to-force and moment receptances for uniform EulerBernoulli beams could be represented by simple closed-form expressions. For a cylindrical free-free beam with coordinates j and k identified at each end, the frequency-dependent direct and cross receptances are given by: − F5 EI共1 + i␩兲␭3F3 h jk = hkj = F8 EI共1 + i␩兲␭3F3 p jj = pkk = F6 EI共1 + i␩兲␭F3 p jk = pkj = F7 共A4兲 EI共1 + i␩兲␭F3 where E is the elastic modulus, I is the second area moment of inertia, ␩ is the frequency-independent damping coefficient 共damping was not included in Bishop and Johnson, but has been added as part of this analysis兲, and: 共A1兲 F1 = sin ␭L sinh ␭L − F10 EI共1 + i␩兲␭2F3 共A3兲 Appendix: Beam Receptance Modeling h jj = hkk = F10 EI共1 + i␩兲␭2F3 ␭4 = ␻ 2m EI共1 + i␩兲L 共A5兲 F3 = cos ␭L cosh ␭L − 1 F5 = cos ␭L sinh ␭L − sin ␭L cosh ␭L F6 = cos ␭L sinh ␭L + sin ␭L cosh ␭L F7 = sin ␭L + sinh ␭L F8 = sin ␭L − sinh ␭L F10 = cos ␭L − cosh ␭L. 共A6兲 In Eq. 共A5兲, the cylindrical beam mass is given by m= ␲共d2o − di2兲L␳ , 4 where do is the outer diameter, di is the inner diameter 共set equal to zero if the beam is not hollow兲, L is the length, and ␳ is the density; the cylinder’s second area moment of inertia is I= ␲共d4o − di4兲 ; 64 and ␻ is the frequency 共in rad/s兲. The Timoshenko beam model, which includes the effects of rotary inertia and shear, was implemented using finite elements 关34兴. Each four degree-of-freedom 共rotation and displacement at both ends兲 free-free beam section was modeled using appropriate mass, M, and stiffness, K, matrices 关39兴. The mass matrix was: 788 / Vol. 127, NOVEMBER 2005 Transactions of the ASME M= ␳Al 共1 + ␾兲2 + 冤 13 35 + 7␾ 10 + ␾2 3 9 11 ␾ ␾ 3␾ 11␾ + 120 + 24 兲l 共 210 70 + 10 + 6 ␾ 13 ␾ 3␾ + 40 + 24 兲l 共 1051 + 60␾ + 120 兲l2 共 420 2 2 2 2 13 35 + 7␾ 10 + ␾2 3 冉冊 ␳Al rg 共1 + ␾兲2 l 2 冤 共 101 − ␾2 兲l − 共 152 + ␾6 + ␾3 兲l2 2 6 5 − 共 10 − 1 6 5 Symmetric ␾ 2 兲l 兲 兲 兲 兲 ␾2 3␾ 40 + 24 l 1 ␾ ␾2 2 140 + 60 + 120 l 11 ␾2 11␾ 210 + 120 + 24 l 1 ␾ ␾2 2 105 + 60 + 120 l 13 −共 −共 共 Symmetric 6 5 − 共 420 + 共 101 − ␾2 兲l − 共 30 + 1 共 共 ␾ 6 + ␾2 6 兲 l2 兲l 兲 l2 1 ␾ − 10 − 2 2 ␾ ␾2 15 + 6 + 3 冥 冥 where A is the cross-sectional area, l is the section length, rg is the radius of gyration, and ␾ is a shear deformation parameter given by ␾= 12EI共1 + ␩兲 , k⬘GAl2 where G= E 2共1 + ␯兲 is the shear modulus 共␯ is Poisson’s ratio兲 and k⬘ is the shear coefficient which depends on the cross-section shape and ␯ 关40兴. 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