0
Technical Briefs

# Stability Analysis of Milling Via the Differential Quadrature Method

[+] Author and Article Information
Ye Ding

e-mail: y.ding@sjtu.edu.cn

LiMin Zhu

State Key Laboratory of Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China

Han Ding

State Key Laboratory of Digital Manufacturing Equipment and Technology,
Huazhong University of Science and Technology,
Wuhan 430074, China

1Corresponding author.

Contributed by the Manufacturing Engineering Division of ASME for publication in the Journal of Manufacturing Science and Engineering. Manuscript received December 25, 2012; final manuscript received April 18, 2013; published online July 17, 2013. Assoc. Editor: Tony Schmitz.

J. Manuf. Sci. Eng 135(4), 044502 (Jul 17, 2013) (7 pages) Paper No: MANU-12-1368; doi: 10.1115/1.4024539 History: Received December 25, 2012; Revised April 18, 2013; Accepted April 23, 2013

## Abstract

This paper presents a time-domain semi-analytical method for stability analysis of milling in the framework of the differential quadrature method. The governing equation of milling processes taking into account the regenerative effect is formulated as a linear periodic delayed differential equation (DDE) in state space form. The tooth passing period is first separated as the free vibration duration and the forced vibration duration. As for the free vibration duration, the analytical solution is available. As for the forced vibration duration, this time interval is discretized by sampling grid points. Then, the differential quadrature method is employed to approximate the time derivative of the state function at a sampling grid point within the forced vibration duration by a weighted linear sum of the function values over the whole sampling grid points. The Lagrange polynomial based algorithm (LPBA) and trigonometric functions based algorithm (TFBA) are employed to obtain the weight coefficients. Thereafter, the DDE on the forced vibration duration is discretized as a series of algebraic equations. By combining the analytical solution of the free vibration duration and the algebraic equations of the forced vibration duration, Floquet transition matrix can be constructed to determine the milling stability according to Floquet theory. Simulation results and experimentally validated examples are utilized to demonstrate the effectiveness and accuracy of the proposed approach.

<>

## References

Budak, E., 2006, “Analytical Models for High Performance Milling. Part I: Cutting Forces, Structural Deformations and Tolerance Integrity,” Int. J. Mach. Tools Manuf., 46(12–13), pp. 1478–1488.
Budak, E., 2006, “Analytical Models for High Performance Milling. Part II: Process Dynamics and Stability,” Int. J. Mach. Tools Manuf., 46(12–13), pp. 1489–1499.
Altintas, Y., and Weck, M., 2004, “Chatter Stability of Metal Cutting and Grinding,” CIRP Ann.—Manuf. Technol., 53(2), pp. 619–642.
Altintas, Y., 2000, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University, Cambridge.
Schmitz, T. L., and Smith, K. S., 2008, Machining Dynamics: Frequency Response to Improved Productivity, Springer, New York.
Schmitz, T. L., Davies, M. A., and Kennedy, M. D., 2001, “Tool Point Frequency Response Prediction for High-Speed Machining by RCSA,” ASME J. Manuf. Sci. Eng., 123(4), pp. 700–707.
Schmitz, T. L., and Duncan, G. S., 2005, “Three-Component Receptance Coupling Substructure Analysis for Tool Point Dynamics Prediction,” ASME J. Manuf. Sci. Eng., 127(4), pp. 781–790.
Tlusty, J., and Ismail, F., 1981, “Basic Non-Linearity in Machining Chatter,” CIRP Ann.—Manuf. Technol., 30(1), pp. 299–304.
Smith, S., and Tlusty, J., 1991, “Overview of Modeling and Simulation of the Milling Process,” ASME J. Eng. Ind., 113(2), pp. 169–175.
Campomanes, M. L., and Altintas, Y., 2003, “An Improved Time Domain Simulation for Dynamic Milling at Small Radial Immersions,” ASME J. Manuf. Sci. Eng., 125(3), pp. 416–422.
Minis, I., and Yanushevsky, R., 1993, “A New Theoretical Approach for the Prediction of Machine Tool Chatter in Milling,” J. Eng. Ind., 115(1), pp. 1–8.
Altintas, Y., and Budak, E., 1995, “Analytical Prediction of Stability Lobes in Milling,” CIRP Ann–Manuf. Technol., 44(1), pp. 357–362.
Budak, E., and Altintas, Y., 1998, “Analytical Prediction of Chatter Stability in Milling—Part I: General Formulation,” ASME J. Dyn. Syst., Meas., Control, 120(1), pp. 22–30.
Budak, E., and Altintas, Y., 1998, “Analytical Prediction of Chatter Stability in Milling—Part II: Application of the General Formulation to Common Milling Systems,” ASME J. Dyn. Syst., Meas., Control, 120(1), pp. 31–36.
Merdol, S. D., and Altintas, Y., 2004, “Multi Frequency Solution of Chatter Stability for Low Immersion Milling,” ASME J. Manuf. Sci. Eng., 126(3), pp. 459–466.
Bayly, P. V., Halley, J. E., Mann, B. P., and Davies, M. A., 2003, “Stability of Interrupted Cutting by Temporal Finite Element Analysis,” ASME J. Manuf. Sci. Eng., 125(2), pp. 220–225.
Mann, B. P., Young, K. A., Schmitz, T. L., and Dilley, D. N., 2005, “Simultaneous Stability and Surface Location Error Predictions in Milling,” ASME J. Manuf. Sci. Eng., 127(3), pp. 446–453.
Butcher, E. A., Ma, H., Bueler, E., Averina, V., and Szabo, Z., 2004, “Stability of Linear Time-Periodic Delay-Differential Equations Via Chebyshev Polynomials,” Int. J. Numer. Methods Eng., 59(7), pp. 895–922.
Butcher, E. A., Bobrenkov, O. A., Bueler, E., and Nindujarla, P., 2009, “Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels,” ASME J. Comput. Nonlinear Dyn., 4(3), p. 031003.
Olgac, N., and Sipahi, R., 2005, “A Unique Methodology for Chatter Stability Mapping in Simultaneous Machining,” ASME J. Manuf. Sci. Eng., 127(4), pp. 791–800.
Insperger, T., and Stépán, G., 2002, “Semi-Discretization Method for Delayed Systems,” Int. J. Numer. Methods Eng., 55(5), pp. 503–518.
Insperger, T., and Stépán, G., 2004, “Updated Semi-Discretization Method for Periodic Delay-Differential Equations With Discrete Delay,” Int. J. Numer. Methods Eng., 61(1), pp. 117–141.
Insperger, T., Stépán, G., and Turi, J., 2008, “On the Higher-Order Semi-Discretizations for Periodic Delayed Systems,” J. Sound Vib., 313(1–2), pp. 334–341.
Insperger, T., and Stépán, G., 2011, Semi-Discretization for Time-delay Systems: Stability and Engineering Applications, Springer-Verlag, New York.
Ding, Y., Zhu, L., Zhang, X., and Ding, H., 2010, “A Full-Discretization Method for Prediction of Milling Stability,” Int. J. Mach. Tools Manuf., 50(5), pp. 502–509.
Insperger, T., 2010, “Full-Discretization and Semi-Discretization for Milling Stability Prediction: Some Comments,” Int. J. Mach. Tools Manuf., 50(7), pp. 658–662.
Quo, Q., Sun, Y., and Jiang, Y., 2012, “On the Accurate Calculation of Milling Stability Limits Using Third-Order Full-Discretization Method,” Int. J. Mach. Tools Manuf., 62, pp. 61–66.
Liu, Y., Zhang, D., and Wu, B., 2012, “An Efficient Full-Discretization Method for Prediction of Milling Stability,” Int. J. Mach. Tools Manuf., 63, pp. 44–48.
Li, M., Zhang, G., and Huang, Y., 2012, “Complete Discretization Scheme for Milling Stability Prediction,” Nonlinear Dyn., 71(1–2), pp. 187–199.
Khasawneh, F. A., and Mann, B. P., 2011, “A Spectral Element Approach for the Stability of Delay Systems,” Int. J. Numer. Methods Eng., 87(6), pp. 566–592.
Compean, F., Olvera, D., Campa, F., López de Lacalle, L., Elias-Zuniga, A., and Rodriguez, C., 2012, “Characterization and Stability Analysis of a Multivariable Milling Tool by the Enhanced Multistage Homotopy Perturbation Method,” Int. J. Mach. Tools Manuf., 57, pp. 27–33.
Ding, Y., Zhu, L., Zhang, X., and Ding, H., 2011, “Numerical Integration Method for Prediction of Milling Stability,” ASME J. Manuf. Sci. Eng., 133(3), p. 031005.
Ding, Y., Zhu, L., Zhang, X., and Ding, H., 2011, “Milling Stability Analysis Using the Spectral Method,” Sci. China, Ser. E: Technol. Sci., 54(12), pp. 3130–3136.
Eksioglu, C., Kilic, Z. M., and Altintas, Y., 2012, “Discrete-Time Prediction of Chatter Stability, Cutting Forces, and Surface Location Errors in Flexible Milling Systems,” ASME J. Manuf. Sci. Eng., 134(6), p. 061006.
Bellman, R., and Casti, J., 1971, “Differential Quadrature and Long-Term Integration,” J. Math. Anal. Appl., 34(2), pp. 235–238.
Bellman, R., Kashef, B. G., and Casti, J., 1972, “Differential Quadrature: A Technique for the Rapid Solution of Nonlinear Partial Differential Equations,” J. Comput. Phys., 10(1), pp. 40–52.
Quan, J., and Chang, C., 1989, “New Insights in Solving Distributed System Equations by the Quadrature Method—I. Analysis,” Comput. Chem. Eng., 13(7), pp. 779–788.
Shu, C., 1991, “Generalized Differential-Integral Quadrature and Application to the Simulation of Incompressible Viscous Flows Including Parallel Computation,” Ph.D. thesis, University of Glasgow, Glasgow, Scotland.
Shu, C., Yao, Q., and Yeo, K. S., 2002, “Block-Marching in Time With DQ Discretization: An Efficient Method for Time-Dependent Problems,” Comput. Methods Appl. Mech. Eng., 191(41–42), pp. 4587–4597.
Bert, C. W., and Malik, M., 1996, “Differential Quadrature Method in Computational Mechanics: A Review,” Appl. Mech. Rev., 49(1), pp. 1–28.
Shu, C., 2000, Differential Quadrature and Its Application in Engineering, Springer-Verlag, London.
Striz, A., Wang, X., and Bert, C., 1995, “Harmonic Differential Quadrature Method and Applications to Analysis of Structural Components,” Acta Mech., 111(1), pp. 85–94.
Fung, T. C., 2001, “Solving Initial Value Problems by Differential Quadrature Method—Part 1: First-Order Equations,” Int. J. Numer. Methods Eng., 50(6), pp. 1411–1427.
Meyer, C. D., 2000, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia.
Farkas, M., 1994, Periodic Motions, Springer-Verlag, New York.
Wang, Y., 2001, “Differential Quadrature Method and Differential Qudrature Element Method—Principle and Applications,” Ph.D. dissertation, Nanjing University of Aeronautics and Astronautics, Nanjing, China.
Tweten, D. J., Lipp, G. M., Khasawneh, F. A., and Mann, B. P., 2012, “On the Comparison of Semi-Analytical Methods for the Stability Analysis of Delay Differential Equations,” J. Sound Vib., 331(17), pp. 4057–4071.
Mann, B. P., Insperger, T., Bayly, P. V., and Stepan, G., 2003, “Stability of Up-Milling and Down-Milling, Part 2: Experimental Verification,” Int. J. Mach. Tools Manuf., 43(1), pp. 35–40.
Henninger, C., and Eberhard, P., 2008, “Improving the Computational Efficiency and Accuracy of the Semi-Discretization Method for Periodic Delay-Differential Equations,” Eur. J. Mech., A/Solids, 27(6), pp. 975–985.

## Figures

Fig. 1

Two-degrees-of-freedom milling system

Fig. 2

Stability diagrams determined by the first-order (1st) SDM, LPBA, and TFBA. The stability lobes presented by grey color for reference are determined by first-order SDM with n = 200.

Fig. 3

Convergence of the critical eigenvalues for the proposed LPBA, TFBA, the CCM, and the IEBSM with different computational parameters

Fig. 4

Computational time of the proposed LPBA, TFBA with different computational parameters for ap = 1 mm, the spindle speed fixed as 5000 rpm and the radial immersion ratio fixed as ae/D = 1

Fig. 5

Comparison of stability lobes for two-degrees-of-freedom milling system. The stability lobes presented by grey color for reference are determined by the TFEA method with n = 4.

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections