Symbolic Stability and Bifurcation Analysis of Time-Periodic Differential-Delay Equations: Applications to High-Speed Machining Models

an NSF funded project

Personnel (in alphabetical order):

• Viktoria Averina, Ph.D. student in Applied Mathematics, U. Minnesota (UAF M.S. Mathematics 2002)  averina@math.umn.edu
• Ed Bueler, Co-Principal Investigator, Assoc. Prof., Dept. of Mathematics and Statistics, UAF,   elbueler@alaska.edu
• Eric Butcher, Principal Investigator, Assoc. Prof., Dept. of Mechanical Engineering, N.M.S.U.,   eab@nmsu.edu
• Tim Carlson, (UAF M.S. Mathematics 2005)
• Venkatesh S Deshmukh, Asst. Prof., Dept. of  Mechanical Engineering, Villanova University,   venkatesh.deshmukh@villanova.edu
  
• Haitao Ma, (UAF M.S. Mechanical Engineering 2003)
• Praveen K Nindujarla, (UAF M.S. Mechanical Engineering 2004)
• Jacob Stroh, (UAF M.S. Mathematics 2006)
• Sergey Voronov, Bauman Moscow St. Technical Univ., Russia
  

Collaborators

• Alexander Gouskov, Bauman Moscow St. Technical Univ., Russia
• Tony Schmitz, Univ. of Florida
• Subhash Sinha, Auburn Univ.
  
• Zsolt Szabo, Budapest Univ. of Technology and Economics, Hungary
  

Papers:

• E. A. Butcher, O. A. Bobrenkov, E. Bueler, and P. Nindujarla, Analysis of milling stability by the Chebyshev collocation method: Algorithm and optimal stable immersion levels, J. Comput. Nonlinear Dynam., accepted.
• Bueler, E. Error bounds for approximate eigenvalues of periodic-coefficient linear delay differential equations, SIAM J. Num. Analysis 45 no. 6 (2007), 2510--2536.
  
• Deshmukh, V., E.A. Butcher, and E. Bueler, Dimensional reduction of nonlinear delay differential equations with periodic coefficients using Chebyshev spectral collocation, Nonlinear Dynamics 52, 2008, 137--149.
• Gouskov, A. M., S. A. Voronov, E. A. Butcher, and S. C. Sinha, Nonconservative Oscillations of a Tool for Deep Hole Honing, to appear in Comm. Nonlinear Science & Numerical Simulation
• Deshmukh, V., H. Ma, and E. A. Butcher, Optimal control of parametrically excited linear delay differential systems via Chebyshev polynomials, Optimal Control Appl. Methods 27 (2006), no. 3, 123--136.
• Ma, H., V. Deshmukh, E. A. Butcher, and V. Averina, Delayed state feedback and chaos control for time-periodic systems via a symbolic approach, Comm. Nonlinear Science & Numerical Simulation, 10 (2005), no. 5, 479--497.
• Ma, H. and E. A. Butcher, Stability of Elastic Columns Subjected to Periodic Retarded Follower Forces, to appear in Journal of Sound and Vibration.
• Butcher, E. A., H. Ma, E. Bueler, V. Averina, and Z. Szabo, Stability of time-periodic delay-differential equations via Chebyshev polynomials,  International Journal for Numerical Methods in Engineering (Internat. J. Numer. Methods Engrg.), 59 (2004), no.7, 895--922.
• Butcher, E.A., H. Ma, and E. Bueler, Chebyshev expansion of linear dynamic systems with time delay and periodic coefficients under control excitations, Transactions of the ASME Journal of Dynamic Systems, Measurement and Control, 125 (2003), 236--243.
  

Proceedings:

• Butcher, E. A., V. Deshmukh, and E. Bueler, A center manifold algorithm for nonlinear dynamic systems with time-delay and parametric excitation, Proceedings of the 14th International Congress on Sound and Vibration, Cairns, Australia, July 2007.
• Butcher, E. A., V. Deshmukh, and E. Bueler, Center manifold reduction of periodic delay differential systems, Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2007, Las Vegas, Nevada, September 2007.
• Butcher, E. A., P. Nindujarla and E. Bueler, Stability of up- and down-milling using a Chebyshev collocation method, proceedings of ASME DETC 2005, September 2005, Long Beach, CA. (547 KB, PDF)
• Averina, V., I. Kolmanovsky, A. Gibson, G. Song and E. Bueler, Analysis and control of delay-dependent behavior of engine air-to-fuel ratio, Proceedings of the IEEE Conference on Control Applications, Toronto, August, 2005.
• Gouskov, A., S. Voronov, E. A. Butcher, and S. C. Sinha, Dynamics and Stability of a Vibratory Drilling Process Controlled by a Piezoactuator, proceedings of 2004 ASME IMECE, Anaheim, CA, Nov. 13-19, 2004
• Gouskov, A., S. Voronov, E. A. Butcher, and S. C. Sinha, Simulation of Machined Surface Formation while Honing, proceedings of 2004 ASME IMECE, Anaheim, CA, Nov. 13-19, 2004.
• Butcher, E.A., Ma, H., Deshmukh, V., and Averina, V., Chaos Control to Periodic Orbit via Time-Delayed State Feedback, proceedings of 11th International Congress on Sound and Vibration, July 5-8, 2004, St. Petersburg, Russia.
• Gouskov, A., Voronov, S., Butcher, E. A., and Sinha, S. C.,  Influence of Honing Dynamics on Surface Formation,  proceedings of 19th Biennial Conference on Mechanical Vibration and Noise, ASME DETC 03,VIB-,Chicago, IL, Sept. 2-6, 2003.
• Butcher, E. A., H. Ma, E. Bueler, V. Averina, and Z. Szabo,  Stability Analysis of Parametrically Excited Systems with Time-Delay,  proceedings of 19th Biennial Conference on Mechanical Vibration and Noise, ASMEDETC 03, VIB-48574, Chicago, IL, Sept. 2-6, 2003.
• Gouskov, A. M., S. A. Voronov, E. A. Butcher, and S. C. Sinha,  Nonconservative Oscillations of a Tool for Deep Hole Honing,  proceedings of International Conference on Physics and Control, August 20-22, 2003, St. Petersburg, Russia.
  
• Butcher, E. A. and H. Ma,  Stability of Discretized Elastic Columns with Periodic Retarded Follower Forces, proceedings of 10th International Congress on Sound and Vibration, July 7-10, 2003, Stockholm, Sweden.
• Deshmukh, V., Ma, H., and Butcher, E. A., Optimal Control of Parametrically Excited Linear Delay Differential Systems via Chebyshev Polynomials, proceedings of 2003 American Control Conference, June 4-6, 2003, Denver, CO.
• Ma, H., Deshmukh, V., Butcher, E. A., and Averina, V., Controller Design for Linear Time-Periodic Delay Systems via a Symbolic Approach, proceedings of 2003 American Control Conference, June 4-6, 2003, Denver, CO.

Presentations: 

• Butcher, E. A., M. Sari, E. Bueler, and T. Carlson, Magnus' expansion for time-periodic systems, 12th Conference on Nonlinear Vibrations, Dynamics, and Multibody Systems, VPI&SU, Blacksburg, VA, June 1-5, 2008.
• Bueler, E., E. A. Butcher, and V. Averina, Reliable stability charts for linear periodic delay differential equations, SIAM CSE05, Orlando, Feb. 2005. (131 KB, PDF)
• Nindujarla, P., Stability of Milling using the Chebyshev Collocation Method, M.S. Mechanical Engineering Thesis defense, December 2, 2004.
  
• Bueler, E., V. Averina, and E. A. Butcher, Periodic Linear DDEs: Collocation Approximation to the Monodromy Operator, SIAM annual meeting,, Portland, OR, July 12-16, 2004.  Slides: Collocation approximation of the monodromy operator for linear, periodic DDEs (341 KB, PDF)
• Slides for Ed's 5/4/04 UAF Department of Mathematical Sciences Colloquium talk (also given at CSU, Chico colloquium 2/6/04): Stability and Delay. (359 KB, PDF)
• Ma, H., V. Deshmukh, E. A. Butcher and V. Averina, Controller design for linear time-periodic delay systems via a symbolic approach, American Control Conference (ACC) 2003, Denver, Colorado.
• Deshmukh, V., H. Ma, and E. A. Butcher, Optimal control of parametrically excited linear delay differential systems via Chebyshev polynomials, American Control Conference (ACC) 2003, Denver, Colorado.
• Szabo, S., and E. A. Butcher, Stability Analysis of Delayed 2nd Order ODEs Based on the Method of Chebyshev Polynomials, Euro Conference on Numerical Methods and Computational Mechanics, Miskolc, Hungary, July 15-19, 2002.

Technical Reports:

• Bueler, E., Chebyshev collocation for linear, periodic ordinary and delay differential equations: a posteriori estimates, preprint 2004 (http://arxiv.org/abs/math.NA/0409464).
  
• Bueler, E., and E. A. Butcher, Stability of periodic linear delay-differential equations and the Chebyshev approximation of fundamental solutions.  UAF Dept. of Mathematical Sciences Technical Report 02-03, 2002 (PDF ).

Theses:

• Jacob Stroh, Non-normality in scalar delay differential equations, UAF MS Mathematics Thesis 2006.
• Carlson, T., Magnus' expansion as an approximation tool for ordinary differential equations.  UAF MS Mathematics Thesis 2005.
  
• Nindujarla, P., Stability of milling using the Chebyshev collocation method, UAF MS Mechanical Engineering Thesis 2004.
• Ma, H., Analysis and control of time-periodic systems with time delay via Chebyshev polynomials.  UAF MS Mechanical Engineering Thesis 2004.
• Averina, V., Symbolic stability of delay differential equations.  UAF MS Mathematics Thesis 2002

Also of interest:

• Butcher, Eric A. and D. J. Segalman, Suppression of regenerative chatter via impedance modulation, 35th Annual Technical Meeting of Society of Engineering Science, WSU, Pullman, WA, Sept. 27-30, 1998

Software:

• ddec, a suite of Matlab codes for stability information and charts on linear periodic DDEs

NSF links:

• Directorate for Engineering of the National Science Foundation
• NSF Award Abstract #0114500 on Fastlane

Subject--related sites: Delay differential equations

• DDE-BIFTOOL: a Matlab package for bifurcation analysis of delay differential equations
  
• Solving DDEs with MATLAB (Shampine and Thompson: "dde23")
• Functional Differential Equation Laboratory (Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ural Branch)