Machine tool dynamics and metal cutting vibrations
Motivated by the collaboration in the BMBF-Verbundprojekt VispaB and the phenomena of regenerative chatter at machine tools I become interested in machining, machine tool dynamics and metal cutting vibrations.
Delay differntial equation (DDE)
One of my research interests are delay differential equations. Some selected topics are:
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Dimension of asymptotic solutions of DDEs with time-varying delay
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Equivalence between DDEs with variable coefficients and DDEs with variable delay
Methods
For the solution and analysis of DDEs with arbitrary delays we use the following methods:
| Name | Application | Type | Delay | References |
|
Method of steps |
anal. |
l, nl |
c, v |
[1], [2], [3] |
|
D-subdivision |
anal. |
l |
c, d |
[4], [5] |
|
Multifrequency approach |
anal., num. |
l |
c, v, d |
[6], [7], [8], [9] |
|
Semidiscretization |
num. |
l |
c, v, d |
[10], [11] |
|
RADAR5 (DDE-Solver) |
num. |
l, nl |
c, v, s |
[12] |
|
Matlab/Simulink (time domain simulation) |
num. |
l, nl |
c, v, s |
[13] |
Symbols: l=linear, nl=nonlinear DDEs; c=constant, v=variable, d=distributed and s=state-dependent delay.
| [1] | R. Bellmann: On the computational solution of differential-difference equations, J. Math. Anal. Appl. 2, 108-110 (1961). |
| [2] |
R. Bellmann, K.L. Cooke: On the computational solution of a class of functional differential equations, J. Math. Anal. Appl. 12, 495-500 (1965).
|
|
[3]
|
A. Bellen, M. Zennaro: Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford (2003).
|
|
[4]
|
V.B. Kolmanovskii, V.R. Nosov: Stability of functional differential equations, Academic Press, London (1986).
|
| [5] |
W. Michiels, S.-I. Niculescu: Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Bases Approach, SIAM, Philadelphia (2007)
|
| [6] |
S. Jayaram, S.G. Kapoor, R.E. DeVor: Analytical Stability Analysis of Variable Spindle Speed Machining, J. Manuf. Sci. Eng. Trans. ASME 122, 391-397 (2000).
|
| [7] |
S. Sastry, S.G. Kapoor, R.E. DeVor: Floquet Theory Based Approach for Stability Analysis of the Variable Speed Face-Milling Process, J. Manuf. Sci. Eng. Trans. ASME 124(1), 10-17 (2002).
|
|
[8]
|
M. Zatarain, I. Bediaga, J. Munoa, R. Lizarralde: Stability of milling processes with continuous spindle speed variation: Analysis in the frequency and time domains, and experimental correlation, CIRP Annals 57(1), 379-384 (2008).
|
| [9] |
S. D. Merdol, Y. Altintas: Multi Frequency Solution of Chatter Stability for Low Immersion Milling, J. Manuf. Sci. Eng. Trans. ASME 126(3), 459-467 (2004).
|
| [10] |
T. Insperger, G. Stepan: Semi-discretization method for delayed systems, Int. J. Numer. Meth. Eng. 55, 503-518 (2002).
|
| [11] | T. Insperger, G. Stepan: Stability analysis of turning with periodic spindle speed modulation via semidiscretization, J. Vibr. Control, 10(12), 1835–1855 (2004). |
| [12] | N. Guglielmi: Open issues in devising software for the numerical solution of implicit delay differential equations, J. Comp. Appl. Math. 185(2), 261-277 (2006). |
| [13] |
L.F. Shampine: Solving ODEs and DDEs with residual control, J. Appl. Num. Math. 52(1), (2005).
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