TY - GEN
T1 - A center manifold algorithm for nonlinear dynamic systems with time-delay and parametric excitation
AU - Butcher, Eric A.
AU - Deshmukh, Venkatesh
AU - Bueler, Ed
PY - 2007
Y1 - 2007
N2 - A technique for center manifold reduction of nonlinear delay differential equations with time-periodic coefficients is presented. The DDEs considered here have at most cubic nonlinearities multiplied by a perturbation parameter. The periodic terms and matrices are not assumed to have predetermined norm bounds, thus making the method applicable to systems with strong parametric excitation. Perturbation expansion converts the nonlinear response problem into solutions of a series of non-homogenous linear ordinary differential equations with time periodic coefficients. One set of linear non-homogenous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. The accuracy of the method is demonstrated with a nonlinear delayed Mathieu equation, a milling model, and a single inverted pendulum with a periodic retarded follower force and nonlinear restoring force in which the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation.
AB - A technique for center manifold reduction of nonlinear delay differential equations with time-periodic coefficients is presented. The DDEs considered here have at most cubic nonlinearities multiplied by a perturbation parameter. The periodic terms and matrices are not assumed to have predetermined norm bounds, thus making the method applicable to systems with strong parametric excitation. Perturbation expansion converts the nonlinear response problem into solutions of a series of non-homogenous linear ordinary differential equations with time periodic coefficients. One set of linear non-homogenous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. The accuracy of the method is demonstrated with a nonlinear delayed Mathieu equation, a milling model, and a single inverted pendulum with a periodic retarded follower force and nonlinear restoring force in which the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation.
UR - http://www.scopus.com/inward/record.url?scp=84881461268&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84881461268&partnerID=8YFLogxK
M3 - Conference contribution
AN - SCOPUS:84881461268
SN - 9781627480000
T3 - 14th International Congress on Sound and Vibration 2007, ICSV 2007
SP - 876
EP - 883
BT - 14th International Congress on Sound and Vibration 2007, ICSV 2007
T2 - 14th International Congress on Sound and Vibration 2007, ICSV 2007
Y2 - 9 July 2007 through 12 July 2007
ER -