Root
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Conference Proceedings
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6th EUROMECH Nonlinear Dynamics Conference (ENOC 2008)
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Order Reduction of Nonlinear Delay-Differential Equations with Periodic Coefficients
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A technique for order reduction of nonlinear delay differential equations with timeperiodic

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.

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.