Linearized Galerkin and artificial parameter techniques for the determination of periodic solutions of nonlinear oscillators. (English) Zbl 1135.65344
Summary: Linearized harmonic balance-Galerkin methods for the determination of the periodic solutions of nonlinear oscillators are presented and the results are compared with those of an artificial parameter method based on the introduction of a linear stiffness term and the Linstedt-Poincaré technique. The linearized harmonic balance methods presented in this paper are not iterative and are based on Taylor series expansions, the neglect of nonlinear terms in the resulting series, a Fourier approximation to the solution, and orthogonality conditions.
It is shown that the first approximation provided by the linearized harmonic balance technique is identical to that of the artificial parameter method, whereas the second one depends on how the equation is linearized and written, and the discrepancies have been found to be caused by the neglect of the nonlinear terms in the Taylor series expansion because a Fourier series expansion of these terms introduces harmonics which are not accounted for in the linearization procedure. It is also shown that the first approximation of the artificial parameter method and the linearized harmonic balance procedure are asymptotically equivalent to that derived from a modified Linstedt-Poincaré method, when the nonlinearities are small, and that these techniques may not be applicable to some oscillators.
It is shown that the first approximation provided by the linearized harmonic balance technique is identical to that of the artificial parameter method, whereas the second one depends on how the equation is linearized and written, and the discrepancies have been found to be caused by the neglect of the nonlinear terms in the Taylor series expansion because a Fourier series expansion of these terms introduces harmonics which are not accounted for in the linearization procedure. It is also shown that the first approximation of the artificial parameter method and the linearized harmonic balance procedure are asymptotically equivalent to that derived from a modified Linstedt-Poincaré method, when the nonlinearities are small, and that these techniques may not be applicable to some oscillators.
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
Keywords:
nonlinear oscillations; linearized harmonic balance method; Galerkin’s method; artificial parameter; Linstedt-Poincaré technique; periodic solutions; Taylor series expansion; Fourier series expansionReferences:
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