Analysis of forced non-linear undamped oscillators by a time transformation method. (English) Zbl 1235.70088

Summary: The recently developed time transformation method [T. D. Burton, Int. J. Nonlinear Mech. 17, 7–19 (1982; Zbl 0499.70031); J. Sound Vib. 87, 535–541 (1983; Zbl 0565.70028); the authors, J. Sound Vib. 87, 543–554 (1983; Zbl 0562.70024)] is extended to the analysis of forced vibrations of the strongly nonlinear undamped oscillator, \(\ddot u+u+\varepsilon f(u)=P(t)\), where \(f(u)\) is a nonlinear function of the displacement \(u(t)\) and \(\varepsilon\) is not necessarily small. Two types of external loading \(P(t)\) are considered: a step input and a simple harmonic input. The frequency-amplitude relation is obtained through a slight modification of the results developed by the authors [op. cit.] for the unforced oscillator. A comparison with the results of numerical and other analytical methods is made. It can be deduced that the time transformation approach offers simple and accurate approximations for the oscillators considered.


70K40 Forced motions for nonlinear problems in mechanics
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