A new
method for the solution of non-sinusoidal periodic states in linear
fractionally damped oscillators is presented. The oscillator is forced by a
periodic discontinuous waveform and a viscous element is taken into account.
The presented method avoids completely the Fourier series calculations of the
input and output oscillator waveforms. In the proposed method, the steady-state
response of fractionally damped oscillator is formulated directly in the time
domain as a superposition of the zero-input and forced responses for each
continuous piecewise segments of the forcing waveform, separately. The whole
periodic response is reached by taking into account the continuity and
periodicity conditions at instants of discontinuities of the excitation and
then using the concatenation procedure for all segments. The method can be
applied efficiently to discontinuous and continuous non-harmonic excitations
equally well. Solutions are exact and there is no need to apply any of the
widely up-to-date used frequency approaches. The Fourier series is completely
cut out of the oscillator analysis.
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