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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