All experiments testing time dilation by measuring the spectral shift of a rotating Mössbauer absorber assume that vibrations do not affect the spectral shift because of their purely random nature and claim that the observed shift is due to time dilation only. Our recent experiment using the Synchrotron Mössbauer Source at ESRF revealed, however, a shift due to the non-random periodic vibration patterns caused by the rotor/bearing system. These patterns fit the predictions of the Jeffcott model for such a system with non-zero eccentricity. We have calculated this shift due to the non-random vibrations and the resulting relative shift between two states when the acceleration of the absorber is anti-parallel and parallel to the source. This relative shift exhibits the same behavior as the observed relative shift. Hence, the effect of the spectral shift due to vibrations cannot be ignored in any Mössbauer rotor experiments for testing time dilation. Recommendations for improvement of future rotor experiments testing time dilation are presented.