This paper presented the systematic procedures for finding the critical speed and the bending moment in each segement of the vibratory variable cross-section shaft carrying multi discs. Lateral vibrations result from transversal dynamic excitations. The influence coefficients are obtained by using "Castigliano- Theorem". Samples of .data for the obtained results are presented for a shaft carrying 'three discs with an eccentricity on the second disc for both cases of end supported and over hanged shafts on rigid support ipcluding the variation of disc position. The results of shaft amplitudes and natural frequencies are obtained by computer analysis.