The main objective of this book is a mathematical study for the nonlinear lateral vibrations of rotating machinery. Rotating machinery vibrations are usually simulated using a two-degree-of freedom nonlinear system known as Jeffcott-rotor system. Two basic models of Jeffcott-rotor system have been investigated. The first is a horizontally supported nonlinear Jeffcott-rotor system. The second is a vertically supported nonlinear Jeffcott-rotor system. Influences of geometric nonlinearity, disc eccentricity, and rotor weight on the considered systems vibrations were studied. Different control algorithms were proposed to suppress those nonlinear vibrations. Moreover, bifurcation behaviors for both the horizontally and vertically supported nonlinear Jeffcott rotor systems that have a transversely cracked shaft were discussed for the purposes of crack diagnosis. All different mathematical models were analyzed analytically applying multiple scales perturbation technique. The acquired analytical results were validated numerically utilizing an appropriate numerical algorithm using Matlab software.

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