Transverse vibrations have been considered previously in great detail for several single-mass rotor systems. The thin disc and long rigid rotors were considered with various complexities at supports, for example, the rigid disc mounted on a flexible massless shaft with rigid bearings (e.g. the simply-supported, cantilever, and overhung in Chapter 2), flexible bearings (anisotropic and cross-coupled stiffness and damping properties), and flexible foundations (Chapter 4). Higher order effects, i.e. the gyroscopic moment on rotors for the simple and most general cases of motion, were also described in detail (Chapter 5). However, in the actual case, as we have seen in the previous two chapters for torsional vibrations, the rotor system can have several masses (e.g. turbine blades, propellers, flywheels, gears, etc.) or distributed mass and stiffness properties, and multiple supports, and other such components like coupling, seals, etc. While dealing with torsional vibrations, we did consider multi-degree-of-freedom (DOF) rotor systems and continuous rotor systems. Mainly five methods were dealt with, that is, the Newton's second law of motion (or the D'Alembert principle), Lagrange's equations, extended Hamilton's principle, the transfer matrix method, and the finite element method. We will be extending the idea of these methods from torsional vibrations to transverse vibrations along with some additional methods, which are suitable for the analysis of multi-DOF rotor system transverse vibrations. In the present chapter, we will consider the analysis of multi-DOF rotor systems by the influence coefficient method, the transfer matrix method, and Dunkerley's approximate method. The main focus of these methods is to estimate the rotor system natural frequencies, mode shapes, and forced responses. The relative merit and demerit of these methods are discussed. The continuous rotor system with analytical approach and discretized multi-DOF rotor systems through the finite element method without and with gyroscopic effects will be treated in Chapters 9 and 10, respectively. Conventional methods of vibrations like the modal analysis, Rayleigh–Ritz method, weighted sum approach, collocation method, mechanical impedance (or receptance) method, and dynamic stiffness method are not covered exclusively in the present textbook because this information is readily available elsewhere (Meirovitch, 1986; Thomson and Dahleh, 1998). However, the basic concepts of these will be used directly whenever it is required with proper referencing.