In the previous chapter, we studied transverse vibrations of multi-DOF rotor systems by considering the shaft as massless (or lumped mass) and flexible by using the influence coefficient and transfer matrix methods. While dealing with torsional vibrations, we learned that rotors could be modeled accurately, if treated as a continuous system (i.e. distributed inertia and elastic properties). The present chapter considers transverse vibrations of the shaft as having distributed mass and stiffness properties. Equations of motion are obtained by using Newton's second law of motion and Hamilton's principle. These equations are partial differential equations as we have seen in the case of torsional vibration. For simple boundary conditions, closed-form solutions are obtained by the method of separation of variables. However, the continuous system approach becomes impracticable for more complex boundary conditions. Hence, we need to resort to some approximate methods, for example, the finite element method (FEM). For the finite element (FE) analysis of rotor systems, the Euler–Bernoulli beam model is considered initially for developing the elemental mass and stiffness matrices. Both the free and forced vibrations are analyzed for a variety of conditions at supports using the FEM. In the FEM, the overall size of matrices increases with the number of degrees of freedom (DOFs) of the system. It is often required to reduce the size of the actual matrices to be solved to save computational time. Such cases can be handled using static and dynamic reductions (or condensations). These reductions introduce some amount of approximation in solutions. Subsequent chapters will discuss higher order effects such as rotary inertia, shear deformation, and gyroscopic moment in the dynamic analysis of rotor systems using the FEM.