Rotor bending dynamics

Rotating machines are used in a wide range of industries. They are very diverse: compressors, turbines, aircraft engines, pumps, alternators... In the interests of quality, efficiency and safety, they must be carefully studied at project level.

In the first stage, machine rotors are dimensioned on the basis of material strength: the aim is to achieve a minimum shaft radius capable of withstanding the rated torque. The dynamic torsional behavior is then studied, with the aim of avoiding operation in a speed range containing one or more critical speeds. Furthermore, if transient conditions exist, for example in the case of an electric motor during start-up or in the event of an accidental short-circuit, the transient behavior must be studied; it then provides a shaft radius greater than the minimum radius defined statically. The dynamics of rotors in bending must then be considered. Two particular effects are generally present: the gyroscopic effect (Coriolis) due to the discs and the damping effect, which can be very significant in the case of hydrodynamic bearings. The first step is to predict the evolution of natural frequencies as a function of rotation speeds: this enables us to determine critical speeds and possible instabilities due to bearings. Secondly, we calculate the steady-state response to unbalance effects and possibly to an asynchronous force.

Section 1 concerns the determination of rotor element characteristics. This involves determining the expressions for the kinetic and deformation energies, as well as the virtual work corresponding to the basic elements: disk, shaft, bearing, unbalance. Two numerical methods are used: the Rayleigh-Ritz method for demonstrating phenomena, and the finite element method for predicting the dynamic behavior of real machines. The application of Lagrange's equations then leads to the equations of motion.

In the 2