Modeling of dynamic rotors with flexible bearings due to the use of viscoelastic materials

Nowadays rotating machines produce or absorb large amounts of power in relatively small physical packages. The fact that those machines work with large density of energy and flows is associated to the high speeds of rotation of the axis, implying high inertia loads, shaft deformations, vibrations and dynamic instabilities. Viscoelastic materials are broadly employed in vibration and noise control of dynamic rotors to increase the area of stability, due to their high capacity of vibratory energy dissipation. A widespread model, used to describe the real dynamic behavior of this class of materials, is the fractional derivative model. Resorting to the finite element method it is possible to carry out the modeling of dynamic rotors with flexible bearings due to the use of viscoelastic materials. In general, the stiffness matrix is comprised of the stiffnesses of the shaft and bearings. As considered herein, this matrix is complex and frequency dependent because of the characteristics of the viscoelastic material contained in the bearings. Despite of that, a clear and simple numerical methodology is offered to calculate the modal parameters of a,simple rotor mounted on viscoelastic bearings. A procedure for generating the Campbell diagram (natural frequency versus rotation frequency) is presented. It requires the embedded use of an auxiliary (internal) Campbell diagram (natural frequency versus variable frequency) in which the stiffness matrix as a frequency function is dealt with. A simplified version of that procedure, applicable to unbalance excitations, is also presented. A numerical example, for two different bearing models, is produced and discussed.