1.3 Internal damping in rotors
1.3.2 Hysteretic and other form of internal damping
Melanson and Zu (1998) performed a modal analysis based on Timoshenko beam theory by incorporating the internal damping, such as the internal viscous and the hysteretic damping in the rotating shaft. They studied the stability of the rotating shaft due to internal damping through numerical simulations. Genta (2004) studied the effect of hysteretic damping in the dynamics of rotors at the subcritical and supercritical speeds for the case of the forward and backward whirl modes. The dynamic behavior of the rotating shaft was obtained with random variations of the
internal damping that causes dynamic instability of the system (Dimentberg, 2005; Vatta and Vigliani, 2008). Montagnier and Hochard (2007) did a comparative instability study between the viscous and hysteretic internal damping models using the Euler-Bernoulli beam theory in a rotor system. The model contains the internal viscous (or hysteretic damping), translational and rotating inertia and gyroscopic effect.
Chouksey et al. (2010) studied the effect of stationary and rotating damping on rotor behavior. The source of the rotating damping is considered due the material damping of shaft.
Chouksey et al. (2012) used the modal analysis to estimate the internal damping and fluid-film forces on a rotor system supported on journal bearings. The result showed that the internal damping of shaft material reduced the stability threshold speed considerably. Bucciarelli (1982) analysed the phenomenon of dynamic instability effects due to internal damping above critical speeds of a shaft system. He observed that the whirling of spinning shaft along with occurrence of unbounded growth in axial displacement. Chandra and Sekhar (2016) studied the nonlinear internal and external damping based on Krylov–Bogoliubov method. The method was used to find solution of a nonlinear equation of motion based on an approximate method including of the continuous wavelet transform based approach for the identification of rotor system parameters, both numerically and experimentally. Vervisch et al. (2014) designed an experimental setup to verify a theoretical modeling of internal (rotating) damping and studied its effect on the stability of rotor system. The stability threshold speed is estimated and investigated the parameters that the threshold speed. The damping is considered proportional to the stiffness in this model. In a separate work, the authors (Vervisch, 2016 and Vervisch et al. 2016) presented experimental details to investigate rotating damping in a thin shaft of rotating machinery. A disc was attached at middle of a shaft and the damping was considered due to rubbing between disc and shaft.
Prediction of the stability threshold of the rotating machinery was performed and validated through measurements.
Many authors have worked on the finite element modelling of rotor-bearing system.
Nelson and McVaugh (1976) presented a rotor-bearing model based on the finite element method to study the nature of whirl speeds and modes for the different damping and stiffness conditions.
In this study, they considered the effect of rotary inertia, gyroscopic and axial load effect. Further the linear finite element model was extended by Zorzi and Nelson (1977) to obtain the detailed analysis of damped rotor stability of a rotor-bearing system in presence of internal viscous and hysteretic damping effects. Nelson (1980) extended the work using the Timoshenko beam theory including of the internal damping. Chen and Ku (1991) and Ku (1998) studied the instability due to hysteretic damping in the forward and backward whirling of a rotor-bearing system. Forrai (2000) presented the stability analysis in a symmetrical rotor-bearing systems based on the finite element method and incorporating the internal damping. The uniform circular shaft in the rotor system modelled using Rayleigh beam theory and additionally including of both internal viscous and hysteretic damping, rigid disc, and discrete isotropic damped bearings. In the mathematical model, the rotatory inertia and gyroscopic moment effects were also considered. It was observed that the whirling motion grew towards unstable form beyond the threshold spin speed and further found that it improved the stability condition by increasing bearing damping.
Sino et al. (2007) dealt with studies of dynamic instabilities in a rotating system due to internal damping based on finite element beam approach by considering of transversal shear effect. Lemahieu et al. (2012) proposed that the internal damping is one of the main root cause of instability in a rotor system. The methodology was constructed in two steps. Firstly, they developed a finite element model of the beam by considering the viscous and hysteretic damping.
This model was utilized to obtain the threshold spin speed for instability and shaft resonance frequencies. Further, they studied the unbalance responses for different damping conditions to compare the relation between damping properties and instability. In second step, the work was extended through an experimental approach to test the damping model. Ren et al. (2017) studied the internal damping effect of a rotating composite shaft at high spin speed. It is found that at a supercritical speed shaft it achieved unstable self-excited vibration under the action of material internal damping. Herein, using the relations of strain-displacement of composite material, the potential energy, the kinetic energy, and the dissipative energy due to internal damping of the rotor shaft are combined for formation of equations of motion. The internal damping dissipative energy was developed based on the Euler- Bernoulli beam theory of the rotor system also for composite shaft, which considers the dissipative characteristics of viscoelastic damping. Finally, they developed the equations of motion applying of Hamilton principle on these combining form of energies. After that they used of the Galerkin method to solve the equations of motion. Jha and Dasgupta (2019) presented an unbalanced flexible shaft with an eccentric disc mounted at the mid-span by considering of the internal damping. Herein, they attenuate the Sommerfeld effect by linearized active magnetic bearing (AMB).
Other than the internal damping, the study of crack model is also important for the identification of crack parameters, so in the next section discusses the details of crack models in literatures.