טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentKligerman Yuri
SubjectDynamics and Stability of Rotating Systems with
Electromagnetic Damping
DepartmentDepartment of Mechanical Engineering
Supervisors Dr. Itzhak Porat
Professor Mark Darlow
Dr. Gavril Solomon


Abstract

An analytical model of an electromagnetic non-contact eddy-current damper for reducing lateral vibration of rotating machinery is developed in the present study.  The damper dissipates energy through induced eddy currents generated in a small disk mounted to and rotating with the shaft.  The eddy-current damper is modeled by a thin nonmagnetic disk translating and rotating in an air gap of a direct current electromagnet.


         The force acting on the rotor in an eddy-current damper is obtained analytically as nonlinear function of the rotor lateral displacement and velocity.  The expression of this force includes a viscous damping term and a cross coupling term.  The cross coupling term was shown to be the source of the rotor equilibrium state instability.  The apparent source of instability is related to the rotation of an eccentric eddy-current damper in the electromagnetic field.  The analysis yields that the rotor equilibrium state becomes unstable via a Hopf bifurcation when reaching a specific supercritical angular velocity.  The threshold of the rotor equilibrium state stability is obtained in a closed form and depends on the natural frequency of the undamped system and all kinds of damping.


         Rubber o-rings supporting the shaft bearings were found to be the main source of the external nonmagnetic damping in the system investigated.  Their equivalent linear viscous characteristics were evaluated from the experiment. The models of frequency dependent damping and frequency independent damping in rubber o-rings were considered. The experimental tests corroborate the analytical approach.


         The linear model of the rotor does not explain the supercritical operation above the threshold of instability.  The rotating system model was extended by inclusion of a cubic restoring force representing nonlinear behavior of shaft supports (e.g., rubber o-rings).  A self excited limit cycle emerges around the unstable equilibrium state.  The radius and frequency of the limit cycle are obtained in a closed form.  Self-excited oscillation represents nonsynchronous whirl of the rotor.


         Forced vibration induced by the rotating system mass imbalance is also investigated analytically and numerically.  The analysis of the forced system yields that the unbalance response of the low speed operating nonlinear rotating system represents synchronous whirl whereas high speed operation is governed by coexisting periodic and quasiperiodic solutions.  Analytical solution is obtained for the periodic unbalance response of the rotating system with a cubic restoring force and an eddy-current damper.  Aperiodic solutions are analyzed using a fourth order Runge-Kutta integration scheme.  Stability of the periodic solutions obtained for the self-excited oscillation and the unbalance forced response are analyzed by use of Floquet theory.


         Significance of nonlinearity associated with the eddy-current damping in the system is analyzed.  The influence of nonlinearity due to eddy-current damping is negligible near the Hopf bifurcation point and builds up slightly with increasing angular velocity of the shaft.


         Analysis presented in this work enables an explanation of the nonlinear dynamics and stability phenomena documented for the rotating systems controlled by the electromagnetic eddy-current dampers.  The alternative eddy-current damper design approaches that could be considered to provide effective damping at all speeds and avoid the instability problems are discussed.