Sommerfeld effect in rotationally symmetric planar dynamical systems

Sommerfeld effect concerns the non-linear jump phenomena induced due to the influence of the unbalance response on a non-ideal drive around the critical speed of the excited structure. In this work, we study the influence of external and internal dampings and gyroscopic forces on the Sommerfeld effect in rotationally symmetric planar dynamical systems. The rotational symmetry assumption allows us to obtain neat analytical results for the steady state dynamics. We show that the rotating material or internal damping and the gyroscopic forces influence the spin rate of the non-ideal system and the former changes the system dynamics in an unexpected manner. In particular, we show that the stability threshold may restrict the jump phenomena due to the Sommerfeld effect for larger values of internal damping. Moreover, it is also shown that the Sommerfeld effect would cease to exist under certain conditions. A stability condition for various steady-state equilibriums (branches of steady-state solutions) is derived. A rotor dynamics problem and a structural dynamics problem where the systems interact with a non-ideal source are considered as illustrative examples. A few numerical results are given to validate the analytical solutions.