Effects of imperfection on the non-linear oscillations of circular plates spinning near critical speed

Abstract

Motivated by the well-known experiments of Tobias and Arnold (Proc. Inst. Mech. Engrs. 171 (1957) 669–690), we study the large-amplitude oscillations of an imperfect, flexible circular plate spinning near a critical speed resonance. A weakly non-linear theory for the transverse oscillations of the rotating, imperfect plate is derived and the near resonant response of a repeated pair of backward and forward travelling waves is analysed. The experimental results of Tobias and Arnold are easily captured within the present analysis. In addition, several new dynamical phenomena are predicted as the imperfection parameter is increased, such as emergence of pseudo-standing waves; their loss of stability through Hopf bifurcations leading to backward and forward travelling waves with slow, periodically modulated amplitudes and phases; period doubling sequences leading to Rössler-type chaotic attractors and boundary crises phenomena; and emergence of isolas of steady wave motions. Sufficiently high asymmetry can reduce by half, the vibration energy near resonant collapse phenomena. The results presented can provide guidelines for designing imperfections in circular saws and hard disk drives to reduce vibrations near critical speed.