A new look on the stabilization of inverted pendulum with parametric   excitation and large random frequencies: analytical and numerical approaches
  In this paper we explore the stability of an inverted pendulum with generalized parametric excitation described by a superposition of $N$ sines with different frequencies and phases. We show that when the amplitude is scaled with the frequency we obtain the stabilization of the real inverted pendulum, i.e., with values of $g$ according to planet Earth ($g\approx 9.8$m/s$^{2}$) for high frequencies. By randomly sorting the frequencies, we obtain a critical amplitude in light of perturbative theory in classical mechanics which is numerically tested by exploring its validity regime in many alternatives. We also analyse the effects when different values of $N$ as well as the pendulum size $l$ are taken into account. 
Roberto da Silva, Sandra D. Prado, Henrique A. Fernandes
physics.class-ph