Parametric double pendulum
José Carlos Sartorelli, Bruno Serminaro
We have studied a parametric double pendulum which is driven by a pulsating
support motion $y_{p}=a\cos (2\pi ft)$ in the vertical direction (see Fig. 1). The arm 1
is attached to the oscillating pivot and is heavier than the arm 2 $(M_{1}\sim 3.5\,M_{2}).$ Both arms have approximatelly the same length.
The frequencies of the two fundamental modes are $f_{in}\approx 1.4$Hz (in
phase) and $f_{out}=2.4$Hz (out of phase). By letting $\theta_1$ and $\theta_2$ denote the angles that arm 1 and 2 make with the vertical, respectively, and by direct integration of the
equations of motion,
we have determined the borders of the stability region of the four fixed
points $(\theta _{1}^{\ast }\ ,\theta _{2}^{\ast })=(0,0),(0,\pi ),(\pi ,0)$
and $(\pi ,\pi )$ in the parameter space $(f,a).$ We have observed a small region
where all the fixed points are stable and a large one where all of them are
unstable.