  For a pendulum suspended below a vibrating block with white noise forcing,
the solution in which the pendulum remains vertical is called the single mode
solution. When this solution becomes unstable there is energy transfer from the
block to the pendulum, helping to absorb the vibrations of the block. We study
the Lyapunov exponent governing the almost-sure stability of the process
linearized along the single mode solution. The linearized equation is excited
by a combination of white and colored noise processes, which makes the
evaluation of the Lyapunov exponent non trivial. We obtain an explicit third
order asymptotic expression for the Lyapunov exponent as the intensity of the
white noise forcing tends to zero.
