ess_quantity">dimensional
      analysis</a> (using the natural frequency $\sqrt{\frac{g}{l}}$ to match
      units) occurs when $b \sqrt\frac{l}{g} \gg ml^2$. This is the case of
      heavy damping, for instance if the pendulum was moving in molasses.  In
      this case, the damping term dominates the acceleration term, and we have:
      $$ml^2 \ddot\theta + b\dot\theta \approx b\dot\theta = u_0 -
      mgl\sin\theta.$$ In other words, in the case of heavy damping, the system
      looks approximately f