  The amplitude of self-oscillating mechanical resonators in cavity
optomechanical systems is typically limited by nonlinearities arising from the
cavity's finite optical bandwidth. We propose and demonstrate a feedback
technique for increasing this limit. By modulating the cavity input field with
a signal derived from its output intensity, we increase the amplitude of a
self-oscillating GHz frequency mechanical resonator by $22\%$ (increase in
coherent phonon number of $50\%$) limited only by the achievable optomechanical
cooperativity of the system. This technique will advance applications dependent
on high dynamic mechanical stress, such as coherent spin-phonon coupling, as
well as implementations of sensors based on self-oscillating resonators.
