  We demonstrate the generation of a strong mechanical squeezing in a
dissipative optomechanical system by introducing a periodic modulation in the
amplitude of a single-tone laser driving the system. The mechanical oscillator
is quadratically coupled to the optical mode, which contributes to a strong
squeezing exceeding the 3-dB standard quantum limit. The Bogoliubov mode of the
mechanical oscillator also cools down to its ground state due to sideband
cooling. We further optimize this ratio of sideband strengths to introduce
enhanced squeezing. We also compare our results with the analytical (under
adiabatic approximation) and the exact numerical solution. Even for a thermal
occupancy of 10^4 phonons, mechanical squeezing beyond 3 dB and a strong
optomechanical entanglement is observed.
