In an optomechanical system with double-sided cavities, it was shown that squeezing and entanglement can be dramatically enhanced by a suitable periodically-modulated field driving either or both ends of the cavity.But, whether two-fold amplitude modulations can have interference effects and further enhance the desired effect is still not known. Here, we revisit this question and show that the relative phase of the two external lasers plays an important role in controlling the classical dynamics of the optical and mechanical modes, leading to irregular oscillating behaviors. Furthermore, the asymptotic dynamics of the mechanical mean values can transitfrom one-period oscillation to period-doubling oscillation, which is a nonlinear process associated with chaos. Except for changes in the oscillation period, the oscillation amplitudes of the classical mechanical mean values can be enhanced by increasing the relative phase between two modulation driving lasers. The enlarged oscillation amplitudes of the classical mechanical mean values could destroy the condition of the effective detuningto generate entanglement. As a consequence, the entanglement between two cavity fields gradually decreases and disappears entirely when the relative phase between two modulation driving lasers is large enough. Although the entanglement between two cavity fields can be hardly enhanced by the cooperative effect of two-fold modulations even when the relative phase is chosen properly, this investigation may provide a new approach for harnessing optomechanical nonlinearity to manipulate rich dynamics.