<p>In this paper, bifurcations of homoclinic torus for a SD oscillator under quasiperiodical excitations are discussed for the purpose of regulation of dynamics and optimal design in engineering. The Melnikov method for a class of bistable oscillators under quasiperiodical excitations is simply described by perturbing the homoclinic manifolds in the high-dimensional phase space and employed in order to obtain the threshold of parameters for the tangency of the perturbed homoclinic structure for the SD oscillator. A semi-analytic and semi-numerical technique is carried out to overcome the difficulty of the improper integration for the corresponding Melnikov function. Under the dual-frequency quasiperiodical excitations, a comprehensive bifurcation analysis is carried out on the bifurcation sets in the five-dimensional parameter space of the SD oscillator, and the theoretical prediction is further verified by numerical simulations. Due to the increase in frequency and the strong nonlinearity of the SD oscillator, a more complex phenomenon occurs.</p>