Starting from the Gyrator trnasform formula, the closed-form filed distribution of generalized sinh-Gaussian beam passing through such a transform system is derived, and the intensity distribution and the corresponding phase distribution associated with the transforming generalized sinh-Gaussian beams are analyzed. Based on the numerical method, the distributions are graphically drawn and it is found that, for appropriate beam parameters and by carefully adjusting the transform angle of Gyrator transform, the dark hollow beam with topological charge index being unity can be realized with sinh-Gaussian beam carrying an edge dislocation. And the influences of the beam parameters and the transform angle on the formation of perfect dark hollow beams are analyzed. However, it is impossible to obtain dark hollow beam with Gyrator transforming cosh-Gaussian beam.