Abstract: The fractional generalized Bagley-Torvik equation as a generalized form of relaxation-vibration model,it has better physical background,and it has very good applications on investigating the problem of vibration in complex medium and modeling motion of a rigid plate immersed in a Newtonian viscous fluid.By using the Laplace transform,exact solutions of the fractional generalized Bagley-Torvik equation are studied,under different parametric conditions,different kinds of analytic solutions of this model are obtained,the dynamical behaviors of these exact solutions are also discussed.In order to show dynamical properties of these exact solutions intuitively,by using software of Maple,the graphs according to time evolution of several representative exact solutions are illustrated.