Abstract: The fractional-order pendulum model under free vibration is a generalization of the classical integer-order pendulum model, which has a good application in the study of vibration problems in complex media with viscous characteristics. The solutions and dynamic properties of fractional linear pendulum model and integer order nonlinear pendulum model are systematically studied by Laplace transform method and dynamic system phase portrait analysis method. Especially, in the investigation of fractional model, a series of exact solutions of the types of Mittag-Leffler function are obtained. Furtherly, by comparing with the dynamic properties of the two kinds of solutions, the relevant conclusions are given. These results are valuable for similar study work on vibration problems in complex media.