Metastructures can be designed to exhibit extraordinary properties and thus are widely applicable in various fields, such as vibration suppression, wave control and energy harvesting. In the present work, a dynamic homogenization model is strictly derived for a metabeam (a beam with periodic resonators) by using the asymptotic homogenization method. Based on the homogenization model, the flexural wave dispersion relation and vibration modes of the metabeam are analyzed and validated, and analytical solutions for its linear and nonlinear vibration responses are also presented. Furthermore, the effect of periodic resonators on the vibration suppression performance of the metabeam is discussed. It is demonstrated that such kind of dynamic homogenization model is of great convenience to predict the vibration responses of metastructures besides the widely studied dispersion relation. Thus it is expected to be useful for the vibration suppression and control of engineering structures.