<p>Many multibody systems in engineering exhibit characteristics of periodic response. When differential-algebraic equations (DAEs) are used to describe these multibody systems, the modeling process becomes more universal and programmatic. However, solving for the periodic response of DAEs directly remains a challenging problem. To address this issue, this paper proposes a symplectic finite element method in time (FET) based on the generalized variational principle. The general variational principle corresponding to DAEs is obtained through the method of Lagrange multipliers. By discretizing the variational principle using the preserved symplectic FET, the corresponding nonlinear algebraic equations are derived. During the iterative solution process, periodic boundary conditions are imposed, thereby obtaining the periodic response of the original DAEs. Three numerical examples, including single rigid body, rigid, and flexible multibody systems, are provided to validate the proposed method. The results demonstrate that this method can significantly reduce computation time compared to the step-by-step integration method while maintaining comparable accuracy. Additionally, during the computation process, sparse matrix storage can be employed to save memory. The main contribution of the paper is proposing an efficient, accurate, and universal algorithm for the periodic response of multibody systems governed by DAEs.</p>