A Mixed Finite Element Method for Vibration Problems of Non-Homogeneous Damped Beams

Beam is one of the common structures in engineering, with the development of technology, homogeneous beams no longer meet the needs of engineering structural design, for this reason, people have researched the non-homogeneous beams. In this paper, we study the mixed finite element method for the vibration problem of non-homogeneous damped beams. The fourth-order differential equations are transformed into a system of low-order partial differential equations by introducing intermediate variables, constructing a semi-discrete extended mixed finite element format, proving the existence and uniqueness of the solution of the format, and utilizing the elliptic projection operator for the error estimation. The time derivative term is discretized by the central difference, and the fully discrete mixed element format is given to prove the stability and convergence of the format. The feasibility and effectiveness of the mixed method are verified by numerical examples, and the effects of different damping coefficients μ on beam vibration are investigated.