Numerical analysis of axial vibrations for a continuum model of a one-dimensional rod with a tuned mass damper

In structural vibration control, tuned mass dampers (TMDs) are widely equipped, typically at the top of structures, to alleviate potential vibration. To evaluate the effectiveness of TMDs, scholars have proposed discretized physical models to simulate the interaction between structures and TMDs. However, few efforts have been made to establish and solve more accurate structure-TMD continuum models whose equations of motion (EOMs) for the main structures are represented by partial differential equations (PDEs) with dynamic boundary conditions (DBCs). Therefore, the authors aim to (1) set up EOMs for a continuous model dynamically affected by a moving body on the boundary and (2) theoretically validate the feasibility of solving such EOMs. To ease equation derivation and numerical calculation, a toy model of a one-dimensional (1D) rod with a TMD is established instead. It is the simplest structure-TMD model and its EOMs are derived by Hamilton’s Principle. Without the availability of exact solutions, a series of discretization numerical methods including Mean Weighted Residual (MWR), Ritz Method, Finite Element Method (FEM), and Lumped Mass Method (LMM) are discussed. Finally, numerical experiments are carried out, and it turns out that FEM and LMM can give satisfying numerical solutions with high efficiency for the problem under consideration.