Spatial Pendulum TMD With Two Tuning Frequencies

Some structures have different natural frequencies in the two lateral X and Y-directions. Tuned damping of such structures require using either a) the less attractive option of two regular PTMDs each tuned to one of the natural frequencies of the structures’ lateral modes or b) one pendulum TMD with two different tuning frequencies (one for each lateral directions), necessitating two different swinging lengths. Pendulum TMDs with two different tuning frequencies in the two lateral X and Y directions, are realized by constraining the swinging length of the pendulum in one direction but not in the other direction. Such two degree-of-freedom pendulum tuned mass damper, is called Bi-PTMD. In this work, the dynamics of a two degree-of-freedom pendulum tuned mass damper (Bi-TMD) appended to a structure with two low-frequency, lateral degrees of freedom (representing the first two modes of a tall structure) is studied and the nonlinear differential equations of motion are derived using the Lagrangian mechanics approach. The equations of motion are simplified using small angle and slow motion assumptions. The system of nonlinear differential equations are numerically simulated in Matlab/Simulink environment and the responses of the structure without and with the pendulum Bi-TMD to a number of different perturbations in the lateral directions are evaluated. The numerical model is verified by comparing its simulation results with the outcomes of SimScape Multibody physical model of the same system.