Optimal parameters of tuned mass dampers for an inverted pendulum with two degrees of freedom

In reality, an inverted pendulum can be used to model many real structures as the fluid tower, super-tall buildings, or articulated tower in the ocean, etc. However, for the inverted pendulum with two degrees of freedom, to the best knowledge of the author, there is no study to determine optimal parameters of two tuned mass dampers (TMD) by using the maximization of equivalent viscous resistance method. Therefore, the current study presents the analytical solutions to the optimization of two orthogonal TMDs, which is used to eliminate vibration of the inverted pendulum with two degrees of freedom. The parameters considered in optimizing are the natural frequency ratios and damping ratios of the two TMDs. The new results of this paper can be summarized as follows: Firstly, the equivalent resistance forces of the two TMDs acting on the inverted pendulum with two degrees of freedom are established. Secondly, the quadratic torque matrices of the vibration response of the inverted pendulum attached with two TMDs is revealed. Thirdly, the optimal expressions are derived using the maximization of equivalent viscous resistance method. The obtained formulae provide exact solutions for the proposed problem. Finally, to confirm the effectiveness of the obtained formulae, parametric studies on vibration are performed for sample articulated tower in the ocean with and without optimal TMDs. Numerical results show that vibrations of the articulated tower attached with optimal TMDs are effectively eliminated. This confirms that the optimal parameters of the two TMDs are determined in this paper are reliable and accurate.