Physics, Tuning, and Performance of the TMD-Inerter for Harmonic Vibrations

This paper analyzes the physics of the TMD-Inerter for harmonic vibrations. The basic TMD-Inerter layout is assumed, where the inerter is installed between the TMD mass and the structural mass. For harmonic vibrations, the inerter force can be formulated as a function of terminal displacements. This formulation demonstrates that the inerter force is, in fact, a negative stiffness force with frequency-dependent negative stiffness coefficient. Based on this finding, the optimal stiffness tuning of the TMD-Inerter is derived. As this stiffness tuning can only be realized by a controlled actuator, the tuning of the spring of the TMD-Inerter is presented. As this spring is a passive element, its optimum tuning must be made at a selected frequency of vibration. It is shown that the average of the TMD natural frequency and structural eigenfrequency leads to a close to optimal spring tuning. This approach needs to be combined with increased damping of the TMD-Inerter to minimize the structural displacement response. Despite the close to optimal tunings of stiffness and damping, the resulting primary structure displacement response is approximately 41.6% greater than that due to the classical TMD. The reason for this lies in the fact that the passive spring of the TMD-Inerter cannot compensate for the frequency-dependent negative stiffness of the inerter within the entire frequency range.