初期変位付与型TMDの同調比の設定

 A TMD (Tuned Mass Damper) is a passive-type control device that consists of a small mass, a spring and a damper. And it absorbs the oscillation energy of structures as the kinetic energy of its mass and disperses it via its damper. The TMD shows high control performance for harmonic responses. On the other hand, the TMD has limited capacity to suppress transient responses. The reason is that there is some time delay before the TMD becomes fully effective because they are initially at rest.<br> Thus, to control the transient response more effectively, we propose TMDs with initial displacement, that is, dampers whose springs are stretched until the release moment.<br> In our previous study, we focused on the considerably high modal damping ratio of the second mode compared to the first mode based on the relationship between the TMD damping ratio and the modal damping ratio of a two-degree-of-freedom model. And we proposed the design formulas for the optimal tuning and damping ratios and the initial displacement to attain high control performance under impulse loading. The proposed design formulas are based on the principle that by giving the specific TMD initial displacement under the specific structural initial condition the structural response of the first mode with low modal damping is eliminated while the structural response of the second mode with high modal damping is only oscillated. But the physical meaning of the design formula for TMD initial displacement is not clear because it is an approximate solution based on the perturbation method.<br> Then, we formulated the equation for initial conditions to release TMD initial displacement from the theoretical free vibration solution and showed its physical meaning clearly. We studied about TMD initial displacement and structural conditions to release initial displacement using the complex plane, and we showed that the initial structural condition to oscillate only the second mode was limited to a neighborhood of ẋ₀≠0, x₀=0. On the other hand, we proved that, by dividing one TMD into plural TMDs that have different natural frequencies, any initial structural conditions to release TMDs became possible.<br> In this paper, firstly, we analytically study the effect of different settings of two TMDs tuning ratios on TMD initial displacements to eliminate the first modal response with the lowest modal damping and structural responses for free vibration using a basic three-degree-of-freedom model. Next, we decompose the initial condition to modal eigenvectors on the complex plane and study the interrelationship between each initial condition of a main mass and two TMDs. Then, we show the reason why different settings of TMDs tuning ratios affect the TMD initial displacements to eliminate the first modal response and the structural responses. Finally, by using the energy indicator “TMD power flow” that shows an additional TMD damping effect proposed by Soong, T.T. and Dargush, G.F., we study the control performance of TMDs with different tuning ratios condition.