To investigate vibration control and optimal design of transmission lines with tuned-mass-damper-inerter (TMDI), the motion equation of transmission lines with TMDI is established in the paper, and the closed-form solutions of the response spectrum of transmission line displacement are derived by the frequency domain analysis method. The design parameters of TMDI are optimized by fixed-point theory, and the vibration control performance of TMDI is discussed. The results show that the increase in apparent mass ratio has a positive effect on the vibration control performance of TMDI; the vibration control performance is greatly affected by frequency ratio and limited by damping ratio; the increase in both mass ratio and apparent mass ratio reduces the peak values of the displacement response spectra of transmission line with TMDI; however, blindly increasing the apparent mass and mass ratio (β > 0.2 or μ > 0.4) has a limited effect on improving the vibration control performance of TMDI; compared with conventional TMD, the peak values of the controlled displacement response spectrum of the transmission line with TMDI can be reduced by about 12%, and TMDI has a better vibration suppression effect on the transmission lines.

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This paper proposes a novel ungrounded TMDI to improve the vibration suppression performance of the transmission line under harmonic excitation. This type of inerter-based damper may transform a translational motion into a rotational motion, greatly increasing the efficiency of vibration suppression. In the present study, the differential equations of motion are first derived based on the transmission line with an ungrounded TMDI structure. Then the closed-form solution of the displacement response spectrum considering the influence of the suspension location of the inerter is developed. The impact of the inerter location on vibration suppression performance is investigated in depth by defining the suspension location factor (υ) and tuning the damping ratio and frequency ratio. The results demonstrate that the suspension location of the inerter has a substantial impact on the damping ratio, frequency ratio, and vibration suppression performance. When the connection location of the inerter is near to the mass of the damper, it degrades the vibration suppression performance of the system. The failure phenomenon of the inerter occurs in the range of 0.2 < υ < 0.3, indicating that the presence of the inerter in this range does not enhance vibration suppression performance. The modal coordinate difference has a considerable impact on the vibration suppression efficacy of the TMDI. With increasing modal coordinate differences, the vibration suppression performance of the TMDI grows dramatically.