## The Closed-Form Theory of Tuned Mass Damper with Hysteretic Friction

**Authors:** Roman Vinokur

The classical theory of lumped mechanical systems employs the viscous friction mechanisms (dashpots), while the loss factors of most solid structures are controlled by hysteresis. This paper presents analysis of the forced vibration of 2-DOF in-series systems with hysteretic friction where one of the partial 1-DOF systems plays the role of tuned (or auxiliary) mass damper (TMD). The assumption of hysteretic damping is acceptable if the loss factor remains about stable at least in the frequency range containing the resonance peaks. The closed-form simple relationships for the transmissibility at the resonance frequencies are derived in the “nearby” case where (1) the natural undamped frequencies of 2-DOF in-series system are most close to each other and (2) the loss factors of the 2-DOF system become similar and equal to the arithmetic average of the partial loss factors. The independent parameters are the mass ratio and partial loss factors. The relationships become very simple if the mass ratio is high or low compared to the square of each partial loss factor. In cases where the ratio of natural undamped frequencies of 2-DOF system are moderately lower or higher than in the “nearby case”, the transmissibility peak magnitudes are about similar to those for the “nearby” case. The results can be utilized for the noise and vibration control in machinery and buildings.

**Comments:** 24 Pages.

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### Submission history

[v1] 2017-12-16 16:46:07

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