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Technical Brief

# A Closed-Form Optimal Tuning of Mass Dampers for One Degree-of-Freedom Systems Under Rotating Unbalance Forcing

[+] Author and Article Information
T. Argentini

Department of Mechanical Engineering,
Politecnico di Milano,
via La Masa 1,
Milano 20156, Italy
e-mail: tommaso.argentini@polimi.it

M. Belloli

Department of Mechanical Engineering,
Politecnico di Milano,
via La Masa 1,
Milano 20156, Italy
e-mail: marco.belloli@polimi.it

P. Borghesani

School of Chemistry,
Physics and Mechanical Engineering,
Queensland University of Technology,
#2 George Street,
Brisbane, Queensland 4000, Australia
e-mail: p.borghesani@qut.edu.au

Given a quadratic equation ax2 + bx + c = 0 with solutions x1 and x2, then $(1/x1)+(1/x2)=-(b/c)$.

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 27, 2014; final manuscript received January 5, 2015; published online February 18, 2015. Assoc. Editor: Yukio Ishida.

J. Vib. Acoust 137(3), 034501 (Jun 01, 2015) (6 pages) Paper No: VIB-14-1101; doi: 10.1115/1.4029576 History: Received March 27, 2014; Revised January 05, 2015; Online February 18, 2015

## Abstract

This paper is focused on the study of a vibrating system forced by a rotating unbalance and coupled to a tuned mass damper (TMD). The analysis of the dynamic response of the entire system is used to define the parameters of such device in order to achieve optimal damping properties. The inertial forcing due to the rotating unbalance depends quadratically on the forcing frequency and it leads to optimal tuning parameters that differ from classical values obtained for pure harmonic forcing. Analytical results demonstrate that frequency and damping ratios, as a function of the mass parameter, should be higher than classical optimal parameters. The analytical study is carried out for the undamped primary system, and numerically investigated for the damped primary system. We show that, for practical applications, proper TMD tuning allows to achieve a reduction in the steady-state response of about 20% with respect to the response achieved with a classically tuned damper.

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## Figures

Fig. 1

Damped TMD connected to a primary system forced by a rotating unbalance

Fig. 2

Magnitude of X˜ versus g with nonoptimal TMD parameters (f = 1 and μ = 0.05)

Fig. 3

Optimal TMD parameters as a function of mass parameter μ

Fig. 4

Magnitude of X˜ versus g with optimal TMD parameters, using μ = 0.05

Fig. 5

Magnitude of Y˜ versus g with optimal TMD parameters, using μ = 0.05

Fig. 6

Ratio between the maxima of the frequency response functions (infinity norm) obtained with classical and optimal parameters as a function of the mass ratio

Fig. 7

Numerically optimized values of ζd and f as a function of ζ for μ = 0.05: solution 1 and solution 2

Fig. 8

Ratio of maxima ||X˜num||∞ as a function of ζ for μ = 0.05: solution 1 and solution 2

Fig. 9

Optimized frequency response functions |X˜num| for solutions 1 and 2, for increasing values on ζ. For this example, with μ = 0.05, the threshold is ζ = 0.012. Solutions 1 and 2—before threshold, solution 1—after threshold, and solution 2—after threshold.

Fig. 10

ζlim as a function of the mass ratio μ

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