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Research Papers

# Effective and Robust Vibration Control Using Series Multiple Tuned-Mass Dampers

[+] Author and Article Information
Lei Zuo

Department of Mechanical Engineering, State University of New York at Stony Brook, Stony Brook, NY 11794-2300lei.zuo@stonybrook.edu

J. Vib. Acoust 131(3), 031003 (Apr 07, 2009) (11 pages) doi:10.1115/1.3085879 History: Received March 19, 2007; Revised February 09, 2008; Published April 07, 2009

## Abstract

Various types of tuned-mass dampers (TMDs), or dynamic vibration absorbers, have been proposed in literature, including the classic TMD, (parallel) multiple TMDs, multidegree-of-freedom (DOF) TMD, and three-element TMD. In this paper we study the characteristics and optimization of a new type of TMD system, in which multiple absorbers are connected to the primary system in series. Decentralized $H2$ and $H∞$ control methods are adopted to optimize the parameters of spring stiffness and damping coefficients for random and harmonic vibration. It is found that series multiple TMDs are more effective and robust than all the other types of TMDs of the same mass ratio. The series two TMDs of total mass ratio of 5% can appear to have 31–66% more mass than the classical TMD, and it can perform better than the optimal parallel ten TMDs of the same total mass ratio. The series TMDs are also less sensitive to the parameter variance of the primary system than other TMD(s). Unlike in the parallel multiple TMDs where at the optimum the absorber mass is almost equally distributed, in the optimal series TMDs the mass of the first absorber is generally much larger than the second one. Similar to the 2DOF TMD, the optimal series two TMDs also have zero damping in one of its two connections, and further increased effectiveness can be obtained if a negative dashpot is allowed. The optimal performance and parameters of series two TMDs are obtained and presented in a form of ready-to-use design charts.

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## Figures

Figure 1

System with series multiple TMDs

Figure 2

Frequency responses of systems with series two TMDs of m1=m2 (solid), with classic TMD (dash), with series two TMDs of m1=m2 optimized when negative damping is allowed (dash-dot), and without TMD (dot), where the mass ratio μ=5% and primary damping ζs=0

Figure 3

The minimal peak magnitude in the frequency domain of the series two TMD of μ=5% at varied mass distribution for undamped (ζs=0, solid line) and damped (ζs=1%, dashed line) primary systems

Figure 4

H∞ optimal tuning ratios and damping ratios of the series two TMD of μ=5% at varied mass distribution, where ζ=0. Solid lines are for f1 and ζ1: dashed lines are for f2 and ζ2.

Figure 5

(a) System with four-element tuned-mass damper when m1∕(m1+m2)=0. (b) System with classic tuned-mass damper when m1∕(m1+m2)=1.

Figure 6

Frequency responses of systems with optimal series two TMDs with m1∕(m1+m2)=0.892 (thicker solid), parallel two TMDs (dot), 2DOF TMD (dash-dot) of optimal ρ=0.751(8), optimal three-or four-element TMD (dash), and classic TMD (thinner solid), where μ=5% and ζs=0

Figure 7

Three types of tuned-mass dampers: (a) series multiple TMDs, (b) parallel multiple TMDs, and (c) multiple DOF TMD

Figure 8

Robustness of system with optimal series two TMDs to the parameter changes of the primary system: nominal system (thicker solid), ms decreases by 10% (thinner solid), ms increases by 10% (dash), ks decreases by 10% (dash-dot), and ks increases by 10% (dot), where μ=5% and ζs=0

Figure 9

Robustness of the system with (a) optimal classic TMD, (b) optimal parallel two TMDs, (c) optimal three-or four-element TMD, and (d) optimal 2DOF TMD to the parameter changes of the primary system: nominal system (thicker solid), ms decreases by 10% (thinner solid), ms increases by 10% (dash), ks decreases by 10% (dash-dot), and ks increases by 10% (dot), where μ=5% and ζs=0

Figure 10

Robustness of the system with optimal series two TMDs to the parameter changes (a) of the first absorber and (b) of the second absorber in comparison to that of the system with (c) optimal classic TMD nominal system (thicker solid), when the mass of the said absorber decreases by 10% (thinner solid) or increases by 10% (dash), when the connection stiffness of the said absorber decreases by 10% (dash-dot) or increases by 10% (dot), when the connection damping of the said absorber decreases by 10% (thicker dash) or increases by 10% (thicker dash-dot), where μ=5% and ζs=0

Figure 11

The H∞ optimal mass distribution m1∕(m1+m2) of the series two-TMD system as a function of the primary damping ratios ζs, where the total mass ratio μ=(m1+m2)∕ms=5%

Figure 12

The H∞ optimal tuning ratios f1 (sold) and f2 (dash) and optimal damping ratios ζ1 (solid) and ζ2 (dash) for series two TMDs at the optimal mass distributions m1∕(m1+m2) as the function of primary damping ratios ζs in comparison with those of H∞ optimal classic TMD (dash-dot), where the total mass ratio μ=5%

Figure 13

Minimal peak magnitude of H∞ optimal series two-TMD system at the optimal mass distributions m1∕(m1+m2) as a function of the primary damping ratio ζs (solid) in comparison with that of H∞ optimal classic TMD (dash-dot), where the total mass ratio μ=5%

Figure 14

The H∞ optimal mass distribution m1∕(m1+m2) of series two-TMD system for various total mass ratio μ, where primary damping ζs=0

Figure 15

Minimal peak magnitude of series two-TMD system at the optimal mass distributions m1∕(m1+m2) as a function of the mass ratio μ (solid) in comparison with minimal peak magnitude of the classic TMD system (dash), where ζs=0

Figure 16

The H∞ optimal tuning ratios f1 (sold) and f2 (dash) and optimal damping ratios ζ1 (solid) and ζ2 (dash) for series two TMDs of the optimal mass distributions m1∕(m1+m2) at various mass ratios μ

Figure 17

The minimal normalized rms response from x0 to xs of the primary system with series two TMDs as a function of the mass distribution m1∕(m1+m2), where the total mass ratio μ=5% and the primary damping ζs=0

Figure 18

The H2 optimal tuning ratios f1 (solid) and f2 (dash) and damping ratios ζ1 (solid) and ζ2 (dash) for the series two TMDs as a function of the mass distribution m1∕(m1+m2), where the total mass ratio μ=5% and the primary damping ζs=0

Figure 19

Frequency responses of H2 optimal series two TMDs with m1∕(m1+m2)=0.909 (thicker solid), H2 optimal parallel two TMDs (dot) and parallel ten TMDs (thinner solid), H2 optimal 2DOF TMD of ρ∕d=0.780 (dash), and H2 optimal classic TMD (dash-dot), where the total mass of absorbers is 0.05ms for all the cases, and the primary system has no damping

Figure 20

Impulse responses of systems with H2 optimal series two TMDs of m1∕(m1+m2)=0.909 (solid), H2 optimal parallel two TMDs (dot), H2 optimal 2DOF TMD of ρ∕d=0.780 (dash), and H2 optimal classic TMD (dash-dot), where the total mass of absorbers is 0.05ms for all the cases, and the primary system has no damping

Figure 21

Impulse responses of systems with H2 optimal series two TMDs of m1∕(m1+m2)=0.909 (thicker line) and H2 optimal parallel ten TMDs (thinner line), where the total mass ratio is 0.05ms for both cases, and the primary system has no damping

Figure 22

The H2 optimal mass distribution m1∕(m1+m2) of series two-TMD system for various mass ratios μ, where ζs=0

Figure 23

Minimal rms responses normalized by ωs of series two-TMD system of the optimal mass distribution m1∕(m1+m2) at various mass ratios μ (solid), compared with optimal parallel two-TMD system (dash-dot), optimal parallel ten-TMD system (dot), and the classic TMD system (dash), where ζr=0

Figure 24

The H2 optimal tuning ratios f1 (sold) and f2 (dash) and optimal damping ratios ζ1 (solid) and ζ2 (dash) for series two TMDs at the optimal mass distributions m1∕(m1+m2) at various mass ratios μ, where ζs=0

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