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Research Papers

Transient Response of MDOF Systems With Inerters to Nonstationary Stochastic Excitation

[+] Author and Article Information
Sami F. Masri

Viterbi School of Engineering,
University of Southern California,
Los Angeles, CA 90089-2531
e-mail: masri@usc.edu

John P. Caffrey

Viterbi School of Engineering,
University of Southern California,
Los Angeles, CA 90089-2531

Hui Li

School of Civil Engineering,
Harbin Institute of Technology,
Harbin 150090, China
e-mail: lihui@hit.edu.cn

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 29, 2017; final manuscript received August 2, 2017; published online August 21, 2017. Assoc. Editor: Walter Lacarbonara.

J. Appl. Mech 84(10), 101003 (Aug 21, 2017) (13 pages) Paper No: JAM-17-1283; doi: 10.1115/1.4037551 History: Received May 29, 2017; Revised August 02, 2017

Explicit, closed-form, exact analytical expressions are derived for the covariance kernels of a multi degrees-of-freedom (MDOF) system with arbitrary amounts of viscous damping (not necessarily proportional-type), that is equipped with one or more auxiliary mass damper-inerters placed at arbitrary location(s) within the system. The “inerter” is a device that imparts additional inertia to the vibration damper, hence magnifying its effectiveness without a significant damper mass addition. The MDOF system is subjected to nonstationary stochastic excitation consisting of modulated white noise. Results of the analysis are used to determine the dependence of the time-varying mean-square response of the primary MDOF system on the key system parameters such as primary system damping, auxiliary damper mass ratio, location of the damper-inerter, inerter mass ratio, inerter node choices, tuning of the coupling between the damper-inerter and the primary system, and the excitation envelope function. Results of the analysis are used to determine the dependence of the peak transient mean-square response of the system on the damper/inerter tuning parameters, and the shape of the deterministic intensity function. It is shown that, under favorable dynamic environments, a properly designed auxiliary damper, encompassing an inerter with a sizable mass ratio, can significantly attenuate the response of the primary system to broad band excitations; however, the dimensionless “rise-time” of the nonstationary excitation substantially reduces the effectiveness of such a class of devices (even when optimally tuned) in attenuating the peak dynamic response of the primary system.

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References

Figures

Grahic Jump Location
Fig. 1

Model of MDOF system with an inerter m* interposed between masses mn−2 and mn

Grahic Jump Location
Fig. 2

Comparison of the mean-square transient response E[q12(t)] of the primary system, without a damper, under two different nonstationary excitations, and two different ratios of critical damping ζ1 = 0.01 and 0.05. LHS panel shows the response under an excitation having a step-function envelope and RHS panel shows the transient response when the envelope function is g(t)=e−1.5t−e−3.5t. For added resolution different amplitude scales are used in the two panels. The abscissas in the plots show normalized time in terms of the primary system period T1.

Grahic Jump Location
Fig. 3

Model of 2DOF system with inerter

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Fig. 4

Comparison of the normalized nonstationary m.s. E[q12(t)] of the response of the primary system m1 in a coupled 2DOF system that is subjected to different types of random excitation. LHS panel corresponds to the case where the envelope of the excitation is a step-function, while in the RHS panel the excitation has an envelope function g(t)=e−1.5t−e−3.5t. In each panel, two different primary system damping ratios ζ1 are shown (0.01 and 0.05). In all cases shown, m2/m1 = 0.10, ω2/ω1 = 1, and ζ2 = ζ1. A normalized time period of duration 8t/T1 is shown. For added resolution, different amplitude scales ζ1 = 0.05.

Grahic Jump Location
Fig. 5

Comparison of the normalized nonstationary m.s. response E[q12(t)] of the primary system m1 with a DVN that is subjected to different types of random excitation. LHS panel corresponds to the case where the envelope of the excitation is a step-function, while in the RHS panel the excitation has an envelope function g(t)=e−1.5t−e−3.5t. In each panel, the primary system damping ζ1 = 0.01. Four cases are considered: (1) SDOF system without damping devices, (2) SDOF system when provided with an optimally tuned DVN with mass ratio μ = 0.01, (3) SDOF system when provided with an optimally tuned DVN with mass ratio μ = 0.02, and (4) SDOF system when provided with an DVN with mass ratio μ = 0.01 and an inerter with mass ratio μ* = 0.01, in which the tuning parameters of the combined DVN + inerter are optimized so as to make the transfer function of the primary system nearly flat around the natural frequency ω1. For added resolution, different amplitude scales are used in the two panels.

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Fig. 6

Potential locations of attaching a DVN with an inerter to the top floor m4 of a building-like 4DOF primary system model. Four scenarios represent the choices of attaching the RHS node of the inerter m* to the DVN device m5, and the LHS node to one of four possible locations: (a) nodes (3, 5), (b) nodes (2, 5), (c) nodes (1, 5), and (d) nodes (0, 5).

Grahic Jump Location
Fig. 7

Comparison of the normalized nonstationary m.s. response E[qi2(t)] of the primary system under stationary excitation, without and with an optimized DVN having a mass ratio μ = 0.01. In each panel, the primary system damping ζ1 = 0.01. (a) 4DOF reference system m.s. response of mi, i = 1,…, 4 and (b) 5DOF augmented system m.s. response of mi, i = 1,…, 5, with E[q52(t)] corresponding to the DVN response. For ease of comparison, identical amplitude and time scales are used for both panels. Time period covered spans a range of 25t/T1.

Grahic Jump Location
Fig. 8

Comparison of the normalized nonstationary m.s. response E[qi2(t)] of the primary system under nonstationary excitation, without and with an optimized DVN having a mass ratio μ = 0.01. LHS panel corresponds to the case where the envelope of the excitation is a step-function, while in the RHS panel the excitation has an envelope function g(t)=e−1.5t−e−3.5t. In each panel, the primary system damping ζ1 = 0.01. (a) 4DOF reference system m.s. response of mi, i = 1,…, 4 and (b) 5DOF augmented system m.s. response of mi, i = 1,…, 5, with E[q52(t)] corresponding to the DVN response. For ease of comparison, identical amplitude and time scales are used for both panels. Time period covered spans a range of 10t/T1.

Grahic Jump Location
Fig. 9

Spectral characteristics of reference 4DOF primary system and controlled 5DOF system with a DVN and inerter. (a) LHS panel shows log-linear plot of the magnitude of the complex transfer functions H44, H33, H22, and H11 of the reference 4DOF system, over a frequency band covering its four modal frequencies that are listed in Table 2. Note that the main peaks of the dimensionless response functions all occur at the fundamental frequency ω1 = 2π, with the corresponding modal damping being ζ1 = 0.01. (b) RHS panel shows the transfer functions of the augmented system under five scenarios, all of them with a DVN of μ = 0.01: (1) Reference 4DOF system without any control devices, (2) 4DOF + optimized DVN, (3) 4DOF + DVN + inerter with μ* = 0.01, (4) 4DOF + DVN + inerter with μ* = 0.05, and (5) 4DOF + DVN + inerter with μ* = 0.10. All cases with inerter have nodes (5, 0). For enhanced readability, different amplitude and frequency scales are used in the LHS and RHS panels.

Grahic Jump Location
Fig. 10

Influence of inerter nodes on the mean-square response E[q42(t)] when used in the reference 4DOF primary system that is equipped with a DVN of mass ratio μ = 0.01 attached to the top mass m4, and that is provided with an inerter of mass ratio μ* = 0.01 whose tuning parameters are optimized to satisfy Den Hartog’s criteria. Five normalized curves are shown in the LHS and RHS panels of the figure: (1) the m.s. response E[q42(t)] of m4 when only the DVN is used, (2) E[q42(t)] when adding an inerter with nodes {3, 5}, (3) E[q42(t)] when adding an inerter with nodes {2, 5}, (4) E[q42(t)] when adding an inerter with nodes {1,5}, and (5) E[q42(t)] when adding an inerter with nodes {0, 5}. LHS panel of plots covers the m.s. response under stationary excitation for normalized time period of 25 t/T1 and RHS panel of plots covers the m.s. response under nonstationary excitation for normalized time period of 10 t/T1. For enhanced readability and resolution, different time and amplitude scales are used in each panel.

Grahic Jump Location
Fig. 11

Influence of inerter mass on the mean-square response E[q42(t)] when used in the reference 4DOF primary system that is equipped with a DVN of mass ratio μ = 0.01 attached to the top mass m4, and that is provided with an inerter of different mass ratios μ* whose tuning parameters are optimized to satisfy Den Hartog’s criteria. In all cases where the inerter is used, its nodes are {0, 5}. Five normalized curves are shown in the LHS and RHS panels of the figure: (1) the m.s. response E[q42(t)] of m4 without any structural control devices, (2) the m.s. response E[q42(t)] of m4 when only the DVN is used, (3) E[q42(t)] when adding an inerter with μ* = 0.01, (4) E[q42(t)] when adding an inerter with μ* = 0.05, and (5) E[q42(t)] when adding an inerter with μ* = 0.10. LHS panel of plots covers the m.s. response under stationary excitation for normalized time period of 25 t/T1 and RHS panel of plots covers the m.s. response under nonstationary excitation for normalized time period of 10 t/T1. For enhanced readability and resolution, different time and amplitude scales are used in each panel.

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