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Research Papers

Transient Response of a SDOF System With an Inerter to Nonstationary Stochastic Excitation

[+] Author and Article Information
Sami F. Masri

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

John P. Caffrey

Viterbi School of Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: jpcengr@aol.com

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 9, 2016; final manuscript received February 3, 2017; published online February 22, 2017. Assoc. Editor: Walter Lacarbonara.

J. Appl. Mech 84(4), 041005 (Feb 22, 2017) (10 pages) Paper No: JAM-16-1550; doi: 10.1115/1.4035930 History: Received November 09, 2016; Revised February 03, 2017

An analytical study is presented of the covariance kernels of a damped, linear, two-degrees-of-freedom (2DOF) system which resembles a primary system that is provided with an auxiliary mass damper (AMD), in addition to an “inerter” (a device that imparts additional inertia to the vibration damper, hence magnifying its effectiveness without a significant damper mass addition). The coupled 2DOF system is subjected to nonstationary stochastic excitation consisting of a modulated white noise. An exponential function, resembling the envelope of a typical earthquake, is considered. 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

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Figures

Grahic Jump Location
Fig. 1

Model of 2DOF system with inerter

Grahic Jump Location
Fig. 2

Comparison of the mean-square transient response E[x12(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. The LHS panel shows the response under an excitation having a step-function envelope; the 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 identical abscissas in the plots show normalized time in terms of the primary system period T1.

Grahic Jump Location
Fig. 3

Comparison of a 2DOF system with two base-excitation types and two damping levels. Normalized transient m.s. response E[x12(t)] of the primary system, the m.s. response E[x22(t)] of the auxiliary mass, and the m.s. relative displacement E[(x2−x1)2] between m2 and m1 are shown over a time span covering eight periods T1 of the primary system. In all displayed cases, m2/m1=0.10,ω2/ω1=1, and ζ2=ζ1. The LHS column of panels corresponds to the case where the primary system damping ratio ζ1=0.01, while in the RHS column of plots, ζ1=0.05. Top row of panels corresponds to the case where the envelope of the excitation is a step-function, while in the lower row of panels the excitation has an envelope function g(t)=e−1.5t−e−3.5t. For added resolution, different amplitude scales are used in the four panels.

Grahic Jump Location
Fig. 4

High-resolution comparison of the normalized m.s. E[x12(t)] of the four cases shown Fig. 3. The 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. For added resolution different amplitude scales (different from the corresponding ones used in Fig. 3) are used in the two panels.

Grahic Jump Location
Fig. 5

Plot of normalized dynamic amplification ratio versus normalized excitation frequency for an SDOF system, subjected to harmonic excitation, under three scenarios: (a) a case in which no auxiliary damper is used (i.e., μ≡m2/m1=0); (b) a case in which a conventional DVN with mass ratio μ=0.05 is used, and (c) a case where a DVN with μ=0.05 and an inerter with μ*≡m*/m1=0.05 are simultaneously used. The abscissa shows the exciting harmonic frequency Ω normalized by the natural frequency ω1 of the primary system. The ordinate shows x1max, the peak steady-state amplitude of the primary system, normalized by its static deflection f1/k1. The primary system damping ratio is ζ1=0.01. The frequency and damping parameters for the auxiliary system with and without an inerter are selected based on the optimized tuning values provided by Eqs. (31)(34). Notice that the peak dynamic amplification ratio (in the absence of any dampers) for this case is equal to 1/(2ζ1)=50, but is clipped as shown for enhanced resolution.

Grahic Jump Location
Fig. 6

Sample impulse response functions hij(t) for m1 and m2. First row shows h11,h12, and h22 for a primary system m1 when m2≈0; Second row shows the same functions when using a DVN having μ=0.05; and the third row with a μ=0.05 and inerter having μ*=0.05. Damping for m1 in all cases is ζ1=0.01; damping ζ2 for m2 is set for an optimized DVN system. Amplitude scales are the same in each column except for the undamped system without a DVN. A time span of eight natural periods T1 is shown.

Grahic Jump Location
Fig. 7

Comparison of time evolution E[x12(t)] for SDOF subjected to stationary base excitation and having different levels of primary system damping ζ1 and with different types of damping devices. First row: with no DVN; second row: with a DVN having μ=0.05; and third row with a DVN and inerter having μ*=0.05. The LHS column for ζ1≈0, the middle column for ζ1=0.01, and the RHS column for ζ1=0.05. Used damper parameters have optimized damping and frequency ratios. Amplitude scales are the same except for the undamped system without a DVN.

Grahic Jump Location
Fig. 8

Comparison of the evolution of the diagonal terms of Kx1,x1(t1,t2) and the antidiagonal terms of the same covariance matrix at diagonal sample time t1=t2=t*=2. Primary system ζ1=0.01, with μ=0.01 and μ*=0.01. Excitation envelope is step-function. Tuning parameters are optimized.

Grahic Jump Location
Fig. 9

Comparison of the evolution of the covariance kernels of three different functions for the 2DOF system: (a) the primary system Kx1,x1(t1,t2); (b) the cross-covariance function Kx1,x2(t1,t2); (c) Kx2,x2(t1,t2), the covariance for the auxiliary mass m2. The system parameters are ζ1=0.01, ω1=2π, μ=0.01, and μ*=0.01, and with the tuning parameters being optimized, and with the nonstationary excitation having an envelope defined by g(t)=e−10t−e−15t. The time spans shown for t1 and t2 cover four natural periods T1 of the primary system, and the amplitude scales are different and selected to enhance resolution. Same perspective view is used for all plots, with the origin of the time axes in the LHS corner of the respective surface plots.

Grahic Jump Location
Fig. 10

Comparison of the normalized transient m.s. response, E[x12(t)], of a SDOF system with two base-excitation types and two inerter sizes, shown over a time span covering eight periods T1 of the primary system. In all displayed cases, m2/m1=0.01, ζ1=0.01, with an optimized choice for the tuning parameters ω2/ω1, and ζ2. The LHS column of panels corresponds to the case where the exponential envelope function is changing relatively rapidly in accordance with g1(t)=e−1.5t−e−3.5t, while in the RHS panel, a relatively slower evolving envelope g2(t)=e−0.25t−e−3.5t is applied. For added resolution, different amplitude scales are used in the two panels shown in (a) and (b), but the peak ordinate values are normalized to be unity.

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