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Research Papers

Effective Stiffening and Damping Enhancement of Structures With Strongly Nonlinear Local Attachments

[+] Author and Article Information
Themistoklis P. Sapsis

 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 sapsis@mit.edu

D. Dane Quinn

Department of Mechanical Engineering,  The University of Akron, Akron, OH 44325 quinn@uakron.edu

Alexander F. Vakakis

Department of Mechanical Science and Engineering,  University of Illinois at Urbana-Champaign, Urbana, IL 61801 avakakis@illinois.edu

Lawrence A. Bergman

Department of Aerospace Engineering,  University of Illinois at Urbana-Champaign, Urbana, IL 61801 lbergman@illinois.edu

J. Vib. Acoust 134(1), 011016 (Jan 09, 2012) (12 pages) doi:10.1115/1.4005005 History: Received December 13, 2010; Revised July 11, 2011; Published January 09, 2012; Online January 09, 2012

We study the stiffening and damping effects that local essentially nonlinear attachments can have on the dynamics of a primary linear structure. These local attachments can be designed to act as nonlinear energy sinks (NESs) of shock-induced energy by engaging in isolated resonance captures or resonance capture cascades with structural modes. After the introduction of the NESs, the effective stiffness and damping properties of the structure are characterized through appropriate measures, developed within this work, which are based on the energy contained within the modes of the primary structure. Three types of NESs are introduced in this work, and their effects on the stiffness and damping properties of the linear structure are studied via (local) instantaneous and (global) weighted-averaged effective stiffness and damping measures. Three different applications are considered and show that these attachments can drastically increase the effective damping properties of a two-degrees-of-freedom system and, to a lesser degree, the stiffening properties as well. An interesting finding reported herein is that the essentially nonlinear attachments can introduce significant nonlinear coupling between distinct structural modes, thus paving the way for nonlinear energy redistribution between structural modes. This feature, coupled with the well-established capacity of NESs to passively absorb and locally dissipate shock energy, can be used to create effective passive mitigation designs of structures under impulsive loads.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Essentially nonlinear energy sinks (NESs) considered: (a) type-I NES, (b) type-II NES, and (c) type-III NES. All linear springs and viscous dampers are uncompressed when horizontal.

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Figure 2

SDOF linear oscillator with type-I NES attached

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Figure 3

Averaging of the time series of the kinetic energy of the linear oscillator in Eq. 1. (a) Averaged kinetic energy Ek(t) using spline interpolation of local maxima. (b) Total mechanical energy in the linear oscillator. (c) Instantaneous kinetic energy.

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Figure 4

Weighted-averaged effective measures for varying initial energy E0 for the impulsively excited system (1): (a) keff¯/k, (b) λeff¯/λ

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Figure 5

Damped transition of system (1) for initial energy E0=9×10-3. (a) Velocity of the linear oscillator with NES attached (——), of the effective oscillator (- - - -), and of the linear oscillator with no NES attached (-·-·-·-·-). (b) Instantaneous normalized effective damping λeff(t)/λ. (c) Instantaneous energies of the linear oscillator and the NES.

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Figure 6

Damped transition of system (1) for initial energy E0=0.3. (a) Velocity of the linear oscillator with NES attached (——), of the effective oscillator (- - - -), and of the linear oscillator with no NES attached (-·-·-·-·-). (b) Instantaneous normalized effective damping λeff(t)/λ. (c) Instantaneous energies of the linear oscillator and the NES.

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Figure 7

Transient response of system (12) for impulsive excitation of the lowest linear mode 1. (a),(d) Velocity time series, (b),(e) normalized instantaneous effective stiffness, and (c),(f) normalized effective damping measure of modes 1 and 2, respectively; ki and λi denote the modal stiffness and damping, respectively, of mode i.

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Figure 8

Weighted-averaged (global) effective measures for the case of a single type-I NES attached to the first floor. Stiffness and damping measures for (a),(b) mode 1 and (c),(d) mode 2; ki and λi denote the modal stiffness and damping, respectively, of mode i.

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Figure 9

Transient response of system (12) for impulsive excitation of impulse excitation with initial energy E(0)=0.4. (a),(e) Velocity time series, (b),(f) normalized instantaneous effective stiffness, (c),(g) normalized effective damping measure, and (d),(h) percentage of instantaneous total energy of modes 1 and 2, respectively; ki and λi denote the modal stiffness and damping, respectively, of mode i.

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Figure 10

Weighted-averaged (global) effective measures for the case of a single type-II NES attached to the second floor. Stiffness and damping measures for (a),(b) mode 1 and (c),(d) mode 2; ki and λi denote the modal stiffness and damping, respectively, of mode i.

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Figure 11

Weighted-averaged (global) effective measures for the case of two type-III NESs attached to the first and second floors. (a) Configuration of the system. Stiffness and damping measures for (b),(c) mode 1 and (d),(e) mode 2; ki and λi denote the modal stiffness and damping, respectively, of mode i; -·-·-·-·- ρ=4, - - - - ρ=2, —— ρ=1.

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Figure 12

Damped transition of the system with two type-III NESs for initial energy E(0)=1.2. Velocity time series of (a) mode 1 and (b) mode 2. Wavelet transform spectra of (c) x1-y1, (d) y1-y2, (e) x2-y3, and (f) y3-y4; dashed lines indicate the natural frequencies of the linear structure.

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