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Research Papers

Leveraging the Equivalence of Hysteresis Models From Different Fields for Analysis and Numerical Simulation of Jointed Structures

[+] Author and Article Information
T. J. Royston

 University of Illinois at Chicago, Chicago, IL 60607troyston@uic.edu

J. Comput. Nonlinear Dynam 3(3), 031006 (Apr 30, 2008) (8 pages) doi:10.1115/1.2908348 History: Received April 03, 2007; Revised January 28, 2008; Published April 30, 2008

An important problem that spans across many types of systems (e.g., mechanical and biological) is how to model the dynamics of joints or interfaces in built-up structures in such a way that the complex dynamic and energy-dissipative behavior that depends on microscale phenomena at the joint/interface is accurately captured, yet in a framework that is amenable to efficient computational analyses of the larger macroscale system of which the joint or interface is a (spatially) small part. Simulating joint behavior in finite element analysis by meshing the joint regions finely enough to capture relevant micromechanics is impractical for large-scale structural systems. A more practical approach is to devise constitutive models for the overall behavior of individual joints that accurately capture their nonlinear and energy-dissipative behavior and to locally incorporate the constitutive response into the otherwise often-linear structural model. Recent studies have successfully captured and simulated mechanical joint dynamics using computationally simple phenomenological models of combined elasticity and slip with associated friction and energy dissipation, known as Iwan models. In the present article, the author reviews the relationship, and in some cases equivalence, of one type of Iwan model to several other models of hysteretic behavior that have been used to simulate a wide range of physical phenomena. Specifically, it is shown that the “parallel-series” Iwan model has been referred to in other fields by different names, including “Maxwell resistive capacitor,” “Ishlinskii,” and “ordinary stop hysteron.” Given this, the author establishes the relationship of this Iwan model to several other hysteresis models, most significantly the classical Preisach model. Having established these relationships, it is then possible to extend analytical tools developed for a specific hysteresis model to all of the models with which it is related. Such analytical tools include experimental identification, inversion, and analysis of vibratory energy flow and dissipation. Numerical case studies of simple systems that include an Iwan-modeled joint illustrate these points.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

The parallel-series Iwan system

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Figure 2

Functional relationships of classical Preisach representation of Iwan hysteresis model. (a) Weighting function μ[x,y]. (b) Everett function F[x,y] based on first order transition curves. Stepped nature of functions is evident for the case of finite N.

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Figure 3

Everett function evaluated along the line z, which is x=−y. (a) F[z]. (b) dF[z]/dz.

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Figure 4

Jointed system for example case study. The vibratory “source” is the force excitation on the right side of the joint. The joint, modeled as a discretized parallel-series Iwan system, is the “path” and the damper is the “receiver.” In the numerical studies presented here, the four parameters of the Iwan system are Fs=1, K=1, β=0, and χ=−1∕2. The damper coefficient is b=1. The discretized Iwan model has N=50 and α=1.2.

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Figure 5

Time domain responses for one cycle of uJ[t], uJR[t], and uJL[t] normalized by uJ0 for uJ0=0.1 (——), 1 (–––), and 10 (----) for ω=1

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Figure 6

uJ[t]∕uJ0 versus FJ[t]∕uJ0 for uJ0=0.1 (——), 1 (–––), and 10 (------) for ω=1

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Figure 7

Dissipative (damping) receiver. Vibratory energy input per cycle (Pin) in dB reference ∣uJ0∣2 for uJ0 from 0.01 to 100 (−40to40dB) and for ω from 0.01 to 100 (−40to40dB). (a) Pin occurring at the fundamental frequency ω. (b) Pin occurring at higher harmonics of ω, summed from 2ω to 254ω.

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Figure 8

Dissipative (damping) receiver. Vibratory energy dissipation per cycle at the joint (Pdis) in dB reference ∣uJ0∣2 for uJ0 from 0.01 to 100 (−40to40dB) and for ω from 0.01 to 100 (−40to40dB). Pdis only occurs at frequency ω as the relative motion across the joint uJ[t] is specified to be harmonic.

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Figure 9

Dissipative (damping) receiver. Vibratory energy transmission to and dissipation per cycle in the receiver (damper) (Ptran) in dB reference ∣uJ0∣2 for uJ0 from 0.01 to 100 (−40to40dB) and for ω from 0.01 to 100 (−40to40dB). Ptran occurring at the fundamental frequency ω is shown. Ptran occurring at higher harmonics of ω, summed from 2ω to 524ω, is identical to Fig. 7.

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Figure 10

The amplitude of FJ[t] versus uJ0. (——) calculated using Eq. 20, (----) fundamental harmonic component at ω calculated from direct time simulation and (–––) sum of higher harmonic components from 2ω to 254ω calculated from direct time simulation.

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Figure 11

Resonant receiver. Vibratory energy input per cycle (Pin) in dB reference ∣uJ0∣2 for uJ0 from 0.01 to 100 (−40to40dB) and for ω from 0.01 to 10 (−40to20dB). (a) Pin occurring at the fundamental frequency ω. (b) Pin occurring at higher harmonics of ω, summed from 2ω to 254ω.

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Figure 12

Resonant receiver. Vibratory energy transmission to and dissipation per cycle in the receiver (damper) (Ptran) in dB reference ∣uJ0∣2 for uJ0 from 0.01 to 100 (−40to40dB) and for ω from 0.01 to 10 (−40to20dB). Ptran occurring at the fundamental frequency ω is shown. Ptran occurring at higher harmonics of ω, summed from 2ω to 524ω, is identical to Fig. 1.

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