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TECHNICAL PAPERS

# Convergence Behaviors of Reduced-Order Models For Frictional Contacts

[+] Author and Article Information
D. V. Deshmukh

Department of Civil Engineering,  University of Virginia, Charlottesville, VA 22904

E. J. Berger1

Department of Civil Engineering,  University of Virginia, Charlottesville, VA 22904berger@virginia.edu

T. J. Mackin, H. Inglis

Department of Mechanical and Industrial Engineering,  University of Illinois, Urbana, IL 61802

1

Corresponding author

J. Vib. Acoust 127(4), 370-381 (Dec 13, 2004) (12 pages) doi:10.1115/1.1924645 History: Received February 11, 2004; Revised December 13, 2004

## Abstract

Numerical models to simulate interface behavior of friction connections under cyclic loading are investigated. The question of validity of lower-order models in successfully capturing response of friction joints under cyclic loading is addressed. Single-element macroslip models are not capable of capturing localized interface behavior prior to gross interfacial slip. This paper focuses on the convergence behavior of a multipoint contact microslip model comprised of Iwan-type elements for different physical parameters such as system response amplitude and kinematic state of the friction joint. System dynamics play a significant role in determining the convergence of structural behavior, especially for tuned damper sets in the nonzero damper mass case. This behavior is explored using simple linearized models that explain the response sensitivity in terms of the overall modal density near the forcing frequency. Convergence of the interface response kinematics is also considered, with a focus on the number of damping elements operating in the stick, stick-slip, and slip regimes at steady state. Energy dissipation scaling under light forcing is also examined, with the class of models considered here yielding scaling exponents consistent with experimental observations and analytical predictions from the literature. We show that the interface kinematic behavior converges at a slower rate than the structural response and therefore requires a higher-order interface model. These results suggest that extremely low-order models (i.e., $<5$ damping elements) provide predictions that are model order dependent, while higher-order models (i.e., $>50$ damping elements) are not. This result impacts model development and calibration approaches, as well as providing clues for appropriate model reduction strategies.

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## Figures

Figure 10

Dissipation scaling exponent under light forcing in the kd−md parameter space for n=75 damping elements

Figure 11

Frequency response plots versus number of damping elements: (a) system amplitude, xo, and (b) frictional work, E

Figure 12

Candidate fitting functions and their resulting (ki,Ni) parameters identified using n=50 damping elements. All candidate functions possess the same total stiffness kd and same total normal load Nd, but their discretizations span the parameter space in various patterns. All ki and Ni are scaled by their respective mean values; km=(1∕n)∑nki and Nm=(1∕n)∑nNi for presentation purposes.

Figure 5

Amplitude convergence for various damper set tuning cases and four different forcing levels

Figure 6

Response time histories for md=0.5, kd=0.5, γ=0.67, and n=5: (a) system displacement and (b) damper velocities showing three kinematic states

Figure 7

Convergence of system response amplitude under various damper configurations and strong forcing γ=0.1

Figure 8

Convergence of damper kinematic states for kd=0.5, γ=0.67, and various md

Figure 9

Convergence of frictional energy dissipation under various damper configurations and strong forcing γ=0.1

Figure 1

Monotonic pullout experiments in a plane stress pullout specimen (26). The interface completely slips at the breakaway point x=xbr, and sliding friction has magnitude μNd. The slope of the F−x curve through the origin is the total interface stiffness kd.

Figure 2

Normalized mathematical model of single-DOF system with multipoint contact friction damper set attached

Figure 3

Friction behavior of multidamper sets: (a) softening under monotonic loading, (b) hysteresis under cyclic loading, and (c) convergence of frictional work with increasing number of dampers

Figure 4

Frequency response functions for the structure and damper elements for a tuned damper set, kd=md=0.5, using a linearized (no friction) model with forcing on the structural mass. The eigenvector corresponding to structure response for the repeated eigenvalue is identically 0.

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