A reduced-order model from high-dimensional frictional hysteresis

Abstract

Hysteresis in material behaviour includes both signum nonlinearities as well as high dimensionality. Available models for component-level hysteretic behaviour are empirical. Here, we derive a loworder model for rate-independent hysteresis from a high-dimensional massless frictional system. The original system, being given in terms of signs of velocities, is first solved incrementally using a linear complementarity problem formulation. From this numerical solution, to develop a reduced-order model, basis vectors are chosen using the singular value decomposition. The slip direction in generalized coordinates is identified as the minimizer of a dissipation-related function. That function includes terms for frictional dissipation through signum nonlinearities at many friction sites. Luckily, it allows a convenient analytical approximation. Upon solution of the approximated minimization problem, the slip direction is found. A final evolution equation for a few states is then obtained that gives a good match with the full solution. The model obtained here may lead to new insights into hysteresis as well as better empirical modelling thereof.

A two-state hysteresis model from high-dimensional friction

“…Differing somewhat from Biswas & Chatterjee [ 1 ], here we consider the intuitively simpler high-dimensional frictional system sketched in figure 1 . In this n -dimensional model (with n large), each spring has stiffness 1/ n , and friction coefficients at the slip sites are As indicated in the figure, u ( t ) is a displacement input to the system, for which a force f ( t ) is needed.…”

“…We solve equation ( 2.1 ) incrementally by casting it first into an LCP (as described in Biswas & Chatterjee [ 1 ]) and then by using Lemke's algorithm (as implemented by Miranda & Fackler [ 13 ]). There is in fact a large literature on solving friction problems using the LCP; readers interested in the theory may consult, e.g.…”

“…With this background, we recently studied [ 1 ] a frictional hysteretic system given by where x is high-dimensional; μ is diagonal; K is symmetric and positive definite; b is a column matrix; f ( t ) is scalar and differentiable; and the signum function ‘sgn’ is defined elementwise as follows: Equation ( 1.1 ) can be solved incrementally via a linear complementarity problem (LCP) [ 12 ] or, less efficiently, using other means as described later. The solution of equation ( 1.1 ) captures important aspects of hysteresis including formation of minor loops.…”

Modelling and analysis of dynamic frictional interactions of vibro-driven capsule systems with viscoelastic property

“…However, the experimental observations are not always aligned with the consideration above, particularly in the unsteady environmental conditions such as the oscillations in relative velocity. Therefore, some engineering investigations have revealed that the frictional dynamics can be multi-valued, which means the friction force travels along different paths during the DS and the AS, and forms a non-reversible curve as reported in (Becker and Mahin, 2013;Biswas and Chatterjee, 2014;Neis et al, 2011;Outirba and Hendrick, 2014;Stefański et al, 2006;Wojewoda et al, 2008). The main reason for this phenomenon is the temporal lag between the variations of the friction force and the relative velocity.…”

Multi-scale concrete model with rate-dependent internal friction

“…Many formulations have been proposed to account for the dissipation in solid media (see e.g. [12,10,2,47,8]), however, herein, with aiming at analyzing the internal mechanisms of dissipation in concrete when no relevant damage phenomena occur at the macroscopic level, the internal dissipation is assumed to be related to the relative motion of micro-crack faces following the same spirit of several papers [46,43,59,36,81,84,78]. The basic idea was to introduce a new (scalar) variable characterizing, in an average way, these micro-sliding along the micro-crack faces with a proper constitutive relation describing the evolution of this variable and its coupling with the Coulomb friction force.…”