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Research Papers

# Stability Analysis of Sliding–Grazing Phenomenon in Dry-Friction Oscillator Using Takagi–Sugeno Fuzzy Approach

[+] Author and Article Information
Kamyar Mehran

School of Engineering,
University of Warwick,
Coventry CV4 7LW, UK
e-mail: k.mehran@warwick.ac.uk

Bashar Zahawi

Department of Electrical and
Computer Engineering,
Khalifa University, PO Box 127788,
Abu Dhabi, UAE
e-mail: bashar.zahawi@kustar.ac.ae

Damian Giaouris

Chemical Process Engineering
Research Institute,
Centre for Research and
Technology Hellas,
Thermi,
Thessaloniki GR57001, Greece
e-mail: giaouris@cperi.certh.gr

Jihong Wang

School of Engineering,
University of Warwick,
Coventry CV4 7LW, UK
e-mail: jihong.wang@warwick.ac.uk

We refer to the proposed TS fuzzy model able to represent a nonsmooth system as a nonsmooth TS fuzzy model.

We mean smooth TS fuzzy model as the typical TS fuzzy model able to approximate smooth nonlinear functions.

The essential part of the proof is similar to the proof presented for nonsliding Filippov systems [21] (DoS of 1). The extended detail of the proof for sliding dynamics including the existence and uniqueness of the solution when there is a rapid switching of sliding motion near the switching manifold is omitted here due to insufficient space.

Manuscript received May 14, 2014; final manuscript received December 31, 2014; published online April 9, 2015. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 10(6), 061010 (Nov 01, 2015) (7 pages) Paper No: CND-14-1127; doi: 10.1115/1.4029663 History: Received May 14, 2014; Revised December 31, 2014; Online April 09, 2015

## Abstract

Dry-friction oscillators are mechanical systems with dry friction and stick-slip vibrations. In the context of control theory, the stability analysis of this type of dynamical systems is important since they exhibit nonsmooth bifurcations, or most famously a sliding–grazing bifurcation inducing abrupt chaos. This paper develops a Lyapunov-based framework to study the so-called structural stability of the system, predicting the onset of such unique bifurcations. To achieve this, the nonlinear system is first represented as a nonsmooth Takagi–Sugeno (TS) fuzzy model, and the structural stability is then formulated as linear matrix inequalities (LMI) feasibility problems with less conservative formulation. Solving the resulting LMI problem, the onset of sliding–grazing bifurcation can be accurately predicted.

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## Figures

Fig. 1

Model of a forced oscillator with dry friction

Fig. 2

(a) 4T (8π/ϖ) periodic orbit grazes the boundary of sliding region (grazing–sliding event) at ϖ = 1.7077997 and turns to a (b) chaotic orbit. Dotted-dashed line in (a) shows a stable, nonsliding periodic orbit for ϖ = 1.7082 before sliding–grazing event, and the solid line in (a) shows the sliding orbit near the tangency of attracting switching manifold.

Fig. 3

The bifurcation diagram shows the grazing–sliding bifurcation where there is a sudden transition to a chaotic attractor

Fig. 4

With small parameter variation, periodic (sliding) orbit ab, touches the switching manifold Σ at x* undergoing a sliding–grazing bifurcation, and bc. In fact, periodic orbits a and c might exist for the same parameters values.

Fig. 5

Bifurcation diagram obtained from the direct simulation of the nonsmooth TS fuzzy model (11), Eq. (12) shows the grazing–sliding DIB at ϖ=1.70778. Insets show steady-state time responses.

Fig. 6

An illustration of region partitioning and estimated energy levels of the nonsmooth Lyapunov function candidate in the local regions (dotted-dashed curves) when sliding (local) orbit grazes the switching manifold

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