BIFURCATION BEHAVIORS OF THE TWO-STATE VARIABLE FRICTION LAW OF A ROCK MASS SYSTEM

2013 ◽  
Vol 23 (11) ◽  
pp. 1350184 ◽  
Author(s):  
GAO XUEJUN
Keyword(s):  
Rock Mass ◽  
Equilibrium Position ◽  
Continuation Method ◽  
Chaotic Motions ◽  
Mass System ◽  
Stick Slip ◽  
Hopf Bifurcation Point ◽  
Friction Law ◽  
State Variable ◽  
Variable Friction

Based on the stability and bifurcation theory of dynamical systems, the bifurcation behaviors and chaotic motions of the two-state variable friction law of a rock mass system are investigated by the bifurcation diagrams based on the continuation method and the Poincaré maps. The stick-slip of the rock mass is formulated as an initial values problem for an autonomous system of three coupled nonlinear ordinary differential equations (ODEs) of first order. The results of linear stability analysis indicate that there is an equilibrium position in the rock mass system. Furthermore, numerical results of nonlinear analysis indicate that the equilibrium position loses its stability from a sup-critical Hopf bifurcation point, and then the bifurcating periodic motion evolves into chaotic motion through a series of period-doubling bifurcations with the decreasing of the control parameter. The stick-slip and chaotic motions evolve into infinity in the end with some unstable periodic motions.

scholarly journals A phase-plane analysis of localized frictional waves

2017 ◽  
Vol 473 (2203) ◽  
pp. 20160606 ◽  
Author(s):  
T. Putelat ◽  
J. H. P. Dawes ◽  
A. R. Champneys
Keyword(s):  
Travelling Waves ◽  
Ground Surface ◽  
State Diagram ◽  
Friction Model ◽  
Phase Plane Analysis ◽  
Thin Elastic Plate ◽  
Stick Slip ◽  
State Laws ◽  
Friction Law ◽  
Numerical Bifurcation

Sliding frictional interfaces at a range of length scales are observed to generate travelling waves; these are considered relevant, for example, to both earthquake ground surface movements and the performance of mechanical brakes and dampers. We propose an explanation of the origins of these waves through the study of an idealized mechanical model: a thin elastic plate subject to uniform shear stress held in frictional contact with a rigid flat surface. We construct a nonlinear wave equation for the deformation of the plate, and couple it to a spinodal rate-and-state friction law which leads to a mathematically well-posed problem that is capable of capturing many effects not accessible in a Coulomb friction model. Our model sustains a rich variety of solutions, including periodic stick–slip wave trains, isolated slip and stick pulses, and detachment and attachment fronts. Analytical and numerical bifurcation analysis is used to show how these states are organized in a two-parameter state diagram. We discuss briefly the possible physical interpretation of each of these states, and remark also that our spinodal friction law, though more complicated than other classical rate-and-state laws, is required in order to capture the full richness of wave types.

Model-Based Investigation of Friction During Start-Up of Hydrodynamic Journal Bearings

1995 ◽  
Vol 117 (4) ◽  
pp. 667-673 ◽  
Author(s):  
A. Harnoy
Keyword(s):  
Friction Model ◽  
Journal Bearings ◽  
Time Variable ◽  
Stribeck Curve ◽  
Dynamic Friction ◽  
Stick Slip ◽  
Friction Characteristics ◽  
Start Up ◽  
Variable Friction ◽  
Steady Velocity

An analysis is developed for the time-variable friction during the start-up of a rotor system. The analysis is based on a dynamic friction model that has been developed from the theory of unsteady lubrication and can describe the observed friction characteristics. The model reduces to the Stribeck curve of friction versus steady velocity, and shows hysteresis curves in oscillating velocity. The “Dahl effect” of a presliding displacement before the breakaway is also included. The results indicate that the friction characteristics and energy friction losses, during the start-up, depend on a set of dimensionless parameters that represent the bearing as well as the dynamic system. The study shows that appropriate design and operation can prevent stick-slip friction and minimize wear during start-up.

Multiple Solutions in an Amplitude Limited Jeffcott Rotor Including Rubbing and Stick-Slip Effect

2005 ◽  
Author(s):  
Jan-Olov Aidanpa¨a¨
Keyword(s):  
Bifurcation Diagram ◽  
Contact Stiffness ◽  
Damping Ratio ◽  
Slip Effect ◽  
Chaotic Motions ◽  
Stick Slip ◽  
Diagram Method ◽  
Machine Failure ◽  
Jeffcott Rotor ◽  
Stable Solutions

The non-linear behaviour of rub-impact rotors have been studied in several papers. In such systems rich dynamics have been found together with the coexistence of solutions within some specific parameter ranges. In this paper an attempt is made to find all stable solutions for an amplitude limited Jeffcott rotor including rubbing and stick-slip effect. The recently suggested “multi bifurcation diagram method” is used to find and extract stable sets of bifurcation diagrams. A system is chosen where the linear stationary amplitude only exceeds the clearance in a narrow region near the natural frequency. Therefore large regions in frequency are expected to have only the linear stationary response. The results show that it is only for very low frequencies that one single solution exists. Even though periodic motions are dominant, there exist large ranges in frequency with quasi-periodic or chaotic motions. For the studied cases, three coexisting stable solutions are most common. In one case as many as four stable solutions was found to coexist. For rotors with large clearances (no impacts necessary) it is still possible to find several coexisting motions. For all cases the stick motion is the most severe one with large amplitudes and high backward whirl frequencies. In real situations the consequence of this stick motion is machine failure. These high amplitude motions were found to be stable over large frequency ranges. From the stability analysis it was found that this rolling motion can be avoided by low spin speed, low contact stiffness, low coefficient of friction, small ratio of disc radius/clearance or high damping ratio. In a design situation the parameters are seldom known with high accuracy. Therefore, it is of interest to know all solutions for parameter intervals. The multi-bifurcation diagram can be used in such situations to design a robust machine or at least be prepared for unwanted dynamics.

The melody of a failing peak – seismic constraints on rock damaging and stick-slip motion at the Hochvogel (DE/AT Alps)

2021 ◽  
Author(s):  
Michael Dietze ◽  
Michael Krautblatter ◽  
Johannes Leinauer ◽  
Luc Illien ◽  
Niels Hovius
Keyword(s):  
Rock Mass ◽  
Land Use Planning ◽  
Large Scale ◽  
Seismic Analysis ◽  
Early Stage ◽  
Spectral Ratio ◽  
Slope Instability ◽  
High Mountain ◽  
Seismic Wave Velocity ◽  
Stick Slip

<p>Large rock slope failures play a pivotal role in long-term landscape evolution and are a major concern in land use planning and hazard aspects. While the failure phase and the time immediately prior to failure are increasingly well studied, the nature of the preparation phase remains enigmatic. This knowledge gap is to a large degree related to challenges in collecting appropriate data in such high mountain terrain. Classic monitoring techniques provide detailed data but mostly of point character and only reflecting the surface expression of processes within the rock mass. Thus, the integral behaviour of a peak, at the surface and at depth remains elusive.</p><p>Here, we present results from a continuous multi-sensor seismic analysis of the Hochvogel summit, a 2592 m high Alpine peak, which is deemed to fail in the near future, as a 5 m wide and 40 m long crack is progressively opening and mobilising up to 260,000 cubic metres of rock. The seismic network consisted of up to seven sensors, installed during July--October 2018 (with 43 days of data loss). We develop and discuss proxy time series indicative of cyclic and progressive changes of the summit.</p><p>Modal analysis, horizontal-to-vertical spectral ratio data and end-member modelling analysis reveal diurnal cycles of increasing and decreasing coupling stiffness of the fragmented rock volume, due to thermal forcing. Relative seismic wave velocity changes mimic this pattern but also reveal the release of stress within the rock mass. At longer time scales, there is a superimposed pattern of stress evolution, which increases for five to seven days and suddenly drops within a few days, also expressed in an increased emission of short seismic pulses indicative of rock cracking. Our data provide essential first order information on an early stage of a large-scale slope instability, which evolves towards a catastrophic failure.</p>

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