Path-Following Bifurcation Analysis of Church Bell Dynamics

2017 ◽  
Vol 12 (6) ◽  
Author(s):  
Antonio Simon Chong Escobar ◽  
Piotr Brzeski ◽  
Marian Wiercigroch ◽  
Przemyslaw Perlikowski
Keyword(s):  
Bifurcation Analysis ◽  
Applied Mathematics ◽  
Path Following ◽  
Engineering Society ◽  
Effective Performance ◽  
Computational Science And Engineering ◽  
Novel Method ◽  
Direct Numerical Integration ◽  
Path Following Methods ◽  
Church Bell

In this paper, we perform a path-following bifurcation analysis of church bell to gain an insight into the governing dynamics of the yoke–bell–clapper system. We use an experimentally validated hybrid dynamical model based on the detailed measurements of a real church bell. Numerical analysis is performed both by a direct numerical integration and a path-following methods using a new numerical toolbox ABESPOL (Chong, 2016, “Numerical Modeling and Stability Analysis of Non-Smooth Dynamical Systems Via ABESPOL,” Ph.D. thesis, University of Aberdeen, Aberdeen, UK) based on COCO (Dankowicz and Schilder, Recipes for Continuation (Computational Science and Engineering), Society for Industrial and Applied Mathematics, Philadelphia, PA). We constructed one-parameter diagrams that allow to characterize the most common dynamical states and to investigate the mechanisms of their dynamic stability. A novel method allowing to locate the regions in the parameters' space ensuring robustness of bells' effective performance is presented.

scholarly journals Bifurcation analysis of a vibro-impact experimental rig with two-sided constraint

Meccanica ◽  
2020 ◽  
Vol 55 (12) ◽  
pp. 2505-2521 ◽  
Author(s):  
Yang Liu ◽  
Joseph Páez Chávez ◽  
Bingyong Guo ◽  
Rauf Birler
Keyword(s):  
Bifurcation Analysis ◽  
Period Doubling ◽  
Path Following ◽  
External Excitation ◽  
Directional Control ◽  
Period 2 ◽  
Rectangular Waveform ◽  
Nonsmooth Dynamical Systems ◽  
Path Following Methods ◽  
Period Doubling Bifurcations

AbstractIn this paper we carry out an in-depth experimental and numerical investigation of a vibro-impact rig with a two-sided constraint and an external excitation given by a rectangular waveform. The rig, presenting forward and backward drifts, consists of an inner vibrating shaft intermittently impacting with its holding frame. Our interests focus on the multistability and the bifurcation structure observed in the system under two different contacting surfaces. For this purpose, we propose a mathematical model describing the rig dynamics and perform a detailed bifurcation analysis via path-following methods for nonsmooth dynamical systems, using the continuation platform COCO. Our study shows that multistability is produced by the interplay between two fold bifurcations, which give rise to hysteresis in the system. The investigation also reveals the presence of period-doubling bifurcations of limit cycles, which in turn are responsible for the creation of period-2 solutions for which the rig reverses its direction of progression. Furthermore, our study considers a two-parameter bifurcation analysis focusing on directional control, using the period of external excitation and the duty cycle of the rectangular waveform as the main control parameters.

Path-Following Methods for Linear Programming

SIAM Review ◽  
1992 ◽  
Vol 34 (2) ◽  
pp. 167-224 ◽  
Author(s):  
Clovis C. Gonzaga
Keyword(s):  
Linear Programming ◽  
Path Following ◽  
Path Following Methods
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