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Research Papers

# Path-Following Bifurcation Analysis of Church Bell Dynamics

[+] Author and Article Information
Antonio Simon Chong Escobar

Centre for Applied Dynamics Research,
University of Aberdeen,
Kings College,
Aberdeen AB24 3UE, UK;
Facultad de Ciencias Naturales y Matemáticas,
Escuela Superior Politécnica del Litoral,
Km. 30.5 Vía Perimetral,
e-mail: a.chong@abdn.ac.uk

Piotr Brzeski

Division of Dynamics,
Lodz University of Technology,
Stefanowskiego 1/15, Lodz 90-924, Poland;
Potsdam Institute for Climate
Impact Research (PIK),
Potsdam 14473, Germany;
Department of Physics,
Humboldt University,
Berlin 12489, Germany
e-mail: piotr.brzeski@p.lodz.pl

Marian Wiercigroch

Centre for Applied Dynamics Research,
University of Aberdeen,
Kings College,
Aberdeen AB24 3UE, UK
e-mail: m.wiercigroch@abdn.ac.uk

Przemyslaw Perlikowski

Division of Dynamics,
Lodz University of Technology,
Stefanowskiego 1/15,
Lodz 90-924, Poland
e-mail: przemyslaw.perlikowski@p.lodz.pl

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 19, 2016; final manuscript received February 4, 2017; published online September 7, 2017. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 12(6), 061017 (Sep 07, 2017) (8 pages) Paper No: CND-16-1630; doi: 10.1115/1.4036114 History: Received December 19, 2016; Revised February 04, 2017

## Abstract

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.

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## References

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Brzeski, P. , Perlikowski, P. , and Kapitaniak, T. , 2015, “ Analysis of Transitions Between Different Ringing Schemes of the Church Bell,” Int. J. Impact Eng., 85, pp. 57–66.
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## Figures

Fig. 1

(a) The Heart of Lodz' bell and (b)–(d) its physical model with the geometry in two planes used to develop its lump mass mathematical model. Figure adapted from Ref. [14].

Fig. 3

Operation modes and events of the considered yoke–bell–clapper system. Details of each event are given in Table 1.

Fig. 2

(a) and (c) Schematics and (b) and (d) phase protraits of the symmetrical flying clapper and falling clapper both with two impacts per period. In the phase portraits, the arrows indicate the effects of collisions.

Fig. 6

Phase portraits of the displacements of the bell and the clapper (panels (a)) and displacement versus velocity of the clapper (panels (b)). Numbers of solutions correspond to the number along branches of stable periodic motions marked in Figs. 4 and 5.

Fig. 4

Bifurcation diagrams obtained by (a) a direct numerical integration and (b) a path-following for fixed yoke geometry, lr = –1.25 m and T∈〈100, 650〉 N⋅m. Different types of solution are marked with different numbers and colors. Solid curves mark stable solutions, where dashed ones unstable parts of branches. Different shapes are used to distinguish different bifurcations (see legend). Vertical lines mark the position of bifurcation points in which the stability changes (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

Fig. 5

Zoom-ups of bifurcation diagram shown in Fig. 4(b). Panel (a) shows in details subcritical pitchfork bifurcation that occurs for T = 616.2 N·m, while subplots (b) and (c) present zoom of each branch close to the bifurcation points.

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