Nonlinear Dynamics of Swinging Clapper Bells under Arbitrary or Resonant Forcing Functions
Study of swinging clapper bells involves aspects encompassing sound and acoustic engineering, mechanical engineering, and structural engineering. From the musical point of view, clapper bells are directly played idiophone instruments, where the playing device, the clapper, although directly excited,...
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| Vydáno v: | Applied sciences Ročník 10; číslo 16; s. 5528 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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MDPI AG
01.08.2020
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| ISSN: | 2076-3417, 2076-3417 |
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| Abstract | Study of swinging clapper bells involves aspects encompassing sound and acoustic engineering, mechanical engineering, and structural engineering. From the musical point of view, clapper bells are directly played idiophone instruments, where the playing device, the clapper, although directly excited, is not explicitly controlled by the bell ringer. The achievement of a clear and optimal sound mainly depends on the acoustic characteristics of the bell and on the regularity of the clapper strokes, which is not only governed by the ringing style and the relevant parameters of clapper and bell but also by the real time corrections to the excitation introduced by trained bell ringers. In fact, despite centuries of experience allowed to optimize the bell performances, standardizing proportions and mounting arrangements, effective sound control requires some fine tuning of the forcing function. Another crucial topic, especially in view of assessing existing structures, regards the evaluation of time histories of the actions transmitted by the bell to the pivots and the study of the interactions between the bell and the supporting structures, belfries, and bell-towers. “Ringability” of swinging bells and bell-structure interactions are usually tackled in the framework of rigid body dynamics, so arriving at an initial value problem, governed by a system of two second order nonlinear ordinary differential equations (ODEs), whose solutions are piecewise-defined functions. In the relevant literature, numerical solutions of the system are commonly sought using built-in algorithms provided in advanced software packages; since the use of such general algorithms is subject to some restrictions, especially regarding the forcing functions, validity of the results is often limited. The present study focuses on an innovative procedure to solve the equations of motion. The method, extremely fast and effective, is based on original numerical explicit-implicit predictor-corrector integration algorithms with constant time step, duly validated reproducing the outcomes of relevant reference case studies. Each time the clapper strikes the bell a new “piece” of the solution is initialized, so avoiding user interventions in the elaboration phase. Independently on the oscillation amplitude and on the duration of the considered time interval, the algorithms can successfully manage undamped oscillations; friction and viscosity damped oscillations; free oscillations in transient and stationary phases; and can be applied also to solve stiff equations. Furthermore, the capability of the proposed methods to deal with arbitrary forcing functions is particularly innovative. The outcomes of relevant case studies, regarding the oscillations of the old tenor bell of the Great St. Mary church in Cambridge, confirm the potentialities of the method, also highlighting some topical issues, involving, for example, the assessment of damping equivalence. Finally, a pioneering feature of the algorithms is their ability to handle and to define “resonant” forcing functions, continuously tuning the frequency of the excitation to the natural frequency of the oscillation, according to the oscillation amplitude. |
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| AbstractList | Study of swinging clapper bells involves aspects encompassing sound and acoustic engineering, mechanical engineering, and structural engineering. From the musical point of view, clapper bells are directly played idiophone instruments, where the playing device, the clapper, although directly excited, is not explicitly controlled by the bell ringer. The achievement of a clear and optimal sound mainly depends on the acoustic characteristics of the bell and on the regularity of the clapper strokes, which is not only governed by the ringing style and the relevant parameters of clapper and bell but also by the real time corrections to the excitation introduced by trained bell ringers. In fact, despite centuries of experience allowed to optimize the bell performances, standardizing proportions and mounting arrangements, effective sound control requires some fine tuning of the forcing function. Another crucial topic, especially in view of assessing existing structures, regards the evaluation of time histories of the actions transmitted by the bell to the pivots and the study of the interactions between the bell and the supporting structures, belfries, and bell-towers. “Ringability” of swinging bells and bell-structure interactions are usually tackled in the framework of rigid body dynamics, so arriving at an initial value problem, governed by a system of two second order nonlinear ordinary differential equations (ODEs), whose solutions are piecewise-defined functions. In the relevant literature, numerical solutions of the system are commonly sought using built-in algorithms provided in advanced software packages; since the use of such general algorithms is subject to some restrictions, especially regarding the forcing functions, validity of the results is often limited. The present study focuses on an innovative procedure to solve the equations of motion. The method, extremely fast and effective, is based on original numerical explicit-implicit predictor-corrector integration algorithms with constant time step, duly validated reproducing the outcomes of relevant reference case studies. Each time the clapper strikes the bell a new “piece” of the solution is initialized, so avoiding user interventions in the elaboration phase. Independently on the oscillation amplitude and on the duration of the considered time interval, the algorithms can successfully manage undamped oscillations; friction and viscosity damped oscillations; free oscillations in transient and stationary phases; and can be applied also to solve stiff equations. Furthermore, the capability of the proposed methods to deal with arbitrary forcing functions is particularly innovative. The outcomes of relevant case studies, regarding the oscillations of the old tenor bell of the Great St. Mary church in Cambridge, confirm the potentialities of the method, also highlighting some topical issues, involving, for example, the assessment of damping equivalence. Finally, a pioneering feature of the algorithms is their ability to handle and to define “resonant” forcing functions, continuously tuning the frequency of the excitation to the natural frequency of the oscillation, according to the oscillation amplitude. |
| Author | Croce, Pietro |
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| Cites_doi | 10.1088/1751-8113/41/5/055205 10.1121/1.1575748 10.1007/s11012-005-7973-y 10.1016/j.engstruct.2005.09.019 10.1016/j.engstruct.2010.07.010 10.1016/j.ymssp.2017.11.025 10.1007/978-0-387-21603-4 10.4028/www.scientific.net/KEM.628.73 10.1002/9781119121534 10.1155/2012/681787 10.1016/j.ijimpeng.2011.12.006 10.1016/0020-7403(76)90020-5 10.1016/j.engstruct.2018.07.104 10.1680/stbu.9.00030 10.1016/j.ijimpeng.2015.06.008 10.1680/eac.16842 10.1016/j.engstruct.2009.01.002 10.1525/9780520911079 10.1080/13632460109350386 10.1016/j.ijimpeng.2015.03.001 10.1016/j.ijimpeng.2012.08.001 10.1016/S0020-7462(01)00033-6 |
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| References | Meneghetti (ref_38) 2010; 32 Ahmad (ref_41) 2016; 11 Montagu (ref_3) 2009; 1 Ivorra (ref_13) 2006; 41 Ivorra (ref_10) 2011; 164 Klemenc (ref_43) 2012; 44 ref_34 ref_31 Brzeski (ref_35) 2015; 85 ref_30 McCombie (ref_25) 2019; 193 Veltmann (ref_32) 1876; 220 Stronge (ref_42) 2013; 51 Musielak (ref_39) 2008; 41 Sachs (ref_4) 1914; 46 ref_19 Ivorra (ref_11) 2006; 28 ref_18 ref_17 ref_15 King (ref_7) 1965; 61 Selby (ref_21) 1997; 26 Lund (ref_20) 1995; 15 Blakeborough (ref_26) 2001; 5 Brzeski (ref_36) 2015; 85 Ivorra (ref_12) 2009; 31 ref_24 ref_22 ref_44 Udwadiaa (ref_40) 2002; 37 ref_1 Threlfall (ref_33) 1976; 18 Brzeski (ref_37) 2018; 104 ref_2 ref_29 ref_28 Cantieni (ref_23) 2014; 628 ref_27 McLachlana (ref_16) 2003; 114 ref_9 ref_8 ref_5 Rossing (ref_14) 1984; 72 ref_6 |
| References_xml | – ident: ref_28 – volume: 41 start-page: 055205 year: 2008 ident: ref_39 article-title: Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients publication-title: J. Phys. A: Math. Theor. doi: 10.1088/1751-8113/41/5/055205 – ident: ref_30 – volume: 114 start-page: 505 year: 2003 ident: ref_16 article-title: The design of bells with harmonic overtones publication-title: J. Acoust. Soc. Am. doi: 10.1121/1.1575748 – ident: ref_5 – ident: ref_24 – ident: ref_34 – volume: 41 start-page: 47 year: 2006 ident: ref_13 article-title: Dynamic forces produced by swinging bells publication-title: Meccanica doi: 10.1007/s11012-005-7973-y – volume: 28 start-page: 660 year: 2006 ident: ref_11 article-title: Dynamic investigations on a masonry bell tower publication-title: Eng. Struct. doi: 10.1016/j.engstruct.2005.09.019 – volume: 32 start-page: 3363 year: 2010 ident: ref_38 article-title: An analytical model based on lumped parameters for the dynamic analysis of church bells publication-title: Eng. Struct. doi: 10.1016/j.engstruct.2010.07.010 – volume: 1 start-page: 7 year: 2009 ident: ref_3 article-title: It’s time to look at Hornbostel-Sachs again publication-title: Muzyka – volume: 220 start-page: 481 year: 1876 ident: ref_32 article-title: Über die Bewegung einer Glocke publication-title: Dinglers Polytech J. – volume: 104 start-page: 716 year: 2018 ident: ref_37 article-title: Impact adding bifurcation in an autonomous hybrid dynamical model of church bell publication-title: Mech. Syst. Signal Process. doi: 10.1016/j.ymssp.2017.11.025 – ident: ref_15 doi: 10.1007/978-0-387-21603-4 – volume: 628 start-page: 73 year: 2014 ident: ref_23 article-title: One-year monitoring of a historic bell tower publication-title: Key Eng. Mater. doi: 10.4028/www.scientific.net/KEM.628.73 – ident: ref_6 – ident: ref_44 doi: 10.1002/9781119121534 – ident: ref_8 – ident: ref_18 doi: 10.1155/2012/681787 – volume: 44 start-page: 29 year: 2012 ident: ref_43 article-title: Dynamics of a clapper-to-bell impact publication-title: Int. J. Impact Eng. doi: 10.1016/j.ijimpeng.2011.12.006 – volume: 46 start-page: 553 year: 1914 ident: ref_4 article-title: Systematik der Musikinstrumente publication-title: Ein Versuch. Z. Ethnol. – volume: 18 start-page: 161 year: 1976 ident: ref_33 article-title: Inertia forces due to bell-ringing publication-title: Int. J. Mech. Sci. doi: 10.1016/0020-7403(76)90020-5 – volume: 193 start-page: 271 year: 2019 ident: ref_25 article-title: Rocking of a bell tower—Investigation by non-contact video measurement publication-title: Eng. Struct. doi: 10.1016/j.engstruct.2018.07.104 – ident: ref_31 – volume: 164 start-page: 3 year: 2011 ident: ref_10 article-title: Masonry bell towers: Dynamic considerations publication-title: Struct. Build. doi: 10.1680/stbu.9.00030 – ident: ref_29 – ident: ref_27 – ident: ref_2 – volume: 85 start-page: 57 year: 2015 ident: ref_35 article-title: Analysis of transitions between different ringing schemes of the church bell publication-title: Int. J. Impact Eng. doi: 10.1016/j.ijimpeng.2015.06.008 – ident: ref_9 doi: 10.1680/eac.16842 – volume: 11 start-page: 6550 year: 2016 ident: ref_41 article-title: Impact models and coefficient of restitution publication-title: A review. ARPN J. Eng. Appl. Sci. – ident: ref_17 – volume: 31 start-page: 1085 year: 2009 ident: ref_12 article-title: Dynamic behaviour of a modern bell tower—A case study publication-title: Eng. Struct. doi: 10.1016/j.engstruct.2009.01.002 – ident: ref_19 – ident: ref_1 doi: 10.1525/9780520911079 – volume: 26 start-page: 321 year: 1997 ident: ref_21 article-title: The structural safety and acceptability of bell towers publication-title: Trans. Built Environ. – ident: ref_22 – volume: 5 start-page: 69 year: 2001 ident: ref_26 article-title: An analytical response of church bells to earthquake excitations publication-title: J. Earthq. Eng. doi: 10.1080/13632460109350386 – volume: 85 start-page: 177 year: 2015 ident: ref_36 article-title: Experimental verification of a hybrid dynamical model of the church bell publication-title: Int. J. Impact Eng. doi: 10.1016/j.ijimpeng.2015.03.001 – volume: 61 start-page: 2846 year: 1965 ident: ref_7 article-title: The clappering of a bell publication-title: Ringing World – volume: 72 start-page: 440 year: 1984 ident: ref_14 article-title: The Acoustics of Bells publication-title: Am. Sci. – volume: 51 start-page: 36 year: 2013 ident: ref_42 article-title: Smooth dynamics of oblique impact with friction publication-title: Int. J. Impact Eng. doi: 10.1016/j.ijimpeng.2012.08.001 – volume: 37 start-page: 1079 year: 2002 ident: ref_40 article-title: On the foundations of analytical dynamics publication-title: Int. J. Non-Linear Mech. doi: 10.1016/S0020-7462(01)00033-6 – volume: 15 start-page: 46 year: 1995 ident: ref_20 article-title: The dynamics of bell towers—A survey in northeast England publication-title: Trans. Built Environ. |
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| Title | Nonlinear Dynamics of Swinging Clapper Bells under Arbitrary or Resonant Forcing Functions |
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