Nonlinear Dynamics of Swinging Clapper Bells under Arbitrary or Resonant Forcing Functions

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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“…In the kinematic analysis, energy dissipation, which is generally disregarded, can be considered hypothesizing suitable frictional forces between the crack surfaces [30]. In contrast, the rocking analysis allows the energy dissipation due to impacts to be considered, by means of suitable coefficients of restitution, which can be derived using kinematic, kinetic or energetic approaches [31], which are equivalent only in the Although out of the scope of the paper, it must be underlined that a relevant issue in the structural analysis of historical masonry structures is the recognition of recurrent relevant elements and associated collapse mechanisms, which are studied adopting static or dynamic approaches, often in the framework of rigid body mechanics. The kinematic analysis is a typical static approach [28,29], while the rocking analysis is a typical dynamic approach [28,29].…”

“…In the kinematic analysis, energy dissipation, which is generally disregarded, can be considered hypothesizing suitable frictional forces between the crack surfaces [30]. In contrast, the rocking analysis allows the energy dissipation due to impacts to be considered, by means of suitable coefficients of restitution, which can be derived using kinematic, kinetic or energetic approaches [31], which are equivalent only in the case when the impact is frictionless [32]. For the rocking analysis, an energetic coefficient of restitution can be defined as a function of the block slenderness [33], but in some cases it leads to unrealistic results [34].…”

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“…This investigates the dynamic behavior of the rigid block according to the Housner's equation [12], duly considering the energy dissipated over motion. A synthetic parameter describing this energy dissipation is the coefficient of restitution [13], [14]. The coefficient of restitution can be defined in several ways [8], but they are equivalent only in case of frictionless impact [9].…”

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“…Apart from the frequency analysis, in the literature one can also find works on the analytical approach to bell dynamics [ 31 , 32 , 33 ]. In the article [ 31 ], a dynamic model of the church bell was developed, depending on the Lagrange equations and the model of concentrated parameters.…”

“…The change in stability and its individual ranges were determined. A mathematical approach using nonlinear ordinary differential equations during bell dynamic analysis was included in [ 33 ]. The developed algorithm enables the analysis of free, damped and undamped oscillations.…”