INTRODUCTIO NThe resurgence of interest in medieval music has led to a revival also in the making of many long-abandon ed instruments such as the gemahorn.which is a traditional folk recorder dating back to before 1300 A.D. It ceased to be used about 1500 A.D. Originally made from the horn of the Chamois it later became more common to use the horns of

It is now well over a century since Lord Rayleigh published his model for western-style bells. He used a hyperboloid o f revolution p lus a flat circular plate for the c rown. By limiting himself to inextensional m odes of a very restricted type, and exploiting the hyperbola's parametric form, he produced an equation whose roots give the locations of nodal circles. Remarkably this equation involves neither the wall thickness nor physical properties of the bell m aterial and this approach remains the only available a nalytical way o f making such predictions. Although he gave adequate accounts of the derivation and method of solution of his equation, Rayleigh did not present m uch in the way o f comparison o f its predictions with experiment. Rather he focussed on using it t o explain the fact that the Hum note never has any nodal circles. In the present paper we consider how well profiles of some modern church and handbells can be fitted by hyperbolae. We compare the model's predictions for these bells with data for a range of inextensional modes and report a new, surprisingly accurate, approximate analytical solution of Rayleigh's equation.

A strike note, characteristic of the particular bell, is heard when a bell is struck. It is observed that the pitch of this note sometimes does not correspond to the frequency of any one of the bell’s normal modes. The origin of this strike note has been a subject of controversy for over 100 years. Previous empirical investigations have mainly made use of musically trained listeners and real or recorded bell sounds. The short duration of the actual strike note and possible influence of musical training on observations may invalidate the conclusions of previous investigators. In this work use has been made of computer generated simulated bell sounds and untrained listeners. It is demonstrated that a strike note may be isolated by beating with a pure test tone and this technique is used to investigate 28 bell-like sounds using a total of 60 listeners. It is concluded that virtual pitch theory provides the best method of predicting the presence or absence of a strike note and its frequency, but that it does not work in every case.

Changes in the frequencies of the musical partials of various types of bells following debossing dismounting/mounting and annealing/quench annealing are reported. Debossing, dismounting and quench annealing lead to frequency drops, while mounting gives rises. Annealing can lead to frequency increases or decreases depending upon the maximum temperature employed and the initial residual stress. Qualitative explanations of these phenomena are given in terms of changes in crown stiffness, internal stress and alloy phase structure. These are supported by the results of X-ray diffraction measurements. Although the effects are all small they can be large enough to be detected by a reasonably musical car. This, together with the fact that the effects cannot be controlled, gives a plausible explanation of why modern bellfounders use vertical lathes for tuning, even with small carillon bells, and do not anneal bells when trying to control warble.

The whole area of the tuning of church and carillon bells is reviewed in an historical context. Experimental results on the tuning of a small carillon bell are reported and compared with the results of van Heuven for the ‘average Hemony bell’. Surprisingly, in view of the very different geometries of the bells, there is excellent agreement. This fact, coupled with unexpected nodal patterns for some of the carillon bell's modes, seems to shed some light on the classification of church bell modes.

The problem of the uniform circular membrane is reviewed in the light of whether or not evanescent-type behaviour can arise in the separated radial r and transverse theta parts of the solution. It is shown on phyiscal grounds that such behaviour inevitably must arise in the radial direction but cannot do so in the transverse. The consequences for the modal functions are considered and shown to be reflected in various well known, but otherwise physically unexplained, properties of the Bessel functions Jm(kr). It is hoped that these physical explanations will make Bessel functions seem more accessible to students.

It is pointed out that the number of zero eigenfrequencies previously predicted by group theoretical techniques for the in-plane modes of a vibrating regular polygon, consisting of identical particles at the vertices connected along the sides by identical springs, seems excessive. The apparent excess is due to unstable modes. Similar systems with the same symmetries but different details for inter-particle forces will not necessarily exhibit such extra zeros.

The modal frequencies of a spiral clock gong have been measured. The mode numbers and forms of vibrations have been identified by finite element modelling and photographic techniques; very good agreement between modelling and experiment was achieved. All modes were found to be either purely in-plane or out-of-plane. In both cases plots of log (2n - 1) vs. log ƒ yielded curves with two distinct regions, the transition occurring where the wavelength corresponded to one turn of the spiral. Both sets of lower modes were very complex but the modal frequencies above n = 11 were found to fit a modified form of Chladni's law. It is suggested that the modified Chladni law may form the basis of many musical sounds.

The basis of conventional musical scales is the octave, which is a natural consequence of the use of wind and stringed instruments whose partial frequencies are integer multiples of a fundamental. However the vibrational modes of bells, gongs, and cymbals do not in general have partials that are all integer multiples of a fundamental. These modes may be described by a modified Chladni law, which has been used in this work as an alternative basis or musical scales.The resulting music is believed to be a sucessful demonstration of generalized anharmonic scales

It is speculated that warble in carillon bells can be caused by mode splitting as a result of internal strains produced by non-uniform cooling during casting. The neutron diffraction technique makes possible the measurement of internal strain and is here applied to a 612 in (166 mm) carillon bell with a pronounced 3 Hz warble. Internal strain variations with a 180° periodicity about the bell circumference were detected. From symmetry considerations this periodicity requires that all partials (other than m = 0) should have their doublets split. Measurements of the bell partials confirmed this splitting. The metallographic phases present in the bell metal were also determined and found to agree with expectations.

The “classical” theory of thin ring vibrations given by Love, which only applies to circular cross-sections, has been modified to cover general shapes of cross-section. A large amount of experimental data has been obtained for all four types of normal modes of thin circular rings with rectangular cross-sections. Wide ranges of ring heights, thicknesses and nodal diameter numbers have been covered. This new data has been used to test the modified formulae, as applied to rectangular cross-sections. It has also been used to test the predictions of the theories of Kirkhope for the four types of mode, and the theory of Bickford and Maganty for axial and torsional modes. Recommendations are made as to which formulae should be used for various aspect ratios and types of modes.

The forms and vibrational modes of church bells, carillon bells and handbells are discussed and compared. When the principal modes are classified in groups according to the numbers and locations of the nodal circles and meridians, the vibrational behaviours of church bells and carillon bells show similarities to those of the much smaller handbells. The different timbres are traced to the differences in the vibrational behaviour and the resulting spectra of partial tones. A subjective strike note, heard in the sound of church bells and carillon bells, does not appear in handbell sound, but a prominent octave partial in handbell sound is radiated indirectly by the fundamental mode of vibration. Modal frequencies in all three types of bells can be fitted to a suitably modified form of Chladni's law. Large church bells and carillon bells follow a 1/f scaling law; small carillon bells scale as f−11, where n is slightly less than 1. In handbell sets, n may vary from about 12 in the larger bells to 13 in the smallest bells.

Finite element calculations of the normal modes of church, carillon, hand and fire-alarm bells have been made. Some of the results are presented and comparisons made which shed new light on mode classification.

New finite element calculations for a modern D5 Taylor church bell show that the regime change previously postulated at about m = 7 for RIR modes does not occur. The experimental frequency data for these modes have therefore been re-examined and found to fit fm = 256 (m − 1)1·404 Hz.

Extensive previously reported experimental data on the normal modes of an English church bell is analyzed with use of a modified version of Chladni's law. Excellent fits are obtained in almost every case provided certain conditions, for which a theoretical basis is presented, are satisfied. Some conclusions are drawn about the validity of a previously proposed theory of mode production mechanisms for the church bell.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Thomas D. Rossing, Robert Perrin, H. John Sathoff, Richard W. Peterson; Vibrational modes of a tuned handbell. J. Acoust. Soc. Am. 1 October 1984; 76 (4): 1263–1267. https://doi.org/10.1121/1.391382 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAcoustical Society of AmericaThe Journal of the Acoustical Society of America Search Advanced Search |Citation Search