Bass Handbells of Aluminum
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a description becomes complex because of the large number of normal modes of diverse character that contribute to the motion. The acoustically important partials (single-frequency components) in the sound of a bell result from vibrationai modes in which the motion is primarily normal to the bell's surface. It has been customary to classify these modes into families or groups with some common property of the nodal pattern for this component. The most important families
m=2
m=3
are those that have an antinode where the clapper strikes (a short distance above the lip for handbells).1 Families of modes in a handbell are arranged in a periodic table in Figure 1. We designate each mode by two numbers, m and n, where m is the number of complete nodal meridians, and n is the number of nodal circles. The fundamental (2,0) mode determines the musical pitch of a handbell. In most carefully tuned handbells, the (3,0) mode has a frequency three times that of the (2,0) mode, and it therefore radiates a note that is a twelfth above the fundamental. Earlier, we noted a second harmonic in the sound of tuned handbells and attributed it to parametric radiation from the mouth as the bell vibrates in the (2,0) mode.2 Parametric radiation also occurs at twice the frequency of the (3,0) mode (in other words, at six times the fundamental frequency), but in handbells of small to medium size, this partial tone has a very small amplitude and is scarcely noticed. In past studies on handbells (see Reference 3, for example), we have used several different methods for modal analysis of large and small handbells. Scanning the near-field sound with a microphone
m=4
m=6
Group 0 (3,0) 3.0
Group
Group III
End view
Figure 1. Periodic table of vibrationai modes in a handbell. Below each drawing are the relative modal frequencies in a Malmark C5 handbell. At lower left (m,n) gives the number of nodal meridians 2m and nodal circles n.8
MRS BULLETIN/MARCH 1995
Bass Handbells of Aluminum
or the surface of the bell with an accelerometer is a direct method that serves as a good starting point. Holographic interferometry provides better spatial resolution of modes, and experimental modal testing with impact excitation provides the means for creating animated drawings of the vibrational modes. In the present studies on bass handbells, we have employed both direct scanning and holographic interferometry. The handbells were driven sinusoidally either using a Bruel & Kjaer
minishaker or by attaching a small NdFeB magnet to which a sinusoidal magnetic field was applied by means of a small solenoid, as previously described.4 The sound field near the surface of the bell was scanned with a 4-mm-diameter microphone whose output was displayed on the vertical axis of a cathode-ray oscilloscope while the driving voltage was displayed on the horizontal axis. The resulting Lissajous pattern changed abruptly in phase when a nodal line was crossed. Alternatively, a small accelerom-
ft ff ff\ (2,0) 49 Hz
(2,0) 49 Hi
(3.0) 148 Hz
(4,0) 389 Hz
(3,1) 593
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