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(Bates et al., 1999) and a drop of 70 Hz for an aluminum fork as the temperature increased by 280°C (Blodgett, 2001).
The dependence of frequency on the properties of the mate- rial from which a tuning fork is made can be a useful means of giving students a tangible experience with the properties of various metals and other materials. Burleigh and Fuierer (2005) and Laughlin et al. (2008) manufactured different collections of 17 tuning forks with identical dimensions but made from a variety of metals, polymers, acrylics, and woods and used them with students in a materials course to explore how the frequency, duration, and amplitude of the tuning fork sound depends on material properties. The material from which a tuning fork is constructed is also important for noneducational applications; MacKechnie et al. (2013) found that a steel tuning fork was more likely to produce a negative test result than an aluminum fork when administering the Rinne test for clinical assessment of conductive hearing loss.
Flexural Bending Modes and Natural Frequencies
A tuning fork that is freely suspended (not held at the stem) will exhibit a number of flexural bending modes similar to those of a free-free bar. Figure 2, a and b, shows the first two out-of-plane flexural bending modes of a free tuning fork, and Figure 2, c and d, shows the first two in-plane bending modes. Because the fork does not have a uniform cross section along its length, the displacement amplitudes and the node positions are not symmetrical about the midpoint of the fork, something that is similar to the bending modes of a nonuniform baseball bat (Russell, 2017). However, these free-free mode shapes are not typically observed when the fork is held at the stem. Instead, the normally observed vibra- tional mode shapes, the shapes that give rise to the sound of the fork, are symmetrical modes in which the tines move in opposite directions (Rossing et al., 1992), as shown in Figure 2, e and f.
When vibrating in the fundamental mode, the tines of a handheld fork flex in opposite directions, like a cantilever beam. The second mode has a node roughly one-fourth of the tine length from the free end. An impact at this loca- tion will excite the fundamental but not the second mode; this is where to strike the fork to produce a pure tone. A fork should be impacted with a soft rubber mallet or struck against a relatively soft body part, like the knee or the pisi- form bone at the base of the palm opposite the thumb (Watson, 2011). A fork should never be struck against a hard tabletop or hit with a metal object; doing so will excite other vibrational modes besides the fundamental and it could possibly dent the fork, changing its frequency.
The Frequencies of the Fundamental and
the “Clang” Mode
When a tuning fork is struck softly, the resulting sound is a pure tone at the frequency of the fundamental symmetrical mode of the tines, as shown in Figure 2e. The spectrum in Figure 3a is for a soft impact on the tines of a 432-Hz tuning fork and shows a single, narrow peak at 432 Hz, 60 dB above the noise floor. Figure 3b shows that when this same 432-Hz fork is given a slightly harder impact at the tip of the tine, both the fundamental and also the second mode are excited. The second mode, called the “clang” mode, has a frequency of 2,605 Hz for this fork, which is slightly more than six times the frequency of the fundamental. The overtones of a tuning fork are not harmonics.
 Figure 2. Flexural bending modes for a tuning fork. Red, antinodes with maximum amplitude; dark blue, nodes with zero amplitude. Top: out-of-plane bending modes for a 430- Hz tuning fork. a: first bending mode at 1372 Hz. b: Second bending mode at 3,731 Hz. Center: in-plane bending modes for a 430-Hz tuning fork. c: First bending mode at 1,974 Hz. d: Second bending mode at 4,285 Hz. Bottom: symmetrical in-plane modes of a 430-Hz tuning fork. e: Fundamental mode at 430 Hz. f: “Clang” mode at 2,612 Hz.
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