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Acoustics of Baseball Bats
 Figure 1. Examples of the variety in baseball and softball bat construction. Left: Baseball bats. Left to right: pro-stock wood, replica of Heinie Groh’s wood bottle bat, large knob wood bat for swing training, wood with composite coating, two piece with aluminum handle and lami- nated bamboo barrel, single-piece aluminum, two-piece aluminum, two-piece stiff composite handle with aluminum barrel, two-piece flex- ible composite handle with aluminum barrel, single-piece composite, composite with double-walled barrel, composite with very stiff handle, aluminum with vibration absorber in knob, aluminum with electronic vibration dissipation circuit on handle, and aluminum with aerody- namic holes in taper. Right: Softball bats. Left to right: wood, 1972 single-walled aluminum, 1993 graphite, 1993 titanium, single-walled aluminum, double-walled aluminum, triple-walled aluminum, two-piece composite handle with aluminum barrel, composite, composite high performance, multiwall (aluminum exterior with composite inner shell), high-performance aluminum double-walled barrel, two-piece antivibration joint with aluminum handle and triple-walled aluminum barrel, two-piece composite handle with aluminum double-walled barrel, two-piece composite handle joined to composite barrel, two-piece stiff handle with composite barrel, and two-piece composite handle with steel single-walled barrel.
 of bat dimensions and construction. For all bats, the handle end of the bat is much thinner than the barrel end. Figure 2a compares the diameter profiles of a baseball bat and softball bat of the same length.
Vibrational Mode Shapes and Frequencies
The violent collision between a baseball and bat can cause postimpact flexing and vibration of the bat. The frequency of the vibration and the corresponding stand- ing wave patterns (mode shapes) depend on the materials and dimensions of the bat. Figure 2, b and c, shows the first two bending mode shapes for a baseball and softball bat compared with the mode shapes for a uniform beam with free ends. The mode shapes for the bats are simi- lar to those of the free-free beam, except that the nodal points (where the displacement is zero) for the bats are shifted toward the thinner handle end and the vibration amplitude is not symmetric but is larger in the handle (Video 1 at
Vibrational mode shapes and frequencies for a baseball or softball bat are obtained by experimental modal analysis, in which a hammer instrumented with a force gauge is used to
tap the bat at one location while the resulting acceleration is measured with an accelerometer at another location, pro- ducing a frequency-response function for that pair of input/ output locations. If the accelerometer is held at a fixed loca- tion while the hammer impacts are moved along the length of the bat, the total set of frequency-response functions may be curve fit to extract vibrational mode shapes (representing the normalized displacement of each point relative to all of the other points), the resonance frequencies for those mode shapes, and the damping decay rates for the modes. For such an experiment, the bat is suspended on rubber bands in a free-free condition.
One might question whether a baseball bat, gripped in the hands, is best compared with a free-free beam instead of be- ing clamped at the handle end. To answer, the frequencies for a handheld baseball bat are much closer to those of a free- free bat than they are for a bat clamped at the handle (Brody, 1990). Free-free boundary conditions provide a good approxi- mation for the measurement and modeling of other handheld sports implements as well, including cricket bats (Brooks et al., 2006), golf clubs (Wang and Wu, 2005), and tennis rackets (Banwell et al., 2014).
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