Page 40 - 2017Winter

Acoustics of Baseball Bats
tion and impulse in the hands is minimized, one still finds a fair amount of disagreement in the literature. The sweet spot for feel has been defined as the center of percussion, the node for the first bending mode, a location between the center of percussion and the node of the first bending mode, or the node of the second bending mode.
Vibrational Modes and the Sweet Zone
Cross (1998) identifies the sweet zone on a baseball bat as the impact location that minimizes the vibration felt by the hands. This is a narrow zone in the barrel of the bat, ap- proximately 5-7 inches (12.7-17.8 cm) from the barrel end. As seen in Figure 4, this definition of the sweet zone falls between the nodes of the first and second bending modes. There is some disagreement in the literature as to whether the sting in the hands is due to the vibration of the first bend- ing mode or a combination of the second bending mode and the initial impulse, having a time duration nearly the same as the period of the second bending mode, traveling along the bat immediately after the collision (Adair, 2001a; Cross, 2001). The first bending mode, with a frequency around 150 Hz, is easily felt by holding the bat lightly at the handle and tapping the barrel end of the bat on the ground. It is easy to locate the node for the first bending mode by lightly hold- ing the bat at the handle and tapping the barrel to find the location where no vibration is felt. However, what a player feels during the violent collision between bat and ball is not the same as this simple test. High-speed video shows that the vibration in the handle can be large enough to cause the player’s top hand to completely lose contact with the bat as the flexural impulse travels down the bat after impact (Video 2 at http://acousticstoday.org/russell-media), and the vibra- tion can even be large enough for a wood bat to splinter or break (https://goo.gl/BZZrA3).
The painful sting resulting from an impact away from the sweet zone is most frequently felt in the fleshy region be- tween thumb and forefinger in the top hand, the hand far- thest from the knob of the handle. Figure 4 shows that this sting location on the handle of a bat corresponds to an anti- node for the second bending mode and a node for the first bending mode, suggesting that the second bending mode is more important to feel. Players tend to show a strong prefer- ence for bats in which the second bending mode has been heavily damped through the use of a vibration absorber tuned to the frequency of the second bending mode (Rus- sell, 2006).
Figure 4. Mode shapes for the first three bending modes of a slow- pitch softball bat, with relative displacements mapped to the physical dimensions of the bat. Solid lines, displacement of the bat at one extreme of the vibration; dashed lines, displacement half a period later, at the other extreme of the vibration cycle; black dots, nodes, locations where the vibrational amplitude is zero. The sweet zone is a region approximately 5-7 inches from the barrel end of the bat; impacts in this zone will minimally excite the bending modes into vibration. Impacts away from the sweet zone will cause the bat to vibrate, and the location where pain is most often felt in the top hand aligns with an antinode (maximum displacement and acceleration) of the second bending mode.
Figure 5. Microphone responses for the impact of a baseball with a wood bat (Sound File 1 at http://acousticstoday.org/russell-media) and an aluminum bat (Sound File 2 at http://acousticstoday.org/russell-media). The high peak at approximately 2,200 Hz represents the character- istic “ping” produced by the aluminum bat and is due to the (n = 2, m = 1) cylindrical hoop mode in the hollow barrel. The smaller peak near 2,800 Hz is due to the (n = 2, m = 2) cylinder mode.
38 | Acoustics Today | Winter 2017