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``````Acoustics of Baseball Bats
these cylinder modes only involve vibration in the hollow barrel, they have no influence on the percep- tion of feel in the handle.
The Hoop Mode and the
Trampoline Effect
The lowest frequency (n = 2, m =
1) cylinder mode of a bat barrel is
called the “hoop mode” (Video 3 at http://acousticstoday.org/russell-
media) and is responsible for the
ping sound of the aluminum bat
in Figure 5. This mode is also re-
sponsible for the potentially per-
formance-enhancing trampoline
effect in a hollow bat, similar to
the trampoline effect provided by
the titanium faceplate of a hollow golf club driver. The tram- poline effect is so named because the elastic barrel of the bat does most of the work during the collision, elastically deform- ing and returning stored potential energy to the ball, much like when a person jumps on a trampoline. The frequency of the hoop mode depends on the material, thickness, and length of the barrel walls, and there is considerable variation in the hoop frequency. The frequency of the hoop mode for a softball bat, shown on the x-axis in Figure 3, ranges from 1,000 Hz to 2,500 Hz, which is more than a musical octave. Figure 7 illustrates that it is possible to find softball bats with hoop frequencies roughly corresponding to the notes of a mu- sical scale. With the right bats, one can even make a bat piano to play “Take Me Out to the Ball Game” (Video 4 at http:// acousticstoday.org/russell-media; for a longer explanation of this video, see http://y2u.be/r4KTGj-2trQ).
The location of maximum amplitude for the (n = 2, m = 1) hoop mode, as measured from the barrel end of the bat, near- ly coincides with the sweet zone. Not only will ball impacts in this region not sting the hands, but the ball could come off the bat faster if the trampoline effect has the right fre- quency. The effective elastic property of the barrel, due to the trampoline effect of the hoop mode, is measured in terms of the bat-ball coefficient of restitution (BBCOR), from which a number of other performance metrics, including batted- ball speed, may be calculated (Smith, 2001, 2008; Nathan, 2003; ASTM, 2014). The frequency of the (n = 2, m = 1) hoop mode correlates rather well with measured BBCOR values and batted-ball speeds for both softball bats (Russell, 2004) and baseball bats (Sutton and Sherwood, 2010; Nathan et al., 64 | Acoustics Today | Spring 2020, Special Issue
40 | Acoustics Today | Winter 2017 Reprinted from volume 13, issue 4
Figure 7. A selection of softball bats with hollow metal or composite cylindrical barrels having hoop frequencies that form a good approximation of a musical scale. This “bat piano” can be used to play tunes (Video 4 at http://acousticstoday.org/russell-media).
2011a). All other properties being equal, a bat with a lower hoop frequency will hit the ball faster and farther than a bat with a higher hoop frequency. The highest performing slow- pitch softball bat manufactured to date is the original com- posite Miken Velocit-E Ultra introduced in 2002, with a very low hoop frequency around 1,000 Hz (this bat was quickly banned and is not currently legal for play).
Aluminum and composite baseball bats currently used for college and high-school play are regulated by a performance standard that limits the BBCOR value to 0.5, which is essen- tially the maximum value for a wood bat. Aluminum and composite BBCOR 0.5 baseball bats have hoop frequencies above 1,800 Hz; at a frequency this high, the trampoline effect is too small to improve the batted-ball speed. There are still advantages to using a nonwood bat, such as increased dura- bility, increased swing speed, and better bat control. However, since 2011, when the BBCOR standard was adopted, college baseball performance metrics (home runs per game, runs per game, batting average) have dropped to 1972 levels when only wood bats were used (https://goo.gl/YafZhz).
A relatively simple model of the bat-ball collision, adapted from a model of the golf ball-club impact, treats the hoop mode of the hollow bat as a linear mass-spring and the ball as a nonlinear mass-spring (to account for hysteresis and energy lost to friction during the collision). Analysis of col- lisions between these two mass-spring systems captures the essential physics of the collision between a ball and a hollow bat (Nathan et al., 2004), explains the correlation between low hoop frequency and high batted-ball speed, and predicts observed trends for bat performance (Russell, 2004).

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