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Table 1. Dimensions and barrel constructions for baseball bats for various groups
Table 1. Dimensions and barrel constructions for baseball bats for various groups.
Figure 2. a: Radius profiles for a baseball bat (wood and aluminum) and slow-pitch softball bat (composite). Measured mode shapes for a wood base- ball and composite softball bat compared with a uniform beam: first bending mode (b); second bending mode (c). In both plots, the handle is at right and the barrel is at left.
Figure 3. The range of frequencies for the lowest flexur- al bending mode and lowest cylindrical barrel mode for slow-pitch softball bats of a variety of constructions. Sin- gle-walled aluminum bats entered the market in the early 1970s and older bats have higher frequencies, although the barrel frequencies moved to lower values as improve- ments in aluminum alloys allowed for thinner barrel walls without sacrificing durability. Double-walled aluminum bats entered the market in the mid-1990s and introduced a significant improvement in performance. In 1993, Easton, Louisville Slugger, and Worth introduced single-walled ti- tanium alloy bats that hit balls so much faster that they were immediately banned. Composite graphite bats were introduced as early as 1993, but only after 2000 did carbon fiber composite bats begin to dominate the market. Modi- fied from Russell (2004).
Bat Length
Barrel Diameter
Barrel Length
Material
Barrel Type
Baseball, professional
31-34 in. (79-86 cm)
2.625 in. (6.7 cm)
3-5 in. (8-13 cm)
Wood
Solid
Baseball, college and high school
31-34 in. (79-86 cm)
2.625 in. (6.7 cm)
3-5 in. (8-13 cm)
Aluminum or composite
Hollow
Softball
33-34 in. (84-86 cm)
2.25 in. (5.7 cm)
10-14 in. (25-36 cm)
Aluminum or composite
Hollow
Y outh baseball
18-30 in. (46-76 cm)
2.25 in. (5.7 cm)
8-10 in. (20-25 cm)
Aluminum or composite
Hollow
A surprising validation of the free-free condition is the fact that a player’s hands do not affect the bat-ball collision. The duration of the bat-ball collision, approximately 0.0007 s for baseball and 0.001 s for softball, is shorter than the time for bending vibrations to travel from the impact point on the bar- rel down to the handle and back. This means that the ball does not know that the handle of the bat exists because the ball has already broken contact with the bat before any vibration re- turns from the handle. The batted ball speed is the same if the bat handle is clamped in a rotating pivot, gripped by a player, or freely supported (Koenig et al., 2004). Brody (1990) even predicted that a batter could completely release the bat just before impact and the resulting ball trajectory would be the same as if the batter was firmly gripping the bat during the entire swing. This actually happened on May 12, 2012, when high-speed video captured Major League Baseball (MLB) player Todd Frazier hitting a home run even though the bat was slipping out of his hands and was completely free at the instant of impact with the ball (https://goo.gl/hQsB1Z).
The frequencies of the first two bending modes are im- portant to the perception of feel because the human hand is most sensitive to vibrations with frequencies between 150 Hz and 550 Hz, with a peak sensitivity around 250 Hz (Reynolds et al., 1977). Variations in shape profiles, differ- ences in material properties between wood and aluminum,
and the fact that composite materials may be manipulated to design bat handles with varying degrees of flexibility allow a bat’s bending frequencies to cover a fairly wide range. For softball and baseball bats, the frequency of the first bend- ing mode typically falls between 80 Hz and 215 Hz and the second bending mode between 350 Hz and 750 Hz. Figure 3 shows the range of vibrational frequencies for a collection of approximately 60 softball bats of various constructions. The y-axis of the plot shows the spread of frequencies for the lowest flexural bending mode in the handle; the x-axis of the plot shows the variation in frequency for the cylindrical shell vibrations in the hollow barrels of aluminum and com- posite bats (these are discussed in the Hoop Modes, Ping, and the Trampoline Effect section).
Bat Vibration: Sting and Sweet Spots
The definition of the sweet spot of a baseball bat is prob- lematic because, as is the case for tennis rackets, the term sweet spot could refer to the location that minimizes the vi- bration and impulse felt by the hands, the location where the maximum amount of energy is transferred from bat to ball, or the location where the ball leaves the bat with maximum velocity, and these three locations do not coincide (Brody, 1986). Furthermore, even if the definition of the sweet spot is limited only to the location where the sensation of vibra-
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