Acoustics and Vibration of Baseball and Softball Bats Daniel A. Russell Vibrational modes of a bat explain the sweet spot, sting for mishits, a metal bat’s “ping,” and the trampoline effect. Postal: 201 Applied Science Building Introduction Pennsylvania State University Baseball and softball are sports, similar to cricket, golf, tennis, hockey, and hurling, University Park that involve a player swinging a handheld bat to hit a ball. In all of these sports, Pennsylvania 16802 the impact between the ball and the bat, racket, stick, or club produces a sound USA heard by both players and fans. It also results in a postimpact vibration felt by the Email: player holding the implement. A surprising fact is the extent to which the sound [email protected] and vibration feedback matter to the player and especially how that feedback influ- ences the perception of performance. The sensation of “feel” perceived by a player depends on the tactile sensation in the hands during and after the contact with the ball, as one would expect, but it is also strongly influenced by the sound of the impact. This has been studied extensively in the game of golf (Roberts et al., 2001, 2006) but also applies to baseball and any other game where a player holds an im- plement used to strike the ball. The perception of feel is composed of (1) vibration sensations in the hands, (2) sound of the impact, and (3) the perceived trajectory of the ball in flight (Hocknell et al., 1996). Popular Bat Wars (http://www.batwars. com) events held across the United States allow players at all levels to try out new bat models from manufacturers, and the ranking of player preference of softball bats is based on performance (distance), feel, balance, sound, and logo design and color. Four of those rankings are personal preferences, whereas two depend on acoustics and vibration. This article describes the flexural bending and cylindrical barrel vibrations of base- ball and softball bats. The flexural bending modes are used to identify the so-called “sweet spot” where impacts do not sting the hands. The source of a metal bat’s “ping” is related to cylindrical modes in the barrel of an aluminum or composite bat, which give rise to a “trampoline effect.” The vibration and acoustic properties of bats are discussed in relation to the development of performance standards. Finally, the myth of the corked wood bat is addressed. Flexural Bending Vibrations in a Bat Figure 1 shows a sampling of the variation available in baseball (left) and softball (right) bats from the author’s laboratory collection of over 120 bats. Profession- al Major (and Minor) League Baseball (MLB) players exclusively use bats made from a single piece of solid wood, with maple and ash being the two most popular woods. College and high-school players primarily use aluminum and/or compos- ite bats with a hollow barrel, although these bats must conform to a performance standard that regulates their performance to be essentially the same as a wood bat. Softball bats, used both for men’s slow pitch and women’s fast pitch, are almost all aluminum or composite hollow-barrel bats. Youth bats used by Little League Baseball are also aluminum or composite but with a much greater variation in length and weight than their adult counterparts. Table 1 summarizes the variation ©2017 Acoustical Society of America. All rights reserved. volume 13, issue 4 | Winter 2017 | Acoustics Today | 35 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 tap the bat at one location while the resulting acceleration end of the bat is much thinner than the barrel end. Figure 2a is measured with an accelerometer at another location, pro- compares the diameter profiles of a baseball bat and softball ducing a frequency-response function for that pair of input/ bat of the same length. output locations. If the accelerometer is held at a fixed loca- tion while the hammer impacts are moved along the length Vibrational Mode Shapes and Frequencies of the bat, the total set of frequency-response functions may The violent collision between a baseball and bat can be curve fit to extract vibrational mode shapes (representing cause postimpact flexing and vibration of the bat. The the normalized displacement of each point relative to all of frequency of the vibration and the corresponding stand- the other points), the resonance frequencies for those mode ing wave patterns (mode shapes) depend on the materials shapes, and the damping decay rates for the modes. For such and dimensions of the bat. Figure 2, b and c, shows the an experiment, the bat is suspended on rubber bands in a first two bending mode shapes for a baseball and softball free-free condition. bat compared with the mode shapes for a uniform beam with free ends. The mode shapes for the bats are simi- One might question whether a baseball bat, gripped in the lar to those of the free-free beam, except that the nodal hands, is best compared with a free-free beam instead of be- points (where the displacement is zero) for the bats are ing clamped at the handle end. To answer, the frequencies for shifted toward the thinner handle end and the vibration a handheld baseball bat are much closer to those of a free- amplitude is not symmetric but is larger in the handle free bat than they are for a bat clamped at the handle (Brody, (Video 1 at http://acousticstoday.org/russell-media). 1990). Free-free boundary conditions provide a good approxi- mation for the measurement and modeling of other handheld Vibrational mode shapes and frequencies for a baseball or sports implements as well, including cricket bats (Brooks et softball bat are obtained by experimental modal analysis, in al., 2006), golf clubs (Wang and Wu, 2005), and tennis rackets which a hammer instrumented with a force gauge is used to (Banwell et al., 2014). 36 | Acoustics Today | Winter 2017 Table 1. Dimensions and barrel constructions for baseball bats for various groups Table 1. Dimensions and barrel constructions for baseball bats for various groups. Bat Length Barrel Diameter Barrel Length Material Barrel Type Baseball, 31-34 in. 2.625 in. 3-5 in. Wood Solid professional (79-86 cm) (6.7 cm) (8-13 cm) Baseball, 31-34 in. 2.625 in. 3-5 in. Aluminum or college and Hollow (79-86 cm) (6.7 cm) (8-13 cm) composite high school 33-34 in. 2.25 in. 10-14 in. Aluminum or Softball Hollow (84-86 cm) (5.7 cm) (25-36 cm) composite Youth 18-30 in. 2.25 in. 8-10 in. Aluminum or Hollow baseball (46-76 cm) (5.7 cm) (20-25 cm) composite 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- Figure 2. a: Radius profiles for a baseball bat (wood and aluminum) and tanium alloy bats that hit balls so much faster that they slow-pitch softball bat (composite). Measured mode shapes for a wood base- were immediately banned. Composite graphite bats were ball and composite softball bat compared with a uniform beam: first bending introduced as early as 1993, but only after 2000 did carbon mode (b); second bending mode (c). In both plots, the handle is at right and fiber composite bats begin to dominate the market. Modi- the barrel is at left. fied from Russell (2004). A surprising validation of the free-free condition is the fact and the fact that composite materials may be manipulated to that a player’s hands do not affect the bat-ball collision. The design bat handles with varying degrees of flexibility allow duration of the bat-ball collision, approximately 0.0007 s for a bat’s bending frequencies to cover a fairly wide range. For baseball and 0.001 s for softball, is shorter than the time for softball and baseball bats, the frequency of the first bend- bending vibrations to travel from the impact point on the bar- ing mode typically falls between 80 Hz and 215 Hz and the rel down to the handle and back.