Page 41 - 2017Winter

Hoop Modes, Ping, and the
Trampoline Effect
The Ping of an Aluminum Bat
The sound of the impact between a baseball and a wood bat at the sweet spot produces a “crack” (Sound File 1 at http:// acousticstoday.org/russell-media), and experienced profes- sional players use the impact sound as a clue to decide which way to begin running to catch the ball before it has traveled far enough for eye tracking to determine its trajectory (Adair, 2001b). An aluminum bat produces a distinct ping (Sound File 2 at http://acousticstoday.org/russell-media), a sound that many baseball purists decry. Figure 5 shows the strong tonal component of the aluminum bat ping (the peak at 2,200 Hz) completely dominates the sound spectrum. The sound of an aluminum bat is distinctive enough that a forensic acoustic analysis of a recorded police emergency 911 call was able to identify a baseball bat found at a homicide crime scene as the probable murder weapon (Marr and Koenig, 2007).
The impact from the bat-ball collision can be loud enough to raise concerns about hearing health for softball players who are subjected to repeated loud impacts (Okuma and Takinami, 1994; Cook and Atcherson, 2014). Although indi- vidual ball impacts with an aluminum bat can produce levels as high as 124.6 dB at the approximate location of a batter’s left ear, the normalized 8-hour equivalent A-weighted level is low enough to be pose little risk during a typical game. However, repeated exposure for a catcher during a game or a batter during batting practice could warrant the use of hear- ing protection.
Hoop Modes in Hollow Cylindrical Barrels
The barrel of a wood baseball bat is solid, but the barrels of aluminum and composite baseball and softball bats are hollow cylindrical shells. A hollow cylindrical shell exhibits several families of mode shapes expressed in terms of the angular position θ around the circumference of the barrel and the distance x along the length of the barrel according to
φ(x, θ) = cos(nθ) sin(mx/L) (1)
where φ represents the normalized radial displacement and L is the barrel length. The mode shape designations (n and m) indicate the number (2n) of axial nodal lines encoun- tered as one traverses the circumference of the barrel and the number (m) of circumferential nodal circles encoun- tered as one traverses the length of the barrel. The n nodes are actually diameters for the circular barrel cross section, and as one traverses the circumference, each diameter is en-
Figure 6. a: Circumferential modes in the hollow cylindrical barrel of an aluminum or composite bat (Video 5 at http://acousticstoday. org/russell-media). Solid and dashed lines, extremes of the vibra- tional displacement, separated by half a cycle. The (n = 0) modes are breathing modes and are not observed. The (n = 1) modes are the flexural bending modes. The lowest frequency (n = 2) mode is called the “hoop mode” of the bat and is used to model the essential physics of the bat-ball collision. b: A spatial Fourier synthesis of the (n = 2, 3, 4, 5) modes produces the “kidney bean” shape that corresponds to the initial deformation of the barrel cross section on impact with a ball. c: A finite element model of the ball-bat collision shows the same ini- tial bat deformation with a slight outward bulge (red) at the top and bottom of the bat cross section and a concave inward compression (blue). c modified from Mustone (2003).
countered twice, so if n = 2, a total of 4 axial node lines are encountered around the circumference. Figure 6a illustrates the circumferential variation of the cylinder radius corre- sponding to different values of n. All of the mode shapes in the same family (same n value but different m values) have the same circumferential displacement but differ in radial displacement along the axial length of the barrel. The (n = 0) modes involve a uniform expansion of the barrel and are not observed in a bat. The (n = 1) modes are the flexural bend- ing modes. The families of cylinder modes with n > 2 are all involved in the deformation and effective elastic property of the bat barrel during the collision with the ball. A spatial Fourier synthesis (Figure 6b) of the (n = 2, 3, 4, 5) modes closely resembles the initial deformation of the bat barrel during impact with the ball. This Fourier reconstruction agrees well with a finite element analysis model of the bat- ball collision (Figure 6c). The higher order cylinder modes are not easily observed for most bats and tend to have reso- nance frequencies high enough, with periods short enough, that the time duration of the impact between bat and ball prevent these modes from adding significantly to the acous- tic or vibrational signature of the bat. Furthermore, because
Winter 2017 | Acoustics Today | 39