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Forensic Science Communications January 2007 – Volume 9 – Number 1 
Case Report

Fundamental Frequency Analysis of a Metal Baseball Bat

Kenneth W. Marr
Audio Forensic Expert
Forensic Audio, Video and Image Analysis Unit
Operational Technology Division
Federal Bureau of Investigation
Quantico, Virginia

Bruce E. Koenig
Audio/Video Forensic Expert
BEK TEK LLC
Clifton, Virginia

Introduction/Background | Acoustic Theory/Limitations | Test Procedures | Results | Discussion and Conclusions | Acknowledgments | References

Introduction/Background

The FBI’s Forensic Audio, Video and Image Analysis Unit conducted a unique signal analysis examination for a sheriff’s department from the Northwest United States. During a homicide investigation, the department determined that the suspect had probably hit the victim on the head with an aluminum baseball bat. The department requested an acoustic analysis to determine if the recovered bat had produced the five “ping” sounds that had occurred during the assault and had been recorded by the department’s 911 system through an open telephone line at the scene. The recording unit was an analog Dictaphone (Nuance, Burlington, Massachusetts) logging system, with storage on a 10.5-inch-diameter reel of 0.5-inch-wide audiotape. Laboratory comparisons were made between the submitted bat, two other test bats, and the 911 recording using narrow-band spectrum analysis. The examination showed that the resonant frequency of the submitted bat was generally consistent with the 911 ping recordings. This case report presents acoustic theory, along with test procedures, results, and conclusions.

Acoustic Theory/Limitations

Aluminum baseball bats have an irregular longitudinal shape, a nonuniform cross section, walls that vary in thickness, unspecified materials alloyed with the aluminum, and various types of filler material in the center. For these and a variety of other reasons, no known accurate acoustic models exist for aluminum baseball bats (Adair 1994; Van Zandt 1992). However, the acoustic theory for metal bars and cylinders can provide some insight into the resonance sounds produced when a bat strikes an object. The sports enthusiast would recognize these resonances as metallic “ping-like” events that occur when a hitter strikes the ball. These sounds are the result of the bat’s oscillation when it hits a hard object, producing a specific vibration (called the fundamental frequency, f0) and a series of harmonic, or nearly harmonic, overtones. Many musical instruments operate in the same manner; for example, middle C on a piano has an f0 of 261.63 hertz (Hz), which means that its lowest frequency of vibration is 261.63 times per second (Berg and Stork 1995). Its overtones are approximately integral multiples of this 261.63 frequency and can include both odd and even harmonics.

The formulas for the expected fundamental frequency of resonance for a metal bar that is free on both ends (free-free bar), a bar that is free on one end and securely clamped on the other (free-fixed bar), and a cylinder with a fixed cap at one end are as follows (Kinsler et al. 1982):

f0 = c/2L   for a free bar,
f0 = c/4L   for a fixed bar,
and
f0 = c/4L   for a cylinder with rigid cap at one end,
where

c = the speed of sound in the bar

and

L = the length of the bar/cylinder.

If a vibrating baseball bat behaved in the same manner as generic aluminum bars and cylinders, its fundamental frequency could be calculated. For example, because the speed of sound in an aluminum bar is 5150 meters/second (Kinsler et al. 1982), using a bat length of 34.0 inches (0.864 meter), the fundamental frequency would be:

f0 = c/2L = 5150/2(0.864) = 2980 Hz
for a free-free bar and
f0 = c/4L = 5150/4(0.864) = 1490 Hz
for both a free-fixed bar and a cylinder with a rigid cap at one end.

The fundamental frequency of resonance was used for analysis in this case, but it may not always be the best parameter for correlation of recorded ping sounds. In this case, the forensic conditions limited the selection of other parameters. Specifically, the object(s) the bat struck, causing the ping sounds, is (are) unknown; the forensic environment is not ideal (the distance from the microphone, background sounds, how the bat was held, and so on are unknown); and the telephone system has limitations, including the microphone, telephone transmission bandwidth, and communication recording system.

The forensic analysis of the 911 recording involved too many unknown factors to make it definitive; therefore, only two test bats were used to conduct the basic resonant-frequency analyses.

Test Procedures

The following procedures were followed to allow resonant-frequency comparisons between the submitted baseball bat, the two test bats, and the recording on the 911 logging tape:

  1. The logging tape was played back on a specialized laboratory unit, and the speed error in the original recording process was determined.


  2. The values of the discrete frequencies present during the five ping sounds in the designated recording on the 911 tape were measured.


  3. The ping frequencies were adjusted to their correct values using the recorded speed error of the 911 system.


  4. The submitted aluminum bat and the two other bats were tested to determine their discrete frequencies when hitting a hard object.


  5. A frequency comparison was made between the ping sounds on the 911 tape and the values of the tested baseball bats.


Most analog recording systems have tape-speed errors when recorded on one unit and then played back on a second system, because of the limitations of the mechanical and electronic components. In the 911 system used by the sheriff’s department, the recorder operated at a very low tape speed (nominally 15/32 inches per second) and did not contain sophisticated speed-control circuits. However, the recording system had a very accurate time-code generator (published error rate of ±0.001%), which recorded a signal on the tape that could be used to determine the playback-speed error. To determine this error, the 911 tape was played back on the laboratory unit, accurately timed with an electronic stopwatch (manual stop and start), and then compared with the actual elapsed time from the time-code signal. The following results were obtained:

Total of real-time playback (stopwatch): 20 minutes no seconds (1200 seconds);
 
Elapsed coded time: 19 minutes 10.0 seconds (1150 seconds);
 
Playback time difference: 50.0 seconds too slow;
 
Correction factor: 1200/1150 = +1.0435.

The 911 tape was then played back on an FBI Laboratory unit, and the five ping sounds were visualized with a laboratory-grade fast Fourier transform (FFT) analyzer. An FFT device converts a signal from the time domain into the frequency domain, producing a narrow-band spectrum display of frequency versus amplitude (Bracewell 2000; Wallace and Koenig 1989). The five designated signals on the 911 tape were analyzed using the FFT unit to determine which discrete frequencies were present. Table 1 reflects the measured f0 frequencies for the five designated ping sounds and their values after being adjusted by the correction multiplier of 1.0435. This table reflects that the range of the corrected f0 frequencies for the five events is 2222.66 Hz to 2306.14 Hz, or 83.48 Hz.

Table 1: Fundamental Frequencies (f0) and Their Corrected Values for Five Ping Sounds from the 911 Tape

Ping Event (sec) Measured f0 (Hz) Corrected f0 (Hz)
95.40 2130.00 2222.66
100.10 2150.00 2243.52
100.97 2210.00 2306.14
101.70 2200.00 2295.70
110.70 2150.00 2243.52

 

The three aluminum bats—two test aluminum bats and the aluminum bat submitted by the sheriff’s department—were then tested. Test bat number one was a Steele-Lite brand (Steele’s Sports, Grafton, Ohio, no longer in business), test bat number two was a Ten Pro (Bombat Sports, Inc., Faith, North Carolina), and the evidence bat was a Tennessee Thumper (Worth Sports, Fenton, Missouri). Because aluminum baseball bats of the same manufacturer, model, and size were expected to produce nearly identical results, the test bats were chosen to reflect the variances, if any, between different types and sizes. Table 2 lists the physical dimensions of the three bats, including the maximum value for their circumferences.

Table 2: Physical Dimensions of the Three Aluminum Baseball Bats Tested

Bat Length
(in.)
Circumference
(in.)
Handle Grip Type
and Length (in.)
Weight
(lb)
Test #1 33.0 7.188 Plastic/10 1.95
Test #2 31.5 6.375 Plastic/8 1.80
Evidence 34.0 7.188 Plastic/10 2.35

Because it was not known how the suspect held the bat during the assault or how the bat contacted the victim and/or nearby objects, the resonance behavior of the bats was tested by striking them at locations along the long axis of the bat and against different materials. The tests were all conducted by holding the bats firmly with two hands at the end of the handle and striking against known materials at measured locations along the bat. The ping sounds were sensed by a professional microphone and recorded on a professional digital audiotape (DAT) recorder, which has a speed error of less than ±0.001%. Narrow-band spectrum analysis was then used to determine the discrete frequencies generated by each bat during each separate test. Table 3 reflects the resultant average f0 when the bats were struck repeatedly against hardwood at various positions along the length of the bat, with an accuracy of 6.25 Hz.

Table 3: Average f0 Values for Test Bats Struck with Hardwood

Distance
from
Handle Tip
(in.)
Test Bat #1 (Hz) Test Bat #2 (Hz) Evidence Bat (Hz)
20–21 2075.00 2837.50 2237.50
22–23 2075.00 2837.50 2250.00
24–25 2062.50 2850.00 2250.00
26–27 2062.50 2837.50 2250.00
28–29 2062.50 2837.50 2250.00
30–31 2062.50 2837.50 2250.00
32 NA NA 2250.00

The bats then were struck in various ways against three different materials: hardwood, concrete, and asphalt pavement. Because the ping sounds on the 911 recordings could have been caused by the bat’s hitting a human head or other objects, these materials were chosen to provide somewhat similar conditions and to determine if the type of material hit by the bats produced different discrete frequencies. Table 4 lists the results of these tests, with the average f0 values having an accuracy of 6.25 Hz.

Table 4: Average f0 Values of Three Test Bats Struck Against Wood, Concrete, and Pavement

Material Test Bat #1
(Hz)
Test Bat #2
(Hz)
Evidence Bat
(Hz)
Wood 2075.00 2837.50 2237.50
Concrete 2075.00 2837.50 2237.50
Pavement 2075.00 2837.50 2237.50

Results

A comparison between the sounds on the 911 tape and the three bats revealed the following results:

  1. The f0 range for test bat one was 2062.50 to 2075.00 Hz; for test bat two, 2837.50 to 2850.00 Hz; and for the evidence bat, 2237.50 to 2250.00 Hz (all with an accuracy of 6.25 Hz).


  2. The speed-corrected f0 values of the ping sounds on the 911 recording had a fairly wide range of 2222.66 to 2306.14 Hz. Only the ping events at 100.10 and 110.70 seconds fell within the f0 range of the evidence bat tests.


  3. A bat held on one end when striking a material would be expected to have an f0 between the free-free and free-fixed bar formulas. Calculating the values from the free-free and free-fixed bar formulas, as listed previously, results in a range of 1536 to 3073 Hz for test bat one, 1607 to 3215 Hz for test bat two, and 1490 to 2980 Hz for the evidence bat. The measured test values for all three bats fell within these quite wide ranges. The formulas also reflect that the longer the bar, the lower the f0; however, the shortest bat (test number two) had the highest f0, and the middle-length bat (test number one) had the lowest f0. Factors that may have contributed to this result include bat shape and the differences in metal alloys, which are beyond the scope of this investigation.

Discussion and Conclusions

  1. The measured aluminum bat resonances have similar characteristics to the theoretical properties of aluminum bars and cylinders, and it may be possible to develop a computer model that could accurately predict those values for a particular baseball bat. However, the general formulas for metal bars produce too wide a range of values to be useful under most forensic applications.


  2. Without specific modeling, the bat(s) in question should be properly tested and measured. Characteristics that should be considered include the physical properties of the bat, the material the bat strikes, the location on the length of the bat where struck, and where the bat was held when striking the object.


  3. In this particular case, the specific f0 values of all three tested bats fell within a fairly narrow range based upon where and what material the bats struck.


  4. The listed variances between the tested resonant values of the evidence baseball bat and the five recorded ping sounds could be due to a number of factors, including how the bat was held during the incident compared with the testing and the possibility that some of the ping sounds originated from different sources.


  5. The measured resonant values of the bats indicate a range of variability expected under normal forensic conditions and, not unexpectedly, do not match the theoretical values for resonating bars and cylinders.

Acknowledgments

The authors thank the following individuals who provided assistance during the various test procedures: David James Snyder, Douglas S. Lacey, Dr. Hirotaka Nakasone, and James John Ryan, Jr.

References

Adair, R. K. The Physics of Baseball. 2nd ed., Perennial-HarperCollins Publisher, Inc., New York, 1994, pp. 108–138.

Berg, R. E. and Stork, D. G. The Physics of Sound. 2nd ed., Prentice Hall, Englewood Cliffs, New Jersey, 1995, pp. 358–361.

Bracewell, R. N. The Fourier Transform and Its Applications. 3rd ed., McGraw-Hill, New York, 2000.

Kinsler, L. E., Frey, A. R., Coppens, A. B., and Sanders, J. V. Fundamentals of Acoustics. 3rd ed., John Wiley and Sons, Hoboken, New Jersey, 1982, pp. 62, 201, 461.

Van Zandt, L. L. The dynamical theory of the baseball bat, American Journal of Physics (1992) 60:172–181. [This article contains an excellent computer-based modeling program for a wooden bat.]

Wallace, A. Jr. and Koenig, B. E. An introduction to single channel FFT analysis, Crime Laboratory Digest (1989) 16:33–39.