|Mode shapes for 30-inch Little League wood bat|
Fundamental bending mode (215 Hz)
The animations at left show the first three bending modes of a freely supported baseball bat. The handle end of the bat is at the right, and the barrel end is at the left. The numbers on the axis represent inches (this data is for a 30 inch Little League wood baseball bat). The animations are experimental results from an measurement called modal analysis, from which the relative motions of every point on the bat may be determined compared to every other point. The amplitude of the vibration is greatly exaggerated for clarity. When excited by an impact force, such as a baseball striking the bat, all of these modes, (as well as some additional higher frequency modes) are excited and the bat vibrates. These vibrational modes can be heard quite clearly by holding the bat between finger and thumb about 5-6 inches from the barrel end and striking the bat end with a hard object. Holding the baseball bat at the handle reduces the amplitude of vibration considerably and causes the vibration to decay very quickly, but does not significantly affect the frequencies or the mode shapes of the vibration.
The fundamental bending mode has two nodes, or positions of zero displacement). One is about 6-1/2 inches from the barrel end close to the sweet spot of the bat. The other at about 24 inches from the barrel end (6 inches from the handle) at approximately the location of a right-handed hitter's right hand.
The second bending mode has three nodes, about 4.5 inches from the barrel end, a second near the middle of the bat, and the third at about the location of a right-handed hitter's left hand.
These animations were obtained from a modal analysis experiment. The bat was suspended at each end with rubber bands. The bat was tapped with a special force hammer at each of 30 points along the bat's length. An accelerometer was fixed to the barrel end of the bat, and a measurement of the ratio acceleration/force (called a Frequency Response Function) was recorded for each of the 30 locations. A modal analysis software package (STAR Modal 5.2) was used to curve fit the FRFs so as to determine mode shapes, frequencies and damping constants. The animations at left were obtained for one of the wood bats in our laboratory.
The figure below compares sweet zones for three Little League (30-inch) bats: two wood bats (ash and maple) and a single-walled aluminum bat (with no fancy technological improvements). The sweet zone (distance between nodes for modes 1 and 2) is wider for the aluminum bat than it is for the two wood bats. So, by this definition of the sweet zone, the aluminum bat has a wider sweet spot. However, the natural damping in the aluminum bat is so small that when a freely-supported bat is struck an impact at the barrel, the aluminum bat continues vibrating more than 10-times as long as a wood bat. In addition, the vibration amplitude felt by the hands is much higher for a single-walled aluminum bat than it is for a wood bat. So, while the sweet zone is wider, the resulting vibration for impacts outside of this zone is actually worse for this aluminum bat. Older aluminum bats (ie., those made in the late 70's and early 80's) actually sting worse than wood bats. The rubber handle wraps and the recent technological modifications to aluminum bats reduce the vibration so that modern aluminum bats now sting less than most wood bats.
 H. Brody, "The sweet spot of a baseball bat," American Journal of Physics, 54(7), 640-643 (1986)
 R. Cross, "The sweet spot of a baseball bat," American Journal of Physics, 66(9), 771-779 (1998)
 H. Brody, "Models of baseball bats," American Journal of Physics, 58(8), 756-758 (1990)
 J. Braham, "Keep your eye on the bat," Machine Design, 69(13), 56-66 (July 10, 1997)
 "Batter up! Piezo dampers take the sting out of swing," Machine Design, 70(15), 46-47 (August 20, 1998)
 J. R. Fricke, "Lodengraf Damping - An Advanced Vibration Damping Technology," Sound and Vibration, 34(7) 22-27 (2000)