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Systems Engineering of the Baseball Bat

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									Systems Engineering of the Baseball Bat
Terry Bahill Systems and Industrial Engineering University of Arizona Tucson, AZ 85721-0020 (520) 621-6561 terry@sie.arizona.edu http://www.sie.arizona.edu/sysengr/ Copyright  1989-2007 Bahill



R. G. Watts and A. T. Bahill, Keep Your Eye on the Ball: Curve Balls, Knuckleballs and Fallacies of Baseball, W. H. Freeman, 2000, ISBN 0-7167-37175. The slides of this presentation are available at http://www.sie.arizona.edu/sysengr/slides/baseballB at.ppt Bahill, A.T. and Gissing, B. "Re-evaluating systems engineering concepts using systems thinking," IEEE Transactions on Systems, Man, and Cybernetics, Part C Applications and Reviews, 28(4), 516-527, 1998. A. T. Bahill, “The ideal moment of inertia for a baseball or softball bat,” IEEE Transactions on System, Man, and Cybernetics – Part A: Systems and Humans, 34(2), 197-204, 2004. Terry Bahill


Definition of engineering
Engineers use principles from basic science (like physics, physiology and psychology) and design things that are useful to people.


Terry Bahill

The SIMILAR process
State the problem Investigate alternatives Model the system Integrate Launch the system Assess performance Re-evaluate.


Terry Bahill

The systems engineering process
Customer Needs State the Problem Re-evaluate Investigate Alternatives Re-evaluate Model the System Re-evaluate Integrate Launch the System Re-evaluate Assess Performance Re-evaluate Outputs Re-evaluate

But, it is not a serial process. It is parallel and highly iterative.

This talk is mostly about modeling.


Terry Bahill

Tasks in the modeling process*
Describe the system to be modeled Determine the purpose of the model Determine the level of the model Gather experimental data describing system behavior Investigate alternative models Select a tool or language for the simulation Make the model Validate the model*  Show that the model behaves like the real system  Emulate something not used in the model’s design  Perform a sensitivity analysis  Show interactions with other models Integrate with models for other systems Analyze the performance of the model Re-evaluate and improve the model Suggest new experiments on the real system Terry Bahill State your assumptions



Terry Bahill

We teach modeling by example
Modeling is usually taught by example. Here is an example.


Terry Bahill

The ideal bat
There is an ideal baseball and softball bat for each individual. To determine the ideal bat for each player we need to consider
 the coefficient of restitution of the bat-ball collision,  the sweet spot of the bat,  the ideal bat weight, and  the weight distribution of the bat.


Terry Bahill

The coefficient of restitution*
batted ball speed - bat speed after CoR= pitch speed - bat speed before
CoR of a baseball-concrete floor collision is about 0.55 Drop a baseball onto a concrete floor; it will rebound CoR2 of the height.

rebound height CoR  original height

Drop it from 3 feet, it will rebound about 11 inches. CoR of a softball-concrete floor collision is around 0.47. Drop it from 3 feet, it will rebound about 8 inches. Most of the CoR of a bat-ball collision is supposed to be due to the ball. However, drop a bat from 3 feet, it will rebound 10 inches. How come?
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The CoR depends on
the ball and the bat, collision speed, shape of the objects, where the ball hits the bat, and temperature.
 (Putting bat ovens in the dugout would help!)


Terry Bahill

We use
CoR = 1.17 (0.56 - 0.001 CollisionSpeed) for an aluminum bat and a softball and CoR = 1.17 (0.61 - 0.001 CollisionSpeed) for a wooden bat and a hardball where CollisionSpeed is in mph.


Terry Bahill

The sweet spot of the bat*
We would like the bat-ball collision to occur near the sweet spot of the bat. The sweet spot has been defined as the  center of percussion  node of the fundamental vibration mode  antinode of the hoop node  maximum energy transfer area  maximum batted-ball speed area  maximum coefficient of restitution area  minimum energy loss area  minimum sensation area  joy spot The sweet spot is centered 5 to 7 inches from the fat end of the bat.
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The center of percussion1
When the ball hits the bat, it produces a translation that pushes the hands backward and a rotation that pulls the hands forward. When a ball is hit at the center of percussion (CoP) for the pivot point, these two movements cancel out, and the batter’s hands feel no sting.


Terry Bahill

The center of percussion2
To find the CoP, hang a bat by the knob (or if possible a point 6 inches from the knob) with 2 or 3 feet of string. Hit the bat with an impact hammer. Hitting it off of the CoP will make it flop like a fish out of water.* Hitting it on the CoP will make it swing like a pendulum.**


Terry Bahill

The center of percussion3
knob pivot cm CoP


dpivot-cm dpivot-cop


d cm cop

I cm  mbat d pivot cm


Terry Bahill

Parameters of C243 wooden bat Stated Length (inches) 34 Period (sec) 1.634 Mass (kg) 0.905 2 Iknob (kg-m ) 0.342 Ipivot (kg-m2) 0.208 Icm (kg-m2) 0.048 Measured dknob-cm (cm) 57 Measured dknob-cop (cm) 62 Calculated dknob-cop (cm) 70 Measured dpiviot-cop (cm) 47 Calculated dpivot-cop (cm) 54 Measured dknob-firstNode (cm) 66
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Stated Weight (oz)

Length (in)

Moment Distance of inertia from knob Period Mass with to center (sec) (kg) respect to of mass, knob, Iknob dk-cm (m) (kg-m2) 1.420 1.570 1.669 1.584 1.667 1.674 1.654 1.634 0.478 0.634 0.764 0.651 0.731 0.810 0.920 0.905 0.346 0.448 0.510 0.477 0.505 0.506 0.571 0.570 0.083 0.174 0.269 0.193 0.255 0.285 0.356 0.342

Tee ball Little League High school Softball Softball, end-loaded Softball, end-loaded Major league, R161 (wood) Major league, C243 (wood)

17 22 26 23 26 29 32 32

25 31 32 33 34 34 34 34

Moment of inertia with respect to center of mass, Icm (kg-m2) 0.026 0.047 0.070 0.045 0.069 0.078 0.056 0.048


Terry Bahill

The node of the fundamental mode
The node of the fundamental vibration mode is the point where the fundamental vibration mode of the bat has a null point. To find this node, with your fingers and thumb grip a bat about six inches from the knob. Tap the barrel at various points with an impact hammer. The point where you feel no vibration and hear almost nothing is the node.


Terry Bahill

Hoop mode
Only hollow metal & composite bats Trampoline effect
 Wood bats don’t deform. All of the energy is stored in the ball.  Most of the losses are in the ball.  A ball has both a contribution to CoR and a stiffness.  A stiff ball will deform the bat more, and therefore store more energy in the bat. BPFs of 1.2 are common.


Terry Bahill

How big is the sweet spot?
The node of the fundamental vibration mode is about a ¼ of an inch wide. The center of percussion is a few inches wide. The least vibrational sensation point is a few inches wide.


Terry Bahill

Distance from barrel end to center of sweet spot for a 34 -inch wooden bat (cm) Definition of sweet spot This study of a Literature wooden C243 bat Center of percussion for a 16 calculated, 14 for a uniform rod (a lower 15 cm pivot point 18 experimental limit), Goldsmith (1960) method 1, 16 Vedula & Sherwood (2004) 15 experimental method 2, 14 experimental method 3 Node of fundamental 20 measured 17 Nathan (2003) vibration mode 17 Cross (2001) 17 Vedula & Sherwood (2004) Maximum energy transfer 20 Brancazio (1987) area Maximum batted-ball speed 14 Nathan (2000 and 2003) area 18 Noble (web site) 17 Vedula & Sherwood (2004) Maximum coefficient of 15 Nathan (2000) restitution area Minimum energy loss area 26 Cross (2001) Minimum sensation area 17 Adair (2001) Joy spot 13 Williams (1982)
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Do batters bunt the ball at the end of the bat rather than at the sweet spot in order to deaden the bunt?


Terry Bahill

Ideal bat weight™*
There is an ideal bat weight™ for each baseball and softball player. It makes the ball go the fastest. Measure bat swings. Make a model for the human. Couple the model to equations of physics. Compute ideal bat weight.


Terry Bahill


Terry Bahill

Ideal bat weight™
90 80 70

Batted-ball Speed Ideal Bat Weight Bat Speed

Speed (mph)

60 50 40 30 20 10 0 0 10 20






Bat Weight (oz)
26 Terry Bahill

Outlawing aluminum bats^
For most college batters, outlawing aluminum bats would produce faster battedball speeds, thus endangering pitchers.


Terry Bahill

System Design
Hardware Software

The NCAA forgot the wetware



Terry Bahill

Rules of thumb for recommending bats
Group Recommended Bat Weight Baseball, major league Height/3 + 7 Baseball, amateur Height/3 + 6 Softball, fast-pitch Height/7 + 15 Softball, slow-pitch Weight/115 + 24 Junior League (13 & 15 years) Height/3 + 1 Little League (11 & 12 years) Weight/18 + 16 Little League (9 & 10 years) Height/3 + 4 Little League (7 & 8 years) Age*2 + 4
Recommended bat weight in ounces, age in years, height in inches, body weight in pounds.
29 Terry Bahill


Terry Bahill

Sweet spot versus center of mass*
speedsweet-spot = 1.15 * speedcenter-of-mass

But the standard deviation is large: 0.06


Terry Bahill

The variable moment of inertia bat experiments


Terry Bahill

The beginnings^
In 1988 we conducted our first variable moment of inertia bat experiments. Lack of funding and the large variability in the data caused us to quit doing those experiments. With retrospective analysis we now know that most of the variability was due to player life experiences:
 the Chinese students who had never played baseball fell into one group,  the Americans who grew up playing baseball fell in to another group, and  the university women softball players fell into another group.


Terry Bahill

The beginnings (continued)
Furthermore, the first three UofA women softball players we measured turned out to have the biggest positive slope, the biggest negative slope and a zero slope (on the next slide). What bad luck!


Terry Bahill

Results of the variable moment of inertia experiments
0.53 60 Distance knob to disk (dk-disk) (m) 0.73 0.89

Bat speed (mph)

50 45 40 35 30 25 0.18 0.25 0.32 15 20

Moment of inertia with respect to the knob (kg-m2)
35 Terry Bahill

Bat speed (m/s)



Moment of inertia experiments
Weight distribution is characterized by moment of inertia. It takes a sweet-spot speed of 50 mph, producing a batted-ball speed of 71 mph, to drive a perfectly hit softball over the left field fence (200 feet) of Hillenbrand stadium. About half of these players can doing this. Which of these batters would profit from using an end-loaded bat?


Terry Bahill

Batted-ball speed


 v ball before ball  after

1  CoR   v bat before  v ball before  

m 1 m

ball bat

m d  I


cm  ss



Terry Bahill

End-loaded bats
• The data for each player can be fit with a line of the form • vbat-before = slope Iknob + intercept • Batters with positive slopes would definitely profit from using end-loaded bats.


Terry Bahill

Optimal inertia
30 25

batted-ball speed (m/s)





0 0 0.5 1 distance knob to disk (m)
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Moment of inertia summary*
Optimal disk placement for university softball players Player Slope y-axis Optimal dk-disk -1 (kg-m-s) intercept (m/s) (m) Amy -35.6 31.7 0.9 Jennie -22.0 30.3 1 Erika -15.2 21.9 1 Leneah -15.2 30.8 1.1 April -11.8 20.5 1.1 Nancy -11.9 22.4 1.1 Lisha -10.2 28.1 1.1 Allison -8.5 22.3 1.1 Katie -5.1 19.2 1.2 Lindsay -5.1 21.0 1.2 Lindsay -5.1 23.2 1.2 Chrissy -3.4 22.3 1.2 Teresa -3.4 22.7 1.2 Nicole -3.4 23.2 1.2 Toni -1.7 21.0 1.3 Mac -1.7 25.5 1.3 Kim 1.7 22.7 More than 1.5 Lindsey 1.7 20.5 More than 1.5 Jenny 15.2 18.9 More than 1.5 40 Susie 18.6 11.2 More than 1.5

Terry Bahill

Moment of inertia conclusions*
Batters with positive slopes should definitely use end loaded bats. We calculated the optimal moment of inertia for the 40 batters in our study. They would all profit from using endloaded bats.


Terry Bahill

Assess performance
The University of Arizona softball team has won the collegiate world series six times in the last dozen years.


Terry Bahill

The distance the ball travels depends on where the ball hits the bat*

Sweet Spot

For Terry use minimax: design the system to minimize the Loss if the Error is maximum Loss


For A-Rod use minimin

43 Terry Bahill

Outlawing aluminum bats would endanger pitchers. All of the batters in our study would profit from using an end-loaded bat. There is an ideal bat (weight & moment of inertia) for each person.


Terry Bahill


Terry Bahill

Seminar materials
Convict of SE cards bat on wire ball impact hammer


Terry Bahill

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