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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 Copyright  1989-2009 Bahill. References.

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systems engineering of the baseball bat

Systems Engineering of the Baseball Bat

Terry Bahill

Systems and Industrial Engineering

University of Arizona

Tucson, AZ 85721-0020

(520) 621-6561

Copyright  1989-2009 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 0-7167-3717-5.
  • The slides of this presentation are available at
  • 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.

© 2009 Bahill

definition of engineering
Definition of engineering

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

© 2009 Bahill

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

© 2009 Bahill

the systems engineering process
The systems engineering process

But, it is not a serial process.

It is parallel and highly iterative.

This talk is mostly about modeling.

© 2009 Bahill

tasks in the modeling process
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
  • State your assumptions

© 2009 Bahill

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

© 2009 Bahill

the ideal bat
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.

© 2009 Bahill

the coefficient of restitution
The coefficient of restitution*

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.

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?

© 2009 Bahill

the cor depends on
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!)

© 2009 Bahill

we use
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.

© 2009 Bahill

the sweet spot of the bat
The sweet spot of the bat*

We would like the bat-ball collision to occur near the sweet spotof 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.

© 2009 Bahill

the center of percussion1
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.

© 2009 Bahill

the center of percussion2
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.**

© 2009 Bahill

the node of the fundamental mode
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.

© 2009 Bahill

hoop mode
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.

© 2009 Bahill

how big is the sweet spot
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.

© 2009 Bahill

  • Do batters bunt the ball at the end of the bat rather than at the sweet spot in order to deaden the bunt?

© 2009 Bahill

ideal bat weight
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.

© 2009 Bahill

ideal bat weight26
Ideal bat weight™

© 2009 Bahill

outlawing aluminum bats
Outlawing aluminum bats^
  • For most college batters, outlawing aluminum bats would produce faster batted-ball speeds, thus endangering pitchers.

© 2009 Bahill

system design





forgot the


System Design

© 2009 Bahill

sweet spot versus center of mass
Sweet spot versus center of mass*

speedsweet-spot = 1.15 * speedcenter-of-mass

But the standard deviation is large: 0.06

© 2009 Bahill

the beginnings
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.

© 2009 Bahill

the beginnings continued
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!

© 2009 Bahill

moment of inertia experiments
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?

© 2009 Bahill

batted ball speed
Batted-ball speed

© 2009 Bahill

end loaded bats
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.

© 2009 Bahill

optimal inertia
Optimal inertia

© 2009 Bahill

moment of inertia conclusions
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 end-loaded bats.

© 2009 Bahill

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

© 2009 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.

© 2009 Bahill

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

© 2009 Bahill