8006 solution
The “S weet S pot pot”
”on a B aseball B at Abstract
This paper presents how to locate the “sweet spot”on a baseball bat and the impact of the material out of which the baseball bat is constructed on the “sweet spot”.As it refers to the knowledge of physics,we use physical models to solve the problem.We get four conclusions as follow:
1.The location of the “sweet spot”is not at the end of the bat.The establishment of a model for vibration of wave shows that the sweet zone was the region located between the nodes of the first and second modes of vibration,that is to say,between about 4-7inches from the barrel end of a 30-inch Little League bat.
2.“Corking”a bat doesn’t enhance the “sweet spot”effect.Apply and analyse the
formula of and ,the moment of inertia,and the ()bat A ball A f v e 1 v e v ++=r
1r -e e A +=rebound theory in Elasticity shows that corked bats don't hit balls further or faster.
3.The material out of which the bat is constructed is relevant to the performance of bat.This model can predict the different behavior for wood or metal bats.On the one hand,from the experiment of the model for vibration of wave,we can get a conclusion that the sweet zone is wider for the aluminum bat than it is for the wood bat.On the other hand,take aluminum for example,aluminum bats can be swung faster,aluminum bats have the “trampoline effect”,aluminum is much stronger than ash wood,and the handle of an aluminum bat is a thin cylindrical tube with reasonably thick walls.
4.The “sweet spot”should also take into account the location where the coefficient of restitution is max,the center of percussion and so on.For most bats,all of these "sweet spots"are at different locations on the bat,so one is often forced to define the sweet spot as a region.
Keywords:sweet spot
physical model vibration collision efficiency moment of
inertia For office use only
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80080066Problem Chosen A office use only F1________________F2________________F3________________F4________________
1Introduction
1.1Statement of Problem
This model aims at using intelligent traffic signaling system,based on fuzzy logic that can organize the vehicles entering a roundabout according to both traffic needs and vehicles priority in those areas.
Fuzzy traffic signal control is an alternative to conventional traffic lights control that can be used for a wider array of traffic patterns at a roundabout.A fuzzy logic controlled traffic light uses sensors that count cars to change the signal light accordingly.
1.2Background
Given the popularity of baseball,and the natural curiosity of physicists to understand collision phenomena,it is surprising that very few measurements have ever been made on the behavior of a baseball bat when it impacts with a ball.There are different types of bats at almost every weight.Both heavy and light bats are used by professional players.Batters know from experience that there is a sweet spot on the bat,about17cm from the end of the barrel,where the shock of the impact,felt by the hands,is reduced to such an extent that the batter is almost unaware of the collision.At other impact points,the impact is usually felt as a sting or jarring of the hands and forearm,particularly if the impact occurs at a point well removed from the sweet spot.
The sweet spot is an impact point,or a narrow impact zone,where the shock of the impact,felt by the hands,is reduced to such an extent that the batter is almost unaware of the collision.The sweet spot of a bat exists because bat vibrations are not
COP excited significantly at that spot.The spot is close to the centre of percussion() for a freely supported bat,so it was thought at one time that the sweet spot might be associated with the.At other impact points,the impact may be felt as a painful COP
sting or jarring of the hands,particularly if the impact occurs at a point well removed from the sweet spot.It has been proposed that two such spots exist,one corresponding to a vibration node and the other corresponding to the centre of percussion.However, it appears that no direct measurements have previously been made to determine which of the two spots is the more significant or whether the sweet spot is perhaps an impact zone encompassing both sweet spots.The two proposed sweet spots are typically only a few centimeters apart,so it is not an easy task to separately identify the two spots in terms of differences in the qualitative feel of the impact.
Almost every bat on the market(especially newer aluminum bats)is claimed to have a"wider"sweet spot.Newer technologies such as triple walled barrels,nitrogen gas bladders,piezoelectric active damping control devices,rubber hand grips,rubber end plugs,etc.,help to minimize the resulting vibration of an aluminum bat.
1.3Assumptions
1.3.1An object is continuous,that is,the whole object is filled up with the
material out of which it is constructed,without leaving any gaps.
1.3.2The object is isotropic,that is,flexibility of a point within the object is the same in all directions.
1.3.3Displacement and deformation are very weak.
1.3.4has no effect on .The effect isn’t considered because it’s very weak.rel v A e ()
bat ball rel v v v +=2General Ball-bat Collision Situation
The impact between a baseball and a bat is an extremely violent one,in which the bat imparts a huge force on the ball thereby causing it to change directions and gain speed.The baseball and the bat are both stiff.As a result,the impact on the bat is very short,about 1msec.During the collision,the ball compresses to a fraction of its undistorted radius,comes to a momentary halt,reverses direction and then expands to its original shape.
This process is inherently inefficient,with a large fraction of the original kinetic energy dissipated in the internal structure of the ball.This inefficiency is characterized by the “coefficient of restitution”or .For a two-body collision,the (COR )e COR is defined as the ratio of the relative velocity after the collision to that before the collision.For a perfectly elastic collision,no energy is dissipated,the two bodies recede with the initial relative velocity,and .For a perfectly inelastic collision,1=e the two bodies stick together,all the initial energy is dissipated,and .CM 0=e The for a collision between the bat and the ball is less than 0.59when energy COR is lost to vibrations.But it can be greater than 0.59if the bat has a trampoline effect on the ball.A high performance bat has a strong trampoline effect and hence a relatively large .
COR 3The Physical Model
Batters know from experience that there is a sweet spot on the bat ,about 17cm from the end of the barrel,where the shock of the impact,felt by the hands,is reduced to such an extent that the batter is almost unaware of the collision.At other impact points,the impact is usually felt as a sting or jarring of the hands and forearm,particularly if the impact occurs at a point well removed from the sweet spot.Therefore,we analyze two aspects as follow:
?Analyze the problems in terms of Wave-Induced Vibration produced by solid
collision in physics.
?Define the batting where the transfer of energy will reach the maximum as the
sweet point.
In brief,we'd better analyze problems based on momentum conservation,Elastic Mechanics (such as Bounce Factor ,,collision efficiency and )and COR ACOR Lever Principle (such as moment of inertia)in physics.In that way,we will get the best result.
3.1Introduction of Vibrational Bending Modes
Mode shapes for30-inch Little League wood bat
Figure1.Fundamental bending mode(215Hz)
Figure2.Second bending mode(670Hz)
Figure3.Third bending mode(1252Hz)
Figure4.Baseball bat
The figures above 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 30inch Little League wood baseball bat).The figures 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 5-6inches 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 6-1/2inches from the barrel end close to the sweet spot of the bat.The other at about 24inches from the barrel end (6inches from the handle)at approximately the location of a right-handed hitter's right hand.
The second bending mode has three nodes,about 4.5inches 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.
3.2M 3.2Momentum
omentum Conservation For a system of interacting objects,the total momentum remains constant,provided no external resultant force acts on the system or the total external force on the system is zero.This is a conservation law,called the conservation of momentum.
Mathematical expressions of momentum conservation:
(1).That is,the total momentum at the beginning is equal to the total p p ′=momentum of the end (or of an intermediate state)for a system of interacting objects.
(2).Namely,the change in the system's total momentum is zero.If the 0=?p system under study is composed of two objects that can be expressed as:
'22'112211v m v m v m v m +=+()
1(On both sides of the equation are the vector addition)；
(3).That is to say,if the system consists of two objects,the changes in 21p p ??=?the momentum of two objects are equal and the momentums of two objects in the end are in the opposite direction.In that case,we should pay attention to the vector property of changes in momentum.In the process of interaction between two objects,the momentum of these two objects may decrease or may also increase,but total momentum remains the same.For a system of a bat and a ball,the
momentum is conserved after they collided with each other.But part of the energy is converted into internal energy.So it doesn’t satisfy the conservation of mechanical energy.
3.3Inference Process and Conclusion
spot””is not at the end of the bat.
3.3.1The location of the“sweet spot
Step1.A model for vibration of wave
Defines the sweet zone as the region located between the nodes of the first and second modes of vibration(between about4-7inches from the barrel end of a30-inch Little League bat).Since the vibrational motion of the bat is very small in this region, an impact in this region will result in very little vibration of the bat(no stinging a player's hands)and a very solid hit will result with maximum energy being given to the ball.An impact to the outside(towards the barrel end)or inside(towards the handle)of this zone will result in a much more significant vibration of the bat,often felt as a painful sting.And the ball will not travel as far because some of the energy is now being stored(or dissipated)in the bat's vibration.Howard Brody has convincingly shown that gripping the bat with the hands adds a significant amount of damping, causing the bat's natural vibrations to decay very quickly.However,gripping the bat with the hands has very little effect on the frequencies or the mode shapes.Thus, suspending the bat from rubber bands to obtain a free-free boundary condition is a legitimate approach to studying a bat's behavior.
Figure5.The first node and the second node of waves
Figure 6.Baseball,baseball bat and the corresponding waves
Step2.Physical Analysis in the Collision
For a sufficiently short or a sufficiently stiff bat,the collision between a bat and a ball can be analyzed using a rigid body approximation,as illustrated in Fig.2.If there is no external reaction force from the hand,and if the bat is initially at rest,then conservation of linear and angular momentum is described by the relations
2
..1v m MV v m b m c b ?=()
2and b v m I b v m b m c b 2..1?=ω()
3where is the bat mass,is the recoil speed of the center of mass of the
M ..m c V ()..m c bat,is the moment of inertia of the bat about its ,is the angular velocity
..m c I ..m c ωof the bat after the collision,and is the impact parameter.For a uniform bat of b length ,
L 12
/2..ML I m c =()4Conservation of energy is described by
c m c m c b b E I MV v m v m 22..2..2221+++=ω()
5where is the energy dissipated in the ball plus the energy stored as vibrational c E energy in the bat.The vibrational energy is eventually dissipated in the bat,or in the hands,well after the collision is over.Inclusion of a vibrational loss term allows one to relax the assumption that the bat is perfectly rigid,but rigid body dynamics alone
does not provide any clues as to the magnitude of .The vibrational losses must c E be determined experimentally or by a flexible bat analysis.
The solution of Eqs --is given by
()2()5()[]R R R f v v e +?+?==1112121()6where
()
21/2v m E f b c =()
7is the fractional energy loss,E
b M m R /=()
8and ()
..2/1m c E I Mb M M +=()9is the equivalent mass of the bat.Equation is therefore the same as that for a head-()6on inelastic collision between point masses and .The equivalent mass is b m E M equal to the actual mass for an impact at the where ,and decreases CM 0=b toward the end of the bat.At the end of the bat,where ,.
2/L b =4/M M E =
Figure 7.Collision of a ball with a freely suspend rigid bat initially at rest.
Figure 8.Experimental arrangement used to measure the incident and rebound speeds
of a ball on an aluminum beam.
3.3.2“Corking Corking””a bat doesn doesn’’t enhance the “sweet spot spot”
”effect Step 1.Field Measurements of Batted-Ball Speed
Let's start with the assumption that Batted-Ball Speed is the desired end ()f v quantity to regulate,and see what it depends on from a physics perspective.If one starts from basic physics conservation laws (conservation of linear and angular momentum,and the conservation of energy),it is relatively simple to derive the equation for batted-ball speed to be:
()bat A ball A v e 1 v e ++=f v ()10where is the pitched speed of the ball,is the linear speed of the bat at the ball v bat v impact location,and is a term called the collision efficiency (sometimes referred to A e as the "apparent coefficient of restitution").The collision efficiency depends on A e the elastic properties of both the bat and ball,as well as on the moment-of-inertia of the bat,and the location of the impact on the bat barrel.It is the most important quantity we can measure regarding bat performance.
Figure 9.The collision between the baseball and the baseball bat
Step 2.Analysis of collision efficiency
“collision efficiency”A e = =??
?
????21BESR 1
e 1A ≤≤?At “sweet spot”,.
()7.02.0≈≈BESR e A r
r
e e A +?=1()11so depend on
A e bat recoil factor (inertial properties)
≡r ball-bat ≡e 0
5.0BPFe COR =≈ball-wall .
≡0e COR There are two kinds of ,one is ,the other is r free r pivot r CM
bat,2
ball bat ball free I b m
M m r +=()12pivot
bat,2
ball pivot I x m r =)
13(
I b m M m r CM
bat,2
ball
ball free +
=
Free
Figure 10.The recoil factor of the free baseball
x
Pivoted
I x m r pivot
bat,2ball pivot
=Figure11.The recoil factor of the pivoted baseball.
is the moment-of-inertia of the bat ,including and I cm
bat,I pivot bat,I For example:Free Wood Bat
-0.10
0.1
0.2
0.30.40.5
distance from tip (inches)Figure12.The Contrast graphics of ,r and e.
A e What we encounter in practice is that,the formula is only relevant to r,since e is a constant.We get x from the problem above.Then,we could just pay attention to the changes of .The material of the corked bat can be regarded as unchanging.pivot bat,I So is almost unchanging.So “corking”a bat doesn’t enhance the “sweet spot”pivot bat,I effect.
3.3.3The material out of which the bat is constructed matter
One the one hand ,according to the analysis of the model in 3.3.1,we know that it has a "wider"sweet spot.The analysis is as follows ：
Figure13.3.below compares sweet zones for three Little League(30-inch)
(1)The Figure1
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 modes1 and2)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 than10-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 late70's and early80'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.
Figure133.Sweet zones for three Little League(30-inch)bats
Figure1
(2)According to the analysis in3.3.2,the bat performance is related to the moment of inertia.We can get the following data on the basis of calculation of moment of inertia.
T able1
Bat B1-B4are34-inch wood baseball bats and B5-B7are33-inch aluminum or composite bats.Bats Y1-Y3are the same make and model wood bats.Bats Y4-Y6are different aluminum bats.Here is a sampling of weight,balance point,and moment of
inertia about the6-in point on the handle.
Baseball Bat Length(in)Weight(oz)MOI6(oz-in2)
B13431.211239
B23436.512283
B33437.311836
B43431.910127
B53331.49325
B63331.09590
B73330.58664
Y13019.75029
Y23022.75800
Y33025.26425
Y43027.26139
Y53017.14420
Y63022.15675 From the data in Table1,we can find that the MOI of wood and aluminum is different.
(3)Besides,the most comprehensive study comparing metal and wood bats under realistic playing conditions was published by Crisco and Greenwald.Their study involved actual players swinging bats at pitched balls in a batting cage.A total of19 players(nine minor-leaguers,six NCAA college,four high school)participated in the study,swinging two wood bat models,and5different metal bat models(6bats of each model).High speed video was used to obtain a3-D map of the bat swing,locating the trajectory of several points on the bat and ball before,during,and after impact.Video data was then analyzed to obtain batted ball speeds.
The results,get from an article named“Why Aluminum Bats Can Perform Better than Wood Bats”by Daniel A.Russell,summarized in Fig.1,show conclusively that metal bats outperform wood bats.The bars in the chart represent the average and spread for the top10%(i.e.,fastest10%)of all hits across all players for each bat.The data shows an average batted ball speed for wood bats(W)around98.6-mph,while balls hit with metal bats(M1-M5)left the bat with speeds averaging between100.3 mph and106.5-mph,up to8-mph faster than wood bats.The data also shows that not all metal bats are the same.In the complete analysis of all of the data,bat M1was found to perform statistically the same as a wood bat,while bats M3and M4were noticeably better and bat M2was dramatically better.
Figure144.Measured batted-ball speed for baseball from a batting cage study by Figure1
Crisco&Greenwald.
Reasons are:
1)Aluminum bats can be swung faster.
Because the barrel of an aluminum bat is hollow,the distribution of mass along the length of a metal bat is considerably different than it is for a solid wood bat.Specially, the difference shows up in the location of the center-of-mass()-otherwise known
CM
CM
as the balance point.The closer the to the handle of the bat,the easier it is to swing the bat.Figure2compares the balance points of four30"youth bats:three wood bats of weights26oz,23oz,and20oz,and an27oz aluminum bat.The balance point for all three of the wood bats is located at the same place-since the profile shapes of the bats are the same and they are all made from solid wood,the balance point is the same regardless of the total weight.In contrast the aluminum bat is actually heavier,but since its balance point is more than an inch closer to the handle it will be easier to swing.This is directly related to the swing weight of a bat-the reason that not all28oz softball bats swing the same.An end-loaded bat can have the same weight as a normal bat,but will feel heavier because more of the mass is distributed towards the barrel end of the bat.
https://www.wendangku.net/doc/353278153.html,paring the location of the for three different weight wood and
CM
one metal youth bats.
Technically we are talking about the moment-of-inertia of the bat.MOI is the product of mass and the square of a distance -which while not the same as the CM location is strongly influenced by the balance point.The closer the is to the CM handle,the lower the MOI will be.Several studies have shown that swing speed depends strongly on the moment-of-inertia of the bat;a player can swing a lower inertia bat faster.This affects performance because higher bat speed is directly related to higher batted ball speed.The faster a player can swing a bat,the higher the final speed of the ball.Lowering the inertia of the bat too much will result in a lower amount of momentum that the bat carries into the collision,reducing the batted ball speed.Ideally a player should use a bat with a high moment of inertia and swing it really fast -but this is difficult unless you are really strong (which is why many baseball and softball players undergo weight training regimens during the off-season in order to bulk up their strength and increase swing speed).Getting back to the comparison of wood and aluminum bats,an aluminum bat with the same weight as a wood bat will have a significantly lower inertia and can thus be swung faster than the wood bat.
Table2.
Data for two of the baseball bats in the Crisco-Greenwald batting cage study
The table above shows data for two of the bats used in the Crisco-Greenwald batting cage study,wood bat (W)and metal bat (M2)from Fig.1above.The aluminum bat was 1.7oz lighter and its was 2.3"closer to the handle,resulting in a CM significantly lower MOI.As a result the average bat swing speed for the top 10%of hits,as measured at the impact location,was 3mph faster for metal bat M2than it was for wood bat W.Criso and Greenwald estimate that this increase in swing speed is responsible for about 4.5mph of the measured increase in batted ball speed.
2)Aluminum bats have the "trampoline effect"
When a ball hits a wood bat,it compresses to nearly half its original diameter,losing up to 75%of its initial energy to internal friction forces during this compression.In a hollow bat,however,the bat barrel compresses somewhat like a spring,when the ball impacts it.This means that the ball is not compressed as much and therefore loses less energy to internal friction forces.Furthermore,most of the energy temporarily stored in the barrel is returned to the ball,and the energy which is lost in the bat compression is a small fraction of what would have been lost in the ball if it had impacted a wood bat instead.The physics behind the trampoline effect is somewhat complex,though a simple model can be used to illustrate the main concepts,as explained elsewhere on this website.What the author wants to do here is discuss Bat Length (in)Weight (oz)CM (in)MOI oz-in 2Swing Speed (mph)
BBS (mph)Wood (W)3430.923.01151667.998.6Metal (M2)
3329.220.7928270.9106.5
experimental evidence that a trampoline effect really does seem to be partly responsible for improvement in performance of aluminum bats over wood.
Again,the only published study to date which offers evidence of an enhancement in performance for metal bats due to an elastic property of the bat is the Crisco-Greenwald study.Figure15compares batted ball speeds for balls hit with a wood bat (orange dots)and the highest performing metal bat(blue dots)used in their study.The horizontal axis(Impact Speed)represents the swing speed of the bat.Plotting the data this way normalizes the results so as to remove the effect of different moments-of-inertia.The figure shows that for a given swing speed,the metal bat can potentially hit the ball5-7mph faster than the wood bat.This can be explained if the metal bat has a trampoline effect which returns more of the energy to the ball.
Crisco&Greenwald were only able to conclude that an"inherent elastic property" of the bat was most likely present to explain the difference in batted-ball speeds shown in Figure16.They were not able to explain differences in this effect between the five metal bats they tested,and so could not explain how the trampoline effect improves performance.During the summer of2004,he was granted access to the five metal bats used in their batting cage study.He measured the frequency of the mode of vibration in the barrel,called the hoop mode,which gives rise to the trampoline effect.From his analysis of various hollow softball bats he has shown that the frequency of the hoop mode correlates pretty well with performance.All other bat parameters being equal,the bat with the lowest hoop frequency will have the highest performance.Figure 17.shows the Bat-Ball Coefficient-of-Restitution(which measures the combined elastic properties of the bat-ball system)as a function of hoop frequency.The solid curve is a theoretical prediction from my simple mass-spring model of the trampoline effect,and the data points represent the measured values(extracted from the
BBCOR
original field study data by my friend and colleague Alan Nathan)for the five metal bats used in the Crisco-Greenwald batting cage study.The plot clearly shows that the higher performing bats have a lower hoop frequency,which indicates that the simple mass-spring model of the trampoline effect captures the essential physics.
Figure1515..Performance of hollow metal bats is due to the trampoline effect.Bats with lower hoop frequency(more compliant barrels)have increased performance.
Figure16.Width of the sweet spot for metal and wood bats is the same.But,batted ball speed is higher with metal bats for impacts away from the sweet spot.
3)Wood bats,especially the thin handled ash bats used by a majority of today's players,have a tendency to break upon impact with a ball near the middle of the bat.
In almost every professional game one or more players break a bat on an inside pitch.One memorable example happened during2000World Series between the New York Yankees and the New York Mets.Yankees pitcher Roger Clemens delivered an inside pitch to Mets catcher Mike Piazza.Piazza's bat broke and the barrel portion bounced out towards Clemens on the mound.Clemens picked up the broken bat and threw it towards Piazza as he ran towards first.There was already some bad feelings between the two players because Clemens had hit Piazza with an inside pitch the previous year,and the broken bat incident only increased the animosity between the two players.The point he wants to make,however,is that Piazza's bat broke.While there are exceptions(Hall of Famer Joe Sewell supposedly used only one bat during his entire14year career)professional players typically go through several dozen bats each during a typical baseball season.Similarly,college teams which used wood bats often broke more than two dozen during a season.Because of shortages of quality wood,and the higher cost of good quality bats,most college teams completely switched over to aluminum bats during the1980's.Most college teams can get by with12aluminum bats per season,as opposed to more than5dozen wood bats per season.
Aluminum is much stronger than ash wood,and the handle of an aluminum bat is a thin cylindrical tube with reasonably thick walls(thicker than the barrel portion).As such,the bat handle is very strong and will not break.The barrel,where the walls are thinner,may dent-or even crack if the walls are too thin,the bat will not break.New alloys and treatment processes(cryogenic treatment)even improve the strength of the aluminum further.It is not,however,impossible for aluminum bats to break-but usually when they do it is not the handle but the barrel that breaks.While watching ESPN Sports Center one night towards the end of May2003,he saw a clip from a college game in which a metal bat shattered leaving the batter holding the handle and half the barrel while the other half of the barrel flew out and landed next to the shortstop.Apparently a player in the2003Women's College Softball World Series also broke an aluminum bat.So,it can happen.
Many people have argued that the fact that aluminum bats don't break has a noticeable impact on the game of baseball.Because college players use aluminum bats,they can hit inside pitches for base hits.As a result,fewer and fewer younger professional pitchers throw inside pitches than used to be the norm in years past.They learn while playing in high school and college that inside pitches get hit for runs.Likewise hitters often find it to be considerably difficult to make the change from aluminum to wood,especially when dealing with inside pitches.
Based on the three points above,we can know why Major League Baseball prohibits metal bats.
4Further Discussions
The model is based on finding the sweet spot on a baseball bat,where maximum power is transferred to the ball when hit.But a multitude of definitions of the sweet spot should be considered synthetically as follow:
a)the location which produces least vibrational sensation (sting)in the batter's
hands
b)the location which produces maximum batted ball speed
c)the location where maximum energy is transferred to the ball
d)the location where coefficient of restitution is maximum
e)the center of percussion
f)the node of the fundamental vibrational mode
g)the region between nodes of the first two vibrational modes
h)the region between center of percussion and node of first vibrational mode
For most bats all of these "sweet spots"are at different locations on the bat,so one is often forced to define the sweet spot as a region,approximately 5-7inches from the end of the barrel,where the batted-ball speed is the highest and the sensation in the hands if minimized.
4.1The center of percussion ()
COP The center of percussion is the point on an object where a perpendicular impact will produce translational and rotational forces which perfectly cancel each other out at some given pivot point,so that the pivot will not be moving momentarily after the impulse.The same point is called the center of oscillation for the object suspended from the pivot as a pendulum.
Centers of percussion are often discussed in the context of a bat,racquet,sword or other long thin objects.The center of percussion may or may not be the "sweet spot"depending on the pivot point chosen.
For a free,rigid beam,a force F applied at right angle at a distance b from the center of gravity (CoG)will result in the CoG moving at a velocity V according to the relation:
dt
dV M F =()14where M is the mass of the beam.Similarly the torque exerted will be as per the relation:
dt
d I
Fb ω=()15where I is the moment of inertia around the CoG and ωis the angular velocity.
For any point P on the opposite side of the CoG from the point of impact,the velocity of point P is ω
A V v ?=()
16where A is the distance of P from the CoG.Hence:F I Ab M dt dv ??
?????=1()17The velocity v is then given by:
dt F I Ab M v ∫???????=1()18The axis of rotation is situated where v =0and the corresponding center of percussion is at distance b from the CoG,with
AM
I b =()194.2The coefficient of restitution (or )
COR e For two objects colliding with each other,the speed of separation after the impact is proportional to the velocity of approach before the impact.The ratio is called the coefficient of restitution ().For a perfectly elastic collision,and it COR 1=COR satisfies the conservation of the conservation of mechanical energy.For an inelastic collision,,it doesn’t satisfy the conservation of the conservation of 1 0=COR A baseball bat is 7to 15times as stiff as a tennis racket,and when it is hit by the bat,it deforms much less.Therefore,the amount of energy that goes into the bat deformation is very small compared to the very large direct energy lost in the ball-bat collision.If this is the case,then the variation of this deformation energy loss with position along the bat can be neglected,and the of the bat can be considered COR essentially constant.Is there then an optimum location along the bat to hit the ball to maximize the “power”---to maximize the rebound velocity of the struck ball if the is considered to be constant?In the literature it is said that there is such a point COR and it is located at the center of percussion.In Sec.V the maximum power point will be found and it will be shown that it is not at the . COP 4.3Conclusions So,where is the Sweet Spot?In some sense we have come full circle to the problem stated at the beginning of this paper,that there is no single definition of the sweet spot for a hollow baseball or softball bat.There are locations on the barrel which result in maximum performance and there are locations which result in minimal discomfort in the hands.These locations are not the same for a given bat,and there is considerable variation in locations between bats.Hopefully this summary will enhance the understanding of what the sweet spot is and what it is not,as well as encouraging further research into the quest for the"perfect bat." 5References [1]Daniel A.Russell,“The sweet spot of a hollow baseball or softball bat",July20, 2009 [2]R.Cross,"The sweet spot of a baseball bat",January22,1998 [3]K.Koenig,N.Mitchel,T.Hannigan,and J.Clutter,"The influence of moment of inertia on baseball/softball bat swing speed",June,2004. [4]Daniel A.Russell,"Are Composite Bats better than Aluminum Bats?",March21, 2005 [5]Daniel A.Russell,"Bat Weight,Swing Speed and Ball Velocity",March27,2008 [6]Daniel A.Russell,"Should Metal Baseball Bats Be Banned Because They are Inherently Dangerous?",April9,2008 [7]Daniel A.Russell,"What about corked bats?",October7,2004 [8]Daniel A.Russell,"Why Aluminum Bats Can Perform Better than Wood Bats", October18,2006 [9]H.Brody,”The sweet spot of a baseball bat”,June27,1985 [10]Daniel A.Russell,“Why Aluminum Bats Can Perform Better than Wood Bats", October18,2006 [11]Daniel A.Russell,"Are Composite Bats better than Aluminum Bats?",March21, 2005 [12]Daniel A.Russell,"Physics and Acoustics of Baseball and Softball Bats",July20, 2009 [13]Daniel A.Russell,"Should Metal Baseball Bats Be Banned Because They are Inherently Dangerous?",April9,2008 [14]Daniel A.Russell,"Are Composite Bats better than Aluminum Bats?",March21, 2005 [15]Daniel A.Russell,"Bat Weight,Swing Speed and Ball Velocity",March27, 2008 [16]Daniel A.Russell,"Should Metal Baseball Bats Be Banned Because They are Inherently Dangerous?",April9,2008 [17]Daniel A.Russell,"Why Aluminum Bats Can Perform Better than Wood Bats", April9,2008 [18]Daniel A.Russell,"Impact of a ball with a bat or racket",January25,1999 [19]Dan Russell,"Swing Weights of Baseball and Softball Bats",April17,2009