A Study of the Barrel Constructions of Baseball Bats (June 9, 2003)
Thanks to wcw. This is a PDF file.
Finally, the question is addressed as to whether a corked wooden bat really outperforms a solid wood bat. Although commonly thought that “corking” a bat provides hitters with better control but no additional power, the results of this study show a slight increase, on the order of 1%, in batted-ball speed with the corked bats in comparison to their solid-wood counterparts.
--posted by TangoTiger at 04:31 PM EDT
Posted 6:10 p.m.,
June 9, 2003
(#1) -
Steve Rohde
If it is accurate that a corked bat will tend to increase bat speed by 1%, what are the implications for how much further the ball would travel if hit by a corked bat. Is it a 1% increase, or is the relationship more complicated than that?
Posted 9:04 p.m.,
June 9, 2003
(#2) -
Gary L
I love rutabaga!
Posted 10:51 p.m.,
June 9, 2003
(#3) -
Chris Black
As I understand it, a 1% increase in velocity (at a cost of roughly 1% mass, don't forget that) will increase the kinetic energy by 1%. Now I'm no physicist, but just applying E=0.5mv^2 and presuming loss of one ounce of weight off a 34 ounce bat. The results don't vary appreciably with the common range of bat weights (30-36 ounces). I'm not sufficiently familiar with how much energy is lost in batting a ball to be able to comment on that, but presuming a linear relationship, a corked bat would seem to make a 400 foot uncorked homer travel 404 feet.
Posted 9:34 a.m.,
June 10, 2003
(#4) -
David Shapiro
Chris, you haven't accounted for the fact that the ball's velocity is not purely horizontal, and that (setting aside drag) its time in the air is a square function based on acceleration due to gravity. So in fact the distance increase will be less than 1% on a bat speed increase of 1%. I really don't think there would be any appreciable effect, even before accounting for the fact that decreased weight would reduce momentum, offsetting at least in part, increased momentum associated with increased velocity.
Posted 9:41 a.m.,
June 10, 2003
(#5) -
Patriot
Whatever advantage the hitter gets from a corked bat must pale in comparison to the added movement on a spitball. But yet Sammy Sosa is the cause of evil in the modern world and Gaylord Perry makes for some funny anecdotes.
Posted 11:02 a.m.,
June 10, 2003
(#6) -
Chris Black
D'oh, you're right David. Makes the difference even smaller. In any event, corking does not seem to have a substantial effect.
Posted 8:47 a.m.,
June 13, 2003
(#7) -
Sylvain(e-mail)
So, as part of may grunt work, I decided to look at the advantage of using a corked bat, using very basic approaches.
Here is the first step: what is the relation between the mass of a bat and its speed? Ok, since it's a basic model it has flaws, I'll try to show them. If anybody has some ameliorations, suggestions...
Since I am bad at using html, I apologize for the formatting.
Suppositions (each is is subject to amelioration, which could be implemented more or less easily):
- the bat follows a circular trajectory around the player. The axis of rotation is vertical (imagine yourself holding a bat horizontally and turning on yourself). The center of the rotation is O. A point of the bat is M.
- The axis of the bat is orthogonal to the axe; and the distance from the bat to the axis is constant.
- the drag is negligible
- the player exerces on the bat a force F which application point is the center of gravity of the bat. I'll later make the assumption that the component horizontal and orthogonal to the bat of this force is constant. (in polar coordinates, the component of F along Utheta is constant).
I also use polar coordinates (r, theta, z, with the associated vectors Ur, Utheta, Uz). Since I don't know how to type theta or omega in HTML, I will use H for theta and w for omega.
df/dt is the derivate of f by t.
d2f/dt2 is the derivate second of f by t.
So: the bats rotates around the axis (read player) with an angular speed w (and the associated vector W). We have w = dH/dt.
The referential is in rotation around a static (galilean) referential; therefore in polar coordinates we have, with the following notations (Speed: v, vector V, Vectorial product sign: ^):
m*dV/dt = mG - 2*m*W^V - m*[dW/dt^OM] - m*[W^(W^V)] + F
The polar coordinates of the vectors are (dr/dt = 0 and dz/dt = 0)
dV/dt ( -r*w*w ; r*dw/dt ; 0)
mG (0;0;-m*g)
W^V (r*w*w)
dW/dt^OM (0; r*dw/dt; 0)
W^(W^V) (r*w*w; 0; 0)
F (Fr, FH, Fz)
Which gives:
-mrww = -2mrww - mrww + Fr
mrdw/dt = -mrdw/dt + FH
0=-mg+Fz
Or:
Fr = 2*m*r*w^2
FH = 2*m*r*dw/dt
Fz = mg
Besides the speed is V = dr/dt*Ur + r*dH/dt*UH + dz/dt*Uz = r*w*UH
So, applicating this to G, center of gravity of the bat (let's call d the distance from g to the axis of rotation):
dw/dt = FH/(2*m*d); supposing that FH is constant (in other words the player's efforts to accelerate the rotation speed of the bat are constant):
w-w(t=0) = FH/((2*m*d)* (t-0); I here suppose that t = 0 at the beginning of the swing.
Let's call w(t=0) wo. A discussion on wo will follow.
So: we now can write that:
v = [FH/(2*m)]*t + wo*d
Suppose that wo = 0; the whole swing takes place at a belt or knee or chest high level. we have v*m = constant (t), which means that a corked bat with the same FH will have a greater speed than a normal bat (decrease in m, increase in v). And in a same amount of time, a corked bat will make more way than a normal bat. Finally, a decrease of 10% of the weight of the bat will make the speed increase by exactly 10%. But the quantity of movement (momentum in English I guess; p = m*v) will stay the same at a given time, if the bat is corked or not. And momentum is important: if one considers the schock between the ball and the bat as elastic, then there is conservation of the kinetic energy of the system (bat+ball), and of its momentum as well.
So what is the advantage of a corked bat ? (based on these results) If a batter uses a corked bat knowing that while making the same effort he can have a greater bat speed: with a corked bat he can:
- hit the ball earlier on the ball's trajectory (i.e. further from the catcher); this can be useful is the batter's swing is slow.
- begin to swing later and have a little more time to recognize the pitch, but still hit it (a corked bat needs less time to make the same way as a normal bat); this can be useful if the batter doesn't lack swinging speed but doesn't see the pitches very well.
I'll add more commentaries later.
By the way some people with a better knowledge of who used corked bats and when might definitely help here.
BUT I made the assumption that wo is equal to zero. The point is that it is not: in fact a better modelisation of the swing of the batter would include the fact that at the beginning of the swing, the bat is higher than the point at where it hit (or miss) the ball; the center of the gravity of the bat goes from head high to belt high, and not belt high during the whole swing as ín my model. And during this "descending" part of the swing, the heavier the bat is, the more the speed the bat takes, and consequently the bigger wo is.
(I'll look at the wo gain during the descending part in a later post).
So the speed gain made by using a corked bat is not as easy as 10% less weight means 10% more speed, what my model would say.
Sylvain
Posted 10:53 a.m.,
June 16, 2003
(#8) -
Sylvain(e-mail)
(homepage)
See homepage: very good article on corked bats (and other stuff).
The next steps will come later this week (I hope).
S
Posted 8:38 a.m.,
June 19, 2003
(#9) -
Sylvain(e-mail)
Looking at the advantage of using a corked bat: linking the mass of the bat and its speed.
Warning: this study is based on very simplistic approaches and approximations! Some better studies (more accurate models) have been and can be performed.
Step #2: in post #7 I considered a swing that was purely lateral. This step is an improvement as I added a vertical component to the swing in order to find out the effect of gravity on a swing. The notations I am going to use are same ones.
First a correction (oups...): instead of Fr = 2*m*r*w^2, once projected on Ur, the principle of Newton gives: Fr = 0 ( in fact Fr = m*d2r/dt2 but d2r/dt2 = 0). My bad.
The centre of gravity (G) of the ball is supposed to be in rotation around a vertical axis (z), with the distance from G to the axis being constant and equal to R. O is the projection of G on the axis.
The coordinates are still polar coordinates: r, theta = H, z. The corresponding vectors are Ur, UH and UZ,
The swing is decomposed in two phases:
1) from H = Ho to H = H1, a descending and rotating one, z varying from zo to z1;
2) a rotating one, H varying from H1 to ? and with z = z1. For this phase I will use the results from post #7.
Ho, H1, zo, z1 are function of:
- the player: how he swings (I'd say that if a hitter is a ground ball hitter than it would imply that the second phase is almost non-existant, Line drive hitter would have a big second phase, and a fly ball hitter would also have a third ascending phase).
- where he wants to swing (low, high in the strike zone).
I also pose that at t = 0, H = Ho and both dz/dt and dH/dt = 0.
I define t1 as the instant when H = H1 and z = z1.
So, in the general case, the equations give:
- on Ur: m*(d2r/dt2 - r*(dH/dt)^2) = -2*m*r*(dH/dt)^2 + m*r*(dH/dt)^2 + Fr
- on UH: m*(r*d2h/dt2 + 2*dr/dt*dh*dt) = - 2*m*dH/dt*dr/dt - m*r*d2H/dt2 + FH
- on Uz: m*d2z/dt2 = -m*g + Fz
Since I supposed that R = constant, dr/dt = d2r/dt2 = 0. Which gives:
- Fr = 0
- 2*m*r*d2H/dt2 = FH
- d2z/dt2 = -m*g + Fz
I now suppose that FH and Fz are constant. Since the first part of the swing was a descending one, it implies dz/dt negative. Or dz/dt = Vz = (-g + Fz/m) * t + Voz, with t positive and Voz = 0 => Fz - mg has to be negative.
It makes sense, Fz has to be inferior to the weight.
And I also have dH/dt = (FH/(2*m*R))*t + dH/dt (at t = 0).
So that dH/dt = w can be written w = constant/m*t
Let's integrate this:
- z(t)-zo = 1/2*(-g + Fz/m)*t^2
- H(t) - Ho = 1/2*A/m*t^2
This two equations can also give z as a function of H .
Remember t1: we have:
- z1 - zo = 1/2*(-g+Fz/m)*(t1)^2
- H1 - Ho = 1/2*A/m*(t1)^2
=> (z1 - zo)/(H1-Ho) = (-g + Fz/m)/(A/m)
This equation gives the relation between the force to apply and the swing chosen/type of swing.
After t = t1, we find ourselves in the case I exposed in post #7. In post #7 I looked at wo; since I changed the origin of time, the wo in post #7 is now w1 or w (t = t1) or dH/dt (t = t1).
Rewritting it gives:
w1 = dH/dt (t = t1) = FH/(2*m*R)*SquareRoot[2*(H1-Ho)*m*2*R/FH]
=> w1 = SquareRoot[FH * (H1 - Ho) / ( m * R )] = B/SquareRoot[m*R]
By replacing t1 by its expression obtained in the z(t) equation, you can also obtain w1 as a function of zo - z1 .
In post #7 we had w(t) = FH/((2*m*d)*t + wo. Using the new notations give, for the second phase:
w(t) = FH/(2*m*R)*t + B/SquareRoot(m*R); since the velocity v in the second phase is equal to w*R we can write:
V = A*t/m + B/SquareRoot(m)
W don't have m*v = constant (t) any more !
What does this mean ?
- If a batter uses a corked bat, the impact of the corked bat will depend on his swing and this advantage is bigger for line drive hitters than for a ground ball hitter.
- the relation between the loss of mass of the bat and the speed is not linear. A gain of mass of 10% will produce a gain of speed inferior to 10%, the factor of "speed restitution" depending on the swing (player + where in the zone).
Further comments and questions are wel(l?)come, as usual.
Sylvain
Posted 5:35 a.m.,
June 20, 2003
(#10) -
Sylvain(e-mail)
Another link that, well, explains it all:
http://www.kettering.edu/~drussell/bats.html
See the article concerning corked bats.
He does everything I did, but better.
Sylvain
Posted 7:58 a.m.,
June 20, 2003
(#11) -
Sylvain(e-mail)
So, let's go to step 3 (the last but most exciting one): the collision between the bat and the ball. The homepage links says it all, but I still have some points to add. There will also be less equations.
As noted in the link posted in #10 (article on bat weight, swing speed and ball velocity), the speed of the batted ball depends on:
- the weight of the bat,
- the speed of the bat,
- the speed of the ball before the collision,
- where the ball has been hit on the bat.
These parameters and the speed of the batted ball are all linked by the following equation, derived from the conservation of the momentum for the system (ball + bat) and introducing a "speed restitution" coefficient e:
V(batted ball)*(mbat + mball) = (mball - e*mbat)*V(pitched ball) + (mbat - e*mbat)*V(bat before collision)
with e = - (V(batted ball)-V(bat after))/(V(pitched ball)-V(bat before))
e depends on the speed of the incoming ball and is about 0.55 for a pitched ball velocity of 90 mph; it actually is a consequence of the conservation of the enery of the whole system bat+ball. The article doesn't precise it but I am pretty sure it also depends on the point of impact between the ball and the bat; see the article on bending modes and the sweet spot.
Remember that all values are algebric ones!
The batter has control on:
- V(bat before),
- mbat, linked to V(bat before) (see step 1 and 2),
- where the ball is hit.
Now we have to make some suppositions: if the ball hits the bat on the very same point whether the bat is corked or not (thus supposing e = constant), Then:
As we've seen, the relation between the speed of the bat and its weight can be written V = constant*t/m + constant/SquareRoot(m).
With t = time spent since the beginning of the swing.
First case: the batter begins to swing at the same time, whether the bat is corked or not. Then V(corked is bigger), then the corked bat will hit the ball earlier. Therefore, even without considering the second term, the momentum of the corked bat will be smaller for the collision ! But, on the other hand, the speed of the pitched ball will be higher (very very slighty) at the time of the collision (for more infos on this, see the thread on loss of velocity of a pitched ball) thus increasing the speed of the batted ball.
All in all, I'd say that the speed of batted in such a case in about the same if the bat is corked or not, the gain of power is therefore negligible, if there is any.
Now case #2: knowing that he has a bat speed advantage, the batter begins to swing later, but the contact with the ball happens after a same rotation of Theta (or in other words the point of contact bat-ball has the same spatial coordinates).
Then during the contact the corked bat will have a higher speed. However the momentum of the bat will also be lower (momentum = constant(t) but here tcorked inferior to tnoncorked and constant is growing with t). No gain at all, a loss I'd even say.
BUT remember that I supposed e = constant. Part of the aim of the batter is to maximize e by optimizing the point of contact. He can do it by lowering his swing, effect on z, or extending R. The equations (posted in previsous posts) show that with a similar force, the deviation will be higher if using a corked bat (m*d2r/dt2 = Fr and m*d2z/dt2 = -m*g + Fz). What happens in our two cases?
Case 1: both swings begin at the same time
Then of course the batter can deviate the bat using the same force. However because of the bat speed, these modifications will take place also "further" in space, at a theta bigger. The batter won't be able to play that much on the point of contact.
Case 2: corked bat swing begins later
The batter will be able to better recognize the pitch, optimize the point of contact, e, and still hit the ball, compared to a normal bat. The gain of speed of the batted ball depends on this optimization. So that there might be a gain in batted ball speed, depending on the point of contact .
So, why use a corked bat? I'd say:
- don't use a corked bat if you're in a power slump, unless you begin to swing later and can hit the ball really better (big e difference) and don't get caught.
- in a hitting slump due to lack of swing speed: by beginning the swing at the same time, a corked bat will help have more contact, more times on base due to errors, and perhaps more hits, but you won't have more power. Still, don't get caught.
- in a hitting slump because of bad pitch recognition: a corked bat will allow you begin to swing later and help you make a better contact, if any. But, just as in the other two cases, don't get caught.
These were my thoughts.
For comments, questions....
Thanks to those courageous and brave who took the time to read, thou shall go back to your castle, find and kill the dragon, and marry the princess.
Sylvain