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The bat has mass M and moment of inertia I about its center of mass. From the reference frame of the center of mass of the bat just before the collision, the ball has initial velocity in the positive -direction while the bat has initial angular velocity . In our setup, and have opposite signs when the batter is swinging at the ball as in figure 1,in which arrows point in the positive directions for the corresponding parameters.

The ball collides with the bat at a distance
from the center of mass of the bat. We assume that the collision is head-on and view the event such that all the
-component velocities are zero at the moment of the collision. After the collision, the ball has a final velocity and the bat has a final linear velocity
and an angular velocity at the center of mass.

Figure1. The collision.

When the ball hits the bat, the ball briefly compresses and decompresses, converting kinetic energy to potential energy and back. However, some energy is lost in the process that is, the collision is inelastic. The ratio of the relative speeds of the bat and the ball before and after the coefficient of restitution, customarily desinelastic collision, and means a perfectly elastic one. In this basic model, we make two simplifying assumptions:

. is constant along the length of the bat, and

. is constant for all .

Given our pre-collision conditions, we can write:

Conservation of linear momentum:

Conservation of angular momentum:

,

Definition of the coefficient of restitution:

.

Solving for gives

,

Where

is the effective mass of the bat.

For calibration purposes we use the following data, which are typical of a regulation bat connecting with a fastball in Major League Baseball. The results are plotted in Figure 2.

m
0.145kg
5.1 oz

M
0.83kg
29 oz

L
0.84m
33 in

I
0.039kg.m2

m/s
150mph

-60rad/s

0.55

Figure 2. Final velocity (solid arc at top), swing speed (dotted rising line), and effective mass(dashed falling curve) as a function of distance (in meters) from center of mass.

The maximum exit velocity is 27m/s, and the sweet spot is 13cm from the center of mass. Missing the sweet spot by up to 5cm results in at most 11m/s difference from the maximum velocity, implying a relatively wide sweet spot.

From this example, we see that the sweet spot is determined by a multitude of factors, including the length, mass, and shape of the baseball bat; the mass of the baseball; and the coefficient of restitution between bat and ball. Furthermore, the sweet spot is not uniquely determined by the bat and ball: It depends also on the incoming baseball speed and the batter’s swing speed.

Figure 2 also shows intuitively why the sweet spot is located somewhere between the center of mass and the end of the barrel. As the point of collision moves outward along the bat, the effective mass of the bat goes up, so that a greater fraction of the initial kinetic energy is put into the bat’s rotation. At the same time, the rotation in the bat means that the barrel of the bat is moving faster than the center of mass (or handle). These two effects work in opposite directions to give a unique sweet spot that’s not at either endpoint.

However, this model tells only part of the story. Indeed, some of our starting assumptions contradict each other.

. We treated the bat as a free body because the collision time was so short. In essence, during the 1ms of the collision, the ball “sees” only the local geometry of the bat, not the batter’s hands on the handle. On the other hand, we assumed that the bat was perfectly rigid-but that means that the ball “sees” the entire bat.

. We also assumed that is constant along the length of the bat and for different collision velocities. Experimental evidence [Adair1994] suggests that neither issue can be ignored for an accurate prediction of the location of the sweet spot.

We need a more sophisticated model to address these shortcomings.

Our Model

We draw from Brody’s rigid-body model but more so from Cross [1999]. One could describe our work as an adaptation of Cross’s work to actual baseball bats. Nathan [2000] attempted such an adaptation but was misled by incorrect intuition about the role of vibrations. We describe his approach and error as a way to explain Cross’s work and to motivate our work.

Previous Models

Brody’s rigid-body model correctly predicts the existence of a sweet spot not at the end of the bat. That model suffers from the fact that the bat is not a rigid body and experiences vibrations. One way to account for vibrations is to model the bat as a flexible object. Beam theories (of varying degrees of accuracy and complication) can model a flexible bat. Van Zandt [1992] was the first to carry out such an analysis, modeling the beam as a Timoshenko beam, a fourth-order theory that takes into account both shear forces and tensile stresses. The equations are complicated and we will not need them. Van Zandt’s model assumes the ball to be uncoupled from the beam and simply takes the impulse of the ball as a given. /the resulting vibrations of the bat are used to predict the velocity of the beam at the impact point (by summing the Brody velocity with the velocity of the displacement at the impact point due to vibrations) and thence the exit velocity of the ball from the equations the coefficient of restitution [Van Zandt1992].

Cross [1999] modeled the interaction of the impact of a ball with an aluminum beam, using the less-elaborate Euler-Bernoulli equations to model the propagation of waves. In addition, he provided equations to model the dynamic coupling of the ball to the beam during the impact. After discretizing the beam spatially, he assumed that the ball acts as a lossy spring coupled to the single component of the region of impact.

Cross’s work was motivated by both tennis rackets and baseball bats, which differ importantly in the time-scale of impact. He baseball bat’s collision lasts only about 1ms, during which the propagation speed of the wave is very important. In this local view of the impact, the importance of the baseball’s coupling with the bat is increased.

Cross argues that the actual vibrational modes and node points are largely irrelevant because the interaction is localized on the bat. The boundary conditions matter only if vibrations reflect off the boundaries; an impact not close enough to the barrel end of the bat will not be affected by the boundary there. In particular, a pulse reflected from a free boundary returns with the same sign (deflected away from the ball, decreasing the force on the ball, decreasing the exit velocity), but a pulse reflected from a fixed boundary returns with the opposite sign (deflected towards the ball, we expect the exit velocity to be uniform along a non-rotating bat. Cross’s model predicts all of these effects, and he experimentally verified them. In our model, we expect similar phenomena, plus the narrowing of the barrel near the handle to act somewhat like a boundary.

  Nathan’s model also attempted to combine the best features of Van Zandt and Cross [Nathan2000]. /his theory used the full Timoshenko theory for the beam and the Cross model for the ball. He even acknowledged the local nature of impact. So where do we diverge from him? His error stems from an overemphasis on trying to separate out the ball’s interaction with each separate vibrational mode.

The first sign of inconsistency comes when he uses the “orthogonality of the eigenstates” to determine how much a given impulse excits each mode. The eigenstates are not ortheogonal. Many theories yield symmetric matrices that need to be diagonalized, yielding the eigenstates; but Timoshenko’s theory does not, due to the presence of odd-order derivatives in its equations. Nathan’s story plays out beautifuuly if only the eigenstates were actually orthogaonal; but we have numerically calculated the eigenstates, and they are not even approximately orthogonal. He uses the orthogonality to draw important conclusions.

. The location of the nodes of the vibrational modes are important.

. High-frequency effects can be completely ignored.

We disagree with both of these.

The correct derivation starts with the following equation of motion, where k is the position of impact, is the displacement and is the external force on the ith segment of the bat, and is an asymmetric matrix:

.

We write the solutions as , where the rows of are eigenmodes with eigenvalues
. Explicitly, , and indicates the kth component of the nth eigenmode. Then we write the equations of motion:

,

.

In the last step, we used the fact that the eigenmodes form a complete basis.

Nathan’s ** uses on the right-hand side simply scaled by normalization constant. At first glance, this seems like a minor the technical detail, but the physics here is important. We calculate that the
terms stay fairly large for even high values of n, corresponding to the high-frequency modes (k is just the position of the impact). This means that there are significant high-frequency modes are necessary for the impulse to propagate slowly as a wave packet. In

Nathan’s model, only the lowest standing modes are excited; so the entire bat starts vibrating as soon as the ball hits. This contradicts his earlier belief in localized collision (which we agree with), that the collision is over so quickly that the ball “sees” only part of the bat. Nathan also claims that the sweet spot is related to the nodes of the lowest mode, which contradicts locality: The location of the lowest-order nodes depends on the geometry of the entire bat, including the boundary conditions at the handle.

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