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 of the coefficient of restitution [van Zandt 1992].
Cross [1999] modeled the interaction of the impact of a ball with an alu-minum 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 dis-cretizing 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. The baseball bat’s colli-sion lasts only about 1 ms, 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 bound-ary conditions matter only if vibrations reflect off the boundaries; an im-pact 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, pushing it back, increasing the exit velocity). Away from the boundary, 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 [Nathan 2000]. His theory used the full Timoshenko the-ory 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 excites each mode. The eigenstates are not orthogonal. Many theories yield symmetric matrices that need to be diagonalized, yielding the eigenstates; but Tim-oshenko’s theory does not, due to the presence of odd-order derivatives in its equations. Nathan’s story plays out beautifully if only the eigen-states were actually orthogonal; 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, wherekis the position of impact,yiis the displacement andFiis the external force on theith segment of the bat, andHij is an asymmetric matrix:
y00k(t) = Hkjyj(t) + Fk(t).
We write the solutions as yk(t) = Φknan(t), where the rows of Φkn are eigenmodes with eigenvalues−ωn2. Explicitly,HjkΦkn = −ωn2Φjn, andΦkn
indicates the kth component of the nth eigenmode. Then we write the equation of motion:
Φkna00n(t) + Φknωn2an(t) = Fk = ΦknΦ−1njFj, a00n(t) + ωn2an(t) = Φ−1nkFk.
In the last step, we used the fact that the eigenmodes form a complete basis.
Nathan’s paper uses on the right-hand side simply ΦknFk scaled by a normalization constant. At first glance, this seems like a minor techni-cal detail, but the physics here is important. We techni-calculate that theΦ−1nkFk terms stay fairly large for even high values ofn, corresponding to the high-frequency modes (k is just the position of the impact). This means that there are significant high-frequency components, at least at first. In fact, the 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 contra-dicts 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.
While the inconsistencies in the Nathan model may cancel out, we build our model on a more rigorous footing. For simplicity, we use the Euler-Bernoulli equations rather than the full Timoshenko equations. The dif-ference is that the former ignore shear forces. This should be acceptable;
Nathan points out that his model is largely insensitive to the shear modu-lus. We solve the differential equations directly after discretizing in space rather than decomposing into modes. In these ways, we are following the work of Cross [1999].
On the other hand, our model extends Cross’s work in several key ways:
• We examine parameters much closer to those relevant to baseball. Cross’s models focused on tennis, featuring an aluminum beam of width 0.6 cm being hit with a ball of 42 g at around 1 m/s. For baseball, we have an aluminum or wood bat of radius width 6 cm being hit with a ball of 145 g traveling at 40 m/s (which involves 5,000 times as much impact energy).
• We allow for a varying cross-section, an important feature of a real bat.
• We allow the bat to have some initial angular velocity. This will let us scrutinize the rigid-body model prediction that higher angular velocities lead to the maximum power point moving farther up the barrel.
To reiterate, the main features of our model are
• an emphasis on the ball coupling with the bat,
• finite speed of wave propagation in a short time-scale, and
• adaptation to realistic bats.
These are natural outgrowths of the approaches in the literature.