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Another model predicts the sweet spot be between nodes of the two lowest natural frequencies of the bat [Nathan 2000]. Given a free bat allowed to oscillate, its oscillations can be decomposed into fundamental nodes of various frequencies. Different geometries and materials have different natural frequencies of oscillation. The resulting wave shapes suggest how to excite those modes (e.g , plucking a string at the node of a vibrational mode will not excite that mode). It is ambiguous which definition of sweet spot this model uses. Using the first definition, it would focus on the uncomfortable excitations of vibrational modes: Choosing the impact location to be near nodes of important frequencies, a minimum of uncomfortable vibrations will result. Using the second definition, the worry is that energy sent into vibrations of the bat will be lost. This model assumes that the most important energies to model are those lost to vibration.

This model raises many questions. Which frequencies get excited and why? The Fourier transform of an impulse in general contains infinitely many modes. Furthermore, wood is a viscoelastic material that quickly dissipates its energies. Is the notion of an oscillating bat even relevant? What types of oscillations are relevant? A cylindrical structure an support numerous different types of modes beyond the transverse modes usually assumed by this model [Graff 1975].

Following the center-of-percussion line of reasoning, how do we model the recoil of the bat? Following the vibrational-nodes line of reasoning, how do we model the vibrations of the bat? In the general theory of impact mechanics [Goldsmith 1960], these two effects are the main ones
(assuming that the bat does not break or deform permanently). Brody [1986] ignores vibrations, Cross [1999] ignores bat rotation but studies the propagation of the impulse coupled with the ball, and Nathan [2000] emphasizes vibrational modes. Our approach reconciles the tension among these approaches while emphasizing the crucial role played by the time-scale of the collision.

Our main goal is to understand the sweet spot. A secondary goal is to understand the differences between the sweet spots of different bat types. Although marketers of bats often emphasize the sweet spot, there are other relevant factors: ease of swing, tendency of the bat to break, psychological effect, and so on. We will argue that it doesn’t matter to the collision whether the batter’s hands are gripping the handle firmly or if the batter follows through on the swing; these circumstances have no bearing on the technique required to swing the bat or how the bat’s properties affect it.

Our ** is organized as follows. First, we present the Brody rigid-body model, illuminating the recoil effects of impact. Next we present a full computational model based on wave propagation in an Euler-Bernoulli beam coupled with the ball modeled as a lossy spring. We compare this model with others and explore the local nature of impact, the interaction of recoil and vibrations, and robustness to parameter changes. We adjust the parameters of the model to comment on the sweet spots of corker bats and aluminum bats. Finally, we investigate the effect of hoop frequencies on aluminum bats.

A ** Example

We begin by considering only the rigid recoil effects of the bat-ball collision, much as in Brody [1986]. For simplicity, we assume that the bat is perfectly rigid. Because the collision happens on such a short time-scale (around 1 ms), we threat the bat as a free body. That is to say, we are not concerned with the batter’s hands exerting force on the bat that may be transferred to the ball.

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