continue

miaoshi

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.

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 difference is that the former ignore shear forces. This should be acceptable; Nathan points out that his model is largely insensitive to the shear modulus. 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 0.6cm being hit with a ball of 42g at around 1m/s. For baseball, we have an aluminum or wood bat of radius width 6cn bing hit with a ball of 145g traveling at 40m/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.

Mathematics of Our Model

Our equations are a discretized version of the Euler-Bernoulli equations:

,

Where

is the mass density,

is the displacement,

is the external force(in our case, applied by the ball),

is the Young’s modulus of the material(a constant),and

is the second moment of area( for a solid disc).

We discretize z in steps of . The only force is from the ball, in the negative direction to the kth segment. Our discrtized equation is:

Our dynamic variables are through . For a fixed left end, we pretend that . For a free left end, we pretend that

.

The conditions on the right end are analogous. These are the same conditions that Cross uses.

Finally, we have an additional variable for the ball’s position (relative to some zero point) .Initially, is positive and is negative, so the ball is moving from the positive direction towards the negative. Let . This variable represents the compression of the ball, and we replace with .Initially, and . The force between the ball and the bat takes the form of hysteresis curves such as the ones shown in Figure 3.

Figure 3. A hysteresis curve used in our modeling, with maximum compression 1.5cm

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foreveringxq

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zhang0108795

求翻译。。。。。。。。。。。。

喇叭

什么都看不懂

weiyi0822

求翻译。。。。。。。。。。。。

linmatsas

OMLDGG…………

小草远在天涯

全英文哪。。。好长啊。。。是在发表什么观点？不同意什么正交？