【正文】 two bending modes of a freely supported baseball bat. The handle end of the bat is at the right, and the barrel end is at the left. The numbers on the axis represent inches (this data is for a 30 inch Little League wood baseball bat). These figures were obtained from a modal analysis experiment. In this opinion we prefer to follow the convention used by Rod Cross[2] who defines the sweet zone as Team 8038 Page 11 of 20 the region located between the nodes of the first and second modes of vibration (between about 47 inches from the barrel end of a 30inch Little League bat). Fig. 46 The figure of “Sweet Zone 2” The solving time in accordance with the searching times and backtrack times. It is objective to consider the two indices together. Optimization Model Based on TOPSIS Method Table42 swing period T bat mass M bat length S CM position d coefficient of restitution BBCOR initial velocity inv swing speed batv ball massballm wood bat (ash) cm Adopting the parameters in the above table and based on the quantitative regions in sweet zone 1 and 2 in , the following can be drawn:[2] Sweet zone 1 is ),( LL = ),50( cmcm Sweet zone 2 is ),( *2*1 LL = ),( cmcm As shown in Fig 43, define the position of Block 2 which is the pivot as the origin of the number axis, and x as a random point on the number axis. 1) Optimization modeling[2] The TOPSIS method is a technique for order preference by similarity to ideal solution whose basic idea is to transform the integrated optimal region problem into seeking the difference among evaluation objects—“distance”. That is, to determine the most ideal position and the acceptable most unsatisfactory position according to certain principals, and then calculate the distance between each evaluation object and Team 8038 Page 12 of 20 the most ideal position and the distance between each evaluation object and the acceptable most unsatisfactory position. Finally, the “sweet zone” can be drawn by an integrated parison. Step 1 : Standardization of the extent value Standardization is performed via range transformation, minmaxmin* xx xxx ??? , *x is a dimensionless quantity， and ]1,0[*?x ),(}, a x {}, i n { m a xm i n*2m a x*1m i n xxxLLxLLx ??? 。 Step 3: Calculating the distance The Euclidean distance of the positive ideal position is: ? ?? ?? )( ** xxd The Euclidean distance of the negative ideal position is: ? ?? ?? )( ** xxd Step 4: Seeking the integrated optimal region The integrated evaluation index of the evaluation object is: ????? dd db… …………………………（ 45） 2) Optimization positioning Considering bat material physical attributes of normal wood, when the period is sT ? and the vibration frequency is 520?f HZ, the ideal “sweet zone” extent can be drawn as ? ?cmcm 0 4 , . As this consequence showed, the “sweet spot” cannot be at the end the bat. This conclusion can also be verified by the model for problem II. Verifying the “sweet spot” is not at the end of the bat 1) Analyzed from the hitting effect According to Formula 411 and Table 42, the maximum battedballspeed of Team 8038 Page 13 of 20 the “sweet spot” can be calculated as smBBS sweet /? , and the maximum battedballspeed of the bat end can be calculated as smBBS end /? . It is obvious that the “sweet spot” is not at the end of the bat. 2) Analyzed from the energy According to the definition of “sweet spot” and the method of locating the “sweet spot”, energy loss should be minimized in order to transfer the maximum energy to the ball. When considering the “sweet spot” region from angle of torque, the position for maximum torque is no doubt at the end of the bat. But this position is also the maximum rebounded point according to the theory of force interaction. Rebound wastes the energy which originally could send the ball further. To sum up the above points: it can be proved that the “sweet spot” is not at the end of the bat by studying the quantitative relationship of the hitting effect and the inference of the energy transformation. Modeling and Solution to Problem II Model Preparation 1) Introduction to corked bat[5][6]: Fig 47 As shown in Fig 47, Corking a bat the traditional way is a relatively easy thing to do. You just drill a hole in the end of the bat, about 1inch in diameter, and about 10inches deep. You fill the hole with cork, super balls, or styrofoam if you leave the hole empty the bat sounds quite different, enough to give you away. Then you glue a wooden plug, like a 1inch dowel, in to the end. Finally you sand the end to cover the evidence. Some sources suggest smearing a bit of glue on the end of the bat and sprinkling sawdust over it so help camouflage the work you have done. 2) Situation studied： Situation of the best hitting effect: vertical collision occurs between the bat and the ball, and the energy loss of the collision is less than 10% and more than 90% of the momentum transfers from the bat to the ball (the hitting point is the “sweet spot”). Team 8038 Page 14 of 20 3) Analysis of COR After the collision the ball rebounded backwards and the bat rotated about its pivot. The ratio of ball speeds (outgoing / ining) is termed the collision efficiency, Ae . A kinematic factor k , which is essentially the effective mass of the bat, is defined as batballI zmk 2? …………………………………………………