【正文】 stitution BBCOR and the momentofinertia of the bat batI . Controlling variable method analysis M is the mass of the object； d is the location of the centerofmass relative to the pivot point； g is the gravitational field strength； batI is the momentofinertia of the bat as measured about the pivot point on the handle。s left hand. Fig45. Second bending mode 2 (670 Hz) The figures show the 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 ??? 。 e. distance between the undetermined COP and the pivot L (the distance between Block 2 and Block 4 in Fig. 43, that is the turning radius) . Fig. 43 Table 41 Block 1 knob Block 2 pivot Block 3 the centerofmass（ CM） Block 4 the center of percussion (COP) Block 5 the end of the bat ? Calculation method of COP[1][4]: distance between the undetermined COP and the pivot: 224?gTL? (g is the gravity acceleration） （ 43） moment of inertia: 220 4?MgLTI ? (L is the turning radius,M is the mass) （ 44） Team 8038 Page 10 of 20 ? Results: The reaction force on the pivot is less than 10% of the batandball collision force. When the ball falls on any point in the “sweet spot” region, the area where the collision force reduction is less than 10% is ),( LL cm, which is called “Sweet Zone 1”. 2) Determining the vibrational node The contact between bat and ball, we consider it a process of wave the bat excited by a baseball of rapid flight, all of these modes, (as well as some additional higher frequency modes) are excited and the bat vibrates .We depend on the frequency modes ,list the following two modes: The fundamental bending mode has two nodes, or positions of zero displacement). One is about 61/2 inches from the barrel end close to the sweet spot of the bat. The other at about 24 inches from the barrel end (6 inches from the handle) at approximately the location of a righthanded hitter39。 c. distance between the pivot and the centerofmass d ( the distance between Block 2 and Block 3 in Fig. 43)。 3) Both the bat and the ball discussed are under mon conditions. Symbols Table 31 Symbols Instructions k a kinematic factor 0I the rotational inertia of the object about its pivot point M the mass of the physical pendulum d the location of the centerofmass relative to the pivot point L the distance between the undetermined COP and the pivot g the gravitational field strength batI the momentofinertia of the bat as measured about the pivot point on the handle T the swing period of the bat on its axis round the pivot S the length of the bat z the distance from the pivot point where the ball hits the bat f vibration frequency ballm the mass of the ball and Solution Modeling and Solution to Problem I Model Preparation 1) Analysis of the pushing force or pressure exerted on hands[1] Team 8038 Page 7 of 20 Fig. 41 As showed in Fig. 41: ? If an impact force F were to strike the bat at the centerofmass (CM) then point P would experience a translational acceleration the entire bat would attempt to accelerate to the left in the same direction as the applied force, without rotating about the pi