20xx年美国数学建模竞赛获奖论文英文版

20xx年美国数学建模竞赛获奖论文英文版内容摘要：

f inertia batI is, the faster the swing speed is) but also the sweet spot effect of the ball which can be reflected by the maximum batted ball speed (BBS). The BBS of different material can be got by analyzing the material parameters that affect the moment of inertia. Then, it can be proved that the hitting effects of different bat material are different. Team 8038 Page 6 of 20 Assumptions and Symbols Model Assumptions 1) The collision discussed in this paper refers to the vertical collision on the “sweet spot”。 2) The process discussed refers to the whole continuous momentary process starting from the moment the bat contacts the ball until the moment the ball departs from the bat。 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 pivot point. If a player was holding the bat in his/her hands, this would result in an impulsive force felt in the hands.  If the impact force F strikes the bat below the centerofmass, but above the centerofpercussion, point P would experience both a translational acceleration in the direction of the force and a rotational acceleration in the opposite direction as the bat attempts to rotate about its centerofmass. The translational acceleration to the left would be greater than the rotational acceleration to the right and a player would still feel an impulsive force in the hands.  If the impact force strikes the bat below the centerofpercussion, then point P would still experience oppositely directed translational and rotational accelerations, but now the rotational acceleration would be greater.  If the impact force strikes the bat precisely at the centerofpercussion, then the translational acceleration and the rotational acceleration in the opposite direction exactly cancel each other. The bat would rotate about the pivot point but there would be no force felt by a player holding the bat in his/her hands.  Define point O as the centerofpercussion（ COP） 1） Locating the COP According to physical knowledge, it can be determined by the following Team 8038 Page 8 of 20 method: Instead of being distributed throughout the entire object, let the mass of the physical pendulum M be concentrated at a single point located at a distance L from the pivot point. This point mass swinging from the end of a string is now a simple pendulum, and its period would be the same as that of the original physical pendulum if the distanceL was MdIL bat （ 41） This locationL is known as the centerofoscillation. A solid object which oscillates about a fixed pivot point is called a physical pendulum. When displaced from its equilibrium position the force of gravity will attempt to return the object to its equilibrium position, while its inertia will cause it to overshoot. As a result of this interplay between restoring force and inertia the object will swing back and forth, repeating its cyclic motion in a constant amount of time. This time, called the period, depends on the mass of the object M , the location of the centerofmass relative to the pivot point d , the rotational inertia of the object about its pivot point 0I and the gravitational field strengthg according to MgdIT 02 (42) 2) Analysis of the vibration:[1] Fig. 42 As showed in Fig. 42, mechanical vibration occurs when the bat hits the ball. Hands feel fortable only when the holding position lies in the balance point. The batting point is the vibration source. Define the position of the vibration source as the vibrational node. Now this vibrational node is one of the optional “sweet spots”. Solutions to the two “sweet spot” regions 1) Locating the COP[1][4] Team 8038 Page 9 of 20  Determining the parameters： a. mass of the bat M。 b. length of the bat S (the distance between Block 1 and Block 5 in Fig 43)。 c. distance between the pivot and the centerofmass d ( the distance between Block 2 and Block 3 in Fig. 43)。 d. swing period of the bat on its axis round the pivot T (take an adult male as an example: the distance between the pivot and the knob of the bat is (the distance between Block 1 and Block 2 in Fig. 43)。 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 Ta。