ME304 Dynamics of Machinery Design Problem 1 Winter, 1998-1999 Grade__ __

Objective:

Your group is to design a baseball bat that is optimized for young children.

Background:

The purpose of a baseball bat is to store the energy supplied by a batter and then deliver that stored energy to a baseball. The amount of energy stored by a bat is given by Eq. 1,

, Eq 1

where *I* is the mass moment of inertia of the bat about the swing axis and v
is the angular velocity of the bat. From Eq.1 it’s apparent that an optimum bat design balances the need to have both a large mass moment of inertia, while not being so large that the batter is unable to produce an large angular velocity of the bat. Another characteristic of an optimum bat is that when a ball is struck with the sweet spot of the bat, the reaction force at the grip is very small. This is achieved by designing the bat so that the center of percussion is at the point of impact with the ball, and the center of rotation is located at the grip. This characteristic of an optimum bat is sometimes lost for young players when they are instructed to move their hand further up the grip (choking-up) on bats designed for older, stronger players. By choking-up on the bat, the player is able to effectively reduce the mass moment of inertia, *I*, so that a larger angular acceleration is possible. However the center of percussion will move for the new grip position. In an effort to explore this problem, your project is to determine the optimum bat design for children.

For the purposes of this project, children are differentiated from adults by the amount of torque they can apply about the swing axis. You are to design optimum bats for children of three different strengths, ones that can produce 5 ft-lb, 15 ft-lb, and 25 ft-lb of torque about the swing axis. Also for the purpose of this project, assume that the bat swings through an arc of 90 degrees in a plane that is parallel to the ground. The bat starts from rest, and has a constant torque applied about the swing axis (see Fig. 1) until the ball is struck.

Design Specifications:

Approximate the bat as two solid cylinders, a small cylinder for the handle and a cylinder with a larger diameter for the barrel where the ball is struck as shown in Fig. 1.

Figure 1

Bat Geometry and Nomenclature

The bat you design should satisfy the following design specifications

- center of rotation is 3.00 inches from the end of the handle (l
_{g}) - center of percussion is 24.00 +/-1.00 inches from handle end of the bat (l
_{p}) - the barrel is at least 10.00 inches long (l
_{b}) - the maximum barrel diameter is 2.75 inches long (d
_{b}) - the handle is at least 8.00 inches long (l
_{h}) - the bat is at least 30.00 inches long (l
_{b}+l_{h}) - the diameter of the handle is 1.00 inch (d
_{h}) - assume that the bat rotates about an axis that is 10.00 inches from the handle end of the bat (note that this axis does not pass through the bat) (l
_{r}) - the bat is made of solid wood with a density of 42 lb/ft
^{3}

__Project Deliverables__:

- Add a cover page to this handout that includes the group number, group member names, date, the class title, class number, and section.
- Fill out the following chart for your optimum bat designs for the three different applied torques (attach all supporting calculations):

Applied Torque (in lb) |
l (in) |
l (in) |
l (in) |
d (in) |
E,impact (in lb) |

5 |
|||||

15 |
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25 |

Also include:

- a plot the stored energy at impact as a function of the mass moment of inertia about the swing axis of a general bat for each of the three torques,
- a short discussion of the results.