Introduction and Purpose
Even though the sport of bungee cord jumping is now on the wane, partially due to the rise in legal cases as a result of injuries and partially due to the fad running its course. However, physics, math, and engineering are involved in successfully designing a "good" bungee jump. The purpose of this project is to integrate these three disciplines together to understand better how they relate to one another.
In this project your team will be expected to develop a bungee jump design using theory coupled with experimental measurements of the parameters or variables necessary for accurately modeling. Your team's first prototype will be the one which is actually used in the competition, to be described later. This is in keeping with the new decision paradigm which says that to be globally competitive, one must strive to model the complete artifact and its manufacturing process so that the first one manufactured can go to the customer.
The goal of this project is to design a quality bungee cord experience that will allow a payload to come as close to the ground as possible without damage. Damage can be incurred by:
- actually impacting the ground, or
- decelerating the payload too rapidly
The bungee designs will be demonstrated in a competition to see which team's design performs the best. During the competition, visual and video tape evidence will be used to judge the closeness to the ground and to determine the maximum deceleration of the payload.
Bungee Cord Jumping
Bungee cord jumping attracts those persons who like the strong feeling of danger mixed with their entertainment. Jumps have been made from high bridges, tall cranes, and hot air balloons. Bungee cord jumpers step, dive, fall, or are pushed off these high perches with only one end of an elastic cord fastened to their bodies. The other end of the cord is supposed to be fastened to the device from which the jump is made.
During the first part of the jump, jumpers are pulled toward earth by gravity, accelerating in free fall, much like a sky diver. For the jumper, this free fall is related to the quality of the jump - the desire is to free fall for as long as possible and to reach a speed that is as high as possible. But, the free fall continues only as long as the bungee cord remains slack. When the slack is gone and the bungee cord begins to stretch, the cord applies an upward force that begins to decelerate the jumper. This decelerating force increases as the bungee cord is stretched farther and farther. If all goes well, jumpers are brought to a stop before the space between them and the ground shrinks to zero.
For this project, the "jumper" will consist of an uncooked egg and any ancillary weight you care to add to it. The egg is vulnerable - if it strikes the ground, it will experience the HD syndrome.
Your team's performance model should aid you in picking a design that best meets the constraints and objectives of this project. The model will contain a variety of variables and parameters which your team must specify.
Modeling the physics of the jump
There are several variables of importance in the jump. This figure may help in relating the various quantities in the equations you will have to solve. At this stage in your physics education, you have learned about Newton's Laws of Motion. This project will provide you an excellent opportunity to apply these laws in an engineering context.
To model the physics of the jump, your team must begin with Newton's Second Law applied to the jumper. The forces in this equation will be somewhat complicated, since the force applied by the bungee cord will act only when the cord is stretched. But, at this stage of your mathematics education, you have learned about the use of Excel spreadsheets to solve a variety of problems, so here is an excellent opportunity to apply this knowledge (and the the use of some new Excel functions) in an engineering context, again.
A quantity of importance is the maximum deceleration encountered by jumpers as the bungee cord stops them. If this deceleration is too great, it may cause damage to the jumper. The maximum deceleration is reached when the net force on the jumper is a maximum. The force must not exceed four times the weight of the jumper.
Modeling the properties of the cord
Elastic materials such as rubber consist of long chain molecules that deform when stretched and recover when released. The processing of these materials determines, among other properties, the relationship between the amount of stretch and the force causing the stretch. The number of cross-links formed through covalent bonding between the long chain molecules can be increased, for example, to decrease the amount of stretch for a given force. Elastic materials with few covalent bonds can easily stretch in one direction to three or four times their original length without damaging the material. Rubber materials change their properties over time and with use as the bonds change.
The bungee cord enters into the modeling equations of the previous section through the relationship between the force used to stretch the cord and the amount it stretches. It is quite common to use the model of a linear spring having a constant "spring constant" which is the proportionality between the force causing the stretch and the amount of the stretch. This linear model is known as Hooke's Law. However, rubber is not truly a linear substance and you will want to describe fully the functional relationship between the force and the displacement (stretch). Your modeling must include this functional relationship.
Your team must make a series of measurements on one or more of the one meter lengths of cord which will be provided to you. You should use a variety of weights to apply static loads to stretch the cord being tested to find the functional relationship between the force and the stretch. A weight (or weights), which act as the stretching force, is (are) to be tied or fastened to the samples of latex cord. The unstretched lengths are measured, then the stretched lengths for various values of the force are measured so that the functional relationship is deduced. This spreadsheet may assist your team in carrying out this portion of the design process.
Solving the models
Your team will combine the physics model and the bungee cord model into one model that you will solve with an Excel spreadsheet. You are to use Euler's method to solve the equations. These are described in the physics text (Selway) on page 133. Additional help will be provided in class.
What you are to design
The release device
Your team must design a device that will hold the egg on the end of the patented Foundation Coalition Bungee Omelet Swingarm. The design and dimensions of this swingarm will be given to your team. Your release device must allow you to release the egg from rest, remotely. A limited set of materials will be provided. Complete sketches of your design must be approved by the engineering instructors before you build it. The release device must then be built according to your plans and will be used by your team in the bungee drop competition. It must be compatible with the FCBO Swingarm and be installable in five muntes or less.
The principle design variables that you will ultimately have to specify are:
- The unstretched length of the cord that you will use in the competition, including the extra length needed at the ends for fastening
- The number of strands of the cord you will use
What you are to have ready on competition day
- The number of strands and their length
- An operational release mechanism, including a plastic bag to reduce the HD factor
- A video tape to record the jump
From previous experience the engineering instructors have discovered that
- You should not over-stretch the strands given for testing. That is, you should not stretch the samples to more than three times their original length. In fact, exceed two times their original length as infrequently as possible.
- You should avoid touching the strands with your bare hands. Oils on your skin have a deleterious effect on the latex strands. Gloves will be provided for handling the samples.
- You should keep the samples out of contact withe the air - keep them in the brown bag provided.
- Although you should stretch each cord a few times before testing, do not put a lot of test cycles on the long strands given to you for testing.