Well, you did it! You finally graduated and got a job! Too bad it's working for minimum as the ride designer for Flip & Flop's Ride Company (Company motto: If you're taken for a ride by us, it's a Flop!) One day, your boss comes to you with a terrible problem. The company's best customer, the mysterious ``Mr. J.'' (rumored to have been a mathematician before he got rich in shadowy circumstances) needs a new ride for his theme park immediately. It's a good thing that you studied pages 179 through 198 of Boyce & DiPrima, your differential equations text. You remember, the section that explains analyzing mechanical vibrations with differential equations and forced vibrations.
 

The only parts in the warehouse (you knew it was a crummy company) are:

1. a spring with a spring constant of k = 10,

2. three shock absorbers (dashpots) with values b = 1, b = 0.2, and b = 0.04.

3. a ride bucket with extra lead weights that can be added or subtracted to adjust the mass m of the bucket and rider so that 10 < m < 30.
 and
4. a mechanical driver for the system.
 

Your plan is to have the ride as a spring mass system with a periodic external force, of the form F₀ cos(g t). The spring-mass system is supposed to scare the living daylights out of the riders by bouncing the riders (safely?) as much as possible. This bungee jumping like ride will be the new rage!
 

When you examine the parts however, you discover that the mechanical driver is damaged (you knew it was a really crummy company). The amplitude F₀ is fixed at 5, but you can still change the frequency (related to g) with a screwdriver; however once the ride has been assembled, no one will be able to change the frequency (Geez!). You must therefore decide what value of g should be used.
 

Your task is to determine the best setting for g, and the best choice of the shock absorber for the ride. You must also report to the boss explaining what you did, so that he can file it in company records, should there ever be a lawsuit. Be sure to include plenty of appropriate graphs, which make it easier for the boss and the lawyers to understand the report.
 

1. Write down the differential equation for y(t), the movement of the basket with F(t) = 5cos(g t), and k = 10.
 

2. Determine the steady-state solution to your equation. The result is given on page 196 of the text -- Show how the result is derived. Since the riders want more bounce for the buck, find the amplitude A of the steady-state solution (how much the basket will bounce). Hint: Your result for A should have the property that A² is a ratio of polynomials in g depending on the parameters b and m. The numerator should be a constant and the denominator a fourth degree polynomial in g.
 

3. Assuming that the mass of the bucket and rider is m = 20, determine what value is to be preferred for g? The answer should depend on b. (Hint: As often occurs in maximization problems, whatever value of g which maximizes A will also, of course, maximize A² and conversely. You will find it easier to work with A²).
 

4. Given your choice for g, determine which of the three shock absorbers will safely do the best job of creating an exciting ride for all whose mass m (of the bucket and riders) satisfies 10 < m < 30. Hint: Graph the solutions for m = 10, 20, and 30. Which weight group will have the most fun?
 
 
 

Note: This material was adapted from the Tuner laboratory material developed by S. Dunbar at the University of Nebraska.