Volcano Bungee Simulation (Second order Ordinary Differential Equations)

A group of Mechanical Engineering students is tasked to model and simulate an epic volcano bungee jumping adventure. Bungee jump from a helicopter into the caldera of an active volcano over a bubbling lava lake. https://www.bungee.com/bzapp/volcano/index.html

An average person can weigh between 50 kg (110 lbs.) to 90 kg (200 lbs.). This is a dangerous adventure. The purpose of the simulation is to analyze the bungee jumping details for people weigh between 50-90 kg using the 30-meter (100 feet) long bungee line prior to jumping over the bubbling lava.

The equation to use for the analysis is Newtons Second Law, F = ma

where F is the sum of the gravitational, aerodynamic drag, and bungee forces acting on the jumper, m is the mass of the jumper, and a is the acceleration.

Define the distance the jumper falls as the variable x. Distance is a function of time, x(t). The jumpers velocity and acceleration are then represented as x and x, respectively. The Newtons equation to solve for acceleration:

x = F / m

Next, determines the forces making up F. The gravitational force will be the jumpers weight, which is:

W = m g

g is about 9.8 m/s2 at the surface of the earth.

The aerodynamic drag, D, will be proportional to the square of the jumpers velocity, D = c (x) 2

The constant c can be computed from the 55 m/s free-fall terminal velocity. At 55 m/s, the aerodynamic drag is equal to the weight of the jumper, so c can be determined using: c = D / (x) 2 = W / (55 m/s)2

Finally, after the jumper has fallen beyond the bungee cord length, the slack in the bungee will be eliminated, and it will begin to exert an arresting force, B, of 20 N for every meter that it is stretched beyond 30 m. B = 20 (x - 30)

The bungee also has a viscous friction force, R, once it begins to stretch, which is given by:

R = 1.5 x

Thus, there will be two equations for computing the acceleration. The first equation will be used when the distance x is less than or equal to 30 m: x = F / m = (W D) / m

A second equation will be used when x is greater than 30 m: x = F / m = (W D B R) / m

Simulation:

? Create one Simulink embedded function model (using the MATLAB function block) to simulate a 50 kg person volcano bungee jumping off a helicopter for the first 500 seconds.

? Add mux, display, scope, and To File blocks as needed in the Simulink model to save the simulation result, the simulation time (t), distance (x), velocity (x) and acceleration (x) to a .mat file.

? Adjust the Simulink models maximum step size to 0.1 or smaller to ensure the simulation will compute enough points for obtaining accurate maximum and minimum values of the simulation.

? Create another Simulink model to simulate a 90 kg person volcano bungee jumping off a helicopter for the first 500 seconds.

? Add To File, mux, display, and scope blocks as needed to collect the Simulink simulation data.

? Adjust the Simulink models maximum step size to 0.1 or smaller to ensure the simulation will compute enough points for obtaining accurate maximum and minimum values of the simulation.

? Create a MATLAB program that will automatically open both of the Simulink models and run both Simulink models. Load the .mat files generated by the Simulink models to the MATLAB program.

? Add an ode45 solution to simulate the 50 kg person volcano bungee jumping distance (x), velocity (x), and acceleration (x) for the first 500 seconds.

? Add another ode45 simulation to simulate the 90 kg person volcano bungee jumping for the first 500 seconds.

? Generate one figure that contains 3x2 subplots to plot and compare the 50 kg person volcano bungee jumping Simulink solution and the 50 kg person volcano bungee jumping ode45 solution side-by-side.

The figure should include the following:

o Plot the bungee jumpers distance (x) vs. t chart with the Simulink Solution on the left, and plot the x vs. t chart using the ode45 solution on the right.

o Plot the bungee jumpers velocity (x) vs. t chart with the Simulink Solution on the left, and plot the x vs. t chart using the ode45 solution on the right.

o Plot the bungee jumpers acceleration (x) vs. t chart with the Simulink Solution on the left, and plot the x vs. t chart using the ode45 solution on the right.

Generate a second figure that also contains 3x2 subplots to plot and compare the 90 kg person volcano bungee jumping Simulink solution and the 90 kg person bungee jumping ode45 solution side-by-side.

The figure should include the following:

o Plot the bungee jumpers distance (x) vs. t chart with the Simulink Solution on the left, and plot the x vs. t chart using the ode45 solution on the right.

o Plot the bungee jumpers velocity (x) vs. t chart with the Simulink Solution on the left, and plot the x vs. t chart using the ode45 solution on the right.

o Plot the bungee jumpers acceleration (x) vs. t chart with the Simulink Solution on the left, and plot the x vs. t chart using the ode45 solution on the right.

? Title and label plots clearly.

? Answer the following simulation analyses questions and print answers to a text file.

? For the 50 kg person bungee jump:

1. What is the estimated peak acceleration values of the jump? Consider the peak acceleration values in both directions (free fall down and going up).

2. What is the estimated peak velocity of the jump?

3. What is the estimated maximum drop distance of the jump? (How far will the jumper fall before the person starts backup?)

4. How close can the helicopter come within the actual pool of molten lava to ensure the helicopter is flying 3 times the maximum drop distance above the lava?

5. The bungee jump simulation starts at 0 seconds. How many seconds will the jumper fall to reach the maximum drop distance?

6. After the jump, when the person dangles on the bungee below the helicopter, what is the approximate distance between the person and the helicopter?

? For the 90 kg person bungee jump:

1. What is the estimated peak acceleration values of the jump? Consider the peak acceleration values in both directions (free fall down and going up).

2. What is the estimated peak velocity of the jump?

3. What is the estimated maximum drop distance of the jump? (How far will the jumper fall before the person starts backup?)

4. How close can the helicopter come within the actual pool of molten lava to ensure the helicopter is flying 3 times the maximum drop distance above the lava?

5. The bungee jump simulation starts at 0 seconds. How many seconds will the jumper fall to reach the maximum drop distance?

6. After the jump, when the person dangles on the bungee below the helicopter, what is the approximate distance between the person and the helicopter?