I need help with two DVQ MatLab problems

timer Asked: Nov 22nd, 2016

Question description


I need help with two different MATLAB projects.

Please check the attachments and Make sure to follow the guideline.

Thank you.

MAT 276 MATLAB 2 Names: Neeley, M Score: This is based on problem number 20 out of section 3.2 from the Zill book, where we have a hemispherical pool being filled by a pipe at the bottom. From class, we discussed how we can form a differential equation for the height (or depth) of the water at time t taking into account both the rate of water flowing in and the rate of water being lost due to evaporation. The hemisphere has radius 10 feet (which is a pretty deep pool, if you think about it). As we saw, if the rate of evaporation is constantly 1% of the exposed surface area of the water in the pool, and water is flowing it at π cubic feet per minute, then dh 1 = − .01 dt 20h − h2 and the height of water in the pool never actually reaches 10 feet due to the limit of dh as h approaches 10. dt However, it seems unlikely that the evaporation rate stays the same as time goes by. If the pool was in direct sunlight during part of the day, more water would evaporate. If we started trying to fill the pool at 8 AM, the rate of evaporation would slowly increase as the sun rose in the sky, and then would likely decrease as the sun set. It’s also likely that the temperature of the air would have an effect on the evaporation rate. Starting from http://www.noaa.gov and looking for our ZIP code, it seems that the time of day when the humidity was at its lowest is about 3 PM. Let’s make the evaporation model a bit more sophisticated than saying it’s at a constant 1%: let’s have it range between a low of .5% and a high of 2%; let’s put the high value at 3 PM and the low at 3 AM, and assume that at least for a while, the temperatures and humidities repeat day to day. So at 8 AM we begin trying to fill the pool. Evaporation works against us at different rates during the day, predicted by a function we’ll call B(t) which we must construct. At 8AM when we begin this process, the pool 2 ft of water standing in it. Let’s increase the flow coming into the pool to 2π cubic feet per minute to try to overcome evaporation. 1. Can you create a sine or cosine wave that has a period of 1440 minutes (24 hours), and a maximum value of .02 occurring at 3 PM and a minimum value of .005 occurring at 3 AM? Depending on what time of day you want to be time t = 0, you might get different functions. MATLAB or Excel would not figure very prominently here unless you want to use it to graph your proposed function to verify that it has the desired properties. This will be what we call B(t). 2. Use your B(t) to modify the differential equation and create a new initial value problem dh 2 = − B(t) dt 20h − h2 h(0) = 2 The 2 in the numerator of the fraction is due to the fact that we’ve doubled the rate of flow from the original problem, and we’re replacing the constant .01 with the function B(t) to simulate changing evaporation rates throughout the day. We should have a short MATLAB M-file which uses Euler’s method to approximate solutions to a DE by this time; you may adapt this to our purposes, or use Excel, or create your own M-file. Keep in mind that the name of the M-file should match the label after function in the first line of the M-file, so be careful about changing the name of the sample file. To get full credit, I’d want to see a graph of the height of water t minutes after the pipe is opened, and a prediction on when the pool reaches a depth of 10 feet, or a discussion of why this cannot ever happen (depending on what your model shows). If you use Excel, do not print out the sheet but submit it in Canvas. If you edit or create an M-file, include that in your submission.
Math 276 MATLAB Project 3 Neeley, M Suppose we want to design a ride for the AZ state fair. Have you heard of a ride named the “Reverse Bungee”, or sometimes referred to as a “Catapult Bungee”? From Wikipedia: “The ride consists of two telescopic gantry towers mounted on a platform, feeding two elastic ropes down to a two person passenger car constructed from an open sphere of tubular steel. The passenger car is secured to the platform with an electro-magnetic latch as the elastic ropes are stretched. When the electromagnet is turned off, the passenger car is catapulted vertically with a g-force of 3 to 5, reaching an altitude of between 50 and 80 meters (180 to 260 ft). The passenger sphere is free to rotate between the two ropes, giving the riders a chaotic and disorienting ride. After several bounces, the ropes are relaxed and the passengers are lowered back to the launch position.” Let’s try to design something using our model for springs. 00 0 mx + βx + kx = 0 This ride wouldn’t allow the riders to tumble around quite as chaotically; it would just shake them up a bit. Imagine a gondola attached to a spring. The spring goes from the top of the gondola upward to an anchor that is at an as-of-yet undetermined point above the ground. (This is different from what we described in class, where the spring was anchored to the bottom.) People are supposed to enter the gondola while it is pinned to the ground, be strapped in, and then enjoy a bouncy ride as the gondola is released and the spring recoils. Clamps on the back of the gondola keep it running along a guide pillar next to the ride, so the motion is straight up and down (rectilinear motion). For now let’s assume: 1. The combined weight of the people and the gondola is 1000 lbs. (Don’t forget to change this into mass! Mass times g would be weight.) 2. The initial position is 75 feet below the equilibrium point of the spring and 1000 lbs gondola-passenger payload. (x(0) = 75) The anchor for the spring is much higher than that, and hopefully won’t have to be considered. 0 3. The initial velocity of the gondola is 0. (x (0) = 0) Other than that, we have control over most of the other variables by using different springs (with different values of k), and controlling how tightly the clamps grip the guiding pillar (influencing β, the drag constant). Try to find values for β and k that are positive and would result in a ride that meets the following criteria: The system should be underdamped. An overdamped ride would not be very bouncy or exciting. Oscillations greater than 10 feet above or below equilibrium should continue occurring for at least 30 seconds. There should be at least 6 times before those 30 seconds where the gondola reaches a turning point (3 minimums, 3 maximums). 00 The acceleration (x (t)) should stay less than 5g. In a written document, please provide: 1. The values you used for β and k and provide a graph of the position of a typical ride for a 1000 lbs payload. 2. Include some convincing evidence that the riders won’t experience more than 5g of force. 3. For safety reasons, also provide graphs and a small analysis of what would happen for an unusually small payload (700 lbs) or an unusually large one (1300 lbs). Determine the maximum altitude reached and estimate the time it takes for the oscillations to shrink to less than 10 feet (at which point we stop everything by reactivating our anchor and dragging the gondola back to earth). In class we might discuss script files and the diff command, which might help with some of this.
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