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MAE384 Matlab functions
创建日期：2024-07-19 17:05:08
所属分类：数学mathematics
标签： MAE384
Matlab functions
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MAE384 Take Home Exam 

submit to Gradescope your complete solution to this take home exam, including a printout of all func- tions/scripts used to solve the take homeexam;

due Tuesday, December 3rd, 2019 at 12:01am (this is the night from Monday to Tuesday).

checkpoint due date: Friday, November 29th, 2019 at 10:00am 

-The exam is open book, open notes (posted notes and solutions on the class’ Canvas site) and you may use Matlab’sbuild-in help system, but only to look up Matlab syntax questions;

-no collaboration isallowed;

-no outside help, including the tutoring center, may be sought to solve theproblems;

-donot use any Matlab build-in functions to find the inverse of a matrix, the norm, or the condition  Doing so will result in a zero for the problem.  

-for each problem, document all steps you took to solve the problem. This can be handwritten, but  must be legible for credit. If the problem states ’By hand’, do not use any program/function to solve the problem,however, you may use a non-programmable calculator to help in determining the numerical results of individual steps;

-given point values per problem are subject tochange;

-submit to Gradescope your complete solution to this take home exam, including a printout of all func- tions/scripts used to solve the take homeexam;

-uploadall Matlab function and script files used to solve the exam to the Take Home Exam 3 link on  This includes any command window commands that must be saved as a script file and submitted;

-all code submitted in your Gradescope and Canvas submissions must beidentical;

for all code results, use format long; and formatcompact;  

-onGradescope associate your answer pages with the corresponding problem  Failure to do so may result in no points given initially and will require a re-grade request to fix.

-on Gradescope and Canvas you will find a checkpoint Take Home Exam 4 assignment. Please submit all your work by the checkpoint due date to these links as if you did a regular submission. You will receive feedbackfor each problem you submitted on whether your answers are good, need more work, or are on the wrong track. The checkpoint submission is voluntary and will have no direct impact on your exam score (other than provide you with feedback to help you improve your final submission).

Problem 0 (0 points)  

How many hours/minutes did it take you to solve all problems of this take home exam? Do not count time spend on solving the bonus problems/honors problem.

Problem 0 required submission:

  • Submithours/minutes you took to solve problems 1-3 on Gradescope;

Problem 1 (15 points) (Core Course Outcome #3)

By hand, derive the general analytical solution to the following PDE

using separation of variables.

Problem 1 required submission:

  • Handwritten(or printed) all steps to obtain the analytical solution.

Problem 2 (40 points) (Core Course Outcomes #2 & #5)  

The temporal evolution of the height of water h in a cylindrical tank of radius r = 1m with periodic inflow and two outlet pipes of equal radius rpipe = 0.15m can be determined by solving the following ODE,

where

c = 75kg/s, ω = 2π Hz, ρ = 1000kg/m3, g = 9.81m/s2, h1 = 0.1m, h2 = 0.2m, t is time, and Atank and Apipe are the cross-sectional areas of the tank and each outlet pipe. Note that the sine in f (t) is inside the cosine. 

At the initial time of t = 0s, the tank is empty, i.e. h(t = 0s) = 0m. Please note the differences of the given values and functions to the setup of homework 5.

Using Nystrom’s RK-3, determine the height h in the water tank with an absolute estimated relative error of less than 10−6 at t = 250 (note these values are different from homework 5). Nystrom’s RK-3 method is given by c1 = 2/8, c2 = c3 = 3/8, a2 = 2/3, a3 = 2/3, b21 = 2/3, b31 = 0, and b32 = 2/3. Solutions used to calculate the estimated relative error must use step sizes that are a factor of 2 different.

Problem 2 required submission:  

  • Softcopy printout of well commented functions and script you coded to solve this problem;
  • Softcopy printout of all results incl.  demonstration that answer has the required  estimated relative  error or less;
  • Uploadof all your Matlab functions and scripts to the Take Home Exam 5 link on 

Problem 3 (10 points) (Core Course Outcomes #6)

Plot the height in the water tank as a function of time in a well annotated plot using the results obtained from problem 2. State the time step size used to generate the data.  

Problem 3 required submission:

  • Softcopy printout of well annotated graph showing h vs. t for 0 ≤ t ≤250s;
  • Statethe time step size used to generate the graph;
  • Uploadof all your Matlab functions and scripts to the Take Home Exam 5 link on 

Problem 4 (25 points) (Core Course Outcomes # 4 & #6) 

For the time evolution of h(t) that satisfies the requirement of problem 2, perform a discrete Fourier transform for h(t) for the time range of 75s  t   250s. In a semilogy plot, show the Power spectrum Pk vs the frequency f , for all allowed frequencies, and in a second plot show the power spectrum for the frequencies from 0 to 0.5Hz and limit the power shown to 1e-20 to 1e-5. List the 3 most important frequencies other than f = 0 in the signal, their period, and their power.

Problem 4 required submission:

  • Softcopyprintout of well commented script you coded to solve this problem;
  • Softcopy printout of well annotated graph showing Pk f for all allowed frequencies;
  • Softcopyprintout of well annotated graph showing Pk  f for 0 ≤ f ≤ 0.5 frequencies;
  • 3 most important frequencies (other than f = 0), their period, and their
  • Uploadof all your Matlab functions and scripts to the Take Home Exam 5 link on 

Problem 5 (10 points) (Core Course Outcomes # 4 & #6)

Plot in one graph the time evolution of h(t) that satisfies the requirement of problem 2 in the time range of 75s t 250s together with data calculated using the DFT ”interpolation” function if you are allowed to use only 3 discrete frequencies (waves) for the interpolation and your ”interpolated” data must be as close as possible to h(t).  

Problem 5 required submission:

  • Softcopy printout of well annotated graph showing h vs t for both the original data from problem 2 and the result of the interpolation function using 3 discrete frequencies only.  The graph  should contain  both curves;
  • Uploadof all your Matlab functions and scripts to the Take Home Exam 5 link on Canvas.
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