MATH 373 Numerical Analysis Homework #1 Assigned: January 24, 2018 Due: February 2, 2018

Transcription

1 MATH 373 Numerical Analysis Homework #1 Assigned: January 24, 2018 Due: February 2, Develop an M-file for the Bisection Method. Command-line usage is [root,func val,error approx,num iterations] = bisection(func,x min,x max,error desired,max iterations) Program the default values error desired = and max iterations = 50. Generate errors if x min x max or if there is no sign change over the interval. Be sure to follow the template on the website for your code. Your code should be submitted on the MCS website using the Submit It! feature. 2. Develop an M-file for the False Position Method. You will need to read 5.5 in order to understand this method. Command-line usage is [root,func val,error approx,num iterations] = false position(func,x min,x max,error desired,max iterations) Program the default values error desired = and max iterations = 50. Generate errors if x min x max or if there is no sign change over the interval. Be sure to follow the template on the website for your code. Your code should be submitted on the MCS website using the Submit It! feature. 3. The secant formula, used to calculate the load that a column can withstand before buckling, is given by [ ( ( σ max = P ) )] L P 1 + ε r sec. A 2k EA If σ max = 200, 000 MPa/m 2 is the maximum stress for the material used to make the column, ε r = 0.25 is the eccentricity ratio, E = 150, 000 MPa is the modulus of elasticity, and L/k = 30 is the slenderness ratio, then determine the smallest stress, P/A, that satisfies the secant formula. I recommend using MPa as your units, instead of Pascals, as a way of keeping the numbers smaller. Also, you will probably need to graph this problem in order to determine where the roots are. Finding an interval with a sign change does take a bit of thought. Use both Bisection and False Position to solve this problem. Report the outputs for the two methods in a properly formatted table. You do not need to include a plot for this report, but if you do not include a plot, then make sure you address how you determined an interval for your methods. This solution must be typeset using L A TEX and a hard copy should be submitted in class. Be sure to follow the template for homework on the course website. 1

2 MATH 373 Numerical Analysis Homework #2 Assigned: February 7, 2018 Due: February 16, Develop an M-file for the Newton-Raphson method. Command-line usage is [x root,func val,error approx,num iterations] = newton raphson(func,dfunc,x guess,error desired,max iterations) Program the default values error desired = and max iterations = 50. Be sure to follow the template outlined in the syllabus for code. Your code should be submitted on the MCS website using the Submit It! feature. 2. Develop an M-file for the Secant method. Command-line usage is [x root,func val,error approx,num iterations] = secant(func,x guess one,x guess two,error desired,max iterations) Program the default values error desired = and max iterations = 50. Structure your code so that the first of the inputted guesses to be deleted is x guess two. Be sure to follow the template outlined in the syllabus for code. Your code should be submitted on the MCS website using the Submit It! feature. 3. Develop an M-file for the Modified-Secant method. Command-line usage is [x root,func val,error approx,num iterations] = secant modified(func,x guess,delta,error desired,max iterations) Program the default values delta=0.1, error desired = , and max iterations = 50. Be sure to follow the template outlined in the syllabus for code. Your code should be submitted on the MCS website using the Submit It! feature. 4. Consider a toy sailboat that will be constructed by hand out of concrete. The shape of the hull will be cylindrical in order to simplify the casting process by using a two-liter soda bottle. The freeboard, f, of a boat is the distance from the surface of the water to the top of the hull, or the amount of the hull that sits above the water. The freeboard is directly related to the thickness, t, of the hull, through the principle of buoyancy, by [ 1 2 πr2 L 1 2 π(r t)2 (L 2t) ] γ concrete +0.2 = 12 [ ( f r2 2 cos 1 r ) ( ( ))] f sin 2 cos 1 Lγ water, r where r is the radius of the outside of the cylindrical hull, L is the length of the hull, γ concrete is the specific weight of concrete, and γ water is the specific weight of water. For the toy sailboat assume r = 2.15 in, L = 7.5 in, γ concrete = 140 lbf/ft 3, and γ water = 62.4 lbf/ft 3. Note that an additional 0.2 lbf has been added to the weight of the sailboat to account for the addition of a keel, sail, and bow in the final design. A thickness for the walls of the sailboat must be chosen, but the choice must balance two factors. (1) If the walls are too thin, then the sailboat could break, since concrete does not perform well in tension. (2) If the walls are too thick, then the sailboat will be too heavy, the freeboard will be too small, and the sailboat could be swamped by a passing wave. Your job is to propose a thickness for the hull of the sailboat. Your solution should consist of three parts, and each part of your solution should be clearly called out in your L A TEX report using the subsection option. 2

3 (a) Provide a table that reports a single value for the freeboard for hull thicknesses of 1/8 in, 3/16 in, 1/4 in, 5/16 in, and 3/8 in. (b) Provide a design recommendation based on the table in the previous subsection and the dimensions of the sailboat, as well as any additional information you think will strengthen your case. This recommendation may be a single recommended thickness or a list of acceptable thicknesses. In either case, justify your choice. (c) Provide a justification for the entries in your table. Use Bisection and Secant to calculate f for a given t, and include the usual information for each run. Be sure to explain why the output from these methods provides reasonable values for the table. Also, include code for Secant in the Code section of your report, but you do not need to provide code for Bisection. You may be able to automate this process by storing the values of t in an array and by using a for loop to step through the same process for each value of t. As always, any code you write to solve the problem should be included in your report. Your L A TEX report is due in class, and it should conform to the standards listed elsewhere. 3

4 MATH 373 Numerical Analysis Homework #3 Assigned: February 16, 2018 Due: February 26, Develop an M-file for the Inverse Quadratic Interpolation method. Command-line usage is [x root,func val,error approx,num iterations] = IQI(func,x guess one,x guess two,x guess three,error desired,max iterations) Program the default values error desired = and max iterations = 50. Structure your code so that the first of the inputted guesses to be deleted is x guess three, followed by x guess two. Be sure to follow the template outlined in the syllabus for code. Your code should be submitted on the MCS website using the Submit It! feature. 2. The equation for the current, i, in a circuit with a resistor, capacitor, and inductor is given by ( ) i = e Rt/(2L) 1 R 2 cos t i 0 LC, 2L where i 0 is the initial current in the circuit, R is the resistance in the circuit, L is the inductance in the circuit, C is the capacitance in the circuit, and t is the duration of the discharge. Overall, the charge stored in the capacitor drives the current through the circuit, while the resistor and inductor dissipate the current. Determine the half-lie of the current (i.e., the time until i/i 0 = 0.5) if R = 100 Ω, L = 5 H, and C = 10 4 F using IQI and the MATLAB command fzero. Report the outputs for the two methods in a properly formatted table. You do not need to include a plot for this report, but if you do not include a plot, then make sure you address how you determined an interval or initial guesses for your methods. This solution must be typeset using L A TEX and a hard copy should be submitted in class. Be sure to follow the template for homework on the course website. 4

5 MATH 373 Numerical Analysis Homework #4 Assigned: February 26, 2018 Due: March 12, Develop an M-file for the Trapezoidal Rule. Test it by solving several integrals for which you know the answers. Command-line usage is as follows. Set the default number of slices to 50. Be sure to follow the template outlined elsewhere for code. Your code should be submitted on the MCS website using the Submit It! feature. integral = trapezoid(func,x min,x max,num slices) 2. Develop an M-file for the Trapezoidal Rule to be used with data sets. Testing this program requires known data sets and the results. The book does have an example you can test your code against, but I needed to create my own data set, integrate it by hand, and then compare the results to my code for a second test. Command-line usage is as follows. Two tests need to be run on the inputs. First, test that x vals and y vals are column vectors. I recommend using the size command. Second, make sure that x vals and y vals have the same length. Be sure to follow the template outlined elsewhere for code. Your code should be submitted on the MCS website using the Submit It! feature. integral = trapezoid data(x vals,y vals) 3. Develop an M-file for Simpson s 1/3 rule. Test it by solving several integrals for which you know the answers. Command-line usage is below. Make sure your code checks to see if num slices is even. Set the default number of slices to 50. Be sure to follow the template outlined elsewhere for code. Your code should be submitted on the MCS website using the Submit It! feature. integral = simpsons third(func,x min,x max,num slices) 4. We are going to check the claims of Estes Industries, which manufactures rocket engines for low-powered rockets. For instance, Estes Industries manufactures a C11 engine, which should produce an average thrust of 11 N and a total impulse of 5 10 N s. The letter in the engine name designates a class of engine that produces a certain impulse. Table 1 lists some of the classes that Estes Industries manufactures, along with the associated range of total impulses. The number in the name states the average thrust of the engine over its burn time. Table 1: Engine class and total impulse. Class Impulse (N s) A B C 5 10 D E The website has tested many rocket engines, and it provides data for each engine as thrust (in Newtons) versus time (in seconds). We will use this data to test the 5

6 claims of the C11, D11, D12, and E12 engines. First, impulse is defined as I = tf t 0 F (t) dt, where F (t) is the thrust as a function of time, t 0 is the start time, and t f is the final time. Second, the average value of a function, f(x), over the interval [a, b] is defined as f ave = 1 b a b a f(x) dx, which we can use to compute the average thrust of each engine. Note that since we only have data about the thrust, we will need to use the program trapezoid data to compute these integrals. For each engine, complete the following steps. I recommend writing a MATLAB function that performs these steps for a given engine, and then you can run the function for each engine. I also recommend that the only input to this function be the variable filename, which contains the name of the file to be loaded. The outputs would be the total impulse and the average thrust. (a) Load the.mat file with the necessary data using the command load(filename), where filename is a string containing the name of the file. The.mat files for the engines are in a.zip files on the course website. The load command will create a variable called thrust data, which will be loaded with the information for the rocket engine. This variable will not appear in the Workspace window, but it will exist. I suggest calling the variable in your M-file without a semicolon after you load it so that you can see what it looks like. Determine the dimensions of thrust data using the size command. For instance, to load the information for the C11 engine, use the command load( C11 thrust curve.mat ); though if you use the input filename, all you would use is load(filename); and the string would be the input to the function. (b) Plot the thrust data using the plot command. Include a title and axis labels using the title, xlabel, and ylabel commands. The : operate will be helpful. (c) Use trapezoid data to compute the total impulse and average thrust of the engine. (d) Plot the average thrust as a dashed horizontal line on the same plot as the thrust data. The hold command will be needed for this. (e) Evaluate the claims of the engine designation. If the engine designation is incorrect, what should it be? Provide a subsection for each of the four engines that includes all of the information produced above. The report must be typeset using L A TEX and a hard copy should be submitted in class. Be sure to follow the template for homework on the course website. 6

7 MATH 373 Numerical Analysis Homework #5 Assigned: March 12, 2018 Due: March 19, Develop an M-file for Romberg integration. Test it by solving several integrals for which you know the answers. Command-line usage is as follows. Set the default error to and the maximum number of iterations to 50. Make sure that you calculate the approximate error by comparing the entries in the first row between iterations. Be sure to follow the template outlined elsewhere for code. Your code should be submitted on the MCS website using the Submit It! feature. [integral,error approx,num iterations] = romberg(func,x min,x max,error desired,max iterations) 2. Develop an M-file for Gauss Quadrature. Test it by solving several integrals for which you know the answers. Command-line usage is as follows. Your code should handle 1 5 points. Set the default number of points to 5. Use a switch statement with num points to choose which formula to implement. Be sure to follow the template outlined elsewhere for code. Your code should be submitted on the MCS website using the Submit It! feature. integral = gauss quad(func,x min,x max,num points) 3. Develop an M-file for Adaptive Quadrature. Test it by solving several integrals for which you know the answers. Command-line usage is as follows. Set the default tolerance to Be sure to follow the template outlined elsewhere for code. Your code should be submitted on the MCS website using the Submit It! feature. integral = quad adapt(func,x min,x max,tolerance) Use a subfunction to handle the recursive part of the algorithm, and use the command-line usage given below. integral = quad step(func,x min,x max,tolerance) 4. Solve Problem 19.10, using the following methods: (a) Use Romberg integration to find both F and d. (b) Use Gauss Quadrature with four points to find both F and d. (c) Use Gauss Quadrature with five points to find both F and d. (d) Use Adaptive Quadrature to find both F and d. Assume that units for H are in feet. This solution must be typeset using L A TEX and a hard copy should be submitted in class. Be sure to follow the template for homework found elsewhere. 7

8 MATH 373 Numerical Analysis Homework (Extra) Assigned: March 23, 2018 Due: Never 1. Find a centered-difference approximation for the first derivative from the tables in Section 21.2 with an error of O(h 4 ). (a) Use this formula to approximate f (4) with h = 0.1, where f(x) = x. Maintain four decimal places throughout your calculations. Answer: f (4) (b) Give a rough estimate of the error associated with your answer in (a) without calculating the actual value. Answer: The error is roughly (0.1) 4. 8

9 MATH 373 Numerical Analysis Homework #6 Assigned: April 6, 2018 Due: April 13, Read Section concerning linear interpolation. Develop an M-file for linear interpolation. Test it by interpolating several known values. Note that Example 17.2 provides two test cases. Check that x one x two, x one < x interpolated, and x interpolated < x two. Note that this method has no default inputs. Your code should be submitted on the MCS website using the Submit It! feature. y interpolated = linear interpolation(x one,y one,x two,y two,x interpolated) 2. Develop an M-file for Euler s method. Test it by solving several ODEs for which you know the answers. Command-line usage is as follows. Use the default h = Check that h works for the interval, and return a suggestion for h if it does not. Make sure that your code outputs column vectors for t vals and y vals. Your code should be submitted on the MCS website using the Submit It! feature. [t vals,y vals] = euler ode(dydt,t start,t final,y 0,h) 3. Develop an M-file for the fourth-order Runge-Kutta method. Test it by solving several ODEs for which you know the answers. Command-line usage is as follows. Use the default h = Check that h works for the interval, and return a suggestion for h if it does not. Make sure that your code outputs column vectors for t vals and y vals. Your code should be submitted on the MCS website using the Submit It! feature. [t vals,y vals] = RK4(dydt,t start,t final,y 0,h) 4. For Homework #7 we will be writing code to simulate the vertical launch of a single-stage rocket. This will be an involved problem, and so I am giving you two weeks to complete it. See Homework #7 for details. 9

10 MATH 373 Numerical Analysis Homework #7 Assigned: April 13, 2018 Due: April 20, Develop an M-file for the fourth-order Runge-Kutta method for systems. Test it by solving several systems of ODEs for which you know the answers. Command-line usage is as follows. Use the default h = Check that h works for the interval, and return a suggestion for h if it does not. Make sure that your code outputs column vectors for t vals and y vals. Also, design your code to work with a dydt function that returns column vectors. Your code should be submitted on the MCS website using the Submit It! feature. [t vals,y vals] = RK4 sys(dydt,t span,y 0,h) 2. Given information about thrust vs. time for a rocket engine in a.mat file, estimate the maximum altitude of the rocket. This is an involved problem that will require several steps. I recommend creating a program with the following definition. [time vals,output vals] = rocket simulator(filename,time final,delta t) The input filename is a string containing the name for the.mat file that contains the data about the rocket engine. The.mat files can be found on the website. The final time and the step size are also inputs. The outputs are an array of time values and an array of output values, which consists of altitude in the first column and velocity in the second column. To aid in the writing of the code, I have split the process into the following steps. (a) Input the constants needed for this problem. The mass of the rocket is 200 g, the diameter of the rocket is 6.6 cm, the density of air, ρ air, is kg/m 3, the unitless drag coefficient is C d = 0.4, and gravity for this problem is m/s 2. (b) Load the.mat file with the necessary data using the command load(filename). The.mat file contains three variables: thrust data, mass initial, and mass final. The variable thrust data is the same array from previous homework. The other two variables are the initial and final masses of the rocket engine. Determine the dimensions of thrust data using the size command. (c) We need to approximate the mass of the rocket engine at every time in thrust data. We will determine the mass by assuming that the mass of propellant consumed at any given time is proportional to the total impulse generated at that time. For instance, if 40% of the total impulse has been generated at a given time, then 40% of the propellant has been consumed at that time. We will do this by creating a function with the commandline usage below. The input is the same filename as above, and the output is an array that contains the mass of the rocket engine for each time value in thrust data. Note that this function will require trapezoid data. Three checks can be made. (1) Check that the final mass is equal to the variable mass final, and (2) plot thrust and mass vs. time for the C11 engine by using the plot and yyaxis commands and comparing the results to Figure 1. (3) Table 2 contains the actual output for the mass of the C11 engine. mass data = rocket engine mass(filename) 10

11 Thrust (N) mass (g) Thrust and mass vs. time. Thrust mass Time (s) Figure 1: Thrust and mass vs. time for the C11 engine. 11

12 Table 2: Thrust and mass data for the C11 engine. Time (s) Thrust (N) Mass (g)

13 Thrust (N) (d) We need to define a number of anonymous functions to describe the various quantities that will appear in the system of ODEs. i. thrust func: this function calculates the thrust at any given time. Note that we only know the thrust at the specific times available in the data set in the associated.mat file. This means that we need to calculate the thrust values for times we do not have in the data set. We can do this using linear interpolation and the linear interpolation function written for the homework. I have written a function, called data calculator, which will take a data set and a time and use linear interpolation to calculate an interpolated value. This function is in the same.zip file as the engine data. Note that data calculator will call your linear interpolation function. I used the following definition. thrust_func data_calculator(thrust_data,time); I recommend testing thrust func by plotting it with the fplot command along with the data set. See Figure 2 as an example. (I have no idea why there is a vertical dashed line at about t = 0.3 s.) Note how the interpolated values lie on the data set. Also, Table 3 shows sample outputs from the function. 25 Thrust vs. Time Data Interpolated Time (s) Figure 2: Thrust (data and interpolated) vs. time for the C11 engine. ii. mass engine func: this function calculates the mass of the engine at any time. This function will also need data calculator and linear interpolation. See Figure 3 as an example. Also, Table 3 shows sample outputs from the function. 13

14 Mass (g) 32 Mass vs. Time Data Interpolated Time (s) Figure 3: Mass (data and interpolated) vs. time for the C11 engine. iii. mass total func: this function calculates the total mass of the rocket (rocket and engine) at any given time. This function will call mass engine func. Also, Table 3 shows sample outputs from the function. iv. drag func: this function calculates the aerodynamic drag on the rocket at any given time, which is given by F drag (v) = 1 2 C d ρ air A v 2, where A is the cross-sectional area of the rocket and v is the velocity of the rocket at the given time. Also, Table 4 shows sample outputs from the function. 14

15 Table 3: Output from the various functions for the C11 engine. Time (s) Thurst (N) Engine Mass (g) Total Mass (kg) Table 4: Output from the drag function. Velocity (m/s) Drag (N)

16 (e) We can now derive the ODE that describes the motion of the rocket in flight. We start with Newton s Law. Note that we are simplifying the calculation by treating the mass, m, as a constant in the first step, and then treating it like a function, m(t), after the first step: ma = F ( d 2 ) y m(t) dt 2 = T (t) W rocket W engine (t) F drag (v(t)) d 2 y dt 2 = 1 m(t) (T (t) W rocket W engine (t) F drag (v(t))). Note that we have defined all of the necessary quantities in the previous steps. Also, since this is a second-order ODE, we will need to use a system of first-order ODEs to solve the problem in MATLAB. Define the system in MATLAB as described in class, using the command-line usage below. dydt vals = dydt rocket(time,y vals) Writing this function poses a problem. Note that the various quantities we need to define the system of ODEs are defined in rocket simulator, which means that we cannot access them in dydt rocket. We can get around this by defining the necessary pieces as global variables, using the global command. For instance, if I want to use thrust func in both rocket simulator and dydt rocket, then I include the code global thrust_func at the beginning of both functions, and then MATLAB will allow both functions to use thrust func. In this way we can bring all of the necessary pieces into dydt rocket. (f) Solve the system using RK4 sys, as described in class. You can see a sample plot in Figure 4, and some sample outputs with a small time interval can be seen in Table 5. We can now use rocket simulator to evaluate the performance of the four rocket engines in the.zip file. We will evaluate the performance of each engine in the following ways. (a) Use the plot and yyaxis commands to plot altitude and velocity vs. time on the same plot, as in Figure 4. Table 5: Sample output with [0, 0.5] and h = 0.1. Time (s) Altitude (m) Velocity (m/s) (b) Our goal is to build a rocket that will reach an altitude of about 150 m, or 500 ft. What is the maximum altitude achieved by each engine? 16

17 Altitude (m) Velocity (m/s) 60 Altitude and Velocity vs. Time Altitude Velocity Time (s) 0 Figure 4: Altitude and velocity vs. time for the C11 engine. (c) We need to make sure that the rocket velocity is at least 13 m/s by the time it reaches 1.5 m in altitude to make sure that the rocket is aerodynamically stable as it leaves the launch pad. (d) We also need to determine the time between the end of the engine burn and the maximum altitude. This will be used to determine the delay charge for the engine that deploys the parachute for the rocket descent. We do not want the delay charge to ignite before the engine reaches maximum altitude, but we also do not want the delay charge to ignite too long after reaching the maximum altitude. 17

18 For each engine, gather the data needed for the previous steps. Provide a table for each engine that includes the burn time, the velocity at an elevation of 1.5 m, the maximum altitude, and the time between the end of the engine burn and the maximum altitude. While the plots may provide an approximation of the necessary information, remember that you have arrays with actual numbers that will provide better results. Also, you can use linear interpolation to arrive at even more precise values. Once you have the information, provide an evaluation of each engine, which will include whether or not the engine will suit our purposes, and if it does, what should the delay be for the engine? Table 6 provides a list of available delays for each engine. Table 6: Available delay times for each engine. Engine Delays (seconds) C11 0, 3, 5, 7 D11 0 D12 0, 3, 5, 7 E12 0, 4, 6, 8 This solution must be typeset using L A TEX and a hard copy should be submitted in class. You do not need to include the full arrays for each rocket. Use the subsection command for each engine to organize your solution. Be sure to follow the template for homework on the course website. 18

19 MATH 373 Numerical Analysis Homework (Extra) Assigned: April 20, 2018 Due: Never 1. Convert the ODE ( y ) 2 + y y = t to a system of first-order ODEs and use RK4 sys to solve the original ODE. Use the initial conditions y(0) = 3, y (0) = 1, and y (0) = 4. Also, use the time interval [0, 5]. Be careful with step size because this particular system of ODEs easily becomes unstable with fourthorder Runge-Kutta. After you have solved the problem, you can check your solution with the following information. Table 7 shows some sample outputs for the solution for the time interval [0, 1] and h = 0.2. Figure 5 shows a plot of the entire solution over [0, 5]. Finally, the code to solve this problem can be found in Listing 1. Table 7: Solution for Problem 1. t y

20 y 20 Another example of a higher-order ODE t Figure 5: Plot of the solution for Problem 1. 20

21 Listing 1: Code for Problem 1. f u n c t i o n [ t v a l s, y v a l s ] = e x h i g h e r ( ) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Brent Deschamp % April 9, 2018 % An a d d i t i o n a l example o f a higher order ODE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The ODE % ( y ) ˆ 2 + y y = g ( t ) % y ( 0 ) = 3, y ( 0 ) = 1, y ( 0 ) = 4 % l e t u=y % l e t w = u % w = 1/u ( g ( t) wˆ2) % y v a l s has the form % [ y, u, w ] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Define the i n puts f o r RK4 sys t s t a r t = 0 ; t f i n a l = 5 ; h =0.1; y 0 = [ ] ; % Save the c u r r e n t d i r e c t o r y c u r r e n t f o l d e r = pwd ; % Change the d i r e c t o r y to where RK4 sys i s s t o r e d cd C: \ Users \bdescham\ Numerical \ODE % Solve the system [ t v a l s, y v a l s ] = RK4 sys higher ode, [ t s t a r t t f i n a l ], y 0, h ) ; % Change the d i r e c t o r y back cd ( c u r r e n t f o l d e r ) ; %Plot the s o l u t i o n p l o t ( t v a l s, y v a l s ( :, 1 ), ok ) ; t i t l e ( Another example o f a higher order ODE ) ; x l a b e l ( t ) ; y l a b e l ( y ) ; g r i d end 21

22 f u n c t i o n d y d t v a l s = e x h i g h e r o d e ( t v a l, y v a l s ) % Define v a r i a b l e s y = y v a l s ( 1, 1 ) ; u = y v a l s ( 1, 2 ) ; w = y v a l s ( 1, 3 ) ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Define the f o r c i n g f u n c t i o n g t ) t ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Define the system o f ODEs d y d t v a l s = [ u ; w; (1/ u ) ( g ( t v a l ) wˆ 2 ) ] ; end 22

23 MATH 373 Numerical Analysis Homework #8 Assigned: April 20, 2018 Due: April 27, The temperature distribution in the wall of a pipe, through which a hot liquid is flowing, is given by the ODE d 2 T dr dt r dr = 0, T (r inner) = T inner. If the pipe described is cooled by convection on the outer surface, then the heat conduction, q conduction, at the outer wall is equal to the heat convection, q convection, to the surroundings. Note that q conduction = ka dt dr, and q convection = ha(t T a ), where k is the thermal conductivity, h is the convective cooling coefficient, A is the exposed surface area, and T a is the ambient temperature. Use finite-difference methods to determine the temperature distribution in the wall if r inner = 1 cm, r outer = 2 cm, T inner = 100 C, T a = 10 C, k = 100 J/(s m K), and h = 500 J/(s m 2 K). Write an M-file that will calculate the temperature distribution in the wall of the pipe for any number of slices, which will be submitted on the MCS website using the Submit It! feature. Use the values above for your solution, but do not hard code these values into your equations, though they should be hard coded into your M-file. The command-line usage is below. Determine the minimum number of slices. The default number of slices is 50. Make sure your code returns column vectors. [r vals,t vals] = hwk8 pipe(num slices) You will also write a L A TEX report, which will be submitted in class. Include a plot of the final temperature distribution in your L A TEX report, but you do not need to report r vals or T vals in the report. You can complete by hand most of the work on paper that you attach to your report, but your report should include the final equations for each index, which have been typeset using L A TEX. This means that Steps 1 6 will be completed by hand, Step 6 will be typeset in the report, Step 7 will be completed by your M-file, and the results of Step 7 will be reported using L A TEX. Each problem on Homework 8 should be typeset in its own report so that the grader can return each problem as soon as it is graded. 2. This problem comes to us courtesy of Dr. Romkes of Mechanical Engineering. Consider a beam of length L resting on a linearly elastic foundation with a constant spring coefficient c (force/length 2 ), which corresponds to modeling part of a foundation on a linear elastic soil. The ODE that governs the deflection of this beam with respect to position, u(x), is given by EI d4 u + cu = w(x), dx4 23

24 where E is the modulus of elasticity, I is the moment of inertia, and w(x) (force/length) is the load applied to the beam. Assume that u (0) = 0, u (0) = 0, u(l) = 0, and u (L) = 0. We will also assume that the load is a constant, w(x) = w. Solve this boundary value problem using finite-difference methods. Table 8: Parameters for Problem 2. L = 3 m w = 15, 000 N/m E = 200 GPa I = m 4 c = 65, 000 N/m 2 Write an M-file that will calculate the deflection of the beam for any number of slices, which will be submitted on the MCS website using the Submit It! feature. Use the values in Table 8 for your solution, but do not hard code these values into your equations, though they should be hard coded into your M-file. The command-line usage is below. Determine the minimum number of slices. The default number of slices is 50. Make sure your code returns column vectors. [x vals,u vals] = hwk8 beam(num slices) You will also write a L A TEX report, which will be submitted in class. Include a plot of the deflection in your L A TEX report, but you do not need to report x vals or u vals in the report. You can complete by hand most of the work on paper that you attach to your report, but your report should include the final equations for each index, which have been typeset using L A TEX. This means that Steps 1 6 will be completed by hand, Step 6 will be typeset in the report, Step 7 will be completed by your M-file, and the results of Step 7 will be reported using L A TEX. Each problem on Homework 8 should be typeset in its own report so that the grader can return each problem as soon as it is graded. 24