MIE334H1S_2019_NUMERICALMETHODSI_E_1

pdf

School

University of Toronto *

*We aren’t endorsed by this school

Course

334

Subject

Mechanical Engineering

Date

Jan 9, 2024

Type

pdf

Pages

Uploaded by ColonelSummerCat37

UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, Friday, April 26, 2019 DURATION: 2 and ½hrs Third Year - Mechanical Engineering MIE3 34111 - Numerical Methods I Calculator Type: 2 Exam Type: C Examiner - D.A. Steinman Rules and Guidelines: The examination is 150 minutes, and is comprised of six (6) questions, worth a total of 100 points. You should attempt all questions. The exam contains 4 pages, including this one. This exam is closed-book. A one-page (double-sided) aid sheet is allowed. No other external material is allowed, such as lecture or tutorial notes, quizzes, midterms, final exams, or books. You must answer all questions in the separate booklets provided. Write your name and student number on each booklet used. Clearly highlight all final answers, and show all intermediate workings. We cannot give partial marks if you do not. Question 1 / 18 Question 4 /18 Question 2 / 18 Ques ti on 5 /10 Question 3 /14 Question 6 /22 1 of

1. (Short answers ONLY) (a) [3 points] The coefficient of determination (R 2 ) used in least squares regression quantifies the difference between two quantities. Identify and describe these two quantities (b) [3 points] For the following numerical integration rules: (a) Trapezoidal rule, (b) Simpson's 3/8 rule, (c) Gauss quadrature (n points). What are the degrees of the interpolation polynomial used to approximate an integral? (c) [3 points] For fitting a cubic spline to a set of n+1 data points what is: The number of interior knots? The minimum number of equations in the system of equations to be solved? The "special" form of the matrix that defines this system of equations? (d) [3 points] For Romberg numerical integration: What other numerical integration method is it based on? What is the order of the error for the first iteration of the Romber g method? What type of numerical integration problem is it best suited for; Integrating tabulated data or a known equation? (e) [6 points] Which of the following sets of linear equations are solvable using Guass-Seidel method? Which ones are not? Of the ones that are solvable using Gauss-Seidel method, which one(s) will definitely converge and which ones will not? What modifications are required to solve the ones that do not converge? Do not solve the equations. Set I Set Set I 5x—y+2z=5 I 2x+4y—z5 I x+4y—z4 3x-8y-2z1 I —6x+7z1 3x+y—z=3 x+y+4z3 I 4x+8y-2z=10 x+y+5z=7 2. [18 points] Reaction rate constants k can be represented as exponential functions of temperature: k=k ° expI RT where k 0 is a pre-exponential factor, Ea is the activation energy, R= 8.314J/mo/Kis the gas constant, and T is the reaction temperature in degrees Kelvin. Consider the rate constant versus temperature data as follows: k (s -1 ) 2.905 5.181 8.901 14.78 T(K) 290 300 1 310 320 Write the linearized form of the reaction rate equation given above Use linear regression to estimate the rate constant at T = 316 K Estimate the activation energy E (in units of kJ/mol) [Note: Retain 4 significant figures in your calculations] 2 of 4

3. [14 points] The following summarizes temperature measurements taken with high precision at different points along a heated steel beam: (m) 0 1 0.5 J 1.0 1 1.5 2.0 ("C) 90.00 164.49 148.90 138.78 1 35.00 Use a Newton cubic interpolating polynomial to estimate the temperature of the beam at a position of 0.7 m. [Note: Retain 4 significant figures in your calculations] If a Lagrange cubic interpolating polynomial was used instead, would the estimation of the temperature be the same or different? Explain. 4. [18 points] Use numerical methods to evaluate the "exact" value for the following double integral: for which 0:!~x:~0.6 and 0:!~y:!~1.0. The following five conditions must be met for your integration: For each variable, x and y, you must choose the most efficient method out of: Trapezoidal, "1/3" or "3/8" Simpson's rules. You must use the least possible number of points in each direction. Your numerical estimation must have zero error (exact solution). You must start by drawing a figure of the domain, show your points and their values in the figure, and state which Method(s) you are using. You must retain 4 significant figures in your calculations [10 marks] The Taylor series expansion can also be used to approximate derivatives for variable step sizes in order to decrease the number of steps in a solution. Use the Taylor series expansion to derive a second order accurate h ah central difference approximation off (x 1 ) for the variable step size i j4-1 case in which the step size is increasing by a factor of a> 1 as we move from pointx,.j toward x, 1 [Hint: when writing the Taylor series expansion forj(xc.i) andj(x 1 +i) neglect all the terms after the second derivative term and replace them with 0(h 3 ) or 0(a 3 0) where necessary] [22 points] A chunk of building material about the size of a basketball breaks off the side of a tall building and falls 40 meters downward to the ground. The governing ODE for the motion is derived from F=ma, i.e.: A[pv2J= ma Mg - C d weight -, drag force 3 of

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

where v(t) and a(t) are the velocity and acceleration in the downward direction, mass m=8kg, gravitational constant g=9.81 m/s 2 , drag coefficient cd=0.55, cross-sectional area A0.075 m 2 and air density p=1.18 kg/m 3 . (a) If y(t) is positive in the downwards direction, show that this 2nd order ODE can be converted into the following set of two 1st order ODEs dv cAll — =g — - --- I —pv' di' m2 (b) Assuming an initial height y(0)=O and initial velocity v(0)=O, tabulate the distance fallen and the speed of the falling chunk, using a constant time step bt=0.5 sec up to a time t=1 sec with the following Methods (retaining 4 significant figures in your calculations): Euler's Method Heun's Method, without iteration [Hint: at each step, solve both predictor equations first, then use those results to solve both corrector equations.] (c) For Heun's Method, given the choice between halving the timestep size and keeping the timestep size the same but using an extra iteration, which would you choose, and why? I 4 of 4