THE UNIVERSITY OF AUCKLAND SEMESTER ONE 2019 Campus: City ENGINEERING SCIENCE Mathematical Modelling 3ES (Time Allowed: TWO hours) ENGSCI 314

NOTE: • Section A consists of 1 Question. Section A is worth a total of 10 marks • Section B consists of 2 Questions. Answer ALL questions. Section B is worth a total of 30 marks • Section C consists of 3 Questions. Answer ALL questions. Section C is worth a total of 40 marks • The examination is out of 80 marks. • Answers for Sections A and B should be written on the ANSWER BOOKLET. • Answers for Section C should be written on the special ANSWER SHEETS provided on pages 7-12 of this exam. A separate Appendices Booklet contains DATA ANALYSES Page 1 of 12 SECTION A Ordinary Differential Equations Answer question 1 from section A Question 1 (10 marks) Find all equilibrium points and use linearisation to determine the type of each equilibrium point (asymptotically stable node, unstable node, unstable saddle, neutrally stable centre, asymptotically stable spiral or unstable spiral) for the following nonlinear system: ���� �� ������ �� 1�� ���� ��4��������1�� Reminder: • Answers for Sections A and B should be written on the ANSWER BOOKLET. • Answers for Section C should be written on the special ANSWER SHEETS provided on pages 7-12 of this exam. Page 2 of 12 ENGSCI 314 SECTION B Answer ALL questions from section B Partial Differential Equations Question 2 (12 marks) (a) What are the basic steps for PDE problem solution? Write them down and be brief. (3 marks) (b) Thetransversevibrationofalongandslender“slinky”springgivenaverysmallamplitude shake from one side can be described using the following equation: ��������, ���� �� ������ ��4������ ������������ ��4�������� �� ���������� ��4���������� ������ (i) Sketch this mode of vibration. (1 mark) (ii) If this spring has mass 200 g and is stretched to an overall length of 5 m, what will its linear density be? (2 marks) (c) The conduction of heat through a wall of thickness 1m is governed by the one dimensional heat equation: ���� �� ���� ������ ���� ������ If the wall has fixed temperature boundary conditions (with zero degrees at each side of the wall) the solution to the heat equation will be: �� ������, ���� �� �� ���� ���������������������������������� ������ Briefly show how we find the solution describing temperatures in the wall for the following initial conditions and write this solution down: ������, 0�� �� 100∘��������2������ (6 marks) (Hint: remember that ���� �� �� ���� ������, 0��sin�������������� ) Page 3 of 12 ������ ENGSCI 314 Question 3 (18 marks) (a) A wall of thickness 1 m has the following boundary conditions: ����0,���� �� 50°; ������1,���� �� 0 (i) (ii) ���� Derive an expression for the steady-state temperature distribution us(x) across the wall. (3 marks) Assume that the wall has an initial temperature distribution of u(x,0)=100oC. Calculate an expression for the initial conditions affecting the transient solution U(x,0). (2 marks) (b) We derived Laplace’s equation for 2D heat flow (in x and y directions only) from the 3 dimensional heat equation: This was done in 2 steps. Very briefly write down the 2 steps and show how we got Laplace’s equation in two dimensions. (2 marks) (c) We have a square cross-section bar for which the boundary conditions are defined by f(x), g(x), h(y) and l(y), the boundary temperatures along the bottom, top, left and right faces respectively. We wish to know the temperatures within. How can we reconstitute this problem so that we can solve it using separation of variables? Be brief; you can use sketches to illustrate your answer. (2 marks) (d) The kinetic energy T and potential energy V equations for a harmonic oscillator of mass m attached to a spring with stiffness K can be written: �� �� ���� �� �� �� �� , �� �� ���� �� �� �� (i) Write down the Lagrangian for this. (2 marks) (ii) Introduce terms to the Euler Lagrange equation* and find the equation of motion. (2 marks) (iii) Show how we can get an expression for the natural frequency of the mass/spring system and calculate the natural frequency for m= 1kg and K= 2 N/m. (4 marks) * The Euler-Lagrange equation is ���� �� �� �� ���� �� �� 0 (e) For the calculus of variations, what is meant by the term functional? Be brief. (1 mark) ENGSCI 314 ���� ���� ������ Page 4 of 12 SECTION C Answer ALL questions in Section C. Answers for Section C should be written on the special ANSWER SHEETS provided. Statistics This question refers to the Alzheimer's Data in APPENDIX A of the Appendices booklet. (a) Write 2-3 comments on what can be seen from the four plots of the data at the start of the analysis (on pages 3 and 4 of the appendices). (2 marks) (b) Writedownthestraight-lineequationforthefittedlineoflog(Ventricles)asaresponseto Age for healthy people (group H). Use numbers rounded to 3 decimal places. (4 marks) (c) Looking at the output for summary(Ventriclesfit1), say what the coefficient of Age:ADH is estimating, in terms of slopes and/or intercepts. (2 marks) (d) For each of the following, either write a sentence interpreting a confidence interval to estimate the requested information, or state why we cannot quantify the requested item using the R output provided. (i) The effect on the size of ventricles for each additional years aging on healthy people. (ii) The effect on the size of ventricles for each additional years aging on people exhibiting dementia symptoms. (iii) In general, regardless of age, the difference in size of ventricles between healthy people and those exhibiting dementia symptoms. (6 marks) (e) Look at the final plot with fitted lines added (on page 9 of the appendices). In 1-2 sentences of plain English, describe what overall conclusion this model seems to be leading to. (2 marks) Question 4 (16 marks) Page 5 of 12 ENGSCI 314 Question 5 (19 marks) This question refers to the Chemical Yields Data in APPENDIX B of the Appendices booklet. (a) Comment on what the interaction plot of the data reveals. (3 marks) (b) Write brief METHODS AND ASSUMPTION CHECKS notes of the analysis of the Chemical Yields Data in APPENDIX B of the APPENDICES BOOKLET. Remember to include the equation of the best model in the analysis. You also need to justify your choice of your final model. (6 marks) (c) Write a brief Executive Summary of the main conclusions you would draw from the analysis of the Chemical Yields Data in APPENDIX B of the APPENDICES BOOKLET. Note: remember to address the questions the Engineer has asked. (10 marks) Question 6 (5 marks) For each of the following statements, say whether or not they are true or false. If they are false, either say why they are false or correct the statement. Statement 1: The P-value is the probability of getting sample data at least as unusual as that observed given that the null hypothesis is true. Statement 2: A t-test for H0: β1 = 0 in the model Y = β0 + β1×X1 will always have a similar level of significance to a t-test for H0: β1 = 0 in the model Y = β0 + β1×X1 + β2×X2. Statement 3: The profiles of an interaction plot need to cross over to indicate interaction between the two factors. Statement 4: Transforming an explanatory variable in regression will only affect the scatter about the trend. Statement 5: When we fit a factor to a Regression Model, we fit dummy variables for all levels of the factor, except one. Answer Booklet for Section C Follows Page 6 of 12 ENGSCI 314 (5 marks) ANSWER BOOKLET: SECTION C ENGSCI 314 Final Examination, First Semester, 2019 Surname: _________________ Forenames: ___________________ ID Number____________ Section C Page 7 of 12 ANSWER BOOKLET: SECTION C ENGSCI 314 Surname: _________________ Forenames: ___________________ ID Number____________ Section C (Continued) Page 8 of 12 ANSWER BOOKLET: SECTION C ENGSCI 314 Surname: _________________ Forenames: ___________________ ID Number____________ Section C (Continued) Page 9 of 12 ANSWER BOOKLET: SECTION C ENGSCI 314 Surname: _________________ Forenames: ___________________ ID Number____________ Section C (Continued) Page 10 of 12 ANSWER BOOKLET: SECTION C ENGSCI 314 Surname: _________________ Forenames: ___________________ ID Number____________ Section C (Continued) Page 11 of 12 ANSWER BOOKLET: SECTION C ENGSCI 314 Surname: _________________ Forenames: ___________________ ID Number____________ Section C (Continued) Page 12 of 12