Question Mat Lab Q You have been supplied with a set of measurements for the lifetime of a bearing in the file ass3q1data.csv. You should use this data to construct a model for the behaviour of the real lifetime

(the lifetime of the population). It has been well-established that the Weibull distribution is the best model for the reliability of objects (Juvinall & Marshek 2011).

For quality control and defining warranties, models for population lifetimes are used to determine how many products will fail within a stipulated period. Using the model for the population, determine which samples are in the bottom 2% of the population. Also determine the fraction of samples which do not last the desired minimum of 950 hours and which fraction of the population would not last 950 hours.

Requirements For this assessment item, you must produce MATLAB code which: 1. Loads the data file ass3q1data.csv and verifies that it has been loaded correctly (first 5 values)

2. Determines the estimated parameters for the Weibull distribution and reports the values to the Command Window. 3. Compares the sample pdf with the population pdf graphically. 4. Compares the mean and standard deviation computed from the samples with the mean and

standard deviation computed from the Weibull distribution's parameters. Discuss why the results are the same or different. This comparison and discussion is to be reported to the Command Window.

5. Verifies the mean and standard deviation calculations by comparing the results for the first 5 values with hand calculations. Be careful how you calculate the standard deviation.

6. Computes the fraction of the samples that are in the bottom 2% of the population and reports

the result to the Command Window. 7. Computes the fraction of the samples and fraction of the population that do not satisfy the minimum of 950 hours and reports the result to the Command Window.

8. Has appropriate comments throughout. Assessment Criteria Your code will be assessed using the following scheme. Note that you are marked based on how well you perform for each category, so the correct answer determined in a basic way will receive half

marks and the correct answer determined using an excellent method/code will receive full marks. Quality of the code 5 marks Quality of header(s) and comments 10 marks

Quality of the data loading and verification 10 marks Quality of the Weibull distribution parameter estimation 10 marks Quality of the pdf comparison 15 marks Quality of the statistics verification 10 marks

Quality of the bottom 2% calculation 20 marks Quality of minimum 950 hours calculation 20 marks Reference

Juvinall, RC & Marshek, KM 2011, Fundamentals of Machine Component Design, 5th edn, Wiley, USA.ENG3104 Engineering Simulations and Computations Semester 2, 2014 Page 2 of 5 Question 2 (150 marks)

Introduction a) It's time to do assignment 1 properly. To begin with, Eq. (1) in assignment 1 is incorrect; this

is the correct derivation:         0

0 0 0 0 0

v t t v t

t dv

a t dt dv adt dv adt

v t v adt

v t v adt     

     

(1) It is therefore necessary to add the initial condition to the integral (as defined previously). For this assignment, load the data in ass1data.csv and integrate it accurately to determine the

position as a function of time. Graphically compare the x-position and y-position as functions of time for the method used in assignment 1 and a better method. Also graphically compare

the heart-shapes that are produced. Compare the position at 6.28 s for both methods. b) Taking the acceleration formulae from Assignment 2 Question 3, solve the ODEs for xposition and y-position to determine what the position should be at 6.28 s using:

i. Euler's method ii. A MATLAB ode solver (programmed in MATLAB) iii. Simulink iv. Validation with the analytical solution

Verify your code for Part (b)(i) by performing hand calculations for the first 5 timesteps and comparing to the result from the code. Compare the four results from Part (b) with each other and the two results from Part (a).

c) Theoretically, the heart shape should be closed at 2 seconds. Prove that this is true analytically. Determine how closely the following numerical methods come to achieving this:

i. Euler's method (choose 2 different timesteps) ii. ode23 (report any changes to the settings from the default)

iii. ode45 (report any changes to the settings from the default)

iv. ode113 (report any changes to the settings from the default)

Compare the results by using these four methods programmed in MATLAB and also using

Simulink (you should have a total of 8 simulations: do not create a Simulink model for Euler's method)