ENG3104 - Engineering Simulations And Computations Assignment

Assignment Task

1 Introduction

Consider a planar truss schematically outlined in Figure 1. The x and y coordinates of points A and C, measured in metres, are (0,0) and (0,2), respectively. In the design, point B can be located at any grid point (black square) on a uniform grid defined over the region 0.2 m ≤ x ≤ 2.4 m, −0.2 m ≤ y ≤ 2.2 m with intervals ∆x = ∆y = 0.2 m. A force W in the vertical direction is applied at point B.

Since a pinned support allows the structural member to rotate, but not to translate in any direction, one can replace it with two forces in the x and y directions (forces from the ground to the structure). Figure 2 shows the applied force W~ and the reaction forces: F~A = (FAx , FAy ) and F~C = (FCx , FCy ).

Assume that the masses of member AB and member BC are negligible and the two pinned supports at A and C have the same strength. Your tasks are to find the reaction forces at A and C for different locations of point B on the grid, and a grid point at which the applied force limit is maximum.

Requirements

For this assessment item, you must perform hand calculations:

1. Express the system (3) in matrix form.

2. Solve the system for point B: (2.4, 0.8) m. Report the force magnitudes FA and FC rounded to three significant figures. You must also produce MATLAB code which:

3. Repeats the hand calculations. Verify the answers by using the reported results from Requirement 2.

4. Numerically solves (3) for each grid point B on the grid. Find the reaction force magnitudes and verify the answers by using the results from Requirement 3.

5. Plots contours of the force magnitude FA over the grid (FA is a function of (xB, yB)).

6. Plots contours of the force magnitude FC over the grid (FC is a function of (xB, yB)).

7. For the location of point B in Requirement 2, numerically determines how large the applied force W can be if the reaction force magnitudes at A and C cannot exceed 2000 N. The value of W found here is called the applied force limit.

8. Repeats the calculation of Requirement 7 for the other locations of point B. Report a grid point at which the applied force limit is maximum.

9. Has appropriate comments throughout your Matlab program. 

2  Introduction

File contains a set of data points (xi , yi), i = (1, 2, · · · , N), where N is the number of data points, x (in meters) is the independent variable and y (in meters) is the dependent variable. Due to measurement error, the values of y have been contaminated. You will need to find some analytical functions representing the underlying behaviour of the variable y by means of linear regression discussed in Module 8, ENG3104 Study Guide. The analytical functions to be considered here are linear, exponential and power functions. For the data set considered in this question (Figure 3), all y values are positive but there are some negative x values. In finding the coefficients of power function, it requires x > 0 and one can try to shift the y-axis to the left and then works on a new coordinate system, i.e. O¯x¯y¯. You will also need to predict the function at the following x value (this value is not to be shared with anyone else):

2.1 Requirements

For this assessment item, you must perform hand calculations:

1. Find the coefficients of the three standard curve-fitting functions: linear, exponential and power using the following two data points: the kth and (k + 1)th rows of the numeric data part (i.e. the kth and (k + 1)th data values), where k is obtained by using Matlab function to round the following value to the nearest integer:

[20 + 2.446]

(this value is not to be shared with anyone else). Report the answers rounded to three significant figures. The purpose of this requirement is to help you get used to regression analysis in this question. You must also produce MATLAB code which:

2. Repeats the hand calculations, where the input data are entered into your Matlab program by typing/copying relevant values from the data file. Verify the answers by using the reported results from Requirement 1.

3. Loads the data file into MATLAB. Verifies that the load has been successful by comparing the obtained first and last rows of the numeric data part with those from copy-and-paste.

4. Finds the coefficients of linear and exponential functions, where all data points are used.

5. Finds the coefficients of power function, where only data points with x > 0 are used.

6. Finds the value of x ∗ in the range of -2 to -1 for which the power function defined in the coordinate system O¯x¯y¯ has a maximum r-squared value. The r-squared values used in a comparison here must be computed using all data points. Plot the r-squared as a function of x^∗ 

7. From the curve-fits found from Requirements 4 and 6, determines which one is the best. Demonstrate it graphically and numerically. Use the best curve fit to estimate the function at x^† 

8. Has appropriate comments throughout your Matlab program.

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