SUBJECT: NUMERICAL METHODS CODE: BUM2313 FACULTY OF INDUSTRIAL SCIENCES & TECHNOLOGY INSTRUCTION: Use MATHEMATICAL SOFTWARE such as EXCEL/ MATLAB/ MAPLE/ C to facilitate the computation. SUBMIT the solution in HARDCOPY & SOFTCOPY. Please save the solution in CD for softcopy. Do the assignment in group as allow by your lecturer. QUESTION 1 The nonlinear resistive circuit shown below is described by the nonlinear equation f ( x)  g ( x)  ( E  x) 0 R

TOPIC: CHAPTER 1, 2,3 & 4 DUE/DURATION:

MARKS:

ASSESSMENT: ASSIGNMENT

2nd May 2014 (before 5 P.M) WEEK 11

100

The function g ( x) gives the current through the nonlinear resistor as a function of the voltage x cross its terminals as shown in the following Figure 1.

Figure 1 Assuming that g ( x)  9sin( x  5)  10

and consider the three following cases:

Case 1: E  5, R  1, Case 2: E  15, R  3, Case 3: E  4, R  0.5. (a) (b) (c) By using an appropriate method that you have learned in this course, find all the solutions of the nonlinear resistive circuit equation for the all cases. Select suitable starting points for xl and xu by plotting f over the interval [0,4] for the all cases, and visually selecting a good starting point. Find the lowest root over the the interval [0,4] by using (i) Bisection method and (ii) False position method. Use the starting points xl and xu in (b) and terminate the computation if  a  104. (For (a) and (b) use two decimal places, for (c) use eight decimal places) (20 Marks)

QUESTION 2 In the design of all – terrain vehicles (ATV), it is necessary to consider the failure of the vehicle when attempting to negotiate two types of obstacles. One type of failure is called hang-up failure and occurs when the vehicle attempts to cross on obstacle that causes the bottom of the vehicle to touch the ground. The other type of failure is called nose-in failure and occurs when the vehicle descends into a ditch and its nose touches the ground. Given that the maximum angle  that can be negotiated by a vehicle when  is the maximum angle at which hang-up failure does not occur satisfies the equation

Asin cos   Bsin2  Ccos   Esin  0 where A  l sin  , B  l cos  , C  (h  0.5D)sin   0.5D cot  and E  (h  0.5D) cos   0.5D cot 

Find  for the situation when l  90, h  50, D  80 and  

. Use Newton Raphson 2 method with 0  0.5 to approximate the solutions until  a   s  0.5%. (Use seven decimal places) (15 Marks)



QUESTION 3 Three-phase loads are common in AC systems. When the system is balanced the analysis can be simplified to a single equivalent circuit model. However, when it is unbalanced the only practical solution involves the solution of simultaneous linear equations. In one model the following equations need to be solved.

120  0.0080( I bi  I ci )  0.4516 I ai  0.7460 I a r  0.0100( I br  I cr )  0 0.0100( I bi  I ci )  0.0080( I br  I cr )  0.4516 I a r  0.7460 I ai  0 60.00  0.7787 I br  0.5205I bi  0.0100( I a r  I c r )  0.0080( I ai  I ci )  0 0.5205I br  0.7787 I bi  0.0100( I ai  I ci )  0.0080( I a r  I c r )  103.9  0 0.6040 I ci  0.0080( I c r  I ai  I b r )  0.0100( I ar  I br )  60.00 0.0100( I ai  I bi )  0.0080( I a r  I b r  I ci )  0.6040  103.9

Find the values of I ar , I ai , I br , I bi , I cr , and I ci using Cramer’s rule method. Use SOFTWARE to solve the above problem. (Use four decimal places) (15 Marks)

QUESTION 4

Trusses are lightweight structures capable of carrying heavy loads. In bridge design, the individual members of the truss are connected with rotatable pin joints that permit forces to be transferred from one member of the truss to another. The accompanying figure shows a truss that is held stationary at the lower left endpoints 1, is permitted to move horizontally at the lower right endpoint 4, and has pin joints at 1,2,3,and 4. A load of 10,000 newtons(N) is placed at joint 3, and the resulting forces on the joints are given by f1 , f2 , f3 , f4 and f5 , as shown. When positive, these forces indicate tension on the truss elements, and when negative, compression. The stationary support member could have both a horizontal force component F1 and a vertical force component F2 , but the movable support member has only a vertical force component F3 . If the truss is in static equilibrium, the forces at each joint must add to the zero vector, so the sum of the horizontal and vertical components at each joint must be zero. This will produces the system of linear equations as shown in the accompanying table.
Joint 1 2 3 4 Horizontal Component Vertical Component

2 f1  f 2  0 2 2 3  f1  f4  0 2 2  f 2  f5  0  F1   3 f 4  f5  0 2

2 f1  F2  0 2 2 1  f1  f3  f 4  0 2 2 f3 10000  0

(a)

1 f 4  F3  0 2 Rewrite the above system of linear equations in matric form AX  b

(b)

Solve the system in (a) using Crout decomposition methods (use four decimal places) (15 Marks)

QUESTION 5 The normal distribution function with   0 and standard deviation  is given as follows

 1  x 2  1 f ( x)  exp       2    2  
(a) By using normal distribution function calculate eight equally spaced points in the intervals (i) [  2 ,   2 ] and (ii) [  3 ,   3 ] for   3 . (b) Find the Lagrange interpolation polynomial that curve fits the data for each interval. Output the coefficients of the polynomial. (c) Illustrate the individual points and Lagrange polynomial function in (a) and (b) on a single plot. (d) By using normal distribution function calculate fourteen equally spaced points in the intervals (i) [  2 ,   2 ] and (ii) [  3 ,   3 ] for   3 . (e) Find the Lagrange interpolation polynomial that curve fits the data for each interval. Output the coefficients of the polynomial. (f) Illustrate the individual points and Lagrange polynomial function in (d) and (e) on a single plot. (g) Fit the data obtained in (a) and (d) by using quadratic spline for the intervals (i)
[  2 ,   2 ] and (ii) [  3 ,   3 ] . Let   3 .

(h)

Illustrate the individual points and quadratic spline interpolation for the data in (a) and (c) on a single plot.

(Use eight decimal places) (35 Marks)