MCD4140 Computing for Engineers Self Study Exercise 10
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MCD4140 Computing for Engineers
Self Study Exercise 10
Note: Tasks below can use for both hand calculation practice and programming practice.
Note: You might want to use extra sheet for hand calculation practice.
Task 1
Write the following set of equations in matrix form:
50 = 5x3 − 7x2
4x2 + 7x3 + 30 = 0
x1 − 7x3 = 40 − 3x2 + 5x1
Use MATLAB to solve for the unknowns. In addition, use it to compute the transpose and the inverse of the coefficient matrix.
Task 2
Considered a simplified electrical circuit and showed how we can apply Kirchoff’s law to derive a set of linear algebraic equations describing the current through each segment. Apply this methodology to the circuit below and write out the set of linear algebraic equations. Arrange these equations in matrix form.
Task 3
Consider now the slightly more complicated electrical circuit below. Apply the same methodology as for tutorial question 4 and write out the set of linear algebraic equations. Arrange these equations in matrix form.
Task 4
Figure above shows three reactors linked by pipes. The rate of transfer of chemicals through each pipe is equal to a flow rate (Q, m3/s) multiplied by the concentration of the reactor from which the flow originates (c, mg/m3).
∑ Qxi = ∑ Qxi
一軒一 一軒一
input mass output mass
If the system is at a steady state, the transfer into each reactor will balance the transfer out. Develop mass-balance equations for the reactors and represents the equations in matrix form, Ax = b.
Task 5
Continue from task 4, solve for the concentrations of each reactor using Naïve Gauss Elimination. Show all steps of the computation. Substitute your results into the original equations to check your answers.
Task 6
Given the equations
「0
8
3
3
L3
Solve by Gauss elimination with partial pivoting. Show all steps of the
computation. Substitute your results into the original equations to check your answers.
Task 7
A builder goes to work very early but he encountered a problem. He forgot the five digit access code, [n1 n2 n3 n4 n5], to his high-tech workshop. However, he remembered the five clues to get the access code. Hereunder are the clues:
1. The 5th number plus the 3rd number of the access code equals 14.
2. The 4th number is one more than the 2nd number.
3. The 1st number is one more than twice the 2nd number.
4. The 2nd plus the 3rd number equals 10.
5. The total of all the five numbers is 32.
(a) Derive a set of equations to compute n 1, n2, n3, n4, and n5. (b) Write this set of equations in matrix form.
(c) Use Gauss Elimination to solve this problem. Examine the numerical values for the coefficient matrix. Consider whether you need to include the capability of partial pivoting in your code. Write a MATLAB to solve this problem using GaussNaive() or GaussPivot() (whichever appropriate). Explain your choice.
Task 8
The torque equations of a robot manipulator are given below:
Torque for Joint 1: τ1 = ( − )fz − 2fz − 2fz
Torque for Joint 2: τ2 = fy + fx + 3fy
Torque for Joint 3: τ3 = fy
Use MATLAB command window only to solve this problem. Remember to
express the set of equations above in the matrix form, [A]{f } = {τ} , where A is the matrix of coefficient, τ = [τ1 τ2 τ3 ]T , f = fx fy fz T .
(a) What is the torques experience by each joint, when the load on the end effector is f = [20N 10N 60N]T ?
(b) Given the maximum allowable torque for each joint are
τ = [60Nm 50Nm 75Nm]T , solve for the maximum allowable force, f .
Task 9
Consider the triple mass-spring system shown in the figure below. Determining the equations of motion from Newton’s second law for each mass using its free-body diagram results in the following differential equations:
x..1 + | 1 2 |x1 − | 2 | x2 = 0
( m1 ) (m1 )
( k ) ( k + k ) ( k )
(m2 ) ( m2 ) (m2 )
( k ) ( k + k )
(m3 ) ( m3 )
where k1 = k2 = k3 = k4 = 40 N/m, and m1 = m2 = m3 = m4 = 1 kg. The three equations can be written in matrix form:
0 = {Acceleration Vector} + [k/m matrix] [dispalcement vector x]
At a specific time when x1 = 0.05m, x2 = 0.04m, and x3 = 0.03m, this forms a tri-diagonal matrix. Use MATLAB to solve for the acceleration of each mass.
Task 10
Consider the truss shown below. In civil engineering it is very important to understand well the forces in such a static truss. The sum of the forces in both the horizontal and vertical directions must be zero at each node, for the system to be in static equilibrium.
For example,
∑ FH = 0 = − F1 cos (30。) + F3 cos(60。) + F1,h
∑ FV = 0 = − F1 sin (30。) + F3 sin(60。) + F1,v
∑ FH = 0 = F2 + F1 cos (30。) + F2,h + H2
∑ FV = 0 = F1 sin (30。) + F2,v + V2
∑ FH = 0 = − F2 + F3 cos (60。) + F3,h
∑ FV = 0 = F3 sin (60。) + F3,v + V3
where Fi,h and Fi,v are the external horizontal and vertical forces applied at node i. Positive sign convention is left to right for horizontal forces and upwards for vertical forces. In the above example, F1,v = -1000lb. All other Fi,h and Fi,v = 0. Express this set of linear algebraic equations in matrix form. Use MATLAB command window to solve for the unknown forces.
2023-01-13