UNIVERSITI TEKNOLOGI MALAYSIA
FACULTY OF MECHANICAL ENGINEERING
SKMM 3023 Applied Numerical Methods
Tutorial 3
Solution of Simultaneous Linear Equations
1. Show a Matlab session to determine the tension in each cable used to support the
(a) 40-lb crate shown in Figure 1 (a), and
(b) 100-kg crate shown in Figure 1 (b).
(a)
(b)
Figure 1: Crate supported by cables.
2. A steel cylinder, of weight W = 500 lb and diameter 1 ft, is placed in a steel V-block as shown
in Figure 2. When a moment M is applied about the axis of the cylinder, the cylinder starts
to rotate. The force equilibrium equations along the x- and y-directions and the moment
equilibrium equation about the axis of the cylinder, respectively, are given by the following
system of simultaneous linear equations
( R1 + µR1 ) − R2 + µR2 ) sin 45◦ = 0
( R1 − µR1 + R2 + µR2 ) cos 45◦ = W
6(µR1 + µR2 ) − M = 0
(E1)
(E2)
(E3)
where R1 and R2 are the normal reactions at the points of contact 1 and 2, respectively, and µ
is the coefficient of friction.
Figure 2: Cylinder in a V-block.
Code the Gauss elimination method in Matlab, C or Fortran to find the magnitude of the
moment M and the reactions at the points of contact between the cylinder and the V-block
(i.e. R1 and R2 ). Use Matlab to code a comparison. Assume that the coefficient of friction
between the cylinder and the V-block is µ = 0.25.
3. The Euler rotations φ, θ and ψ for the coordinate transformation to ( x, y, z)-system from
( X, Y, Z )-system can be expressed as

 




1
0
0
cos θ sin θ 0
1
0
0  X 
 x 
y
= 0
cos φ sin φ   − sin θ cos θ 0   0
cos ψ sin ψ 
Y




z
0 − sin φ cos φ
0
0 1
0 − sin ψ cos ψ
Z
(a) Without resorting to any programming tool, use the above relation to find the values of
X, Y and Z when x = 1, y = 2 and z = 3 with φ = 30◦ , θ = 20◦ and ψ = 10◦ .
Hint: You need to find an inverse of a matrix.
(b) Write a simple Matlab script to solve the same.
4. The nodal displacement of the crane shown in Figure 3 can be found by solving the equilibrium equations
 



u1 
P1 
170.4105
37.9473 −113.8420 −37.9473 






 


u2
P2
37.9473
69.2176 −37.9473 −12.6491 


=
 −113.8420 −37.9473
u  
P 
120.9131
45.0184  

 3 
 
 3 

u4
P4
−37.9473 −12.6491
45.0184
19.7202
where ui , i = 1, 2, 3, 4, are the components of nodal displacement (inch) and Pi is the load
applied (lb) along the direction ui , i = 1, 2, 3, 4.
Code the Gauss-Siedel iteration method in Matlab, C or Fortran to find the nodal displacements when the load lifted is given by 1000 lb (i.e. P1 = P2 = P3 = 0, P4 = −1000). Compare
your results with Matlab output.
Figure 3: Crane.