SME3701 Assignment 01 Due May 2026 |Solid Mechanics IV|

UNIVERSITY OF SOUTH AFRICA
College of Science, Engineering and Technology


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SME3701: Solid Mechanics IV

Assignment 01 — Semester 1, 2026

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SME3701
Module Code:
Solid Mechanics IV
Module Name:
01
Assignment Number:
22
Total Marks:




Submitted in partial fulfilment of the requirements for Solid Mechanics IV – UNISA 2026

,UNISA | SME3701 Solid Mechanics IV – Assignment 01



Question 1: Undamped Free Vibration of a Mass–Spring System [22 marks]


Consider a single-degree-of-freedom (SDOF) mass–spring system undergoing undamped free
vibration, as illustrated in Figure 1. The system consists of a mass of 5 kg attached to a spring
with stiffness 2000 N/m. The system is initially displaced by 0.02 m and released from rest
with zero initial velocity. The motion of the system is described by the equation:



x(t) = x0 cos(ωn t)


where ωn is the natural angular frequency.




k




m x(t)

m = 5 kg, k = 2000 N/m, x0 = 0.02 m


Figure 1: Undamped Free Vibration of a Mass–Spring System.




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, UNISA | SME3701 Solid Mechanics IV – Assignment 01



1.1 Natural Angular Frequency ωn [3]


Question 1.1: Determine the natural angular frequency ωn of the system using MATLAB.


Step-by-step solution:

Step 1: Recall the formula for natural angular frequency.

For an undamped SDOF mass–spring system, the natural angular frequency is defined as
(Rao, 2018:45):

r
k
ωn = (1)
m

where:

• k = spring stiffness [N/m]
• m = mass [kg]

Step 2: Substitute the given values.

Given: k = 2000 N/m, m = 5 kg.


2000 √
r
ωn = = 400 = 20 rad/s (2)
5

Therefore, the natural angular frequency of the system is:



ωn = 20 rad/s (3)


Step 3: MATLAB implementation.

1 % Define system parameters
2 m = 5; % Mass [ kg ]
3 k = 2000; % Spring stiffness [ N / m ]
4

5 % Calculate natural angular frequency
6 omega_n = sqrt ( k / m ) ;
7


8 % Display result


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, UNISA | SME3701 Solid Mechanics IV – Assignment 01



9 fprintf ( ’ Natural angular frequency : omega_n = %.4 f rad / s \ n ’ , omega_n ) ;

Listing 1: MATLAB – Natural Angular Frequency


MATLAB output:


Natural angular frequency: omega_n = 20.0000 rad/s

Implementation Insight
The natural angular frequency ωn = 20 rad/s implies a natural frequency of fn =
ωn /(2π) ≈ 3.18 Hz and a natural period of Tn = 1/fn ≈ 0.314 s. Implying that the
mass completes one full oscillation every 0.314 seconds.




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, UNISA | SME3701 Solid Mechanics IV – Assignment 01



1.2 MATLAB Script for Displacement x(t) [2]


Question 1.2 (restated): Develop a MATLAB script to compute the displacement x(t) of
the system.


Step-by-step solution:

Step 1: State the displacement equation.

For an undamped SDOF system released from rest with initial displacement x0 , the equation
of motion yields the solution (Inman, 2014:22):



x(t) = x0 cos(ωn t) (4)


where:

• x0 = 0.02 m (initial displacement)
• ωn = 20 rad/s (natural angular frequency)
• t = time [s]

Step 2: MATLAB script.

1 % System parameters
2 m = 5; % Mass [ kg ]
3 k = 2000; % Spring stiffness [ N / m ]
4 x0 = 0.02; % Initial displacement [ m ]
5 v0 = 0; % Initial velocity [ m / s ]
6


7 % Natural angular frequency
8 omega_n = sqrt ( k / m ) ; % omega_n = 20 rad / s
9

10 % Time vector
11 t = linspace (0 , 5 , 1000) ; % 0 to 5 seconds , 1000 points
12


13 % Displacement equation : x ( t ) = x0 * cos ( omega_n * t )
14 x = x0 .* cos ( omega_n .* t ) ;
15


16 % Display first five values for verification
17 disp ( ’ Time [ s ] Displacement [ m ] ’) ;


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