Diagram adopted from Thomson WT, Dahleh MD 1993, 'Free Vibration' in Theory of Vibration with Applications, 5th edn, Prentice Hall, USA, p17
Vibration is the periodic motion of a body or system of connected bodies displaced from a position of equilibrium. Free vibration is maintained by gravitational or elastic restoring forces and undamped vibrations are indefinite motions of a body or system where frictional effects are neglected. Thus, a theoretical pendulum may be considered as a system experiencing undamped free vibration. However, in reality, just as with many other scientific theories, this is impossible because friction is present both internally and externally.
Figure below represents the simplest type of vibrating motion - an undamped spring-mass system which is assumed to move only along the vertical direction and to have 1 degree of freedom (DOF).
Diagram adopted from Thomson WT, Dahleh MD 1993, 'Free Vibration'
in Theory of Vibration with Applications, 5th edn, Prentice Hall, USA,
p17
When the system is in motion it will oscillate about the horizontal. The displacement from this equilibrium position is denoted by x and the deformation from an unstretched position is labelled with delta. The spring force, Δk is equal to the gravitational force acting downwards on the mass so we can write that as Δk = w = mg. The elastic restoring force F = kx also acts on the system and is always directed towards the equilibrium position. If we take down to be positive, then by Newton's second law we can sum the forces in following manner:
And since
Δk = w, this expression can further be reduced
to 
---(eq1). Additionally, we will introduce
a constant ωn
which is the circular frequency or more commonly known as the natural frequency
(expressed in rad/s) and is given by
Substituting ωn into eq1 we obtain a "standard form" of a simple harmonic motion:
---(eq2)
eq2 is a homogenous second-order linear differential equation and it can be mathematically proven that its general solution is:
x = A sin ωnt + B cos ωnt ---(eq3) [mathematical proof]
where A and B are the two constants of integration. By taking two successive time derivatives of f(x), we obtain velocity and acceleration, respectively:
If the above second derivative is substituted into eq2 together with eq3, it becomes clear that the differential equation is satisfied.
A and B are the constants of integration which we are trying to find. They are easily derived from initial conditions when we take time to be zero. That is:
Substituting these new values into eq3, we obtain:
Furthermore, eq3 can be written in the following, simplified form:
This is a convenient way to write the general solution because we can plot it on an x vs. ωnt axis to obtain the following graph
From the mathematical proof for the simplified version of eq3, we know the values of A and B, so we can then square and add these two equations to obtain the amplitude of displacement, C
And for the phase difference of displacement, φ, we get
The time taken for the sine curve to complete one cycle is 2π or in other words, the period is
And finally, the frequency, which is the reciprocal of the period is given by:
Example 1
Determine the period of vibration for the simple pendulum shown below. The bob has a mass m and is attached to a cord of length l. Neglect the size of the bob and mass of cable.
Solution
As the pendulum swings about the normal, the mass of the bob (neglecting the mass of the cable) will try to counterbalance it. This restoring force will be related to the angular displacement theta (and arc length s). If we only consider the forces acting on the bob (pictured on the right), we will notice there are two: one is the tension in the cord, acting through the cord towards the anchorage and the other is the weight of the bob acting straight down.
The tangential component of the tension force will include the restoring force in the following manner:
Notice how the period does not depend on the mass of the bob or its velocity, but rather just the length of the cord. This was first discovered by Galileo Galilei in 1638.
Example 2
For the following pulley system turning clockwise, write down the equation of motion and find its natural frequency. Neglect friction.
Solution
If the system is rotating clockwise, then mass 1 will be moving up and mass 2 down. Let's say up is positive and down is negative.
In order to write the equation of motion for the whole system, we'll need to investigate each component individually. First, let us consider masses 1 and 2 and write down their equations of motion:
T signifies the tension of the cable. Now, we write the equation for pulley:
Now we simply add the two equations together to get the required solution:
Finally, for the natural frequency, we need the total mass of the system which is everything in the parentheses preceding acceleration in our equation of motion.
[
