1. Mechanical Vibrations
Dr. Adnan Dawood Mohammed
(Professor of Mechanical Engineering)

2. Free vibration of single degree of freedom system
vibrations .
Introduction 1

3. Free vibration of single degree of freedom system
vibrations .
Examples

4. Free vibration of single degree of freedom system
vibrations .
Examples

5. Free vibration of single degree of freedom system
vibrations .
Equation of motion
Equation of motion
Newton's 2nd law of motion
Other methods
D’Alembert’s principle
Virtual displacement
Energy conservation

6. Free vibration of single degree of freedom system
vibrations .
Newton's 2nd law of motion
Select a suitable coordinate
Determine the static equilibrium configuration of the system
Draw the free-body diagram
Apply Newton s second law of motion
The rate of change of momentum of a mass is equal to the force acting on it.

7. Free vibration of single degree of freedom system
vibrations .
Newton's 2nd law of motion
Energy conservation
Potential energy
Kinetic energy

8. Free vibration of single degree of freedom system
vibrations .
Vertical system

9. Free vibration of single degree of freedom system
vibrations .
Solution to the equation of motion:

10. Free vibration of single degree of freedom system
vibrations .

11. Free vibration of single degree of freedom system
vibrations .

12. Free vibration of single degree of freedom system
vibrations .

13. Free vibration of single degree of freedom system
vibrations .
Graphical representation
x(t ) = A cos(ωnt – ϕ )

14. Free vibration of single degree of freedom system
vibrations .
Torsional vibration

15. Free vibration of single degree of freedom system
vibrations .
Natural Frequency (ωn) :
It is a system property. It depends, mainly, on the stiffness and the mass of vibrating system.
It has the units rad./sec, or cycles/sec. (Hz)
It is related to the natural period of oscillation (τn) such that, τn = 2π/ωn and ωn = 2 π fn where fn is the natural frequency in Hz.

16. Free vibration of single degree of freedom system
vibrations .
Example 2.1
The column of the water tank shown in Fig. is 90m high and is made of reinforced concrete with a tubular cross section of inner diameter 2.4m and outer diameter 3m. The tank mass equal 3 x 105 kg when filled with water. By neglecting the mass of the column and assuming the Young’s modulus of reinforced concrete as 30 Gpa. determine the following:
the natural frequency and the natural period of transverse vibration of the water tank
the vibration response of the water tank due to an initial transverse displacement of tank of 0.3 m and zero initial velocity.
the maximum values of the velocity and acceleration experienced by the tank.

17. Free vibration of single degree of freedom system
vibrations .
Example 2.1 solution:
Initial assumptions:
the water tank is a point mass
the column has a uniform cross section
the mass of the column is negligible

18. Free vibration of single degree of freedom system
vibrations .
Example 2.1 solution: a. Calculation of natural frequency: 1. Stiffness: , But: So: 2. Natural frequency :

19. Free vibration of single degree of freedom system
vibrations .
Example 2.1 solution: b. Finding the response: 1. x(t ) = A cos (ωn t - ϕ ) So, x(t ) = 0.3 cos ( t )

20. Free vibration of single degree of freedom system
vibrations .
Example 2.1 solution: c. Finding the max. velocity: Finding the max. acceleration :

21. Free vibration of single degree of freedom system
vibrations .
Example 2.2 :

22. Free vibration of single degree of freedom system
Example 2.2 : solution

23. Free vibration of single degree of freedom system
vibrations .
Simple pendulum
Governing equation:
Assume θ is very small
Natural frequency (ωn)

24. Free vibration of single degree of freedom system
vibrations .
Solution

25. Free vibration of single degree of freedom system
vibrations .
Example 2.3
Any rigid body pivoted at a point other than its center of mass will oscillate about the pivot point under its own gravitational force. Such a system is known as a compound pendulum (as shown). Find the natural frequency of such a system.

26. Free vibration of single degree of freedom system
vibrations .
Solution
the governing equation is found as:
Assume small angle of vibration:
So:

27. Free vibration of single degree of freedom system
vibrations .
Example 2.5: Q2.45
Draw the free-body diagram and derive the equation of motion using Newton s second law of motion for each of the systems shown in Fig