Mechanical Vibration - MDPN 471
Instructor
Dr. Mostafa Gamal Abdelmageed
T.A.
Eng. Mohamed Hassan
1
Course Content
• Introduction and Fundamental of Vibrations
• Free Vibration of Single-DOF Systems
• Forced Vibration of Single-DOF Systems
• Vibration-Measuring Instruments
• Undamped Free Vibration of Two-DOF Systems
• Forced Vibration of Two-DOF Systems
• Vibration Control
Free damped vibration
Vibration
Single degree Two degrees
of freedom of freedom
system system
Free Forced Free Forced
Undamped Damped Undamped damped Undamped Undamped
Free damped vibration
viscous damping force F is proportional to the
velocity or v and can be expressed as:
Examples are the oscillations of the pendulum in
viscus medium.
Free damped vibration
Free damped vibration
Assumed solution:
where C and s are constants to be determined.
Free damped vibration
These roots give two solutions:
Free damped vibration
The general solution is given by a combination of the
two solutions s1 and s2:
where and are arbitrary constants to be determined
from the initial conditions of the system.
Free damped vibration
Critical Damping Constant and the Damping
Ratio:
Damping ratio
Damping constant
Free damped vibration
The solution:
Considering that ζ ≠ 0, the following 3 cases exist:
Case 1. Underdamped system (ζ < 1)
Case 2. Critically damped system (ζ = 1)
Case 3. Overdamped system (ζ > 1)
Free damped vibration
Case 1. Underdamped system (ζ < 1)
For this condition, (ζ2 - 1) is negative and the roots s1
and s2 can be expressed as:
Response will be:
Free damped vibration
Case 1. Underdamped system (ζ < 1)
In another form:
For the initial conditions:
Free damped vibration
Case 1. Underdamped system (ζ < 1)
Constants C1’ and C2’ will be:
Final form:
Free damped vibration
Case 1. Underdamped system (ζ < 1)
Damped harmonic motion of angular frequency:
Free damped vibration
Case 1. Underdamped system (ζ < 1)
Damped frequency is always less than the undamped natural
frequency.
It is the only case that leads to an oscillatory motion.
Free damped vibration
Case 1. Underdamped system (ζ < 1)
Free damped vibration
Logarithmic decrement:
It represents the rate at which the amplitude of a
free-damped vibration decreases.
It is defined as the natural logarithm of the ratio of
any two successive amplitudes.
Free damped vibration
Logarithmic decrement:
Free damped vibration
Case 2. Critically damped system (ζ = 1)
For this condition, (ζ2 - 1) is zero and the roots s1 and
s2 can be expressed as:
Response will be:
Free damped vibration
Case 2. Critically damped system (ζ = 1)
The application of the initial conditions:
Response will be:
Free damped vibration
Case 3. Overdamped system (ζ > 1)
For this condition, (ζ2 - 1) is positive and the roots s1
and s2 can be expressed as:
Response will be:
Free damped vibration
Case 3. Overdamped system (ζ > 1)
The application of the initial conditions:
Sub. With C1 and C2 to get response:
Free damped vibration
Example 1
The anvil of a forging hammer
weighs 5,000 N and is mounted on a
foundation that has a stiffness of
5x10e6 N/m and a viscous damping
constant of 10,000 N-s/m. During a
particular forging operation, the tup,
weighing 1,000 N, is made to fall
from a height of 2 m onto the anvil.
If the anvil is at rest before impact
by the tup, determine the response
of the anvil after the impact.
