Introduction to Mechanical Vibration

1. Introduction to
Mechanical Vibrations
Saurabh Tiwari (Asst.
Prof.)
Dept of Mechanical Engg
Saurabh23gsti@gmail.co

2. What is Vibration ??
❖Vibration:- To and fro periodic motion about the mean position.
When the motion is repeated in equal intervals of time, it is known as
periodic motion.
Simple harmonic motion is the simplest form of periodic motion.
Let a body having simple harmonic motion is represented by the
equation
x = A cos ωt
where A is the amplitude of oscillation measured from the equilibrium
position of the mass.

3. Let a body having simple harmonic motion is represented by the
equation
x = Acos ωt or Asin ωt
where A is the amplitude of oscillation measured from the equilibrium
position of the mass.

4. ❖ A vibratory system basically consist
of three elements, namely the mass,
the spring and damper.
1) Kinetic energy storing device (Mass = m)
2) Potential energy storing device(Stiffness = S)
3) Energy Dissipating device (Damper = C)
Kinetic friction ≠ 0
Unbalanced Force Fun≠ 0
Elementary parts of a vibrating System
3
2
1

5. ❖ Vibrations can be classified into three categories:
1) free vibration,
2) Forced vibration, and
3) self-excited vibration.
Classification of Vibration
In a vibrating body, there is exchange of energy from one
form to another.
Energy is stored by mass in the form of kinetic energy (1/2
mv2
), in the spring in the form of potential energy (1/2 kx )
and dissipated in the damper in the form of heat energy
which opposes the motion of the system.

6. Definition:-“The vibration in which there is no friction at all as well as
there is no external force after the release of the system are known as
natural vibration”
At unstretched position;
kΔ=W=mg……(1)
Natural Vibration

7. mẍkx=0
At static equilibrium;
-k(Δ+x) = m(g+a)
-kx = mẍ or mẍ+kx=0
ẍ+{k/m}x=0….... (2)
Eq. 2 is the equation of natural system.
Eq. 2 is the second order linear differential equation.
The solution of this eq. is
x = Rsin{√(k/m)*t}+ Φ Where; R and Φ are constant
On comparing with simple harmonic function x = A cos ωt the natural
frequency of the system ,
ɷn = √(k/m) rad/sec
Tn = 2∏/ɷn sec