Dynamics of Machines Unit III - Long Vibration * Introduction on Vibration * Longitudinal vibration- Short notes and MCQ

1. ME8594 - DYNAMICS OF MACHINES
UNIT-III-FREE VIBRATION
(LONGITUDINAL VIBRATION)
By,
Dr.S.SURESH,
Assistant Professor,
Department of Mechanical Engineering
Jayalakshmi Institute of Technology.

2. INTRODUCTION
When elastic bodies such as a spring, a beam and a shaft are
displaced from the equilibrium position by the application of
external forces, and then released, they execute a vibratory
motion.
Causes of vibration
The causes of vibration are unbalanced forces, elastic nature of
the system, self excitations, winds and earthquakes.
Effects of vibration
 Stress
 Noise
 Wear

3. TERMS USED IN VIBRATORY MOTION
1. Frequency
Frequency is the number of cycles described in one second.
Its unit is Hz.
2. Period
Period is the time interval after which the motion is
repeated itself.
3. Cycle of vibration
Cycle is defined as the motion completed during one time
period.

4. DAMPER OR SHOCK ABSORBERS
 Damping: The resistance against the vibration is
called damping.
Viscous Damping is the damping provided by fluid
resistance.
Coloumb damping is the damping results from two
dry or unlubricated surfaces rubbing together.

5. Different types of vibrations
1. Free vibrations
a) Longitudinal vibration,
b) Transverse vibration, and
c) Torsional vibration.
2. Forced vibrations, and
3. Damped vibration.

6. Free vibration
When no external force acts on the body, after
giving it an initial displacement, then the body is
said to be under free or natural vibration.
The frequency of the free vibrations is called free
or natural frequency.
Example: Simple Pendulum
Forced vibrations
When the body vibrates under the influence of
external force, then the body is said to be under
forced vibrations.
Example : Electric bell
Damped vibrations
When there is a reduction in amplitude over
every cycle of vibration, the motion is said to be
damped vibration.

7. FREE VIBRATION
Types of Free Vibrations
Tensile or Compressive stress
Shear stress

8. 1. LONGITUDIONAL VIBRATION
1.1 Natural Frequency of Free Longitudinal Vibrations
S = Stiffness of the spring (N/m)
m = Mass of the body (Kg)
δ = Static deflection (m)
x = Displacement (m)
Methods:
1) Equilibrium method
2) Energy method
kinetic energy and potential energy must be a constant quantity
which is same at all the times
3) Rayleigh's method
maximum kinetic energy at mean position is equal to maximum
potential energy at extreme position

9.  Accelerating force = Mass x Acceleration
 Restoring force = W – s (δ+x)
= s x
Pull force = W = mg
Spring force = W = s δ
Fundamental equation of simple harmonic motion
(Differential equation of motion)
1) Equilibrium method

10.  Natural circular frequency
 Time period
 Natural Frequency
(or)

11. 1.2 FREQUENCY OF FREE DAMPED VIBRATIONS
• Damping force :
• Accelerating force :
• Spring force :
Differential equation of motion

12.  Damping Coefficient
The damping force per unit velocity is known as damping
coefficient.
C= Damping force / Velocity
 Critical damping coefficient
The critical damping coefficient is the amount of damping required
for a system to be critically damped.
 Damping Factor or Damping Ratio
The ratio of the actual damping coefficient (c) to the critical
damping coefficient (c) is known as damping factor
=ζ (zeta)

13.  Periodic time of damped free vibration
 Circular frequency
 Natural frequency of damped free vibration

15.  Logarithmic Decrement
It is defined as the natural logarithm of the amplitude reduction
factor.
The amplitude reduction factor is the ratio of any two successive
amplitudes on the same side of the mean position.

16. RESONANCE
 If the frequency of the external force is the
same as the natural frequency, the amplitude
becomes quite large. This is called resonance.
 Resonance is useful in paddleball, microwaves,
music, Tv/radio receivers

17. Effect of Inertia of the Constraint in
Longitudinal and Transverse Vibration
Considering the effect of inertia of the constraint (Shaft)

18. LONGITUDINAL VIBRATION – IMPORTANT FORMULAS
UNDAMPED DAMPED
Circular Frequency (Ѡ)
Natural Frequency (f)
Time period (tp)
Damping Coefficient (C)
Critical Damping
Coefficient (Cc)
Damping factor
(Damping ratio)
Logarithmic Decrement

19. OBJECTIVE TYPE QUESTIONS