Free vibrations

 At the end of this unit, the students should be able to compute the frequency of free vibrations.
SYLLABUS
Basic features of vibratory systems – Degrees of freedom – single degree of freedom – Free
vibration– Equations of motion – Natural frequency – Types of Damping – Damped vibration–
Torsional vibration of shaft – Critical speeds of shafts – Torsional vibration – Two and three
rotor torsional systems.

i. Vibrations: When elastic bodies such as a spring, a beam and shaft are displaced from the
equilibrium position by the application of external forces, and then released, they execute a
vibratory motion.

i. Period of vibration or time period: It is the time
interval after which the motion is repeated itself.
The period of vibration is usually expressed in
seconds.
ii. Cycle: It is the motion completed during one time
period.
iii. Frequency: It is the number of cycles described in
one second. In S.I. units, the frequency is expressed
in hertz (briefly written as Hz) which is equal to one
cycle per second.

i. Causes of Vibrations:
 Unbalanced forces: Produced within the machine due to
wear and tear.
 External excitations: Can be periodic or random
ii. Resonance:
 When the frequency of the external or applied force is
equal to the natural frequency resonance occurs.
Vertical Shaking Accident and Cause Investigation of 39-story Office Building

Components of vibratory system:
i. Spring/Restoring element:
 Its denoted by k or s;
 SI unit – N/m
ii. Dashpot/Damping component
 Its denoted by c;
 SI unit – N/m/s
iii. Mass/Inertia component
 Its denoted by m;
 SI unit – kg

 The minimum number of independent
coordinates required to determine completely
the position of all parts of a system at any
instant of time defines the degree of freedom of
the system.
1 DOF 2 DOF

1. Free or Natural Vibrations: 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 vibrations. The frequency of the free vibrations is called free or
natural frequency.
2. Forced vibrations When the body vibrates under the influence of external
force, the the body is said to be under forced vibrations.The vibrations have
the same frequency as the applied force
3. Damped vibrations: When there is a reduction in amplitude over every
cycle of vibration, due to frictional resistance, the motion is said to be
damped vibration.

1. Longitudinal Vibrations:
Parallel to axis of shaft
2. Transverse Vibrations:
Approx. Perpendicular to axis
of shaft
3. Torsional Vibrations:
Moves in circles about axis of
shaft
Longitudinal Transverse Torsional

d = Static deflection of spring
in meters.

Equilibrium method – Longitud. Vibrations
 Restoring force
 W – (sd + sx)
 - sx
 Accelerating force
m(d2x/dt 2)
 Equating both
m(d2x/dt 2) + sx = 0
d2x/dt 2 + (s/m) x = 0
 SHM equation
d2x/dt 2 + w2 x = 0
 Angular Velocity:

W = mg =
sd
SF = 0

Equilibrium method
 Time period
 tp = 2p/w
 Natural frequency
 A
 B
 Deflection
 s= W/A = Ee = E x (d/l)
 d = Wl/ AE

Rayleigh’s method
In this method, the maximum kinetic energy at the mean position is equal to the maximum
potential energy (or strain energy) at the extreme position. Assuming the motion executed by
the vibration to be simple harmonic, then

Transverse vibrations
<= Same as Longitudinal
Vibrations

1. Natural frequency of longitudinal and transverse vibrations:
Where,
fn = Natural frequency. (Hz)
tp = Time period (s)
s = Stiffness (N/m)
m = Mass. (kg)
g = acceleration due to gravity (m/s2)
d = Static deflection. (m)

A cantilever shaft 50 mm diameter and 300 mm long has a disc of mass
100 kg at its free end. The Young's modulus for the shaft material is 200
GN/m 2 . Determine the frequency of longitudinal and transverse
vibrations of the shaft

2. Static deflection in beams,
Where,
fn = Natural frequency. (Hz)
tp = Time period (s)
s = Stiffness (N/m)
m = Mass. (kg)
g = acceleration due to gravity (m/s2)
d = Static deflection. (m)

2. Static deflection in beams,

1. A shaft of length 0.75 m, supported freely at the ends, is carrying a body of
mass 90 kg at 0.25 m from one end. Find the natural frequency of transverse
vibration. Assume E = 200 GN/m2 and shaft diameter = 50mm.
Given
l = 0.75 m ;
m = 90 kg ;
a = AC = 0.25 m ;
E = 200 GN/m2 = 200 × 109 N/m2
d = 50 mm = 0.05 m

 Moment of inertia of shaft
 Static deflection
 Natural frequency

A flywheel is mounted on a vertical shaft as shown in Fig. 23.8.
The both ends of the shaft are fixed and its diameter is 50 mm.
The flywheel has a mass of 500 kg. Find the natural frequencies of
longitudinal and transverse vibrations. Take E = 200 GN/m2
.

Free vibrations

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Free vibrations