Dom unit 3 slides

Dynamics of Machines (ME 8594)
Unit-3
by
Dr. B. Janarthanan
Professor
Department of Mechanical Engineering
Mohamed Sathak A.J. College of Engineering

ME8594 DYNAMICS OF MACHINES
OBJECTIVES:
1. To understand the force-motion relationship in components subjected
to external forces and analysis of standard mechanisms.
2. To understand the undesirable effects of unbalances resulting from
prescribed motions in mechanism.
3. To understand the effect of Dynamics of undesirable vibrations.
4. To understand the principles in mechanisms used for speed control and
stability control.
OUTCOMES:
Upon the completion of this course the students will be able to
CO1 Calculate static and dynamic forces of mechanisms.
CO2 Calculate the balancing masses and their locations of reciprocating
and rotating masses.
CO3 Compute the frequency of free vibration.
CO4 Compute the frequency of forced vibration and damping coefficient.
CO5 Calculate the speed and lift of the governor and estimate the
gyroscopic effect on automobiles, ships and airplanes.
11 October 2020 B. Janarthanan Dynamics of Machines 2

ME8594 DYNAMICS OF MACHINES
UNIT III FREE VIBRATION
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.
BOOKS
1. Rattan, S.S, “Theory of Machines”, 4th Edition, Tata
McGraw-Hill, 2014.
2. Khurmi R.S., Theory of Machines, 14th Edition, S
Chand Publications, 2005

Some terms used in vibration
• Displacement, velocity and acceleration
• Periodic motion
• Time period
• Frequency
• Amplitude
• Natural frequency
• Degree of freedom – minimum number of independent
coordinates required to specify the motion of a system at
any instant
• Fundamental mode of vibration – mode having lowest
natural frequency

Fundamentals of vibration
Types of vibrations
• The vibratory motion is classified in three
ways as
1. According to actuating forces on the body
2. According to direction of motion
3. According to damping property

• According to actuating forces on the body
1. Free vibration – oscillates under initial
disturbance with no external force. E.g.,
simple pendulum
2. Forced vibration – subjected to periodic
external force. E.g., machine tools, electric
bells, diesel engines

• According to direction of motion
1. Longitudinal vibration (linear or axial)
2. Transverse vibration
3. Torsional vibration

• According to damping property
1. Damped vibration – energy is dissipated in
friction or other resistance during oscillation
2. Undamped vibration – no energy is lost
during vibratory motion

Simple Harmonic Motion
One of the simplest examples of mechanical
vibration is the classical simple pendulum
For small amplitudes of vibration the periodic
motion can be shown to be simple harmonic motion

Basics of vibration
B. Janarthanan Dynamics of Machines 1111 October 2020
• Basic elements in any vibratory system
1. Mass (or inertia) – means for storing kinetic energy
2. Spring (or elasticity) – for storing potential energy
3. Damper – for dissipating energy

Undamped free longitudinal vibration

Equation of motion

Free Vibration of Single Degree of Freedom System
Degrees-of-freedom:- The number of independent co-ordinates required to
describe the motion of a system
Applying D’Alembert’s principle to convert dynamic problem into
equivalent static problem
This is the equation of motion for
undamped free vibration of SDOF
system
Free body
diagram

Undamped Free Vibration - SHM
( )



+=−=






+==
=
tASintASinx
tASintACosx
tASinx
22
2


Displacement
Velocity
Acceleration
f
T
m
k
or
m
k
km
tkAtAm






2
2
)(
0
0SinSin
2
2
2
==
==
=+−
=+−
A is the amplitude of oscillation
T is the period of oscillation
f is frequency of oscillation in cycles per second or
Hz
ω is the frequency of oscillation in radians per
second

Undamped Free Vibration
The solution is
0=+ kxxm The governing differential equation:
For Torsional vibration
txtSin
x
x nn
n


cos)0(
)0(
+=

Response
0=+
••
 qI

Damped free vibration

Free body
diagram
If ζ<1, under damped
If ζ=1, critically
damped
If ζ>1, over damped

Over damping
Courtesy : S.S. Rattan, Theory of machines
The solution is simply a sum of two decaying
exponentials with no oscillation

Underdamping

Underdamping
Which shows that the ratio of amplitudes of two
successive oscillation

Critical damping
This turns out to be a desirable outcome in many cases
where engineering design of a damped oscillator is
required (e.g., a door closing mechanism).

Critical damping

➢ Large guns are critically damped
➢ Automatic door and window closer mechanism

Logarithmic decrement
The natural logarithm of the ratio of two successive amplitudes is
called logarithmic decrement
( )
( ) ( )22
1
1
2
1
22
lnln








 
−
=
−
==
==





=
+
n
n
d
n
dn
dTn
n
n
Te
X
X
If the system executes n cycles






=
nX
X
n
0
ln
1


A mass of 1 kg is to be supported on a spring having a stiffness
of 9800 N/m. The damping coefficient is 5.9 N-sec/m.
Determine the natural frequency of the system. Find also the
logarithmic decrement and the amplitude after three cycles if
the initial displacement is 0.003m.

Torsional vibrations
Courtesy : R.S. Khurmi

Single rotor system

Two rotor system

Torsionally equivalent shaft

Two rotor system
A steel shaft 1.5 m long is 95 mm in diameter for the first 0.6 m of its length, 60
mm in diameter for the next 0.5 m of the length and 50 mm in diameter for the
remaining 0.4 m of its length. The shaft carries two flywheels at two ends, the
first having a mass of 900 kg and 0.85 m radius of gyration located at the 95
mm diameter end and the second having a mass of 700 kg and 0.55 m radius
of gyration located at the other end. Determine the location of the node and the
natural frequency of free torsional vibration of the system. The modulus of
rigidity of shaft material may be taken as 80 GN/m2.

Three rotor system

Torsional vibrations (3-rotor system)
Three rotors A, B and C having moment of inertia of 2000 ; 6000 ;
and 3500 kg-m2 respectively are carried on a uniform shaft of
0.35 m diameter. The length of the shaft between the rotors A and
B is 6 m and between B and C is 32 m. Find the natural frequency
of the torsional vibrations. The modulus of rigidity for the shaft
material is 80 GN/m2.
[Ans. 6.16 Hz ; 18.27 Hz]

Torsional vibrations (3-rotor system)
A 4-cylinder engine and flywheel coupled to a propeller are
approximated to a 3-rotor system in which the engine is
equivalent to a rotor of moment of inertia 800 kg-m2, the
flywheel to a second rotor of 320 kg-m2 and the propeller to a
third rotor of 20 kg-m2. The first and the second rotors being
connected by 50 mm diameter and 2 metre long shaft and the
second and the third rotors being connected by a 25 mm
diameter and 2 metre long shaft. Neglecting the inertia of the
shaft and taking its modulus of rigidity as 80 GN/m2,
determine:
1. Natural frequencies of torsional oscillations, and
2. The positions of the nodes.

Contact
Dr. B. Janarthanan
Professor
Department of Mechanical Engineering
Mohamed Sathak A.J. College of Engineering
Email : vbjana@gmail.com,
mech.janarthanan@msajce-edu.in

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