2. Unit 1_Vibrations for MTech students eng

© MIT-ADT University, Pune 2023
Tuesday, December 31, 2024
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Lecture by Abhijeet G. Chavan
MIT School of Engineering, Pune
Course Code: 22MEVT432
Course: Noise, Vibration and Harshness (NVH) in Electric Vehicles
Name of the Faculty: Prof. Abhijeet G. Chavan
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a b h i j e e t . c h a v a n @ m i t u n i v e r s i t y. e d u . i n

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Unit No 1
Introduction to Vibration and Noise
Syllabus:
Natural vibrations, SDOF and MDOF concept and modeling,
Undamped, damped and forced vibrations, resonance,
transmissibility, modes of vibrations.
Basics of sound propagation, Sound levels and spectra,
Quantification of sound, Noise sources, generation and
radiation, Noise source identification and noise induced hearing
loss.

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Introduction to Vibration and Noise
Elements of a vibratory system:
• Mass (Inertia)
• Spring (Stiffness / Energy storage)
• Damper (Energy dissipation)
• External Excitation (Forcing functions)
Essential Elements for vibratory motion…!
What is a vibratory motion…??

Natural vibrations (free Vibrations)
Degrees of Freedom
DOF: Number of independent parameters required to ‘completely’ describe the
motion/position of the body
Free Vibrations: Vibrations of a body because of its own physical properties when
there is no external excitation

Tuesday, December 31, 2024 Lecture by Abhijeet G.Chavan 5
How many maximum degrees of freedom we can have…??
Food for thought…

Tuesday, December 31, 2024 Lecture by Abhijeet G.Chavan 6
Infinite………..!
Food for thought… (ans)

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Simple Pendulum Spring Mass System Rotor System
One (single) DOF Systems

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Two Mass System
Two Rotor System
Combined Mass and
Pendulum System
Two DOF Systems

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Three Mass System
Three Pendulum System
Three Rotor System
Three DOF Systems

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Discrete (Lumped Mass Systems) Vs Continuous systems

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Free Vibrations and Forced Vibrations
Un-damped Vibrations and Damped Vibrations
Linear Vibrations and Non Linear Vibrations
Deterministic and Random Vibrations
Classification of Vibrating systems

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Classification of Vibrating systems
Longitudinal
Vibrations
Transverse
Vibrations
Torsional
Vibrations

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Frequency: Number of cycles per unit time.
Units: Hz, RPM, rad/s
Time period: Time required to complete one cycle
Units: seconds, minutes
Amplitude: Maximum displacement of vibrating body from its mean
(equilibrium) position
Units: mm, microns
Remember: Frequency and Time Period are reciprocals of each other
Basic Terminology

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S.H.M.:- A “periodic” motion in which
• Acceleration is directly proportional to distance (displacement) from
mean position
•Acceleration is always directed towards mean position
•Examples: Motion imparted by Scotch Yoke Mechanism
Simple Harmonic Motion (S.H.M.)

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SHM obtained by Scotch Yoke Mechanism

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Harmonic motion can be represented
conveniently by means of a vector OP of
magnitude ‘A’ rotating at a constant
angular velocity v.
projection of tip
of the vector on
the vertical axis
projection of tip of
the vector on the
horizontal axis
Vector representation of SHM

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Displacement
Velocity
Acceleration
SHM Differential Equations

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Displacement
Velocity
Acceleration
“Things to remember”
In harmonic motion
Velocity and Acceleration
lead the displacement by
π/2 and π
Displacement, Velocity & Acceleration of
SHM

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Modeling of SDOF System
m= mass of body (Kg)
k= stiffness of spring (N/mm)
x= displacement of body from
mean position (mm)
Typical Spring Mass System
Any assumptions
required…??

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Typical Spring Mass System
Can you
imagine???

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Can you form
the equation by
applying
Newton’s 2nd
Law…???
This is how it actually looks…!

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Maybe this will help…!

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… By applying Newton’s 2nd Law of motion
… Rearranging the terms we get differential equation of
motion for spring mass system
How to solve
this…???
Equation of motion for SDOF System

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… By rearranging the previous equation we get
… replacing k/m by omega^2
Are you familiar
with this
equation…???
Solution of Equation of motion for
SDOF System

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Spring Mass System performing SHM
Yes… this is
equation of
S.H.M. ….!!!
SDOF System

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Natural
Frequency of
the system..
SDOF System

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What will happen if
stiffness of spring is
changed…
What will happen if
value of mass is
changed…
How to
change..???
Factors affecting natural frequency of
system

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Damping is resistance offered by a body to motion of vibrating system
1. It may be fluid or solid, internal or external.
2. Generally this isn't inherent property of system.
3. Small amount of damping does not have significant effect on performance
of system but as the value of damping increases the effect is significant.
4. Damping is used in various mechanical systems to control the amplitude of
vibrations.
Damping..

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1. Viscous Damping: Occurs when system vibrates in viscous fluid. Viscosity of fluid is
offering resistance to motion. Amount of damping resistance will depend upon
velocity and damping coefficient.
2. Dry friction (Coulomb’s) damping: When two machine parts rub against each other in
dry or un-lubricated conditions. Damping resistance is constant and is independent of
rubbing velocity.
3. Solid (structural) damping: This is due to internal friction of molecules. Energy is
dissipated due to molecular friction in each cycle.
4. Slip (Interfacial) damping: Energy is dissipated due to microscopic slip on the
interfaces of machine parts. This is non linear damping.
5. Eddy current damping: This is achieved by inducing an eddy current in moving plate
thereby providing resistance to motion of plate. Damping obtained is similar to viscous
damping.
Types of Damping

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…. Equation of motion
Free Vibration with Viscous Damping

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Solution in the
form
Substituting in equation
Characteristic equation is defined as
Roots of
characteristic
equation are
Solution

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1. If ξ is greater than one, the roots P1 and P2 are real and distinct. (Over damped)
2. If ξ is equal to one, the root P1 = P2 and both roots are real. (Critically damped)
3. If ξ is less than one, the roots P1 and P2 are complex conjugates. (Under damped)
Types of cases

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In the over damped case the roots P1 and P2 are real. The response of the single degree
of freedom system can be written as
A1 and A2 are arbitrary constants.
The solution is sum of two exponential functions
Motion of the system is non oscillatory
Overdamped case

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The roots P1 and P2 of the characteristic equation are equal and are given by
Solution is
A1 and A2 are arbitrary constants.
Solution x(t) is non oscillatory
Solution is product of a linear function of time and an exponential decay.
Critically damped case

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New term is defined “Damped natural frequency”
Solution
Roots of the characteristic equations P1 and P2 are complex conjugates.
Under damped case

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Comparison of motions with different
types of damping

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Shock absorber of a motorcycle Recoil of cannon
Real Life examples

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• The logarithmic decrement 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.
Under damped
response
For non successive cycles
Logarithmic Decrement

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Forced Vibrations
Forcing
Functions
Periodic
Harmonic
Non
Harmonic
Impulsive Random

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Free Body
Diagram
Steady State
response
Total response
Equation is non homogeneous:
Its general solution x(t) is given
by the sum of the
homogeneous solution xh(t)
and the particular solution xp(t)
EOM for system excited with force F(t)

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Amplitude
(displacement)
Phase Angle
Equation of Motion
EOM for system excited with harmonic
force

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1. For an undamped system M
approaches ∞ as r approaches 1.
2. Any amount of damping reduces
the magnification factor (M) for all
values of the forcing frequency.
3. For any specified value of r, a higher
value of damping reduces the value of
M.
4. In the degenerate case of a constant
force (when r = 0), the value of M = 1.
5. The reduction in M in the presence
of damping is very significant at or
near resonance.
6. The amplitude of forced vibration
becomes smaller with increasing
values of the forcing frequency
Graph of Amplitude ratio and
Frequency Ratio

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Graph of Phase angle and Frequency
Ratio

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Real life examples
Rotating Unbalance
Reciprocating Unbalance
Base Excitation

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The ratio of the amplitude of the response to that of the base motion y(t), X/Y,
is called the displacement transmissibility
Displacement Transmissibility

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1. The value of Td is unity at r = 0 and close to unity for small values of r.
2. For an undamped system,
3. The value of Td is less than unity
4. The value of Td is unity
5.
6.
Displacement Transmissibility

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The ratio FT/kY is known as the force transmissibility.
The transmitted force is in phase with the motion of the mass x(t)
Force Transmissibility

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MCQ Test:
1.
https://docs.google.com/forms/d/e/1FAIpQLSdnh7tVMNfscuNKieV4CWM3z4uCdyMlYjQqS
U4iZaSIKJkMug/viewform?usp=sf_link
• Mechanical Vibrations By G K Grover
• Mechanical Vibrations By S S Rao
• Theory of Vibrations By William Thompson
• Mechanical Vibrations By V P Singh
Further readings and resources

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2. Unit 1_Vibrations for MTech students eng

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