Dr. Millerjothi, BITS Pilani, Dubai Campus
Chapter:1 ▪ Brief review of fundamental concepts of vibration ▪ Vibration Analysis ▪ Analysis of Simple vibrating systems
Dr. Millerjothi, BITS Pilani, Dubai Campus
✓Vibrations can lead to excessive deflections and failure on the machines and structures.
✓ To reduce vibration through proper design of machines and their mountings.
✓ To utilize profitably in several consumer and industrial applications.
✓ To improve the efficiency of certain machining, casting, forging & welding processes.
✓ To stimulate earthquakes for geological research and conduct studies in design of nuclear reactors.
Importance of the Study of Vibration
Dr. Millerjothi, BITS Pilani, Dubai Campus
!✓Vibrational problems of prime movers due to
inherent unbalance in the engine. ✓Wheel of some locomotive rise more than
centimeter off the track – high speeds – due to imbalance.
✓ Turbines – vibration cause spectacular mechanical failure.
EXAMPLE OF PROBLEMS
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DISADVANTAGES
!✓Cause rapid wear. ✓Create excessive noise. ✓ Leads to poor surface finish (eg: in metal
cutting process, vibration cause chatter). ✓Resonance – natural frequency of vibration of a
machine/structure coincide with the frequency of the external excitation (eg: Tacoma Narrow Bridge – 1948)
Dr. Millerjothi, BITS Pilani, Dubai Campus
Applications
Dr. Millerjothi, BITS Pilani, Dubai Campus
Basic Concepts of Vibration
❑ Vibration = any motion that repeats itself after an interval of time. !
❑ Vibratory System consists of: 1) spring or elasticity 2) mass or inertia 3) damper !!❑ Involves transfer of potential energy to kinetic energy and vice
versa.
Dr. Millerjothi, BITS Pilani, Dubai Campus
❑ Degree of Freedom (d.o.f.) = min. no. of independent coordinates required to determine completely the positions of all parts of a system at any instant of time
❑ Examples of single degree-of-freedom systems:
Basic Concepts of Vibration
Dr. Millerjothi, BITS Pilani, Dubai Campus
Basic Concepts of Vibration
❑ Examples of single degree-of-freedom systems:
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Basic Concepts of Vibration
❑ Examples of Two degree-of-freedom systems:
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Basic Concepts of Vibration
❑ Examples of Three degree of freedom systems:
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Basic Concepts of Vibration
❑ Example of Infinite number of degrees of freedom system: !!!
❑ Infinite number of degrees of freedom system are termed continuous or distributed systems.
❑ Finite number of degrees of freedom are termed discrete or lumped parameter systems.
❑ More accurate results obtained by increasing number of degrees of freedom.
Dr. Millerjothi, BITS Pilani, Dubai Campus
Classification of vibration
❑ Free Vibration:A system is left to vibrate on its own after an initial disturbance and no external force acts on the system. E.g. simple pendulum
❑ Forced Vibration:A system that is subjected to a repeating external force. E.g. oscillation arises from diesel engines ➢Resonance occurs when the frequency of the external
force coincides with one of the natural frequencies of the system
Dr. Millerjothi, BITS Pilani, Dubai Campus
Dr. Millerjothi, BITS Pilani, Dubai Campus
❑ Undamped Vibration:When no energy is lost or dissipated in friction or other resistance during oscillations
❑ Damped Vibration: When any energy is lost or dissipated in friction or other resistance during oscillations
❑ Linear Vibration:When all basic components of a vibratory system, i.e. the spring, the mass and the damper behave linearly
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Damped and Undamped vibrations
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❑ Nonlinear Vibration:If any of the components behave nonlinearly
❑ Deterministic Vibration:If the value or magnitude of the excitation (force or motion) acting on a vibratory system is known at any given time
❑ Nondeterministic or random Vibration: When the value of the excitation at a given time cannot be predicted
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❑ Examples of deterministic and random excitation:
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Harmonic motion
❑ Periodic Motion: motion repeated after equal intervals of time
❑ Harmonic Motion: simplest type of periodic motion ❑ Displacement (x): !❑ Velocity: !
❑ Acceleration:
τπt
Ax 2sin=
)2/sin(cos πωωωω +=== tAtAdtdx
x!
)(sinsin 222
2
πωωωω +=−== tAtAdtxd
x!!
xx 2ω−=!!
Dr. Millerjothi, BITS Pilani, Dubai Campus
The similarity between cyclic (harmonic) and sinusoidal motion.
Dr. Millerjothi, BITS Pilani, Dubai Campus
Dr. Millerjothi, BITS Pilani, Dubai Campus
The trigonometric functions of sine and cosine are related to the exponential function by Euler’s equation.
exp(iθ) = cos(θ) + i sin(θ) The vector P rotating at constant angular Speed ω can be represented as !!!!Where x =real component and y = imaginery component.
Dr. Millerjothi, BITS Pilani, Dubai Campus
Dr. Millerjothi, BITS Pilani, Dubai Campus
Vibrations of several different frequencies exist simultaneously. Such vibrations result in a complex waveform which is repeated periodically as shown
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A periodic function:
Harmonic Analysis
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Fourier Series Expansion: If x(t) is a periodic function with periodic τ, its Fourier Series representation is given by
∑∞
=
++=
+++
+++=
1
0
21
210
)sincos(2
...2sinsin
...2coscos2
)(
nnn tnbtna
a
tbtb
tataatx
ωω
ωω
ωω
Dr. Millerjothi, BITS Pilani, Dubai Campus
•Complex Fourier Series: The Fourier series can also be represented in terms of complex numbers.
titetite
ti
ti
ωω
ωωω
ω
sincos
sincos
−=
+=−
Also,
ieet
eet
titi
titi
2sin
2cos
ωω
ωω
ω
ω
−
−
−=
+=
and
Dr. Millerjothi, BITS Pilani, Dubai Campus
•Frequency Spectrum: Harmonics plotted as vertical lines on a diagram of amplitude (an and bn or dn and Φn) versus frequency (nω).
Dr. Millerjothi, BITS Pilani, Dubai Campus
Even and odd functions:
∑∞
=
+=
=−
1
0 cos2
)(
)()(
nn tna
atx
txtx
ω
Even function & its Fourier series expansion
Odd function & its Fourier series expansion
∑∞
=
=
−=−
1
sin)(
)()(
nn tnbtx
txtx
ω
Dr. Millerjothi, BITS Pilani, Dubai Campus
Half-Range Expansions:
The function is extended to include the interval – τ to 0 as shown in the figure. The Fourier series expansions of x1(t) and x2(t) are known as half-‐range expansions.
Dr. Millerjothi, BITS Pilani, Dubai Campus
Example 1Addition of Harmonic Motions
Find the sum of the two harmonic motions
!
Solution: Method 1: By using trigonometric relations:
Since the circular frequency is the same for both x1(t) and x2(t), we express the sum as
).2cos(15)( and cos10)(21
+== ttxttx ωω
E.1)()()()cos()(21txtxtAtx +=+= αω
Dr. Millerjothi, BITS Pilani, Dubai Campus
That is, !!That is, !!By equating the corresponding coefficients of cosωt and sinωt
on both sides, we obtain
( )E.2)()2sinsin2cos(cos15cos10
)2cos(15cos10sinsincoscosttt
ttttAωωω
ωωαωαω
−+=
++=−
E.3)()2sin15(sin)2cos1510(cos)sin(sin)cos(cos
ttAtAtω
ωαωαω
−
+=−
( )1477.14
)2sin15(2cos1510
2sin15sin2cos1510cos
22
=
++=
=
+=
A
AA
α
α
Dr. Millerjothi, BITS Pilani, Dubai Campus
and
E.5)(5963.742cos1510
2sin15tan 1
°=
!"#
$%&
+= −α
Dr. Millerjothi, BITS Pilani, Dubai Campus
Method 2: By using complex number representation:: the two harmonic motions can be denoted in terms of complex numbers:
!!!!The sum of x1(t) and x2(t) can be expressed as
[ ] [ ][ ] [ ] E.7)(15ReRe)(
10ReRe)()2()2(
22
11
++ ≡=
≡=titi
titi
eeAtxeeAtx
ωω
ωω
[ ] E.8)(Re)( )( αω += tiAetx
Example 2
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where A and α can be determined using the following equations
!!!!!!and A = 14.1477 and α = 74.5963º
2,1;)( 22 =+= jbaA jjj
2,1;tan 1 =!!"
#$$%
&= − j
ab
j
jjθ
Dr. Millerjothi, BITS Pilani, Dubai Campus
Vibration Terminology
❑ Definitions of Terminology: ➢Amplitude (A) is the maximum displacement of a vibrating
body from its equilibrium position
➢Period of oscillation (T) is time taken to complete one cycle of motion !
➢Frequency of oscillation (f) is the no. of cycles per unit time
ωπ2
=T
πω2
1==
Tf
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➢Natural frequency is the frequency which a system oscillates without external forces
➢Phase angle (φ) is the angular difference between two synchronous harmonic motions
( )φω
ω
+=
=
tAxtAx
sinsin
22
11
Dr. Millerjothi, BITS Pilani, Dubai Campus
➢Beats are formed when two harmonic motions, with frequencies close to one another, are added
Dr. Millerjothi, BITS Pilani, Dubai Campus
➢The peak value generally indicates the maximum stress that the vibrating part is undergoing.
➢The average value indicates a steady or static value. It is found from
!!For example, the average value of complete A sin t is zero, but
of a half cycle
Dr. Millerjothi, BITS Pilani, Dubai Campus
➢The mean square value of a time function x(t) is found from average of the squared values, integrated over some time interval T
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➢Decibel is originally defined as a ratio of electric powers. It is now often used as a notation of various quantities such as displacement, velocity, acceleration, pressure, and power
!!"
#$$%
&=
!!"
#$$%
&=
0
0
log20dB
log10dB
XX
PP
where P0 is some reference value of power and X0 is specified reference voltage.
Dr. Millerjothi, BITS Pilani, Dubai Campus
Problem 3
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Solution
Dr. Millerjothi, BITS Pilani, Dubai Campus
Dr. Millerjothi, BITS Pilani, Dubai Campus
Dr. Millerjothi, BITS Pilani, Dubai Campus
Dr. Millerjothi, BITS Pilani, Dubai Campus