fundamental concepts of vibration-presentation
TRANSCRIPT
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Dr. Millerjothi, BITS Pilani, Dubai Campus
Chapter:1 ▪ Brief review of fundamental concepts of vibration ▪ Vibration Analysis ▪ Analysis of Simple vibrating systems
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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
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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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Dr. Millerjothi, BITS Pilani, Dubai Campus
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)
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Dr. Millerjothi, BITS Pilani, Dubai Campus
Applications
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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.
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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
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Dr. Millerjothi, BITS Pilani, Dubai Campus
Basic Concepts of Vibration
❑ Examples of single degree-of-freedom systems:
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Dr. Millerjothi, BITS Pilani, Dubai Campus
Basic Concepts of Vibration
❑ Examples of Two degree-of-freedom systems:
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Dr. Millerjothi, BITS Pilani, Dubai Campus
Basic Concepts of Vibration
❑ Examples of Three degree of freedom systems:
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Dr. Millerjothi, BITS Pilani, Dubai Campus
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.
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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
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❑ 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−=!!
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The similarity between cyclic (harmonic) and sinusoidal motion.
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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.
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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
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•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
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•Frequency Spectrum: Harmonics plotted as vertical lines on a diagram of amplitude (an and bn or dn and Φn) versus frequency (nω).
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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
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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.
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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
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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
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and
E.5)(5963.742cos1510
2sin15tan 1
°=
!"#
$%&
+= −α
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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 =!!"
#$$%
&= − j
ab
j
jjθ
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Vibration Terminology
❑ Definitions of Terminology: ➢Amplitude (A) is the maximum displacement of a vibrating
body from its equilibrium position
➢Period of oscillation (T) is time taken to complete one cycle of motion !
➢Frequency of oscillation (f) is the no. of cycles per unit time
ωπ2
=T
πω2
1==
Tf
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➢Natural frequency is the frequency which a system oscillates without external forces
➢Phase angle (φ) is the angular difference between two synchronous harmonic motions
( )φω
ω
+=
=
tAxtAx
sinsin
22
11
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➢Beats are formed when two harmonic motions, with frequencies close to one another, are added
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➢The peak value generally indicates the maximum stress that the vibrating part is undergoing.
➢The average value indicates a steady or static value. It is found from
!!For example, the average value of complete A sin t is zero, but
of a half cycle
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➢The mean square value of a time function x(t) is found from average of the squared values, integrated over some time interval T
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➢Decibel is originally defined as a ratio of electric powers. It is now often used as a notation of various quantities such as displacement, velocity, acceleration, pressure, and power
!!"
#$$%
&=
!!"
#$$%
&=
0
0
log20dB
log10dB
XX
PP
where P0 is some reference value of power and X0 is specified reference voltage.
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Problem 3
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Solution
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