Mechanical Vibrations
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Brief History
Vibration study started from about 6000 years ago when the 1st musicalinstrument was invented.
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Importance of the study of vibration
Vibration is founded all around us In past, vibration was studied to understand the physical phenomena and derive
a mathematical model to describe it . In recent times, the motivation of studying vibration is the engineering
applications.
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Vibration in engineering :
1. In machines
2. Through structure
Vibration causes fatigue stresses in structures and wear in rotating parts of machine.
Importance of the study of vibration
Resonance is one of the most devastating effects of vibration on machines and structures.
Resonance happen when the natural frequency of the system equals the excitation frequency of the external excitation.
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Tacoma Narrows bridge failure due to wind excitation (July 1,1940 – November
7,1940)
Importance of the study of vibrationv
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Advantages
Health therapy
Domestic applicatio
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Industrial applicatio
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Music Disadvantages
Machine failure
Noise Structure failure
Vibration system elementary parts v
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Potential energy
Kinetic energy
Energy dissipation
Elastic element
Inertial element
Damping element
Examples
Rotational vibration
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Translational vibration
Toriosnalvibration
Degree of Freedom (DoF)
Definition
The minimum number of independent coordinates required to
determine completely the positions of all parts of the system at
any instant.
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Degree of Freedom (DoF)
Examples (single DoF)
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Degree of Freedom (DoF)
Examples (Two DoF)
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Degree of Freedom (DoF)
Examples (Three DoF)
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Degree of Freedom (DoF)
Examples (infinite DoF)
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Discrete and continues systems v
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Vibrating system
Discrete Continues
Finite DoF Infinite DoF
Vibration classifications v
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Vibration
Excitation
Free
Applying initial
excitation only and leave the system to
vibrate
Forced
Continues repeating
force acting on
the system
Damping
Damped
Energy dissipation
after a period of
time
Un-damped
No energy dissipation
and system
will vibrate for ever
Linearity
Linear
All the basic
components (spring, mass and damper ) all have linear
behaviors
None-linear
If any of the
vibration system
elements act
nonlinearly
Determent
Deterministic
If the magnitude
of excitation is known
at any time
Random
If the magnitude
of excitation can not be predicted at a given
time
Deterministic and random vibrationsv
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Vibration analysis procedures v
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Mathematical modeling
Derivation of governing equations
Solution of the governing equations
Interpretation of the results
Vibration system elements v
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Vibration system
elements
Spring Inertia Damper
Has stiffness (k)
Neglected mass and
damping effect.
Has mass (M,I)
Neglected stiffness and
damping effect.
Has damping effect (C)
Neglected mass and stiffness
Vibration system elements : spring
LoLo
L
F
δ
Applying Load
deflection (δ) = L – Lo
F = K δ where K is thespring stiffness
increasing K makes thespring stiffer.
Stiffer springs needs moreforce to deflect it
Potential energy (U):U = 0.5 k.x2
K=(d4G/8D3n)
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Wire diameter (d)Shear modules (G)
Mean coil diameter (D)Number of turns (n)
Spring special case: rod S
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l
AEk
Modules of elasticity
Cross sectional area
Spring special case: cantilever beam S
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Moment of inertia
Modules of elasticity
Cross sectional area
Spring special case: torsional stiffness of shafts S
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1
4
12
4
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1
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3212 l
dDG
l
GJkt
2
4
23
4
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2
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3223 l
dDG
l
GJkt
Spring special case: parallel connection S
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.keq = k1 + k2
Spring special case: series connection S
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21
111
kkkeq
ExampleS
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ExampleS
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Vibration system elements : mass or inertia S
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In translational motion systems, we use the mass, M (kg) .
In rotational and torsional vibration systems, we use the mass moment of inertia, I (kg.m2).
Newton’s law of motion:
Translational system:∑forces = Mass * acceleration
Rotational system:∑Moment = mass moment of
inertia *angular acceleration
Case1: translational masses connected to rigid barS
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3
2
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2
1
21 m
l
lm
l
lmmeq
Case2: translational and rotational masses coupled together S
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Equivalent translational mass
2R
Jmm o
eq
Equivalent rotational mass
2mRJJ oeq
2.2.
2
1
2
1oJxmT
Damping effect Viscous damping
Dry friction Material or solid or hysteretic damping
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Energy dissipation due to motion of mechanical parts in fluids
Amount of dissipated energy depends on:Size and shapes of vibrating bodies Fluid viscosity Vibration frequency Vibrating body velocity
The damping force is proportional to the velocity of the vibrating body
Harmonic motion S
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Periodic motion is the motion
that repeats it self after a period
of time
Harmonic motion is the
simplest type of periodic motion
x = Asin(θ) = Asin(ωt)
Velocity:
Acceleration:
tAdt
dx cos
xtAdt
xd 22
2
2
sin
Scotch mechanism
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Vector notation
y=A sin(ωt)
Complex notation
iAeiAbiaX ))sin()(cos(
P
End of chapter
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