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MAE 340 – Vibrations
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Introduction to Mechanical
Vibrations
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Instruments Ltd.
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MAE 340 – Vibrations
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Vibration
• Vibration is the study of:
the repetitive motion of objects relative to a stationary
frame of reference or nominal position (e.g., the equilibrium
position)
• Vibration is an important factor in many designs:
Products that break if they vibrate too much:
Buildings and bridges
MAE 340 – Vibrations
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Vibration
Products that can’t be used if they vibrate too
much:
Power/machine tools
Robots
Products that customers don’t like to vibrate too
much:
Seat for automobile/tractor/airplane
Products that have to vibrate in a specific way:
MAE 340 – Vibrations
Free vs. Forced Vibration
• Free Vibration vs. Forced Vibration
Free Vibration
All interfaces of the body with the
environment are static.
Forced Vibration
At least one point of the body is
subjected to periodic forces or
displacements.
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MAE 340 – Vibrations
SDOF vs. MDOF Systems
• In a Single-Degree-of-Freedom (SDOF)
System we study the motion of a rigid body
in one direction.
The motion may be rectilinear or rotational.
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MAE 340 – Vibrations
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SDOF vs. MDOF Systems
• In a Multi-Degree-of-Freedom (MDOF)
System we study the independent motions
of multiple rigid bodies or one rigid body
in multiple directions.
x4
x1
x2
k1
k2
Fcos ωt
m4
x3
m1
m2
k4
m3
x2
k3
m2
x1
k2
k
c
k
m1
c
k1
MAE 340 – Vibrations
Discrete vs. Continuous Systems
• A discrete system has rigid (lumped)
masses connected by massless, flexible
members (e.g., massless springs and dampers)
x
k
m
c
k
c
k
c
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MAE 340 – Vibrations
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Discrete vs. Continuous Systems
• A continuous system has flexible
members whose distributed mass is
significant to the vibrations
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MAE 340 – Vibrations
Spring-Mass System
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MAE 340 – Vibrations
Spring-Mass System
• Solving differential equation
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MAE 340 – Vibrations
Spring-Mass System
• How do we find A and φ?
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MAE 340 – Vibrations
Graphing x (t )
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MAE 340 – Vibrations
Working with Vibration Amplitudes
• If we have any two of:
Natural frequency =
Displacement amplitude =
Velocity amplitude =
Acceleration amplitude =
then we can get the other two.
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MAE 340 – Vibrations
Vibration Nomograph
• Use it to specify
limits on vibration:
Frequency
Displacement
amplitude
Velocity amplitude
Acceleration
amplitude
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