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MAE 340 – Vibrations
Introduction to Mechanical Vibrations
Image from E. KlingeléImage from Sound by Singer
Image from dub-connection
Image from FDR Library and Museum
Image from Hunter Engineering Co.
Image from A-Tech Instruments Ltd.
Image from Giant Bicycle Inc.
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MAE 340 – Vibrations 2
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
�
�
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MAE 340 – Vibrations 3
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:�
�
�
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MAE 340 – Vibrations 4
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 5
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.
Image from D Inman, Engineering Vibration.
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MAE 340 – Vibrations 6
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.
k1
m1
x1
k2
x2
m2
k3
m3
k4
m4
x4
x3
Fcos ωt
k2
m2
k1
m1
x2
x1
k ck c
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Discrete vs. Continuous Systems
• A discrete system has rigid (lumped)
masses connected by massless, flexible members (e.g., massless springs and dampers)
MAE 340 – Vibrations 7
k
m
x
c
k ck c
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Discrete vs. Continuous Systems
• A continuous system has flexible members whose distributed mass is significant to the vibrations
MAE 340 – Vibrations 8
Image from Dlubal Software GmbHImage from www.jeffkemp.com
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MAE 340 – Vibrations 9
Spring-Mass System
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MAE 340 – Vibrations 10
Spring-Mass System
• Solving differential equation
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MAE 340 – Vibrations 11
Spring-Mass System
• How do we find A and φ?
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MAE 340 – Vibrations 12
Graphing x (t )
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MAE 340 – Vibrations 13
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 14
Vibration Nomograph• Use it to specify
limits on vibration:
� Frequency
� Displacement amplitude
� Velocity amplitude
� Acceleration amplitude