me 392 chapter 8 modal analysis & some other stuff april 2, 2012 week 12 joseph vignola
TRANSCRIPT
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ME 392Chapter 8
Modal Analysis& Some Other Stuff
April 2, 2012week 12
Joseph Vignola
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Assignments
I would like to offer to everyone the extra help you might need to catch up.
Assignment 5 was due March 28Lab 3 is due April 6
Lab 4 is due April 20 (this Friday)Assignment 6 was due April 27
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Use Templates
Please make and use templates in Word for the assignments and lab reports
You can make a template out of lab 1
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Assignments
I have gotten several write-ups that don’t include plots or results
I sent out an example for assignment 5
Please make sure you include Matlab plots and results
Please find the slopes and the linear limit of the shaker
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Calendar for the Rest of the Semester
We are running out of timeMonday Tuesday Wednesday Thursday Friday
11 12 13 14 15 Odyssey Day
18 19 20 21 Easter break
22 Easter break
25 Easter breakJV in Florida
26JV in Florida
27JV in Florida
28 29
Monday Tuesday Wednesday Thursday Friday
2 3 Finals week 4 Finals week 5 Finals week 6 Finals week
April
May
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Calendar for the Rest of the Semester
We are running out of timeMonday Tuesday Wednesday Thursday Friday
11 12 13 14 15 Odyssey Day
18 19 20 21 Easter break
22 Easter break
25 Easter breakJV in Florida
26JV in Florida
27JV in Florida
28 29
Monday Tuesday Wednesday Thursday Friday
2 3 Finals week 4 Finals week 5 Finals week 6 Finals week
April
May
Please keep the lab clear particularly for Odyssey Day
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Calendar for the Rest of the Semester
We are running out of timeMonday Tuesday Wednesday Thursday Friday
11 12 13 14 15 Odyssey Day
18 19 20 21 Easter break
22 Easter break
25 Easter breakJV in Florida
26JV in Florida
27JV in Florida
28 29
Monday Tuesday Wednesday Thursday Friday
2 3 Finals week 4 Finals week 5 Finals week 6 Finals week
April
May
Please keep the lab clear particularly for Odyssey DayThen we have Easter break
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Calendar for the Rest of the Semester
We are running out of timeMonday Tuesday Wednesday Thursday Friday
11 12 13 14 15 Odyssey Day
18 19 20 21 Easter break
22 Easter break
25 Easter breakJV in Florida
26JV in Florida
27JV in Florida
28 29
Monday Tuesday Wednesday Thursday Friday
2 3 Finals week 4 Finals week 5 Finals week 6 Finals week
April
May
Please keep the lab clear particularly for Odyssey DayThen we have Easter break Then I am out of town for a few days
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Calendar for the Rest of the Semester
We are running out of timeMonday Tuesday Wednesday Thursday Friday
11 12 13 reading day 14 15 Odyssey Day
18 19 20 21 Easter break
22 Easter break
25 Easter breakJV in Florida
26JV in Florida
27JV in Florida
28 29
Monday Tuesday Wednesday Thursday Friday
2 Senior Design Day 3 Finals week 4 Finals week 5 Finals week 6 Finals week
April
May
Please keep the lab clear particularly for Odyssey DayThen we have Easter break Then I am out of town for a few daysSenior Design DayThen finals week
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Calendar for the Rest of the Semester
We are running out of timeMonday Tuesday Wednesday Thursday Friday
11 today 12 13 14 15
18 19 20 21 Easter break
22 Easter break
25 Easter breakJV in Florida
26JV in Florida
27JV in Florida
28 29
Monday Tuesday Wednesday Thursday Friday
2 3 Finals week 4 Finals week 5 Finals week 6 Finals week
April
May
hard due dateYou must have turned in by these dates so that I have time to grade them
Assignment 2 Friday, April 15
Assignment 3 Friday, April 15
Assignment 4 Friday, April 15
Assignment 5 Friday, April 15
Lab report 2 Monday, April 18
Lab report 3 Monday, April 18
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Citations
Please use proper citations
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Citations
Please use proper citations
[1] J. F. Vignola, et al., Shaping of a system's frequency response using an array of subordinate oscillators, J. Acoust. Soc. Am., vol. 126, pp. 129-139, 2009.
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Citations
Please use proper citations
I don’t care what format you use, just pick one and be consistent
[1] J. F. Vignola, et al., Shaping of a system's frequency response using an array of subordinate oscillators, J. Acoust. Soc. Am., vol. 126, pp. 129-139, 2009.
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Citations
Please use proper citations
I don’t care what format you use, just pick one and be consistent
It is NOT ok to copy text from a source and follow it with a citation
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Citations
Please use proper citations
I don’t care what format you use, just pick one and be consistent
It is NOT ok to copy text from a source and follow it with a citation
Never use anyone else’s text without quotes
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Citations
Please use proper citations
I don’t care what format you use, just pick one and be consistent
It is NOT ok to copy text from a source and follow it with a citation
Never use anyone else’s text without quotes
And, oh, by the way, don’t quote
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Citations
Please use proper citations
All figure taken from outside must have a citation
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Citations
Please use proper citations
Taken from Vignola [1] 2009
All figure taken from outside must have a citation
[1] J. F. Vignola, et al., Shaping of a system's frequency response using an array of subordinate oscillators, J. Acoust. Soc. Am., vol. 126, pp. 129-139, 2009.
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Citations
Please use proper citations
Taken from Vignola [1] 2009
All figure taken from outside must have a citation
[1] J. F. Vignola, et al., Shaping of a system's frequency response using an array of subordinate oscillators, J. Acoust. Soc. Am., vol. 126, pp. 129-139, 2009.
… and the figure must have a taken from label
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Plagiarism
Please write your lab reports on your own
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Plagiarism
Please write your lab reports on your own
I will help you and you can also ask the TAs for help
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Plagiarism
Please write your lab reports on your own
I will help you and you can also ask the TAs for help
Don’t paraphrase someone else’s work
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Plagiarism
Please write your lab reports on your own
I will help you and you can also ask the TAs for help
Don’t paraphrase someone else’s work……not from someone in this class or anyone who took the class sometime in the past or anyone.
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Plagiarism
Please write your lab reports on your own
I will help you and you can also ask the TAs for help
Don’t paraphrase someone else’s work……not from someone in this class or anyone who took the class sometime in the past or anyone. The University of Wisconsin has an excellent web page that discuss plagiarism in detail
http://students.wisc.edu/saja/misconduct/UWS14.html
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Plagiarism
Please write your lab reports on your own
I will help you and you can also ask the TAs for help
Don’t paraphrase someone else’s work……not from someone in this class or anyone who took the class sometime in the past or anyone. The University of Wisconsin has an excellent web page that discuss plagiarism in detail
http://students.wisc.edu/saja/misconduct/UWS14.html
What constitutes plagiarism does not change from instructor to instructor
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Block Diagrams
Please don’t include the LabView block diagram in your lab reports
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Block Diagrams
Please don’t include the LabView block diagram in your lab reports
Taken from Vignola 1991
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Block Diagrams
Please don’t include the LabView block diagram in your lab reports
yes
Taken from Vignola 1991
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Block Diagrams
Please don’t include the LabView block diagram in your lab reports
yes
Taken from Vignola 1991
Taken from Vignola 2011
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Block Diagrams
Please don’t include the LabView block diagram in your lab reports
yes
noTaken from Vignola 1991
Taken from Vignola 2011
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Matlab Code
Matlab code NEVER belongs in the body of a lab report
Put it in the appendix if you think it is really necessary
L = .7;x = L*[0:.005:1];h = .003;[f,t] = freqtime(.0004,1024); n=1:15;k = pi*n'/L;An = (-4*h./((pi*n).^2)); a = 1/7;Bn = (2*h./((pi*n).^2))*(1/(a*(1-a))).*sin(a*n*pi);c = 100;omega = n*pi*c/L;
Don’t include
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Equations
Please use the equation editor in MS Word for all equations
It is easy to use but if you don’t know how to use it please let me help you learn it
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Equations
Please use the equation editor in MS Word for all equations
It is easy to use but if you don’t know how to use it please let me help you learn it
yes
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Equations
Please use the equation editor in MS Word for all equations
It is easy to use but if you don’t know how to use it please let me help you learn it
yes
no Bn = (2*h./((pi*n).^2))*(1/(a*(1-a))).*sin(a*n*pi);
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Single Degree of Freedom Oscillator
m
k b
x(t)
Last week we looked at the single degree of freedom
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Single Degree of Freedom Oscillator
And can predict its behavior in either time
m
k b
x(t)
Last week we looked at the single degree of freedom
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Single Degree of Freedom Oscillator
And can predict its behavior in either time or frequency domain
m
k b
x(t)
Last week we looked at the single degree of freedom
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Multi Degree of Freedom Oscillators
…what I didn’t tell you last week was that real structures have more that one resonance
Every one of these resonance has a particular frequency
And a specific shape that describes the way it deforms.
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Multi Degree of Freedom Oscillators
…what I didn’t tell you last week was that real structures have more that one resonance
m1
k1 b1
F(t)
x1(t)
Every one of these resonance has a particular frequency
And a specific shape that describes the way it deforms.
We can model a structure as a collection masses – springs and dampers
m2
k2 b2
m3
k3 b3
x2(t)
x3(t)
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Two Degree of Freedom Oscillator
As an example consider two mass constrained to only move side to side
Taken from Judge, ME 392 Notes
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Two Degree of Freedom Oscillator
As an example consider two mass constrained to only move side to side
Taken from Judge, ME 392 Notes
One mode has the two masses moving together
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Two Degree of Freedom Oscillator
As an example consider two mass constrained to only move side to side
Taken from Judge, ME 392 Notes
One mode has the two masses moving together
The other mode has the masses moving in opposition
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Clamped String
Strings like those on musical instruments respond are discreet frequencies
A mode can only be excited at one frequency
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Clamped String
Strings like those on musical instruments respond are discreet frequencies
The mode shapes are simple sine functions
A mode can only be excited at one frequency
For the guitar string the mode frequencies are integer multiplies of the lowest (the fundamental)
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Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
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Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go
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Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
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Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 49: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/49.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 50: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/50.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 51: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/51.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 52: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/52.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 53: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/53.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 54: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/54.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 55: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/55.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 56: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/56.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 57: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/57.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 58: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/58.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 59: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/59.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 60: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/60.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 61: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/61.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 62: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/62.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 63: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/63.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 64: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/64.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 65: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/65.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 66: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/66.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 67: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/67.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 68: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/68.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 69: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/69.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 70: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/70.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 71: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/71.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 72: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/72.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 73: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/73.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 74: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/74.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 75: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/75.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 76: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/76.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 77: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/77.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 78: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/78.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 79: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/79.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 80: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/80.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 81: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/81.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 82: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/82.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 83: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/83.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 84: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/84.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 85: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/85.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 86: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/86.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 87: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/87.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 88: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/88.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 89: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/89.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 90: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/90.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 91: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/91.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 92: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/92.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 93: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/93.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 94: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/94.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 95: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/95.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 96: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/96.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 97: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/97.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 98: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/98.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 99: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/99.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 100: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/100.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 101: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/101.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 102: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/102.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 103: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/103.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 104: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/104.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 105: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/105.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 106: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/106.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 107: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/107.jpg)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
So if I pull the string at sum location, say the middle and let it go…
You can predict the displacement field by adding up a modes
![Page 108: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/108.jpg)
Cantilever
We used a cantilever for as a single degree of freedom oscillator in lab 3
A simple beam has an infinite number of resonances (this is the same as saying an infinite number of modes)
![Page 109: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/109.jpg)
Cantilever
We used a cantilever for as a single degree of freedom oscillator in lab 3
A simple beam has an infinite number of resonances (this is the same as saying an infinite number of modes) A mode frequencies are
where
See Blevins, Formulas for Natural Frequencies and Mode shapes
![Page 110: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/110.jpg)
Cantilever
We used a cantilever for as a single degree of freedom oscillator in lab 3
A simple beam has an infinite number of resonances (this is the same as saying an infinite number of modes) A mode frequencies are
whereA mode shapes are
See Blevins, Formulas for Natural Frequencies and Mode shapes
![Page 111: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/111.jpg)
Cantilever
We used a cantilever for as a single degree of freedom oscillator in lab 3
A simple beam has an infinite number of resonances (this is the same as saying an infinite number of modes) A mode frequencies are
whereRatio of frequencies
![Page 112: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/112.jpg)
Cantilever
We used a cantilever for as a single degree of freedom oscillator in lab 3
A simple beam has an infinite number of resonances (this is the same as saying an infinite number of modes)
You will put three accelerometers along the length of the beam and drive it with a hammer that has a force sensor on it
![Page 113: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/113.jpg)
Cantilever
We used a cantilever for as a single degree of freedom oscillator in lab 3
A simple beam has an infinite number of resonances (this is the same as saying an infinite number of modes)
You will put three accelerometers along the length of the beam and drive it with a hammer that has a force sensor on it or an electronic pulse from the shaker with at Gaussian pulse that is provided on the class webpage
![Page 114: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/114.jpg)
Making a Pulse
We used a cantilever for as a single degree of freedom oscillator in lab 3
I you use the hammer to strike the you need to look at the time history to be sure you have a clean single strike
Taken from www.pcb.com
We can strike a structure and record a force time history
![Page 115: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/115.jpg)
Making a Pulse
If you use the Gaussian pulse from the class webpage you can program the time width of the pulse
You will notice that as you make the time width wider the frequency band will get narrower
![Page 116: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/116.jpg)
Making a Pulse
My code that makes Gaussian pulses is on the class webpage
There is one for both Matlab and LabView
![Page 117: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/117.jpg)
Cantilever
One thing we’ve learned is that if I force a structure at a specific frequency, it will respond at that frequency… … at least in the linear world
m
k b
p(t)
x(t)So if the driving force is
The steady state displacement response will be something like
Non-linearities will make it something like
![Page 118: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/118.jpg)
Cantilever
One thing we’ve learned is that if I force a structure at a specific frequency, it will respond at that frequency… … at least in the linear world
m
k b
p(t)
x(t)So if the driving force is
The steady state displacement response will be something like
Non-linearities will make it something like
But we’re not going to worry about this
![Page 119: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/119.jpg)
Cantilever
If the frequency in is the frequency out…and amplitude of the output is big at resonance.
m
k b
p(t)
x(t)
![Page 120: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/120.jpg)
Cantilever
If the frequency in is the frequency out…and amplitude of the output is big at resonance.
m
k b
p(t)
x(t)
So let’s imagine that Gaussian impulse again
![Page 121: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/121.jpg)
Cantilever
If the frequency in is the frequency out…and amplitude of the output is big at resonance.
m
k b
p(t)
x(t)
So let’s imagine that Gaussian impulse again
![Page 122: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/122.jpg)
Cantilever
If the frequency in is the frequency out…and amplitude of the output is big at resonance.
m
k b
p(t)
x(t)
So let’s imagine that Gaussian impulse again
Earlier in the semester we talked about Fourier’s theorem
Any function of time can be expressed as a sum of sinusoids
![Page 123: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/123.jpg)
Cantilever
If the frequency in is the frequency out…and amplitude of the output is big at resonance.
m
k b
p(t)
x(t)
So let’s imagine that Gaussian impulse again
Earlier in the semester we talked about Fourier’s theorem
Any function of time can be expressed as a sum of sinusoids
![Page 124: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/124.jpg)
Cantilever
If the frequency in is the frequency out……than the output is made from the same frequencies as the input
m
k b
p(t)
x(t)
So let’s imagine that Gaussian impulse again
![Page 125: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/125.jpg)
Cantilever
If the frequency in is the frequency out……than the output is made from the same frequencies as the input
m
k b
p(t)
x(t)
So let’s imagine that Gaussian impulse again
Where
![Page 126: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/126.jpg)
Cantilever
If the frequency in is the frequency out……than the output is made from the same frequencies as the input
m
k b
p(t)
x(t)
So let’s imagine that Gaussian impulse again
Where
And Bn are system frequency response coefficients
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Cantilever
If the frequency in is the frequency out……than the output is made from the same frequencies as the input
m
k b
p(t)
x(t)
So let’s imagine that Gaussian impulse again
The frequencies in the pulse are pulse are determined by the time width of the pulse
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Cantilever
If the frequency in is the frequency out……than the output is made from the same frequencies as the input
m
k b
p(t)
x(t)
So let’s imagine that Gaussian impulse again
The bandwidth of the response is inversely proportional to the width of the pulse
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Cantilever
Think about a system that has more that one resonance
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Cantilever
Think about a system that has more that one resonance
m1
k1 b1
F(t)
x1(t)
m2
k2 b2
m3
k3 b3
x2(t)
x3(t)
This one has three resonance frequencies
If you drive it at one frequency it will respond at that frequency only
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Cantilever
Think about a system that has more that one resonance
m1
k1 b1
F(t)
x1(t)
m2
k2 b2
m3
k3 b3
x2(t)
x3(t)
This one has three resonance frequencies
If you drive it at one frequency …it will respond at that frequency only
If you excite it with many frequencies, a broad band excitation…it will respond at any of the mode frequencies in the excitation band
![Page 132: ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola](https://reader031.vdocuments.net/reader031/viewer/2022032802/56649e035503460f94aedc07/html5/thumbnails/132.jpg)
Cantilever
Think about a system that has more that one resonance
m1
k1 b1
F(t)
x1(t)
m2
k2 b2
m3
k3 b3
x2(t)
x3(t)
This one has three resonance frequencies
If the pulse is 0.005 s wide the band width will be 31Hz
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Cantilever
So the system’s modal frequencies are 10, 28 & 99Hz
m1
k1 b1
F(t)
x1(t)
m2
k2 b2
m3
k3 b3
x2(t)
x3(t)
We should expect to see the first and second modes But not the third
If the pulse is 0.005 s wide the band width will be 31Hz
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Cantilever
• You will record the from from the hammer as well has three accelerometers responses
• Form three transfer functions in frequency domain
• Extract the magnitude and phase at the first three peaks for the transfer functions
• Normalize the magnitude and phase of the peaks to the magnitude and phase of the first
• For each of the three frequencies plot the normalized responses as a function of accelerometer position.