ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola.
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Transcript of ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola.
ME 392Chapter 8
Modal Analysis& Some Other Stuff
April 2, 2012week 12
Joseph Vignola
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
Use Templates
Please make and use templates in Word for the assignments and lab reports
You can make a template out of lab 1
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
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
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
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
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
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
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
Citations
Please use proper citations
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.
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.
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
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
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
Citations
Please use proper citations
All figure taken from outside must have a citation
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.
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
Plagiarism
Please write your lab reports on your own
Plagiarism
Please write your lab reports on your own
I will help you and you can also ask the TAs for help
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
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.
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
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
Block Diagrams
Please don’t include the LabView block diagram in your lab reports
Block Diagrams
Please don’t include the LabView block diagram in your lab reports
Taken from Vignola 1991
Block Diagrams
Please don’t include the LabView block diagram in your lab reports
yes
Taken from Vignola 1991
Block Diagrams
Please don’t include the LabView block diagram in your lab reports
yes
Taken from Vignola 1991
Taken from Vignola 2011
Block Diagrams
Please don’t include the LabView block diagram in your lab reports
yes
noTaken from Vignola 1991
Taken from Vignola 2011
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
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
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
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);
Single Degree of Freedom Oscillator
m
k b
x(t)
Last week we looked at the single degree of freedom
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
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
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.
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)
Two Degree of Freedom Oscillator
As an example consider two mass constrained to only move side to side
Taken from Judge, ME 392 Notes
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
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
Clamped String
Strings like those on musical instruments respond are discreet frequencies
A mode can only be excited at one frequency
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)
Clamped String
Strings like those on musical instruments respond are discreet frequencies
The response to any excitation can only be composed of those modes
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
Making a Pulse
My code that makes Gaussian pulses is on the class webpage
There is one for both Matlab and LabView
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
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
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)
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
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
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
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
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
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
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
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
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
Cantilever
Think about a system that has more that one resonance
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
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
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
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
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.