me 392 chapter 8 modal analysis some other stuff april 2 2012 week 12 n.
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ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 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 28 Lab 3 is due April 6 Lab 4 is due April 20 (this Friday)

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assignments
Assignments

I would like to offer to everyone the extra help you might need to catch up.

Assignment 5 was due March 28

Lab 3 is due April 6

Lab 4 is due April 20 (this Friday)

Assignment 6 was due April 27

use templates
Use Templates

Please make and use templates in Word for the assignments and lab reports

You can make a template out of lab 1

assignments1
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
Calendar for the Rest of the Semester

We are running out of time

April

May

calendar for the rest of the semester1
Calendar for the Rest of the Semester

We are running out of time

April

May

Please keep the lab clear particularly for Odyssey Day

calendar for the rest of the semester2
Calendar for the Rest of the Semester

We are running out of time

April

May

Please keep the lab clear particularly for Odyssey Day

Then we have Easter break

calendar for the rest of the semester3
Calendar for the Rest of the Semester

We are running out of time

April

May

Please keep the lab clear particularly for Odyssey Day

Then we have Easter break

Then I am out of town for a few days

calendar for the rest of the semester4
Calendar for the Rest of the Semester

We are running out of time

April

May

Please keep the lab clear particularly for Odyssey Day

Then we have Easter break

Then I am out of town for a few days

Senior Design Day

Then finals week

calendar for the rest of the semester5
Calendar for the Rest of the Semester

We are running out of time

April

May

hard due date

You must have turned in by these dates so that I have time to grade them

citations
Citations

Please use proper citations

citations1
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.

citations2
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.

citations3
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

citations4
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

citations5
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

citations6
Citations

Please use proper citations

All figure taken from outside must have a citation

citations7
Citations

Please use proper citations

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.

Taken from Vignola [1] 2009

citations8
Citations

Please use proper citations

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

Taken from Vignola [1] 2009

plagiarism
Plagiarism

Please write your lab reports on your own

plagiarism1
Plagiarism

Please write your lab reports on your own

I will help you and you can also ask the TAs for help

plagiarism2
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

plagiarism3
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.

plagiarism4
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

plagiarism5
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
Block Diagrams

Please don’t include the LabView block diagram in your lab reports

block diagrams1
Block Diagrams

Please don’t include the LabView block diagram in your lab reports

Taken from Vignola 1991

block diagrams2
Block Diagrams

Please don’t include the LabView block diagram in your lab reports

yes

Taken from Vignola 1991

block diagrams3
Block Diagrams

Please don’t include the LabView block diagram in your lab reports

yes

Taken from Vignola 1991

Taken from Vignola 2011

block diagrams4
Block Diagrams

Please don’t include the LabView block diagram in your lab reports

yes

no

Taken from Vignola 1991

Taken from Vignola 2011

matlab code
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

Don’t include

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;

equations
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

equations1
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

equations2
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
Single Degree of Freedom Oscillator

Last week we looked at the single degree of freedom

k

b

m

x(t)

single degree of freedom oscillator1
Single Degree of Freedom Oscillator

Last week we looked at the single degree of freedom

k

b

m

x(t)

And can predict its behavior in either time

single degree of freedom oscillator2
Single Degree of Freedom Oscillator

Last week we looked at the single degree of freedom

k

b

m

x(t)

And can predict its behavior in either time or frequency domain

multi degree of freedom oscillators
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 oscillators1
Multi Degree of Freedom Oscillators

…what I didn’t tell you last week was that real structures have more that one resonance

k1

b1

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

m1

x1(t)

k2

b2

m2

x2(t)

k3

b3

x3(t)

m3

F(t)

two degree of freedom oscillator
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 oscillator1
Two Degree of Freedom Oscillator

As an example consider two mass constrained to only move side to side

One mode has the two masses moving together

Taken from Judge, ME 392 Notes

two degree of freedom oscillator2
Two Degree of Freedom Oscillator

As an example consider two mass constrained to only move side to side

One mode has the two masses moving together

The other mode has the masses moving in opposition

Taken from Judge, ME 392 Notes

clamped string
Clamped String

Strings like those on musical instruments respond are discreet frequencies

A mode can only be excited at one frequency

clamped string1
Clamped String

Strings like those on musical instruments respond are discreet frequencies

A mode can only be excited at one frequency

For the guitar string the mode frequencies are integer multiplies of the lowest (the fundamental)

The mode shapes are simple sine functions

clamped string2
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 string3
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 string4
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 string5
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 string6
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 string7
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 string8
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 string9
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 string10
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 string11
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 string12
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 string13
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 string14
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 string15
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 string16
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 string17
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 string18
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 string19
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 string20
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 string21
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 string22
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 string23
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 string24
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 string25
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 string26
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 string27
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 string28
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 string29
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 string30
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 string31
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 string32
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 string33
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 string34
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 string35
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 string36
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 string37
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 string38
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 string39
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 string40
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 string41
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 string42
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 string43
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 string44
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 string45
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 string46
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 string47
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 string48
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 string49
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 string50
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 string51
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 string52
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 string53
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 string54
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 string55
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 string56
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 string57
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 string58
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 string59
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 string60
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 string61
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 string62
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 string63
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 string64
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
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)

cantilever1
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

cantilever2
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

A mode shapes are

See Blevins, Formulas for Natural Frequencies and Mode shapes

cantilever3
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

Ratio of frequencies

cantilever4
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

cantilever5
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
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

We can strike a structure and record a force time history

Taken from www.pcb.com

making a pulse1
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 pulse2
Making a Pulse

My code that makes Gaussian pulses is on the class webpage

There is one for both Matlab and LabView

cantilever6
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

k

b

So if the driving force is

m

x(t)

p(t)

The steady state displacement response will be something like

Non-linearities will make it something like

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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

k

b

So if the driving force is

m

x(t)

p(t)

The steady state displacement response will be something like

But we’re not going to worry about this

Non-linearities will make it something like

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Cantilever

If the frequency in is the frequency out…

and amplitude of the output is big at resonance.

k

b

m

x(t)

p(t)

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Cantilever

If the frequency in is the frequency out…

and amplitude of the output is big at resonance.

k

b

So let’s imagine that Gaussian impulse again

m

x(t)

p(t)

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Cantilever

If the frequency in is the frequency out…

and amplitude of the output is big at resonance.

k

b

So let’s imagine that Gaussian impulse again

m

x(t)

p(t)

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Cantilever

If the frequency in is the frequency out…

and amplitude of the output is big at resonance.

k

b

So let’s imagine that Gaussian impulse again

m

x(t)

p(t)

Earlier in the semester we talked about Fourier’s theorem

Any function of time can be expressed as a sum of sinusoids

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Cantilever

If the frequency in is the frequency out…

and amplitude of the output is big at resonance.

k

b

So let’s imagine that Gaussian impulse again

m

x(t)

p(t)

Earlier in the semester we talked about Fourier’s theorem

Any function of time can be expressed as a sum of sinusoids

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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

k

b

So let’s imagine that Gaussian impulse again

m

x(t)

p(t)

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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

k

b

So let’s imagine that Gaussian impulse again

m

x(t)

p(t)

Where

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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

k

b

So let’s imagine that Gaussian impulse again

m

x(t)

p(t)

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

k

b

So let’s imagine that Gaussian impulse again

m

x(t)

p(t)

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

k

b

So let’s imagine that Gaussian impulse again

m

x(t)

p(t)

The bandwidth of the response is inversely proportional to the width of the pulse

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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

This one has three resonance frequencies

If you drive it at one frequency it will respond at that frequency only

k1

b1

m1

x1(t)

k2

b2

m2

x2(t)

k3

b3

x3(t)

m3

F(t)

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Cantilever

Think about a system that has more that one resonance

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

k1

b1

m1

x1(t)

k2

b2

m2

x2(t)

k3

b3

x3(t)

m3

F(t)

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Think about a system that has more that one resonance

This one has three resonance frequencies

k1

b1

m1

x1(t)

If the pulse is 0.005 s wide

the band width will be 31Hz

k2

b2

m2

x2(t)

k3

b3

x3(t)

m3

F(t)

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So the system’s modal frequencies are 10, 28 & 99Hz

We should expect to

see the first and second modes

But not the third

k1

b1

m1

x1(t)

If the pulse is 0.005 s wide

the band width will be 31Hz

k2

b2

m2

x2(t)

k3

b3

x3(t)

m3

F(t)

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  • 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.