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

Joseph Vignola



11 12 13 14 15 Odyssey Day


22 Easter break


26JV in Florida

27JV in Florida

28 29



April

May

Please keep the lab clear particularly for Odyssey Day

Joseph Vignola



11 12 13 14 15 Odyssey Day


22 Easter break


26JV in Florida

27JV in Florida

28 29



April

May

Please keep the lab clear particularly for Odyssey DayThen we have Easter break

Joseph Vignola



11 12 13 14 15 Odyssey Day


22 Easter break


26JV in Florida

27JV in Florida

28 29



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

Joseph Vignola



11 12 13 reading day 14 15 Odyssey Day


22 Easter break


26JV in Florida

27JV in Florida

28 29


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

Joseph Vignola



11 today 12 13 14 15


22 Easter break


26JV in Florida

27JV in Florida

28 29



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




Lab report 2 Monday, April 18

Lab report 3 Monday, April 18

Joseph Vignola

Citations

Please use proper citations

Joseph Vignola

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.

Joseph Vignola

Citations


I don’t care what format you use, just pick one and be consistent


Joseph Vignola

Citations



It is NOT ok to copy text from a source and follow it with a citation

Joseph Vignola

Citations




Never use anyone else’s text without quotes

Joseph Vignola

Citations




Never use anyone else’s text without quotes

And, oh, by the way, don’t quote

Joseph Vignola

Citations


All figure taken from outside must have a citation

Joseph Vignola

Citations


Taken from Vignola [1] 2009



Joseph Vignola

Citations


Taken from Vignola [1] 2009



… and the figure must have a taken from label

Joseph Vignola

Plagiarism

Please write your lab reports on your own

Joseph Vignola

Plagiarism


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

Joseph Vignola

Plagiarism



Don’t paraphrase someone else’s work

Joseph Vignola

Plagiarism



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.

Joseph Vignola

Plagiarism



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


Joseph Vignola

Plagiarism



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


What constitutes plagiarism does not change from instructor to instructor


Joseph Vignola

Block Diagrams

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

Block Diagrams


Taken from Vignola 1991

Block Diagrams


yes


Block Diagrams


yes



Block Diagrams


yes

noTaken from Vignola 1991


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



yes

Equations



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


And can predict its behavior in either time

m

k b

x(t)



And can predict its behavior in either time or frequency domain

m

k b

x(t)


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




One mode has the two masses moving together




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


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


The response to any excitation can only be composed of those modes

Clamped String



So if I pull the string at sum location, say the middle and let it go

Clamped String



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





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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


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



whereA mode shapes are

See Blevins, Formulas for Natural Frequencies and Mode shapes

Cantilever



whereRatio of frequencies

Cantilever



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



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


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


m

k b

p(t)

x(t)

So let’s imagine that Gaussian impulse again

Cantilever


m

k b

p(t)

x(t)


Cantilever


m

k b

p(t)

x(t)


Earlier in the semester we talked about Fourier’s theorem

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

Cantilever


m

k b

p(t)

x(t)


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)


Cantilever


m

k b

p(t)

x(t)


Where

Cantilever


m

k b

p(t)

x(t)


Where

And Bn are system frequency response coefficients

Cantilever


m

k b

p(t)

x(t)


The frequencies in the pulse are pulse are determined by the time width of the pulse

Cantilever


m

k b

p(t)

x(t)


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


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


m1

k1 b1

F(t)

x1(t)

m2

k2 b2

m3

k3 b3

x2(t)

x3(t)


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


m1

k1 b1

F(t)

x1(t)

m2

k2 b2

m3

k3 b3

x2(t)

x3(t)


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.

Documents

ME 392 Chapter 8 Modal Analysis & Some Other Stuff April 2, 2012 week 12 Joseph Vignola