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) 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 time MondayTuesdayWednesdayThursdayFriday Odyssey Day Easter break 22 Easter break 25 Easter break JV in Florida 26 JV in Florida 27 JV in Florida 2829 MondayTuesdayWednesdayThursdayFriday 23 Finals week4 Finals week5 Finals week6 Finals week April May

Calendar for the Rest of the Semester We are running out of time MondayTuesdayWednesdayThursdayFriday Odyssey Day Easter break 22 Easter break 25 Easter break JV in Florida 26 JV in Florida 27 JV in Florida 2829 MondayTuesdayWednesdayThursdayFriday 23 Finals week4 Finals week5 Finals week6 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 time MondayTuesdayWednesdayThursdayFriday Odyssey Day Easter break 22 Easter break 25 Easter break JV in Florida 26 JV in Florida 27 JV in Florida 2829 MondayTuesdayWednesdayThursdayFriday 23 Finals week4 Finals week5 Finals week6 Finals week April May Please keep the lab clear particularly for Odyssey Day Then we have Easter break

Calendar for the Rest of the Semester We are running out of time MondayTuesdayWednesdayThursdayFriday Odyssey Day Easter break 22 Easter break 25 Easter break JV in Florida 26 JV in Florida 27 JV in Florida 2829 MondayTuesdayWednesdayThursdayFriday 23 Finals week4 Finals week5 Finals week6 Finals week 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 Semester We are running out of time MondayTuesdayWednesdayThursdayFriday reading day1415 Odyssey Day Easter break 22 Easter break 25 Easter break JV in Florida 26 JV in Florida 27 JV in Florida 2829 MondayTuesdayWednesdayThursdayFriday 2 Senior Design Day 3 Finals week4 Finals week5 Finals week6 Finals week 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 Semester We are running out of time MondayTuesdayWednesdayThursdayFriday 11 today Easter break 22 Easter break 25 Easter break JV in Florida 26 JV in Florida 27 JV in Florida 2829 MondayTuesdayWednesdayThursdayFriday 23 Finals week4 Finals week5 Finals week6 Finals week April May hard due date You must have turned in by these dates so that I have time to grade them Assignment 2Friday, April 15 Assignment 3Friday, April 15 Assignment 4Friday, April 15 Assignment 5Friday, April 15 Lab report 2Monday, April 18 Lab report 3Monday, 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 , 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 , 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 , 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 , … 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

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 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 no Taken 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 m1m1 k1k1 b1b1 F(t) x 1 (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 m2m2 k2k2 b2b2 m3m3 k3k3 b3b3 x 2 (t) x 3 (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 where A 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 where Ratio 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 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 B n 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 m1m1 k1k1 b1b1 F(t) x 1 (t) m2m2 k2k2 b2b2 m3m3 k3k3 b3b3 x 2 (t) x 3 (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 m1m1 k1k1 b1b1 F(t) x 1 (t) m2m2 k2k2 b2b2 m3m3 k3k3 b3b3 x 2 (t) x 3 (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 m1m1 k1k1 b1b1 F(t) x 1 (t) m2m2 k2k2 b2b2 m3m3 k3k3 b3b3 x 2 (t) x 3 (t) This one has three resonance frequencies If the pulse is s wide the band width will be 31Hz

Cantilever So the system’s modal frequencies are 10, 28 & 99Hz m1m1 k1k1 b1b1 F(t) x 1 (t) m2m2 k2k2 b2b2 m3m3 k3k3 b3b3 x 2 (t) x 3 (t) We should expect to see the first and second modes But not the third If the pulse is 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.