me 392 chapter 7 single degree of freedom oscillator march 26 , 2012 week 11
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ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11. Joseph Vignola. Assignments . I would like to offer to everyone the extra help you might need to catch up. Assignment 5 is due today Lab 3 is March 30 ( next Friday ). File Names, Title Pages & Information. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/1.jpg)
ME 392Chapter 7
Single Degree of Freedom Oscillator
March 26, 2012week 11
Joseph Vignola
![Page 2: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/2.jpg)
Assignments I would like to offer to everyone the extra help you might need to catch up.
Assignment 5 is due todayLab 3 is March 30 (next Friday)
![Page 3: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/3.jpg)
File Names, Title Pages & Information
Please use file names that I can search for
For example
“ME_392_assignment_5_smith_johnson.doc”
Please include information at the top of any document you give me. Most importantly:
NameDate What it isLab partner
![Page 4: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/4.jpg)
Single Degree of Freedom OscillatorThe single degree of freedom (SDoF) oscillator is a starting model for many problems
m
k b
F(t)
x(t)
![Page 5: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/5.jpg)
Single Degree of Freedom OscillatorThe single degree of freedom (SDoF) oscillator is a starting model for many problems
A mass, m is free to move along one axes only. Here the x-axis
m
k b
F(t)
x(t)
![Page 6: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/6.jpg)
Single Degree of Freedom OscillatorThe single degree of freedom (SDoF) oscillator is a starting model for many problems
A mass, m is free to move along one axes only. Here the x-axis
What is the first thing you do with a problem like this?
m
k b
F(t)
x(t)
![Page 7: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/7.jpg)
Single Degree of Freedom OscillatorThe single degree of freedom (SDoF) oscillator is a starting model for many problems
A mass, m is free to move along one axes only. Here the x-axis
What is the first thing you do with a problem like this?
Draw a free body diagram
m
k b
F(t)
x(t)
![Page 8: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/8.jpg)
Single Degree of Freedom OscillatorThe single degree of freedom (SDoF) oscillator is a starting model for many problems
A mass, m is free to move along one axes only. Here the x-axis
What is the first thing you do with a problem like this?
m
k b
F(t)
x(t)
m
F(t)
![Page 9: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/9.jpg)
Single Degree of Freedom OscillatorThe single degree of freedom (SDoF) oscillator is a starting model for many problems
A mass, m is free to move along one axes only. Here the x-axis
A spring that pulls the mass back to its equilibrium position
m
k b
F(t)
x(t)
m
F(t)
![Page 10: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/10.jpg)
Single Degree of Freedom OscillatorThe single degree of freedom (SDoF) oscillator is a starting model for many problems
A mass, m is free to move along one axes only. Here the x-axis
A spring that pulls the mass back to its equilibrium position. The spring force is
m
k b
F(t)
x(t)
m
F(t)
![Page 11: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/11.jpg)
Single Degree of Freedom OscillatorThe single degree of freedom (SDoF) oscillator is a starting model for many problems
A mass, m is free to move along one axes only. Here the x-axis
A spring that pulls the mass back to its equilibrium position. The spring force is
A damper slows the mass by removing energy. Force is proportional to velocity
m
k b
F(t)
x(t)
m
F(t)
![Page 12: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/12.jpg)
Single Degree of Freedom OscillatorThe single degree of freedom (SDoF) oscillator is a starting model for many problems
A mass, m is free to move along one axes only. Here the x-axis
A spring that pulls the mass back to its equilibrium position. The spring force is
A damper slows the mass by removing energy. Force is proportional to velocity
A force drives the mass
m
k b
F(t)
x(t)
m
F(t)
![Page 13: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/13.jpg)
Single Degree of Freedom OscillatorThe single degree of freedom (SDoF) oscillator is a starting model for many problems
A mass, m is free to move along one axes only. Here the x-axis
A spring that pulls the mass back to its equilibrium position. The spring force is
A damper slows the mass by removing energy. Force is proportional to velocity
A force drives the mass
m
k b
F(t)
x(t)
m
F(t)
![Page 14: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/14.jpg)
Single Degree of Freedom OscillatorThe single degree of freedom (SDoF) oscillator is a starting model for many problems
A mass, m is free to move along one axes only. Here the x-axis
A spring that pulls the mass back to its equilibrium position. The spring force is
A damper slows the mass by removing energy. Force is proportional to velocity A force drives the mass
m
k b
F(t)
x(t)
m
F(t)
![Page 15: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/15.jpg)
Single Degree of Freedom Oscillator
m
k b
F(t)
x(t)
![Page 16: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/16.jpg)
Single Degree of Freedom Oscillator
This equation can be written as
m
k b
F(t)
x(t)
![Page 17: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/17.jpg)
Single Degree of Freedom Oscillator
This equation can be written as
Let’s solve the inhomogeneous problem
m
k b
F(t)
x(t)
![Page 18: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/18.jpg)
Single Degree of Freedom Oscillator
This equation can be written as
Define two terms
m
k b
x(t)
![Page 19: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/19.jpg)
Single Degree of Freedom Oscillator
This equation can be written as
Define two terms
m
k b
x(t)
is called the natural frequency and has units of radian/second
![Page 20: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/20.jpg)
Single Degree of Freedom Oscillator
This equation can be written as
Define two terms
m
k b
x(t)
is called the natural frequency and has units of radian/second
is the damping ratio and is dimensionless
![Page 21: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/21.jpg)
Single Degree of Freedom OscillatorYou will determine the natural frequency and damping ratio of Lab 3
Define two terms
m
k b
x(t)
is called the natural frequency and has units of radian/second
is the damping ratio and is dimensionless
![Page 22: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/22.jpg)
Single Degree of Freedom Oscillator
The solution to this ODE with initial conditions
is…
m
k b
x(t)
The behavior of the system depends on
![Page 23: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/23.jpg)
Single Degree of Freedom Oscillator
The solution to this ODE with initial conditions
is
m
k b
x(t)
![Page 24: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/24.jpg)
Single Degree of Freedom Oscillator
The solution to this ODE with initial conditions
is
m
k b
x(t)
The period of the oscillation is
![Page 25: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/25.jpg)
Single Degree of Freedom Oscillator
The solution to this ODE with initial conditions
is
m
k b
x(t)
![Page 26: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/26.jpg)
Single Degree of Freedom Oscillator
The solution to this ODE with initial conditions
is
m
k b
x(t)
In this expression the time constantis related to other physical parameters by
![Page 27: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/27.jpg)
The system response is sinusoidal
has natural frequency of
There's an exponential decay Where
So we can extract the damping ratio, ζ if we can measure
Summary of Free Ring-down
m
k b
x(t)
![Page 28: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/28.jpg)
The system response is sinusoidal
has natural frequency of
There's an exponential decay Where
So we can extract the damping ratio, ζ if we can measure
Summary of Free Ring-down
m
k b
x(t)
The greater the damping the wider the resonance peak
![Page 29: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/29.jpg)
Summary of Free Ring-down
m
k b
And plot response as a function of frequency
This leads to another way to estimate the damping ratio, ζ
we can drive the oscillator at a series for frequencies and measure the response amplitude
![Page 30: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/30.jpg)
Single Degree of Freedom OscillatorAnd plot response as a function of frequency
We always assume that there is some error in our measurement.
![Page 31: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/31.jpg)
Single Degree of Freedom Oscillator… so for a plot with perhaps 20 measurements
![Page 32: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/32.jpg)
Single Degree of Freedom Oscillator… so for a plot with perhaps 20 measurements we can curve fit to extract the width of the resonance curve
![Page 33: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/33.jpg)
Details of the Time FitLet’s assume we have noisy ring down data
![Page 34: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/34.jpg)
Details of the Time FitLet’s assume we have noisy ring down data
We can generate the envelope using the magnitude of the Hilbert transform
![Page 35: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/35.jpg)
Details of the Time FitLet’s assume we have noisy ring down data
We can generate the envelope using the magnitude of the Hilbert transform
On a log scale at least the beginning looks linear
This means that we can use polyfit.m
![Page 36: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/36.jpg)
Details of the Time FitLet’s assume we have noisy ring down data
We can generate the envelope using the magnitude of the Hilbert transform
On a log scale at least the beginning looks linear
This means that we can use polyfit.m
![Page 37: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/37.jpg)
Details of the Time FitLet’s assume we have noisy ring down data
We can generate the envelope using the magnitude of the Hilbert transform
On a log scale at least the beginning looks linear
This means that we can use polyfit.m
![Page 38: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/38.jpg)
Details of the Time FitLet’s assume we have noisy ring down data
We can generate the envelope using the magnitude of the Hilbert transform
On a log scale at least the beginning looks linear
This means that we can use polyfit.m
![Page 39: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/39.jpg)
Details of the Time FitLet’s assume we have noisy ring down data
We can generate the envelope using the magnitude of the Hilbert transform
On a log scale at least the beginning looks linear
This means that we can use polyfit.m
Where is the time constant
![Page 40: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/40.jpg)
Details of the Time FitLet’s assume we have noisy ring down data
We can generate the envelope using the magnitude of the Hilbert transform
On a log scale at least the beginning looks linear
This means that we can use polyfit.m
Where is the time constant
![Page 41: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/41.jpg)
Details of the Time Fitsf = 10000;N = 10000;si = 1/sf;k = 1e5;m = 2;x0 = 3;[f,t] = freqtime(si,N); omegac = sqrt(k/m);fc = omegac/(2*pi);zeta = .05;tau= 1 ./(omegac*zeta); env = x0*exp(-t*(1../tau));displacement = env.*(cos(omegac*t)*ones(size(tau))) + .075*randn(size(t));DISPLACEMENT = fft(displacement);[a,b] = max(abs(DISPLACEMENT));fc_data = f(b); env = abs(hilbert(displacement));lenv =log(env);fit_range = [.01 .2];[a,bin_range(1)] = min(abs(t-fit_range(1)));[a,bin_range(2)] = min(abs(t-fit_range(2)));p = polyfit(t(bin_range(1):bin_range(2)),lenv(bin_range(1):bin_range(2)),1); fit = polyval(p,t);tau_from_fit = -1/p(1);
![Page 42: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/42.jpg)
Details of the Frequency Domain FitLet’s assume we have noisy FRF data
![Page 43: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/43.jpg)
Details of the Frequency Domain FitLet’s assume we have noisy FRF data
And we expect the FRF to be of the form
We need you find
![Page 44: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/44.jpg)
Details of the Frequency Domain FitLet’s assume we have noisy FRF data
And we expect the FRF to be of the form
We need you find
That best fit the data
![Page 45: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/45.jpg)
Details of the Frequency Domain FitLet’s assume we have noisy FRF data
And we expect the FRF to be of the form
We need you find
That best fit the data
Use fminsearch.m
![Page 46: ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26 , 2012 week 11](https://reader036.vdocuments.net/reader036/viewer/2022062315/5681611b550346895dd073ac/html5/thumbnails/46.jpg)
Using the Lorentzian Fitfminsearch.m requires that you 1)make a fitting function 2)a guess or a starting point
My fitting program used three additional subroutines there are
Three_parameter_curve_fit_test.m (main program)lorentzian_fit_driver3.mlorentzian3.mlorentzian_fit3.m
These m-files can be found on the class webpage