me 392 chapter 7 single degree of freedom oscillator march 26 , 2012 week 11
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ME 392Chapter 7
Single Degree of Freedom Oscillator
March 26, 2012week 11
Joseph Vignola
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)
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
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Single Degree of Freedom Oscillator
m
k b
F(t)
x(t)
Single Degree of Freedom Oscillator
This equation can be written as
m
k b
F(t)
x(t)
Single Degree of Freedom Oscillator
This equation can be written as
Let’s solve the inhomogeneous problem
m
k b
F(t)
x(t)
Single Degree of Freedom Oscillator
This equation can be written as
Define two terms
m
k b
x(t)
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
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
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
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
Single Degree of Freedom Oscillator
The solution to this ODE with initial conditions
is
m
k b
x(t)
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
Single Degree of Freedom Oscillator
The solution to this ODE with initial conditions
is
m
k b
x(t)
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
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 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
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
Single Degree of Freedom OscillatorAnd plot response as a function of frequency
We always assume that there is some error in our measurement.
Single Degree of Freedom Oscillator… so for a plot with perhaps 20 measurements
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
Details of the Time FitLet’s assume we have noisy ring down data
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
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
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
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
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
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
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
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);
Details of the Frequency Domain FitLet’s assume we have noisy FRF data
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
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
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
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
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