1. ME 392 Chapter 7 Single Degree of Freedom Oscillator ME 392 Chapter 7 Single Degree of Freedom Oscillator March 26, 2012 week 11 Joseph Vignola

2. 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)

3. 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: Name Date What it is Lab partner

4. Single Degree of Freedom Oscillator The single degree of freedom (SDoF) oscillator is a starting model for many problems m k b F(t) x(t)

5. Single Degree of Freedom Oscillator The 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)

6. Single Degree of Freedom Oscillator The 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)

7. Single Degree of Freedom Oscillator The 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)

8. Single Degree of Freedom Oscillator The 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)

9. Single Degree of Freedom Oscillator The 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)

10. Single Degree of Freedom Oscillator The 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)

11. Single Degree of Freedom Oscillator The 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)

12. Single Degree of Freedom Oscillator The 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)

13. Single Degree of Freedom Oscillator The 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)

14. Single Degree of Freedom Oscillator The 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)

15. Single Degree of Freedom Oscillator m k b F(t) x(t)

16. Single Degree of Freedom Oscillator This equation can be written as m k b F(t) x(t)

17. Single Degree of Freedom Oscillator This equation can be written as Let’s solve the inhomogeneous problem m k b F(t) x(t)

18. Single Degree of Freedom Oscillator This equation can be written as Define two terms m k b x(t)

19. 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

20. 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

21. Single Degree of Freedom Oscillator You 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

22. 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

23. Single Degree of Freedom Oscillator The solution to this ODE with initial conditions is m k b x(t)

24. 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

25. Single Degree of Freedom Oscillator The solution to this ODE with initial conditions is m k b x(t)

26. Single Degree of Freedom Oscillator The solution to this ODE with initial conditions is m k b x(t) In this expression the time constant is related to other physical parameters by

27. 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)

28. 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

29. 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

30. Single Degree of Freedom Oscillator And plot response as a function of frequency We always assume that there is some error in our measurement.

31. Single Degree of Freedom Oscillator … so for a plot with perhaps 20 measurements

32. 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

33. Details of the Time Fit Let’s assume we have noisy ring down data

34. Details of the Time Fit Let’s assume we have noisy ring down data We can generate the envelope using the magnitude of the Hilbert transform

35. Details of the Time Fit Let’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

36. Details of the Time Fit Let’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

37. Details of the Time Fit Let’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

38. Details of the Time Fit Let’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

39. Details of the Time Fit Let’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

40. Details of the Time Fit Let’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

41. Details of the Time Fit sf = 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);

42. Details of the Frequency Domain Fit Let’s assume we have noisy FRF data

43. Details of the Frequency Domain Fit Let’s assume we have noisy FRF data And we expect the FRF to be of the form We need you find

44. Details of the Frequency Domain Fit Let’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

45. Details of the Frequency Domain Fit Let’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

46. Using the Lorentzian Fit fminsearch.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.m lorentzian3.m lorentzian_fit3.m These m-files can be found on the class webpage