Nonlinear Dynamics – Phenomena and Applications

1. Nonlinear Dynamics – Phenomena and Applications Ali H. Nayfeh Department of Engineering Science and Mechanics Virginia Tech Lyapunov Lecture The 2005 ASME International Design Engineering Technical Conferences 24-28 September 2005

2. Outline • Parametric Instability in Ships The Saturation Phenomenon • Exploitation of the Saturation Phenomenon for Vibration Control • Transfer of Energy from High-to-Low Frequency Modes • Crane-Sway Control • From theory to laboratory to field • Ship-mounted cranes • Container cranes • Concluding Remarks Lyapunov Lecture 2005

3. Parametric Instability in Ships • A recent accident attributed to parametric instability • A C11 class container ship suffered a parametric instability of over 35 degrees in roll • Many containers were thrown overboard • Shipper sued ship owner for negligent operation • Case was settled out of court Lyapunov Lecture 2005

4. Parametric Instability in a Tanker Model Only pitch and heave are directly excited Virginia Tech 1991 I. Oh • L : 223.5 cm • B : 29.2 cm • D : 19.1 cm • W: 30.5 kg without ballast • W: 54.5 kg with ballast • Roll frequency : 0.32 Hz • Wave frequency: 0.60 Hz Lyapunov Lecture 2005

5. Laboratory Results on a Tanker Model Virginia Tech 1991 Lyapunov Lecture 2005

6. Autoparametric Instability in Ships • In 1863, Froude remarked in the Transactions of the British Institute of Naval Architects that a ship whose frequency in heave (pitch) is twice its frequency in roll has undesirable sea keeping characteristics Lyapunov Lecture 2005

7. Destroyer Model in a Regular Head Wave Only pitch and heave are directly excited Virginia Tech 1991 I. Oh • Model: US Navy Destroyer • Hull # 4794 • Bare Hull Model • Roll freq. : 1.40 Hz • Pitch freq. : 1.65 Hz • Heave freq.: 1.45 Hz • Model with Ballast • Roll freq. : 0.495 Hz • Pitch freq. : 0.910 Hz • Heave freq.: 1.260 Hz • Wave freq. : 0.90 Hz Lyapunov Lecture 2005

8. A Possible Explanation of Froude’s Remark Larry Marshal & Dean Mook • Roll and pitch motions are uncoupled linearly • They are coupled nonlinearly- A paradigm Lyapunov Lecture 2005

9. Perturbation Solution • Method of Multiple Scales or Method of Averaging • Perturbation Methods with Maple: http://www.esm.vt.edu/~anayfeh/ • Perturbation Methods with Mathematica: http://www.esm.vt.edu/~anayfeh/ • Roll response: • Pitch response: Lyapunov Lecture 2005

10. and Equilibrium Solutions • Linear response • Nonlinear response Independent of Excitation Amp. F Lyapunov Lecture 2005

11. Response Amplitudes The Saturation Phenomenon Pitch Amplitude b a Roll Amplitude b Pitch Amplitude a Wave Height Linear Response Response after Saturation Lyapunov Lecture 2005

12. Exploitation of the Saturation Phenomenon for Vibration Control • The ship pitch is replaced with a mode of the plant • The ship roll is replaced with an electronic circuit • The mode of the plant is coupled quadratically to the electronic circuit • The coupling is effected by an actuator and a sensor • Actuator • Piezoceramic or magnetostrictive or electrostrictive material • Sensor • Strain gauge or accelerometer Shafic Oueini, Jon Pratt, and Osama Ashour Lyapunov Lecture 2005

13. Absorber • Plant model • Equations of controller and control signal Lyapunov Lecture 2005

14. Perturbation Solution Lyapunov Lecture 2005

15. and Equilibrium Solutions • Linear response • Nonlinear response Independent of Excitation Amp. F Lyapunov Lecture 2005

16. Bifurcation Analysis a,b b a a F Linear Response Response after Saturation (Region of Control) Lyapunov Lecture 2005

17. Optimal Absorber Frequency Plant Amplitude Controller Damping Feedback Gain Plant Response Amplitude Lyapunov Lecture 2005

18. Experiments • Beams and Plates • Actuators • Piezoceramic patches • Magnetostrictive unbiased Terfenol-D • Sensors • Strain gauge • Accelerometer • Implementation • Analog • Digital Lyapunov Lecture 2005

19. Sensor and ActuatorConfiguration Strain Gauge Shaker Fixture Piezoceramic Actuators Lyapunov Lecture 2005

20. Single-Mode ControlW =11.5Hz Lyapunov Lecture 2005

21. Amplitude-Response CurveW = 10.95Hz Lyapunov Lecture 2005

22. Frequency-Response CurveF = 30mg Lyapunov Lecture 2005

23. Control of Plates A schematic of a cantilever plate with a PZT actuator Lyapunov Lecture 2005

24. Response Curves Frequency -response curves Force-response curves Lyapunov Lecture 2005

25. Zero-to-One Internal Resonance T. Anderson, B. Balachandran, Samir Nayfeh, P. Popovic, M. Tabaddor, K. Oh, H. Arafat, and P. Malatkar • Natural frequencies: 0.65, 5.65, 16.19, 31.91 Hz f = 16.23 Hz Lyapunov Lecture 2005

26. Zero-to-One Internal ResonanceExternal Excitation • Natural frequencies: 0.70, 5.89, 16.75, 33.10, 54.40 Hz f = 32.20 Hz Lyapunov Lecture 2005

27. Zero-to-One Internal ResonanceParametric Excitation • Natural frequencies: 0.65, 5.65, 16.19, 31.91 Hz f = 32.289 Hz Lyapunov Lecture 2005

28. Simultaneous One-to-Oneand Zero-t-one Resonances • Natural Frequencies: 1.303, 9.049, 25.564, 50.213, 83.105 Hz • Excitation frequency: • 83.5 Hz near the fifth • natural frequency • Large response at • 1.3 Hz : first-mode • frequency Lyapunov Lecture 2005

29. One-to-One Internal ResonanceWhirling Motion • Natural Frequencies: 1.303, 9.049, 25.564, 50.213, 83.105 Hz • Excitation frequency: • 84.9 Hz near the fifth • natural frequency Lyapunov Lecture 2005

30. One-to-One Internal ResonanceWhirling MotionNote the reverse in the direction of whirl • Natural Frequencies: 1.303, 9.049, 25.564, 50.213, 83.105 Hz • Excitation frequency: • 84.5 Hz near the fifth • natural frequency Lyapunov Lecture 2005

31. Simultaneous One-to-Oneand Zero-t-one Resonances • Natural Frequencies: 1.303, 9.049, 25.564, 50.213, 83.105 Hz • Excitation frequency: • 84.98 Hz near the fifth • natural frequency • Large response at 1.3 Hz : • first-mode frequency Lyapunov Lecture 2005

32. Simultaneous One-to-Oneand Zero-t-one Resonances Natural Frequencies: 1.303, 9.049, 25.564, 50.213, 83.105 Hz f = 83.5 Lyapunov Lecture 2005

33. A Paradigm for Zero-to-One Resonance Samir Nayfeh Lyapunov Lecture 2005

34. Nondimensionalization • We introduce a small parameter • We introduce nondimensional quantities • Nondimensional equations Lyapunov Lecture 2005

35. Variation of Parameters • We let • Detuning from resonance Lyapunov Lecture 2005

36. Variational Equations Lyapunov Lecture 2005

37. Averaged Equations--Modulation Equations Lyapunov Lecture 2005

38. Equilibrium Solutionsor Fixed Points Lyapunov Lecture 2005

39. Two Possible Fixed Points • First • Second mode oscillates around an undeflected first mode • Second • Second mode oscillates around a statically deflected first mode Lyapunov Lecture 2005

40. Frequency-Response Curves Lyapunov Lecture 2005

41. Ship-Mounted Crane Uncontrolled Response Ziyad Masoud • Animation is faster than real time • 2° Roll at wn • 1° Pitch at wn • 1 ft Heave at 2wn Lyapunov Lecture 2005

42. Control Strategy • Control boom luff and slew angles, which are already actuated • Time-delayed position feedback of the load cable angles. For the planar motion, Lyapunov Lecture 2005

43. Damping Lyapunov Lecture 2005

44. Controlled Response • Animation is faster than real time • 2° Roll at wn • 1° Pitch at wn • 1 ft Heave at 2wn Lyapunov Lecture 2005

45. Controlled vs. Uncontrolled Response(Fixed Crane Orientation) Lyapunov Lecture 2005

46. Controlled vs. Uncontrolled Response(Fixed Crane Orientation) Lyapunov Lecture 2005

47. Controlled Response Slew Operation • Animation is faster than real time • 2° Roll at wn • 1° Pitch at wn • 1 ft Heave at 2wn Lyapunov Lecture 2005

48. Controlled vs. Uncontrolled Response(Slewing Crane) Lyapunov Lecture 2005

49. Controlled vs. Uncontrolled Response(Slewing Crane) Lyapunov Lecture 2005

50. Performance of Controllerin Presence of Initial Disturbance • Animation is faster than real time • 2° Roll at wn • 1° Pitch at wn • 1 ft Heave at 2wn Lyapunov Lecture 2005