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

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

4. Lyapunov Lecture 2005 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 Parametric Instability in a Tanker Model Only pitch and heave are directly excited Virginia Tech 1991 I. Oh

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

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

7. Lyapunov Lecture 2005 Destroyer Model in a Regular Head Wave 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 Only pitch and heave are directly excited Virginia Tech 1991 I. Oh

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

9. Lyapunov Lecture 2005 Perturbation Solution Pitch response: 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:

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

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

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

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

14. Lyapunov Lecture 2005 Perturbation Solution

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

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

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

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

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

20. Lyapunov Lecture 2005 Single-Mode Control  z

21. Lyapunov Lecture 2005 Amplitude-Response Curve  z

22. Lyapunov Lecture 2005 Frequency-Response Curve F = 30mg

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

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

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

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

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

28. Lyapunov Lecture 2005 Simultaneous One-to-One and Zero-t-one Resonances Natural Frequencies: 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

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

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

31. Lyapunov Lecture 2005 Natural Frequencies: 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 Simultaneous One-to-One and Zero-t-one Resonances

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

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

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

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

36. Lyapunov Lecture 2005 Variational Equations

37. Lyapunov Lecture 2005 Averaged Equations-- Modulation Equations

38. Lyapunov Lecture 2005 Equilibrium Solutions or Fixed Points

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

40. Lyapunov Lecture 2005 Frequency-Response Curves

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

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

43. Lyapunov Lecture 2005 Damping

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

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

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

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

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

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

50. Lyapunov Lecture 2005 Performance of Controller in Presence of Initial Disturbance Animation is faster than real time Animation is faster than real time 2° Roll at  n 2° Roll at  n 1° Pitch at  n 1° Pitch at  n 1 ft Heave at 2  n 1 ft Heave at 2  n

51. Lyapunov Lecture 2005 Experimental Demonstration A 3 DOF ship-motion simulator platform is built: A 3 DOF ship-motion simulator platform is built: It has the capability of performing general pitch, roll, and heave motions A 1/24 scale model of the T-ACS (NSWC) crane is mounted on the platform A PC is used to apply the controller and drive the crane A 3 DOF ship-motion simulator platform is built: A 3 DOF ship-motion simulator platform is built: It has the capability of performing general pitch, roll, and heave motions A 1/24 scale model of the T-ACS (NSWC) crane is mounted on the platform A PC is used to apply the controller and drive the crane Ziyad Masoud and Ryan Henry

52. Lyapunov Lecture 2005 Uncontrolled Response 1° Roll at  n 1° Roll at  n 0.5° Pitch at  n 0.5° Pitch at  n 0.5 in Heave at 2  n 0.5 in Heave at 2  n

53. Lyapunov Lecture 2005 Controlled Response 2° Roll at  n 2° Roll at  n 1° Pitch at  n 1° Pitch at  n 0.5 in Heave at 2  n 0.5 in Heave at 2  n

54. Lyapunov Lecture 2005 Controlled Response Slewing Crane 2° Roll at  n 2° Roll at  n 1° Pitch at  n 1° Pitch at  n 0.5 in Heave at 2  n 0.5 in Heave at 2  n

55. Lyapunov Lecture 2005 Performance of Controller (in Presence of Initial Conditions)

56. Lyapunov Lecture 2005 Container Cranes

57. Lyapunov Lecture 2005 65-Ton Container Crane Commanded Cargo Trajectory

58. Lyapunov Lecture 2005 65-Ton Container Crane Uncontrolled Simulation The animation is twice as fast as the actual speed The animation is twice as fast as the actual speed

59. Lyapunov Lecture 2005 65-Ton Container Crane Controlled Simulation The animation is twice as fast as the actual speed The animation is twice as fast as the actual speed

60. Lyapunov Lecture 2005 65-Ton Container Crane Full-Scale Simulation Results

61. Lyapunov Lecture 2005 Experimental Validation on IHI 1/10 th Scale Model Load Path

62. Lyapunov Lecture 2005 IHI Model Ziyad Masoud and Nader Nayfeh

63. Lyapunov Lecture 2005 Experimental Results IHI Model

64. Lyapunov Lecture 2005 Manual Mode - Uncontrolled IHI Model Half Speed

65. Lyapunov Lecture 2005 Manual Mode - Controlled IHI Model

66. Lyapunov Lecture 2005 Experimental Validation Virginia Tech Model Ziyad Masoud and Muhammad Daqaq

67. Lyapunov Lecture 2005 Manual Mode - Uncontrolled Virginia Tech Model Half Speed

68. Lyapunov Lecture 2005 Manual Mode - Controlled Virginia Tech Model

69. Lyapunov Lecture 2005 Pendulation Controller Controller can suppress cargo sway in Controller can suppress cargo sway in Commercial cranes Commercial cranes Military cranes Military cranes Effectiveness of the Controller has been demonstrated using computer models of Effectiveness of the Controller has been demonstrated using computer models of Ship-mounted boom cranes Ship-mounted boom cranes Land-based rotary cranes Land-based rotary cranes 65-ton container crane 65-ton container crane Telescopic crane Telescopic crane Controller has been validated experimentally on scaled models of Controller has been validated experimentally on scaled models of Ship-mounted boom crane Ship-mounted boom crane Land-based rotary crane Land-based rotary crane Container crane in an industrial setting Container crane in an industrial setting Full-scale container crane Full-scale container crane

70. Lyapunov Lecture 2005 Concluding Remarks Nonlinearities pose challenges and opportunities Nonlinearities pose challenges and opportunities Challenges Challenges Design systems that overcome the adverse effects of nonlinearities Design systems that overcome the adverse effects of nonlinearities Develop passive and active control strategies to expand the design envelope Develop passive and active control strategies to expand the design envelope Opportunities Opportunities Exploit nonlinearities for design Exploit nonlinearities for design

71. Is nonlinear thinking in order ? Lyapunov Lecture 2005

72. Controller Nonlinear delay feedback control Nonlinear delay feedback control PID Plant Gain Calculator Controller + + + - T k 

73. Lyapunov Lecture 2005 Typical Terfenol-D Strut Prestress housing Prestress spring Solenoid Magnet Terfenol-D

74. Lyapunov Lecture 2005 Terfenol-D Constitutive Law  Field (H) Bias line Linear operation Nonlinear operation Nonlinear operation

75. Lyapunov Lecture 2005 Setup Shaker Excitation Terfenol-D Actuator Shaker Accelerometer Shafic Oueini & Jon Pratt

76. Lyapunov Lecture 2005 Single-Mode Control  z

77. Lyapunov Lecture 2005 Required Luff Rate Using the motions of the Bob Hope obtained with the integrated Stabilization System, we calculated the crane luff rates demanded by the controller and compared them with the rates supplied by MacGregor

78. Lyapunov Lecture 2005 Summary Anti-Roll Tanks Anti-Roll Tanks Demonstrated the benefits of active anti-roll tanks in regular and irregular seas (for all headings) Demonstrated the benefits of active anti-roll tanks in regular and irregular seas (for all headings)  A thirty-fold roll reduction with a tank mass= 0.6 % ship mass for all headings in SS5  Less than 0.5° roll Fender and Mooring Subsystem Fender and Mooring Subsystem Developed a control strategy to maintain a skin-to-skin configuration between two ships Developed a control strategy to maintain a skin-to-skin configuration between two ships Prevents metal-on-metal contact between two ships Prevents metal-on-metal contact between two ships Minimizes the motions of the Bob Hope and the Argonaut Minimizes the motions of the Bob Hope and the Argonaut Limits the motion of the Argonaut relative to the Bob Hope Limits the motion of the Argonaut relative to the Bob Hope Reduces the demand on cranes Reduces the demand on cranes Enables operations in SS4 & SS5 Enables operations in SS4 & SS5 Decreases the transfer time Decreases the transfer time

79. Lyapunov Lecture 2005 Effectiveness of Mooring System

80. Lyapunov Lecture 2005 Controller Nonlinear delay feedback control Nonlinear delay feedback control PID Plant Gain Calculator Controller + + + - T k 

81. Lyapunov Lecture 2005 The Control Unit Trolley Hoist 1 Hoist 2 Sway Joystick - Trolley Joystick - Hoist Quadrature Encoder Input ADC Trolley Motor Hoist 2 Motor Hoist 1 Motor DAC Control Unit

82. Lyapunov Lecture 2005 Controller Circuit Piezoceramic Actuator System     

83. Lyapunov Lecture 2005 Nonresonance Interaction Zero-to-One Internal Resonance Natural frequencies: 0.65, 5.65, 16.19, 31.91 Hz Natural frequencies: 0.65, 5.65, 16.19, 31.91 Hz f = 16.25 Hz

84. Lyapunov Lecture 2005 Comparison between Responses of Beam and Hubble Telescope

85. Lyapunov Lecture 2005 IHI Scale Model Profile