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 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 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 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 Laboratory Results on a Tanker Model Virginia Tech 1991 Lyapunov Lecture 2005 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 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 A Possible Explanation of Froude’s Remark Larry Marshal & Dean Mook Roll and pitch motions are uncoupled linearly • They are coupled nonlinearly- A paradigm 2 r 0 2 r 2 p F cos(t ) 2 p 2 p 2r and p Lyapunov Lecture 2005 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: 1 a cos t 1 2 2 • Pitch response: b cos ( t 2 ) Lyapunov Lecture 2005 Equilibrium Solutions • Linear response F a 0 and b 2 p • Nonlinear response a f ( , , r , p , F , r , p ) ( 2 r ) b r 2 r 2 r 2 Independent of Excitation Amp. F Lyapunov Lecture 2005 Response Amplitudes The Saturation Phenomenon b Pitch Amplitude a Roll Amplitude b Pitch Amplitude a Linear Response Wave Height Response after Saturation Lyapunov Lecture 2005 Exploitation of the Saturation Phenomenon for Vibration Control Shafic Oueini, Jon Pratt, and Osama Ashour 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 Lyapunov Lecture 2005 Absorber • Plant model u 2 p u u F cos(t ) Fc 2 p p • Equations of controller and control signal v 2 c v v u v 2 c 1 2 c and Fc v 2 Lyapunov Lecture 2005 Perturbation Solution u b cos( t ) 2 1 v a cos t 2 1 2 Lyapunov Lecture 2005 Equilibrium Solutions • Linear response F a 0 and b 2 p • Nonlinear response a f ( , , c , p , F , c , p ) ( 2 c ) b c 2 c 2 c 2 Independent of Excitation Amp. F Lyapunov Lecture 2005 Bifurcation Analysis a,b b a ( 2 c ) b c 2 c 2 c 2 a Linear Response Response after Saturation (Region of Control) F Lyapunov Lecture 2005 Optimal Absorber Frequency Plant Amplitude 1 c 2 ( 2 c ) b c 2 c 2 Controller Damping c b Plant Response Amplitude 2 c Feedback Gain b0 Lyapunov Lecture 2005 Experiments Beams and Plates Actuators Piezoceramic patches Magnetostrictive unbiased Terfenol-D Sensors Strain gauge Accelerometer Implementation Analog Digital Lyapunov Lecture 2005 Sensor and Actuator Configuration Strain Gauge Shaker Fixture Piezoceramic Actuators Lyapunov Lecture 2005 Single-Mode Control 11.5Hz F5.8mg8.9mg Strain (V) 0.50 0.00 -0.50 0 100 200 300 Time (sec) Lyapunov Lecture 2005 Amplitude-Response Curve Strain (V) 10.95Hz 10.00 Open-Loop Closed-Loop 0.00 0.00 20.00 40.00 60.00 80.00 Forcing Amplitude (mg) Lyapunov Lecture 2005 Frequency-Response Curve F = 30mg Strain (V) 6.00 4.00 Open-Loop 2.00 Closed-Loop 0.00 10.00 10.40 10.80 11.20 11.60 Forcing Frequency (Hz) Lyapunov Lecture 2005 Control of Plates A schematic of a cantilever plate with a PZT actuator Lyapunov Lecture 2005 Response Curves 5 10 4 Strain (mV) Strain (dB) 0 -10 3 2 -20 1 0 -30 17.2 17.6 18 Frequency (Hz) 18.4 18.8 Frequency -response curves 0 4 8 12 Input Shaker Acceleration (mg) 16 20 Force-response curves Lyapunov Lecture 2005 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 Zero-to-One Internal Resonance External Excitation Natural frequencies: 0.70, 5.89, 16.75, 33.10, 54.40 Hz f = 32.20 Hz Lyapunov Lecture 2005 Zero-to-One Internal Resonance Parametric Excitation Natural frequencies: 0.65, 5.65, 16.19, 31.91 Hz f = 32.289 Hz Lyapunov Lecture 2005 Simultaneous One-to-One and 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 One-to-One Internal Resonance Whirling 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 One-to-One Internal Resonance Whirling Motion Note 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 Simultaneous One-to-One and 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 Simultaneous One-to-One and Zero-t-one Resonances Natural Frequencies: 1.303, 9.049, 25.564, 50.213, 83.105 Hz f = 83.5 Lyapunov Lecture 2005 A Paradigm for Zero-to-One Resonance Samir Nayfeh 2 u1 1 u1 2 u2 2 u2 3 21u1 1u1 2 2u1u2 3 2 2u2 3u2 2 4u1 u2 f cos t 1 2 Lyapunov Lecture 2005 Nondimensionalization We introduce a small parameter 1 / 2 We introduce nondimensional quantities t t / 2 , u1 c1u1, u2 c2u2 , 2 Nondimensional equations u1 u1 21u1 2 2 3 (41u1 3 u2 u2 2 2 u2 ( 3u2 2 2u1u2 ) 2 4u1 u2 f cos t ) Lyapunov Lecture 2005 Variation of Parameters We let u1 v1 u2 a (t ) cos[t (t )] u2 a (t ) sin[ t (t )] t (t ) Detuning from resonance 1 2 Lyapunov Lecture 2005 Variational Equations u1 v1 3 2 2 v1 (u1 21v1 41u1 2u1a cos ) 3 3 2 a sin (a cos 3a cos 4u1 a cos 2 2 a sin f cos t ) 3 3 2 a cos (a cos 3a cos 4u1 a cos 2 2 a sin f cos t ) Lyapunov Lecture 2005 Averaged Equations-Modulation Equations u1 v1 v1 3 (u1 2 1v1 41u1 1 2 2u1a ) 2 1 a ( 2 a f sin ) 2 1 1 3 f 2 2 ( 4u1 3a cos ) 2 2 8 2a Lyapunov Lecture 2005 Equilibrium Solutions or Fixed Points v1 0 3 u1 41u1 1 2 2u1a 0 2 2 2 a f sin 0 3 f 2 2 4u1 3a cos 0 4 a Lyapunov Lecture 2005 Two Possible Fixed Points First u1 0 2 3 f 3a 2 2 4 22 4 a Second mode oscillates around an undeflected first mode Second 2 2a u1 81 2 2 3 f 2 2 2 3a 4u1 2 4 2 4 a Second mode oscillates around a statically deflected first mode Lyapunov Lecture 2005 Frequency-Response Curves 1 1, 2 2 3 1, 4 3 Lyapunov Lecture 2005 Ship-Mounted Crane Uncontrolled Response Ziyad Masoud Animation is faster than real time 2° Roll at n 1° Pitch at n 1 ft Heave at 2n Lyapunov Lecture 2005 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, x p (t ) x0 (t ) kl sin[ in (t )] y p (t ) y0 (t ) kl sin[ out (t )] where x0 and y0 are some reference position, k is a gain, and is the time delay Lyapunov Lecture 2005 Damping Lyapunov Lecture 2005 Controlled Response Animation is faster than real time 2° Roll at n 1° Pitch at n 1 ft Heave at 2n Lyapunov Lecture 2005 Controlled vs. Uncontrolled Response (Fixed Crane Orientation) Lyapunov Lecture 2005 Controlled vs. Uncontrolled Response (Fixed Crane Orientation) Lyapunov Lecture 2005 Controlled Response Slew Operation Animation is faster than real time 2° Roll at n 1° Pitch at n 1 ft Heave at 2n Lyapunov Lecture 2005 Controlled vs. Uncontrolled Response (Slewing Crane) Lyapunov Lecture 2005 Controlled vs. Uncontrolled Response (Slewing Crane) Lyapunov Lecture 2005 Performance of Controller in Presence of Initial Disturbance Animation is faster than real time 2° Roll at n 1° Pitch at n 1 ft Heave at 2n Lyapunov Lecture 2005 Experimental Demonstration Ziyad Masoud and Ryan Henry 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 Lyapunov Lecture 2005 Uncontrolled Response 1° Roll at n 0.5° Pitch at n 0.5 in Heave at 2n Lyapunov Lecture 2005 Controlled Response 2° Roll at n 1° Pitch at n 0.5 in Heave at 2n Lyapunov Lecture 2005 Controlled Response Slewing Crane 2° Roll at n 1° Pitch at n 0.5 in Heave at 2n Lyapunov Lecture 2005 Performance of Controller (in Presence of Initial Conditions) Lyapunov Lecture 2005 Container Cranes Lyapunov Lecture 2005 65-Ton Container Crane Commanded Cargo Trajectory Lyapunov Lecture 2005 65-Ton Container Crane Uncontrolled Simulation The animation is twice as fast as the actual speed Lyapunov Lecture 2005 65-Ton Container Crane Controlled Simulation The animation is twice as fast as the actual speed Lyapunov Lecture 2005 65-Ton Container Crane Full-Scale Simulation Results Lyapunov Lecture 2005 Experimental Validation on IHI 1/10th Scale Model Load Path Lyapunov Lecture 2005 IHI Model Ziyad Masoud and Nader Nayfeh Lyapunov Lecture 2005 Experimental Results IHI Model Lyapunov Lecture 2005 Manual Mode - Uncontrolled IHI Model Half Speed Lyapunov Lecture 2005 Manual Mode - Controlled IHI Model Lyapunov Lecture 2005 Experimental Validation Virginia Tech Model Ziyad Masoud and Muhammad Daqaq Lyapunov Lecture 2005 Manual Mode - Uncontrolled Virginia Tech Model Half Speed Lyapunov Lecture 2005 Manual Mode - Controlled Virginia Tech Model Lyapunov Lecture 2005 Pendulation Controller Controller can suppress cargo sway in Commercial cranes Military cranes Effectiveness of the Controller has been demonstrated using computer models of Ship-mounted boom cranes Land-based rotary cranes 65-ton container crane Telescopic crane Controller has been validated experimentally on scaled models of Ship-mounted boom crane Land-based rotary crane Container crane in an industrial setting Full-scale container crane Lyapunov Lecture 2005 Concluding Remarks Nonlinearities pose challenges and opportunities Challenges Design systems that overcome the adverse effects of nonlinearities Develop passive and active control strategies to expand the design envelope Opportunities Exploit nonlinearities for design Lyapunov Lecture 2005 Is nonlinear thinking in order ? Lyapunov Lecture 2005 Controller Nonlinear delay feedback control + + + - PID k, Controller Plant Gain Calculator T Lyapunov Lecture 2005 Typical Terfenol-D Strut Prestress spring Magnet Terfenol-D Prestress housing Solenoid Lyapunov Lecture 2005 Terfenol-D Constitutive Law Nonlinear operation Bias line Nonlinear operation Linear operation Field (H) Lyapunov Lecture 2005 Setup Shaker Excitation Shafic Oueini & Jon Pratt Shaker Accelerometer Terfenol-D Actuator Lyapunov Lecture 2005 Single-Mode Control 47.5Hz Acceleration (g) 0.50 0.00 -0.50 0 10 20 30 40 Time (sec) 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 Jib angular rate vs maximum controlled rate 12.00 max crane rate max control commanded rate 10.00 jib rate (deg/s) 8.00 6.00 4.00 2.00 0.00 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 jib angle (deg) Lyapunov Lecture 2005 Summary Anti-Roll Tanks 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 Developed a control strategy to maintain a skin-to-skin configuration between two ships Prevents metal-on-metal contact between two ships Minimizes the motions of the Bob Hope and the Argonaut Limits the motion of the Argonaut relative to the Bob Hope Reduces the demand on cranes Enables operations in SS4 & SS5 Decreases the transfer time Lyapunov Lecture 2005 Effectiveness of Mooring System 12.00 max crane speed SS5 controller requirements - 20° following off stern of Bob Hope 10.00 SS4 controller requirements - 15° off head seas SS5 controller requirements - 15° off head seas Rate (deg/s) 8.00 6.00 4.00 2.00 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 Commanded Crane Angle (degrees) Lyapunov Lecture 2005 Controller Nonlinear delay feedback control + + + - PID k, Controller Plant Gain Calculator T Lyapunov Lecture 2005 The Control Unit Trolley Hoist 1 Quadrature Encoder Input Hoist 2 Control Unit Sway ADC Trolley Motor DAC Hoist 1 Motor Hoist 2 Motor Joystick - Trolley Joystick - Hoist Lyapunov Lecture 2005 Controller Circuit Piezoceramic Actuator KD uv S K 1 K2 1 2 s s v P v P u System v 2 Lyapunov Lecture 2005 Nonresonance Interaction Zero-to-One Internal Resonance Natural frequencies: 0.65, 5.65, 16.19, 31.91 Hz f = 16.25 Hz Lyapunov Lecture 2005 Comparison between Responses of Beam and Hubble Telescope Lyapunov Lecture 2005 IHI Scale Model Profile Lyapunov Lecture 2005