Nonlinear Dynamics – Phenomena and Applications

85
Nonlinear Dynamics – Phenomena and Applications Ali H. Nayfeh Ali H. Nayfeh Department of Engineering Science and Department of Engineering Science and Mechanics Mechanics Virginia Tech Virginia Tech Lyapunov Lecture The 2005 ASME International Design Engineering Technical Conferences 24-28 September 2005

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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. - PowerPoint PPT Presentation

Transcript of Nonlinear Dynamics – Phenomena and Applications

Nonlinear Dynamics – Phenomena and Applications

Ali H. NayfehAli H. Nayfeh

Department of Engineering Science and MechanicsDepartment of Engineering Science and Mechanics

Virginia TechVirginia Tech

Lyapunov Lecture

The 2005 ASME International Design Engineering Technical Conferences

24-28 September 2005

Lyapunov Lecture 2005

Outline

Parametric Instability in Ships The Parametric Instability in Ships The Saturation PhenomenonSaturation Phenomenon

Exploitation of the Saturation Phenomenon Exploitation of the Saturation Phenomenon for Vibration Controlfor Vibration Control

Transfer of Energy from High-to-Low Transfer of Energy from High-to-Low Frequency ModesFrequency Modes

Crane-Sway ControlCrane-Sway Control From theory to laboratory to fieldFrom theory to laboratory to field Ship-mounted cranesShip-mounted cranes Container cranesContainer cranes

Concluding RemarksConcluding Remarks

Lyapunov Lecture 2005

A recent accident attributed to A recent accident attributed to parametric instabilityparametric instability A C11 class container ship suffered a A C11 class container ship suffered a

parametric instability of over 35 degrees in parametric instability of over 35 degrees in rollroll

Many containers were thrown overboardMany containers were thrown overboard Shipper sued ship owner for negligent Shipper sued ship owner for negligent

operationoperation Case was settled out of court Case was settled out of court

Parametric Instability in Ships

Lyapunov Lecture 2005

L : 223.5 cmB : 29.2 cm D : 19.1 cmW: 30.5 kg without ballastW: 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 1991I. Oh

Lyapunov Lecture 2005

Laboratory Results on a Tanker ModelVirginia Tech 1991

Lyapunov Lecture 2005

Autoparametric Instability in Ships

In 1863, Froude remarked in the In 1863, Froude remarked in the Transactions of the British Institute of Transactions of the British Institute of Naval Architects thatNaval Architects that

a ship whose frequency in heave (pitch) is a ship whose frequency in heave (pitch) is twice its frequency in roll has undesirable twice its frequency in roll has undesirable sea keeping characteristicssea keeping characteristics

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 1991I. Oh

Lyapunov Lecture 2005

A Possible Explanation of Froude’s Remark

Roll and pitch motions are uncoupled linearlyRoll and pitch motions are uncoupled linearly

22 0r r

2 22 cos( )p p F t

• They are coupled nonlinearly- A paradigm

Larry Marshal & Dean Mook

p2 andp r

Lyapunov Lecture 2005

Perturbation Solution

212

1cos ta

)(cos 2 tb

• 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:

Lyapunov Lecture 2005

Equilibrium Solutions

• Linear response

0a2 p

Fb

and

• Nonlinear response

( , , , , , , )r p r pa f F

2 2 2( 2 )r r r

r

b

Independent of

Excitation Amp. F

Lyapunov Lecture 2005

Response Amplitudes The Saturation Phenomenon

LinearResponse

Response after Saturation

b

a

a

Pitch Amplitude

Roll Amplitude

Wave Height

b Pitch Amplitude

Lyapunov Lecture 2005

Exploitation of the Saturation Phenomenon for Vibration Control

The ship pitch is replaced with a mode of the plantThe ship pitch is replaced with a mode of the plant The ship roll is replaced with an electronic circuitThe ship roll is replaced with an electronic circuit The mode of the plant is coupled quadratically to The mode of the plant is coupled quadratically to

the electronic circuitthe electronic circuit The coupling is effected by an actuator and a The coupling is effected by an actuator and a

sensorsensor ActuatorActuator

Piezoceramic or magnetostrictive or electrostrictive Piezoceramic or magnetostrictive or electrostrictive materialmaterial

SensorSensor Strain gauge or accelerometerStrain gauge or accelerometer

Shafic Oueini, Jon Pratt, and Osama Ashour

Lyapunov Lecture 2005

Absorber

• Plant model22 cos( )p p cu u u F t F

p • Equations of controller and control signal

22 c cv v v u v 21

and2c cF v

Lyapunov Lecture 2005

Perturbation Solution

1 2

1cos

2v a t

2cos( )u b t

Lyapunov Lecture 2005

Equilibrium Solutions

• Linear response

0a2 p

Fb

and

• Nonlinear response

( , , , , , , )c p c pa f F

2 2 2( 2 )c c c

c

b

Independent of

Excitation Amp. F

Lyapunov Lecture 2005

Bifurcation Analysis

a,b

FLinear

ResponseResponse after Saturation

(Region of Control)

b

a

a

2 2 2( 2 )c c c

c

b

Lyapunov Lecture 2005

Optimal Absorber Frequency

1

2c

c b

ControllerDamping

FeedbackGain

0bPlant Response

Amplitude

2 2 2( 2 )c c c

c

b

Plant Amplitude

Lyapunov Lecture 2005

Experiments

Beams and PlatesBeams and Plates ActuatorsActuators

Piezoceramic patchesPiezoceramic patches Magnetostrictive unbiased Terfenol-DMagnetostrictive unbiased Terfenol-D

SensorsSensors Strain gaugeStrain gauge AccelerometerAccelerometer

ImplementationImplementation AnalogAnalog DigitalDigital

Lyapunov Lecture 2005

Sensor and ActuatorConfiguration

PiezoceramicActuators

Strain Gauge

ShakerFixture

Lyapunov Lecture 2005

Single-Mode Controlz

0 100 200 300

Time (sec)

-0 .50

0.00

0.50

Str

ain

(V

)

mgmgF 9.88.5

Lyapunov Lecture 2005

Amplitude-Response Curve

z

0.00 20.00 40.00 60.00 80.00

Forcing Am plitude (m g)

0.00

10.00

Str

ain

(V

)

Open-Loop

Closed-Loop

Lyapunov Lecture 2005

Frequency-Response Curve

F = 30mg

10.00 10.40 10.80 11.20 11.60

Forcing Frequency (Hz)

0.00

2.00

4.00

6.00

Str

ain

(V

)

Open-Loop

Closed-Loop

Lyapunov Lecture 2005

Control of Plates

A schematic of a cantilever plate with a PZT actuator

Lyapunov Lecture 2005

17.2 17.6 18 18.4 18.8

F requency (Hz)

-30

-20

-10

0

10

Str

ain

(d

B)

0 4 8 12 16 20

Inpu t Shaker Acce le ra tion (m g)

0

1

2

3

4

5

Str

ain

(m

V)

Frequency -response curves Force-response curves

Response Curves

Lyapunov Lecture 2005

Zero-to-One Internal Resonance

Natural frequencies: 0.65, 5.65, 16.19, 31.91 HzNatural 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

Lyapunov Lecture 2005

Natural frequencies: 0.70, 5.89, 16.75, 33.10, 54.40 HzNatural frequencies: 0.70, 5.89, 16.75, 33.10, 54.40 Hz

f = 32.20 Hz

Zero-to-One Internal ResonanceExternal Excitation

Lyapunov Lecture 2005

Zero-to-One Internal ResonanceParametric Excitation

Natural frequencies: 0.65, 5.65, 16.19, 31.91 HzNatural frequencies: 0.65, 5.65, 16.19, 31.91 Hz

f = 32.289 Hz

Lyapunov Lecture 2005

Simultaneous One-to-Oneand Zero-t-one Resonances

Natural Frequencies:Natural Frequencies:1.303, 9.049, 25.564,1.303, 9.049, 25.564,50.213, 83.105 Hz50.213, 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 ResonanceWhirling Motion

Natural Frequencies:Natural Frequencies:1.303, 9.049, 25.564,1.303, 9.049, 25.564,50.213, 83.105 Hz50.213, 83.105 Hz

• Excitation frequency:

84.9 Hz near the fifth

natural frequency

Lyapunov Lecture 2005

Natural Frequencies:Natural Frequencies:1.303, 9.049, 25.564,1.303, 9.049, 25.564,50.213, 83.105 Hz50.213, 83.105 Hz

• Excitation frequency:

84.5 Hz near the fifth

natural frequency

One-to-One Internal ResonanceWhirling Motion

Note the reverse in the direction of whirl

Lyapunov Lecture 2005

Natural Frequencies:Natural Frequencies:1.303, 9.049, 25.564,1.303, 9.049, 25.564,50.213, 83.105 Hz50.213, 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-Oneand Zero-t-one Resonances

Lyapunov Lecture 2005

Natural Frequencies: 1.303, 9.049, 25.564, 50.213, 83.105 Hz

f = 83.5

Simultaneous One-to-Oneand Zero-t-one Resonances

Lyapunov Lecture 2005

A Paradigm for Zero-to-One Resonance

21

tfuuuuuu

uuuuuu

cos2

2

2214

323222

222

2212

311111

211

Samir Nayfeh

Lyapunov Lecture 2005

Nondimensionalization

We introduce a small parameterWe introduce a small parameter

We introduce nondimensional quantitiesWe introduce nondimensional quantities

Nondimensional equationsNondimensional equations

21 /

22221112 ,,,/ ucuucutt

)cos(2

)4(2

2214

3232222

2212

311

2111

21

tfuuuuuu

uuuuuu

Lyapunov Lecture 2005

Variation of Parameters

We letWe let

Detuning from resonanceDetuning from resonance

)(

)](sin[)(

)](cos[)(

2

2

11

tt

tttau

tttau

vu

12

Lyapunov Lecture 2005

Variational Equations

)cossin2

coscoscos(cos

)cossin2

coscoscos(sin

2

214

333

2

214

333

tfa

auaaa

tfa

auaaa

)cos42( 2212

3111111

11

auuvuv

vu

Lyapunov Lecture 2005

Averaged Equations--Modulation Equations

)cos28

321

21

(

)sin21

(

23

214

2

af

au

faa

)21

42( 212

3111111

11

auuvuv

vu

Lyapunov Lecture 2005

Equilibrium Solutionsor Fixed Points

021

4

02

123111

1

auuu

v

0cos43

0sin22

3214

2

af

au

fa

Lyapunov Lecture 2005

Two Possible Fixed Points

FirstFirst

Second mode oscillates around an undeflected first modeSecond mode oscillates around an undeflected first mode

SecondSecond

Second mode oscillates around a statically deflected first modeSecond mode oscillates around a statically deflected first mode

01 u 222

22

3 443

a

fa

222

2214

23

1

22

1

443

82

a

fua

au

Lyapunov Lecture 2005

Frequency-Response Curves

3,1

2,1

43

21

Lyapunov Lecture 2005

Ship-Mounted CraneUncontrolled ResponseUncontrolled Response

Animation is faster Animation is faster than real timethan real time

2° Roll at 2° Roll at nn

1° Pitch at 1° Pitch at nn

1 ft Heave at 21 ft Heave at 2nn

Ziyad Masoud

Lyapunov Lecture 2005

Control Strategy

Control boom luff and slew angles, which Control boom luff and slew angles, which are already actuatedare already actuated

Time-delayed position feedback of the Time-delayed position feedback of the load cable angles. For the planar motion,load cable angles. For the planar motion,

delay time theis and gain, a is

position, reference some are and where

)](sin[)()(

)](sin[)()(

00

0

0

k

yx

tkltyty

tkltxtx

outp

inp

Lyapunov Lecture 2005

Damping

Lyapunov Lecture 2005

Controlled Response

Animation is faster Animation is faster than real timethan real time

2° Roll at 2° Roll at nn

1° Pitch at 1° Pitch at nn

1 ft Heave at 21 ft Heave at 2nn

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 OperationSlew OperationAnimation is faster Animation is faster

than real timethan real time2° Roll at 2° Roll at nn

1° Pitch at 1° Pitch at nn

1 ft Heave at 21 ft Heave at 2nn

Lyapunov Lecture 2005

Controlled vs. Uncontrolled Response (Slewing Crane)

Lyapunov Lecture 2005

Controlled vs. Uncontrolled Response (Slewing Crane)

Lyapunov Lecture 2005

Performance of Controllerin Presence of Initial Disturbance

Animation is faster Animation is faster than real timethan real time

2° Roll at 2° Roll at nn

1° Pitch at 1° Pitch at nn

1 ft Heave at 21 ft Heave at 2nn

Lyapunov Lecture 2005

Experimental Demonstration

A 3 DOFA 3 DOF ship-motionship-motion simulator simulator platform is built: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 DOFA 3 DOF ship-motionship-motion simulator simulator platform is built: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

Lyapunov Lecture 2005

Uncontrolled Response

1° Roll at 1° Roll at nn

0.5° Pitch at 0.5° Pitch at nn

0.5 in Heave at 20.5 in Heave at 2nn

Lyapunov Lecture 2005

Controlled Response

2° Roll at 2° Roll at nn

1° Pitch at 1° Pitch at nn

0.5 in Heave at 20.5 in Heave at 2nn

Lyapunov Lecture 2005

Controlled Response Slewing Crane

2° Roll at 2° Roll at nn

1° Pitch at 1° Pitch at nn

0.5 in Heave at 20.5 in Heave at 2nn

Lyapunov Lecture 2005

Performance of Controller(in Presence of Initial Conditions)

Lyapunov Lecture 2005

Container Cranes

Lyapunov Lecture 2005

65-Ton Container CraneCommanded Cargo

TrajectoryCommanded Cargo

Trajectory

Lyapunov Lecture 2005

65-Ton Container Crane

Uncontrolled SimulationUncontrolled Simulation

The animation is The animation is twice as fast as the twice as fast as the actual speedactual speed

Lyapunov Lecture 2005

65-Ton Container Crane

Controlled SimulationControlled SimulationThe animation is The animation is

twice as fast as the twice as fast as the actual speedactual speed

Lyapunov Lecture 2005

65-Ton Container Crane

Full-Scale Simulation ResultsFull-Scale Simulation Results

Lyapunov Lecture 2005

Experimental Validation on IHI 1/10th Scale Model

Load PathLoad Path

Lyapunov Lecture 2005

IHI Model

Ziyad Masoud and Nader Nayfeh

Lyapunov Lecture 2005

Experimental ResultsIHI Model

Lyapunov Lecture 2005

Manual Mode - UncontrolledIHI Model

HalfSpeed

Lyapunov Lecture 2005

Manual Mode - ControlledIHI Model

Lyapunov Lecture 2005

Experimental ValidationVirginia Tech Model

Ziyad Masoud and Muhammad Daqaq

Lyapunov Lecture 2005

Manual Mode - Uncontrolled Virginia Tech Model

HalfSpeed

Lyapunov Lecture 2005

Manual Mode - Controlled Virginia Tech Model

Lyapunov Lecture 2005

Pendulation ControllerController can suppress cargo sway inController can suppress cargo sway in

Commercial cranesCommercial cranesMilitary cranesMilitary cranes

Effectiveness of the Controller has been Effectiveness of the Controller has been demonstrated using computer models of demonstrated using computer models of Ship-mounted boom cranesShip-mounted boom cranesLand-based rotary cranesLand-based rotary cranes65-ton container crane65-ton container craneTelescopic craneTelescopic crane

Controller has been validated experimentally on Controller has been validated experimentally on scaled models ofscaled models ofShip-mounted boom craneShip-mounted boom craneLand-based rotary craneLand-based rotary craneContainer crane in an industrial settingContainer crane in an industrial settingFull-scale container craneFull-scale container crane

Lyapunov Lecture 2005

Concluding Remarks

Nonlinearities pose challenges and Nonlinearities pose challenges and opportunities opportunities

ChallengesChallenges Design systems that overcome the adverse Design systems that overcome the adverse

effects of nonlinearitieseffects of nonlinearities Develop passive and active control strategies Develop passive and active control strategies

to expand the design envelopeto expand the design envelope

OpportunitiesOpportunities Exploit nonlinearities for designExploit nonlinearities for design

Is nonlinear thinking in order

?Lyapunov Lecture 2005

Lyapunov Lecture 2005

Controller

Nonlinear delay feedback controlNonlinear delay feedback control

PID Plant

GainCalculatorController

+

+

+

-

T

k

Lyapunov Lecture 2005

Typical Terfenol-D Strut

Prestress housing

Prestress spring

Solenoid

Magnet

Terfenol-D

Lyapunov Lecture 2005

Terfenol-DConstitutive Law

Field (H)

Bias line

Linear operation

Nonlinearoperation

Nonlinearoperation

Lyapunov Lecture 2005

SetupShaker Excitation

Terfenol-DActuator

Shaker

Accelerometer

Shafic Oueini & Jon Pratt

Lyapunov Lecture 2005

Single-Mode Control

z

0 10 20 30 40

Time (sec)

-0 .50

0.00

0.50

Acc

eler

atio

n (

g)

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

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00

jib angle (deg)

jib r

ate

(deg

/s)

max crane rate

max control commanded rate

Lyapunov Lecture 2005

Summary

Anti-Roll TanksAnti-Roll Tanks Demonstrated the benefits of active anti-roll tanks in regular and irregular seas Demonstrated the benefits of active anti-roll tanks in regular and irregular seas

((for all headingsfor all headings)) A thirty-fold roll reduction with a tank mass= 0.6 % ship mass for all headings in SS5A thirty-fold roll reduction with a tank mass= 0.6 % ship mass for all headings in SS5 Less than 0.5Less than 0.5° roll° roll

Fender and Mooring SubsystemFender and Mooring Subsystem Developed a control strategy to maintain a skin-to-skin configuration between Developed a control strategy to maintain a skin-to-skin configuration between

two shipstwo ships

Prevents metal-on-metal contact between two shipsPrevents metal-on-metal contact between two ships

Minimizes the motions of the Bob Hope and the ArgonautMinimizes the motions of the Bob Hope and the Argonaut

Limits the motion of the Argonaut relative to the Bob HopeLimits the motion of the Argonaut relative to the Bob Hope

Reduces the demand on cranesReduces the demand on cranes

Enables operations in SS4 & SS5Enables operations in SS4 & SS5

Decreases the transfer timeDecreases the transfer time

Lyapunov Lecture 2005

Effectiveness of Mooring System

0.00

2.00

4.00

6.00

8.00

10.00

12.00

10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00

Commanded Crane Angle (degrees)

Ra

te (

de

g/s

)

max crane speed

SS5 controller requirements - 20° following off stern of Bob Hope

SS4 controller requirements - 15° off head seas

SS5 controller requirements - 15° off head seas

Lyapunov Lecture 2005

Controller

Nonlinear delay feedback controlNonlinear delay feedback control Nonlinear delay feedback controlNonlinear delay feedback control

PID Plant

GainCalculatorController

+

+

+

-

T

k

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 MotorDACControl Unit

Lyapunov Lecture 2005

Controller Circuit Piezoceramic Actuator

System

1K 2Ks1

s2

DK

2vu

v

vvu

Lyapunov Lecture 2005

Nonresonance InteractionZero-to-One Internal Resonance

Natural frequencies: 0.65, 5.65, 16.19, 31.91 HzNatural 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