nonlinear dynamics – phenomena and applications ali h. nayfeh department of engineering science...

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


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


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


Laboratory Results on a Tanker ModelVirginia Tech 1991


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


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


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


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:


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


Response Amplitudes The Saturation Phenomenon

LinearResponse

Response after Saturation

b

a

a

Pitch Amplitude

Roll Amplitude

Wave Height

b Pitch Amplitude


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


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


Perturbation Solution

1 2

1cos

2v a t

2cos( )u b t


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


Bifurcation Analysis

a,b

FLinear

ResponseResponse after Saturation

(Region of Control)

b

a

a

2 2 2( 2 )c c c

c

b


Optimal Absorber Frequency

1

2c

c b

ControllerDamping

FeedbackGain

0bPlant Response

Amplitude

2 2 2( 2 )c c c

c

b

Plant Amplitude


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


Sensor and ActuatorConfiguration

PiezoceramicActuators

Strain Gauge

ShakerFixture


Single-Mode Controlz

0 100 200 300

Time (sec)

-0 .50

0.00

0.50

Str

ain

(V

)

mgmgF 9.88.5


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


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


Control of Plates

A schematic of a cantilever plate with a PZT actuator


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


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


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


Zero-to-One Internal ResonanceParametric Excitation


f = 32.289 Hz


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


One-to-One Internal ResonanceWhirling Motion




natural frequency





natural frequency

One-to-One Internal ResonanceWhirling Motion

Note the reverse in the direction of whirl





natural frequency

• Large response at 1.3 Hz :

first-mode frequency



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

f = 83.5



A Paradigm for Zero-to-One Resonance

21

tfuuuuuu

uuuuuu

cos2

2

2214

323222

222

2212

311111

211

Samir Nayfeh


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


Variation of Parameters

We letWe let

Detuning from resonanceDetuning from resonance

)(

)](sin[)(

)](cos[)(

2

2

11

tt

tttau

tttau

vu

12


Variational Equations

)cossin2

coscoscos(cos

)cossin2

coscoscos(sin

2

214

333

2

214

333

tfa

auaaa

tfa

auaaa

)cos42( 2212

3111111

11

auuvuv

vu


Averaged Equations--Modulation Equations

)cos28

321

21

(

)sin21

(

23

214

2

af

au

faa

)21

42( 212

3111111

11

auuvuv

vu


Equilibrium Solutionsor Fixed Points

021

4

02

123111

1

auuu

v

0cos43

0sin22

3214

2

af

au

fa


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


Frequency-Response Curves

3,1

2,1

43

21


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


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


Damping


Controlled Response






Controlled vs. Uncontrolled Response (Fixed Crane Orientation)


Controlled Response

Slew OperationSlew OperationAnimation is faster Animation is faster

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




Controlled vs. Uncontrolled Response (Slewing Crane)


Performance of Controllerin Presence of Initial Disturbance






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


Uncontrolled Response


0.5° Pitch at 0.5° Pitch at nn

0.5 in Heave at 20.5 in Heave at 2nn


Controlled Response





Controlled Response Slewing Crane





Performance of Controller(in Presence of Initial Conditions)


Container Cranes


65-Ton Container CraneCommanded Cargo

TrajectoryCommanded Cargo

Trajectory


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



Controlled SimulationControlled SimulationThe animation is The animation is

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



Full-Scale Simulation ResultsFull-Scale Simulation Results


Experimental Validation on IHI 1/10th Scale Model

Load PathLoad Path


IHI Model

Ziyad Masoud and Nader Nayfeh


Experimental ResultsIHI Model


Manual Mode - UncontrolledIHI Model

HalfSpeed


Manual Mode - ControlledIHI Model


Experimental ValidationVirginia Tech Model

Ziyad Masoud and Muhammad Daqaq


Manual Mode - Uncontrolled Virginia Tech Model

HalfSpeed


Manual Mode - Controlled Virginia Tech Model


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


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


Controller

Nonlinear delay feedback controlNonlinear delay feedback control

PID Plant

GainCalculatorController

+

+

+

-

T

k


Typical Terfenol-D Strut

Prestress housing

Prestress spring

Solenoid

Magnet

Terfenol-D


Terfenol-DConstitutive Law

Field (H)

Bias line

Linear operation

Nonlinearoperation

Nonlinearoperation


SetupShaker Excitation

Terfenol-DActuator

Shaker

Accelerometer

Shafic Oueini & Jon Pratt


Single-Mode Control

z

0 10 20 30 40

Time (sec)

-0 .50

0.00

0.50

Acc

eler

atio

n (

g)


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


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


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


Controller

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

PID Plant

GainCalculatorController

+

+

+

-

T

k


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


Controller Circuit Piezoceramic Actuator

System

1K 2Ks1

s2

DK

2vu

v

vvu


Nonresonance InteractionZero-to-One Internal Resonance


f = 16.25 Hz


Comparison between Responses of Beam and Hubble Telescope


IHI Scale Model Profile

nonlinear dynamics – phenomena and applications ali h. nayfeh department of engineering science...

Documents

lyapunov lecture

hz pitch

hz parametric instability

hz model

hz heave

heave pitch

ballast roll frequency

hz wave frequency