Nonlinear Dynamics-Phenomena and Applications
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Transcript Nonlinear Dynamics-Phenomena and Applications
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 uv
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