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  2r 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

b0
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
F5.8mg8.9mg
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
u1  1 u1
2
u2   2 u2
3
 21u1  1u1
2
  2u1u2
3
 2 2u2   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
u1   u1  21u1  
2
2
3
(41u1
3
u2  u2  2 2 u2   ( 3u2
2
  2u1u2 )
2
  4u1 u2
 f cos t )
Lyapunov Lecture 2005
Variation of Parameters
 We let
u1   v1
u2  a (t ) cos[t   (t )]
u2  a (t ) sin[ t   (t )]
  t   (t )
 Detuning from resonance
  1  
2
Lyapunov Lecture 2005
Variational Equations
u1   v1
3
2
2
v1   (u1  21v1  41u1   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
u1   v1
v1 
3
 (u1  2 1v1  41u1
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  41u1
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  
81
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 2n
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 2n
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 2n
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 2n
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 2n
Lyapunov Lecture 2005
Controlled Response
2° Roll at n
1° Pitch at n
0.5 in Heave at 2n
Lyapunov Lecture 2005
Controlled Response
Slewing Crane
2° Roll at n
1° Pitch at n
0.5 in Heave at 2n
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