Nonlinear Dynamics and Resonances: a surveyO.V.Gendelman
Technion – Israel Institute of Technology
Technion
1924 Einstein Tree
Technion
Faculty of Mechanical Engineering
- Biomechanics- Optomechaincs- Robotics- Dynamical Systems- Control Theory- Hydrodynamics and
Microfluidics- CAD/CAM
Resonance in Linear Systems
Resonance in Linear Systems –No Damping
00, = =00, 1.1 = =
Beatings
1 1 1 2
2 2 2 1
( ) 0
( ) 0
u u u u
u u u u
+ + − =
+ + − =
Nonlinear oscillator (weak nonlinearity)
3( cos )u u u u t + = − + +
- Frequency close to natural- Co-existence of the response
regimes- Hystheresis
Co-existence of responses
Simple Hystheresis3( cos(1 cos ) )u u u u t t + = − + + +
Frequency versus time
Resonance – idea (not definition!)SMALL REASONS (FORCING, COUPLING) LEAD TO LARGE CONSEQUENCES
Why we see only one side of the moon?
Why we see only one side of the moon?
Equal frequencies of rotation – a variant of the resonance.
3
3( ) sin 2 0
2
GMC B A
r + − =
Capture into the resonance
qL = − L – Phase on the Kepler orbitq – rational number
02sin =+ QC
Capture into the resonance
qL = − L – Phase on the Kepler orbitq – rational number
02sin =+ QC
PENDULUM!
Pendulum –most famous oscillatory system
17
The first dynamical system - pendulum
Galileo Galilei (1564-1642) The famous lamp(Pisa cathedral)
Pendulum-oscillatory regimes
18
The first dynamical system – pendulumLinear – weakly nonlinear –strongly nonlinear
Frequency of oscillations depends on the energy
Strong nonlinearity versus weak nonlinearity
19
The first dynamical system – pendulumLinear – weakly nonlinear –strongly nonlinearFrequency of oscillations depends on the energy
0 1.98E =
0 2.02E =
Capture into the resonance –account of tidal forces
baQC −=+ 2sin
No damping
Q a Q a
Capture into the resonance –account of tidal forces
baQC −=+ 2sin With the damping
Balthazar Van der Pol (1889 –1959)
- Limit cycle oscillations (electric circuits)
- Relaxation oscillations (1929)
- Chaos in driven autoresonant systems
- Oscillations in biological systems
- Heart pacemakers
Van der Pol oscillator (modern version)
Van der Pol oscillator (modern version)
3
0 0( ) ( ) ( )i v v E v E = = − − −Characteristics of the tunneling diode (negative damping)
Van der Pol oscillator (modern version)
3
0 0( ) ( ) ( )i v v E v E = = − − −Characteristics of the tunneling diode (negative damping)
0
1( ( ) )
1
V V E WC
W VL
= − + −
=
Van der Pol oscillator (modern version)
3
0 0( ) ( ) ( )i v v E v E = = − − − 0
1( ( ) )
1
V V E WC
W VL
= − + −
=
21 1( 3 ) 0V V V V
C LC − − + =
Van der Pol oscillator (modern version)
3
0 0( ) ( ) ( )i v v E v E = = − − − 0
1( ( ) )
1
V V E WC
W VL
= − + −
=
21 1( 3 ) 0V V V V
C LC − − + =
Rescaling:
3, t ,
t Lx V
CLC
= → =
2(1 ) 0x x x x− − + =Canonic form of Van der Pol (VdP)
Equation
VdP equation
2(1 ) 0x x x x− − + =
The case of small ε
Time series for ε=0.1
Phase portrait
VdP equation
2(1 ) 0x x x x− − + =
The case of small ε - analysis
2
3
( ) exp( )
1( ) 0
2 4
( )exp( ( ))
1( )
2 4
x ix t it
N t i t
N N N
+ =
− − =
=
= −
VdP equation
2(1 ) 0x x x x− − + =
The case of small ε - analysis
2
3
( ) exp( )
1( ) 0
2 4
( )exp( ( ))
1( )
2 4
x ix t it
N t i t
N N N
+ =
− − =
=
= −
Solution
02
0
2
0
2( ) , (0)
41 exp( )
N t N NN
tN
= =−
+ −
• AND WHAT ABOUT LARGE FORCING AND LARGE DAMPING?
VdP equation
2(1 ) 0x x x x− − + =
The case of large ε – simulation
ε=10
VdP equation
2(1 ) 0x x x x− − + =
The case of large ε – phase portrait
VdP equation
2(1 ) 0x x x x− − + =
The case of large ε – Lienard variables
3
3
/ 3 /
( / 3 )
/
y x x x
x x x y
y x
= − + +
= − +
= −
VdP equation
2(1 ) 0x x x x− − + =
2 2(1 ) 0x x x x − − + =
The case of large ε – reduction of the VdP
equation
1/ 1
/d d
tdt d
=
= =
VdP equation
2(1 ) 0x x x x− − + =
2 2(1 ) 0x x x x − − + =
The case of large ε – reduction of the VdP
equation
1/ 1
/d d
tdt d
=
= =
Small parameter multiplies the term with the highest derivative!
Singularly perturbed problem.
Van der Pol model
2
3
3
(1 ) 0
/ 3 /
( / 3 )
/
x x x x
y x x x
x x x y
y x
− − + =
= − + +
= − +
= −
Lienard system, ε>>1
Equation for x – “fast” dynamics
The system “tries” to nullify the velocity in x direction and does it fast!
Van der Pol model
2
3
3
(1 ) 0
/ 3 /
( / 3 )
/
x x x x
y x x x
x x x y
y x
− − + =
= − + +
= − +
= −
Lienard system, ε>>1
Equation for x – “fast” dynamics
Equation for y – “slow” dynamics
Slow evolution of x and y provided that the fast evolution is absent
Van der Pol model
2
3
3
(1 ) 0
/ 3 /
( / 3 )
/
x x x x
y x x x
x x x y
y x
− − + =
= − + +
= − −
= −
3( / 3 )
0
x x x y
y y const
= − −
= =
Fast dynamics (principal approximation)
Van der Pol model
2
3
3
(1 ) 0
/ 3 /
( / 3 )
/
x x x x
y x x x
x x x y
y x
− − + =
= − + +
= − +
= −
3( / 3 )
0
x x x y
y y const
= − +
= =
Fast dynamics (principal approximation)
The system is attracted to the “slow” manifold
3 / 3y x x= − +
Van der Pol model
3 / 3y x x= − +
Slow manifold
Stable branches
Unstable branch
Van der Pol model
3 / 3y x x= − +
Slow manifold
Stable branches
Unstable branch
Fast motion
Van der Pol model2
3
3
(1 ) 0
/ 3 /
( / 3 )
/
x x x x
y x x x
x x x y
y x
− − + =
= − + +
= − +
= −
3
2
2
/ 3( 1)
1/
y x x x xx x x
xy x
= − = − − = −
−= −
Lienard system, ε>>1
“Slow” motion:
Van der Pol model
“Slow” motion
Van der Pol model
Complete cycle of the relaxation oscillation!
Van der Pol model
Complete cycle of the relaxation oscillation!
Let us compute the period.
Van der Pol model
2
2
1
12 ( ) (3 2ln 2)
1slow
xx T x dx
x x = − = − = −
−
Complete cycle of the relaxation oscillation!
Slow part:
Van der Pol model
(3 2ln 2)slowT = −
3( / 3 )
0
x x x y
y y const
= − +
= =
23
3
1
( / 3 ) 2(1/ )
/ 3 2 / 32 / 3fast
x x x y dxT O
x xy
−
= − + = =
− + +=
Complete cycle of the relaxation oscillation!
Slow part:
Fast part:
For jump: y=±2/3
Van der Pol model
(3 2ln 2)slowT = −
3( / 3 )
0
x x x y
y y const
= − +
= =
23
3
1
( / 3 ) 2(1/ )
/ 3 2 / 32 / 3fast
x x x y dxT O
x xy
−
= − + = =
− + +=
Complete cycle of the relaxation oscillation!
Slow part:
Fast part:
For jump: y=±2/3
Are there any corrections of lower order?
Van der Pol model
(3 2ln 2)slowT = −
2 1
xx
x = −
−
2
3
1
2(1/ )
/ 3 2 / 3fast
dxT O
x x
−
= =− + +
Slow part:
Are there any corrections of lower order?
Slow –flow equation
Singularity at x=±1!
Van der Pol model
(3 2ln 2)slowT = −
2 1
xx
x = −
−
2
3
1
2(1/ )
/ 3 2 / 3fast
dxT O
x x
−
= =− + +
Slow part:
Are there any corrections of lower order?
Slow –flow equation
Singularity at x=±1!
Approximation breaks down – infinite slow velocity.
Van der Pol model
(3 2ln 2)slowT = −
2 1
xx
x = −
−
2
3
1
2(1/ )
/ 3 2 / 3fast
dxT O
x x
−
= =− + +
Slow part:
Are there any corrections of lower order?
Slow –flow equation
Singularity at x=±1!
Approximation breaks down – infinite slow velocity.
Matching is required
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Matching in physical plane
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 1
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 1
2
0 0 0
2
0 0
0
( 1) 0
ln ( 1)
0, 0, 0
x x x
x x A
x A
− + =
= − − +
=
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 1
2
0 0 0
2
0 0
0
0
( 1) 0
ln ( 1)
0, 0, 0
0, ~ 1 2
Singularity!
x x x
x x A
x A
x
− + =
= − − +
=
→ − + −
Near x=1 separate approximation is required
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 2
4/3 2/3
2
2
2
, 1 ( )
2 1 0
, const
x v
d v dvv
d d
dvv B B
d
= = +
+ + =
+ + = =
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 2
4/3 2/3
2
2
2
, 1 ( )
2 1 0
, const
x v
d v dvv
d d
dvv B B
d
= = +
+ + =
+ + = =
/ ,
( ) (Ai( ) Bi( ))
, const
v z z
z B C B
C
=
= − + −
=
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 2
Matching with Zone 1
, ~
0
( ) Ai ( ) / Ai( )
v
B C
v
→ − −
= =
= − − −
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 2
Matching with Zone 1
, ~
0
( ) Ai ( ) / Ai( )
v
B C
v
→ − −
= =
= − − −
Everything is OK until singularity…
1
( ) 0, 2,33811
In the vicinity:
( ) ~ ( ) simple pole
Ai
v
−
− = =
− −
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 3
4/3 2
22
2
3
( 1) 0
/ 3
d w dww
d d
dww w D
d
= +
+ − =
= − +
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
Zone 3
4/3 2
22
2
3
( 1) 0
/ 3
d w dww
d d
dww w D
d
= +
+ − =
= − +
Matching with Zone 2
4/3
4/3
1 1 2 / 3ln( )
1 3 1
, 2 / 3 (exp( ))
w
w w
w O
+ +− = +
− −
→ = − − + −
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =2
0 0
4/3
4/3
ln ( 1)
3 / 2 ln 2 3 / 2
3 2ln 2 3
x x A
A
T
= − − +
= − +
= − +
Zone 1
Matching with Zone 1 (lower branch) and completing the half – cycle.
4/3 4/3( , ) ( 2 / 3, )x = − −
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
1/3(3 2ln 2) 3 , 2.3381T −= − + =
Expression for the period (initial variables)
Van der Pol model - matching
2 2(1 ) 0x x x x − − + =
1/3(3 2ln 2) 3 , 2.3381T −= − + =
1/3 1
1
4/3
2(3 2ln 2) 3 ln
3
{ln 2 ln 3 3 1 ln 2ln Ai ( )}
( ln ), 2.3381, 0.17235
T
b
O b
− −
−
−
= − + − +
− + − − − − +
+ = =
Expression for the period (initial variables)
After much more painstaking efforts one can obtain:
Phenomenology - 1Concept of fast – slow partition (boundary layer)
Phenomenology - 2Concept of fast – slow partition (boundary layer)
Slow process Fast process
Phenomenology - 3
Synchronization
Synchronization – simple model
),(
),(
21222
21111
Fdt
d
Fdt
d
+=
+=
1 2 1 2 1 2
1 2 1 2
( , ) ( );
( ),
i iF F
dF F F F
dt
= − = −
= − + = −
Synchronization – simple model
The first term of Fourier series
1 2 1 0sin( )d
Fdt
= − + +
Possibility of Stationary Solution
1
~
21
1 −
F
Earthquake
L'Aquila Earthquake, Italy, 200972
Simple Procedure for Seismic Analysis of Liquid-Storage Tanks, Malhotra, Structural Engineering International, 2000
Tanks with liquids
Liquid SloshingWhat is it?
73
Sloshing noun any motion of the free liquid
surface inside its container. It is caused by
any disturbance to partially filled liquid
containers.
(Liquid sloshing dynamics- Theory and Applications, R.A. Ibrahim, 2005)
Sloshing/convective portion
Static portion
Liquid SloshingWhat is it?
74
1. Infinitely many degrees of freedom
2. Substantially nonlinear vibrations
How to explain the dynamics?
"כעש
תה
ר ייא
בד
ו"
75
http://www.mscsoftware.com/en/product/dytran
Reduced-0rder modeling
( ) ( )2 42 2 2
1 2
1 1 1 1 1
2 2 2 2 4tt ttL T V Mu mv k u k v u k v u= − = + − − − − −
( )21
2t tD c v u= −
Analytic treatmentGeneral structure of slow invariant manifold
Possibility of relaxation oscillations? Analysis of slow motion is required.
Numeric experimentsSteady – state and strongly quasiperiodic responses coexist for different IC
A=0.225, λ=0.2, ε=0.05. y1(0)=0.29,dy1/dt(0)=0.25, y2(0)=0, dy2/dt(0)=-0.15
A=0.225, λ=0.2, ε=0.05. y1(0)=0, dy1/dt(0)=0, y2(0)=0, dy2/dt(0)=0.
Numeric experimentsWeakly quasiperiodic response and SMR coexist for different IC
A=0.24, λ=0.2, ε=0.05. y1(0)=0.29, dy1/dt(0)=0.25, y2(0)=0, dy2/dt(0)=-0.15
A=0.24, λ=0.2, ε=0.05. y1(0)=0, dy1/dt(0)=0, y2(0)=0, dy2/dt(0)=0.
Successive captures into the resonance
Captured
Jumps out
Experimentsl and numeric results (full-scale sloshing!)
a) b)
c) d)
Experimental and numeric results (full-scale sloshing1)
83