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