An Introduction to Nonlinearity

Mike Brennan (UNESP)

Bin Tang (Dalian University of Technology, China)

Gianluca Gatti (University of Calabria, Italy)

An Introduction to Nonlinearity • Introduction • Historical perspective • Nonlinear stiffness

• Symmetry • Asymmetry

• Nonlinear damping • Conclusions

Dynamical System

Inertia force

+

Damping force

+

Stiffness force

=

Excitation force

m

k

c

Common stiffness nonlinearities

Common damping nonlinearities

Historical Perspective

Historical Perspective Galileo Galilei, 1564–1642, studied the

pendulum. He noticed that the natural frequency of

oscillation was roughly independent of the amplitude

of oscillation

Christiaan Huygens, 1629–1695, patented the

pendulum clock in 1657. He discovered that wide

swings made the pendulum inaccurate because he

observed that the natural period was dependent on

the amplitude of motion, i.e. it was a nonlinear

system.

Historical Perspective

Robert Hooke, 1635–1703, is famous for

his law which gives the linear relationship

between the applied force and resulting

displacement of a linear spring.

Isaac Newton, 1643–1727, is of course,

famous for his three laws of motion.

Historical Perspective

John Bernoulli, 1667–1748, studied a

string in tension loaded with weights. He

determined that the natural frequency of a

system is equal to the square root of its

stiffness divided by its mass, n k m

sinmy ky F t

Leonhard Euler, 1707–1783, was the first

person to write down the equation of motion of a

harmonically forced, undamped oscillator,

. He was also the first person

to discover the phenomenon of resonance

Historical Perspective

F kyRobert Hooke, 1678

Leonhard Euler, 1750 sinmy ky F t

Isaac Newton, 1687 F my

9 years

63 years

Historical Perspective

2

1 2f k y k y

Hermann Von Helmholtz, 1821–1894, was

the first person to include nonlinearity into the

equation of motion for a harmonically forced

undamped single degree-of-freedom oscillator. He

postulated that the eardrum behaved as an

asymmetric oscillator, such that the restoring force

was which gave rise to additional

harmonics in the response for a tonal input.

John William Strutt, Third Baron Rayleigh,

1842–1919, considered the free vibration of a

nonlinear single-degree-of-freedom in which the

force-deflection characteristic was symmetrical,

given by 3

1 3f k y k y

The Duffing Equation

3

1 3 cosmy cy k y k y F t

Georg Duffing 1861-1944

The Duffing Equation: Nonlinear Oscillators and their Behaviour

Editors: Ivana Kovacic and Michael J. Brennan

Published in 2011

Nonlinear stiffness

linear

hardening

spring

softening

spring

Nonlinear Spring

displacement

x

force

f

k f

x

For a linear system

f kx

stiffnessf

x

3

1 3sF k y k y

Symmetric Nonlinear Spring

Force-deflection

0 =original length of springd

1s sF F k l 0y y d2

3 0 1k d k

3

sF y y

Non-dimensional

Symmetric Nonlinear Spring

3

sF y y

3

sF y y

Note symmetry

sF y

2

2

3

1 3sk yF k y k y

Asymmetric Nonlinear Spring

Force-deflection

0 =original length of springd

2 0 1k d k

32

sF yy y

Non-dimensional

Asymmetric Nonlinear Spring

2 3

sF y y y 2 3

sF y y y

2 3

sF y y y 2 3

sF y y y

linear

0

Asymmetric Nonlinear Spring

2 3

sF y y y 2 3

sF y y y

2 3

sF y y y 2 3

sF y y y

linear

0

Asymmetric Nonlinear Spring

2 3

sF y y y 2 3

sF y y y

2 3

sF y y y 2 3

sF y y y

linear

0

Asymmetric Nonlinear Spring

2 3

sF y y y 2 3

sF y y y

2 3

sF y y y 2 3

sF y y y

linear

0

Note Asymmetry

Pendulum – softening stiffness

22

2sin cos

dml mgl M t

dt

3 5

sin ....3! 5!

Softening

Stiffness Moment

3

stiffness moment 6

mgl

lin

k

k

angle (degrees)

2

lin

12

k

k

2 4

lin

12 24

k

k

0 20 40 60 80 900

0.2

0.4

0.6

0.8

1

Pendulum – softening stiffness

Pendulum – softening stiffness

30 60 120 170

Geometrically nonlinear spring

0 =original length of springd

0

2 22 1s

dF ky

y d

Force-deflection

21 ,

1

s

dF y

y

Non-dimensional

2s sF F kd y y dod d d

Geometrically nonlinear spring

3

1 3sF k y k y

which approximates to

0

2 22 1s

dF ky

y d

1 02 1k k d d 3

3 0k k d d

Geometrically nonlinear spring – non-dimensional

which approximates to

3

sF y y

21 ,

1

s

dF y

y

1 d 2d

Geometrically nonlinear spring – non-dimensional

actual

approximation

Geometrically nonlinear spring – snap through

sF

Geometrically nonlinear spring – snap through

2 41 11 ,

2 8 V d y dy

Potential Energy 22V V kd

V

Example of a snap-through system

. Nonlinear Energy Harvesting Device – snap through

Nonlinear Energy Harvesting Device .

Positive Stiffness

Beam

Negative Stiffness

Magnets

Nonlinear Energy Harvesting Device – snap through

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Stiff vertical spring

Optimal vertical spring

Soft vertical spring

Forc

e

Deflection

Nonlinear Energy Harvesting Device – snap through

-0.5

0

0.5

0.85

0.9

0.95

1

1.05-2

-1

0

1

2

3

4

5

d displacment

potential energy

Nonlinear Energy Harvesting Device – snap through

Vibration Isolation

vk c

m

ef

tf

tx

ex

vk

hk hk

c

m

ef

tf

tx

ex

l

Vibration Isolation

An Achievable Stiffness Characteristic

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-0.2

-0.1

0

0.1

0.2

0.3N

orm

alis

ed s

tiff

ne

ss f

orc

e

Normalised displacement

Low stiffness

Static equilibrium position

Very low stiffness

(natural frequency)

Bubble Mount

Large Deflection of Beams

Large deflection of beams

2

2

32 2

,

1

w

xM EI

w

x

Softening

22 2

2

2

2 2

4

4,

3

2

ww s wEI A

wEI

x x xf x t

x x t

Causes nonlinear effect

Large deflection of beams

-in-plane stiffness

Causes nonlinear effect

2

0

2

4

24 2

2 2( , )s

lEA w

dxl

w w wA EI T f x t

t x xx

Hardening

Large deflection of beams

-in-plane stiffness

2

0

2

4

24 2

2 2( , )s

lEA w

dxl

w w wA EI T f x t

t x xx

Assume simple supports and first mode only

sinx

xl

, ( )w x t x q t , 2 cos f x t F x l t

so 2

3

1 32cos

d qm k q k q F t

dt

2 2 4 3

1= 1 2sk T l EI EI l 4 3

3 8k EA l

Asymmetrical Systems

Bubble Mount

Bubble Mount

Asymmetry

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1

1.5

2

2.5

3

3.5

4

4.5

5

displacement

stiffness

hardening hardening

hardening softening

2

1 3stiffness 3k k x

Asymmetry

3

1 3sF k y k y

-2 -1 0 1 2-10

-5

0

5

10

y

sF

Symmetric system

Asymmetry

3

0 1 3sF F k y k y

-2 -1 0 1 2-10

-5

0

5

10

sF

y

Asymmetric system

-2 -1 0 1 2-10

-5

0

5

10

Asymmetry

3

0 1 3sF F k y k y

sF

y

Asymmetric system

2 3

1 2 3ˆ ˆ ˆ

sF k z k z k z

Asymmetric System - Cable

22 3

1 2 32cos

d qm k q k q k q F t

dx

Mode shapes

Increasing tension

Due to asymmetry

-softening effect

Nonlinear damping

Nonlinear damping

Nonlinear damping

Nonlinear damping

(a)

k

a

c1

O

x

m

fe

yft

ch

For / 0.2x a

2

2d h

xf c x

a

2

2 2

hd

c xf x

a x

Damping force

Damping coefficient is dependent on the square of the amplitude

Summary

• Historical perspective

• Nonlinear stiffness

• Symmetry

• Asymmetry

• Nonlinear damping