an-introduction-to-nonlinearity

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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. 
http://upload.wikimedia.org/wikipedia/commons/c/cc/Galileo.arp.300pix.jpg
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 
http://en.wikipedia.org/wiki/File:Johann_Bernoulli2.jpg
http://upload.wikimedia.org/wikipedia/commons/6/60/Leonhard_Euler_2.jpg
Historical Perspective 
F kyRobert Hooke, 1678 
Leonhard Euler, 1750 sinmy ky F t 
Isaac Newton, 1687 F my
9 years 
63 years 
http://upload.wikimedia.org/wikipedia/commons/6/60/Leonhard_Euler_2.jpg
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 
http://upload.wikimedia.org/wikipedia/commons/c/c7/Hermann_von_Helmholtz.jpg
The Duffing Equation 
   31 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



stiffness
f
x
3
1 3s
F k y k y 
Symmetric Nonlinear Spring 
Force-deflection 
0 =original length of springd
1s sF F k l 0y y d
2
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 3s
k 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 
2
2
2
sin cos
d
ml mgl M t
dt

  
3 5
sin ....
3! 5!
 
    
Softening 
Stiffness Moment 
3
stiffness moment 
6
mgl


 
  
 
lin
k
k
angle (degrees)
2
lin
1
2
k
k

 
2 4
lin
1
2 24
k
k
 
  
0 20 40 60 80 90
0
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 2
2 1s
d
F ky
y d
 
  
  
Force-deflection 
2
1 ,
1
 
  
  
s
d
F y
y
Non-dimensional 
2s sF F kd y y d od d d
Geometrically nonlinear spring 
3
1 3sF k y k y 
which approximates to 
0
2 2
2 1s
d
F 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  
2
1 ,
1
 
  
  
s
d
F y
y
1 d   2d 
Geometrically nonlinear spring 
– non-dimensional 
actual
approximation
Geometrically nonlinear spring 
– snap through 
sF
Geometrically nonlinear spring 
– snap through 
  2 4
1 1
1 ,
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 2
0
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
F
o
rc
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
0.3
N
o
rm
al
is
ed
 s
ti
ff
n
e
ss
 f
o
rc
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
3
2 2
,
1
w
xM EI
w
x


  
  
   
Softening 
 
22 2
2
2
2 2
4
4
,
3
2
ww s w
EI A
w
EI
x x x
f 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
l
EA w
dx
l
w w w
A 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
l
EA w
dx
l
w w w
A EI T f x t
t x xx

 
 
 
   
        

Assume simple supports and first mode only 
  sin
x
x
l


 
  
 
   , ( )w x t x q t    , 2 cos  f x t F x l t
so 
2
3
1 32
cos
d q
m k q k q F t
dt
  
   2 2 4 31= 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
 
Asymmetry3
0 1 3sF F k y k y  
sF
y
Asymmetric system 
2 3
1 2 3
ˆ ˆ ˆ
sF k z k z k z  
 
Asymmetric System - Cable 
2
2 3
1 2 32
cos
d q
m 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
y
ft
ch
For / 0.2x a 
2
2d h
x
f c x
a

 
2
2 2
h
d
c x
f x
a x


Damping force 
Damping coefficient is dependent 
on the square of the amplitude 
 
Summary 
• Historical perspective 
• Nonlinear stiffness 
• Symmetry 
• Asymmetry 
• Nonlinear damping