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 m + Damping force c + Stiffness force k = Excitation force

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  km

Leonhard Euler, 1707–1783, was the first person to write down the equation of motion of a harmonically forced, undamped oscillator, my ky F sin  t . He was also the first person to discover the phenomenon of resonance Historical Perspective

Robert Hooke, 1678 F ky

9 years

Isaac Newton, 1687 F my

63 years

Leonhard Euler, 1750 my ky Fsin t Historical Perspective

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 2 was f  k 12 y k y 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 f k13 y k y The Duffing Equation

3 my cy  k13 y  k y  Fcos 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 Nonlinear Spring

f f k stiffness  x hardening spring x linear force f softening For a linear system spring f kx

displacement x Symmetric Nonlinear Spring

Force-deflection

3 Fs  k13 y k y

Non-dimensional 3 Fs  y y

2 Fss F k1 l y y d0   k3 d 0 k 1

d0 =original length of spring Symmetric Nonlinear Spring

3 Fs  y y

Fys  3 Fs  y y Note symmetry Asymmetric Nonlinear Spring

Force-deflection

2 3 Fs  k13 y ky2  k y

Non-dimensional 2 3 Fs yy  y 

  k2 d 0 k 1

d0 =original length of spring Asymmetric Nonlinear Spring

23 23 Fs  y  y  y Fs  y  y  y linear   0

23 23 Fs  y  y  y Fs  y  y  y Asymmetric Nonlinear Spring

23 23 Fs  y  y  y Fs  y  y  y linear   0

23 23 Fs  y  y  y Fs  y  y  y Asymmetric Nonlinear Spring

23 23 Fs  y  y  y Fs  y  y  y linear   0

23 23 Fs  y  y  y Fs  y  y  y Asymmetric Nonlinear Spring

23 23 Fs  y  y  y Fs  y  y  y linear   0

Note Asymmetry

23 23 Fs  y  y  y Fs  y  y  y Pendulum – softening stiffness

Stiffness Moment

d 2 ml2  mglsin M cos t dt 2

35 sin    .... 3! 5!

 3 stiffness moment mgl  6

Softening Pendulum – softening stiffness

1

0.8 k  2 1 klin 2 k 0.6 klin 0.4 k 24 1   0.2 klin 2 24

0 0 20 40 60 80 90 angle (degrees) Pendulum – softening stiffness

30 60 120 170 Geometrically nonlinear spring

Force-deflection

d F21 ky 0 s 22 yd

Non-dimensional d Fy1, s 2 y 1

Fss F2 kd y y d d do d d0 =original length of spring Geometrically nonlinear spring

d F21 ky 0 s 22 yd

which approximates to

3 Fs  k13 y k y

3 k1021 k d d  k30 k d d Geometrically nonlinear spring – non-dimensional

d Fy1, s 2 y 1 which approximates to

3 Fs  y y

 1 d   d 2 Geometrically nonlinear spring – non-dimensional

approximation

actual Geometrically nonlinear spring – snap through

Fs Geometrically nonlinear spring – snap through

Potential Energy 2 V V2 kd  11 V1,  d y24  dy 28 

V Example of a snap-through system . Nonlinear Energy Harvesting Device – snap through . Nonlinear Energy Harvesting Device Nonlinear Energy Harvesting Device – snap through

Positive Stiffness Negative Stiffness

Beam Magnets Nonlinear Energy Harvesting Device – snap through 2 Stiff vertical spring 1.8 Optimal vertical spring Soft vertical spring

1.6

1.4

1.2

1 Force

0.8

0.6

0.4

0.2

0 0 0.5 1 1.5 2 Deflection Nonlinear Energy Harvesting Device – snap through

5 4 3 potential energy 2 1 0 -1 -2 1.05 1 0.5 0.95 0 0.9 -0.5 0.85 d displacment Vibration Isolation

fe xt m

kv c

xe

ft Vibration Isolation

f e k kh h xt m l

kv c

xe

ft An Achievable Stiffness Characteristic

0.3

0.2

0.1

0

stiffness force stiffness

-0.1

-0.2 Low stiffness Normalised

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Normalised displacement Bubble Mount

Very low stiffness Static(natural equilibrium frequency) position

Large Deflection of Beams Large deflection of beams

2w x2 M EI 3 , 2 w 2 1  x

Causes nonlinear effect

Softening

2 4w  s 2 w 3 22 w  w EI4   A 2 EI 2  2 f x, t x  x  t 2 x  x  x  Large deflection of beams -in-plane stiffness

2 2w 42 wEAl  w  w  A2  EI4  Ts dx 2  f(,) x t t  x2l  x  x 0 

Causes nonlinear effect

Hardening Large deflection of beams -in-plane stiffness 2 2w 42 wEAl  w  w  A2  EI4  Ts dx 2  f(,) x t t  x2l  x  x 0  Assume simple supports and first mode only  x   x  sin w x,() t   x q t f x, t  F x l 2 cos t l dq2 so m k q  k q3  Fcos t dt 2 13 k= 1 T l2 EI 2 EI 4 2 l 3 k  43 EA8 l 1 s     3  

Asymmetrical Systems Bubble Mount Bubble Mount

Asymmetry

stiffnessk 3 k x2 stiffness 13 5

4.5

4

3.5 softening hardening

3

2.5

2

1.5

1 0 0.2 hardening0.4 0.6 0.8 1 hardening1.2 1.4 1.6 1.8 2

displacement

Asymmetry

10 Symmetric system

3 Fs  k13 y k y 5

F s 0

-5

-10 -2 -1 0 1y 2

Asymmetry

10 Asymmetric system

3 5 Fs  F0  k 1 y  k 3 y F s 0

-5

-10 -2 -1 0 1 y 2

Asymmetry

10 Asymmetric system

3 5 Fs  F0  k 1 y  k 3 y F s 0

ˆ ˆ23 ˆ -5 Fs  k1 z  k 2 z  k 3 z

-10 -2 -1 0 1 y 2

Asymmetric System - Cable

Mode shapes

Increasing tension

dq2 m k q  k q23  k q  Fcos t dx2 1 2 3

Due to asymmetry -softening effect

Nonlinear damping

Nonlinear damping

Nonlinear damping

Nonlinear damping

(a) Damping force fe cx2 m fx h d 22 O ax 

c 1 x ch k For xa/ 0.2 x2 fdh c x a2 a y ft Damping coefficient is dependent on the square of the amplitude

Summary

• Historical perspective • Nonlinear stiffness • Symmetry • Asymmetry • Nonlinear damping