Basic phenomenology of simple nonlinear vibration … phenomenology of simple nonlinear vibration!...

Basic phenomenology of simple nonlinear vibration

!

(free and forced)

Manoj Srinivasan (2016)

MassSpring

Damper

x(t)

x(t)

x(t)

e

mass m

length lgravity g

A

O

Hardening

Softening

Nonlinear spring-mass system

No damping

Nonlinear spring-mass systemNo damping

Frequency (time period) of free vibration oscillations

depends on oscillation amplitude-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

xDot

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1oscillation amplitude

6.25

6.3

6.35

6.4

6.45

6.5

6.55

time

perio

d of

osc

illatio

n

softening cubic nonlinearity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1oscillation amplitude

6.05

6.1

6.15

6.2

6.25

6.3

time

perio

d of

osc

illatio

n

hardening cubic nonlinearity

unlike for linear spring-mass system

Timeperiod

Oscillation amplitude Oscillation amplitude

HardeningSoftening Timeperiod

guitar “pitch glide” due to amplitude dependence

of frequency? (assume stiffening)

how would you expect the frequency to change as the oscillation amplitude

decreases?

0 20 40 60 80 100 120 140 160 180 200t

-10

-8

-6

-4

-2

0

2

4

6

8

10

x

Damped Duffingwith alpha>0

stiffening spring

x(t)

t

Damped free vibrations of Duffing equation

https://www.youtube.com/watch?v=VkkOFLouQDs

notice frequency change

MassSpring

Damper

Damping ratio Natural frequency

( Static deflection to

static force F0 )

External forcing

Normalized force amplitude =(units of length)

Linear spring-mass-damper system

Frequency response oflinear spring-mass-damper system

figure source: Wikipedia

MassSpring

Damper

External forcing

Duffing equation with forcing

x(t)

x(t)

x(t)

e

mass m

length lgravity g

A

O

Hardening

Softening

cubic nonlinearity

Cubic nonlinearity with or without quadratic nonlinearity

cubic

When we do Taylor series of an odd function about an equilibrium for with spring force = 0

When we do Taylor series of odd function about an equilibrium with spring force not = 0

quadratic

(Recall HW1 forillustrative example)

cubic

(or) just Taylor series of a not-odd function

Duffing’sequation

Frequency response ofDuffing equation

(cubic nonlinearity)

Primary resonance . Big response amplitude when forcing frequency ωo ~ ‘linear’ natural frequency ωn

Super-harmonic resonance: Big response amplitude when forcing frequency ωo ~ ωn / integer (example: ωo ~ ωn/3)!Sub-harmonic resonance: Big response amplitude when forcing frequency ωo ~ ωn x integer. (example: ωo ~ 3ωn)!!

Secondary resonance

Primary resonance and secondary resonances

Multiple peaks due to nonlinearity even though it is a single DOF system

( Recall that an N DOF linear system (N>1) will have multiple peaks due to there being N modes and corresponding natural frequencies)

Linear vs Hardening vs SofteningHow the primary resonance’s frequency response changes

Linearsystemresponse

Hardening springSoftening spring

forcing frequencyforcing frequency

Response amplitude (y axis) vs forcing frequency (x axis)for 3 cases

arrows indicate response obtained in forward and backward sweeps

Nayfeh and Mook

Hardening vs SofteningAmplitude response by simulation

until steady state

0 0.5 1 1.5 2 2.5 3 3.5Forcing frequency �

0

1

2

3

4

5

6

7

8

9

10

Stea

dy O

scilla

tion

Ampl

itude

Steady response amplitude vs frq. with the points connectedcubic nonlinearity alpha = 0.5

forward frequency sweepbackward frequency sweep

0 0.5 1 1.5 2 2.5 3 3.5Forcing frequency �

0

10

20

30

40

50

60

Stea

dy O

scilla

tion

Ampl

itude

Steady response amplitude vs frq. with the points connectedcubic nonlinearity alpha = -4

forward frequency sweepbackward frequency sweep

Hardening Softening

Forward sweep (magenta) and Backward sweep (blue) shown

see MATLAB program

Frequency response for forced Duffing with

hardening spring

0.5 1 1.5 2 2.5 3 3.5 4forcing frequency

0

0.5

1

1.5

2

2.5

3

3.5

4

resp

onse

am

plitu

de

Frequency response for forced Duffing with

softening spring

Amplitude response obtained by finding fixed points of Poincare maps

(so we can find both stable and unstable motions)

blue = stable periodic responsered = unstable periodic response

forcing frequency forcing frequency

resp

onse

am

plitu

de

resp

onse

am

plitu

desee MATLAB program

Ueda shows that the fully nonlinear forced Duffing(linear stiffness = 0) has many parameter regimes with

many different behavior

Dampingcoeff

c

Forcing amplitude A

See paper uploaded,Ueda 1991.

Following slidesshow some MATLABillustrations of this

complexity

-1 -0.5 0 0.5 1x

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

xdot

Five different co-existing periodic motionsFor the same parameter values: k = 0, A = 0.2, c = 0.08, omega = 1

0 10 20 30 40 50 60 70t

-1.5

-1

-0.5

0

0.5

1

1.5

x(t)x(t)

t

xDot

x

Chaotic motion (unique)For the same parameter values: k = 0, A = 10, c = 0.1, omega = 1

0 20 40 60 80 100 120 140 160 180 200t

-4

-3

-2

-1

0

1

2

3

4

x(t)

-8 -6 -4 -2 0 2 4 6 8x

-6

-4

-2

0

2

4

6

xdot

Chaotic

0 10 20 30 40 50 60 70 80 90 100t

-4

-3

-2

-1

0

1

2

3

4

x(t)

0 10 20 30 40 50 60 70 80 90 100t

-4

-3

-2

-1

0

1

2

3

4

x(t)

Co-existing periodic motion and chaotic motionFor the same parameter values: k = 0, A = 12.5, c = 0.1, omega = 1

-10 -8 -6 -4 -2 0 2 4 6 8 10x

-8

-6

-4

-2

0

2

4

6

8

xdot

periodic motion chaotic motion

0 1 2 3 4 5 6 7 8 9 10Omega (Hz) -- in angular frequency

0

1

2

3

4

Fourier Spectrum of External Forcing function

0 1 2 3 4 5 6 7 8 9 10in angular frequency (Hz)

0

0.5

1

1.5

2Fourier Spectrum of the oscillation Response

Frequency content of response

c = 0.1m = 1, k = 1epsilon = 0.4omega = 1

A = 5

Even when the responsehas the same frequency as the

forcing, there can be other harmonics in response due to

nonlinearity(unlike linear systems)

Primary resonance (forcing ω = 1)

Input

Response

Four

ier

ampl

itude

spe

ctru

m

frequency


0

1

2

3

4



0

0.5

1

1.5

2

2.5

3

3.5Fourier Spectrum of the oscillation Response

c = 0.1m = 1, k = 1epsilon = 0.4omega = 2

A = 5

Frequency content of responseForcing freq ω = 2

Input

Response

Four

ier

ampl

itude

spe

ctru

m

frequency

far fromresonance

we don’t see much higher harmonics in

response


0

2

4

6

8



0

0.2

0.4

0.6

0.8

1


Chaos = broad band frequency content

40 60 80 100 120 140 160t

-4

-3

-2

-1

0

1

2

3

4

x

Phase space in 3D

x(t)

Four

ier

ampl

itude

spe

ctru

m

frequency

chaotic motion


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14Fourier Spectrum of External Forcing function


00.05

0.10.15

0.20.25

0.30.35

Fourier Spectrum of the oscillation Response

Ueda(Manoj

notation)!

c = 0.08k = 0

epsilon = 1; omega = 1A = 0.2;


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14Fourier Spectrum of External Forcing function


0

0.05

0.1

0.15

0.2

0.25


period 2 motion

period 3 motion-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

x

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

xdot

x

xDot

-0.4 -0.2 0 0.2 0.4 0.6x

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

xdotxDot

x

Beware: !softening cubic nonlinearityand purely quadratic nonlinearity!have regimes where the stiffness is ‘negative’ for somelarge amplitudes => system can go to infinity if the forcing is not small enough!Fix:!A quadratic nonlinearity could be accompanied by a stiffeningcubic nonlinearity which keeps the motion bounded!A softening cubic nonlinearity could be accompanied by a stiffening x^5 nonlinearity, which keeps the motion bounded!

Basic phenomenology of simple nonlinear vibration … phenomenology of simple nonlinear vibration!...

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