On the Impulse Response of theOn the Impulse Response of the Duffing Oscillatorg

Mike Brennan (UNESP)Gianluca Gatti (University of Calabria, Italy)Gianluca Gatti (University of Calabria, Italy)Bin Tang (Dalian University of Technology, China)

1

Outline

Objective:(1) Understanding four basic analysis method.(2) Transient response of Duffing oscillator. ( ) p g

2

Outline

1 Review2 Key Features of the Impulse Response3 Conclusion

3

Review-IntroductionFree vibration of a Duffing oscillator Initial displacement / Impulse response Initial displacement / Impulse response

(Initial velocity) ?• The Straightforward expansion

• The Lindstedt-Poincaré Methode dstedt o ca é et od

• The Method of Multiple Scales

• The Method of Harmonic Balance Can these results capture the key features for theCan these results capture the key features for the

impulse response of a damped system?

4

Review-Equation of Motion Impulse excitation

m

Aδ(t)

k k

m

xO

c k1,k3

N di i l

31 3 ( )mx cx k x k x A t

• Non-dimensional32 ( )y y y y

5

2 ( )y y y y

The Method of Multiple Scales (MMS) Damping is small

23sin 1y e e

0 sin 116

y e e

γ→ 0

i

× Not correct for damped system!

0 siny e

2( ) sin 1ey

× Not correct for damped system!

√ Only valid for lightly damped system!

2( )

1y

√ Only valid for lightly damped system!

6

The Modified KBM method (MKBMM)

22

21/2 22

3 11( ) sin 1 ln 1

2 8(1 )

eey

2

22 8(1 )3 1

1 18

e

222 32

3/2 22 23/22

3 112 3 sin 3 1 ln 12 8(1 )16 4 3 3 1

ee

e

21 1

8

3

223 3 11 e 3

16 4

23

23/2 22 2

2

3 11cos3 1 ln 12 8(1 )3 3 1

1 18

ee

e

8

7

The Modified KBM method (MKBMM)

22

21/2 2

3 11( ) sin 1 ln 1

2 8(1 )

eey

1/2 22

22 8(1 )3 1

1 18

e

2 23 11

8

e

2 2

2 2

3( ) sin 1 11 16 1

ey e

γ→ 0

22

( ) sin 11

ey

8

Lindstedt-Poincaré method (LPM)

21 3( ) sin 1y e e

22 2 2

2 22 2 2 2

( ) sin 11 8 1 16 1

6 4 1 2 3 3

y e e

2 2 2 222 2

2

2 3 33 1 1 1 sin 14 31 8 1

3 1 3

e

2 2

2 2

2

3 1 33 3 cos 14 3 8 1

e

23 1 22

22 34 3

e

22 2 2 2

22 2

3 13 31 sin 3 1 cos3 14 38 1 8 1

e

9

Lindstedt-Poincaré method (LPM)

3 2

0 2 2

3sin 11 8 1

ey

Damped frequency isindependent with time d

d

pd

2when 2 1, 1 2e

2 23( ) sin 1 1ey e

MKBMM 2 2

( ) sin 1 11 16 1

y e

MKBMM 23 1

18

e

108

Comparison of the Results – Time Domain

1) 0.6)

γ = 0 2; γ = 0 2;

0 5men

t y(

0 4men

t y( γ = 0.2;

ζ = 0.5γ = 0.2;ζ = 0.05

0.5

ispl

acem

0.4

ispl

acem

0

sion

al d

i

0.2si

onal

di

-0.5

n-di

men

s

0

n-di

men

s

0 5 10 15-1N

on

0 0.5 1 1.5 2 2.5 3

-0.2

di i l i /

Non

11—, Runge-Kutta method ; −−, MMS; ∙∙∙∙∙, MKBMM; −∙−, LPM.

Non-dimensional time / 2Non-dimensional time / 2

Comparison of the Results – Freq. Domain γ = 0.2; ζ = 0.05

101

Envelope of the responseFEM;DSM;

100

10

plitu

de

|y|

0

Envelope of the responseTheory Solution.

1

100

nsio

nal a

mp

-50

Deg

)

10-1

Non

-dim

en

-100

se a

ngle

(D

0 1 2 3 4 510-2

Non-dimensional frequency

-150Pha

0 1 2 3 4 5-200

Non-dimensional frequency

—, Runge-Kutta method ;∙∙∙∙∙, MKBMM;

12−∙−, LPM.

Comparison of the Results – Freq. Domain γ = 0.2; ζ = 0.5

Envelope of the responseFEM;DSM;

100

plitu

de

|y|

0 Envelope of the responseTheory Solution.

10-1

nsio

nal a

mp

-50

eg )

Non

-dim

en

-100

se a

ngle

(De

0 1 2 3 4 510-2

Non-dimensional frequency

N

-150Phas

0 1 2 3 4 5-200

Non-dimensional frequency

—, Runge-Kutta method ;∙∙∙∙∙, MKBMM;

LPM

13

−∙−, LPM.

Key Features Envelope of the response

e

1/222

Envelope3 1

1 1

e

e

1 18

2

When 23 1

18

e

2Envelope

1

e

21

14

Key Features Damped natural frequency which changes with time

2

2 23( ) 1 1

8 8 3 3e

When and nonlinearity is very small

8 8 3 3e

0.1 y y

2 22

3( ) 18 1

e

When

28 10.1

23( ) 1 e

15

( )8

Key Featuresγ = 0.2; ζ = 0.05Envelope IF (Instan Freq )Envelope IF (Instan. Freq.)

1

A( )

1.08

y

()

0.6

0.8

enve

lope

A

1 04

1.06

ous

frequ

ncy

0.4

imen

sion

al

1.02

1.04

inst

anta

neo

0 2 4 6 8 10 120

0.2

Non

-di

0 2 4 6 8 10 12

1

Non

-dim

. i 0 2 4 6 8 10 12

Non-dimensional time /20 2 4 6 8 10 12

Non-dimensional time /2

−∙−,2 2

2

3( ) 18 1

e

−∙−, 2

Envelope1

e

16+ , MKBMM○ , Numerical results + HT

+ , MKBMM○ , Numerical results + HT

Concluding RemarksAnalytical method Method of multiple scales (MMS) √ Good Method of multiple scales (MMS) √ Good Modified Krylov-Bogoliubov-Mitropolskiy method

(MKBMM) √ Good(MKBMM) √ Good Lindstedt-Poincaré method (LPM)

• ×cannot capture time dependent damped natural freq.

Two key features of the impulse responsey p p The envelope of the decay of free vibration

• Lightly damped case: approximately exponential decay• Lightly damped case: approximately exponential decay

The time dependent damped natural frequency

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References [1] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations. Wiley, New York,

1995. [2] J J Th Vib ti d St bilit Ad d Th [2] J.J. Thomsen, Vibrations and Stability, Advanced Theory,

Analysis, and Tools, 2nd ed., Springer, Berlin, 2003. [3] S. Martin. The Volterra and Wiener Theories of Nonlinear [ ]

Systems, John Wiley & Sons, New York, 1980. [4] I. Kovacic, M.J. Brennan, The Duffing Equation: Nonlinear

Oscillators and their Behaviour Wiley Chichester 2011Oscillators and their Behaviour, Wiley, Chichester, 2011.

[5] K.S. Mendelson, Perturbation theory for damped nonlinear [ ] y poscillations. Journal of Mathematical Physics, 11, 3413-3415, 1970.

[6] R.G. White, Effects of non-linearity due to large deflections in the derivation of frequency response data from the impulse response ofderivation of frequency response data from the impulse response of structures. Journal of Sound and Vibration, 29, 295-307, 1973.

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References [7] M. Feldman, Non-linear system vibration analysis using Hilbert

transform-I. Free vibration analysis method ‘FREEVIB’. Mechanical Systems and Signal Processing 8 119 127 1994Systems and Signal Processing, 8, 119-127, 1994.

[8] Bin Tang, M.J. Brennan, On the impulse response of the Duffing oscillator, International Conference on Vibration and Vibro-acoustics (ICVV2014), January 13-15, 2014, Harbin, China.

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Thank You for Your Attention!!!Thank You for Your Attention!!!Any Questions are welcome!y

谢谢 (Xièxiè)！( )

Bin Tang

20Institute of Internal Combustion Engine, Dalian University of Technology, China.