Determine the Parameters of a Systemwith Cubic Stiffness Nonlinearity

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

1

Rubber Bubble Isolator

V l tiffStatic equilibrium position

Very low stiffness (natural frequency) 2

Wire Rope Isolator

3

Geometrically Nonlinear Stiffness

ef

m

eftx

k cvk c

ex

tf

e

tf

4

Geometrically Nonlinear Stiffness

efhk hk

m

eftx

k c

l

vk c

ex

tf

e

tf

High Static Low Dynamic Stiffness (HSLDS) isolatorMany engineering applicationsLow frequency isolation 5

Objective

One estimated method for the cubic stiffness  

6

Duffing Oscillator

3 ( )k k F t 31 3 cos( )mx cx k x k x F t

Non-dimensional Equation3 ˆ2 F 32 cosu u u u F

23 0

0 , , , ,2

k xmg x cx yk k

1 0 1

2 1

2

ˆ

nk x k mk Ft F

1 , , , n n

nt F

m mg

7

Frequency Response Function

22 2 2 2 23 31 2 4 1 +

ˆY Y F

2 2 2 2 21,2 21 2 4 1 +

4 4Y Y

Y

Backbone curve 2 2 231 24d dY 4

413 1

32 2 33 ˆ12u F

Jump-up freq.

1

2 22

ˆ1 31 1 F

Jump-down freq21 1

2 4d

Jump-down freq.8

Frequency Response Curve

40Jump-down freq.

30

40

30

Backbone curve 20Y

10

Jump-up freq.0.5 1 1.50

9

Frequency Response Curve

40Jump-down freq.

30

40

30

Backbone curve 20Y

10

Jump-up freq.0.5 1 1.50

10

Estimated Method

4

Jump-up freq.4

41

32 2 33 ˆ12u F

432

2

1 2 1ˆ 3 uF

2

Jump-down freq.1

2 22

2

ˆ1 31 1dF

2 222

4 2 1 1ˆ3 dF

Backbone curve

22 4d

23 dF

Backbone curve2 2 231 2

4d dY 2 22

4 1 23 dY

4d d 23 dY

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Estimated Error

4

Jump-up freq.90

1005%

4

132 2 33 ˆ1

2u F 70

8090

4%

3%2

4

321 2 506070

erro

r

2%

3%

322

1 2 1ˆ 3 uF

304050

% e

1%

1020

1 1.05 1.1 1.15 1.2 1.25 1.30

uu

12

Estimated Error

1

Jump-down freq.90

1001

2 22

2

ˆ1 31 12 4d

F

70

8090

2 4

2 224 50

6070

erro

r

4%3%

5%

222

4 2 1 1ˆ3 dF

304050

% e

1%2%

3%

1020

1%

1 1.05 1.1 1.15 1.2 1.25 1.30

dd

13

Estimated ErrorBackbone curve

90100

2 2 231 24d dY

708090

4

2 24 506070

erro

r

4%5%

2 22

4 1 23 d

dY

3040%

e 4%

2%3%

1020 1%

1 1.05 1.1 1.15 1.2 1.25 1.30

dd

14

Estimated ErrorBackbone curve

2 2 231 24d dY 4

y a bx y a bx

Line least-squares fitLine least squares fit

15

Test‐rig

16

Stepped Sine200

m/s

2 ) 150ra

tion

(m

100

Acc

eler

5050

20 25 30 35 40 45 50

f (Hz)17

Concluding Remarks The nonlinear stiffness can be estimated from a singlemeasurement, the error from this measurement can potentiallybe large.Exciting the system over a range of amplitudes and using linearl f b h d h lleast‐squares fit is a better method to estimate the nonlinearstiffness of Duffing‐like system.Sl f i b tt th t d i d thSlow frequency sweep is better than stepped sine and thesystem jumps‐down at a frequency closer to the actual jump‐down frequency during slow frequency sweep.down frequency during slow frequency sweep.

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References[1] I. Kovacic, M.J. Brennan, The Duffing Equation: Nonlinear Oscillators and their Behaviour, Wiley, Chichester, 2011.[2] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, New York, 1995.[3] K. Worden, G.R. Tomlinson, Nonlinearity in Structural Dynamics: Detection, Identification and Modelling, Institute of Ph i P bli hi B i t l d Phil d l hi 2001Physics Publishing, Bristol and Philadelphia, 2001.

[4] G K h K W d A F V k ki J C G li l P t[4] G. Kerschen, K. Worden, A.F. Vakakis, J.C. Golinval, Past, present and future of nonlinear system identification in structural dynamics Mechanical Systems and Signal Processing 20(3) (2006)dynamics, Mechanical Systems and Signal Processing 20(3) (2006) 505–592.

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References[5] M.J. Brennan, I. Kovacic, A. Carrella and T.P. Waters, On the jump‐up and jump‐down frequencies of the Duffing oscillator, Journal of Sound and Vibration 318(4–5) (2008) 1250–1261.[6] Y. Benhafsi, J.E.T. Penny, M.I. Friswell, A parameter d f h d f d lidentification method for discrete nonlinear systems incorporating cubic stiffness elements, The International Journal of Analytical and Experimental Modal Analysis 7(3) (1992) 179–of Analytical and Experimental Modal Analysis 7(3) (1992) 179–195. [7] Bin Tang, M.J. Brennan, V. Lopes Jr., S. da Silva, R. Ramlan, An[7] Bin Tang, M.J. Brennan, V. Lopes Jr., S. da Silva, R. Ramlan, An Experimental Study to Determine the Parameters of a System with Cubic Stiffness Nonlinearity, Submitted to Journal of Sound and Vibration 2014.

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

谢谢 (Xièxiè)！( )

Bin Tang

21Institute of Internal Combustion Engine, Dalian University of Technology, China.