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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
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
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
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
11
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
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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
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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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