[PPT] - On the Interaction of an Electro-dynamic On the Interaction of an PowerPoint Presentation

SLIDE 1

On the Interaction of an Electro-dynamic On the Interaction of an Electro-dynamic Shaker and a Beam with Stiffness Nonlinearity

B. Tang1, M.J. Brennan2, G. Gatti3

1Institute of Internal Combustion Engine, Dalian University of Technology, China 2Departamento de Engenharia Mecânica, UNESP, Ilha Solteira, Brazil 3Department of Mechanical, Energy and Management Engineering, University of Calabria, Italy

SLIDE 2

Dalian Dalian

Dalian is a small city in China……, Only 6 million people live in the city  Only 6 million people live in the city 

SLIDE 3

Dalian Dalian

Dalian University of Technology (DUT) Dalian University of Technology (DUT)

SLIDE 4

Dalian

2010, ISVR, Univ Soton, UK 2013-14, UNESP, Brasil

kv y O x

m

cv kh, δ kh, δ l

SLIDE 5

Motivation

40 30 35 40

de softening

20 25 30

t Amplitud linear hardening

Nonlinear Isolators

10 15 20

placement

5 10

Disp

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Non-dimensional frequency

Energy Harvesting Devices

Vibration Jumps

SLIDE 6

Objective

To identify nonlinear stiffness characteristics of a

Objective

To identify nonlinear stiffness characteristics of a

structure experimentally (b)

f(t)

(b) Input f (t) Output x(t)

m O

Input Output

c k1, k3 x(t) O

Grey Box – we know the type of nonlinear stiffness expected

SLIDE 7

Example Structure 1 Example Structure 1 A Vibration Absorber

SLIDE 8

A Vibration Absorber

m= 7 5 g

Deformed brass plate

m 7.5 g

Brass plate (0.15 mm)

Support structure (52 mm diameter) Support structure (52 mm diameter)

SLIDE 9

Vibration Test

x m x1 x2 c k1, k3

Shaker mass

mshaker cshaker kshaker

Shaker suspension Bin Tang, M.J. Brennan, G. Gatti, N.S. Ferguson. Experimental characterization of a nonlinear vibration absorber using free vibration. Journal of Sound and Vibration 367 (2016) 159-169.

SLIDE 10

Vibration Test

shaker absorber

  

shaker EQ

m m m m m   x

Q shaker

m m  mEQ c k1, k3

1, 3

SLIDE 11

Experimental result

x 10

4

Time range over which analysis is conducted

2 3

0 s 0.3 s

1 nt (m)

nt (m)

placemen

148 Hz

placemen

2
1

Disp Low frequency

Disp

0.2 0.4 0.6 0.8

3

High frequency

Ti ( )

t (s)

Steady state

Free vibration

CS

Time (s)

SLIDE 12

System Characteristics

 

( ) ( ) cos ( ) x t A t t  

Create analytic signal Create analytic signal

Hilbert transform

( ) ( ) H[ ( )] w t x t j x t  

) ( ( ) A t w t 

 

angle ) ( ( ) t t w  

B kb

( ) d t 

Envelope

) ( ( )

 

a g e ) ( ( ) t t w 

Backbone curve

( )

d t

dt   

Envelope

SLIDE 13

System Characteristics

2 5 x 10

4

3 x 10

4

Backbone curve Envelope

2 2.5 m) 3

m) )

1 1.5 acement (m 2 velope (m)

cement ( elope (m)

0.5 1 Displa 1 Env

Displac Enve

80 90 100 110 120 130 140

f (Hz) 0.05 0.1 0.15 0.2 0.25 0.3 t (s)

Time (s) Frequency (Hz) Time (s) Frequency (Hz)

SLIDE 14

Backbone Curve

   

2 2 2 3

3 4

dd d

t t k m A    

2

1

dd n

    

1 EQ n

k m  

EQ

4 m

1 EQ n

k m 

 

EQ 1

2 c m k   Backbone curve

7.5 x 10

5

Envelope

8

6 7

9
8.5
g(m))

pe)

Slope

5 6

 2 (rad/s)2

2 d



9.5

9 nvelope) (lo

g(Envelo

4



10.5
10

log(En

Log

1 2 3 4 5 6 x 10

8

3 Displacement2 (m2) 6.5

Displacement2 (m)2

0.05 0.1 0.15 0.2 0.25 0.3

11

t (s)

Time (s)

SLIDE 15

Estimated Parameters

D i Linear Nonlinear Mass (g) Damping Ratio Linear stiffness (N/m) Nonlinear stiffness (N/m3) Shaker and support 351 0.04 17000

support

Vibration 7 55 0 02 Current 2380 5 93×1010 absorber 7.55 0.02 Method 2380 5.93×1010 RSFM 2730 5.17×1010

SLIDE 16

Estimated Parameters

3

cx k x k x m x     

Restoring Force Surface Method (RFSM) Set

1 3 EQ

cx k x k x m x    

EQ

( , , ) x x m x  

Extract a section of the surface between
Plot 3D surface
Set

2

0.1 m/s 0.1 m/s x    

Extract a section of the surface between

1 e (N)

rce (N)

ring force
ring For
1

Resto

3
2
1

1 2 3 x 10

4
2

Displacement (m)

SLIDE 17

Estimated Parameters

3

cx k x k x m x     

Restoring Force Surface Method (RFSM) Set

 

( ) ( ) cos ( ) x t A t t  

1 3 EQ

cx k x k x m x    

EQ

( , , ) x x m x  

Extract a section of the surface between
Plot 3D surface
Set

3 x 10

4

2

0.1 m/s 0.1 m/s x    

Extract a section of the surface between

1 2 (m)

t (m)

1 e (N)

rce (N)

placement

acement

ring force
ring For
2
1

Disp

Displa

1

Resto

0.05 0.1 0.15 0.2 0.25 0.3

3

t (s)

3
2
1

1 2 3 x 10

4
2

Displacement (m)

Displacement (m) Time (s)

SLIDE 18

Example Structure 2 Example Structure 2 A Beam with a Compressive Load Compressive Load

Bin Tang, M.J. Brennan, V. Lopes Jr., S. da Silva, R. Ramlan. Using nonlinear jumps to estimate cubic stiffness nonlinearity: An experimental study Proceedings of the to estimate cubic stiffness nonlinearity: An experimental study, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 230(19) (2016) 3575-3581.

SLIDE 19

Beam Experiment

Axial force

P

Beam

First Mode

3 1 3

cos mq cq k q k q F t       

Accelerometer

2 4 1 2 3

= 1 2 l EI k EI l P         

Shaker

 

4EA

k  

Shaker

Al 

3 3

8 k l  2 Al m  

SLIDE 20

Frequency Sweep (0.05 Hz/s)

leration

50 100 150

Acce

50 100 150

Time (s)

SLIDE 21

Frequency Sweep (0.05 Hz/s)

Increasing excitation amplitude

leration

50 100 150

Acce

50 100 150 50 100 1500 50 100 150 180 50 100 150

Time (s)

50 100 150 0 50 100 1500 50 100 150 180

Increasing excitation amplitude

n

Increasing excitation amplitude

Accelerati

50 100 150 200 50 100 150 200 230 0 50 100 150 200 230 50 100 150 200 230

Time (s)

SLIDE 22

Frequency Sweep (0.05 Hz/s)

 

2 2 3 2 2

3 1 2 4

dd d n

Y k m      

Jump frequency Amplitude at jump frequency

4 m

7 x 10

4

2nd Mode

6 5 7

2nd Mode Slope

6 6.5 ad/s)2

2 dd

 p

5 5 6

 d

2 (ra

2

(rad/s )

5.5

Shaker

0.2 0.4 0.6 0.8 1 x 10

5

5

Y d

2 (mm)

SLIDE 23

Comparison with RFSM

30 20

e (N) 7 43 kN/m k

10

g force

1

7.43 kN/m k 

10

storing

3 3

4.78e8 N/m k 

20

Res

3
2
1

1 2 3

30

( ) Displacement (mm)

SLIDE 24

Frequency Sweep (0.05 Hz/s)

isp.

Increasing excitation amplitude

4 6 4 6 4 5

en Point Di

2 . 2 4 2 2 4 2

Drive i ( )

50 100 150 6 4 2 50 100 150 180 6 4 2 50 100 150 5 4 2 50 100 150

Time (s)

Why are there peaks and bulge?

Disp.

1 2 1 2 2.5

dle Point D

1
2
1

50 100 150 0 50 100 150

Mid Time (s)

Increasing excitation amplitude

50 100 150

2

50 100 150 180

2.5

SLIDE 25

Beam Model

Axial force (a) Beam Accelerometer Shaker Shaker

SLIDE 26

Measured and estimated FRF

101 100

1

10 10-1 10-2

X Y X Y

n

X  

20 30 40 50 60 70 80 90 10-3

11 21 11 sh 11 sh

, 1 1

A B

X Y X Y F Y Z F Y Z    

11 2 2 1

2

n A Ar Ar r b r r r

X Y F j       



   



2 sh sh sh sh

Z k m j c     

21 2 2 1

2

n B Ar Br r b r r r

X Y F j       



   



SLIDE 27

Measured displacement transmissibility

4 5 5 4 4.5 3.5

T

2.5 3 32 34 36 38 40 42 44 2

f (H )

B

X T 

f (Hz)

Beam model shape changing?

A

T X

Beam model shape changing?

SLIDE 28

Measured relative displacement

x 10

3

2 x 10 1.5 mm) 1

XA) (m

0.5 ( XB 32 34 36 38 40 42 44

Relative displacements follow the similar trend when

32 34 36 38 40 42 44

f (Hz)

Relative displacements follow the similar trend when the nonlinearity in the beam is severity

SLIDE 29

Dynamic stiffness of the whole system

Linear case:

cs 11 sh

Z Z Z  

1 F

11 11

1

b A

F Z Y X  

11 A

Nonlinear case:

 

2 2 2

3 3 + Z k k X k X X m jc       

 

cs cs sh3 3 cs cs

+ 4 4

A b B A

Z k k X k X X m jc    

SLIDE 30

Nelder-Mead simplex algorithm

 

2 2 2 cs cs sh3 3 cs cs

3 3 + 4 4

A b B A

Z k k X k X X m jc       

cs

1 = , =

A B A

X X X T F Z F F

2 x 10-3

x 10-3

10-3

2.5 x 10-3

1.5

2 2.5 x 10

2 2.5 x 10 3

2 )

1 / F | (m/N)

1

1 1.5

F | (m/N)

1.5

1 1.5 / F | (m/N)

0.5 |X /

0.5

0.5 1 |X /

0.5 1

0.5 1 |X /

32 34 36 38 40 42

f (Hz)

32 34 36 38 40 42

f (Hz)

32 34 36 38 40 42

f (Hz) f (Hz)

SLIDE 31

Dynamic stiffness of the whole system

2

3 3

N li

 

2 2 2 cs cs sh3 3 cs cs

3 3 + 4 4

A b B A

Z k k X k X X m jc       

Nonlinear case:

1 x 10

4

1 x 10

4

0.5 s 0.5

0.5

g of Zcs

0.5

32 34 36 38 40 42

1

1 x 10

4

1 x 10

4

d Imag

1

0.5 0.5

eal and

0.5

1

0.5

Re

32 34 36 38 40 42

1

32 34 36 38 40 42

1

Freq (Hz)

SLIDE 32

Concluding Remarks

Nonlinearity needs to be known if a system is to be

ti i d d ff t id d

g

ptimised - or adverse affects avoided.
If the type of nonlinearity is known a-priori then

If the type of nonlinearity is known a-priori then tests can be targeted to specifically excite this type

f nonlinearity.

y

Free and forced vibration can be used to determine

h d i ft i li it a hardening or softening nonlinearity.

Difficult to determine nonlinear stiffness accurately

Difficult to determine nonlinear stiffness accurately.

Shaker-structure interaction problem.

32

SLIDE 33

Thank You for Your Attention!!! Thank You for Your Attention!!! Thank You for Your Attention!!! Thank You for Your Attention!!! Any Questions are welcome! Any Questions are welcome! Any Questions are welcome! Any Questions are welcome! 谢谢 (Xiexie)！谢谢 (Xiexie)！