Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008...

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Vibrating Theory Vibrating Theory in Composite Structures in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston
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Page 1: Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston.

Vibrating TheoryVibrating Theoryin Composite Structuresin Composite Structures

DERF November 2008

Jelena Muric-NesicSupervisors: Z.Stachurski, P.Compston

Page 2: Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston.

Vibrating Theory, Jelena MN

Vibration machine for curing laminates

Department of Engineering, ANU

Page 3: Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston.

Vibrating Theory, Jelena MN

Previous Experimental Results

void content for 30 min of vibrationsno cover top cover +vacuum

0

1

2

3

4

5

6

7

8

9

10 30 50 NO 10 30 50 NO 10v 30v 50v NOvfrequency, Hz

Page 4: Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston.

Vibrating Theory, Jelena MN

Composite Structures• Fibre-Resin-Voids

100μm

Page 5: Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston.

Vibrating Theory, Jelena MN

Fibre-resin-bubble system

• Pressure

• Viscosity

• Buoyancy• Bjerknes Forces

• Diffusion

)exp(0 RT

Q

gr

VB

2

9

2

rPPghP vibrag

2

r

P2

r

P2

00

)(

c

c

p

tRp

00 /)( ccptRp

00 /)( ccptRp

Expansion Shrinkage

)()( tptVFB

Bubble shrinks

Bubble grows

Page 6: Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston.

Vibrating Theory, Jelena MN

Vibrations of Viscoelastic Model• Amplitude of vibrations

• E ≈ A2

tAA sin0

A

Resonance Phenomenon

Page 7: Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston.

Vibrating Theory, Jelena MN

Analysis of a bubble in viscous fluid

Bubble subjected to (i) hydrostatic and (ii) oscillating pressures

Fluid velocity potential field (Helmholtz equation):

k - wave vector; v- bubble radius oscillation velocity;u - bubble centre oscillation velocity; r, Θ - spherical coordinates

Assume bubble diameter << distance between bubbles, and

kR << 1, bubble diameter small compared to wavelengthThe solution is:

where the coefficients, ani, are found from boundary conditions, and Pn(cosΘ) are the Legendre polynomials

)exp( kritiAPPPP hah

02 k iiii ur cos/

)(cos)(...)(cos)()( 6661100 jjiiiiiiii PkrhaPkrhakrha

Page 8: Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston.

Vibrating Theory, Jelena MN

Analysis of a bubble in viscous fluid

Interaction forces are obtained from:

Resonance amplitude

Lagrangian:

T - kinetic fluid energy, U - potential fluid energy

Ub - bubble potential energy, c - acoustic wave velocity

The solution for radius oscillations is:

ωb - resonant angular frequency of a bubble,

ω - applied angular frequency

δ - absorption coefficient

lLF /

'2

'

2 02

22 dPPd

c

PdUUTL b

22220 )1/(

bR

PR

2220

222 )/1(4 bfffm

FA

Page 9: Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston.

Vibrating Theory, Jelena MN

Analysis of a bubble in viscous fluid

Doinikov solution for angular resonance frequency of the gas bubble is:

c - sound wave velocity inside the bubbleρ - density inside the bubbleσ - surface tension of the liquid

Dissipated work is transformed into heat, for every cycle the temperature will rise, and viscosity changes:

2/1

00

20

2 2~~31

R

c

Rb

tc

WT

p

d

i = 1

i = 2

RA.A. Doinikov “Acoustic radiation pressure on a compressible sphere in a viscous fluid” J. Fluid Mech (1994)

)exp(0 RT

Q

Page 10: Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston.

Vibrating Theory, Jelena MN

Resonant frequencySimply supported plate with a=10cm and h=3mm• ANUQSM ~ 41Hz• AL mould ~ 730Hz• Glass fibres ~ 10,600Hz• Epoxy resin ~ 2,760Hz• Uncured laminate ~150Hz• Cured laminate ~ 12,300Hz• Bubble (100μm radius) ~ 2,000Hz

h

D

a

2

22

)1(12 2

3

EhD

Page 11: Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston.

Vibrating Theory, Jelena MN

Resonant frequency

water

ANUQSM 41Hz

Al mold 730Hz

Laminate: Cured 12,300Hz Uncured 153Hz

Square plate 10x10x0.3cm Resin 2,800Hz Glass fibres 10,600Hz Glass fibre 6,400Hz

Uncured laminate 10x10x0.3cm 153Hz 20x20x0.3cm 38Hz 30x30x0.3cm 17Hz 50x50x0.3cm 6Hz0

0

00

43

2

1

R

p

Rf

m

kf

2

10

Bubbles 2,000Hz

Page 12: Vibrating Theory in Composite Structures Vibrating Theory in Composite Structures DERF November 2008 Jelena Muric-Nesic Supervisors: Z.Stachurski, P.Compston.

Vibrating Theory, Jelena MN

Conclusions

• Theory of vibrations and resonance in liquid-solid-gas systems still under development

• …