Vibrating TheoryVibrating Theoryin Composite Structuresin Composite Structures

DERF November 2008

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

Vibrating Theory, Jelena MN

Vibration machine for curing laminates

Department of Engineering, ANU

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

Vibrating Theory, Jelena MN

Composite Structures• Fibre-Resin-Voids

100μm

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

Vibrating Theory, Jelena MN

Vibrations of Viscoelastic Model• Amplitude of vibrations

• E ≈ A2

tAA sin0

A

Resonance Phenomenon

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

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

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

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

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

Vibrating Theory, Jelena MN

Conclusions

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

• …