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