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

2. Vibrating Theory, Jelena MN Vibration machine for curing laminates Department of Engineering, ANU

3. Vibrating Theory, Jelena MN Previous Experimental Results

4. Vibrating Theory, Jelena MN Composite Structures Fibre-Resin-Voids 100μm

5. Vibrating Theory, Jelena MN Fibre-resin-bubble system Pressure Viscosity Buoyancy Bjerknes Forces Diffusion Expansion Shrinkage Bubble shrinks Bubble grows

6. Vibrating Theory, Jelena MN Vibrations of Viscoelastic Model Amplitude of vibrations E ≈ A 2 A Resonance Phenomenon

7. 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 wavelength The solution is: where the coefficients, a ni, are found from boundary conditions, and P n (cosΘ) are the Legendre polynomials

8. 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 U b - bubble potential energy, c - acoustic wave velocity The solution for radius oscillations is: ω b - resonant angular frequency of a bubble, ω - applied angular frequency δ - absorption coefficient

9. 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 bubble ρ - density inside the bubble σ - surface tension of the liquid Dissipated work is transformed into heat, for every cycle the temperature will rise, and viscosity changes: i = 1 i = 2 R A.A. Doinikov “Acoustic radiation pressure on a compressible sphere in a viscous fluid” J. Fluid Mech (1994)

10. Vibrating Theory, Jelena MN Resonant frequency Simply 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

11. 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 6Hz  Bubbles 2,000Hz

12. Vibrating Theory, Jelena MN Conclusions Theory of vibrations and resonance in liquid-solid-gas systems still under development …