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

Vibrating Theory, Jelena MN Previous Experimental Results

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

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

Vibrating Theory, Jelena MN Vibrations of Viscoelastic Model Amplitude of vibrations E ≈ A 2 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 wavelength The solution is: where the coefficients, a ni, are found from boundary conditions, and P n (cosΘ) are the Legendre polynomials

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

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)

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

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

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