Stephen Warr

Tracking the translational and rotational motion of granular particles : use of high-speed photography and image processing

Abstract Digital high-speed photography has been used to investigate the dynamics of a two-dimensional model granular system, consisting of steel or [poly(methyl methacrylate)] discs under vibration between glass walls. Images are analysed by computer using Hough transform algorithms to measure the two-dimensional position and angle of discs within the field of view. Two preliminary test problems indicate position errors of 5% to 30% of the pixel separation, and angular errors of 0.017 to 0.025 rad depending on magnification. The combination of all-digital recording system and automated pattern recognition software allows rapid acquisition of high quality data. Translational and rotational velocities, calculated for the vibrated discs, are used to construct speed and velocity distribution functions; the speed distribution closely approximates to the Maxwell—Boltzmann distribution and the velocity distribution is Maxwellian. Granular temperatures are anisotropic over the range of amplitude and frequency of the vibration studied.

Dynamic holography for optical interconnections. II. Routing holograms with predictable location and intensity of each diffraction order

An analysis of dynamic phase-only holograms, described by fractional notation and recorded onto a pixelated spatial light modulator (SLM) in a reconfigurable optical beam-steering switch, is presented. The phase quantization and arrangement of the phase states and the SLM pixelation and dead-space effects are decoupled, expressed analytically, and simulated numerically. The phase analysis with a skip-rotate rule reveals the location and intensity of each diffraction order at the digital replay stage. The optical reconstruction of the holograms recorded onto SLMs with rectangular pixel apertures entails sinc-squared scaling, which further reduces the intensity of each diffraction order. With these two factors taken into account, the highest values of the nonuniform first-order diffraction efficiencies are expected to be 33%, 66%, and 77% for two-, four-, and and eight-level one-dimensional holograms with a 90% linear pixel fill factor. The variation of the first-order diffraction efficiency and the relative replay intensities were verified to within 1 dB by performing the optical reconstruction of binary phase-only holograms recorded onto a ferroelectric liquid crystal on a silicon SLM.

Dynamics of granular stratification

dynamical segregation process—known as ‘‘kinematic sieving,’’ ‘‘free surface segregation,’’ or simple ‘‘percolation’’— contributes to the spontaneous stratification of grains in the case of thick flows. We characterize the dynamical process of stratification by measuring all relevant quantities: the velocity of the rolling grains, the velocity of the kink, and the wavelength of the layers. We also measure other phenomenological constants such as the rate of collision between rolling and static grains, and all the angles of repose characterizing the mixture. The wavelength of the layers behaves linearly with the thickness of the layer of rolling grains ~i.e., with the flow rate!, in agreement with theoretical predictions. The velocity profile of the grains in the rolling phase is a linear function of the position of the grains along the moving layer, which implies a linear relation between the mean velocity and the thickness of the rolling phase. We also find that the speed of the upward-moving kink has the same value as the mean speed of the downward-moving grains. We measure the shape and size of the kink, as well as the profiles of the rolling and static phases of grains, and find agreement with recent theoretical predictions. @S1063-651X~98!07809-X#

Probability distribution functions for a single-particle vibrating in one dimension: experimental study and theoretical analysis

We consider the form of the rebound velocity, ν0, particle velocity, ν, and height, h, probability density functions (PDFs) for the one-dimensional motion of a single particle on a sinusoidally oscillating base. The motion is considered in the limit of high excitation (vibration frequency ⪢ collision rate). Experimentally, we find that these PDFs are well-approximated by Pν0(ν0) ∞ ν0 exp(− αν02), a Gaussian Pν(ν) ∞ exp(− αν2) and a Boltzmann-type function Ph(h) ∞ exp(− 2αgh), where α is a constant and g is the acceleration due to gravity. We develop an analytical model which accurately predicts the general form for the rebound velocity PDF; the other two PDFs are then analytically shown to follow as a consequence. Scaling laws for the particle granular temperature with peak base velocity and particle-base restitution coefficient, determined from previous work, can also be predicted from the PDF. A fine scale “spiky” structure in the rebound velocity PDF is found, using numerical simulations, to be a consequence of resonance phenomena between the particle and vibrating base. Good agreement between scaling laws from the theory and simulation is found but insufficient data is obtainable to derive accuracy exponents experimentally.

FLUIDISATION, SEGREGATION AND STRESS PROPAGATION IN GRANULAR MATERIALS

The paper reviews recent experimental investigations into three areas of the dynamic and static behaviour of model granular materials. In the first, digital high speed photography and computer image processing are used to study the fluidisation of a two-dimensional model powder undergoing vertical vibration. The granular temperature is deduced from measured velocity distribution functions, and the scaling dependence on vibration frequency, amplitude and number of grains is compared with simulations and a simple analytical model. The system has also been used to investigate the size-ratio and acceleration dependence of particle size segregation behaviour for a single intruder in a two-dimensional bed of monodisperse particles at low accelerations. Particle trajectory maps show that, at all base accelerations, the intruder and surrounding particles move upwards at the same speed. Unlike recent suggestions in the literature, there appears to be no fundamental difference in segregation mechanism between the so-called intermittent and continuous regimes. The third area concerns the propagation of stress within a three-dimensional conical pile. A simple elasto-optical method has been used to measure the normal force distribution at the pile base. Experiments on sandpiles confirmed the existence of counter-intuitive pressure dips at the centre. Variations in surface roughness were found to have little effect on the pressure profiles.

POLARISATION INSENSITIVE DIFFRACTIVE FLC SYSTEMS

Abstract We show that when used as diffractive optical elements, FLC displays are inherently independent of polarisation and the overall diffractive effect of the device is identical to an array of imperfect {0, π} phase modulators. This realization opens the path for FLC diffraction to be used in a much wider range of optical systems. In this paper we present detailed experimental results of a polarisation independent single-mode fibre switch for reconfigurable distribution of an optical signal to one of an array of output fibres.