Viscosity measurement in thin lubricant films using shear ultrasonic reflection

When a shear ultrasonic wave is incident on a solid and liquid boundary, the proportion that is reflected depends on the liquid viscosity. This is the basis for some instruments for on-line measurement of bulk liquid viscosity. In machine elements, the lubricant is usually present in a thin layer between two rubbing solid surfaces. The thin film has a different response to an ultrasonic shear wave than liquid in bulk. In this work, this response is investigated with the aim of measuring viscosity in situ in a lubricating film. The proportion of the wave reflected at a thin layer depends on the layer stiffness. A shear wave is reflected by the shear stiffness of the thin layer. For a thin viscous liquid layer, the stiffness is a complex quantity dependent on the viscosity, wave frequency, and film thickness. This stiffness is incorporated into a quasi-static spring model of ultrasonic reflection. In this way, the viscosity can be determined from shear-wave reflection if the oil-film thickness is known. The approach has been experimentally evaluated on some static oil film between Perspex plates. Predictions of the spring model gave good measurement up to layer thicknesses of around 15 μm. For thicker layers, the shear stiffness reduces to such an extent that almost all the wave is reflected and the difference associated with the layer response is hard to distinguish from background noise

A new fabrication method for precision antenna reflectors for space flight and ground test

Communications satellites are using increasingly higher frequencies that require increasingly precise antenna reflectors for use in space. Traditional industry fabrication methods for space antenna reflectors employ successive modeling techniques using high- and low-temperature molds for reflector face sheets and then a final fit-up of the completed honeycomb sandwich panel antenna reflector to a master pattern. However, as new missions are planned at much higher frequencies, greater accuracies will be necessary than are achievable using these present methods. A new approach for the fabrication of ground-test solid-surface antenna reflectors is to build a rigid support structure with an easy-to-machine surface. This surface is subsequently machined to the desired reflector contour and coated with a radio-frequency-reflective surface. This method was used to fabricate a 2.7-m-diameter ground-test antenna reflector to an accuracy of better than 0.013 mm (0.0005 in.) rms. A similar reflector for use on spacecraft would be constructed in a similar manner but with space-qualified materials. The design, analysis, and fabrication of the 2.7-m-diameter precision antenna reflector for antenna ground tests and the extension of this technology to precision, space-based antenna reflectors are described

Spanning tree generating functions and Mahler measures

We define the notion of a spanning tree generating function (STGF) $\sum a_n z^n$, which gives the spanning tree constant when evaluated at $z=1,$ and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hypergeometric functions for all regular two and three dimensional lattices (and one higher-dimensional lattice). This gives closed form expressions for the spanning tree constants for all such lattices, which were previously largely unknown in all but one three-dimensional case. We show for all lattices that these can also be represented as Dirichlet $L$-series. Making the connection between spanning tree generating functions and lattice Green functions produces integral identities and hypergeometric connections, some of which appear to be new.Comment: 26 pages. Dedicated to F Y Wu on the occasion of his 80th birthday. This version has additional references, additional calculations, and minor correction

The phase shift of an ultrasonic pulse at an oil layer and determination of film thickness

An ultrasonic pulse incident on a lubricating oil film in a machine element will be partially reflected and partially transmitted. The proportion of the wave amplitude reflected, termed the reflection coefficient, depends on the film thickness and the acoustic properties of the oil. When the appropriate ultrasonic frequency is used, the magnitude of the reflection coefficient can be used to determine the oil film thickness. However, the reflected wave has both a real component and an imaginary component, and both the amplitude and the phase are functions of the film thickness. The phase of the reflected wave will be shifted from that of the incident wave when it is reflected. In the present study, this phase shift is explored as the film changes and is evaluated as an alternative means to measure oil film thickness. A quas i-static theoretical model of the reflection response from an oil film has been, developed. This model relates the phase shift to the wave frequency and the film properties. Measurements of reflection coefficient from a static model oil film and also from a rotating journal bearing have been recorded. These have been used to determine the oil film thickness using both amplitude and phase shift methods. In both cases, the results agree closely with independent assessments of the oil film thickness. The model of ultrasonic reflection is further extended to incorporate mass and damping terms. Experiments show that both the mass and the internal damping of the oil films tested in this work have a negligible effect on ultrasonic reflection. A potentially v ery useful application for the simultaneous measurement of reflection coefficient amplitude and phase is that the data can be used to negate the need for a reference. The theoretical relationship between phase and amplitude is fitted to the data. An extrapolation is performed to determine the values of amplitude and phase for an infinitely thick layer. This is equivalent to the reference signal determined by measuring the reflection coefficient directly, but importantly does not require the materials to be separated. This provides a simple and effective means of continuously calibrating the film measurement approach

Fluctuations of a long, semiflexible polymer in a narrow channel

We consider an inextensible, semiflexible polymer or worm-like chain, with persistence length $P$ and contour length $L$, fluctuating in a cylindrical channel of diameter $D$. In the regime $D\ll P\ll L$, corresponding to a long, tightly confined polymer, the average length of the channel $$ occupied by the polymer and the mean square deviation from the average vary as $=[1-\alpha_\circ(D/P)^{2/3}]L$ and $<\Delta R_\parallel^{\thinspace 2}\thinspace>=\beta_\circ(D^2/P)L$, respectively, where $\alpha_\circ$ and $\beta_\circ$ are dimensionless amplitudes. In earlier work we determined $\alpha_\circ$ and the analogous amplitude $\alpha_\Box$ for a channel with a rectangular cross section from simulations of very long chains. In this paper we estimate $\beta_\circ$ and $\beta_\Box$ from the simulations. The estimates are compared with exact analytical results for a semiflexible polymer confined in the transverse direction by a parabolic potential instead of a channel and with a recent experiment. For the parabolic confining potential we also obtain a simple analytic result for the distribution of $R_\parallel$ or radial distribution function, which is asymptotically exact for large $L$ and has the skewed shape seen experimentally.Comment: 21 pages, including 4 figure

Supersymmetric Extension of the Quantum Spherical Model

In this work, we present a supersymmetric extension of the quantum spherical model, both in components and also in the superspace formalisms. We find the solution for short/long range interactions through the imaginary time formalism path integral approach. The existence of critical points (classical and quantum) is analyzed and the corresponding critical dimensions are determined.Comment: 21 pages, fixed notation to match published versio

Possible origin of 60-K plateau in the YBa2Cu3O(6+y) phase diagram

We study a model of YBa2Cu3O(6+y) to investigate the influence of oxygen ordering and doping imbalance on the critical temperature Tc(y) and to elucidate a possible origin of well-known feature of YBCO phase diagram: the 60-K plateau. Focusing on "phase only" description of the high-temperature superconducting system in terms of collective variables we utilize a three-dimensional semi microscopic XY model with two-component vectors that involve phase variables and adjustable parameters representing microscopic phase stiffnesses. The model captures characteristic energy scales present in YBCO and allows for strong anisotropy within basal planes to simulate oxygen ordering. Applying spherical closure relation we have solved the phase XY model with the help of transfer matrix method and calculated Tc for chosen system parameters. Furthermore, we investigate the influence of oxygen ordering and doping imbalance on the shape of YBCO phase diagram. We find it unlikely that oxygen ordering alone can be responsible for the existence of 60-K plateau. Relying on experimental data unveiling that oxygen doping of YBCO may introduce significant charge imbalance between CuO2 planes and other sites, we show that simultaneously the former are underdoped, while the latter -- strongly overdoped almost in the whole region of oxygen doping in which YBCO is superconducting. As a result, while oxygen content is increased, this provides two counter acting factors, which possibly lead to rise of 60K plateau. Additionally, our result can provide an important contribution to understanding of experimental data supporting existence of multicomponent superconductivity in YBCO.Comment: 9 pages, 8 figures, submitted to PRB, see http://prb.aps.or

Statistics of nested spiral self-avoiding loops: exact results on the square and triangular lattices

The statistics of nested spiral self-avoiding loops, which is closely related to the partition of integers into decreasing parts, is studied on the square and triangular lattices.Comment: Old paper, for archiving. 7 pages, 2 figures, epsf, IOP macr

The measurement of lubricant-film thickness using ultrasound

Ultrasound is reflected from a liquid layer between two solid bodies. This reflection depends on the ultrasonic frequency, the acoustic properties of the liquid and solid, and the layer thickness. If the wavelength is much greater than the liquid-layer thickness, then the response is governed by the stiffness of the layer. If the wavelength and layer thickness are similar, then the interaction of ultrasound with the layer is controlled by its resonant behaviour. This stiffness governed response and resonant response can be used to determine the thickness of the liquid layer, if the other parameters are known. In this paper, ultrasound has been developed as a method to determine the thickness of lubricating films in bearing systems. An ultrasonic transducer is positioned on the outside of a bearing shell such that the wave is focused on the lubricant-film layer. The transducer is used to both emit and receive wide-band ultrasonic pulses. For a particular lubricant film, the reflected pulse is processed to give a reflection-coefficient spectrum. The lubricant-film thickness is then obtained from either the layer stiffness or the resonant frequency. The method has been validated using fluid wedges at ambient pressure between flat and curved surfaces. Experiments on the elastohydrodynamic film formed between a sliding ball and a flat surface were performed. Film-thickness values in the range 50-500 nm were recorded, which agreed well with theoretical film-formation predictions. Similar measurements have been made on the oil film between the balls and outer raceway of a deep-groove ball bearing

Lattice Green functions in all dimensions

We give a systematic treatment of lattice Green functions (LGF) on the $d$-dimensional diamond, simple cubic, body-centred cubic and face-centred cubic lattices for arbitrary dimensionality $d \ge 2$ for the first three lattices, and for $2 \le d \le 5$ for the hyper-fcc lattice. We show that there is a close connection between the LGF of the $d$-dimensional hypercubic lattice and that of the $(d-1)$-dimensional diamond lattice. We give constant-term formulations of LGFs for all lattices and dimensions. Through a still under-developed connection with Mahler measures, we point out an unexpected connection between the coefficients of the s.c., b.c.c. and diamond LGFs and some Ramanujan-type formulae for $1/\pi.$Comment: 30 page