J. Woodhouse

Wave propagation in two-dimensional periodic lattices

Plane wave propagation in infinite two-dimensional periodic lattices is investigated using Floquet-Bloch principles. Frequency bandgaps and spatial filtering phenomena are examined in four representative planar lattice topologies: hexagonal honeycomb, Kagomé lattice, triangular honeycomb, and the square honeycomb. These topologies exhibit dramatic differences in their long-wavelength deformation properties. Long-wavelength asymptotes to the dispersion curves based on homogenization theory are in good agreement with the numerical results for each of the four lattices. The slenderness ratio of the constituent beams of the lattice (or relative density) has a significant influence on the band structure. The techniques developed in this work can be used to design lattices with a desired band structure. The observed spatial filtering effects due to anisotropy at high frequencies (short wavelengths) of wave propagation are consistent with the lattice symmetries.

On the oscillations of musical instruments

The time‐domain description of musical and other nonlinear oscillators complements the more commonly used frequency‐domain description, and is advantageous for some purposes. It is especially advantageous when studying large‐amplitude oscillations, for which nonlinearity may be severe. It gives direct insight into the physical reasons for the variation of waveform as playing conditions vary, and into certain phenomena which may seem counter‐intuitive from the frequency‐domain viewpoint, such as the musically undesirable flattening in the pitch of a bowed string when the bow is pressed too hard onto the string. It is easy to set up efficient time‐domain simulations on a small computer, a fact that has been surprisingly little exploited in musical acoustics. The simplest relevant model is described here. It demonstrates some of the basic nonlinear behavior of the clarinet, violin, and flute families with very little programming effort. Remarkably, a single set of model equations has relevance to all three c...

Vibration isolation from irregularity in a nearly periodic structure: Theory and measurements

This article describes the theory and a simple experiment carried out to demonstrate the phenomenon of Anderson localization in an acoustical context. This is an effect whereby the propagation of vibration in a structure which is not entirely regular is impeded by the irregularities, giving rise on the average to an exponential decay of vibration level away from the driving point, even in the absence of any dissipation. The structure used in the experiment was a stretched string with masses attached to it. This string was studied with regular spacings of the masses and after the masses had been moved in a controlled way to provide a small degree of irregularity. In both cases, the transmission of energy from end to end of the string was measured as a function of frequency, and also the mode shapes in the second and fourth passbands were measured so as to demonstrate the underlying physics of the localization phenomenon, in which the individual modes making up each passband change from being extended throu...

On measuring the elastic and damping constants of orthotropic sheet materials

Abstract Many common sheet materials, ranging from natural materials such as wood to modern composites, possess approximately orthotropic symmetry. Within the approximations of thin-plate bending theory, the linear vibrational properties of such sheets are governed by four elastic constants and four damping constants (at any given frequency). A simple procedure is presented whereby all four elastic constants may be determined, quickly and with reasonable accuracy, from measurements of the resonant frequencies of low-frequency modes of thin rectangular plates with free edges. Also, at least three of the four damping constants may be determined by measuring the damping factors of the same modes—it turns out that the fourth damping constant does not usually have sufficient influence on the low-frequency modes for a reliable value to be found by this approach. The procedure is illustrated with measurements on a range of different sheet materials: wooden plates cut at different angles from the solid timber, plywood, and two very different fibre-reinforced composites. The discussion of these experimental results suggests that this simple procedure could form a valuable part of any programme of quality control, material selection or non-destructive testing involving orthotropic sheet materials.

Theories of noise and vibration transmission in complex structures

Theories for analysing the vibrational behaviour of complex structures are examined, parallels being drawn with several other areas of physics in which problems of wave propagation in inhomogeneous media are studied. There are three main stages to the investigation. First, the response to random driving of a single, essentially homogeneous, system is examined. The second, and much more detailed, discussion concerns energy transport between discrete coupled subsystems. In particular, the authors investigate an approach to this problem which is known as statistical energy analysis. The third main topic is the phenomenon of Anderson localisation as it applies to certain problems of sound and vibration transmission-the phenomenon is much better known in the field of solid-state physics. Applications of it to vibration are of interest in themselves, and also shed light on the theoretical basis of statistical energy analysis, which is a diffusive transport theory.

An approach to the theoretical background of statistical energy analysis applied to structural vibration

Rayleigh’s classical approach to the study of vibration of systems having a finite number of degrees of freedom is applied to the problem of coupling of subsystems in a complicated structure, in order to probe the regions of applicability of the approach to vibration analysis usually known as statistical energy analysis (SEA). The classical method has advantages of simplicity and rigor over previous approaches to the background of SEA in certain cases, and provides extensions and simplifications in several areas of the theory. It also suggests modifications to SEA modeling strategy depending on the type of coupling involved, even when that coupling is weak, so that earlier analyses might be thought to apply.

The tribology of rosin

Abstract Rosin is well known for its ability to excite stick–slip vibration on a violin string but the precise characteristics of the material which enable it to exhibit this behaviour have not been studied in any detail. A method is described in which the coefficient of friction of rosin is measured during individual cycles of a stick–slip vibration. Friction versus sliding velocity characteristics deduced in this way exhibit hysteresis, similar to that found in other investigations using different materials. No part of the hysteresis loops follow the friction/velocity curve found from steady-sliding experiments. Possible constitutive laws are examined to describe this frictional behaviour. It is suggested by a variety of evidence that contact temperature plays an important role. Friction laws are developed by considering that the friction arises primarily from the shear of a softened or molten layer of rosin, with a temperature-dependent viscosity or shear strength. The temperature of the rosin layer is calculated by modelling the heat flow around the sliding contact. The temperature-based models are shown to reproduce some features of the measurements which are not captured in the traditional model, in which friction depends only on sliding speed. A model based on viscous behaviour of a thin melted layer of rosin gives predictions at variance with observations. However, a model based on plastic yielding at the surface of the rosin gives good agreement with these observations.

The influence of cell geometry on the elasticity of softwood

The cellular microstructure of softwoods such as spruce may be approximated as an irregular two-dimensional honeycomb. The nine macroscopic elastic constants of the wood, regarded as an orthotropic continuum, are governed by the geometric configuration of this honeycomb, together with the intrinsic material properties of the cell walls. Simple modelling is developed to allow all nine of these constants to be estimated from detailed microscopic measurements of the cell geometry, using assumed values for the cell-wall properties. Account is taken both of the cell-to-cell variations in growth and of the larger-scale modulation of cell properties in the annual growth rings. Results based on study of four samples of Norway spruce show very encouraging agreement with published measurements, and allow the relative importance of various effects to be assessed quantitatively.

The low frequency vibration of a ribbed cylinder, part 1: theory

The theory of vibrations of a cylinder braced by circular T-section ribs spaced regularly along its length is presented. Transmission experiments to validate the theory have been carried out on a ribbed cylinder model and these are described in an accompanying paper. Sufficiently detailed theoretical modelling of the ribs has enabled very good agreement between theory and experiment to be obtained.

Enhancing Parametric Sensitivity in Electrically Coupled MEMS Resonators

In an array of identical resonators coupled through weak springs, a small perturbation in the structural properties of one of the resonators strongly impacts coupled oscillations causing the vibration modes to localize. Theoretical studies show that measuring the variation in eigenstates due to such vibration-mode localization can yield orders of magnitude enhancement in signal sensitivity over the technique of simply measuring induced resonant-frequency shifts. In this paper, we propose the application of mode localization for detecting small perturbations in stiffness in pairs of nearly identical weakly coupled microelectromechanical-system resonators and also examine the effect of initial mechanical asymmetry caused by fabrication tolerances in such sensors. For the first time, the variation in eigenstates is studied by coupling the resonators using electrostatic means that allow for significantly weaker coupling-spring constants and the possibility for stronger localization of vibration modes. Eigenstate variations that are nearly three orders of magnitude greater than the corresponding shifts in the resonant frequency for an induced perturbation in stiffness are experimentally demonstrated. Such high electrically tunable parametric sensitivities, together with the added advantage of intrinsic common-mode rejection, pave the way to a new paradigm of mechanical sensing.