When a force deforms an elastic object, practical experience suggests that the resulting displacement will be in the same direction as the force. This property is known as positive stiffness. Less familiar is the concept of negative stiffness, where the deforming force and the resulting displacement are in opposite directions. (Negative stiffness is distinct from negative Poisson's ratio, which refers to the occurrence of lateral expansion upon stretching an object.) Negative stiffness can occur, for example, when the deforming object has stored (or is supplied with) energy. This property is usually unstable, but it has been shown theoretically that inclusions of negative stiffness can be stabilized within a positive-stiffness matrix. Here we describe the experimental realization of this composite approach by embedding negative-stiffness inclusions of ferroelastic vanadium dioxide in a pure tin matrix. The resulting composites exhibit extreme mechanical damping and large anomalies in stiffness, as a consequence of the high local strains that result from the inclusions deforming more than the composite as a whole. Moreover, for certain temperature ranges, the negative-stiffness inclusions are more effective than diamond inclusions for increasing the overall composite stiffness. We expect that such composites could be useful as high damping materials, as stiff structural elements or for actuator-type applications.

Extreme damping in composite materials with negative-stiffness inclusions

https://www.win.tue.nl/~sjoerdr/papers/boek.pdf

An introduction to acoustics

Modelling wave propagation in two-dimensional structures using finite element analysis

The vibrational songs of several species of cydnid bugs and ‘small cicadas’ (leafhoppers and planthoppers) living on various types of plants are recorded by means of laser vibrometry. The recorded vibrational songs are analysed with respect to amplitude, frequency spectrum and structure in the time domain (Figs. 2–5). The emission of vibrational songs from singing insects on plants is simulated. A small magnet is glued to the surface of the plant and moved by means of an electromagnet about one cm away (Fig. 1). The vibrations are recorded by means of laser vibrometry. The propagation velocity of the vibrations increases with the square root of frequency, i.e. in the way expected for bending waves. The mechanical properties of plants ranging from soft bean plants to stiff reeds and maples are measured. The results are used for calculating the theoretical propagation velocities of bending waves. The measured and the calculated values are rather close (Table 1). Although the mechanical properties of the plants studied vary widely, the propagation velocities at a certain frequency are of the same order of magnitude (Table 1). In all the plants studied, only little vibrational energy is lost by friction at frequencies below some kHz. Communication by means of bending waves is possible over distances of some meters. The bending waves are reflected with little loss of energy both from the root and from the top of the plant. The vibration signals may therefore travel up and down the plant several times before decaying completely (Fig. 7). The vibration at a certain spot on the plant depends not only on the distance to and nature of the emitter, but also on the modes of vibration of the plant. The amplitude of vibration does not decrease monotonically with distance from the emitter (Fig. 6). These filtering properties of the plants mean that it is essentially impossible to predict which frequencies in the signals will be amplified or attenuated in the plant at the location of the receiving animal. The vibrational signals recorded from the animals cover wide frequency bandwidths. The signals are therefore well adapted to the filtering properties of the plants, but the signals of the species studied here do not appear to be particularly adapted to specific properties of the host plants. The muscular power needed for communication by means of various types of vibrational signals is calculated. The result of this calculation supports the conclusion that the signals recorded here are carried by means of bending waves. The communication strategies open to small insects are considered. Vibrational signals appear to be an efficient means of communication, but only certain types of signals are suited, because the plants cause a considerable distortion of the signals. One kind of distortion, the dispersive property, may — in theory — be used by the listening animals to obtain information about the direction and distance to the singing animals.

Plants as transmission channels for insect vibrational songs

The reciprocity theorem has been used for over 100 years to establish interesting and useful relations between different loading states of a body. This book discusses current and novel uses of reciprocity relations for the determination of elastodynamic fields. The author, who is internationally distinguished for his contributions to theoretical and applied mechanics, presents a novel method to solve for wave fields, shedding new light on the use of reciprocity relations for dynamic fields in an elastic body. The material presented in the book is relevant to several fields in engineering and applied physics. Examples are ultrasonics for medical imaging and non-destructive evaluation, acoustic microscopy, seismology, exploratory geophysics, structural acoustics, and the response of structures to high-rate loads and the determination of material properties by ultrasonic techniques.

Reciprocity in Elastodynamics

Although sound waves in structures cannot be heard directly, and only be felt at low frequencies, they play an important role in noise control, because many sound signals are generated or transmitted in structures before they are radiated into the surrounding medium. In several respects sound waves in structures and sound waves in gases or liquids are similar, there are, however, also fundamental differences, which are due to the fact that solids have a certain shear stiffness, wheras gases or liquids show practically none. As a consequence acoustic energy can be transported not only by the normal compressional waves but also by shear waves and many combinations of compressional (sometimes loosely called longitudinal) and shear waves . For noise control purposes bending waves (which are a special combination of compressional and shear waves) are of primary importance; for some special cases (quasi-) longitudinal waves and torsional waves also have to be considered.

Thomas D. Northwood

Papers

Structure-borne sound