B.R. Mace

Finite element prediction of wave motion in structural waveguides

A method is presented by which the wavenumbers for a one-dimensional waveguide can be predicted from a finite element (FE) model. The method involves postprocessing a conventional, but low order, FE model, the mass and stiffness matrices of which are typically found using a conventional FE package. This is in contrast to the most popular previous waveguide/FE approach, sometimes termed the spectral finite element approach, which requires new spectral element matrices to be developed. In the approach described here, a section of the waveguide is modeled using conventional FE software and the dynamic stiffness matrix formed. A periodicity condition is applied, the wavenumbers following from the eigensolution of the resulting transfer matrix. The method is described, estimation of wavenumbers, energy, and group velocity discussed, and numerical examples presented. These concern wave propagation in a beam and a simply supported plate strip, for which analytical solutions exist, and the more complex case of a viscoelastic laminate, which involves postprocessing an ANSYS FE model. The method is seen to yield accurate results for the wavenumbers and group velocities of both propagating and evanescent waves.

Wave reflection and transmission in beams

Abstract The vibrational behaviour of beam systems can be expressed in terms of waves of both propagating and near field types. A propagating wave incident upon a discontinuity gives rise to reflected and transmitted waves of both kinds whose amplitudes may be found from well-known reflection and transmission coefficients. In this paper the approach is extended to the case of incident near field waves, reflection and transmission matrices being derived for the cases of a point support and a change in section. Reflection at a boundary and the effects of applied excitations are also considered. It is seen that incident near fields can give rise to substantial propagating components. The application of the results to the analysis of free and forced vibration of beams is then demonstrated. By adopting this approach the effects of the interaction of the near fields with neighbouring discontinuities are fully included.

Periodically stiffened fluid-loaded plates, I: Response to convected harmonic pressure and free wave propagation

Abstract In this paper several aspects of the vibration of and sound radiation from an infinite fluid-loaded plate stiffened periodically by line supports are studied. The supports may exert both forces and moments on the plate. First the response to a convected harmonic pressure is found by using Fourier transforms and the response is seen to consist of an infinite set of space harmonics whose amplitudes are found explicitly. Certain infinite sums arise and when fluid loading is neglected these are evaluated analytically. A condition is derived for the propagation of free waves. For the fluid loaded case an acoustically damped “propagating” wave is seen to exist. An expression for the response to a general excitation is derived and from this the acoustic pressure in the far field is determined.

Statistical energy analysis, energy distribution models and system modes

Abstract Expressions for the energy influence coefficients of a built-up structure are found in terms of the modes of the whole structure. These coefficients relate the time and frequency average energies of the subsystems to the subsystem input powers. Rain-on-the-roof excitation over a frequency band Ω is assumed. It is then seen that the system can be described by an SEA model only if a particular condition involving the mode shapes of the system is satisfied. Broadly, the condition holds if the mode shapes of the modes in the frequency band of excitation are, on average, typical enough of all the modes of the system in terms of the distribution of energy throughout the system. If this condition is satisfied then the system can be modelled using an “quasi-SEA” approach, irrespective of the level of damping or of the strength of coupling. However, the resulting model need not be of a proper SEA form, and in particular the indirect coupling loss factors may not be negligible.

Periodically stiffened fluid-loaded plates, II: Response to line and point forces

Abstract In this paper the response of infinite periodically stiffened fluid-loaded plates is examined for line and point force excitations. The responses are given as a single and double integral, respectively. When fluid loading is neglected the response to a line force is evaluated explicitly by contour integration and the response to a point force then reduces to a single integral. When fluid loading is included the responses are found by numerical integration and efficient methods of evaluating the integrals are discussed. At low frequencies the stiffened plate is seen to behave as an orthotropic plate. The input mobility for both types of excitation is seen to exhibit peaks at certain frequencies due to coherent interference of the reflections from the supports.

Uncertainty in structural dynamics

The effects of uncertainty are of growing concern in the design of engineering structures. The fact that the properties of the structure are uncertain implies that there is consequent uncertainty in the dynamic response. Similarly, there is inevitable manufacturing variability: mass-produced items are never identical. Indeed the properties of an individual system will change with time due to environmental conditions, loads, wear, etc. Uncertainty and variability raise issues concerning safety, reliability, quality of performance, worst-case behaviour and so on, and in turn these issues lead to demands for modelling methods which specifically include uncertainties in the properties of the structure. In the past, factors of safety might be introduced. However, the desire for greater efficiency, improved performance and reduced costs has led to a demand for improved computational methods, especially for high-cost structures. The goal is to apply such methods at the design stage to produce structures which are safe, reliable and have acceptable noise and vibration performance under all environmental and operating conditions which they are expected to encounter, and to produce designs which are robust with respect to manufacturing variability.

A shape memory alloy adaptive tuned vibration absorber: design and implementation

In this paper a tuned vibration absorber (TVA) is realized using shape memory alloy (SMA) elements. The elastic modulus of SMA changes with temperature and this effect is exploited to develop a continuously tunable device. A TVA with beam elements is described, a simple two-degree-of-freedom model developed and the TVA characterized experimentally. The behaviour during continuous heating and cooling is examined and the TVA is seen to be continuously tunable. A change in the tuned frequency of 21.4% is observed between the cold, martensite, and hot, austenite, states. This corresponds to a change in the elastic modulus of about 47.5%, somewhat less than expected. The response time of the SMA TVA is long because of its thermal inertia. However, it is mechanically simple and has a reasonably good performance, despite the tuning parameters depending on the current in a strongly nonlinear way.

Wave Reflection and Transmission in Timoshenko Beams and Wave Analysis of Timoshenko Beam Structures

This paper concerns wave reflection, transmission, and propagation in Timoshenko beams together with wave analysis of vibrations in Timoshenko beam structures. The transmission and reflection matrices for various discontinuities on a Timoshenko beam are derived. Such discontinuities include general point supports, boundaries, and changes in section. The matrix relations between the injected waves and externally applied forces and moments are also derived. These matrices can be combined to provide a concise and systematic approach to vibration analysis of Timoshenko beams or complex structures consisting of Timoshenko beam components. The approach is illustrated with several numerical examples.

Sound radiation from a plate reinforced by two sets of parallel stiffeners

Abstract A solution is developed for the sound radiation from a point-excited infinite fluid-loaded plate which is reinforced by two sets of parallel stiffeners. The stiffeners are intended to represent the bulkheads and intermediate frames of a hull structure. The solution is found by using Fourier wavenumber transforms, and the stationary phase approximation is used to find an expression for the far field pressure. The effects of the two sets of stiffeners on the radiated pressure depend primarily on the number of frames between successive bulkheads and on the point of application of the force. The presence of frames and bulkheads away from the point of excitation becomes less important as the frequency of the excitation increases.

Sound radiation from fluid loaded orthogonally stiffened plates

Abstract The radiation of sound from infinite fluid loaded plates is examined when the plates are reinforced with two sets of orthogonal line stiffeners. The stiffeners are assumed to be equally spaced and exert only forces on the plate. The response to a convected harmonic pressure is found by using Fourier transforms and is given in terms of the harmonic amplitudes of the stiffener forces. These forces satisfy an infinite set of simultaneous equations to which a numerical solution must be found. An expression for the response to a general excitation is derived and from this the acoustic pressure in the far field is determined with particular reference to point force excitation.