On numerical issues for the wave/finite element method

The waveguide finite element (WFE) method is a numerical method to investigate wave motion in a uniform waveguide. Numerical issues for the WFE method are specifically illustrated in this report. The method starts from finite element mass and stiffness matrices of only one element of the section of the waveguide. The matrices may be derived from commercial FE software such that existing element libraries can be used to model complex general structures. The transfer matrix, and hence the eigenvalue problem, is formed from the dynamic stiffness matrix in conjunction with a periodicity condition. The results of the eigenvalue problem represent the free wave characteristics in the waveguide. This reportconcerns numerical errors occurring in the WFE results and proposing approaches to improve the errors.In the WFE method, numerical errors arise because of (1) the FE discretisation error, (2) round-off errors due to the inertia term and (3) ill-conditioning. The FE discretisation error becomes large when element length becomes large enough compared to the wavelength. However, the round-off error due to the inertia term becomes large for small element lengths when the dynamic stiffness matrix is formed. This tendency is illustrated by numerical examples for one-dimensional structures.Ill-conditioning occurs when the eigenvalue problem is formed and solved and the resulting errors can become large, especially for complex structures. Zhong’s method is used to improve the conditioning of the eigenvalue problem in this report. Errors in the eigenvalue problem are first mathematically discussed and Zhong’s method validated. In addition, singular value decomposition is proposed to reduce errors in numerically determining theeigenvectors. For waveguides with a one-dimensional cross-section, the effect of the aspect ratio of the elements on the conditioning is also illustrated. For general structures, there is a crude trade-off between the conditioning, the FE discretisation error and the round-off error due to the inertia term. To alleviate the trade-off, the model with internal nodes is applied. At low frequencies, the approximate condensation formulation is derived and significant errorreduction in the force eigenvector components is observed.Three approaches to numerically calculate the group velocity are compared and the finite difference and the power and energy relationship are shown to be efficient approaches for general structures

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

An experimental investigation of the natural frequency statistics of a beam with spatially correlated random masses

Experimental investigations into the dynamic response of structures with material or geometrical random fields usually depend upon an initial characterization of this variability, with very little control over the statistics at its early manufacturing stage. This provides the need of a minimal number of samples to generate an ensemble of dynamic responses, making such experimental data scarcely found in the literature. In this work, a cantilever beam with small masses attached along its length according to a given discrete random field has an ensemble of natural frequencies measured for a number of correlation lengths. The results can be used to investigate the effects of the correlation length on the subsequent natural frequency statistics. The experimental results are compared with a wave approximation for flexural waves using a continuous random field for the mass density, in order to approximate the mass distribution. Issues concerning this approximation are discussed. In addition, results are also compared with a simple added mass approximation with assumed modes from a FE solution

Initial experimental investigations on natural fibre reinforced honeycomb core panels

The main attention of the present work is on eco-friendly honeycomb cores for sandwich panels. They are manufactured by combining flax fibres with polyethylene matrix; the analyses involve both reinforced and un-reinforced cores. Some experimental tests have been planned and carried out in order to qualify the modal characteristics of this important class of panels. Tests results, herein discussed, report a great improvement of reinforced cores (continuous-unidirectional and short-random) compared to un-reinforced ones in mechanical properties. An improvement in damping value is achieved by filling the core with wool fibres resulting in minimal weight increase. A summary of the impact and acoustic tests results of preview tests are also reported in order to have a global view of the behaviour of these sandwich panels