Energy density equations and power flow in structures

A potential alternative to Statistical Energy Analysis that is gaining increasing interest in recent years is the “thermal” energy flow approach. Its advantage is represented by the possibility of modelling the spatial distribution of energy density at high frequencies, thus yielding a more effective estimate of the system behaviour than the average constant value given by SEA. However, the thermal analogy proposed by the energy flow approach is questionable for any type of wave in any type of structure. To make the analysis more clear, exact equations for power balance in continuous structures are derived. The investigation confirms the questionability of the thermal approach and shows whether and when it is possible to determine exact equations for the energy density. Physical considerations are developed to explain some critical points. For one-dimensional systems a transmission potential is defined in analogy to the temperature in heat conduction problems

An envelope energy model for high frequency dynamic structures

High frequency structural and acoustic problems require prohibitive computational efforts. The tendency, nowadays, is to find a solution in statistical terms (SEA) through an average of the field variables on the space domain. A limitation of SEA is the loss of any local information. In contrast with SEA, a power flow method [1] can describe a trend of the energy density along the structure, thus improving the quality of the solution. However, in dealing with flexural waves, the power flow neglects the near field contribution: the related solution can sometimes differ considerably from the expected trend. In this paper a held trend is obtained in a totally different manner. An envelope energy is used that describes well the exact solution: specifically, only the decaying fields, obtained from the projection on the real axis of the damped bending wavenumbers are accounted for, while the propagating components are omitted. Simulated results are presented and compared with exact and approximate solution

Energy flow uncertainties in vibrating systems: definition of a statistical confidence factor

A method for evaluating the energy flow confidence level in vibrating systems with randomly perturbed parameters is presented. The energy flow is predicted in terms of the mobilities of resonant subsystems or by the solution of the velocity wave field for non-resonant subsystems. The statistical moments of the energy flow are calculated by a perturbation technique and a confidence factor is defined as the ratio between mean and standard deviation. The properties of the confidence factor are investigated by a theoretical analysis as a function of frequency. Three cases are studied to compare the confidence factor obtained theoretically with a prediction provided by a Monte-Carlo simulation

Paper ESDA2008-59092 APPLICATION OF THE COMPLEX ENVELOPE VECTORIZATION TO A BOUNDARY ELEMENT FORMULATION

ABSTRACT The complex envelope vectorization (CEV) is a recent method that has been successfully applied to structural and internal acoustic problems. Unlike other methods proposed in the last two decades to solve high frequency problems, CEV is not an energy method, although it shares with all the other techniques a variable transformation of the field variable. By such transformation involving a Hilbert transform, CEV allows the representation of a fast oscillating signal through a set of low oscillating signals. Thanks to such transformation it is possible to solve a high frequency dynamic problem at a computational cost that is lower than that required by finite elements. In fact, by using finite elements, a high frequency problem usually implies large matrices. On the contrary the CEV formulation is obtained by solving a set of linear problems of highly reduced dimensions. Although it was proved that CEV is in general a successful procedure, it was shown that it is particularly appropriate when the modes of the system have a negligible role on the solution. Moreover, the numerical advantage of the CEV formulation is much more pronounced when full matrices are used. Thus, for the first time it is applied to a boundary element formulation (BEM). Both external and internal acoustic fields of increasing complexity are considered: the internal and external field generated by a pulsating sphere; the external field of a forced box, where the velocity field is determined by finite elements; a set of 4 plates that form an open cavity. The results are compared with those obtained by a BEM procedure (SYSNOISE), highlighting the good quality of the proposed approach. An estimate of the computational advantage is also provided. Finally it is worthwhile to point out that the reduction of the BE matrices allows for an in-core solution even for large problems

Complex envelope displacement analysis: a quasi static approach to vibrations

A new model to analyze high frequency vibrations is presented. Instead of using the physical oscillating displacement, the problem is described in terms of a complex envelope, generated by an appropriate use of the Hilbert transform. The model can be put in the category of those methods that try to describe some representative characteristic of the oscillating solution (average energy level, thermal trend, etc.) rather than the solution itself, avoiding the computational problems connected with high frequency problems. Although the envelope solution, by itself, is sufficient and convenient to deal with structural-acoustic coupling, the proposed model presents the twofold advantage of avoiding computational problem connected with high frequency vibrations while keeping the capability of recovering the oscillating response, when required

New insights on energy methods for high frequency vibrations

Invited plenary lectur