Peter D. Folkow

Dynamic equations for fluid-loaded porous plates using approximate boundary conditions

Systematically derived equations for fluid-loaded thin poroelastic layers are presented for time-harmonic conditions. The layer is modeled according to Biot theory for both open and closed pores. Series expansion techniques in the thickness variable are used, resulting in separate symmetric and antisymmetric plate equations. These equations, which are believed to be asymptotically correct, are expressed in terms of approximate boundary conditions and can be truncated to arbitrary order. Analytical and numerical results are presented and compared to the exact three dimensional theory and a flexural plate theory. Numerical comparisons are made for two material configurations and two thicknesses. The results show that the presented theory predicts the plate behavior accurately.

Approximate boundary conditions for a fluid-loaded elastic plate

Approximate boundary conditions for an infinite elastic layer immersed in a fluid are derived. By using series expansions in the thickness coordinate of the plate fields, the displacements fields are eliminated, adopting the three-dimensional equations of motion. The sums and differences of the boundary pressure fields and their normal derivatives are related through a set of approximate boundary conditions, one symmetric and one antisymmetric. These equations involve powers in the layer thickness together with partial derivatives with respect to time as well as the spatial variables in the plate plane. The approximate boundary conditions can be truncated to an arbitrary order, and explicit relations are presented including terms of order five. Comparisons are made with effective boundary conditions using classical plate theories. The numerical examples involve reflection and transmission of plane waves incident on the plate at different angles, as well as the pressure fields due to a line force. Three fluid-loading cases are studied: modest, heavy, and light loadings. The results using truncated approximate boundary conditions are compared to exact and classical plate solutions. The examples show that the accuracies of the power series approximations of order three and higher are very good in the frequency interval considered.

Direct and inverse problems on nonlinear rods

In this paper a class of models on nonlinear rods, which includes spatial inhomogeneities, varying cross-sectional area and arbitrary memory functions, is considered. The wave splitting technique is applied to provide a formulation suitable for numerical computation of direct and inverse problems. Due to the nonlinearity of the material, there are no well defined characteristics other than the leading edge, so the method of characteristics, highly successful in the computation of linear wave splitting problems, is abandoned. A standard finite difference method is employed for the direct problem, and a shooting method is introduced for the inverse problem. The feasibility of the inverse algorithm is presented in various numerical examples.

Wave propagators for the Timoshenko beam

The propagation and scattering of waves on the Timoshenko beam are investigated by using the method of wave propagators. This method is more general than the scattering operators connected to the imbedding and Green function approaches; the wave propagators map the incoming field at an internal position onto the scattering fields at any other internal position of the scattering region. This formalism contains the imbedding method and Green function approach as special cases. Equations for the propagator kernels are derived, as are the conditions for their discontinuities. Symmetry requirements on certain coupling matrices originating from the wave splitting are considered. They are illustrated by two specific examples. The first being an unrestrained beam with a varying cross-section and the other a homogeneous, viscoelastically restrained beam. A numerical algorithm for solving the equations for the propagator kernels is described. The algorithm is tested for the case of a viscoelastically restrained, homogeneous beam. In a limit these results agree with the ones obtained for the reflection kernel by a previously developed algorithm for the imbedding reflection equation.

Time domain inversion of a viscoelastically restrained Timoshenko beam

This paper deals with the inverse scattering problem for a homogeneous Timoshenko beam suspended on a semi-infinite viscoelastic layer. The purpose is to derive the properties of the suspension from knowledge of the reflection data. The method used is the imbedding technique, which render solutions that are independent of the excitation. From a numerical point of view, the reflection equation in its usual form is inappropriate when solving the inverse problem. In order to bypass such complications, the reflection equation is modified and solved. In the numerical results, the properties of the layer are calculated using noisy reflection data.

Dynamic equations for a fully anisotropic piezoelectric rectangular plate

Equations for an anisotropic piezoelectric plate without any planes of symmetry.A hierarchy of equations based on 3D theory that are asymptotically correct.Analytical expressions for orthotropic plate equations for different orders.Eigenfrequencies and field distributions over the plate cross section. A hierarchy of dynamic plate equations based on the three dimensional piezoelectric theory is derived for a fully anisotropic piezoelectric rectangular plate. Using power series expansions results in sets of equations that may be truncated to arbitrary order, where each order set is hyperbolic, variationally consistent and asymptotically correct (to all studied orders). Numerical examples for eigenfrequencies and plots on mode shapes, electric potential and stress distributions curves are presented for orthotropic plate structures. The results illustrate that the present approach renders benchmark solutions provided higher order truncations are used, and act as engineering plate equations using low order truncation.

Verification of Self-Tuning 4DOF Piezoelectric Energy Harvester with Enhanced Bandwidth

In this paper, we present an analytical model to predict enhanced bandwidth for a piezoelectric energy harvester with self-tuning, accomplished by a sliding mass. The model predicts that by implementing asymmetry of different piezoelectric cantilever lengths , the bandwidth can theoretically approach 60 Hz. Validation measurements demonstrate an increased 3dB bandwidth up to 21 Hz with 150 mW, by configuration 23/17 mm in open length – providing sufficient power for a ZigBee to continually transmit.

Smart design piezoelectric energy harvester with self-Tuning

Piezoelectric energy harvesting on a gas turbine implies constraints like high temperature tolerance, size limitation and a particular range of vibrations to utilise. In order to be able to operate under these conditions a harvester needs to be space effective and efficient and to respond to the appropriate range of frequencies. We present the design, simulation and measurements for a clamped-clamped coupled piezoelectric harvester with a free-sliding weight, which adds self-Tuning for improved response within the range of vibrations from the gas turbine. We show a peak open circuit voltage of 11.7 V and a 3 dB bandwidth of 12 Hz.