Truncated Eigenfunction Expansion Research Articles

Diagonalization of 1-D differential operators with piecewise constant coefficients using the uncertainty principle

A highly accurate and efficient numerical method is presented for computing the solution of a 1-D time-dependent partial differential equation in which the spatial differential operator features a piecewise constant coefficient defined on n pieces, in either self-adjoint and non-self-adjoint form, on a finite interval with periodic boundary conditions. The Uncertainty Principle is used to estimate the eigenvalues of the operator. Then, these estimates are used to construct a basis of eigenfunctions for use with a spectral method. The solution is presented as a truncated eigenfunction expansion, where each eigenfunction is a wave function that changes frequencies at the interfaces between different materials. Numerical experiments demonstrate the accuracy, efficiency and scalability of the method in comparison to other methods.

Eddy-Current Coil Interaction With A Perfectly Conducting Wedge of Arbitrary Angle

In this article, a closed-form expression for the impedance of a tangential eddy-current coil in the presence of an infinite conducting wedge of arbitrary angle is derived. The truncated eigenfunction expansion (TREE) solution given here is valid in the quasi-static frequency regime. The theory was validated via comparison to an independent analytical expression for the impedance change of a horizontal coil over a conducting half-space due to Burke. We present results for three geometries: a conducting quarter-space, a conducting wedge of angle 225 degrees, and a semi-infinite conducting sheet. Our theory predicts a measurable change in the tangent coil reactance in the presence of all three geometries.

Elastic and inelastic atomicscattering in a standing-wave laser field

The motion of a two-level atom in a standing-wave laser fieldis studied using an approximation to the resolvent operator inthe form of a truncated eigenfunction expansion, with the fieldtreated in the occupation-number representation. The formalismprovides expressions for the time-dependent S-matrix fortransitions into both ground and excited atomic states. Theeffective Hamiltonian for purely elastic scattering is constructed, allowing for the direct study of scattering in theBragg regime. Contact with standard treatments of Braggscattering in terms of Mathieu functions is obtained byimposition of additional approximations. Comparison ofnumerical results obtained with and without imposition of theseapproximations provides an indication of their range ofvalidity. The dependence of excitation probabilities on fieldstrength and detuning is studied. These numerical illustrationsprovide evidence that in cases of physical interest excitationprobabilities can be significant and can be reliably estimatedfor a broad range of field parameters using the present formalism.

剛結合された2本のビームのモデリングとロバスト制御

Summary form only given. In this paper, modeling and robust control of a flexible structure with a closed-loop mechanism are discussed. It is important to derive a mathematical model and construct a controller for such complicated space structures. We consider two connected flexible beams as a simple model of the space structures with the closed-loop mechanism. This structure can be regarded as an element of the truss structure. We derive dynamic equations of the connected flexible beams by means of Hamilton's principle. As the obtained boundary conditions of the distributed parameter system are nonhomogeneous, we introduce a change of variables to derive homogeneous boundary conditions. The solution of an eigenvalue problem related to the distributed parameter system yields eigenvalues and corresponding eigenfunctions. On the basis of the truncated eigenfunction expansion, an approximated finite-dimensional modal model are derived. As the original system is infinite dimensional and controllers should be designed on the basis of an approximated finite dimensional model, it is necessary to compensate the spillover instability caused by the residual modes which are neglected at the controller design phase. It is difficult to estimate the physical parameters of the system, for example a flexural rigidity, a damping coefficient, etc., correctly. In order to compensate the unmodeled dynamics and the parameter uncertainty, a robust controller should be constructed. In the paper, an optimal controller with low-pass property and a robust H/sub /spl infin// controller are designed. Simulations have been carried out and responses for the controllers are compared.

Investigation of stress singularity coefficient for a finite plate containing rigid line

The eigenfunction expansion variational method (abbreviated as EEVM) is proposed to determine the stress singularity coefficients for a finite plate containing the rigid line. In the method, the undetermined coefficients in the truncated eigenfunction expansion form are determined by using the variational method. To explain the use of this method, several numerical examples are given.

An investigation of the stress intensity factor for a finite internally cracked plate by using variational method

The eigenfunction expansion variational method (EEVM) is proposed to determine the stress intensity factors for two-dimensional cracked bodies. In the new method, the undetermined coefficients in the truncated eigenfunction expansion form are determined by using the variational method. It is expected that the uncertainty initiated in boundary collocation scheme can be avoided. Several numerical examples are given, which can prove the efficiency of the EEVM method.

Axisymmetric Stress-Wave Propagation across the Common End Face between Two Semi-Infinite Cylinders, Solid to Fluid

When a harmonic wave train traveling in an elastic rod strikes its interface with a contained fluid column of the same radius, some of the energy is transmitted to the fluid and the rest is reflected back down the rod. In the process, elastic energy is temporarily stored by the deformation of the rod's end face. To study these phenomena in detail analytically, a truncated eigenfunction expansion is used. The equations of motion and the transverse boundary conditions are satisfied exactly in the two media. The resulting eigenfunction expansions for the interface boundary conditions are truncated, and imposed at enough points along a radius on the interface to permit a solution of the resulting system of equations for the amplitude coefficients in the truncated series. The stress and displacement fields at any point in either medium, along with intensity, power, and transmission loss, are then reconstructed, using the appropriate truncated series. The computer results obtained for the case of Nylon and water over a limited frequency range validate the approximation procedure and provide new information about the stress and displacement fields as functions of frequency and location. At the frequency of “end resonance,” it is found that the transmission loss curve has a sharp peak, indicating a possible use for this configuration as an acoustic notch filter.