Order Eigenvalue Problems Research Articles

Non-rotating beams isospectral to a given rotating uniform beam

In this paper, we seek to find non-rotating beams with continuous mass and flexural stiffness distributions, that are isospectral to a given uniform rotating beam. The Barcilon–Gottlieb transformation is used to convert the fourth order governing equation of a non-rotating beam, to a canonical fourth order eigenvalue problem. If the coefficients in this canonical equation match with the coefficients of the uniform rotating beam equation, then the non-rotating beam is isospectral to the given rotating beam. The conditions on matching the coefficients leads to a pair of coupled differential equations. We solve these coupled differential equations for a particular case, and thereby obtain a class of non-rotating beams that are isospectral to a uniform rotating beam. However, to obtain isospectral beams, the transformation must leave the boundary conditions invariant. We show that the clamped end boundary condition is always invariant, and for the free end boundary condition to be invariant, we impose certain conditions on the beam characteristics. We also verify numerically that the frequencies of the non-rotating beam obtained using the finite element method (FEM) are the exact frequencies of the uniform rotating beam. Finally, the example of beams having a rectangular cross-section is presented to show the application of our analysis. Since experimental determination of rotating beam frequencies is a difficult task, experiments can be easily conducted on these rectangular non-rotating beams, to calculate the frequencies of the rotating beam.

Eigenstructure assignment in undamped vibrating systems: A convex-constrained modification method based on receptances

The problem of designing vibrating systems having a desired eigenstructure is addressed in this paper using a method based on receptances. The method proposed relies entirely on receptances and hence neither analytical nor modal models are needed. The problem is formulated as a constrained inverse eigenvalue problem in order to ensure the physical feasibility of the solutions computed. The formulation proposed, suitable for linear undamped systems, allows handling an arbitrary number of assigned eigenpairs and of modifiable parameters. Moreover, a regularization term allows biasing the solution towards preferable modifications and can assure good numerical conditioning. Finally, since the problem is formulated as a convex-constrained problem, a unique solution must exist and can be computed through effective and fast convergence rate algorithms. A numerical example and the comparison with a well-established method demonstrate that the proposed method provides effective results. The application of the method to the modification of a laboratory vibrating system gives experimental evidence of its effectiveness.

A perturbed Whittaker–Kotelʼnikov–Shannon sampling theorem

The sampling theorem of Whittaker (1915) [31], Kotelʼnikov (1933) [25] and Shannon (1949) [28] gives cardinal series representations for finite L 2 -Fourier transforms at equidistant sampling points. Here we investigate the situation when the Fourier transform is replaced by a perturbed one. Thus the kernel of the transform will be of the form exp ( − i x t ) + ε ( x , t ) , instead of exp ( − i x t ) in the unperturbed case. The perturbed kernel arises from first order eigenvalue problems with rank one perturbations.

Identification of autoregressive moving average systems based on noise compensation in the correlation domain

This study presents a new scheme for the identification of minimum-phase autoregressive moving average (ARMA) systems from noise-corrupted observations. From the autocorrelation function (ACF) of the observed data, exploiting the characteristics of the zero lag, the authors develop a set of equations containing the lower lags of the ACF and solve the corresponding quadratic eigenvalue problem in order to estimate simultaneously the AR parameters and the observation noise variance. In the proposed identification technique, both the white noise and the periodic impulse-train excitations are considered for the purpose of practical applications. In order to estimate the MA parameters, first, a residual signal is obtained by filtering the noisy observations via the estimated AR parameters. A noise-subtraction algorithm is proposed utilising the estimated noise variance and the AR parameters to reduce the effect of noise from the ACF of the residual signal. The MA parameters are then estimated by using the spectral factorisation corresponding to the noise-compensated ACF of the residual signal. In order to demonstrate the effectiveness of the proposed method, extensive experimentations are performed considering synthetic ARMA systems of different orders in the presence of noise and results are compared with those of some of the existing methods. Computer simulations demonstrate a superior identification results in terms of estimation accuracy and consistency even under heavy noisy conditions. Simulation results are also provided for the identification of a human vocal-tract system using natural speech signals showing a superior performance of the proposed technique in terms of the power spectral density of the synthesised speech signal.

A Born–Oppenheimer Expansion in a Neighborhood of a Renner–Teller Intersection

We perform a rigorous mathematical analysis of the bending modes of a linear triatomic molecule that exhibits the Renner–Teller effect. Assuming the potentials are smooth, we prove that the wave functions and energy levels have asymptotic expansions in powers of \({\epsilon}\) , where \({\epsilon^4}\) is the ratio of an electron mass to the mass of a nucleus. To prove the validity of the expansion, we must prove various properties of the leading order equations and their solutions. The leading order eigenvalue problem is analyzed in terms of a parameter \({\tilde{b}}\) , which is equivalent to the parameter originally used by Renner. For \({0< \tilde{b}< 1}\) , we prove self-adjointness of the leading order Hamiltonian, that it has purely discrete spectrum, and that its eigenfunctions and their derivatives decay exponentially. Perturbation theory and finite difference calculations suggest that the ground bending vibrational state is involved in a level crossing near \({\tilde{b}=0.925}\) . We also discuss the degeneracy of the eigenvalues. Because of the crossing, the ground state is degenerate for \({0< \tilde{b}< 0.925}\) and non-degenerate for \({0.925< \tilde{b}< }\) .

Basis properties of a fourth order differential operator with spectral parameter in the boundary condition

Abstract We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in L p(0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.

A mixed finite element method for fourth order eigenvalue problems

A mixed finite element method for approximating eigenpairs of IV order elliptic eigenvalue problems with Dirichlet boundary conditions has been given. The method can be applied to the vibration analysis of anisotropic/orthotropic/isotropic/biharmonic plates. Computer implementation procedures for this mixed method are given along with the results of numerical experiments.

Linear system solution by null‐space approximation and projection (SNAP)

AbstractSolutions of large sparse linear systems of equations are usually obtained iteratively by constructing a smaller dimensional subspace such as a Krylov subspace. The convergence of these methods is sometimes hampered by the presence of small eigenvalues, in which case, some form of deflation can help improve convergence. The method presented in this paper enables the solution to be approximated by focusing the attention directly on the ‘small’ eigenspace (‘singular vector’ space). It is based on embedding the solution of the linear system within the eigenvalue problem (singular value problem) in order to facilitate the direct use of methods such as implicitly restarted Arnoldi or Jacobi–Davidson for the linear system solution. The proposed method, called ‘solution by null‐space approximation and projection’ (SNAP), differs from other similar approaches in that it converts the non‐homogeneous system into a homogeneous one by constructing an annihilator of the right‐hand side. The solution then lies in the null space of the resulting matrix. We examine the construction of a sequence of approximate null spaces using a Jacobi–Davidson style singular value decomposition method, called restarted SNAP‐JD, from which an approximate solution can be obtained. Relevant theory is discussed and the method is illustrated by numerical examples where SNAP is compared with both GMRES and GMRES‐IR. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Generalized modeling of drilling vibrations. Part II: Chatter stability in frequency domain

A time domain model of the drilling process and hole formation mechanism is presented in Part I, and the general solution of drilling chatter stability in frequency domain is presented in this paper. The drill's flexibility in torsional, axial and lateral directions are considered in determining the regenerative chip thickness. Stability is modelled as a fourth order eigenvalue problem with a regenerative delay term. The critical radial depth of cut and spindle speed are analytically determined from the eigenvalues of the characteristics equation of the dynamic drilling process in frequency domain. The method is compared against the extensive numerical solutions in time domain which were presented in Part I, cutting experiments and previously published partial stability laws. The time domain model presented in Part I of the paper considers tool geometry dependent mechanics, all vibration directions and the true kinematics of drilling, while allowing for nonlinearities such as tool jumping out of cut and nonlinear cutting force models. It is shown that accurate prediction of drilling stability requires modeling of drill/hole surface contact stiffness and damping which is still a research challenge.

On sampling theory and eigenvalue problems with an eigenparameter in the boundary conditions

This paper is devoted to the investigation of sampling theory associated with second order eigenvalue problems with an eigenparameter appearing in the boundary conditions. We study two cases. The first is when the eigen-parameter appears linearly in all boundary conditions and the second is when it appears only in one condition. We closely follow the analysis derived by C. T. Fulton (1977) to establish the needed relations for the derivations of the sampling theorems including the construction of Green’s function as well as the eigenfunction expansion theorem. We derive sampling representations for transforms whose kernels are either solutions or Green’s functions.

Chatter Stability of Plunge Milling

Plunge milling operations are used to remove excess material in boring cylinders, roughing pockets, dies and mold cavities. This paper presents a frequency domain, chatter stability prediction theory for plunge milling. The regenerative chip thickness is modeled as a function of lateral, axial and torsional vibrations. The stability of the plunge milling is formulated as a fourth order eigenvalue problem by relating the regenerative chip thickness, cutting forces and torque, and the structural modes of the cutter. The stability lobes are predicted analytically from the eigenvalue solution. The stability lobes are experimentally proven by conducting over one hundred plunge milling tests.

A parallel mixed finite element implementation for approximation of eigenvalues and eigenvectors of fourth-order eigenvalue problems

The paper deals with a parallel implementation of a mixed finite element method of approximation of eigenvalues and eigenvectors of fourth order eigenvalue problems with variable/constant coefficients. The implementation has been done in Silicon Graphics Origin 3800, a four processor Intel Xeon Symmetric Multiprocessor and a beowulf cluster of four Intel Pentium III PCs. The generalised eigenvalue problem obtained after discretization using the mixed finite element method is solved using the package LANSO. The numerical results obtained are compared with existing results (if available). The time, speedup comparisons in different environments for some examples of practical and research interest and importance are also given.

Finite Element Approximation of a 2D–1D Contact Eigenvalue Problem

We consider an elliptic eigenvalue problem of second order on a two-component domain consisting of an interval and a rectangle with a common point. The interaction between the domains is expressed through a coupling condition in integral form for the components of the vector eigenfunction. We set up finite element methods for this problem, the involved error analysis being affected by the coupling condition, which requires a modification of the vector Lagrange interpolant on the overall mesh. The resulting error estimates however, show the optimal rate of convergence, as compared to those for eigenvalue problems with classical boundary or transition conditions.

Finite Element Approximation of a Contact Vector Eigenvalue Problem

We consider a nonstandard elliptic eigenvalue problem of second order on a two-component domain consisting of two intervals with a contact point. The interaction between the two domains is expressed through a coupling condition of nonlocal type, more specifically, in integral form. The problem under consideration is first stated in its variational form and next interpreted as a second-order differential eigenvalue problem. The aim is to set up a finite element method for this problem. The error analysis involved is shown to be affected by the nonlocal condition, which requires a suitable modification of the vector Lagrange interpolant on the overall finite element mesh. Nevertheless, we arrive at optimal error estimates. In the last section, an illustrative numerical example is given, which confirms the theoretical results.

Isoparametric mixed finite element approximation of eigenvalues and eigenvectors of 4th order eigenvalue problems with variable coefficients

Estimates for the combined effect of boundary approximation and numerical integration on the approximation of (simple) eigenvalues and eigenvectors of 4th order eigenvalue problems with variable/constant coefficients in convex domains with curved boundary by an isoparametric mixed finite element method, which, in the particular case of bending problems of aniso-/ortho-/isotropic plates with variable/constant thickness, gives a simultaneous approximation to bending moment tensor field and displacement field `u', have been developed.

Asymptotic solutions for mixed mode loading of cracks and wedges in power law hardening materials

The effects of arbitrary loading on the asymptotic stress and strain fields in non-linear fracture for a stationary crack tip are investigated. The material behavior is modeled as pure power law hardening within the theory of J 2-deformation plasticity. Of particular interest is the role of two-parameter higher order solutions in “mixed mode” fracture, where the first two terms in the expansion of the displacement field are variable-separable with real, but unequal eigenvalues. A displacement-based finite element method coupled with singular value decomposition, which is presented herein, is used to solve the non-linear higher order eigenvalue problem. The versatility of this method to account for arbitrary loading and geometry is used to investigate asymptotic crack solutions by studying the limit as a wedge becomes a crack. By using this approach it has been determined that in homogeneous plane strain and plane stress fracture, and also plane strain fracture of an interface crack between a hardening material and a rigid substrate, there are two distinct asymptotic solutions. These solutions are referred to as mode I dominant and mode II dominant. Each of the three fracture cases has one higher order solution, where the dominant term in the series is identical to the pure mode dominant term. In these solutions two asymptotic terms are necessary to account for the mixed nature of the far-field loading. In both plane strain fracture problems this higher order solution is mode I dominant, while for plane stress it is mode II dominant. The mode I dominant interface crack higher order solution was first presented by Sharma and Aravas [Int J Solids Struct 30 (1993) 695]. The “mode II dominant” solution for homogeneous plane strain fracture, first presented by Shih [Fract Anal, ASTM STP 560 (1974) 187], is a true mixed mode asymptotic solution governed by a single real eigenvalue, where the mixed nature of the loading enters into the leading term. Contrary to this mixed mode asymptotic solution, there is strong evidence that the mode II dominant plane strain interface crack solution is oscillatory, i.e., non-separable with a form similar to that of the compressible linear elastic case that results from a complex eigenvalue. The mode I dominant plane stress asymptotic solution could not be determined by the current approach.

A numerical investigation of the solution of a class of fourth–order eigenvalue problems

This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfunction is studied since it is known that this has an unbounded number of oscillations when approaching certain types of corners on domain boundaries. Recent computational results of Bjorstad & Tjostheim, using a highly accurate spectral Legendre–Galerkin method, have demonstrated that a number of these sign changes may be accurately computed on a square domain provided sufficient care is taken with the numerical method. We demonstrate that similar accuracy is also achieved using an unstructured finite–element solver which may be applied to problems on domains with arbitrary geometries. A number of results obtained from this mixed finite–element approach are then presented for a variety of domains. These include a family of circular sector regions, for which the oscillatory behaviour is studied as a function of the internal angle, and another family of (symmetric and non–convex) domains, for which the parity of the least eigenfunction is investigated. The paper not only verifies existing asymptotic theory, but also allows us to make a new conjecture concerning the eigenfunctions of the biharmonic operator.

Algorithm 800

This article describes LAPACK-based Fortran 77 subroutines for the reduction of a Hamiltonian matrix to square-reduced form and the approximation of all its eigenvalues using the implicit version of Van Loan's method. The transformation of the Hamiltonian matrix to a square-reduced form transforms a Hamiltonian eigenvalue problem of order 2 n to a Hessenberg eigenvalue problem of order n . The eigenvalues of the Hamiltonian matrix are the square roots of those of the Hessenberg matrix. Symplectic scaling and norm scaling are provided, which, in some cases, improve the accuracy of the computed eigenvalues. We demonstrate the performance of the subroutines for several examples and show how they can be used to solve some control-theoretic problems.

High Precision Solutions of Two Fourth Order Eigenvalue Problems

We solve the biharmonic eigenvalue problem $\Delta^2u = \lambda u$ and the buckling plate problem ${\Delta}^2u = - {\lambda}\Delta u$ on the unit square using a highly accurate spectral Legendre--Galerkin method. We study the nodal lines for the first eigenfunction near a corner for the two problems. Five sign changes are computed and the results show that the eigenfunction exhibits a self similar pattern as one approaches the corner. The amplitudes of the extremal values and the coordinates of their location as measured from the corner are reduced by constant factors. These results are compared with the known asymptotic expansion of the solution near a corner. This comparison shows that the asymptotic expansion is highly accurate already from the first sign change as we have complete agreement between the numerical and the analytical results. Thus, we have an accurate description of the eigenfunction in the entire domain.

On a variational approximation method for 2nd order eigenvalue problems in a multi–component domain with nonlocal dlrichlet transition conditions

We consider a class of eigenvalue problems (EVPs) on a bounded multi-component domain Ω in the plane, consisting of a number of convex polygonal subdomains. Classical mixed Robin-Dirichlet conditions are imposed on the outer boundaries, while on the interfaces between two subdomains we impose nonlocal transition conditions of Dirichlet type. This problem extends the single-component problem, treated in [4]. The aim of the paper is twofold. First, by passing to a product Sobolev space setting, we recast the problem into the framework of abstract variational EVPs, studied e.g.in [7]. This allows us to infer the existence of exact eigenpairs, showing some useful properties. Secondly, the variational EVP may serve as the starting point for setting up variational (internal) approximation methods, such as (conforming) finite element methods (FEMs). For this purpose the proof of a density theorem is crucial. From here on, for clarity in the exposition, we restrict ourselves to a domain Ω composed of 4 rectangles. Introducing suitable (families of) finite element spaces V h1, similar convergence properties and error estimates as in [4] are obtained for the finite element approximations of the eigenpairs, both without and with numerical quadrature. The error analysis will heavily rest upon the properties of a deliberately introduced imperfect Lagrange interpolant on each of the subdomains. Finally, we briefly deal with the corresponding algebraic EVP, identifying a proper basis of the product space V h.