Wittrick–Williams algorithm proof of bracketing and convergence theorems for eigenvalues of constrained structures with positive and negative penalty parameters

Abstract

SUMMARYThe well-established Wittrick-Williams algorithm is used to derive novel and general proofs that show that the eigenvalues of systems with constraints can be bracketed by replacing the constraints by positive and negative pairs of either ordinary or inertial penalty parameters. It is also shown that convergence occurs from both above and below when the numerical values of these parameters are increased towards infinity. The proofs are applicable in many contexts but are derived in that of structural systems, for which the eigenvalues are either buckling load factors or the squares of natural frequencies of vibration; ordinary penalty parameters are stiffnesses of translational and rotational springs; and inertial penalty parameters are either masses or rotary inertias. The penalty parameters can be used to constrain a system or to impose constraints between systems. It is shown that the use of inertial penalty parameters has several advantages compared with using ordinary ones. Then the pth eigenvalue of a system with n constraints is bounded closely from above by the ( p +n)th eigenvalue of the system with very large positive inertial penalty parameters and from below by the pth eigenvalue, when large negative values are used instead. This work is expected to enhance the versatility of numerical eigenproblem methods, e.g. the Rayleigh-Ritz method.

An exact spectral-dynamic stiffness method for free flexural vibration analysis of orthotropic composite plate assemblies – Part I: Theory

“…This has been proved to be a stable and satisfactory method, particularly when applying the Wittrick-Williams algorithm [11,12,14,15,51]. In the current S-DSM, the aforementioned elastic edge constraints directly facilitate the application of the penalty method, with the elastic constants K x ; K y ; R x ; R y of Eq.…”

Power secant method applied to natural frequency extraction of Timoshenko beam structures

“…This work consolidates the possibility of using an iterative process for extracting eigenvalues, without the compulsory use of the Wittrick-Williams algorithm [6,10,11,14]. Ref.…”

“…In Plane Rotation (6) stressing that, for the present theory, the rotation θ(x, t) cannot be directly obtained from the derivative of v(x, t), as it could in the Euler-Bernoulli theory. Substitution of expressions (6) for displacements in the equations from the previous section, after some algebraic transformations, leads to:…”

The use of pseudo‐inertia in asymptotic modelling of constraints in boundary value problems

“…The difficulty in determining an appropriate magnitude for a penalty parameter is that it needs to be sufficiently large to affect a constraint, however, small enough to avoid numerical problems 11–13. This has been the subject of several recent publications 14–22. It has been shown that for linear boundary value problems, the use of positive and negative penalty parameters enables the determination of any error due to constraint violation and that it is possible to obtain good approximations to the constrained solution using interpolation of the penalized solutions with respect to inverse penalty parameters.…”

“…For vibration problems, this limitation has also been overcome by using positive and negative inertial‐type penalty parameters instead of the ordinary stiffness‐type penalty parameters 17, 21. This was recently extended to include any linear eigenvalue problem through the use of a generic eigenpenalty parameter 22. The possibility of using the inertial penalty functions in static stress analysis by introducing a virtual inertia was also explored by Ilanko and Kreuzer 23.…”