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A series method applied to engineering calculations in structural dynamics
Abstract
This paper shows an application of the F-functions series method to calculate the response of structures in face of an earthquake, modelled by a 2DOF. The F-functions series method is an adaptation of the ideas of Scheifele to integrate forced and damped oscillators. This algorithm presents the advantage of integrating precisely the perturbed problem with only two F-functions. Method coefficients are calculated by simple algebraic recurrences in which the perturbation function is involved. Results show the good precision compared to those obtained by other wellknown integrators implemented in MAPLE. Results are also contrasted with classic methods of Structural Engineering.
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In a first part, this article presents the adaptation of Scheifele functions to forced and damped oscillators, designing a series method based on these, which integrates the non perturbed problem with no truncation error. This method is highly accurate, however it is difficult to adapt to each specific problem. In order to overcome this difficulty, in a second part, we describe the transformation of the series method to a multistep scheme. Explicit and implicit methods are formulated and combine to create a predictor-corrector method, which precisely integrates the homogenous problem. The computational algorithm is developed and the results obtained are contrasted by the series method and by the multistep algorithm with other known integrators.
In 1971 Scheifele designed a method for the integration of perturbed oscillators by refining the classical method of power series. This method consists of defining a certain sequence of entire functions and finding the sought solution in the form of a linear combination of them using coefficients obtained through recurrence formulas. In spite of the good behavior of the Scheifele method, it is only practical for use in a few particular cases, in which the force function is very simple, given the complexity of the preliminary calculations needed to obtain the recurrence formulas. This problem is resolved in this article by converting the G-functions method into a multistep scheme. The key property is that those algorithms are able to integrate, without truncation error, harmonic oscillators involving various frequencies, and that, for perturbed problems, the local error contains the (small) perturbation parameter as a factor. Therefore, the methods are useful for finding highly accurate solutions to problems whose solutions are close to quasi-periodic ones. An application is performed in investigating the behavior of the proposed schemes for the integration of the orbital motion of Earth satellites.
We describe a four-step exponential-fitted method for systems of second-order differential equations of the form y″ = ƒ(x,y). This is a sixth-order method depending on five parameters which are automatically adjusted in terms of the equations to be solved. Some other relevant features are as follows: (i) it requires only two solution values to start; (ii) it allows modification of the stepsize during the integration process; (iii) it works in the predictor-corrector mode with only one function evaluation per step; (iv) the whole integration process is controlled in terms of the requested value for the local truncation error.Our method was tested on a representative set of problems taken from physics and found to behave particularly well on the problems involving oscillatory phenomena. A selection of experimental results is given in which our method is compared with a widely used code.
Es wird ein Schrittverfahren zur numerischen Integration von gestörten linearen Differentialgleichungssystemen aufgestellt. Die Grundlage dazu bildet die Definition einer Folge von ganzen Funktionen, die keine Polynome sind, aber eine einfache Potenzreihendarstellung besitzen. Im Gegensatz zu den üblichen Potenzreihenverfahren, wird nicht nach Potenzen der unabhängigen Variabeln, sondern nach diesen Funktionen entwickelt. Das Integrationsverfahren hat die Eigenschaft, dass die ungestörten Differentialgleichungen ohne Diskretisationsfehler integriert werden, und dass die Eigenwerte der Koeffizientenmatrix nicht berechnet werden müssen. Die Verbesserung gegenüber der gewöhnlichen Potenzreihenmethode wird durch asymptotische Formeln für die Residuen und durch numerische Beispiele belegt.
Nach einer Einleitung über Ziel und Zweck der Arbeit wird die Methode anhand eine: gestörten linearen Differentialgleichung zweiter Ordnung illustriert. Anschliessend wird das Integrationsverfahren auf beliebige gestörte lineare Systeme verallgemeinert.
Two piecewise-linearized methods for the numerical solution of nonlinear second-order differential equations which contain the first-order derivative are presented. These methods provide piecewise analytical solutions in open intervals, result in explicit finite difference equations for both the displacement and the velocity or only the displacement, at nodal points, and continuous solutions everywhere. One of the piecewise-linearized methods is a two-level method that provides smooth solutions and is self-starting; the other is a non-self-starting three-level technique. The paper also presents several iterative schemes, a time-linearized three-point technique and a time-linearized Numerov method. An extensive assessment of the piecewise-linearized methods is carried out in order to assess their accuracy in nine examples, and it is shown that, in some cases, these techniques provide as accurate results as exponentially- or trigonometrically-fitting methods that do require an estimate of the dominant frequency of the solution. The piecewise-linearized methods presented here do not require such an estimate at all, but may exhibit spurious fixed points/attractors in nonlinear problems and their accuracy is a strong function of both the nonlinearities and the time step. However, they are applicable to nonlinear second-order differential equations which contain the first-order derivative.
This article presents a new family of real functions with values within the ring of M(m,R) matrices, Φ-functions for perturbed linear systems and a numerical method adapted for integration of this type of problem. This method permits the system solution to be expressed as a series of Φ-functions. The coefficients of this series are obtained through recurrences in which the perturbation intervenes.The Φ-functions series method has the advantage of being exactly integrated in the perturbed problem. For this purpose an appropriate B matrix is selected and used to construct the operator described in this article, thus annihilating the disturbance terms, transforming the system into a homogenous second-order system, which is exactly integrated with the two first Φ-functions.The article ends with a detailed study of four perturbed systems which illustrate how the method is used in stiff problems or in highly oscillatory problems, contrasting its behaviour by studying its accuracy in comparison with other well-known codes.
A new special-purpose method of numerical integration for a perturbed linear system of ordinary differential equations is presented, based on Scheifele’s G-functions method.The new method converts Scheifele’s method in a multistep scheme that conserves the capability of integrating without truncation errors the unperturbed linear system of ODE’s. The new numerical method presents a good property contrasted with other VSVO multistep methods using G-functions; it adds a simple algebraic procedure for the computation of the coefficients irrespective of what the order is, obtained through recurrence formulas.Some numerical examples are included showing the good behaviour of the new code when it is applied to the class of problems to which it is addressed.
Highly accurate long-term numerical integration of nearly oscillatory systems of ordinary differential equations (ODEs) is a common problem in astrodynamics. Scheifele’s algorithm is one of the excellent integrators developed in the past years to take advantage of special transformations of variables such as the K-S set. It is based on using expansions in series of the so-called G-functions, and generalizes the Taylor series integrators but with the remarkable property of integrating without truncation error oscillations in one basic known frequency. A generalization of Scheifele’s method capable of integrating exactly harmonic oscillations in two known frequencies is developed here, after introducing a two parametric family of analytical ϕ-functions. Moreover, the local error contains the perturbation parameter as a factor when the algorithm is applied to perturbed problems. The good behavior and the long-term accuracy of the new method are shown through several examples, including systems with low- and high-frequency constituents and a perturbed satellite orbit. The new methods provide significantly higher accuracy and efficiency than a selection of well-reputed general-purpose integrators and even recent symplectic or symmetric integrators, whose good behavior in the long-term integration of the Kepler problem and the other oscillatory systems is well stated in recent literature.