Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems

Rysak, Andrzej; Gregorczyk, Magdalena

doi:10.3390/app11156955

Cited by 12 publications

(4 citation statements)

References 26 publications

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“…A total of 12 models with different structural properties are intentionally designed, and the fundamental frequencies of each model are shown in Table 1. The system is assumed to be linear elastic and is subjected to a ground acceleration üg (t) of sine-wave form as follows, üg (t) = sin( ωt) (12) and then, F i+1 in Equation ( 1) is expressed as…”

Section: Determination Of ω C Based On Dmfmentioning

confidence: 99%

“…Direct integration algorithms are widely utilized to solve temporally discretized equations of motion (EOMs) in the dynamic analysis of civil engineering structures [1][2][3]. Various integration algorithms have been well established based on different design conceptions in the past decades, such as the finite difference schemes [4][5][6][7], spectral methods of discretization [8,9], and methods based on other ideas [10][11][12]. In general, integration algorithms can be classified as explicit and implicit according to the expressions of displacement and velocity.…”

Section: Introductionmentioning

confidence: 99%

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Choices of the Critical Frequency for φ in TL-φ Algorithms When Applied to Multi-Degree of Freedom Systems

Tang

Luo

2022

Buildings

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TL-φ algorithms are newly developed explicit structure-dependent integration algorithms utilized for solving the temporally discretized equations of motion. In contrast to the existing algorithms, the most significant improvement of TL-φ algorithms is in diminishing the amount of period errors by introducing a precorrection coefficient φ into the integration parameters of TL-φ algorithms, which is related to the critical frequency of a system. In the previous work, the fundamental frequency of the system is deemed to be the critical frequency, so that φ is a constant scaling corresponding to the fundamental frequency for both single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems. However, for a MDOF system, the first mode may not contribute to the total response more than other ones under a given external excitation, calculating φ only by the fundamental frequency will underestimate the contribution of the higher-frequency modes to structural dynamics. In this paper, choices of the critical frequency for φ when applying TL-φ algorithms to MDOF systems are investigated thoroughly. By considering the initial structural properties of the system and the frequency characteristics of the external excitation simultaneously, a calculation criterion of φ for MDOF systems under specific external excitations is proposed. Four numerical examples with different initial structure properties and loading conditions are designed, and the results demonstrate that the proposed criterion can be potentially used to solve structural dynamic problems of MDOF systems with a more desirable numerical dispersion performance.

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Section: Determination Of ω C Based On Dmfmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Choices of the Critical Frequency for φ in TL-φ Algorithms When Applied to Multi-Degree of Freedom Systems

Tang

Luo

2022

Buildings

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“…Mukhtarov and Yücel [14] utilised the ADM to investigate the eigenvalues and eigenfunctions of twointerval SLPs that arise in modelling many phenomena in physics and engineering. Rysak and Gregorczyk [15] have shown the effectiveness of the DTM in solving problems arising in fractional dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

A New Transformation Method for Solving High-Order Boundary Value Problems

Yücel

Muhtarov

Mukhtarov

2022

Journal of New Theory

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The main purpose of this work is to provide a new approximation method, the so-called parameterised differential transform method (PDTM), for solving high-order boundary value problems (HOBVPs). Our method is based on the classical differential transform method and differs from it by calculating the coefficients of the solution, which has the form of a series. We applied the proposed new method to fourth-order boundary value problems to substantiate it. The resulting solution is graphically compared with the exact solution and the solutions obtained by the classical DTM and ADM methods.

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“…Since 1994, the Chebyshev polynomials [7], Legendre polynomials [8], Bessel polynomials [9], Hermite polynomials [10], Laguerre polynomials [11], and matrix methods have been used in many research studies to solve linear and nonlinear equations with high orders including partial differential equations, hyperbolic partial differential equations, delay equations, integral and integro-differential equations, SDOF and MDOF systems, etc. One of the approaches to find the solution to an initial value problem is taking semi-analytical procedures such as the differential transform method (DTM) [12][13][14][15][16][17][18][19][20][21][22][23]. The nature of dynamic equations of motion of SDOF systems makes them differential equations with initial values; hence, DTM has been also applied to solve non-linear SDOF problems [24][25][26][27][28][29][30][31][32][33].…”

Section: Introduction and State-of-artmentioning

confidence: 99%

Dynamic Response Analysis of Structures Using Legendre–Galerkin Matrix Method

Momeni

Beni²,

Bedon

et al. 2021

Applied Sciences

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The solution of the motion equation for a structural system under prescribed loading and the prediction of the induced accelerations, velocities, and displacements is of special importance in structural engineering applications. In most cases, however, it is impossible to propose an exact analytical solution, as in the case of systems subjected to stochastic input motions or forces. This is also the case of non-linear systems, where numerical approaches shall be taken into account to handle the governing differential equations. The Legendre–Galerkin matrix (LGM) method, in this regard, is one of the basic approaches to solving systems of differential equations. As a spectral method, it estimates the system response as a set of polynomials. Using Legendre’s orthogonal basis and considering Galerkin’s method, this approach transforms the governing differential equation of a system into algebraic polynomials and then solves the acquired equations which eventually yield the problem solution. In this paper, the LGM method is used to solve the motion equations of single-degree (SDOF) and multi-degree-of-freedom (MDOF) structural systems. The obtained outputs are compared with methods of exact solution (when available), or with the numerical step-by-step linear Newmark-β method. The presented results show that the LGM method offers outstanding accuracy.

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Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems

Cited by 12 publications

References 26 publications

Choices of the Critical Frequency for φ in TL-φ Algorithms When Applied to Multi-Degree of Freedom Systems

Choices of the Critical Frequency for φ in TL-φ Algorithms When Applied to Multi-Degree of Freedom Systems

A New Transformation Method for Solving High-Order Boundary Value Problems

Dynamic Response Analysis of Structures Using Legendre–Galerkin Matrix Method

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