A canonical form of the equation of motion of linear dynamical systems

Kawano, Daniel T.; Junior, Rubens Gonçalves Salsa; Ma, Fai; Morzfeld, Matthias

doi:10.1098/rspa.2017.0809

Cited by 7 publications

(2 citation statements)

References 15 publications

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“…In a broad sense, when the matrix coefficients are constant, under some circumstances regarding C, it may be possible to convert the original coupled problem into n uncoupled problems [15]. When coefficient matrix C is written as a linear combination of M and K one can use a change of basis given by the eigenvectors of the generalized eigenvalue problem (modal problem), resulting in n independent one-dimensional second order ODEs [16].…”

Section: Introductionmentioning

confidence: 99%

A systematic approach to obtain the analytical solution for coupled linear second-order ordinary differential equations: Part II

Janczkowski Fogaça,

Lenz Cardoso

2024

J Braz. Soc. Mech. Sci. Eng.

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Coupled systems of second order ODEs are found in many areas, such as vibrations and electric circuits. Thus, accurate and efficient solutions are of great interest. This manuscript extends the concept of generalized integrating factor, recently introduced by the authors for one dimensional problems, to coupled n-dimensional systems of second order ODEs. The general formulation for time-varying coefficients is found and particularized to the constant coefficients. Closed-form integrating factors are obtained for the constant coefficient case under mild assumptions about the damping coefficient matrix. Analytical solutions for different types of continuous and discontinuous excitations are presented for problems with constant coefficients and proportional damping, disregarding the damping level. Examples are provided for each type of excitation studied in this manuscript. Results obtained with the proposed approach are compared to traditional numerical methods. Results show that the proposed procedure is accurate, not suffering with interpolation errors found in traditional numerical methods.

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Section: Introductionmentioning

confidence: 99%

A systematic approach to obtain the analytical solution for coupled linear second-order ordinary differential equations: Part II

Janczkowski Fogaça,

Lenz Cardoso

2024

J Braz. Soc. Mech. Sci. Eng.

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“…The reader is referred to any one of the many comprehensive treatises on structural dynamics for a thorough description of the methods listed above, for example [1][2][3]. See also [4][5][6][7][8][9][10][11][12] for further extensions and refinements of these methods.…”

Section: Introductionmentioning

confidence: 99%

Generalization of Duhamel's integral to multi-degree-of-freedom systems

2022

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This article presents a generalization to multi-degree-of-freedom (MDOF) systems of the classical convolution formula for single-degree-of-freedom (SDOF) systems that is widely known as Duhamel's integral. Remarkably, the extended convolution formula for a MDOF system is ultimately cast directly in terms of the system stiffness, mass and damping matrices in a manner that fully parallels the SDOF case, and without resorting to real or complex modal superposition. Still, although the proof based on the use of the Laplace transform might suggest that the obtained formulae are applicable to any type of viscous damping matrix, a more careful evaluation demonstrates that they contain a rather subtle constraint that implies that damping is of the classical type. The free vibration elicited by non-vanishing initial conditions is also considered. As a result, we ultimately obtain the general solution for MDOF systems with arbitrary dynamic loads and initial conditions. For undamped systems, the equivalence of the classical modal analysis with the direct approach proposed herein is demonstrated analytically. Special closed-form solutions, singular solutions and numerical examples are also investigated to a limited extent so as to further illustrate the application of the proposed formulae in the context of practical problems.

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Decoupling of Linear Systems by Phase Synchronization

Junior

2020

Advances in the Theory of System Decoupling

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A canonical form of the equation of motion of linear dynamical systems

Cited by 7 publications

References 15 publications

A systematic approach to obtain the analytical solution for coupled linear second-order ordinary differential equations: Part II

A systematic approach to obtain the analytical solution for coupled linear second-order ordinary differential equations: Part II

Generalization of Duhamel's integral to multi-degree-of-freedom systems

Decoupling of Linear Systems by Phase Synchronization

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