Two-sided bounds for the asymptotic behaviour of free nonlinear vibration systems with application of the differential calculus of norms

Kohaupt, L.

doi:10.1080/00207160802166481

Cited by 6 publications

(13 citation statements)

References 22 publications

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“…This is the key idea of this paper: bounds that solely depend on the solution structure or bounds that incorporate the approximation error. Firstly, we were able to generalize results from the linear time-invariant [17,18] to time-periodic setting and derive a time-varying norm that captures important properties such as decoupling, filtering and monotonicity. Secondly, we used two different methodologies where the approximation error is incorporated in the upper bound.…”

Section: Discussionmentioning

confidence: 99%

“…where (P m A) denotes the component-wise Chebyshev projection of A, see (17). If (P m A)(t 1 ) commutes with (P m A)(t 2 ) for all times t 1 and t 2 , then the solution to the approximated system (18) is given by y(t) = exp t 0 (P m A)(τ )dτ x 0 .…”

Section: Spectral Methods and Spectral Boundmentioning

confidence: 99%

“…Their solution is defined via the matrix exponential and their numerical evaluation can be done by methods to solve ordinary differential equations, e.g., by Runge-Kutta methods [12], or the computation of the matrix exponential [13,21]. Two-sided bounds for the solution of linear time-invariant systems have been investigated in a series of papers [17,18]. A time-varying system in general does not possess a closed form solution (unless the system matrix commutes for any two times).…”

Section: Introductionmentioning

confidence: 99%

“…A time-varying system in general does not possess a closed form solution (unless the system matrix commutes for any two times). Hence, the theory derived in [17,18] cannot easily be extended to a general linear time-varying system with an infinite time horizon. In this paper, we investigate linear time-periodic systems and generalize the theory of bounds to the solution of linear time-periodic systems while using its solution structure defined by Floquet's theory [8].…”

Section: Introductionmentioning

confidence: 99%

“…3.1 we summarize results for linear time-invariant systems [18,19]. Two-sided bounds on the solution with the differential calculus of norms, e.g., in [15][16][17], are shown. In Sect.…”

Section: Introductionmentioning

confidence: 99%

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Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

Benner

Denißen

Kohaupt

2016

J. Appl. Math. Comput.

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Linear time-periodic systems arise whenever a nonlinear system is linearized about a periodic trajectory. Examples include anisotropic rotor-bearing systems and parametrically excited systems. The structure of the solution to linear time-periodic systems is known due to Floquet's Theorem. We use this information to derive a new norm which yields two-sided bounds on the solution and in this norm vibrations of the solution are suppressed. The obtained results are a generalization for linear time-invariant systems. Since Floquet's Theorem is non-constructive, the applicability of the aforementioned results suffers in general from an unknown Floquet normal form. Hence, we discuss trigonometric splines and spectral methods that are both equipped with rigorous bounds on the solution. The methodology differs systematically for the two methods. While in the first method the solution is approximated by trigonometric splines and the upper bound depends on the approximation quality, in the second method the linear time-periodic system is approximated and its solution is represented as an infinite series. Depending on the smoothness of the time-periodic system, we formulate two upper bounds which incorporate the approximation error of the linear time-periodic system and the truncation error of the series representation. Rigorous bounds on the solution are necessary whenever reliable results are needed, and hence they can support the analysis and, e.g., stability or robustness of the solution may be proven or falsified. The theoretical results are illustrated and compared to trigonometric spline bounds and spectral bounds by means of three examples that include an anisotropic rotor-bearing system and a parametrically excited Cantilever beam.

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Section: Discussionmentioning

confidence: 99%

Section: Spectral Methods and Spectral Boundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…3.1 we summarize results for linear time-invariant systems [18,19]. Two-sided bounds on the solution with the differential calculus of norms, e.g., in [15][16][17], are shown. In Sect.…”

Section: Introductionmentioning

confidence: 99%

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Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

Benner

Denißen

Kohaupt

2016

J. Appl. Math. Comput.

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On the vibration-suppression property and monotonicity behavior of a special weighted norm for dynamical systems

Kohaupt

2013

Applied Mathematics and Computation

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Solution of the matrix eigenvalue problem VA+A*V=μV with applications to the study of free linear dynamical systems

Kohaupt

2008

Journal of Computational and Applied Mathematics

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The new idea is to study the stability behavior of the solution x = x(t) of the initial value problemẋ = Ax, t t 0 , x(t 0 ) = x 0 , with A ∈ C n×n , in a weighted (semi-) norm · R where R is taken as an appropriate solution of the matrix eigenvalue problem RA + A * R = R, rather than as the solution of the algebraic Lyapunov matrix equation RA + A * R = −S with given positive (semi-) definite matrix S. Substantially better results are obtained by the new method. For example, if A is diagonalizable and all eigenvalues, which is much more than the old result, which only states that lim t→∞ x(t) = 0. Especially, the semi-norms · R i have a decoupling and filter effect on x(t). Further, new two-sided bounds (depending on x 0 ) for the asymptotic behavior can be derived. The best constants in the bounds are obtained by the differential calculus of norms. Applications are made to free linear dynamical systems, and computations underpin the theoretical findings. : Matrix eigenvalue problem VA + A * V = V ; Algebraic Lyapunov equation RA + A * R = −S; Linear dynamical system; Decoupling and filter effect of weighted semi-norms; Stability; Two-sided bounds depending on the initial conditions; Differential calculus of norms

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Two-sided bounds for the asymptotic behaviour of free nonlinear vibration systems with application of the differential calculus of norms

Cited by 6 publications

References 22 publications

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

On the vibration-suppression property and monotonicity behavior of a special weighted norm for dynamical systems

Solution of the matrix eigenvalue problem VA+A*V=μV with applications to the study of free linear dynamical systems

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