Recent results on stability and response bounds of linear systems - a review.
01 Jan 2006-Vol. 38, Iss: 6, pp 489-496
TL;DR: The literature on linear systems emerging from second order differential equations is extensive because such systems are ubiquitous in modeling, particularly modeling of mechanical systems as mentioned in this paper. But this paper is not a comprehensive overview of the recent research in this field.
Abstract: The literature on linear systems emerging from second order differential equations is extensive because such systems are ubiquitous in modeling, particularly modeling of mechanical systems. This paper offers an overview of some of the recent research in this field, in particular on the subject of stability and response bounds of linear systems. In addition to reporting some interesting recent stability investigations, the basic concepts of stability are reviewed, and a short introduction to Lyapunov's direct method is also presented. Particularly important for applications are response bounds for stable linear systems; therefore a comprehensive section has been devoted to this specific subject.
Citations
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TL;DR: In this article, the authors investigated the influence of gyroscopic effect on stability of the wheelset of a railway vehicle and found that the influence becomes significant with increasing pitch rotor inertia.
Abstract: The wheelset of the railway vehicle is a rotor which itself has gyroscopic effect. Nowadays, the rolling stock has entered the era of high speed, and the wheel rotates faster than in the past. The influence of gyroscopic effect on stability is little understood. Metelitsyn’s inequality theorem for asymptotic stability has some advantages to analyze this problem although this method is sufficient but not necessary condition. Based on its deduction, the extremal eigenvalues criterion and compared with Routh-Hurwitz criterion, both are applied to solve the critical value of speed. Further, according to the instability criterion, gyroscopic contributory ratio is derived to study how the role the gyroscopic effect plays in stability. Moreover, the effect of gyroscopic matrix or gyroscopic terms pitch rotor inertia Iy on stability coefficient is investigated. The results show that Iy is a key factor to wheelset gyroscopic stability. The gyroscopic effect becomes significant, and the stability increases with increasing Iy. The results also indicate that the critical value of speed solved by Metelitsyn theorem is more conservative than the one it solved by Hurwitz criterion, which proves that Metelitsyn inequality theorem for asymptotic stability is a sufficient but not necessary condition in the way of attaining the numerical simulation result. Finally, the test for the influence of gyroscopic effect on stability needs to be further studied.
5 citations
Cites background from "Recent results on stability and res..."
...In view of further development of the nonconservative stability theory of gyroscopic systems, Pommer and Kliem offer an overview of some of the recent researches on the stability and response bounds of linear system [24]....
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TL;DR: In this article, the presence of pure imaginary eigenvalues of the partially damped vibrating systems is treated and the number of such eigen values is determined using the rank of a matrix which is directly related to the system matrices.
Abstract: The presence of pure imaginary eigenvalues of the partially damped vibrating systems is treated. The number of such eigenvalues is determined using the rank of a matrix which is directly related to the system matrices.
5 citations
Cites background from "Recent results on stability and res..."
...A survey of the stability criteria for linear second order systems is given in [2]....
[...]
TL;DR: In this article, linear vibrating systems with symmetric positive definite matrices and damping matrices are studied, in which the inertia and stiffness matrix are symmetric and the damping matrix is symmetric====== positive semi-definite, respectively.
Abstract: In this paper, linear vibrating systems, in which the inertia and stiffness
matrices are symmetric positive definite and the damping matrix is symmetric
positive semi-definite, are studied. Such a system may possess undamped
modes, in which case the system is said to have residual motion. Several
formulae for the number of independent undamped modes, associated with purely
imaginary eigenvalues of the system, are derived. The main results formulated
for symmetric systems are then generalized to asymmetric and symmetrizable
systems. Several examples are used to illustrate the validity and application
of the present results.
3 citations
Cites background from "Recent results on stability and res..."
...A survey of the stability criteria for linear second order systems is given in [3]....
[...]
01 Jan 2012
TL;DR: In this article, a review of some previous work by the author and others on continuous elastic structures subjected to dynamic loads is presented, and bounds on the response are obtained with the use of Liapunov functionals.
Abstract: This chapter contains a review of some previous work by the author and others. Continuous elastic structures subjected to dynamic loads are considered. Results are obtained with the use of Liapunov (Lyapunov) functionals. For cases in which the stability of an equilibrium configuration is of interest, Liapunov’s direct (second) method is applied. For cases in which the structure is in motion due to dynamic loads (e.g., transient loads), bounds on the response are presented.
1 citations
References
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TL;DR: In this article, the authors investigated the influence of gyroscopic effect on stability of the wheelset of a railway vehicle and found that the influence becomes significant with increasing pitch rotor inertia.
Abstract: The wheelset of the railway vehicle is a rotor which itself has gyroscopic effect. Nowadays, the rolling stock has entered the era of high speed, and the wheel rotates faster than in the past. The influence of gyroscopic effect on stability is little understood. Metelitsyn’s inequality theorem for asymptotic stability has some advantages to analyze this problem although this method is sufficient but not necessary condition. Based on its deduction, the extremal eigenvalues criterion and compared with Routh-Hurwitz criterion, both are applied to solve the critical value of speed. Further, according to the instability criterion, gyroscopic contributory ratio is derived to study how the role the gyroscopic effect plays in stability. Moreover, the effect of gyroscopic matrix or gyroscopic terms pitch rotor inertia Iy on stability coefficient is investigated. The results show that Iy is a key factor to wheelset gyroscopic stability. The gyroscopic effect becomes significant, and the stability increases with increasing Iy. The results also indicate that the critical value of speed solved by Metelitsyn theorem is more conservative than the one it solved by Hurwitz criterion, which proves that Metelitsyn inequality theorem for asymptotic stability is a sufficient but not necessary condition in the way of attaining the numerical simulation result. Finally, the test for the influence of gyroscopic effect on stability needs to be further studied.
5 citations
TL;DR: In this article, the presence of pure imaginary eigenvalues of the partially damped vibrating systems is treated and the number of such eigen values is determined using the rank of a matrix which is directly related to the system matrices.
Abstract: The presence of pure imaginary eigenvalues of the partially damped vibrating systems is treated. The number of such eigenvalues is determined using the rank of a matrix which is directly related to the system matrices.
5 citations
TL;DR: In this article, linear vibrating systems with symmetric positive definite matrices and damping matrices are studied, in which the inertia and stiffness matrix are symmetric and the damping matrix is symmetric====== positive semi-definite, respectively.
Abstract: In this paper, linear vibrating systems, in which the inertia and stiffness
matrices are symmetric positive definite and the damping matrix is symmetric
positive semi-definite, are studied. Such a system may possess undamped
modes, in which case the system is said to have residual motion. Several
formulae for the number of independent undamped modes, associated with purely
imaginary eigenvalues of the system, are derived. The main results formulated
for symmetric systems are then generalized to asymmetric and symmetrizable
systems. Several examples are used to illustrate the validity and application
of the present results.
3 citations
01 Jan 2012
TL;DR: In this article, a review of some previous work by the author and others on continuous elastic structures subjected to dynamic loads is presented, and bounds on the response are obtained with the use of Liapunov functionals.
Abstract: This chapter contains a review of some previous work by the author and others. Continuous elastic structures subjected to dynamic loads are considered. Results are obtained with the use of Liapunov (Lyapunov) functionals. For cases in which the stability of an equilibrium configuration is of interest, Liapunov’s direct (second) method is applied. For cases in which the structure is in motion due to dynamic loads (e.g., transient loads), bounds on the response are presented.
1 citations