Lyapunov Inverse Iteration for Computing a Few Rightmost Eigenvalues of Large Generalized Eigenvalue Problems

A distributed Proportional-Retarded (PR) protocol is studied for leaderless coordination of a large scale multi-agent network with undirected links and noisy measurements. The protocol, already developed for SISO systems, introduces a time delay to mimic a derivative action allowing improvement of agents’ transient responses but being insensitive to high-frequency signals. The main contribution of this chapter is the stability analysis of the closed-loop network system of arbitrarily large scale, in terms of PR protocol parameters. Next, this is connected to our recent results in [28] from which we summarize how PR protocol can be tuned for the network system to achieve certain performance as dictated by its rightmost eigenvalues. Over a case study, the results are demonstrated.

Bacterial cells exhibit a diverse array of shapes and sizes, largely influenced by their cell walls in conjunction with cytoskeletal proteins and internal turgor pressure. The present study develops a theoretical framework for modeling the shape dynamics of actively expanding bacterial cell walls, grounded in the concept of minimal energy dissipation. In the context of a bacterial cell wall, dissipative forces are generated through the insertion of peptidoglycan (PG) strands, while driving forces stem from alterations in mechanochemical energy, crucial for sustaining the cell wall's shape. The interplay between mechanical and chemical energies facilitates the evaluation of the free energy landscape and helps in predicting the homeostasis of the bacterial cell size. The size limit derived through linear stability analysis (LSA) of a model system accurately mirrors the phase diagram produced by the theoretical model. Nonetheless, given the cell wall's intricate molecular architecture, a more detailed constitutive model is expected to provide more precise quantitative insights.

Stability analysis of a degenerate hyperbolic system modelling a heat exchanger

The dynamic analysis of a structural system with different material and geometric properties leads to linear eigenvalue problem such as generalized (or standard) eigenvalue problem. In general, the material and geometric properties are assumed to be in the form of crisp (or exact). However, due to several errors and insufficient or incomplete information of data, the uncertainties are assumed to be present in the material and geometric properties. These uncertain material and geometric properties may be modeled through convex normalized fuzzy sets. In standard fuzzy arithmetic, all the operands are assumed to be independent of each other. But when they are partially or completely dependent on each other, the standard fuzzy arithmetic results in a wider range. This situation is known as the “dependency problem” or “overestimation problem”. In this regard, a fuzzy-affine approach is developed to overcome the dependency problem. This proposed approach may improve the outer enclosures and give tighter bounds to the fuzzy solution set. This chapter deals with the dynamic analysis of various structural systems viz. multi degrees-of-freedom spring-mass structural system, multi-storey frame structural system, etc. by adopting the proposed approach. Several numerical examples related to the dynamic analysis of these structural systems have been worked out to illustrate the reliability and efficiency of the present approach.KeywordsAffine arithmeticFuzzy-affine arithmeticDynamic analysis of structural systemFuzzy generalized eigenvalue problemFuzzy standard eigenvalue problem

The dynamics of cylinder in a confined swirling flow with constant vorticity

On the effects of a beam-like piezoelectric passive controller on the linear stability of the visco-elastic Beck’s beam

Friction-induced vibration due to mode coupling is a major concern in a wide variety of mechanical systems. Though there are numerous papers on both linear and non-linear stability analyses of dynamical systems with friction-induced vibrations, the effects of damping on the evolution of the stable–unstable regions is not yet fully understood. The purpose of this work is to study a pin-disc model describing a typical friction-induced vibration due to a geometric constraint in order to illustrate the effects of damping on linear stability analyses. The results not only illustrate the influence of damping for determining the stable and unstable zones, but they also indicate that the “most efficient damping factor of the coupling modes” is an important factor to be taken into account. The physical mechanism causing this change in instability as a consequence of the damping factor is also suggested to help avoid bad design.

An adaptive BDDC method enhanced with prior selected primal constraints

Introduction: This article is the result of the research “Analysis of the dynamic systems component of the engineering programs in control and automation and technology in industrial electronics,” developed at the Francisco José de Caldas District University in the years 2022-2023. Problem: The document addresses the lack of a clear and comprehensive structure in the analysis of linear dynamic systems, as well as the absence of an innovative approach to addressing these topics in traditional courses, which hinders their understanding and application in different disciplines. Objective: The objective is to present a novel approach to analyzing linear dynamic systems, providing a complete framework of concepts, relationships, and important tools in this analysis, as well as a literature review based on over seventeen years of experience in courses and studies of this kind. Methodology: A journey of concepts is proposed from signals and systems, through modeling with methods such as black box, white box, gray box, to linear analysis based on examples with calculation of differential equations, state representation, transfer function, and block diagrams. Results: The document aggregates and articulates all the concepts of dynamic systems, along with the relationships and tools used, offering a more practical and intuitive approach to understanding the material. Conclusion: The document provides a comprehensive and articulated view of key concepts in the analysis of linear dynamic systems, highlighting an innovative approach that facilitates their understanding. Its main contribution is to aggregate and articulate these concepts, along with the tools and relationships used, to offer a more practical and clear approach for their study. Originality: The originality lies in proposing a novel and structured approach to analyzing linear dynamic systems, addressing the lack of clarity and completeness in traditional approaches. Limitations: Although the document proposes a novel approach, it does not delve into specific aspects of some topics covered, which could limit the detailed understanding of certain concepts. Deepening: The document could delve into the practical application of the proposed concepts and tools in real cases of dynamic systems, as well as into the comparison with traditional approaches to highlight the differences and advantages of the proposed new approach.

This study presents a novel reduced-order method to find critical eigenvalues of ultra large-scale power system. First, the numerical solution of matrix exponential is computed by precise time-step integration. Secondly, based on the numerical solution, the numerical curve of the trace of matrix exponential is formed. Thirdly, the candidates of critical eigenvalues are extracted from the numerical curve of the trace by Prony analysis, and finally, a set of weight coefficients is calculated to confirm critical eigenvalues from candidate eigenvalues. Since the trace contains all eigenvalues, no critical eigenvalue can be lost in analysing the numerical curve of the trace. In the later period of time integration, the effect of all non-critical eigenvalues to the trace is decayed and oppositely the effect of all critical eigenvalues is amplified. Thus, several rightmost eigenvalues can only be extracted from the numerical curve of the trace. Case studies for 16 and 9004 order system have validated that the proposed method can be used to find all critical eigenvalues of ultra large-scale power system.

A parallel mixed finite element implementation for approximation of eigenvalues and eigenvectors of fourth-order eigenvalue problems

A hybrid spectral nodal method for multigroup X,Y-geometry discrete ordinates eigenvalue problems

We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems: First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is surprising since their counterparts on nondirected graphs are unstable when system size is large enough. Second, we show that the phase transition between the stable and unstable phase is universal in the sense that it depends only on a few parameters, such as, the mean degree and a degree correlation coefficient. In addition, in the unstable regime we characterize the nature of the destabilizing mode, which also exhibits universal features. These results follow from an exact theory for the leading eigenvalue of infinitely large graphs that are locally tree-like and oriented, as well as, for the right and left eigenvectors associated with the leading eigenvalue. We corroborate analytical results for infinitely large graphs with numerical experiments on random graphs of finite size. We discuss how the presented theory can be extended to graphs with diagonal disorder and to graphs that contain nondirected links. Finally, we discuss the influence of small cycles and how they can destabilize large dynamical systems when they induce strong enough feedback loops.

In this paper, the dynamic stability analysis of an elastic rod subjected to conservative and nonconservative forces by using the integral equation method is presented. The stability analysis of multiple-parameter elastic system is taken consideration with two types damping terms, which are the internal and the exteral damping effects. The numerical method derived from the boundary integral equation in the sub-domain, which was proposed for dynamic stability analysis of oneparameter system subjected to nonconservative force, is applied to the analysis of multiple-parameter systems. The models in numerical examples are cantilever-type elastic rods under the combined action of these forces. The obtained numerical results proved the effectiveness of our method and showed mechanical characteristics for these multiple-parameter systems.

A quadratically converging algorithm for the computation of the distance to instability of a broad class of nonlinear eigenvalue problems is presented, including the polynomial eigenvalue problem and the delay eigenvalue problem. The algorithm is grounded in a recently presented approach for computing the pseudospectral abscissa. The application of the algorithm only relies on the availability of a method to compute the rightmost eigenvalue of perturbed problems obtained by adding rank one perturbations to the coefficient matrices, for which, in case of large and sparse matrices, efficient iterative algorithms can be used.