scholarly journals A Novel Method for Accelerating the Analysis of Nonlinear Behaviour of Power Grids using Normal Form Technique

2019 ◽  
Author(s):  
Nnaemeka Sunday Ugwuanyi ◽  
Xavier Kestelyn ◽  
Olivier Thomas ◽  
Bogdan Marinescu
Keyword(s):  
Normal Form ◽  
Power Grids ◽  
Nonlinear Behaviour ◽  
Novel Method ◽  
Normal Form Technique

The Selection of the Linearized Natural Frequency for the Second-Order Normal Form Method

2011 ◽  
Author(s):  
Zhenfang Xin ◽  
S. A. Neild ◽  
D. J. Wagg
Keyword(s):  
Normal Form ◽  
Dynamic Response ◽  
Natural Frequency ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Second Order ◽  
Nonlinear Dynamic Response ◽  
Weakly Nonlinear ◽  
Selection Of ◽  
Normal Form Technique

The normal form technique is an established method for analysing weakly nonlinear vibrating systems. It involves applying a simplifying nonlinear transform to the first-order representation of the equations of motion. In this paper we consider the normal form technique applied to a forced nonlinear system with the equations of motion expressed in second-order form. Specifically we consider the selection of the linearised natural frequencies on the accuracy of the normal form prediction of sub- and superharmonic responses. Using the second-order formulation offers specific advantages in terms of modeling lightly damped nonlinear dynamic response. In the second-order version of the normal form, one of the approximations made during the process is to assume that the linear natural frequency for each mode may be replaced with the response frequencies. Here we will show that this step, far from reducing the accuracy of the technique, does not affect the accuracy of the predicted response at the forcing frequency and actually improves the predictions of sub and superharmonic responses. To gain insight into why this is the case, we consider the Duffing oscillator. The results show that the second-order approach gives an intuitive model of the nonlinear dynamic response which can be applied to engineering applications with weakly nonlinear characteristics.

scholarly journals Identifying critical higher-order interactions in complex networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Mehmet Emin Aktas ◽  
Thu Nguyen ◽  
Sidra Jawaid ◽  
Rakin Riza ◽  
Esra Akbas
Keyword(s):  
Complex Networks ◽  
Higher Order ◽  
Power Grids ◽  
Giant Component ◽  
Centrality Measures ◽  
First Order ◽  
Pairwise Interactions ◽  
Size Of Giant Component ◽  
Novel Method ◽  
Structure And Functions

AbstractDiffusion on networks is an important concept in network science observed in many situations such as information spreading and rumor controlling in social networks, disease contagion between individuals, and cascading failures in power grids. The critical interactions in networks play critical roles in diffusion and primarily affect network structure and functions. While interactions can occur between two nodes as pairwise interactions, i.e., edges, they can also occur between three or more nodes, which are described as higher-order interactions. This report presents a novel method to identify critical higher-order interactions in complex networks. We propose two new Laplacians to generalize standard graph centrality measures for higher-order interactions. We then compare the performances of the generalized centrality measures using the size of giant component and the Susceptible-Infected-Recovered (SIR) simulation model to show the effectiveness of using higher-order interactions. We further compare them with the first-order interactions (i.e., edges). Experimental results suggest that higher-order interactions play more critical roles than edges based on both the size of giant component and SIR, and the proposed methods are promising in identifying critical higher-order interactions.

scholarly journals Normal Form for High-Dimensional Nonlinear System and Its Application to a Viscoelastic Moving Belt

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
S. P. Chen ◽  
Y. H. Qian
Keyword(s):  
Normal Form ◽  
Nonlinear Dynamical Systems ◽  
High Dimensional ◽  
Computation Method ◽  
Double Zero ◽  
Nonlinear Dynamical ◽  
Novel Method ◽  
Explicit Formulae ◽  
The Stability ◽  
Pure Imaginary Eigenvalues

This paper is concerned with the computation of the normal form and its application to a viscoelastic moving belt. First, a new computation method is proposed for significantly refining the normal forms for high-dimensional nonlinear systems. The improved method is described in detail by analyzing the four-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain three different cases, that are, (i) two pairs of pure imaginary eigenvalues, (ii) one nonsemisimple double zero and a pair of pure imaginary eigenvalues, and (iii) two nonsemisimple double zero eigenvalues. Then, three explicit formulae are derived, herein, which can be used to compute the coefficients of the normal form and the associated nonlinear transformation. Finally, employing the present method, we study the nonlinear oscillation of the viscoelastic moving belt under parametric excitations. The stability and bifurcation of the nonlinear vibration system are studied. Through the illustrative example, the feasibility and merit of this novel method are also demonstrated and discussed.

scholarly journals Structural optimization for nonlinear dynamic response

2015 ◽  
Vol 373 (2051) ◽  
pp. 20140408 ◽  
Author(s):  
Suguang Dou ◽  
B. Scott Strachan ◽  
Steven W. Shaw ◽  
Jakob S. Jensen
Keyword(s):  
Normal Form ◽  
Resonance Condition ◽  
Nonlinear Resonance ◽  
Computational Method ◽  
Frame Structure ◽  
Nonlinear Behaviour ◽  
Coupled Mode ◽  
Resonant Modes ◽  
Order Of Magnitude ◽  
Micro Systems

Much is known about the nonlinear resonant response of mechanical systems, but methods for the systematic design of structures that optimize aspects of these responses have received little attention. Progress in this area is particularly important in the area of micro-systems, where nonlinear resonant behaviour is being used for a variety of applications in sensing and signal conditioning. In this work, we describe a computational method that provides a systematic means for manipulating and optimizing features of nonlinear resonant responses of mechanical structures that are described by a single vibrating mode, or by a pair of internally resonant modes. The approach combines techniques from nonlinear dynamics, computational mechanics and optimization, and it allows one to relate the geometric and material properties of structural elements to terms in the normal form for a given resonance condition, thereby providing a means for tailoring its nonlinear response. The method is applied to the fundamental nonlinear resonance of a clamped–clamped beam and to the coupled mode response of a frame structure, and the results show that one can modify essential normal form coefficients by an order of magnitude by relatively simple changes in the shape of these elements. We expect the proposed approach, and its extensions, to be useful for the design of systems used for fundamental studies of nonlinear behaviour as well as for the development of commercial devices that exploit nonlinear behaviour.

scholarly journals Weighted growth functions of automatic groups

2017 ◽  
Author(s):  
Mikael Vejdemo-Johansson
Keyword(s):  
Normal Form ◽  
Growth Function ◽  
Braid Groups ◽  
Group Element ◽  
Systems Of Equations ◽  
Graphs Of Groups ◽  
Growth Functions ◽  
Automatic Groups ◽  
Novel Method ◽  
Context Free

The growth function is the generating function for sizes of spheres around the identity in Cayley graphs of groups. We present a novel method to calculate growth functions for automatic groups with normal form recognizing automata that recognize a single normal form for each group element, and are at most context free in complexity: context free grammars can be translated into algebraic systems of equations, whose solutions represent generating functions of their corresponding non-terminal symbols. This approach allows us to seamlessly introduce weightings on the growth function: assign different or even distinct weights to each of the generators in an underlying presentation, such that this weighting is reflected in the growth function. We recover known growth functions for small braid groups, and calculate growth functions that weight each generator in an automatic presentation of the braid groups according to their lengths in braid generators.

scholarly journals Weighted growth functions of automatic groups

2017 ◽  
Author(s):  
Mikael Vejdemo-Johansson
Keyword(s):  
Normal Form ◽  
Growth Function ◽  
Braid Groups ◽  
Group Element ◽  
Systems Of Equations ◽  
Graphs Of Groups ◽  
Growth Functions ◽  
Automatic Groups ◽  
Novel Method ◽  
Context Free

The growth function is the generating function for sizes of spheres around the identity in Cayley graphs of groups. We present a novel method to calculate growth functions for automatic groups with normal form recognizing automata that recognize a single normal form for each group element, and are at most context free in complexity: context free grammars can be translated into algebraic systems of equations, whose solutions represent generating functions of their corresponding non-terminal symbols. This approach allows us to seamlessly introduce weightings on the growth function: assign different or even distinct weights to each of the generators in an underlying presentation, such that this weighting is reflected in the growth function. We recover known growth functions for small braid groups, and calculate growth functions that weight each generator in an automatic presentation of the braid groups according to their lengths in braid generators.

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