An algorithm for higher order Hopf normal forms

Normal form theory is important for studying the qualitative behavior of nonlinear oscillators. In some cases, higher order normal forms are required to understand the dynamic behavior near an equilibrium or a periodic orbit. However, the computation of high-order normal forms is usually quite complicated. This article provides an explicit formula for the normalization of nonlinear differential equations. The higher order normal form is given explicitly. Illustrative examples include a cubic system, a quadratic system and a Duffing–Van der Pol system. We use exact arithmetic and find that the undamped Duffing equation can be represented by an exact polynomial differential amplitude equation in a finite number of terms.published_or_final_versio

Three-dimensional mixed mode analysis of a cracked body by fractal finite element method

A semi-analytical method namely fractal finite element method is presented for the determination of stress intensity factor for the straight three-dimenisonal plane crack. Using the concept of fractal geometry, infinite many of finite elements are generated virtually around the crack border. Based on the analytical global displacement function, numerous DOFs are transformed to a small set of generalised coordinates in an expeditious way. No post-processing and special finite elements are required to develop for extracting the stress intensity factors. Examples are given to illustrate the accuracy and efficiency of the present method. Very good accuracy (with less than 3% errors) is obtained for the maximum value of SIFs for different modes.postprin

Wind flow in the recessed cavities of a tall building

Fulltext in: http://www.iawe.org/Proceedings/7APCWE/M3D_3.pdfTechnical Session: M3-D Computational Wind Engineering (2), no.3In a congested city like Hong Kong, residential tall buildings are often built with an irregular plan form and with a number of apartments arranged as wing sections extending from a central core. To provide views and sufficient ventilation to the apartments, deeply recessed cavities are placed between adjacent building wings. This paper reports a CFD study of the wind-induced flow inside a recessed cavity of a tall building with an objective to assess the adequacy of ventilation inside the cavity. The dimensions of the cavity are varied systematically to investigate the flow exchange between the cavity and the outside at different heights. It is found that the flow inside the cavity is not a simple cross flow or a stagnation flow. Flow exchange takes place in different directions along the building heights.link_to_subscribed_fulltex

A new reduced control design based on the theory of wave domaincontrol

The work in this paper is based on the theory discussed previously by Wang et al. (1994). The main idea is to establish a transformation, which changes the original system into an image system, in which the control force is designed in the context of wave domain control and wave control, so that the number of degrees of freedom in the undisturbed state of the image system can be reduced. The design of control in the original system can be derived by inverse transformation. This method, compared with the previous one, is more general and is easy to apply.published_or_final_versio

Robust stabilization of singular-impulsive-delayed systems with nonlinear perturbations

Many dynamic systems in physics, chemistry, biology, engineering, and information science have impulsive dynamical behaviors due to abrupt jumps at certain instants during the dynamical process, and these complex dynamic behaviors can be modeled by singular impulsive differential systems. This paper formulates and studies a model for singular impulsive delayed systems with uncertainty from nonlinear perturbations. Several fundamental issues such as global exponential robust stabilization of such systems are established. A simple approach to the design of a robust impulsive controller is then presented. A numerical example is given for illustration of the theoretical results. Meanwhile, some new results and refined properties associated with the M-matrices and time-delay dynamic systems are derived and discussed.published_or_final_versio

Body-force linear elastic stress intensity factor calculation using fractal two level finite element method

Fractal two level finite element method (F2LFEM) for the analysis of linear fracture problems subjected to body force loading is presented. The main objective here is to show that by employing the F2LFEM a highly accurate and an efficient linear analysis of fracture bodies subjected to internal loading can be obtained as it is hard to find any analytical and exact values of stress intensity factor (SIF) for any kind of geometry subjected to internal loading. Also in this paper, a fast method to transform the body force to the reduced force vector is presented and has been effectively employed. The problems solved here include both the single mode or mixed mode cracks subjected to internal body-force or external loading. In comparison with other numerical algorithms, it seems that with a small amount of computational time and computer storage, highly accurate results can be obtained. © 1995.link_to_subscribed_fulltex

Fractal two-level finite element analysis of cracked Reissner's plate

A cracked thick plate subjected to edge moment and transverse loading was customarily analysed either by a fine finite element mesh or by singular elements. In this paper an alternative method is recommended in which conventional finite elements with infinitesimal mesh are used and the number of unknowns is reduced by interpolating the nodal displacements by means of the global interpolating function around the singular region. The global interpolating function is derived by using eigenfunction technique based on Reissner's transverse shear plate theory. The crack parameters such as stress intensity factor and moment intensity factor can be evaluated directly from the coefficients of the global interpolating function. New elements need not to be generated and integration is avoided completely. Accurate results with error less than 0-5% are achieved with little computational efforts. Examples on edge cracked plate and central cracked plate subjected to both edge moment and transverse loading are considered.link_to_subscribed_fulltex

Fractal two-level finite-element method for two-dimensional cracks

Fractal two-level finite-element method has been extended to mode II crack problems. In order to encourage the me of this versatile tool on crack problems, some user guide-lines for evaluating the stress intensity factor and for improving the convergence of stress intensity factors are suggested for structural engineers. The study is focused on (7) the effect of the numerical integration scheme on the convergency of stress intensity factors, (2) the configuration of fractal meshes, and (3) the influence of the aspect ratio of the fractal mesh on the accuracy of stress intensity factors. Both mode I and mode II two-dimensional crack problems are discussed. Examples for both the lagrangian and the serendipity types of elements are used to demonstrate the numerical efficiency and accuracy of the present method. In summary, it is not difficult to achieve accuracy within 3% error on stress intensity factors with a personal computer in a few seconds of execution time. © 1996 Microcomputers in Civil Engineering. Published by Blackwell Publishers.link_to_subscribed_fulltex

Analytical formulation of dynamic stiffness

The dynamic stiffness matrix method enables one to model an infinite number of natural modes by means of a small number of unknowns. The method has been extended to skeletal structures with uniform or non-uniform, straight or curved, damped or undamped beam members. For two-dimensional structures, if one of the dimensions can be eliminated by means of the Kantorovich method, the method still applies. However, for more complicated systems, analytical formulation of the dynamic stiffness is tedious. A computer assisted analytical method is introduced here for any structural members the differential governing equations of which are expressible in matrix polynomial form. Complex arithmetics are used to cater for all possible classification of the characteristic roots. Numerical examples are given and are compared with existing results

Determination of Jordan chains by extended matrices

The major obstacle to determination of the Jordan chains for a highly degenerated eigenproblem is that the triangular combinations of the principal vectors in a Jordan chain are also principal vectors and the linear combinations of the eigenvectors of all Jordan blocks associated with the same eigenvalue are also eigenvectors. These indeterminate constants will hide the Jordan block structure and make the analysis very difficult. We propose an extended matrix method to find the Jordan chains and eliminate the indeterminate constants so that the Jordan block structure can be computed sequentially. An example with the Segre characteristic [(321)11] is given. © 1998 John Wiley & Sons, Ltd.link_to_subscribed_fulltex