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THE general theorems given in Sections 4 and 6 include, from the fundamental point of view, all that is required for the analysis of redundant structures. However, to facilitate practical calculations it is helpful to develop more explicit methods and formulae. To find these is the purpose of this Section.

The purpose of this paper is to present a nonlinear model along with stability analysis of a flexible supersonic flight vehicle system.

The mathematical state space nonlinear model of the system is derived using Lagrangian approach such that the applied force, moment, and generalized force are all assumed to be nonlinear functions of the system states. The condition under which the system would be unstable is derived and when the system is stable, the region of attraction of the system equilibrium state is determined using the Lyapunov theory and sum of squares optimization method. The method is applied to a slender flexible body vehicle, which is referenced by the other researchers in the literature.

It is demonstrated that neglecting the nonlinearity in external force, moment and generalized force, as it was assumed by other researchers, can cause significant variations in stability conditions. Moreover, when the system is stable, it is shown analytically here that a reduction in dynamic pressure can make a larger region of attraction, and thus instability will occur in a larger angle of attack, greater angular velocity and elastic displacement.

In order to carefully study the behavior of aeroelastic flight vehicle, a nonlinear model and analysis is definitely necessary. Moreover, for the design of the airframe and/or control purposes, it is essential to investigate region of attraction of equilibrium state of the stable flight vehicle.

Current stability analysis methods for nonlinear elastic flight vehicles are unable to determine the state space region where the system is stable. Nonlinear modeling affects the determination of the stability region and instability condition. This paper presents a new approach to stability analysis of the nonlinear flexible flight vehicle. By determining the region of attraction when the system is stable, it is demonstrated analytically, in this research, that decreasing the dynamic pressure can produce larger region of attraction.

The analysis of plate systems (Figure 1) with mixed finite elements is limited to geometric smooth partial systems (Figure 1a) with continuous strains and stresses. This is due to the independent approximation of the parameters of the local generalized displacements (displacements, slopes) and generalized stresses (stress resultants at the cross‐section) in the nodes of the partial systems (Figure 1c). Owing to the simultaneous approximation, an additional static constraint of the generalized stresses, apart from the kinematic constraint of the displacements which is usual in the finite element methods, is needed at the assembly, so that discontinuities of the stresses resulting from the equilibrium conditions cannot be determined.

The analysis of the wing/fuselage and fuselage/tail unit interaction forces is extended to cover the case when the attached component is more conveniently analysed by the Matrix Displacement Method. The flexibility matrix of the complete aircraft, supported on the wing/fuselage attachment points, follows from the results derived in this and previous sections and takes into account the elastic interaction between the various components. The dynamical matrix of the complete free aircraft is set up and for completeness the theory and properties of the normal modes of vibration are given. A final sub‐section discusses some points of detail in the mass distribution and the definition of the forces on the aircraft.

HAVING discussed in the standard longhand notation the main ideas and methods for the calculation of redundant structures on the basis of forces as unknowns we now turn our attention to the matrix formulation of the analysis. Consider a system consisting of s structural elements with a total number n of redundancies which may be forces (stresses), moments or any generalized forces. We select a basic system by ‘cutting’ a number r of redundancies where r<n. Thus, the simple idea of a statically determinate basic system (r=n) is but a particular case of our investigations.

The purpose of this paper is to derive linear modal equations describing the forced liquid sloshing in a rigid truncated (tapered) conical tank, as well as to show how to couple these modal equations with “global” dynamic equations of a complex mechanical system carrying this tank.

Derivation of the modal equations can be based on the Trefftz variational method developed by the authors in a previous paper. Describing the coupled dynamics utilizes Lukovsky' formulas for the resulting hydrodynamic force and moment due to liquid sloshing.

The so‐called Stokes‐Joukowski potentials can be found by using the Trefftz method from the authors' previous paper with the same polynomial‐type functional basis. Coupling the modal equations with the global dynamic equations becomes a relatively simple task facilitated by Lukovsky's formulas. Using the linear multimodal method can be an efficient alternative to traditional numerical and analytical tools employed for studying the coupled vibrations of a tower with a conical rigid tank on the tower top.

The derived modal equations are equipped by tables with the computed non‐dimensional hydrodynamic coefficients. Interested readers (engineers) can incorporate the modal equations into the global dynamic equations of a whole mechanical system without new computations of these coefficients.

The multimodal method can be an alternative to traditional numerical tools. Using the derived modal equations simplifies analytical studies and provides efficient calculations of the coupled dynamics of a mechanical system carrying a rigid tapered conical tank with a liquid.

The purpose of this paper is to find an accurate, efficient and easy-to-implement point estimate method (PEM) for the statistical moments of random systems.

First, by the theoretical and numerical analysis, the approximate reference variables for the frequently used nine types of random variables are obtained; then by combining with the dimension-reduction method (DRM), a new method which consists of four sub-methods is proposed; and finally, several examples are investigated to verify the characteristics of the proposed method.

Two types of reference variables for the frequently used nine types of variables are proposed, and four sub-methods for estimating the moments of responses are presented by combining with the univariate and bivariate DRM.

In this paper, the number of nodes of one-dimensional integrals is determined subjectively and empirically; therefore, determining the number of nodes rationally is still a challenge.

Through the linear transformation, the optimal reference variables of random variables are presented, and a PEM based on the linear transformation is proposed which is efficient and easy to implement. By the numerical method, the quasi-optimal reference variables are given, which is the basis of the proposed PEM based on the quasi-optimal reference variables, together with high efficiency and ease of implementation.

A DSIR Sponsored Research Programme on the Development and Application of the Matrix Force Method and the Digital Computer. This work presents a rational method for the structural analysis of stressed skin fuselages for application in conjunction with the digital computer. The theory is a development of the matrix force method which permits a close integration of the analysis and the programming for a computer operating with a matrix interpretive scheme. The structural geometry covered by the analysis is sufficiently arbitrary to include most cases encountered in practice, and allows for non‐conical taper, double‐cell cross‐sections and doubly connected rings. An attempt has been made to produce a highly standardized procedure requiring as input information only the simplest geometrical and elastic data. An essential feature is the use of the elimination and modification technique subsequent to the main analysis of the regularized structure in which all cutouts have been filled in. Current Summary A critical historical appraisal of previous work in the Western World on fuselage analysis is given in the present issue together with an outline of the ideas underlying the new theory.

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.

This paper presents a novel framework for solving elliptic partial differential equations (PDEs) over irregularly spaced meshes on bounded domains.

Second‐generation wavelet construction gives rise to a powerful generalization of the traditional hierarchical basis (HB) finite element method (FEM). A framework based on piecewise polynomial Lagrangian multiwavelets is used to generate customized multiresolution bases that have not only HB properties but also additional qualities.

For the 1D Poisson problem, we propose – for any given order of approximation – a compact closed‐form wavelet basis that block‐diagonalizes the stiffness matrix. With this wavelet choice, all coupling between the coarse scale and detail scales in the matrix is eliminated. In contrast, traditional higher‐order (n>1) HB do not exhibit this property. We also achieve full scale‐decoupling for the 2D Poisson problem on an irregular mesh. No traditional HB has this quality in 2D.

Similar techniques may be applied to scale‐decouple the multiresolution finite element (FE) matrices associated with more general elliptic PDEs.

By decoupling scales in the FE matrix, the wavelet formulation lends itself particularly well to adaptive refinement schemes.

The paper explains second‐generation wavelet construction in a Lagrangian FE context. For 1D higher‐order and 2D first‐order bases, we propose a particular choice of wavelet, customized to the Poisson problem. The approach generalizes to other elliptic PDE problems.