A Derivation Procedure for the Dynamic Flexibility Matrix of a Triangular Bending Element

Application of the dynamic condensation approach to the hybrid FE-SEA model of mid-frequency vibration in complex built-up systems

The dynamic stiffness matrix of a gradient elastic flexural Bernoulli–Euler beam finite element is analytically constructed with the aid of the basic and governing equations of motion in the frequency domain. The simple gradient theory of elasticity is used with just one material constant (internal length) in addition to the classical moduli. The flexural element has one node at every end with three degrees of freedom per node, i.e., the displacement, the slope, and the curvature. Use of this dynamic stiffness matrix for a plane system of beams enables one by a finite element analysis to determine its dynamic response harmonically varying with time external load or the natural frequencies and modal shapes of that system. The response to transient loading is obtained with the aid of Laplace transform with respect to time. A stiffness matrix is constructed in the transformed domain, the problem is formulated and solved by the finite element method, and the time domain response is finally obtained by a time domain inversion of the transform solution. Because the exact solution of the governing equation of motion in the frequency domain is used as the displacement function, the resulting dynamic stiffness matrices and the obtained structural response or natural frequencies and modal shapes are also exact. Examples are presented to illustrate the method and demonstrate its advantages. The effects of the microstructure on the dynamic behavior of beam structures are also determined.

Dynamic stiffness analysis of toroidal shells

A numerical strategy for predicting the coupling-strength of acoustic fluid-structure interaction

Summary A problem of elastic structures stress-strain field determination is considered in this article. A theoretical background and computation algorithms of equilibrium finite element method are presented there. The different external effects are estimated, namely: load, prestressing, inicial strains and support settlements. The dual relationships (equilibrium and geometrical equations, stiffness and flexibility equations) of equilibrium element, also the expressions of stiffness and flexibility matrices are given. These relationships describe the stress-strain field of the finite element and allow to transforme the flexibility matrix to the stiffness matrix and on the contrary. There are presented direct and variational formulations of the problem. The algorithms of the forces method and displacements method are made for their solution. Easier is the algorithm of displacements method, because making equations of the forces method needs to solve the system of equilibrium equations. But the formulation and solving of the displacements equations can be changed by direct formulation of global flexibility matrix of the structure. In these cases, when the degree of static indeterminacy of the discrete model of structure is quite less than the number of freedom degrees, suth the change can be effective. The computational results illustrate the high accuracy of equilibrium elements not only with respect to forces, but also to displacements. The direct stiffness method in equilibrium elements makes it possible not only to simplify significantly implementation of the equilibrium element method, but also to include equilibrium elements in libraries of geometrically compatible elements.

Formation of a suitable null basis for equilibrium matrix is the main part of finite elements analysis via force method. Foran optimal analysis, the selected null basis matrices should be sparse and banded corresponding to produce sparse, banded and well-conditioned flexibility matrices. In this paper, an efficient method is developed for the formation of null bases of finite element models (FEMs) consisting of rectangular plane stress and plane strain serendipity family elements, corresponding to highly sparse and banded flexibility matrices. This is achieved by associating special graphs with the FEM and selecting appropriate subgraphs and forming the self-equilibrating systems (SESs) on these subgraphs. The efficiency of the present method is illustrated through three examples.

Efficient finite element analysis using graph-theoretical force method; rectangular plane stress and plane strain Lagrange family elements

The dynamic stiffness matrices have been widely used to give a better approximation of the model, both in linear and non-linear analysis. The derivatives of the dynamic stiffness matrices are required frequently in a number of numerical procedures, such as inverse iterations and Newton-Rapson iterations. It is well known that the mass matrix is equal to the first derivative of the dynamic stiffness matrix with respect to the minus of the square of the vibration frequency. In this paper, the above statement is generalized for general system parameters. It can be used to show that the geometric matrix is equal to the first derivative of the dynamic stiffness matrix with respect to the axial force.

Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method

Chapter Five - FLEXIBILITY, STIFFNESS AND MASS MATRICES

Vibration analysis of combined conical-cylindrical shells based on the dynamic stiffness method

Natural frequencies and harmonic responses of multi-story frames using single variable shear deformation theory

Investigation of bar system modal characteristics using Dynamic Stiffness Matrix polynomial approximations

An improved numerical method evaluating exact static element stiffness matrices of thin-walled beam-columns on elastic foundations

A two-dimensional consistent approach for static and dynamic analyses of uniform beams