Characterization and synthesis of Rayleigh damped elastodynamic networks

Correction of stiffness and mass matrices utilizing simulated measured modal data

<div class="htmlview paragraph">With highly complex structures, it is sometimes desirable to derive a dynamic model of the system from experimental vibration data. This paper presents algorithms for synthesizing the mass and stiffness matrices from experimentally derived modal data in a way which preserves the physical significance of the individual mass and stiffness elements. The synthesizing procedures allow for the incorporation of other mass and stiffness data, whether empirical or based on the analyst's insight. The mass and stiffness matrices are derived for a cantilever beam example and are compared with those obtained using earlier techniques.</div>

This paper proposes a method to identify mass and stiffness matrices of structures. Element mass and stiffness matrices have mass and stiffness distributions. Therefore, a ratio between two elements in these matrices has a proper value. Element mass and stiffness matrices are divided into several groups considering this relation. A group is represented by a sparse matrix (called group matrix) with only elements in the group matrix. Mass and stiffness matrices are modified group by group. The objective function to be minimized is the sum of each norm of difference between analytical and identified group matrices. A numerical example is given to show the effectiveness of the method.

In order to present an efficient, practical technique to determine progressive failure mechanism of structures, modelling of member deterioration by using a spring system is outlined. The procedure uses updates of member stiffness and mass matrices as well as the random load vector in incremental forms. In this procedure, the assembly process produces redistributions of the system stiffness and mass matrices, and the load vector. In the calculation of response spectral values, the original forms remain unchanged. Inversion of the stiffness matrix is calculated by using the Neumann expansion solution in which the original stiffness matrix is inverted only once so that a considerable computation time is saved in the whole calculation process. An incremental solution technique is presented for spectral analyses of both static and dynamic sensitive structures. In the case of dynamic analysis, special attention is paid to estimations of modified natural frequencies and mode shapes of deteriorated structures, which may affect response spectral values considerably. The technique, which is presented in the paper, is attractive in practical applications and can be efficiently used in the reliability calculation as well, and also it can be successfully used to determine a progressive failure mechanism of the structure.

A two-stage approach to stochastic finite element model updating using FRF data

In this study, we propose effective formulations for modal, frequency response, and transient analyses using condensed matrices of the finite element (FE) model. Employing the iterated improved reduced system (IIRS) method, a transformation matrix is defined that condenses the stiffness and inertial effects of nodes to be neglected into nodes of interest. Using this, the condensed mass and stiffness matrices are derived. With these two condensed matrices, a condensed damping matrix is derived using the Rayleigh damping. By considering the condensed matrices in the original structural dynamic formulation based on the global FE matrices, the condensed structural dynamic formulation is derived, and the approximated solutions are calculated from this condensed formulation. To verify the performance of the proposed formulation, we perform a structural dynamic analysis on a stiffened plate, and by comparison with the solutions calculated from the global FE matrices, the proposed formulations have been found to provide highly accurate solutions with an excellent computational efficiency.

Dynamic modeling and simulation of deploying process for space solar power satellite receiver

Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solutions

In this work, we choose the points and weights of the Gauss–Jacobi, Gauss–Radau–Jacobi and Gauss–Lobatto–Jacobi quadrature rules that optimize the number of operations for the mass and stiffness matrices of the high‐order finite element method. The procedure is particularly applied to the mass and stiffness matrices using the tensor‐based nodal and modal shape functions given in (Int. J. Numer. Meth. Engng 2007; 71(5):529–563). For square and hexahedron elements, we show that it is possible to use tensor product of the 1D mass and stiffness matrices for the Poisson and elasticity problem. For the triangular and tetrahedron elements, an analogous analysis given in (Int. J. Numer. Meth. Engng 2005; 63(2):1530–1558) was considered for the selection of the optimal points and weights for the stiffness matrix coefficients for triangles and mass and stiffness matrices for tetrahedra. Copyright © 2009 John Wiley & Sons, Ltd.

Component mode synthesis (CMS) is a widely employed model reduction technique used to reduce the computational cost associated with the dynamic analysis of complex engineering structures. To generate CMS models, specifically the formulation of Craig and Bampton, both normal fixed-interface modes and constraint modes of the component’s structure are calculated. These modes are used in conjunction with the component level mass and stiffness matrices to generate reduced mass and stiffness matrices used in the final analyses. For some component models, the most computationally expensive part of this procedure is calculating the component normal modes information. Several different approaches are utilized to investigate the sensitivity of system level responses to variations in several aspects of the CMS models. One approach evaluates changes due to modifications of the reduced mass and stiffness matrices assuming that the mode shapes do not change. The second approach assumes that the mode shapes change but the reduced mass and stiffness matrices do not change. An example is presented to show the influence of these two approaches.

A new Gibbs sampling based algorithm for Bayesian model updating with incomplete complex modal data

Of the two main finite element approaches, the displacement method has progressed more rapidly than the force method in the area of element representation. Many authors have contributed to the displacement method with both static and dynamic stiffness matrices for beam elements, plane stress elements and plate bending elements. The plane stress and bending elements being rectangular, triangular or quadrilateral in form. A dynamic stiffness matrix consists of a static stiffness matrix and a mass matrix. In the force method, .static flexibility matrices have been developed for beam elements and plane stress elements. However, static flexibility matrices for plate bending elements have only recently been published (Kaufman and Hall, Morley, Robinson, Przemieniecki). For the past few years, the author has been investigating the rank force method for structural vibration analysis. In ref. 6 a dynamic flexibility matrix is presented for a beam element. This matrix consists of a static flexibility matrix and an inverse mass matrix. Ref. 5 contains the derivation of a dynamic flexibility matrix for a rectangular plate element in bending, twisting and shear. The author, in collaboration with Petyt, demonstrated how a dynamic stiffness and flexibility matrix can be extended to include material damping.

Development of size-dependent consistent couple stress theory of Timoshenko beams

In the current study, vibration analysis of functionally graded (FG) nano-sized beams resting on a elastic foundation is presented via a finite element method. The elastic foundation is simulated by using one-parameter Winkler type elastic foundation model. Euler-Bernoulli beam theory and Eringen’s nonlocal elasticity theory are utilized to model the functionally graded nano-sized beams with various boundary conditions such as simply supported at both ends (S-S), clamped-clamped (C-C) and clamped-simply supported (C-S). Material properties of functionally graded nanobeam vary across the thickness direction according to the power-law distribution. The vibration behaviors of functionally graded nanobeam composed of alumina (Al2O3) and steel are shown using nonlocal finite element formulation. The importance of this paper is the utilize of shape functions and the Eringen's nonlocal elasticity theory to set up the stiffness matrices and mass matrices of the functionally graded nano-sized beam resting on Winkler elastic foundation for free vibration analysis. Bending stiffness, foundation stiffness and mass matrices are obtained to realize the solution of vibration problem of the FG nanobeam. The influences of power-law exponent (k), dimensionless nonlocal parameters (e0a/L), dimensionless Winkler foundation parameters (KW), mode numbers and boundary conditions on frequencies are investigated via several numerical examples and shown by a number of tables and figures.

This paper deals with the representation of the damping matrix in terms of mass and stiffness matrices. A relationship is proposed that can represent not only Rayleigh damping, but also the general nonproportional damping. This proposed relationship is mathematically based on a theorem of matrix representation that, under certain circumstance (which can be easily checked), a matrix C can be represented accurately by a polynomial of two given matrices M and K. In engineering applications, in the sense of least‐squares approach or series solutions, the first few terms of the polynomial of the given matrices M and K are shown to be sufficiently accurate to represent the damping matrix. An attempt is made to interpret the physical meaning of some of the parameters of the proposed damping matrix. Numerical examples are given to illustrate the simplicity and accuracy of the proposed method as applied to nonproportional damping problems in structural dynamics. The proposed approach can also be applied in system ...