Solution of quadratic matrix equations for free vibration analysis of structures

An efficient digital computer procedure and the related numerical algorithm are presented herein for the solution of quadratic matrix equations associated with free vibration analysis of structures. Such a procedure enables accurate and economical analysis of natural frequencies and associated modes of discretized structures. The numerically stable algorithm is based on the Sturm sequence method, which fully exploits the banded form of associated stiffness and mass matrices. The related computer program written in FORTRAN V for the JPL UNIVAC 1108 computer proves to be substantially more accurate and economical than other existing procedures of such analysis. Numerical examples are presented for two structures: a cantilever beam and a semicircular arch.     

Development of a unified numerical procedure for free vibration analysis of structures

This paper presents the details of a unified numerical algorithm and the associated computer program developed for the efficient determination of natural frequencies and modes of free vibration of structures. Both spinning and nonspinning structures with or without viscous and/or structural damping may be solved by the current procedure; in a addition, the program is capable of solving static problems with multiple load cases as well as the quadratic matric eigenproblem associated with a finite dynamic element structural discretization. A special symmetric matrix decomposition scheme has been adopted for matrix triangularization, which renders the program rather efficient and economical. Also, a novel bisection scheme is described that further accelerates the solution convergence rate, particularly for the case of repeated roots. The associated computer program adopts and out-of-core solution strategy, thereby enabling effective solutions to be achieved for large, complex, practical structures. A complete listing of the program written in FORTRAN V, for the UNIVAC 1100/82 computer, along with the source deck is available for ready use.     

Development of a finite dynamic element for free vibration analysis of two-dimensional structures

The paper develops an efficient free-vibration analysis procedure of two-dimensional structures. This is achieved by employing a discretization technique based on a recently developed concept of finite dynamic elements, involving higher order dynamic correction terms in the associated stiffness and inertia matrices. A plane rectangular dynamic element is developed in detail. Numerical solution results of free-vibration analysis presented herein clearly indicate that these dynamic element combined with a suitable quadratic matrix eigenproblem solution technique effect a most economical and efficient solution for such an analysis when compared with the usual finite element method.     

Geometrically nonlinear analysis of circulant structures using an efficient eigensolution method

In this paper, an efficient method is presented for the geometrically nonlinear analysis of circulant structures. A structure is called regular if its model can be formed as the product of two subgraphs, and if one of the subgraphs is a cycle then it is termed circulant. In the present method, we deal with the eigensolution of circulant structures by simply finding the eigenvalues and eigenvectors of the blocks of the stiffness matrix rather than those of the entire structural matrix, leading to a considerable reduction in the computational time. The developed method utilizes concepts from product graphs and linear algebra for the nonlinear analysis of structures. Graph products are used to perform the eigensolution of circulant structures through those of its constituting subgraphs. Here, using the existing numerical methods, nonlinear static and dynamic analyses can be performed. In the presented method instead of solving the characteristic equation in each iteration of each step, the eigensolution of the structure is obtained using the eigensolution of the primary structure (the structure of the first iteration of the first step) together with some simple mathematical operations. In the present method instead of heavy matrix operations such as inverting a matrix and solution of the corresponding equations, one needs only to perform simple matrix multiplications and additions. The advantage of the presented method becomes more apparent when it is applied to the nonlinear analysis of a structure, where analysis should be performed many times.     

Dislocation structures in cyclically deformed nickel polycrystals

The effect of the grain orientation and the plastic strain amplitude _pa on the saturated dislocation structure was studied on individual grains of cyclically deformed nickel polycrystals by means of scanning electron microscopy using the electron back scattering pattern technique and the channelling contrast of back scattered electrons. The main features of the dislocation configuration in a grain were found to be essentially determined by the crystallographic axial orientation of the grain. A labyrinth-like dislocation pattern is typical for grains with axial orientations near [001], a patch pattern exists in grains with a loading axis (LA) near [011] and fragmented dislocation walls are dominant in grains with LA near [11]. Grains with axial orientations in the central part of the stereographic standard triangle contain a bundle arrangement of dislocation structures. All four types of dislocation structures, but mostly the bundle type, can occur together with the ladder structure of persistent slip bands. Cell patterns were found to be a result of a modification of the bundle and patch configuration at high deformation amplitudes. The mesoscopic dimensions of the dislocation patterns turned out to depend on _pa in the same way for all grain orientations: while the thickness of regions with high dislocation density is reduced with increasing _pa, the width of regions with low dislocation density remains roughly constant.     

Meshfree natural vibration analysis of 2D structures

Determination of resonance frequencies and vibration modes of mechanical structures is one of the most important tasks in the product design procedure. The main goal of this paper is to describe a pioneering application of the solution structure method (SSM) to 2D structural natural vibration analysis problems and investigate the numerical properties of the method. SSM is a meshfree method which enables construction of the solutions to the engineering problems that satisfy exactly all prescribed boundary conditions. This method is capable of using spatial meshes that do not conform to the shape of a geometric model. Instead of using the grid nodes to enforce boundary conditions, it employs distance fields to the geometric boundaries and combines them with the basis functions and prescribed boundary conditions at run time. This defines unprecedented geometric flexibility of the SSM as well as the complete automation of the solution procedure. In the paper we will explain the key points of the SSM as well as investigate the accuracy and convergence of the proposed approach by comparing our results with the ones obtained using analytical methods or traditional finite element analysis. Despite in this paper we are dealing with 2D in-plane vibrations, the proposed approach has a straightforward generalization to model vibrations of 3D structures.     

Consistent higher-order free vibration analysis of composite sandwich plates

The consistent higher-order dynamic formulation for foam-type (soft) core sandwich beams was extended to the case of composite sandwich plates. Eight dynamic governing equations and the corresponding boundary conditions were derived through the application of Hamilton’s principle. The extended formulation was applied to the free vibration analysis of soft-core and honeycomb-core sandwich plates with anti-symmetric and symmetric lay-ups. The vibration results for the thin and thick composite sandwich plates obtained using the extended formulation were consistent with the predictions of the higher order mixed layerwise theory for laminated and sandwich plates. To simplify the formulation for the case of symmetric sandwich plates, the general dynamic formulation was decoupled into two systems of equations representing symmetric and anti-symmetric vibrations. The numerical study demonstrates the importance of the present formulation for the prediction of higher mode vibration response of composite sandwich plates.     

Nonlinear free vibration analysis of laminated composite shells in hygrothermal environments

The nonlinear free vibration behaviour of laminated composite shells subjected to hygrothermal environments is investigated using the finite element method. The present finite element formulation considers doubly curved shells, and the Green–Lagrange type nonlinear strains are incorporated into the first-order shear deformation theory. The analysis is carried out using quadratic eight-noded isoparametric elements. The validity of the model is demonstrated by comparing the present results with the solutions available in the literature. A parametric study is carried out varying the curvature ratios and side to thickness ratios of composite cylindrical shell, spherical shell and hyperbolic paraboloid shell panels with simply supported boundary conditions.     

Advances in nonlinear vibration analysis of structures. Part-I. Beams

The development of nonlinear vibration formulations for beams in the literature can be seen to have gone through distinct phases — earlier continuum solutions, development of appropriate forms, extra-variational simplifications, debate and discussions, variationally correct formulations and finally applications. A review of work in each of these phases is very necessary in order to have a complete understanding of the process of evolution of this field. This paper attempts to achieve precisely this objective.     

A lanczos-based technique for exact vibration analysis of skeletal structures

This paper presents and discusses a Lanczos-based eigensolution technique for evaluating the natural frequencies and modes from frequency-dependent eigenproblems in structural dynamics. The new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency-dependent displacement fields in formulating the system matrices. The method is well suited to the solution of large-scale problems. The new solution methodology presented here is based on the ability to evaluate a specific set of parameterized non-linear eigenvalue curves at given values of the parameter through an implicitly restarted Lanczos technique. Numerical examples illustrate that the implicitly restarted Lanczos method with secant interpolation accurately evaluates the exact natural frequencies and modes of the non-linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models. In addition, the eigenvalue technique presented here is shown to be far more computationally efficient on large-scale problems than the determinant search techniques traditionally employed for solving exact vibration problems.     

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

In this paper, nonlinear static and free vibration analysis of functionally graded piezoelectric plates has been carried out using finite element method under different sets of mechanical and electrical loadings. The plate with functionally graded piezoelectric material (FGPM) is assumed to be graded through the thickness by a simple power law distribution in terms of the volume fractions of the constituents. Only the geometrical nonlinearity has been taken into account and electric potential is assumed to be quadratic across the FGPM plate thickness. The governing equations are obtained using potential energy and Hamilton’s principle that includes elastic and piezoelectric effects. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect using higher order plate elements. The present finite element is modeled with displacement components and electric potential as nodal degrees of freedom. Results are presented for two constituent FGPM plate under different mechanical boundary conditions. Numerical results for PZT-4/PZT-5H plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response are also studied. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.     

Analysis of structures transformable to circulant form using U-transformation and Kronecker products

In this paper, eigenvalues and eigenvectors of the specific types of structural matrices are studied, and a simple method is presented for calculating their eigenvalues. First, the required formulation to diagonalize circulant and block circulant matrices is presented by using U-matrix transformation. Then utilizing the method of this paper, matrices with non-circulant forms are converted into their circulant counterpart matrices. In order to demonstrate the efficiency of the method, some examples are provided using numerical methods such as finite differences, finite element and finite stripe methods.     

The study of crack-propagation behaviors and dislocation structures in cyclically deformed polycrystalline IF steel

Strain controlled fatigue experiments were employed to evaluate automotive-grade interstitial-free ferrite steels (IF steels). This study demonstrates that the loop-patch structure, dislocation walls and dislocation cells larger than 2 μm have no significant effect on fatigue failure. The cracks prefer to grow near grain boundaries while dislocation cells smaller than 2 μm tend to develop along the grain boundaries and triple junction of the grains. Once the dislocation cells smaller than 2 μm develop from grain boundaries to grain interior, the cracks will propagate near the grain boundaries and through the grain interior simultaneously.     

A simple numerical method of cycle jumps for cyclically loaded structures

A method for accelerated numerical simulations of structures subjected to cyclic loading is investigated. Of particular interest is a class of structures where the structural properties evolve with time. The proposed method is based on conducting detailed finite element analysis for a set of cycles to establish a trend line, extrapolating the trend line spanning many cycles, and use the extrapolated state as initial state for additional FEA simulations. This includes a control function that automatically monitors the length of the cycle jump to ensure a realistic solution. We compare the proposed method to a reference calculation, where all incremental cycles are conducted, and find that the cycle jump solution replicates the true solution.     

Hierarchic models for the free vibration analysis of functionally gradient plates

International Journal of Mechanics and Materials in Design - Hierarchic models for the modal analyses of ceramic–metal functional gradient (FG) plates are introduced. This study was motivated...