Eigenvalue Extraction Research Articles

Nonlinear oscillations of functionally graded microplates

The size-dependent nonlinear oscillation characteristics of a functionally graded microplate is investigated numerically, in which all the displacements, i.e. in-plane as well as out-of-plane, and their inertia are accounted for. The potential energy of the functionally graded microsystem is obtained based on a modified version of the couple stress theory, so as to account for size effects, together with the Mori-Tanaka homogenisation mixture model for the graded material property. The kinetic and size-dependent potential energies of the microsystem are dynamically balanced by the work of an external force via the Lagrange equations and truncated employing an assumed-mode discretization scheme. Extensive numerical simulations are conducted upon the discretised model of the microsystem through use of a continuation technique as well as an eigenvalue extraction method (for the nonlinear and linear studies, respectively). The effect of several functionally graded microsystem parameters, namely the material gradient index, the material length-scale parameter, and the frequency and amplitude of an exciting external force on the response is investigated.

Efficient Computation of Sparse Matrix Functions for Large-Scale Electronic Structure Calculations: The CheSS Library.

We present CheSS, the "Chebyshev Sparse Solvers" library, which has been designed to solve typical problems arising in large-scale electronic structure calculations using localized basis sets. The library is based on a flexible and efficient expansion in terms of Chebyshev polynomials and presently features the calculation of the density matrix, the calculation of matrix powers for arbitrary powers, and the extraction of eigenvalues in a selected interval. CheSS is able to exploit the sparsity of the matrices and scales linearly with respect to the number of nonzero entries, making it well-suited for large-scale calculations. The approach is particularly adapted for setups leading to small spectral widths of the involved matrices and outperforms alternative methods in this regime. By coupling CheSS to the DFT code BigDFT, we show that such a favorable setup is indeed possible in practice. In addition, the approach based on Chebyshev polynomials can be massively parallelized, and CheSS exhibits excellent scaling up to thousands of cores even for relatively small matrix sizes.

An early intelligent diagnosis model for drilling overflow based on GA–BP algorithm

Aiming at the problems existing in the current drilling overflow accident, this paper uses standpipe pressure (SPP) and casing pressure (CP) monitoring data based on the pressure wave transmission to eliminate the impact of pressure wave jitter on the overall trend of SPP and CP monitoring data by eigenvalue extraction of the monitoring data of SPP and CP. The GA–BP (Genetic algorithm and BP neural network) is used to establish the functional relationship between the expected outputs of well overflow diagnostic model and several characteristic parameters. The model reduces the error caused by the selection of single feature parameters, and also the convergence speed of BP neural network is accelerated by genetic algorithm, and avoids falling into local extreme value. The early diagnosis of drilling overflow is achieved, and probability of wrong judgment and miss judgment of drilling kick is reduced. Finally, the drilling data of YY well in a certain oilfield is taken as the research object. The experimental results show that the error of the GA–BP model is 73.73% less and the accuracy is higher than the BP neural network model. The early-stage intelligent diagnosis model of drilling overflow based on GA–BP algorithm can timely and effectively diagnoses the accident of drilling overflow.

Resonance Responses of Geometrically Imperfect Functionally Graded Extensible Microbeams

This paper aims at analyzing the size-dependent nonlinear dynamical behavior of a geometrically imperfect microbeam made of a functionally graded (FG) material, taking into account the longitudinal, transverse, and rotational motions. The size-dependent property is modeled by means of the modified couple stress theory, the shear deformation and rotary inertia are modeled using the Timoshenko beam theory, and the graded material property in the beam thickness direction is modeled via the Mori–Tanaka homogenization technique. The kinetic and size-dependent potential energies of the system are developed as functions of the longitudinal, transverse, and rotational motions. On the basis of an energy method, the continuous models of the system motion are obtained. Upon application of a weighted-residual method, the reduced-order model is obtained. A continuation method along with an eigenvalue extraction technique is utilized for the nonlinear and linear analyses, respectively. A special attention is paid on the effects of the material gradient index, the imperfection amplitude, and the length-scale parameter on the system dynamical response.

Size-dependent internal resonances and modal interactions in nonlinear dynamics of microcantilevers

The size-dependent internal energy transfer in the nonlinear dynamical behaviour of a microcantilever with an intermediate spring-support is investigated. A geometric size-dependent nonlinearity due to large changes in the curvature is taken into account in the longitudinal and transverse motions. Based on the modified couple stress theory, the potential energy of the system is developed; the kinetic energy is also constructed in term of the displacement field. The energy terms are balanced with the potential energy stored in the intermediate spring-support. The centreline-inextensibility assumption is applied leading to the continuous model of the system involving nonlinear inertial components as well as size-dependent nonlinear curvature components. Based on a weighted-residual technique, the continuous model is reduced and the resultant truncated model is solved via use of a continuation technique. The linear component of the truncated model is solved through an eigenvalue extraction method in order to verify the occurrence of internal energy transfer and modal interaction mechanisms. For the system tuned to internal resonances, the highly nonlinear dynamical response is obtained, taking into account both inertial and geometric (due to large rotations) nonlinearities. It is shown that taking into account the length-scale parameter changes the internal energy transfer mechanisms significantly.

Used Eigenvalues and Eigenvectors to Compress the Image

This article is a detailed review and explanation of the field of linear algebra in which all mathematical concepts and structures concerned with linear algebra. The main objective of the paper is to highlight the significant applications of linear algebra in the field of medical engineering. Principal component analysis is one of the most important compression and feature extraction algorithms used in engineering (Santo, 2012). It mainly depends on the calculation and extraction of eigenvalues and eigenvectors which are then used to represent an input; whether it's the image or a simple matrix. In this article, the use of principal component analysis for medical image compression is an important and novel application of linear algebra (Rajendra, 1996).

Complex motion characteristics of three-layered Timoshenko microarches

Numerical modelling and simulations are carried out on the nonlinear size-dependent motion of three-layered Timoshenko microarches. At the first step, the theoretical model of the three-layered microsystem is obtained based on an energy method. The second step involves the numerical simulations on the linear part of the dynamical model of the system in order to examine the possibility of internal energy transfer and modal interactions in the system dynamics. The third step is to analyse the nonlinear motion characteristics of the cases with internal energy transfer via constructing frequency–responses and force–responses. The size effects are modelled via use of the modified couple stress theory. Hamilton’s principle is used for the dynamic energy balance. The geometric imperfection is included in the model by an initial deflection in the transverse direction. The theoretical models developed for all the longitudinal, transverse, and rotational motions involve all the inertial terms. The numerical simulations are performed via a continuation method in conjunction with a direct time-integration technique for the nonlinear analysis and an eigenvalue extraction method for the linear analysis. The main aim is to analyse the level and mechanism on which the energy transfer occurs; it is also examined that how the energy transfer changes the resonant response of the microsystem and if these changes cause drastic influences.

Oscillations of functionally graded microbeams

The size-dependent oscillations of a third-order shear-deformable functionally graded microbeam are investigated taking into account all the longitudinal and transverse displacements and inertia as well as the rotation and rotary inertia. The modified couple stress theory along with the Mori–Tanaka homogenisation technique is employed to develop formulations for the elastic potential energy as well as the kinetic energy of the system. The energy of the system is balanced by the work of a harmonic excitation force via an energy method based on Hamilton's principle, yielding the size-dependent coupled nonlinear continuous models of the functionally graded system for the longitudinal and transverse displacements as well as the rotational motion. A model reduction procedure, on the basis of a weighted-residual method, is applied without any simplifications on the displacement/inertia/rotation. This operation yields three sets of second-order reduced-order coupled model of the functionally graded system for the longitudinal, transverse, and rotational motions. These reduced-order models are solved via use of a continuation method in order to construct the nonlinear frequency-response and force-response curves of the functionally graded system. A linear analysis is also performed by means of an eigenvalue extraction method in order to determine the linear natural frequencies of the system. It is shown that the material gradient index as well as the length-scale parameter of the functionally graded system affects the system dynamics substantially.

Nonlinear dynamics of bi-layered graphene sheet, double-walled carbon nanotube and nanotube bundle

Due to strong van der Waals (vdW) interactions, the graphene sheets and nanotubes stick to each other and form clusters of these corresponding nanostructures, viz. bi-layered graphene sheet (BLGS), double-walled carbon nanotube (DWCNT) and nanotube bundle (NB) or ropes. This research work is concerned with the study of nonlinear dynamics of BLGS, DWCNT and NB due to nonlinear interlayer vdW forces using multiscale atomistic finite element method. The energy between two adjacent carbon atoms is represented by the multibody interatomic Tersoff–Brenner potential, whereas the nonlinear interlayer vdW forces are represented by Lennard-Jones 6–12 potential function. The equivalent nonlinear material model of carbon–carbon bond is used to model it based on its force–deflection relation. Newmark’s algorithm is used to solve the nonlinear matrix equation governing the motion of the BLGS, DWCNT and NB. An impulse and harmonic excitations are used to excite these nanostructures under cantilevered, bridged and clamped boundary conditions. The frequency responses of these nanostructures are computed, and the dominant resonant frequencies are identified. Along with the forced vibration of these structures, the eigenvalue extraction problem of armchair and zigzag NB is also considered. The natural frequencies and corresponding mode shapes are extracted for the different length and boundary conditions of the nanotube bundle.

Parametric instability of microbeams in supercritical regime

The supercritical parametric instability of a microbeam subject to a time-dependent axial load is examined. The axial load is comprised of a constant mean value along with harmonic fluctuations. The mean value is increased from zero and is set to a value in the supercritical regime; the nonlinear parametric instability over the buckled state, due to the axial load variations, is examined. From the modelling perspective, based on the modified couple stress theory, the potential energy of the system is obtained in terms of the system parameters and displacement field. Moreover, the kinetic energy is formulated as a function of system parameters and the displacement field. The continuous model is developed by means of Hamilton’s principle and then truncated into a reduced-order model via a weighted-residual technique. Three different numerical techniques, i.e. the pseudo-arclength continuation method, a direct time-integration scheme, and an eigenvalue extraction, are employed to solve the high-dimensional reduced-order model. A stability analysis is also conducted via the Floquet theory. The nonlinear size-dependent parametric response of the system over the buckled configuration is presented in the form of frequency–response diagrams, force–response curves, time histories, phase-plane portraits, fast Fourier transforms, and Poincare sections.

Exact Beam Element and Algorithms for Stability Analysis of Frame Structures Including Transverse Shear Deformation

Based on the general solutions of the governing equation for linear buckling of Timoshenko beam under axial force, exact shape functions for transverse shear deformable beam are constructed and corresponding exact element stifiness matrix and geometrical stifiness matrix are formulated for stability analysis of frame structures. Two new algorithms are developed for the determination of a given number of critical load factors or all critical load factors in a given range. In addition to the conventional eigenvalue extraction method, the well-known Wittrick-Williams method is employed in both algorithms. Numerical examples are presented to demonstrate the efiectiveness of the proposed element and algorithms.

Chaotic Extension Neural Network-Based Fault Diagnosis Method for Solar Photovoltaic Systems

At present, the solar photovoltaic system is extensively used. However, once a fault occurs, it is inspected manually, which is not economical. In order to remedy the defect of unavailable fault diagnosis at any irradiance and temperature in the literature with chaos synchronization based intelligent fault diagnosis for photovoltaic systems proposed by Hsieh et al., this study proposed a chaotic extension fault diagnosis method combined with error back propagation neural network to overcome this problem. It used the nn toolbox of matlab 2010 for simulation and comparison, measured current irradiance and temperature, and used the maximum power point tracking (MPPT) for chaotic extraction of eigenvalue. The range of extension field was determined by neural network. Finally, the voltage eigenvalue obtained from current temperature and irradiance was used for the fault diagnosis. Comparing the diagnostic rates with the results by Hsieh et al., this scheme can obtain better diagnostic rates when the irradiances or the temperatures are changed.

Application of Nondimensional Dynamic Influence Function Method for Eigenmode Analysis of Two-Dimensional Acoustic Cavities

This paper establishes an improved NDIF method for the eigenvalue extraction of two-dimensional acoustic cavities with arbitrary shapes. The NDIF method, which was introduced by the authors in 1999, gives highly accurate eigenvalues despite employing a small number of nodes. However, it needs the inefficient procedure of calculating the singularity of a system matrix in the frequency range of interest for extracting eigenvalues and mode shapes. The paper proposes a practical approach for overcoming the inefficient procedure by making the final system matrix equation of the NDIF method into a form of algebraic eigenvalue problem. The solution quality of the proposed method is investigated by obtaining the eigenvalues and mode shapes of a circular, a rectangular, and an arbitrarily shaped cavity.

A new approach to stability analysis of frame structures using Trefftz-type elements

This paper presents some new developments in structural stability analysis. A novel Trefftz-type finite element and efficient algorithms are proposed for calculating numerically exact solutions for frame structures. New shape functions are constructed by using general solutions of the homogeneous governing equation and an exact Trefftz element is formulated. An iterative algorithm based on conventional eigenvalue extraction method is then proposed, with which very accurate solutions can be obtained efficiently by using coarse finite element meshes. Further, an efficient method based on the new element matrices and Rayleigh's quotient is developed for a posteriori error estimation and solution improvement. Numerical examples are presented to demonstrate the effectiveness of the proposed element and algorithms. Finally, concluding remarks are made including those on the further extension of the approach.

The evaluation of disc brake squeal propensity through a fully coupled transient thermomechanical model

This paper presents a fully coupled transient thermomechanical analysis of a disc brake that takes into account the so-called ‘rotating-heat-source’ effect. Results from this are used to inform the time-based extraction of system eigenvalues that permit quantification of the squeal propensity of the brake. The model performance is validated against a series of dynamometer tests, based upon the SAE J2521 drag brake test schedule, in which the ability of the model to predict the fugitive nature of the squeal problem is confirmed. Finally, a parametric study is also presented that shows how the technique can be used to desensitise the squeal propensity to both material and geometric design changes.

Dynamic Analysis of Response of Cofferdam with Steel Sheet Piles Induced by Earthquake Excitation

For its construction convenience and structure integrity, cofferdams are widely employed in port engineering. Past experience has shown that cofferdam are subjected to damage due to earthquake excitations. Numerical analyses with both response spectrum and step-by-step integration methods are conducted by using Lanczos eigenvalue extraction technique to obtain natural frequencies and modes, and solving dynamic equations with Newmark implicit method to consider geometric nonlinearity. The computational results show that the natural frequency of cofferdam system is low and the horizontal translation stiffness of cofferdam in positive direction is higher than that in negative direction. Under seismic excitation, the displacement response of inner steel sheet is much more obvious than that of outer one. And the distribution of horizontal displacements in steel sheets presents the characteristics that the corresponding values increase with their heights in the cofferdam system. On the contrary, the deviatonic stresses of cofferdam decrease with the augments of height.

The Application Research of Coating Thickness Detection Using the Tree Classifier

Using Ultrasonic Testing to test the coating thickness of various industrial structure is a commonly used thickness measurement method. But in some special structures, such as steel in the solid rocket motors. insulation structure, the coating thickness could only be tested by the Ultrasonic emitted from the outside of the steel ,This paper presents a method of using resonant relationship of the two detection echo as the delay time of eigenvalue, and make the eigenvalue extraction boil down to a multi-pattern recognition problem. using a self-design binary tree classifier to classify them, effectively improving the feature extraction accuracy and the signal to noise ratio of the thickness detection.

Power secant method applied to natural frequency extraction of Timoshenko beam structures

This work deals with an improved plane frame formulation whose exact dynamic stiffness matrix (DSM) presents, uniquely, null determinant for the natural frequencies. In comparison with the classical DSM, the formulation herein presented has some major advantages: local mode shapes are preserved in the formulation so that, for any positive frequency, the DSM will never be ill-conditioned; in the absence of poles, it is possible to employ the secant method in order to have a more computationally efficient eigenvalue extraction procedure. Applying the procedure to the more general case of Timoshenko beams, we introduce a new technique, named "power deflation", that makes the secant method suitable for the transcendental nonlinear eigenvalue problems based on the improved DSM. In order to avoid overflow occurrences that can hinder the secant method iterations, limiting frequencies are formulated, with scaling also applied to the eigenvalue problem. Comparisons with results available in the literature demonstrate the strength of the proposed method. Computational efficiency is compared with solutions obtained both by FEM and by the Wittrick-Williams algorithm.

Effects of Delamination on the Vibration Charecteristics of Composite Beams

Development of an early damage detection method for delamination is one of the most important keys in maintaining the integrity and safety of composite structures. The present study focuses on the effects of delamination on the vibration characteristics of beams with single through-width delamination. Comparison is made between experimental study on aluminum specimens and corresponding finite element models created in ANSYS using brick elements. Specimens are made by bonding thin aluminum strips using very thin adhesive layers, with delaminations introduced by inserting cellophane paper at pre-determined locations. Vibration tests are carried out on the specimens, the results of which are compared with those of the finite element analysis. The finite element model was then used to demonstrate the comparison of natural frequency, mode shapes and curvature mode shapes of the intact and delaminated models. Further, the analysis is extended into layered composites of orthotropic materials. Comparisons are made between published experimental results of laminates and results of finite element analyses on models with layered brick elements using ANSYS. Eigen value extraction was done on ‘free’ models, with no constrains between the delaminated layers and models with linear spring connecting the delaminated layers.

Friction‐Induced Vibration of Artificial Hip Joints

AbstractA hip replacement system is analysed with respect to its susceptibility to friction induced vibrations under laboratory conditions and in a pseudo‐in vivo setup. The method of complex eigenvalue extraction within the framework of finite‐element modelling and analysis is used to identify the geometric properties and system parameters that might lead to unstable sliding states. The approach presented reveals negatively damped eigenmodes for both the pseudoin vivo setup and the not‐implanted system under laboratory conditions without boundary conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)