Frequency-dependent Shape Functions Research Articles

Dynamics of hierarchical beam lattice structures by an exact reduced-order dynamic-stiffness model

An exact reduced-order dynamic-stiffness model is proposed to study the dynamics of two-dimensional hierarchical beam lattices. Upon modelling every member of the lattice by a single exact two-node beam element, the model retains a few primary nodes of the unit cell and involves an exact dynamic condensation of the nodal degrees of freedom of the lattice structure internal to the unit cell and connected to the primary nodes. It is shown that the modes of the reduced-order model exhibit two orthogonality conditions, with dynamic-stiffness terms pertinent to the condensed lattice structures. The orthogonality conditions are the basis to decouple the motion equations and derive the exact modal response to external loading, in both time and frequency domains, under the assumption of proportional damping. The modal response is obtained in simple and elegant analytical form for the whole lattice, including the condensed lattice structures internal to the unit cells. Moreover, it can be calculated very expeditiously thanks to an exact representation of the eigenfunctions based on frequency dependent shape functions, which applies regardless of geometry and complexity of the hierarchical lattice structure. Numerical results confirm the exactness of the proposed reduced-order dynamic-stiffness model by comparison with standard finite-element solutions in ABAQUS.

Free Vibration of Timoshenko–Ehrenfest Beams and Frameworks Using Frequency-Dependent Mass and Stiffness Matrices

Abstract The frequency-dependent mass and stiffness matrices of a Timoshenko–Ehrenfest beam are developed through extensive application of symbolic computation. Explicit algebraic expressions for the frequency dependent shape functions and each of the independent elements of the frequency-dependent mass and stiffness matrices are presented concisely. The ensuing frequency-dependent mass and stiffness matrices of the Timoshenko–Ehrenfest beam are applied with particular reference to the Wittrick–Williams algorithm to investigate the free vibration characteristics of an individual Timoshenko–Ehrenfest beam and a framework. The results are discussed with significant conclusions drawn. The proposed method retains the exactness of results like the dynamic stiffness method, but importantly, it opens the possibility of including damping in the analysis.

Extended Wittrick–Williams algorithm for eigenvalue solution of stochastic dynamic stiffness method

This paper proposes an efficient and reliable eigenvalue solution technique for analytical stochastic dynamic stiffness (SDS) formulations of beam built-up structures with parametric uncertainties. The SDS formulations are developed based on frequency-dependent shape functions in conjunction with both random-variable and random-field structural parameters. The overall numerical framework is aimed towards representing the broadband dynamics of structures using very few degrees of freedom. This paper proposes a novel approach combining the Wittrick–Williams(WW) algorithm, the Newton iteration method and numerical perturbation method to extract eigensolutions from SDS formulations. First, the eigenvalues and eigenvectors of the deterministic DS formulations are computed by the WW algorithm and the corresponding mode finding technique, which are used as the initial solution. Then, a numerical perturbation technique based on the inverse iteration and homotopy method is proposed to update the eigenvectors and eigenvalues. The robustness and efficiency of the proposed method are guaranteed through several technique arrangements. Through numerical examples, the proposed method is demonstrated to be robust within the whole frequency range. This method provides an efficient and reliable tool for stochastic analysis of eigenvalue problems relevant to free vibration and buckling analysis of built-up structures.

Free vibration of orthotropic Levy-type solution plates by using SEM

Orthotropic Levy-type solution plates considering shear deformation were solved by using the Spectral Element Method (SEM). In SEM, the exact solution of structural dynamic related problems can be solved by using the frequency-dependent shape functions. The solutions of SEM are accurate in decreasing the amount of Degree Of Freedoms (DOF) to discard the cost and computational disadvantages in the Finite Element Method (FEM). This study investigates the free vibration problems of orthotropic Levy-type solution plates solutions based on the two variable Refined Plate Theory (RPT). SEM for orthotropic Levy-type solution plates in the frequency domain was formulated to solve free vibration problems. The differential governing equations of the orthotropic plate element in SEM are formulated into the form of transcendental stiffness matrices. The boundary conditions of the spectral element model consist of four edge-type DOFs at each side of the orthotropic Levy-type solution plates. The natural frequencies of the plate are calculated by means of the Wittrick-Williams procedure. The numerical investigations are presented to show the accuracy, efficiency, and effectiveness of SEM without any discretization scheme.

A dynamic finite element method for the estimation of cable tension

Cable supported structures have been widely used in civil engineering. Cable tension estimation has great importance in cable supported structures\' analysis, ranging from design to construction and from inspection to maintenance. Even though the Bernoulli-Euler beam element is commonly used in the traditional finite element method for calculation of frequency and cable tension estimation, many elements must be meshed to achieve accurate results, leading to expensive computation. To improve the accuracy and efficiency, a dynamic finite element method for estimation of cable tension is proposed. In this method, following the dynamic stiffness matrix method, frequency-dependent shape functions are adopted to derive the stiffness and mass matrices of an exact beam element that can be used for natural frequency calculation and cable tension estimation. An iterative algorithm is used for the exact beam element to determine both the exact natural frequencies and the cable tension. Illustrative examples show that, compared with the cable tension estimation method using the conventional beam element, the proposed method has a distinct advantage regarding the accuracy and the computational time.

A Dynamic Finite Element for Coupled Extensional-Torsional Vibration of Uniform Composite Thin-Walled Beams

A Dynamic Finite Element (DFE) formulation for the free vibration analysis of extension-torsion coupled uniform composite thin-walled beams is presented. Employing the exact solutions of the differential equations governing the uncoupled vibrations of a uniform beam element, the analytical expressions for extensional and torsional dynamic trigonometric shape functions are derived. By exploiting the principle of virtual work and the frequency-dependent shape functions, the element dynamic stiffness matrix is developed. The application of the theory is demonstrated by a Circumferentially Uniform Stiffness (CUS) composite circular tube for which the influence of ply fibre-angle on the natural frequencies is studied. A variety of CUS configurations are studied and the correctness of the theory and the superiority of the proposed DFE over the conventional FEM methods are confirmed by numerical checks and the published results.

Adaptive Frequency-Dependent Shape Functions for Accurate Estimation of Modal Frequencies

AbstractAn adaptive linear FEM for modal analysis of structures is presented in this paper. The method uses frequency-dependent shape functions in addition to conventional low-order polynomial interpolation functions to improve accuracy in natural-frequency and mode-shape estimation. The proposed technique requires the iterative computation of the mass and stiffness matrices because shape functions are adjusted recursively as functions of estimated natural frequencies without mesh adaptation or refinement. Applied to frame structures, the method uses conventional polynomial interpolating functions for axial, bending, and torsional displacements and additional generalized coordinates multiplied by frequency-dependent functions derived from modal analysis of continuous beam models. Because these additional functions or their derivatives do not vanish at nodes located at the element boundaries, linear kinematic constraints are imposed to the augmented set of displacement coordinates to ensure displacement fi...

Frequency-Domain Spectral Element Model of a Uniform Spinning Shaft

This paper presents a spectral element model for the spinning uniform shaft represented by the Timoshenko beam model. The bearing-supports are represented by equivalent springs. The variational approach is used to formulate the spectral element model from the frequency-dependent shape functions derived from exact wave solutions to the governing differential equations.

Spectral finite element and nonlocal continuum mechanics based formulation for torsional wave propagation in nanorods

Because of future promising exploration of nanotechnology, focus is being put in the miniaturization of mechanical and electromechanical devices. Attention is sought toward the development of nanodevices and nanomachines. The length scales associated with nanostructures like are such that to apply any classical continuum techniques, we need to consider the small length scales such as lattice spacing between individual atoms, surface properties, grain size, etc. This makes a physically consistent classical continuum model formulation very challenging. So this work presents Eringen's nonlocal elasticity theory, that has been incorporated into classical torsional rod model to capture unique properties of the nanorods under the umbrella of continuum mechanics theory. The strong effect of the nonlocal scale has been obtained which leads to substantially different torsional wave behaviors of nanorods from those of macroscopic rods. Nonlocal torsional rod model is developed for nanorods. Explicit expressions are derived for torsional wavenumbers and wave speeds of nanorods. The analysis shows that the wave characteristics are highly over estimated by the classical rod model, which ignores the effect of small-length scale. The studies also shows that the nonlocal scale parameter introduces certain band gap region in torsional wave mode where no wave propagation occurs. This is manifested in the spectrum cures as the region where the wavenumber tends to infinite or wave speed tends to zero. Next, the Spectral Finite Element formulation of nanorods is performed. The exact frequency dependent shape functions and the dynamic stiffness matrix for the nanorod are obtained as a function of nonlocal scale parameter. It has been found that the nonlocal small scale has significant effect on the exact shape functions and the elements of the dynamic stiffness matrix. These effects are also captured in the present work. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave dispersion properties of carbon nanotubes.

Spectral element analysis of the blood flow in human arteries

It has been well recognized that cardiovascular diseases are closely related to blood flow characteristics. Thus, how to predict them accurately enough in an efficient way has been an important research issue. This paper introduces a 1D spectral element model for the blood flow in the human arteries with varying cross-sections. The variational approach is used to formulate the 1D spectral element model. The exact wave solutions to the frequency-domain governing differential equations are used to determine the frequency-dependent shape functions. The spectral finite element model is then applied to some example arteries to investigate flow characteristics such as blood flow rate and blood pressure through the arteries.

Spectral Element Modeling for Elastic Two-Layer Beams

This paper develops a spectral element model for elastic-elastic two-layered beams. First, the axial-bending coupled equations of motion for an elastic two-layer laminated beam are derived. The spectral element model is then formulated by using the wave solutions satisfying governing equations in frequency-domain as the frequency-dependent shape functions. The spectral element model is finally applied to a cantilevered elastic-elastic two-layered beam as an illustrative problem. The high accuracy of the present spectral element model is verified by comparing the SEM results with those obtained by conventional FEM.

Dynamic stiffness for thin-walled structures by power series

The dynamic stiffness method is introduced to analyze thin-walled structures including thin-walled straight beams and spatial twisted helix beam. A dynamic stiffness matrix is formed by using frequency dependent shape functions which are exact solutions of the governing differential equations. With the obtained thin-walled beam dynamic stiffness matrices, the thin-walled frame dynamic stiffness matrix can also be formulated by satisfying the required displacements compatibility and forces equilibrium, a method which is similar to the finite element method (FEM). Then the thin-walled structure natural frequencies can be found by equating the determinant of the system dynamic stiffness matrix to zero. By this way, just one element and several elements can exactly predict many modes of a thin-walled beam and a spatial thin-walled frame, respectively. Several cases are studied and the results are compared with the existing solutions of other methods. The natural frequencies and buckling loads of these thin-walled structures are computed.

Transient Dynamics of Stochastically Parametered Beams

The problem of determining the statistics of the transient response of randomly inhomogeneous beams is formulated. This is based on the use of stochastic dynamic stiffness coefficients in conjunction with the fast Fourier transform algorithm. The dynamic stiffness coefficients, in turn, are determined using a stochastic finite-element formulation that employs frequency-dependent shape functions. The approach is illustrated by analyzing the response of a random rod subject to a boxcar type of axial impact and, also, by considering the flexural response of a randomly inhomogeneous beam resting on a randomly varying Winkler's foundation and subjected to the action of a moving force. A discussion on the treatment of system property random fields as being non-Gaussian in nature is presented. Also discussed are the methods for handling nonzero initial conditions within the framework of the frequency domain response analysis employed in the study. Satisfactory comparisons between the analytical results and simulation results are demonstrated.

A Dynamic Finite Element (DFE) method for free vibrations of bending-torsion coupled beams

In this paper, a Dynamic Finite Element (DFE) formulation for the free vibration analysis of bending-torsion coupled beams is presented. First, the exact solutions of the differential equations governing the uncoupled vibrations of a uniform beam are found. The employment of these solutions as basis functions leads to the appropriate frequency dependent shape functions which can then be utilized to find the nodal approximations of variables. By exploiting the Principle of Virtual Work (PVW), the elementary Dynamic Stiffness Matrix (DSM) is then obtained which has both mass and stiffness properties. The implementation of the derived DFE matrices in a program is discussed with a particular reference to the Wittrick–Williams algorithm. The application of the theory is demonstrated by illustrative examples wherein a substantial amount of coupling between bending and torsion is highlighted. The correctness of the theory is confirmed, to a high degree of accuracy, by published results and numerical checks.

Dynamic stiffness of randomly parametered beams

A finite element-based methodology is developed for the determination of the dynamic stiffness matrix of Euler-Bernoulli beams with randomly varying flexural and axial rigidity, mass density and foundation elastic modulus. The finite element approximation made employs frequency dependent shape functions and the analysis avoids eigenfunction expansion which, not only eliminates modal truncation errors, but also, restricts the number of random variables entering the formulations. Application of the proposed method is illustrated by considering two problems of wide interest in engineering mechanics, namely, vibration of beams on random elastic foundation and the problem of seismic wave amplification through randomly inhomogeneous soil layers. Satisfactory agreement between analytical solutions and a limited amount of digital simulation results is also demonstrated.

Dynamic stiffness analysis of laminated composite plates

An analysis is presented for the vibration and stability problem of composite laminated plates by using the dynamic stiffness matrix method. A dynamic stiffness matrix is formed by frequency dependent shape functions which are exact solutions of the governing differential equations. It eliminates spatial discretization error and is capable of predicting several natural modes by means of a small number of degrees of freedom. The natural frequencies and buckling loads of composite laminated plates are calculated numerically. The effects of the boundary conditions, the number of layers, the orthotropicity ratio, the side to thickness ratio, and the aspect ratio are studied. It is also illustrated that connected composite plate structures can be handled without difficulty by the present method.

APPLICATION OFk-STEP LANCZOS METHOD TO EXTRACT THE INTERIOR EIGENVALUES OF SKELETAL SYSTEMS

Unlike analyses performed to determine structural response to environmental loadings (i.e., wind or earthquake), where only a few of the lowest natural modes are assumed to participate in the overall response, mechanically induced loadings can excite higher-order modes and require solution of the interior eigenvalue problem. In such applications, care must be taken to ensure that the finite element model of the vibrating structure effectively models the higher-order modal response and that the eigensolver used to extract the natural frequencies does not encounter convergence problems due to the close spacing of the interior eigenvalues. This paper formulates and applies an implicitly restarted Lanczos-based eigensolver for computing the natural frequencies of skeletal systems from the interior of the eigenvalue spectrum. This interior eigenvalue problem is formulated using a finite element method which utilizes both the polynomial- and frequency-dependent shape functions in representing beam deformation; thus, the resulting eigenvalue problem is nonlinear, or eigenvalue dependent. Planar frame examples are given which illustrate the effectiveness of the proposed modelling technique and eigensolver.

Dynamic stiffness analysis of axially loaded non-uniform Timoshenko columns

A dynamic stiffness method is introduced to analyze axially loaded non-uniform Timoshenko columns. A dynamic stiffness matrix is formed by frequency-dependent shape functions which are exact solutions of the governing differential equations. It eliminates spatial discretization errors and is capable of predicting a number of natural modes by means of a small number of degrees of freedom. The method is now widely used in various dynamic problems. In this study the two coupled governing differential equations of a non-uniform column are solved by power series. The natural frequencies of the non-uniform beams under axial force can be found by equating to zero the determinant of the dynamic stiffness matrix of the system. Numerical examples are studied and the results are compared with those obtained by finite element method.

Dynamic stiffness analysis of non-uniform timoshenko beams

A dynamic stiffness method is introduced to analyze non-uniform Timoshenko beams with various boundary conditions. A dynamic stiffness matrix is formed by using frequency dependent shape functions which are exact solutions of the governing differential equations. This eliminates spatial discretization error and is capable of predicting many natural modes with use of a small number of degrees of freedom. The method is now widely used in various dynamic problems [1-4]. In this study it is not necessary to reduce the two coupled governing differential equations into one equation, as has been done by many authors [5-8]. The natural frequencies of the non-uniform beams can be found by equating to zero the determinant of the dynamic stiffness matrix of the system. Several cases are studied and the results are compared with those obtained by the finite element method.

Frequency-Dependent Element Mass Matrices

This paper considers some of the theoretical aspects of the formulation of frequency-dependent structural matrices. Two types of mass matrices are examined, the consistent mass matrix found by integrating frequency-dependent shape functions, and the mixed mass matrix found by integrating a frequency-dependent shape function against a static shape function. The coefficients in the power series expansion for the consistent mass matrix are found to be determinable from those in the expansion for the mixed mass matrix by multiplication by the appropriate constant. Both of the mass matrices are related in a similar manner to the coefficients in the frequency-dependent stiffness matrix expansion. A formulation is derived which allows one to calculate, using a shape function truncated at a given order, the mass matrix expansion truncated at twice that order. That is the terms for either of the two mass matrix expansions of order 2n are shown to be expressible using shape functions terms of order n. Finally, the terms in the matrix expansions are given by formulas which depend only on the values of the shape function terms at the boundary.