Approximate Natural Frequencies Research Articles

A refined first-order shear deformation theory for large dynamic and static deflection investigations of abruptly/harmonically pressurized/blasted incompressible circular hyperelastic plates

In this paper, nonlinear large-deformation dynamic and static responses of the suddenly and harmonically pressurized or blasted compliant incompressible hyperelastic circular plates are investigated through the development of a novel enhanced circular plate theory that in contrast to the available plate theories, utilizes transverse normal strains whose order is much higher than those of the in-plane strains. For this reason, no shear correction factor is needed for the new plate theory. The deformation gradient and Green's strain tensors are first determined based on the first-order shear deformation plate theory as predictors and then, corrected by simultaneously including the transverse compliance and volume conservation concepts in the neo-Hookean framework of hyperelasticity. The equations of motion are obtained using Hamilton's principle. The resulting highly nonlinear equations that include both large deformation and constitutive nonlinearities are solved by a hybrid iterative/updating technique that utilizes a combination of the differential quadrature (DQ) spatial-discretization and Newmark's time-marching methods, after an exhaustive order analysis for the identification of the extremely small terms. The effects of the changes in geometric parameters, material properties, and loading rate and type are investigated on the dynamic lateral deflection of the plate and the approximate fundamental natural frequency. Results reveal that the contribution of the higher vibration modes in the responses is much more remarkable in the high-rate and high-frequency loadings. Furthermore, the deformed shape of the plate that indicates the superimposed effects of the higher vibration modes is plotted for successive time instants.

Applicability and Limitations of Ru’s Formulation for Vibration Modelling of Double-Walled Carbon Nanotubes

In this paper, a comparison is conducted between two different formulations of the van der Waals interaction coefficient between layers, as applied to the vibrations of double-walled carbon nanotubes (DWCNTs); specifically, the evaluation of the natural frequencies is achieved through Ru’s and He’s formulations. The actual discrete DWCNT is modelled by means of a couple of concentric equivalent continuous thin cylindrical shells, where Donnell shell theory is adopted to obtain strain-displacement relationships. In order to take into account the chirality effect of DWCNT, an anisotropic elastic shell model is considered. Simply supported boundary conditions are imposed and the Rayleigh–Ritz method is used to obtain approximate natural frequencies and mode shapes. A parametric analysis considering different values of diameters and numbers of waves along longitudinal and circumferential directions is performed by adopting Ru’s and He’s formulations. From the comparisons, it is evident that Ru’s formulation provides unsatisfactory results for relatively low values of diameters and relatively high numbers of circumferential waves with respect to the more accurate He’s formulation. This behaviour is observed for every number of longitudinal half-waves. Therefore, Ru’s formulation cannot be used for the vibration modelling of DWCNTs in a large range of diameters and wavenumbers.

Approximate natural frequencies of AFGM beam

This article gives the approximate natural frequencies of axial functionally graded material (AFGM) beams. Based on Matlab software and the traditional finite element method, we can estimate the natural frequencies for this structure under four types of boundary conditions. Some examples are given which prove that this way is simple and applicable.

Nonlocal anisotropic elastic shell model for vibrations of double-walled carbon nanotubes under nonlinear van der Waals interaction forces

In this paper, a novel nonlocal anisotropic elastic shell model is developed to investigate the nonlinear vibrations of double-walled carbon nanotubes (DWCNTs) in the framework of Sanders–Koiter shell theory. Van der Waals interaction forces between the two concentric single-walled carbon nanotubes (SWCNTs) composing a DWCNT are modelled via Lennard-Jones potential and He’s formulation. In the linear vibration analysis, the displacement field of each SWCNT is expanded by means of a double mixed series in terms of Chebyshev orthogonal polynomials along the longitudinal direction and harmonic functions along the circumferential direction, and Rayleigh–Ritz method is considered to get approximate natural frequencies and modal shapes. In the nonlinear vibration analysis, the three displacements of each SWCNT are re-expanded by means of the approximate eigenfunctions derived in the linear analysis, and an energy approach based on Lagrange equations is adopted to obtain a set of nonlinear ordinary differential equations of motion, which is then solved numerically. Molecular dynamics simulations are performed in order to calibrate the proper value of nonlocal parameter to be inserted in the constitutive equations of the proposed elastic continuum model. A simplified linear distribution of van der Waals interaction forces is initially adopted to analyse the nonlinear vibrations of DWCNTs, obtaining a hardening nonlinear behaviour. By considering a more realistic nonlinear distribution of van der Waals interaction forces, a stronger hardening nonlinear behaviour is found.

Applicability and limitations of Donnell shell theory for vibration modelling of double-walled carbon nanotubes

In this paper, the comparison is conducted between two shell theories as applied to the vibrations of double-walled carbon nanotubes (DWCNTs); specifically, the evaluation of the natural frequencies is conducted via Donnell and Sanders shell theories. The discrete DWCNTs are modelled by means of couples of concentric continuous circular cylindrical shells, where van der Waals interaction forces between the two shells are modelled via He’s formulation. In order to take into account the chirality of DWCNTs, an anisotropic elastic shell model is used. Simply supported, clamped and free boundary conditions are applied and Rayleigh–Ritz method is adopted to obtain approximate natural frequencies and mode shapes. At the beginning, comparisons with experimental and molecular dynamics data are made, from which it is confirmed that anisotropic elastic shell model is more accurate than isotropic one and it is obtained that Sanders shell theory is more accurate than Donnell one. Then, a parametric analysis considering different values of aspect ratio and numbers of waves along the longitudinal and circumferential directions is carried out in the framework of the anisotropic elastic shell model. From these simulations, it is found that Donnell shell theory yields unsatisfactory results for relatively low numbers of longitudinal and circumferential waves, and for relatively high values of aspect ratio, with respect to Sanders shell theory. Therefore Donnell shell theory cannot be used for vibration modelling of DWCNTs in a large range of longitudinal and circumferential wavenumbers and aspect ratios.

Computational analysis of naturally oscillating tandem square and circular bluff bodies: a GPU based immersed boundary – lattice Boltzmann approach

ABSTRACT In this study, a fluid-structure interaction solver is employed to explore the flow physics of tandem oscillating square bluff bodies by varying the corner radius. Based on previous research, dynamic behavior, and fluid forces only square and circular bluff bodies are chosen for a supplementary investigation. This study investigates the effects of the streamwise displacement, , of two elastically mounted square and circular bluff bodies in a tandem arrangement. Fluid flows at a Reynolds number of 100 and a fixed mass ratio are used for the upstream and downstream bluff bodies. As the most complex oscillations and wake flows occur adjacent to the approximate natural frequency ratio, thus a constant natural frequency of is defined. To achieve excellent parallel performance, an immersed boundary lattice Boltzmann method code coupled with a structural equation is developed and accelerated using a graphical processing unit. The results for flow-induced vibrations suggest that the flow response characteristics of the tandem oscillating bluff bodies are strong functions of the streamwise displacement. Four different flow regimes of the bluff body arrangements are identified on the basis of response amplitude. Both the square and circular downstream bluff bodies exhibit two peak transverse amplitudes in the existing scrutinized spacing ratios. In addition, the critical spacing ratio values are dissimilar for the two bluff bodies. Moreover, at the end of the spacing ratio (), the wake effects for the downstream square body are amplified, and it oscillates near a single oscillating circular bluff body. Finally, the flow physics, as well as the hydrodynamic force coefficients of the tandem oscillating bluff bodies are also reviewed.

Magnetoelastic nonlinear free vibration of an annular plate under a nonuniform magnetic field of the long straight current‐carrying wire

AbstractIn this article, the magnetoelastic nonlinear free vibration of a thin conductive annular plate under the nonuniform toroidal magnetic field is investigated, where the magnetic field is generated by a long straight current‐carrying wire. Based on electromagnetic theory, expressions of the nonuniform magnetic field, electromagnetic forces are deduced. The nonlinear free vibration equation of plate is derived by using the Hamilton principle. Considering different boundary conditions, the axisymmetric vibration differential equation is achieved by the Galerkin method. The method of multiple scales is employed to obtain the second‐order approximate analytical solutions and natural frequency. In numerical calculation, the characteristics of nonlinear natural vibration and singular points varying with different parameters, for example, current intensity and plate size, are presented, respectively. The results indicate that, with the increase of current, natural frequency increases and then stabilizes in the inner‐clamped and out‐free (C‐F) boundary, but decreases and stabilizes in the cases of the inner‐clamped and out‐clamped (C‐C) boundary or inner‐simply and out‐simply (S‐S) boundary. Moreover, when wire current is off, natural frequency maintains a constant, where the equilibrium solution of the system is a center. In addition, for all boundary conditions, natural frequency increases significantly with increase of internal radius and thickness, but decreases with the increase of outer radius.

Applicability and Limitations of Simplified Elastic Shell Theories for Vibration Modelling of Double-Walled Carbon Nanotubes

The applicability and limitations of simplified models of thin elastic circular cylindrical shells for linear vibrations of double-walled carbon nanotubes (DWCNTs) are considered. The simplified models, which are based on the assumptions of membrane and moment approximate thin-shell theories, are compared with the extended Sanders–Koiter shell theory. Actual discrete DWCNTs are modelled by means of couples of concentric equivalent continuous thin, circular cylindrical shells. Van der Waals interaction forces between the layers are taken into account by adopting He’s model. Simply supported and free–free boundary conditions are applied. The Rayleigh–Ritz method is considered to obtain approximate natural frequencies and mode shapes. Different aspect and thickness ratios, and numbers of waves along longitudinal and circumferential directions, are analysed. In the cases of axisymmetric and beam-like modes, it is proven that membrane shell theory, differently from moment shell theory, provides results with excellent agreement with the extended Sanders–Koiter shell theory. On the other hand, in the case of shell-like modes, it is found that both membrane and moment shell theories provide results reporting acceptable agreement with the extended Sanders–Koiter shell theory only for very limited ranges of geometries and wavenumbers. Conversely, for shell-like modes it is found that a newly developed, simplified shell model, based on the combination of membrane and semi-moment theories, provides results in satisfactory agreement with the extended Sanders–Koiter shell theory in all ranges.

Vibration analysis of rotating functionally graded tapered beams with hollow circular cross-section

The dynamic modeling and free vibrations of rotating functionally graded (FG) tapered cantilever beams with hollow circular cross-section are studied in this paper. To capture the additional dynamic stiffening terms, the axial shrinkage of the beam caused by the transverse displacement is considered. The dynamic equations of the system governing stretching motion, flapwise bending motion, and chordwise bending motion are derived via employing assumed modes method and Lagrange's equations. Based on the first order approximate coupling (FOAC) dynamic model, natural frequencies and mode shapes of the beam system are calculated by solving eigenvalue problem of the deduced dimensionless vibration equations. Influences of the angular speed, the hub radius, the slenderness ratio, the ratio of hollow radius to the root radius, the taper ratio, and the functional gradient index on natural frequencies are studied. Frequency veering and mode shape interaction are discussed when the bending-stretching mode coupling effect of the beam is considered.

Stochastic Natural Frequency Analysis of Composite Structures Based on Micro-Scale and Meso-Scale Uncertainty

Composite structure often shows undesirably significant uncertainty in its mechanical properties, which may consequently result into large stochastic variation of its natural frequency. This study provides stochastic natural frequency analysis of typical composite structures based on micro-scale (constituent-scale) and meso-scale (ply-scale) uncertainty. Uncertainty propagation across micro-scale and meso-scale is investigated. Response surface method (RSM) based on finite element modeling is employed to obtain approximate natural frequency of structures with complex shape or boundary conditions, and mean value and standard deviation of natural frequency of composite plate and cylindrical shell are derived. Differences in natural frequency statistics of composite plates and cylindrical shells derived by considering uncertainty at different scales are quantified and discussed. Significant statistical correlation between ply elastic properties and ply density is observed, and the statistical correlation is demonstrated to lay great influence on the statistics of structure natural frequency.

Linear vibrations of triple-walled carbon nanotubes

In this paper, the linear vibrations of triple-walled carbon nanotubes (TWNTs) are investigated. A multiple elastic thin shell model is applied. The TWNT dynamics is studied in the framework of the Sanders–Koiter shell theory. The van der Waals interaction between any two layers of the TWNT is modelled by a radius-dependent function. The shell deformation is described in terms of longitudinal, tangential and radial displacements. Simply supported, clamped and free boundary conditions are applied. The three displacement fields are expanded by means of a double mixed series based on Chebyshev polynomials for the longitudinal variable and harmonic functions for the tangential variable. The Rayleigh–Ritz method is applied to obtain approximate natural frequencies and mode shapes. The present model is validated in the linear field by means of comparisons with data from the literature. This study is focused on determining the effect of geometry and boundary conditions on the natural frequencies of TWNTs.

High-precision analytic solution of the problem on bending vibrations of a clamped square plate

An original iteration algorithm is used to construct new analytic expressions for computing approximate natural frequencies and shape modes of bending vibrations of a square homogeneous plate clamped along its contour. The errors are estimated by comparing with the results of well-known numerical high-precision computations. The results of analytic computations are also compared with experimental data obtained by the author by the resonance method. The proposed research technique and the obtained high-precision expressions for the natural shape modes can be used in the case of rectangular plates and for other types of boundary conditions. A numerical-analytical method is used to show that the small isoperimetric theorem holds.

Analytically approximate natural sloshing modes and frequencies for an upright circular container with poles

Natural sloshing modes and frequencies are analytically approximated for an upright circular cylindrical tank equipped with an array of vertical poles (pipes), as happens, for example, in shafts of ocean platforms. The method leads to exact analytical solutions when they exist and demonstrates rather good accuracy in other cases. The approximate natural frequencies are compared with an existing asymptotic estimate.

A novel analytical approximation technique for highly nonlinear oscillators based on the energy balance method

In the present paper, a novel analytical approximation technique has been proposed based on the energy balance method (EBM) to obtain approximate periodic solutions for the focus generalized highly nonlinear oscillators. The expressions of the natural frequency-amplitude relationship are obtained using a novel analytical way. The accuracy of the proposed method is investigated on three benchmark oscillatory problems, namely, the simple relativistic oscillator, the stretched elastic wire oscillator (with a mass attached to its midpoint) and the Duffing-relativistic oscillator. For an initial oscillation amplitude A0=100, the maximal relative errors of natural frequency found in three oscillators are 2.1637%, 0.0001% and 1.201%, respectively, which are much lower than the errors found using the existing methods. It is highly remarkable that an excellent accuracy of the approximate natural frequency has been found which is valid for the whole range of large values of oscillation amplitude as compared with the exact ones. Very simple solution procedure and high accuracy that is found in three benchmark problems reveal the novelty, reliability and wider applicability of the proposed analytical approximation technique.

Analytically approximate natural sloshing modes and frequencies in two-dimensional tanks

Analytical approaches to nonlinear and linear sloshing problems need to know approximate natural sloshing modes expressed by continuously-differentiable harmonic functions. A new method for constructing those approximate modes as well as for a fast computing of the corresponding natural sloshing frequencies is proposed in the two-dimensional case. The method facilities a parametric study of the natural sloshing frequencies in a prismatic tank associated with LNG (Liquefied Natural Gas) containers. The results are extensively compared with other approximate analytical solutions.

Frequency Analysis of an Arbitrarily Supported, Exponentially Tapered Beam

In most introductory vibration textbooks, the vibration of non-uniform beams is not covered. In this paper, the free vibration of an arbitrarily supported Euler–Bernoulli beam of constant thickness and exponentially varying width is investigated. The ‘assumed modes‘ method is utilized to obtain the approximate natural frequencies of the tapered beam, where the basis functions used in the expansion correspond to the eigenfunctions of the uniform beam, including its rigid-body modes. Expressing the eigenfunctions in terms of exponentials, the generalized masses and stiffnesses for the non-uniform beam can all be obtained in closed form, which substantially simplifies the analysis. The proposed scheme is simple to code, efficient to apply and can be easily modified to accommodate a tapered beam with various boundary conditions. Additionally, with slight modifications, the proposed method can also be used to analyze the free vibration of a combined system consisting of an exponentially varying beam carrying lumped attachments. Numerical experiments validate the present approach, and show that the approximate natural frequencies converge very quickly to the exact solution.

Closed-form approximate solution for natural frequency of axially moving beams

In this study, the artificial parameter method is utilized to find closed-form, approximate natural frequencies of axially moving beam. The method provides efficient approximate natural frequencies for the beam with different boundary conditions. The validity and accuracy of the obtained solution are examined by comparing with numeric values and data found in the literature. It is shown that the present solution is valid and accurate for both high axial speed and large bending stiffness.

Transverse vibration and stability of spinning circular plates of constant thickness and different boundary conditions

The lower approximate natural frequencies of a spinning circular plate are determined for various boundary conditions. The plate is rotating with constant speed of spin about the axis of symmetry perpendicular to its plane. The boundary conditions treated here are: clamped, simply supported, free and guided. It is assumed that the normal displacement of the plate is small and that it does not affect during the oscillation the angular and radial variable stress distribution due to the rotating of the plate. This divides the problem at hand into two parts, of which the first part determines the radial and angular variable planar stresses in the plate, while the second part requires the solution of the partial differential equation with variable coefficients. The approximate lower natural frequencies are determined as functions of the speed of spin and exhibit in the plates with clamped, guided and simply supported boundary conditions instabilities.

Second order mode-finding method in dynamic stiffness matrix methods

This paper addresses the transcendental eigenvalue problem, which arises for those structures, e.g., frames, for which dynamic member stiffnesses are available and are exact, in the sense that the appropriate differential equations have been solved. Thus it does not relate to traditional finite element vibration (or buckling) methods which are approximate and give a linear eigenvalue problem which can be solved by many excellent methods. The well-established Wittrick–Williams algorithm is a powerful, reliable and efficient means of obtaining the desired number of natural frequencies for transcendental eigenvalue problems but, except for one very recent paper involving explicit derivatives and a recursive method, the corresponding vibration mode computations suffer from various difficulties and far from match the elegance of the frequency computations. This paper presents a newly developed, mathematically elegant and computationally efficient algorithm for accurate and reliable computation of vibration modes. It uses standard inverse iteration with approximate natural frequencies of only first order accuracy to produce the corresponding vibration modes to second order accuracy. Extrapolation can then be used to improve the original natural frequencies to second order accuracy. Moreover, the algorithm automatically provides a helpful so-called “ μ ̃ -check” which enables the acceptability of the mode accuracy to be judged. Numerical examples presented include some demanding ones, e.g., with coincident natural frequencies. These show the excellent performance of the method. Two advantages over the recent recursive paper are that explicit derivatives are replaced by differencing and recursion is avoided. In combination, these two advantages make the new method much more suitable for retrofitting into existing computer programs.

Closed-form fundamental-frequency estimates for polar orthotropic circular plates

This paper provides simple, yet accurate, estimates of the fundamental natural frequencies of vibration of simply supported and clamped polar orthotropic circular plates with and without internal (central) point supports. Such estimates are obtained by means of Rayleigh’s method coupled with the deflection functions for the corresponding statically loaded systems. As will be seen, the present formulation, which generalizes an earlier work dealing with plates composed of isotropic materials, yields compact, closed-form expressions, enabling the ready calculation of (approximate) natural frequencies for arbitrary values of the elastic constants. Furthermore, remarkable simplifications in the derivations for, and form of, some of these expressions are achieved through the application of a computerized symbolic algebra system.