Natural Frequencies of Long Tapered Cantilevers

1967 ◽  
Vol 18 (4) ◽  
pp. 309-320 ◽  
Author(s):  
W. Carnegie ◽  
J. Thomas
Keyword(s):  
Eigenvalue Problem ◽  
Digital Computer ◽  
Natural Frequencies ◽  
Flexural Vibration ◽  
Mode Shapes ◽  
Bernoulli Equation ◽  
Eigenvalues And Eigenvectors ◽  
Wide Range ◽  
Modes Of Vibration ◽  
Theoretical Results

SummaryThe effect of depth taper on the flexural vibration characteristics of a beam of uniform width is investigated in this paper. The frequency parameters and mode shapes for the first five modes of vibration of tapered beams are presented for a wide range of depth taper. The Euler-Bernoulli equation of a beam is reduced to an eigenvalue problem and its eigenvalues and eigenvectors are obtained by using a digital computer. The theoretical results are compared with those of other authors and with the experimental results.

scholarly journals An investigation into the free vibrations of carbon nanotubes using analytical and finite element methods

2021 ◽  
Author(s):  
Ishan Ali Khan
Keyword(s):  
Carbon Nanotubes ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Natural Frequencies ◽  
Nonlinear Eigenvalue Problem ◽  
Dynamic Stiffness ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Wide Range ◽  
Walled Cnts

Since their discovery, immense attention has been given to carbon nanotubes (CNTs), due to their exceptional thermal, electronic and mechanical properties and, therefore, the wide range of applications in which they are, or can be potentially, employed. Hence, it is important that all the properties of carbon nanotubes are studied extensively. This thesis studies the vibrational frequencies of double-walled and triple-walled CNTs, with and without an elastic medium surrounding them, by using Finite Element Method (FEM) and Dynamic Stiffness Matrix (DSM) formulations, considering them as Euler-Bernoulli beams coupled with van der Waals interaction forces. For FEM modelling, the linear eigenvalue problem is obtained using Galerkin weighted residual approach. The natural frequencies and mode shapes are derived from eigenvalues and eigenvectors, respectively. For DSM formulation of double-walled CNTs, a nonlinear eigenvalue problem is obtained by enforcing displacement and load end conditions to the exact solution of single equation achieved by combining the coupled governing equations. The natural frequencies are obtained using Wittrick-Williams algorithm. FEM formulation is also applied to both double and triple-walled CNTs modelled as nonlocal Euler-Bernoulli beam. The natural frequencies obtained for all the cases, are in agreement with the values provided in literature.

scholarly journals Vibrations of Complete Spherical Shells With Imperfections

2007 ◽  
Vol 129 (3) ◽  
pp. 363-370 ◽  
Author(s):  
Thomas A. Duffey ◽  
Jason E. Pepin ◽  
Amy N. Robertson ◽  
Michael L. Steinzig ◽  
Kimberly Coleman
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Spherical Shells ◽  
The Past ◽  
Dynamic Holography ◽  
Theoretical Investigations ◽  
Modes Of Vibration ◽  
Geometrical Imperfections ◽  
Theoretical Results

Numerous theoretical investigations on the natural frequencies for complete spherical shells have been reported over the past four decades. However, attempts at correlating the theoretical results with either experimental or simulated results (both for axisymmetric and nonaxisymmetric modes of vibration) are almost completely lacking. In this paper, natural frequencies and mode shapes obtained from axisymmetric and nonaxisymmetric theories of vibration of complete spherical shells and from finite element computer simulations of the vibrations, with and without geometrical imperfections, are presented. Modal tests reported elsewhere on commercially available, thin spherical marine floats (with imperfections) are then utilized as a basis for comparison of frequencies to both the theoretical and numerical results. Because of the imperfections present, “splitting” of frequencies of nonaxisymmetric modes is anticipated. The presence of this frequency splitting phenomenon is demonstrated. In addition, results of a “whole field” measurement on one of the imperfect shells using dynamic holography are presented.

scholarly journals An investigation into the free vibrations of carbon nanotubes using analytical and finite element methods

2021 ◽  
Author(s):  
Ishan Ali Khan
Keyword(s):  
Carbon Nanotubes ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Natural Frequencies ◽  
Nonlinear Eigenvalue Problem ◽  
Dynamic Stiffness ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Wide Range ◽  
Walled Cnts

Since their discovery, immense attention has been given to carbon nanotubes (CNTs), due to their exceptional thermal, electronic and mechanical properties and, therefore, the wide range of applications in which they are, or can be potentially, employed. Hence, it is important that all the properties of carbon nanotubes are studied extensively. This thesis studies the vibrational frequencies of double-walled and triple-walled CNTs, with and without an elastic medium surrounding them, by using Finite Element Method (FEM) and Dynamic Stiffness Matrix (DSM) formulations, considering them as Euler-Bernoulli beams coupled with van der Waals interaction forces. For FEM modelling, the linear eigenvalue problem is obtained using Galerkin weighted residual approach. The natural frequencies and mode shapes are derived from eigenvalues and eigenvectors, respectively. For DSM formulation of double-walled CNTs, a nonlinear eigenvalue problem is obtained by enforcing displacement and load end conditions to the exact solution of single equation achieved by combining the coupled governing equations. The natural frequencies are obtained using Wittrick-Williams algorithm. FEM formulation is also applied to both double and triple-walled CNTs modelled as nonlocal Euler-Bernoulli beam. The natural frequencies obtained for all the cases, are in agreement with the values provided in literature.

scholarly journals Using Pade´ Approximants and Curve-fitting to Approximate Eigenvalues and Eigenvectors for Large Design Changes

1996 ◽  
Vol 118 (1) ◽  
pp. 151-153 ◽  
Author(s):  
J. M. Vance ◽  
J. E. Bernard
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Solution Process ◽  
Mode Shapes ◽  
Eigenvalues And Eigenvectors ◽  
Computational Burden ◽  
Design Change ◽  
Design Changes ◽  
Develop Software ◽  
Unique Method

Our overall goal is to develop software that facilitates the interactive participation of the designer in the optimization process. We are focusing this research on problems which use finite element solutions as part of the objective function. One challenge to implementing interactive participation in these types of problems is the high computational burden of computing a finite element solution for each design change. The research presented here focuses on a unique method to develop fast approximations for natural frequencies and mode shapes which can be used to avoid the time-consuming re-solution process and which will facilitate interactive design for systems with even large design changes.

Flexural Vibration of Prestressed Concrete Bridges with Corrugated Steel Webs

2017 ◽  
Vol 17 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Xia-Chun Chen ◽  
Zhen-Hu Li ◽  
Francis T. K. Au ◽  
Rui-Juan Jiang
Keyword(s):  
Finite Element ◽  
Shear Deformation ◽  
Prestressed Concrete ◽  
Natural Frequencies ◽  
Sandwich Beam ◽  
Flexural Vibration ◽  
Concrete Bridges ◽  
Mode Shapes ◽  
Beam Model ◽  
Prestressed Concrete Bridges

Prestressed concrete bridges with corrugated steel webs have emerged as a new form of steel-concrete composite bridges with remarkable advantages compared with the traditional ones. However, the assumption that plane sections remain plane may no longer be valid for such bridges due to the different behavior of the constituents. The sandwich beam theory is extended to predict the flexural vibration behavior of this type of bridges considering the presence of diaphragms, external prestressing tendons and interaction between the web shear deformation and flange local bending. To this end, a [Formula: see text] beam finite element is formulated. The proposed theory and finite element model are verified both numerically and experimentally. A comparison between the analyses based on the sandwich beam model and on the classical Euler–Bernoulli and Timoshenko models reveals the following findings. First of all, the extended sandwich beam model is applicable to the flexural vibration analysis of the bridges considered. By letting [Formula: see text] denote the square root of the ratio of equivalent shear rigidity to the flange local flexural rigidity, and L the span length, the combined parameter [Formula: see text] appears to be more suitable for considering the diaphragm effect and the interaction between the shear deformation and flange local bending. The diaphragms have significant effect on the flexural natural frequencies and mode shapes only when the [Formula: see text] value of the bridge falls below a certain limit. For a bridge with an [Formula: see text] value over a certain limit, the flexural natural frequencies and mode shapes obtained from the sandwich beam model and the classical Euler–Bernoulli and Timoshenko models tend to be the same. In such cases, either of the classical beam theories may be used.

Analysis of Modal Parameters of Adhesively Bonded Double-Strap Joints by the Modal Strain Energy Method

1993 ◽  
Author(s):  
Mohan D. Rao ◽  
Krishna M. Gorrepati
Keyword(s):  
Strain Energy ◽  
Natural Frequencies ◽  
Structural Parameters ◽  
Flexural Vibration ◽  
Mode Shapes ◽  
Modal Parameters ◽  
Adhesively Bonded ◽  
Modal Strain Energy ◽  
Joint System ◽  
Damping Material

Abstract This paper presents the analysis of modal parameters (natural frequencies, damping ratios and mode shapes) of a simply supported beam with adhesively bonded double-strap joint by the finite-element based Modal Strain Energy (MSE) method using ANSYS 4.4A software. The results obtained by the MSE method are compared with closed form analytical solutions previously obtained by the first author for flexural vibration of the same system. Good agreement has been obtained between the two methods for both the natural frequencies and system loss factors. The effects of structural parameters and material properties of the adhesive on the modal properties of the joint system are also studied which are useful in the design of the joint system for passive vibration and noise control. In order to evaluate the MSE and analytical results, some experiments were conducted using aluminum double-strap joint with 3M ISD112 damping material. The experimental results agreed well with both analytical and MSE results indicating the validity of both analytical and MSE methods. Finally, a comparative study has been conducted using various commercially available damping materials to evaluate their relative merits for use in the design of these joints.

A Note on the Lowest Natural Frequency of a Square Plate Point-Supported at the Corners

1962 ◽  
Vol 66 (616) ◽  
pp. 240-241 ◽  
Author(s):  
C. L. Kirk
Keyword(s):  
Natural Frequency ◽  
Natural Frequencies ◽  
Present Note ◽  
Flexural Vibration ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Electronic Digital Computer ◽  
Isotropic Plates ◽  
Four Corners ◽  
Square Plate

Recently Cox and Boxer determined natural frequencies and mode shapes of flexural vibration of uniform rectangular isotropic plates, that have free edges and pinpoint supports at the four corners. In their analysis, they obtain approximate solutions of the differential equation through the use of finite difference expressions and an electronic digital computer. In the present note, the frequency expression and mode shape for a square plate, vibrating at the lowest natural frequency, are determined by considerations of energy. The values obtained are compared with those given in reference.

scholarly journals Coupled flexural - torsional vibration and stability analysis of pre-loaded beams using conventional and dynamic finite element methods

2021 ◽  
Author(s):  
Heenkenda Jayasinghe
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Axial Force ◽  
Torsional Vibration ◽  
Natural Frequencies ◽  
Nonlinear Eigenvalue Problem ◽  
Compressive Force ◽  
Mode Shapes ◽  
Dynamic Finite Element ◽  
Starting Point

Dynamic Finite Element (DFE) and conventional finite element formulations are developed to study the flexural - torsional vibration and stability of an isotropic, homogeneous and linearly elastic pre-loaded beam subjected to an axial load and end-moment. Various classical boundary conditions are considered. Elementary Euler - Bernoulli bending and St. Venant torsion beam theories were used as a starting point to develop the governing equations and the finite element solutions. The nonlinear Eigenvalue problem resulted from the DFE method was solved using a program code written in MATLAB and the natural frequencies and mode shapes of the system were determined form the Eigenvalues and Eigenvectors, respectively. Similarly, a linear Eigenvalue problem was formulated and solved using a MATLAB code for the conventional FEM method. The conventional FEM results were validated against those available in the literature and ANSYS simulations and the DFE results were compared with the FEM results. The results confirmed that tensile forces increased the natural frequencies, which indicates beam stiffening. On the contrary, compressive forces reduced the natural frequencies, suggesting a reduction in beam stiffness. Similarly, when an end-moment was applied the stiffness of the beam and the natural frequencies diminished. More importantly, when a force and end-moment were acting in combination, the results depended on the direction and magnitude of the axial force. Nevertheless, the stiffness of the beam is more sensitive to the changes in the magnitude and direction of the axial force compared to the moment. A buckling analysis of the beam was also carried out to determine the critical buckling end-moment and axial compressive force.

scholarly journals UNCOUPLED VIBRATIONS IN FUNCTIONALLY GRADED TIMOSHENKO BEAM

2016 ◽  
Vol 54 (6) ◽  
pp. 785 ◽  
Author(s):  
Nguyen Tien Khiem ◽  
Nguyen Ngoc Huyen
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Flexural Vibration ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Power Law Distribution ◽  
Material Parameters ◽  
Flexural Vibrations ◽  
Fgm Beam

Free vibration of FGM Timoshenko beam is investigated on the base of the power law distribution of FGM. Taking into account the actual position of neutral plane enables to obtain general condition for uncoupling of axial and flexural vibrations in FGM beam. This condition defines a class of functionally graded beams for which axial and flexural vibrations are completely uncoupled likely to the homogeneous beams. Natural frequencies and mode shapes of uncoupled flexural vibration of beams from the class are examined in dependence on material parameters and slendernes

Some Results on Axisymmetric Vibration of Spherical Shells of Variable Thickness

1990 ◽  
Vol 112 (4) ◽  
pp. 432-437 ◽  
Author(s):  
A. V. Singh ◽  
S. Mirza
Keyword(s):  
Variable Thickness ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Axisymmetric Vibration ◽  
Spherical Shells ◽  
Varying Thickness ◽  
Wide Range

Natural frequencies and mode shapes are presented for the free axisymmetric vibration of spherical shells with linearly varying thickness along the meridian. Clamped and hinged edges corresponding to opening angles 30, 45, 60 and 90 deg have been considered in this technical brief to cover a wide range from shallow to deep spherical shells. Variations in thickness are seen to have very pronounced effects on the frequencies and mode shapes.

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