scholarly journals Free and Forced Vibrations of an Axially-Loaded Timoshenko Multi-Span Beam Carrying a Number of Various Concentrated Elements

2012 ◽  
Vol 19 (4) ◽  
pp. 735-752 ◽  
Author(s):  
Yusuf Yesilce
Keyword(s):  
Axial Force ◽  
Natural Frequencies ◽  
Beam Theory ◽  
Coefficient Matrix ◽  
Mode Shapes ◽  
Harmonic Force ◽  
Concentrated Force ◽  
Vibrating System ◽  
Assembly Technique ◽  
Vibrating Systems

In the existing reports regarding free and forced vibrations of the beams, most of them studied a uniform beam carrying various concentrated elements using Bernoulli-Euler Beam Theory (BET) but without axial force. The purpose of this paper is to utilize the numerical assembly technique to determine the exact frequency-response amplitudes of the axially-loaded Timoshenko multi-span beam carrying a number of various concentrated elements (including point masses, rotary inertias, linear springs and rotational springs) and subjected to a harmonic concentrated force and the exact natural frequencies and mode shapes of the beam for the free vibration analysis. The model allows analyzing the influence of the shear and axial force and harmonic concentrated force effects and intermediate concentrated elements on the dynamic behavior of the beams by using Timoshenko Beam Theory (TBT). At first, the coefficient matrices for the intermediate concentrated elements, an intermediate pinned support, applied harmonic force, left-end support and right-end support of Timoshenko beam are derived. After the derivation of the coefficient matrices, the numerical assembly technique is used to establish the overall coefficient matrix for the whole vibrating system. Finally, solving the equations associated with the last overall coefficient matrix one determines the exact dynamic response amplitudes of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force. Equating the determinant of the overall coefficient matrix to zero one determines the natural frequencies of the free vibrating system (the case of zero harmonic force) and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. The calculated vibration amplitudes of the forced vibrating systems and the natural frequencies of the free vibrating systems are given in tables for different values of the axial force. The dynamic response amplitudes and the mode shapes are presented in graphs. The effects of axial force and harmonic concentrated force on the vibration analysis of Timoshenko multi-span beam are also investigated.

scholarly journals Free Vibrations of a Reddy-Bickford Multi-Span Beam Carrying Multiple Spring-Mass Systems

2011 ◽  
Vol 18 (5) ◽  
pp. 709-726 ◽  
Author(s):  
Yusuf Yesilce
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Structural Elements ◽  
Mass System ◽  
Assembly Technique

The structural elements supporting motors or engines are frequently seen in technological applications. The operation of machine may introduce additional dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko single-span beams carrying a number of spring-mass system and multi-span beams carrying multiple spring-mass systems are plenty, but the free vibration analysis of Reddy-Bickford multi-span beams carrying multiple spring-mass systems has not been investigated by any of the studies in open literature so far. This paper aims at determining the exact solutions for the natural frequencies and mode shapes of Reddy-Bickford beams. The model allows analyzing the influence of the shear effect and spring-mass systems on the dynamic behavior of the beams by using Reddy-Bickford Beam Theory (RBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The natural frequencies of Reddy-Bickford single-span and multi-span beams calculated by using the numerical assembly technique and the secant method are compared with the natural frequencies of single-span and multi-span beams calculated by using Timoshenko Beam Theory (TBT); the mode shapes are presented in graphs.

scholarly journals NASTRAN Analysis of a Turbine Blade and Comparison With Test and Field Data

1975 ◽  
Author(s):  
J. M. Allen ◽  
L. B. Erickson
Keyword(s):  
Turbine Blade ◽  
Natural Frequencies ◽  
Beam Theory ◽  
Mode Shapes ◽  
Stress Distributions ◽  
Free Standing ◽  
Element Analysis ◽  
Metallographic Examination ◽  
Stiffening Effect ◽  
Generalized Beam Theory

A NASTRAN finite element analysis of a free standing gas turbine blade is presented. The analysis entails calculation of the first four natural frequencies, mode shapes, and relative vibratory stresses, as well as deflections and stresses due to centrifugal loading. The stiffening effect of the centrifugal force field was accounted for by using NASTRAN’s differential stiffness option. Natural frequencies measured in a rotating test correlated well with computed results. Areas of maximum vibratory stress (fundamental mode) coincided with the three zones of crack initiation observed in a metallographic examination of a fatigue failure. Airfoil stress distributions were found to be significantly different from that predicted by generalized beam theory, especially near the airfoil-platform junction.

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.

An Exact Solution for the Natural Frequencies of Flexible Beams Undergoing Overall Motions

2003 ◽  
Vol 9 (11) ◽  
pp. 1221-1229 ◽  
Author(s):  
Ali H Nayfeh ◽  
S.A. Emam ◽  
Sergio Preidikman ◽  
D.T. Mook
Keyword(s):  
Exact Solution ◽  
Natural Frequencies ◽  
Three Dimensional ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Bernoulli Beam ◽  
Flexible Beams ◽  
Euler Bernoulli Beam

We investigate the free vibrations of a flexible beam undergoing an overall two-dimensional motion. The beam is modeled using the Euler-Bernoulli beam theory. An exact solution for the natural frequencies and corresponding mode shapes of the beam is obtained. The model can be extended to beams undergoing three-dimensional motions.

Thermo-Electro-Mechanical Vibration Characteristics of Graphene/Piezoelectric/Graphene Sandwich Nanobeams

2017 ◽  
Vol 35 (1) ◽  
pp. 65-79 ◽  
Author(s):  
N. Kammoun ◽  
H. Jrad ◽  
S. Bouaziz ◽  
M. B. Amar ◽  
M. Soula ◽  
...  
Keyword(s):  
Nonlocal Elasticity ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Mechanical Vibration ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Mode Shapes ◽  
Governing Equations ◽  
Numerical Examples ◽  
Generalized Differential

AbstractThis paper reports an investigation on thermo-electro-mechanical vibration of graphene/piezoelectric graphene/piezoelectric/graphene sandwich nanobeams. Based on the nonlocal elasticity theory, Timoshenko beam theory and Hamilton's principles, the governing equations are developed and solved using generalized differential quadrature (GDQ) method. The effects of the nonlocal parameter, external electrical voltage, temperature change and axial force on vibration of graphene/piezoelectric/graphene sandwich nanobeams are examined. The performance and the accuracy of the presented model are highlighted through numerical examples with different boundary conditions. This study reports that the nonlocal parameter and thermo-electro-mechanical loadings have important effect on the natural frequencies and the deflection mode shapes of the graphene/piezoelectric/graphene sandwich nanobeam. The present work can serve as guideline for the design of a nanoscale graphene/piezoelectric/graphene beams based electromechanical resonator sensors.

Structural Response of Pipeline Free Spans Based on Beam Theory

2002 ◽  
Author(s):  
Olav Fyrileiv ◽  
Kim Mo̸rk
Keyword(s):  
Natural Frequencies ◽  
Structural Response ◽  
Beam Theory ◽  
Vertical Vibration ◽  
Mode Shapes ◽  
Ocean Current ◽  
Vibration Modes ◽  
Wave Loading ◽  
Vortex Induced Vibrations ◽  
Subsea Pipelines

One of the main risk factors for subsea pipelines exposed on the seabed is fatigue failure of free spans due to ocean current or wave loading. This paper describes how the structural response of a free span, as input to the fatigue analyses, can be assessed in a simple and still accurate way by using improved beam theory formulations. In connection with the release of the DNV Recommended Practice, DNV-RP-F105 “Free Spanning Pipelines”, the simplified structural response quantities have been improved compared to previous codes. The boundary condition coefficients for the beam theory formulations have been updated based an effective span length concept. This concept is partly based on theoretical studies and partly on a large number of FE analyses. The updated expressions are general and fit all types of soil and pipe dimensions for lower lateral and vertical vibration modes. The present paper focus on estimation of simplified response quantities such as lower natural frequencies and associated mode shapes. Hydrodynamical aspects of Vortex Induced Vibrations (VIV) are outside the scope of this paper.

scholarly journals On the Finite Element Free Vibration Analysis of Delaminated Layered Beams: A New Assembly Technique

2021 ◽  
Author(s):  
Nicholas H. Erdelyi ◽  
Seyed M. Hashemi
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Bending Vibration ◽  
Natural Frequencies And Modes ◽  
Layered Beams ◽  
Assembly Technique ◽  
Free Mode ◽  
Element Free

The dynamic analysis of flexible delaminated layered beams is revisited. Exploiting Boolean vectors, a novel assembly scheme is developed which can be used to enforce the continuity requirements at the edges of delamination region, leading to a delamination stiffness term. The proposed assembly technique can be used to form various beam configurations with through width delaminations, irrespective of the formulation used to model each beam segment. The proposed assembly system and the Galerkin Finite Element Method (FEM) formulation are subsequently used to investigate the natural frequencies and modes of 2- and 3-layer beam configurations. Using the Euler-Bernoulli bending beam theory and free mode delamination, the governing differential equations are exploited and two beam finite elements are developed. The free bending vibration of three illustrative example problems, characterized by delamination zones of variable length, is investigated. The intact and defective beam natural frequencies and modes obtained from the proposed assembly/FEM beam formulations are presented along with the analytical results and those available in the literature

scholarly journals Natural Frequencies and Mode Shapes of Drill Pipe in Subsea Xmas Tree Installation

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Wensheng Xiao ◽  
Haozhi Qin ◽  
Jian Liu ◽  
Qi Liu ◽  
Junguo Cui ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Drill Pipe ◽  
Beam Theory ◽  
Transformation Method ◽  
Variable Coefficients ◽  
Mode Shapes ◽  
Order Partial Differential Equation ◽  
Lumped Mass ◽  
Acceleration Sensors

In this study, experimental and numerical investigations on the vibration characteristics of a drill pipe during the lowering of a subsea Xmas tree were presented. A fourth-order partial differential equation with variable coefficients was established based on Euler–Bernoulli beam theory. The natural frequencies and mode shapes are obtained by using the differential transformation method. Four drill pipe models of different sizes were used in the experiments which were measured using piezoelectric acceleration sensors and fiber Bragg grating sensors, respectively. The factors that affect the natural frequencies and mode shapes, such as length, diameter, lumped mass, and boundary conditions, were analyzed. The results show that all factors have remarkable effects on the natural frequency, but changes in the length and diameter of the pipe have little effect on the mode shapes; the main factors affecting the mode shape are the boundary conditions and lumped mass. The results of the numerical calculation were validated by a comparison with the experimental results and showed good agreement.

Transverse Vibration of Two Axially Moving Beams Connected by an Elastic Foundation

2005 ◽  
Author(s):  
Mohamed Gaith ◽  
Sinan Mu¨ftu¨
Keyword(s):  
Elastic Foundation ◽  
Transverse Vibration ◽  
Natural Frequencies ◽  
Flexural Rigidity ◽  
Beam Theory ◽  
Mode Shapes ◽  
Winkler Elastic Foundation ◽  
Canonical State ◽  
Axially Moving Beams ◽  
Axially Moving

Transverse vibration of two axially moving beams connected by a Winkler elastic foundation is analyzed analytically. The system is a model of paper and paper-cloth (wire-screen) used in paper making. The two beams are tensioned, translating axially with a common constant velocity, simply supported at their ends, and of different materials and geometry. Due to the effect of translation, the dynamics of the system displays gyroscopic motion. The Euler-Bernoulli beam theory is used to model the deflections, and the governing equations are expressed in the canonical state form. The natural frequencies and associated mode shapes are obtained. It is found that the natural frequencies of the system are composed of two infinite sets describing in-phase and out-of-phase vibrations. In case the beams are identical, these modes become synchronous and asynchronous, respectively. Divergence instability occurs at the critical velocity; and, the frequency-velocity relationship is similar to that of a single traveling beam. The effects of the mass, flexural rigidity, and axial tension ratios of the two beams, as well as the effects of the elastic foundation stiffness are investigated.

scholarly journals On the Finite Element Free Vibration Analysis of Delaminated Layered Beams: A New Assembly Technique

2021 ◽  
Author(s):  
Nicholas H. Erdelyi ◽  
Seyed M. Hashemi
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Bending Vibration ◽  
Natural Frequencies And Modes ◽  
Layered Beams ◽  
Assembly Technique ◽  
Free Mode ◽  
Element Free

The dynamic analysis of flexible delaminated layered beams is revisited. Exploiting Boolean vectors, a novel assembly scheme is developed which can be used to enforce the continuity requirements at the edges of delamination region, leading to a delamination stiffness term. The proposed assembly technique can be used to form various beam configurations with through width delaminations, irrespective of the formulation used to model each beam segment. The proposed assembly system and the Galerkin Finite Element Method (FEM) formulation are subsequently used to investigate the natural frequencies and modes of 2- and 3-layer beam configurations. Using the Euler-Bernoulli bending beam theory and free mode delamination, the governing differential equations are exploited and two beam finite elements are developed. The free bending vibration of three illustrative example problems, characterized by delamination zones of variable length, is investigated. The intact and defective beam natural frequencies and modes obtained from the proposed assembly/FEM beam formulations are presented along with the analytical results and those available in the literature

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