Effects of Different Models on Natural Frequencies of Short Span Bridges Used in High Speed Rail

2015 ◽  
Author(s):  
Said I. Nour ◽  
Mohsen A. Issa
Keyword(s):  
Timoshenko Beam ◽  
High Speed ◽  
Natural Frequencies ◽  
Bernoulli Beam ◽  
High Speed Rail ◽  
Vertical Stiffness ◽  
Short Span ◽  
Elastically Supported ◽  
Euler Bernoulli Beam ◽  
Track Modulus

The natural frequencies of vibration of short span bridges used in high-speed rail were investigated. Three different models of increasing complexity were evaluated and their effects on the vibration frequency were compared to the first basic model of simply supported Euler-Bernoulli beam. In the second and third cases, the bridge was modeled as an Euler-Bernoulli and Timoshenko beam supported at its two ends by identical spring elements with an equivalent vertical stiffness to simulate elastomeric bearings and soil foundation. The boundary value problem was solved numerically to extract the bridge eigenfrequencies. In the case of Euler-Bernoulli beam, curve fitting techniques were used to deduce accurate simple empirical formulae to calculate the first six natural frequencies of an elastically supported bridge. In the case of a Timoshenko beam, graphical solutions were proposed to compute the fundamental frequency. Results confirmed that the use of Timoshenko beam theory reduces the natural frequency and the consideration of flexible supports further decreases the natural frequency. In the fourth model, the interaction of the track and the bridge was included. The bridge was modeled as an elastically supported beam and the track was modeled as a spring-damper element with an equivalent vertical stiffness resulting from track components like rail pads, cross-ties and ballast. A parametric study was performed to analyze the effects of the track stiffness on the natural frequencies of the bridge. Graphical solutions were presented to quantify the change of the normalized natural frequencies of the system with the increase in the track modulus. Results indicated that the changes in the track modulus have no significant effects in models with rigid supports. A decrease in the fundamental frequency was noticeable with softer track modulus as the support flexibility increased.

Natural frequencies of Euler-Bernoulli beam with open cracks on elastic foundations

2006 ◽  
Vol 20 (4) ◽  
pp. 467-472 ◽  
Author(s):  
Youngjae Shin ◽  
Jonghak Yun ◽  
Kyeongyoun Seong ◽  
Jaeho Kim ◽  
Sunghwang Kang
Keyword(s):  
Natural Frequencies ◽  
Bernoulli Beam ◽  
Elastic Foundations ◽  
Euler Bernoulli Beam

Medium Frequency Vibration Analysis of Beam Structures Modeled by the Timoshenko Beam Theory

2020 ◽  
Author(s):  
Yichi Zhang ◽  
Bingen Yang
Keyword(s):  
Shear Deformation ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Beam Structure ◽  
Medium Frequency ◽  
Beam Structures ◽  
Euler Bernoulli Beam

Abstract Vibration analysis of complex structures at medium frequencies plays an important role in automotive engineering. Flexible beam structures modeled by the classical Euler-Bernoulli beam theory have been widely used in many engineering problems. A kinematic hypothesis in the Euler-Bernoulli beam theory is that plane sections of a beam normal to its neutral axis remain normal when the beam experiences bending deformation, which neglects the shear deformation of the beam. However, as observed by researchers, the shear deformation of a beam component becomes noticeable in high-frequency vibrations. In this sense, the Timoshenko beam theory, which describes both bending deformation and shear deformation, may be more suitable for medium-frequency vibration analysis of beam structures. This paper presents an analytical method for medium-frequency vibration analysis of beam structures, with components modeled by the Timoshenko beam theory. The proposed method is developed based on the augmented Distributed Transfer Function Method (DTFM), which has been shown to be useful in various vibration problems. The proposed method models a Timoshenko beam structure by a spatial state-space formulation in the s-domain, without any discretization. With the state-space formulation, the frequency response of a beam structure, in any frequency region (from low to very high frequencies), can be obtained in an exact and analytical form. One advantage of the proposed method is that the local information of a beam structure, such as displacements, bending moment and shear force at any location, can be directly obtained from the space-state formulation, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated with the FEA in numerical examples, where the efficiency and accuracy of the proposed method is present. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are illustrated through comparison of the Timoshenko beam theory and the Euler-Bernoulli beam theory.

scholarly journals Macaulay's Method for a Timoshenko Beam

2007 ◽  
Vol 35 (4) ◽  
pp. 285-292 ◽  
Author(s):  
N. G. Stephen
Keyword(s):  
Timoshenko Beam ◽  
Bending Moment ◽  
Shearing Force ◽  
Bernoulli Beam ◽  
Deflection Curve ◽  
Axial Coordinate ◽  
Deflection Analysis ◽  
Euler Bernoulli Beam

The Macaulay bracket notation is familiar to many engineers for the deflection analysis of a Euler–Bernoulli beam subject to multiple or discontinuous loads. An expression for the internal bending moment, and hence curvature, is valid at all locations along the beam, and the deflection curve can be calculated by integrating twice with respect to the axial coordinate. The notation obviates the need for matching of multiple constants of integration for the various sections of the beam. Here, the method is extended to a Timoshenko beam, which includes the additional deflection due to shear. This requires an additional expression for the shearing force, also valid at all locations along the beam.

scholarly journals A Structured Approach to Solve the Inverse Eigenvalue Problem for a Beam with Added Mass

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Farhad Mir Hosseini ◽  
Natalie Baddour
Keyword(s):  
Natural Frequencies ◽  
Point Of View ◽  
Inverse Eigenvalue Problem ◽  
Bernoulli Beam ◽  
Combined System ◽  
Vibrational System ◽  
Single Mass ◽  
Inverse Eigenvalue ◽  
Euler Bernoulli Beam ◽  
Structured Approach

The problem of determining the eigenvalues of a vibrational system having multiple lumped attachments has been investigated extensively. However, most of the research conducted in this field focuses on determining the natural frequencies of the combined system assuming that the characteristics of the combined vibrational system are known (forward problem). A problem of great interest from the point of view of engineering design is the ability to impose certain frequencies on the vibrational system or to avoid certain frequencies by modifying the characteristics of the vibrational system (inverse problem). In this paper, a method to impose two natural frequencies on a dynamical system consisting of an Euler-Bernoulli beam and carrying a single mass attachment is evaluated.

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.

Mechanical Characters of Six Engineering Timoshenko Beams

2014 ◽  
Vol 668-669 ◽  
pp. 201-204
Author(s):  
Hong Liang Tian
Keyword(s):  
Timoshenko Beam ◽  
Analytic Solutions ◽  
Timoshenko Beams ◽  
Bernoulli Beam ◽  
Simply Supported ◽  
Normality Assumption ◽  
Length Direction ◽  
The Right ◽  
Euler Bernoulli Beam ◽  
Mechanical Characters

Timoshenko beam is an extension of Euler-Bernoulli beam to interpret the transverse shear impact. The more refined Timoshenko beam relaxes the normality assumption of plane section that remains plane and normal to the deformed centerline. The manuscript presents some exact concise analytic solutions on deflection and stress resultants of NET single-span Timoshenko beam with general distributed force and 6 kinds of standard boundary conditions, adopting its counterpart Euler-Bernoulli beam solutions. Engineering example shows that scale impact would not unveil itself for micro structure with micrometer μm-order length, yet will be prominent for nanostructure with nanometer nm-order length. When simply supported CNTs is undergone to a concentrative force at the median and complete bend moment, scale action is observed along the ensemble CNTs, while it unfurls itself the most at the position of the concentrated strength. When a clamped-free CNTs is exposed to a centralized force at the mesial and distributed force, there is no scale impact about the deflection at all positions on the left border of the concentrated strength position, while such operation inspires at once at all positions on the right margin of the concentrated strength position. When a clamped-clamped CNTs is lain under a concentrative strength at the middle, the deflection of NET Euler-Bernoulli CNTs reflects scale effect completely. Notable differences between the deflection of Euler-Bernoulli CNTs and that of Timoshenko CNTs are reflected at large ratio of diameter versus length. The deflection of NET clamped-free and simply supported Timoshenko beam doesn’t introduce surplus scale process in terms of its counterpart, NET Euler-Bernoulli beam. However, the deflection of NET clamped-clamped Timoshenko beam does involve additional scale impact solely including the method when the concentrated strength position is at the midway in the beam-length direction.

scholarly journals An Accurate Estimation of the Eigenfrequency of an Offshore Wind Turbine Considered as the Stepped Euler-Bernoulli Beam in Three-Spring Flexible Foundation Using the Power Series Method

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mouafo Teifouet Armand Robinson ◽  
Zhenyu Wang
Keyword(s):  
Power Series ◽  
Wind Turbine ◽  
Wall Thickness ◽  
Natural Frequencies ◽  
Offshore Wind ◽  
Series Method ◽  
Bernoulli Beam ◽  
Power Series Method ◽  
Flexible Foundation ◽  
Euler Bernoulli Beam

The present study employs the power series method (PSM) to accurately predict the natural frequencies of eleven offshore wind turbines (OWT). This prediction is very important as it helps in the quick verification of experimental or finite element results. This study idealizes the OWT as a stepped Euler-Bernoulli beam carrying a top mass and connected at its bottom to a flexible foundation. The first part of the beam represents a monopile and the transition piece while its second part is a tower. The foundation is modeled using three springs (lateral, rotational, and cross-coupling springs). This work’s aim is at improving therefore the previous researches, in which the whole wind turbine was taken as a single beam, with a tower being tapered and its wall thickness being negligible compared to its diameter. In order to be closer to real-life OWT, three profiles of the tapered tower are explored: case 1 considers a tower with constant thickness along its height. Case 2 assumes a tower’s thickness being negligible compared to its mean diameter, while case 3 describes the tower as a tapered beam with varying thickness along its height. Next, the calculated natural frequencies are compared to those obtained from measurements. Results reveal that case 2, used by previous researches, was only accurate for OWT with tower wall thickness lower than 15 mm. Frequencies produced in case 3 are the most accurate as the relative error is up to 0.01%, especially for the OWT with thicknesses higher or equal to 15 mm. This case appears to be more realistic as, practically, wall thickness of a wind tower varies with its height. The tower-to-pile thickness ratio is an important design parameter as it highly has impact on the natural frequency of OWT, and must therefore be taken into account during the design as well as lateral and rotational coupling springs.

Effects of Boundary Flexibility in the Moving Oscillator Problem

2001 ◽  
Author(s):  
Chin An Tan ◽  
Yonghong Chen ◽  
Lawrence A. Bergman
Keyword(s):  
Natural Frequency ◽  
Eigenfunction Expansion ◽  
Transverse Direction ◽  
Interaction Force ◽  
Critical Value ◽  
Significant Implication ◽  
Bernoulli Beam ◽  
Elastically Supported ◽  
Euler Bernoulli Beam ◽  
Moving Oscillator

Abstract In this paper, the problem of an oscillator moving across an elastically supported Euler-Bernoulli beam is examined. The oscillator is modeled by a one-degree-of-freedom sprung mass and the end supports are modeled by linear springs in the transverse direction. Solution for the response of the beam is represented by an eigenfunction expansion series. Numerical results are obtained for the eigenvalues and the response of the elastically supported beam, and the interaction force (force in the oscillator spring). To guide the discussion, a critical value of the support stiffness is determined from the plot of the first natural frequency versus the support stiffness. Effects of the boundary flexibility on the maximum beam response and the maximum interaction force are discussed as a function of the speed and the oscillator frequency. The boundary flexibility is shown to have a significant implication in the design analysis of the moving oscillator problem, especially for shorter span beam structures.

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