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.

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.

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.

Dynamics of Eccentric Shafts Under Linearly Varying Tension Rotating in a Viscous Medium

1974 ◽  
Vol 96 (4) ◽  
pp. 1285-1290
Author(s):  
V. Prodonoff ◽  
C. D. Michalopoulos
Keyword(s):  
Steady State ◽  
Beam Theory ◽  
Viscous Medium ◽  
Bernoulli Beam ◽  
Axial Coordinate ◽  
Simply Supported ◽  
Deterministic Function ◽  
Mass Eccentricity ◽  
Bending Stresses ◽  
Euler Bernoulli Beam

Using Euler-Bernoulli beam theory an investigation is made of the dynamic behavior of an eccentric vertical circular shaft rotating in viscous medium. The shaft is subjected to linearly-varying tension and has distributed mass and elasticity. The mass eccentricity is assumed to be a deterministic function of the axial coordinate. The solution is obtained by modal analysis. An example is considered wherein the shaft is simply supported at the top and vertically guided at the bottom. Steady-state deflections and bending stresses are computed for a particular eccentricity function over a range of speeds of rotation which includes a resonant frequency.

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.

Fractional visco-elastic Timoshenko beam from elastic Euler–Bernoulli beam

2014 ◽  
Vol 226 (1) ◽  
pp. 179-189 ◽  
Author(s):  
Antonina Pirrotta ◽  
Stefano Cutrona ◽  
Salvatore Di Lorenzo
Keyword(s):  
Timoshenko Beam ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam

scholarly journals On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Wlodzimierz Czyczula ◽  
Piotr Koziol ◽  
Dorota Blaszkiewicz
Keyword(s):  
Axial Force ◽  
Timoshenko Beam ◽  
Linear Models ◽  
Moving Load ◽  
Railway Track ◽  
Variable Load ◽  
Static Case ◽  
Timoshenko Beams ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam

The paper tries to clarify the problem of solution and interpretation of railway track dynamics equations for linear models. Set of theorems is introduced in the paper describing two types of equivalence: between static and dynamic track response under moving load and between the dynamic response of track described by both the Euler-Bernoulli and Timoshenko beams. The equivalence is clarified in terms of mathematical method of solution. It is shown that inertia element of rail equation for the Euler-Bernoulli beam and constant distributed load can be considered as a substitute axial force multiplied by second derivative of displacement. Damping properties can be treated as additional substitute load in the static case taking into account this substitute axial force. When one considers the Timoshenko beam, the substitute axial force depends additionally on shear properties of rail section, rail bending stiffness, and subgrade stiffness. It is also proved that Timoshenko beam, described by a single equation, from the point of view of solution, is an analogy of the Euler-Bernoulli beam for both constant and variable load. Certain numerical examples are presented and practical interpretation of proved theorems is shown.

On the Curvature of an Euler–Bernoulli Beam

2003 ◽  
Vol 31 (2) ◽  
pp. 132-142 ◽  
Author(s):  
Osman Kopmaz ◽  
Ömer Gündoğdu
Keyword(s):  
Nonlinear Theory ◽  
Quantitative Assessment ◽  
Bending Moment ◽  
Point Of View ◽  
Bernoulli Beam ◽  
Cantilevered Beam ◽  
Single Moment ◽  
The Difference ◽  
Euler Bernoulli Beam ◽  
The Relationship

This paper deals with different approaches to describing the relationship between the bending moment and curvature of a Euler—Bernoulli beam undergoing a large deformation, from a tutorial point of view. First, the concepts of the mathematical and physical curvature are presented in detail. Then, in the case of a cantilevered beam subjected to a single moment at its free end, the difference between the linear theory and the nonlinear theory based on both the mathematical curvature and the physical curvature is shown. It is emphasized that a careless use of the nonlinear mathematical curvature and moment relationship given in most standard textbooks may lead to erroneous results. Furthermore, a numerical example is given for the reader to make a quantitative assessment.

scholarly journals Estimation of Wheelset Natural Vibration Characteristics Based on Transfer Matrix Method with Various Elastic Beam Models

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Pengfei Liu ◽  
Hongjun Liu ◽  
Qing Wu
Keyword(s):  
Transfer Matrix ◽  
Timoshenko Beam ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Natural Vibration ◽  
Elastic Beam ◽  
Vibration Characteristics ◽  
Bernoulli Beam ◽  
Flexible Wheelset ◽  
Euler Bernoulli Beam

The elastic vibration of the wheelset is a potential factor inducing wheel-rail defects. It is important to understand the natural vibration characteristics of the flexible wheelset for slowing down the defect growth. To estimate the elastic free vibration of the railway wheelset with the multidiameter axle, the transfer matrix method (TMM) is applied. The transfer matrices of four types of elastic beam models are derived including the Euler–Bernoulli beam, Timoshenko beam, elastic beam without mass and shearing stiffness, and massless elastic beam with shearing stiffness. For each type, the simplified model and detailed models of the flexible wheelset are developed. Both bending and torsional modes are compared with that of the finite element (FE) model. For the wheelset bending modes, if the wheel axle is modelled as the Euler–Bernoulli beam and Timoshenko beam, the natural frequencies can be reflected accurately, especially for the latter one. Due to the lower solving accuracy, the massless beam models are not applicable for the analysis of natural characteristics of the wheelset. The increase of the dividing segment number of the flexible axle is helpful to improve the modal solving accuracy, while the computation effort is almost kept in the same level. For the torsional vibration mode, it mainly depends on the axle torsional stiffness and wheel inertia rather than axle torsional inertia.

Sliding mode boundary control for vibration suppression in a pinned-pinned Euler–Bernoulli beam with disturbances

2016 ◽  
Vol 24 (6) ◽  
pp. 1109-1122 ◽  
Author(s):  
Dimitri Karagiannis ◽  
Verica Radisavljevic–Gajic
Keyword(s):  
Sliding Mode Control ◽  
Boundary Conditions ◽  
Sliding Mode ◽  
Bending Moment ◽  
Control Technique ◽  
Beam Model ◽  
Bernoulli Beam ◽  
Mode Control ◽  
Control Functions ◽  
Euler Bernoulli Beam

This work addresses the control of a pinned-pinned beam represented by the fourth order partial differential equation commonly known as the Euler–Bernoulli beam model. The system under consideration has pinned boundary conditions on one end (displacement and bending moment fixed at zero) and controlled boundary conditions on the other end (displacement and bending moment are prescribed by control functions). There are also unknown bounded disturbances included on the controlled boundary. A backstepping control technique which introduces arbitrary damping into the system is discussed, and a method for applying this control in the presence of unknown disturbances is developed using sliding mode control theory. Sliding mode controllers are developed in a way that does not create a chattering effect, which is a common issue with sliding mode control. Simulation results are presented to show how the system dampens out vibrations at an arbitrarily determined rate and how the control functions respond to unmodeled disturbances.

On the Resolution of Existing Discontinuities in the Dynamic Responses of an Euler-Bernoulli Beam Subjected to the Moving Mass

2006 ◽  
Author(s):  
M. H. Kargarnovin ◽  
K. Saeedi
Keyword(s):  
Contact Point ◽  
Bending Moment ◽  
Dynamic Responses ◽  
Distributed Parameter ◽  
Moving Mass ◽  
Bernoulli Beam ◽  
Concentrated Mass ◽  
Uniform Cross Section ◽  
Support Conditions ◽  
Euler Bernoulli Beam

The dynamic response of a one-dimensional distributed parameter system subjected to a moving mass with constant speed is investigated. An Euler-Bernoulli beam with the uniform cross-section and finite length with specified boundary support conditions is assumed. In this paper, rather a new method based on the time dependent series expansion for calculating the bending moment and the shear force due to motion of the mass is suggested. Governing differential equations of the motion are derived and solved. The accuracy of the numerical results primarily is verified and further the rapid convergence of this new technique was illustrated over other existing methods. Finally, it is shown that a considerable improvement is obtained in capturing the incurred discontinuities at the contact point of traveling concentrated mass.

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