Beams Of Uniform Cross-section Research Articles

ANSYS based Modal Analysis of a Fixed Beam of Uniform Cross Section with a Concentrated Mass at Mid Span

The report outlines ANSYS based modal and structural analysis of a fixed beam of dimension 20mm*20mm*1000mm that was constructed in SOLIDWORKS. Structural steel was selected as the material for analysis. The analysis was done in 3 phases, the first analysis was done without any crack and the other two analysis involved cracked beams with cracks created at different lengths. Suitable boundary and loading conditions were used to find the modes of frequencies. The resulting variation in static deflection, position of mode shapes representing transverse deflection and variation in position of equivalent stress corresponding to varying beam conditions with and without cracks are studied.

АВТОМАТИЗАЦИЯ МЕТОДИКИ РАСЧЕТА СОБСТВЕННОЙ ЧАСТОТЫ ПОПЕРЕЧНЫХ КОЛЕБАНИЙ ГРЕБНОГО ВАЛА

Improving the designing quality of engineering systems and accuracy of their calculations is the basis for increasing the technical and economic indicators of modern engineering. Using automated calculations during the design phase raises labour productivity and product quality. There have been discussed the stages of software implementation of engineering calculations of the natural frequency of transverse vibrations of the ship propeller shafts. The problem of determining the first natural frequency of transverse vibrations of the shaft depending on the method of its fixing has been solved. A computational scheme has been build, which presents a uniform cross-section beam based on a point elastic support with a concentrated mass at its end. Differential equations for oscillations have been derived; the method of finding the natural frequencies has been described. For proper calculating natural frequency there has been developed a sequence and its software implementation in Maple environment with application of computer algebra package. The effect of the method of fixing on natural frequency of shaft transverse vibrations has been estimated. The problem of finding natural frequency of transverse vibrations on the extended elastic stern shaft bearing has been considered. The algorithm presented in the paper serves as a basis for formalizing and automating the stage of calculating transverse vibrations and is also applicable for calculating the natural frequencies of various elements, including the shaft and the sliding bearing.

Confined focused sheet beam flow of a thermionic linear electron source

A magnetically confined sheet beam has been designed and tested for its uniform transmission. The beam is fully immersed in a uniform magnetic field of 50 Gauss which bends and focuses the beam throughout in a uniform laminar flow at 180̂. A sheet beam of uniform cross section (0.35 cm × 12.5 cm) successfully has been tested; the width of the beam is many times the thickness. The gun is designed and simulated in CST software. The beam is uniform throughout with parameters; current density, emittance and perveance values; 1.04 A/cm2, 14 π-mm-mR and 5 μAV−3/2, respectively. The simulation and test results are in agreement. The first application is the evaporation of refractory metals for coating in the field of nanotechnology and also expected to be used in klystrons for advanced particle accelerators and colliders at higher acceleration potentials.

Modeling Large Deflections of Initially Curved Beams in Compliant Mechanisms Using Chained Beam Constraint Model

Understanding and analyzing large and nonlinear deflections are the major challenges of designing compliant mechanisms. Initially, curved beams can offer potential advantages to designers of compliant mechanisms and provide useful alternatives to initially straight beams. However, the literature on analysis and design using such beams is rather limited. This paper presents a general and accurate method for modeling large planar deflections of initially curved beams of uniform cross section, which can be easily adapted to curved beams of various shapes. This method discretizes a curved beam into a few elements and models each element as a circular-arc beam using the beam constraint model (BCM), which is termed as the chained BCM (CBCM). Two different discretization schemes are provided for the method, among which the equal discretization is suitable for circular-arc beams and the unequal discretization is for curved beams of other shapes. Compliant mechanisms utilizing initially curved beams of circular-arc, cosine and parabola shapes are modeled to demonstrate the effectiveness of CBCM for initially curved beams of various shapes. The method is also accurate enough to capture the relevant nonlinear load-deflection characteristics.

A Versatile 3R Pseudo-Rigid-Body Model for Initially Curved and Straight Compliant Beams of Uniform Cross Section

Rigid-body discretization of continuum elements was developed as a method for simplifying the kinematics of otherwise complex systems. Recent work on pseudo-rigid-body (PRB) models for compliant mechanisms has opened up the possibility of using similar concepts for synthesis and design, while incorporating various types of flexible elements within the same framework. In this paper, an idea for combining initially curved and straight beams within planar compliant mechanisms is developed to create a set of equations that can be used to analyze various designs and topologies. A PRB model with three revolute joints is derived to approximate the behavior of initially curved compliant beams, while treating straight beams as a special case (zero curvature). The optimized model parameter values are tabled for a range of arc angles. The general kinematic and static equations for a single-loop mechanism are shown, with an example to illustrate accuracy for shape and displacement . Finally, this framework is used for the design of a compliant constant force mechanism to illustrate its application, and comparisons with finite element analysis (FEA) are provided for validation.

Post-buckling response of non-uniform cross-section bilaterally constrained beams

Buckling and post-buckling behaviors of structural elements have been widely used to create multi-stable mechanisms that have shown a great efficiency in many applications such as sensing, actuation and energy harvesting. Under an increasing axial loading, the strain energy stored in a buckled bilaterally constrained elastica is suddenly released, through a snap-through transition. These transitions can be used to convert low-rate and low-frequency excitations into high-rate motions that are converted into electrical signals using piezoelectric transducers. However, for efficient sensing and energy harvesting, these transitions have to be controlled. It has been shown that the spacing between the transitions cannot be controlled just by tuning the geometry properties of a uniform cross-section beam. This paper investigates the effect of different non-uniform cross-sections on the post-buckling response of a bilaterally constrained beam. An energy-based model that takes into account the shape variation is herein presented. The variation of the beam’s cross-section area can either be continuous or piecewise continuous. The total potential energy of the system is minimized under constraints that represent the physical confinement of the beam between the lateral boundaries. Results demonstrate that the spacing ratio between buckling-mode transitions can be efficiently controlled by the beam’s shape and geometry dimensions.

Mechanics of energy transfer and failure of ductile microscale beams subjected to dynamic loading

The mechanical response of microelectromechanical systems (MEMS) under impulse loading conditions has not been thoroughly studied to date, partially because of the lack of means to provide such extreme loading rates to miniature devices. However, the increasing use of MEMS-based sensors and actuators in adverse environments, which include extreme strain rate loading, has motivated the investigation of the response of MEMS components under these conditions. In this work, basic and mostly commonly employed Au MEMS components were subjected to impulse loads of 40 ns in duration, which were generated by a high power pulsed laser in order to achieve acceleration levels on the order of 10 9 g. This allowed for the microdevice mechanical/structural response to be investigated at time scales that were of the order of wave transit times in the substrate and the devices. Basic microscale structures, such as cantilevers and fixed-fixed beams of uniform cross-section, were employed to facilitate comparisons with companion finite element simulations in order to gain insight into the mechanisms responsible for impulsive deformation at the microscale. The simulations investigated the effect of loading rate, boundary conditions, beam length, material constitutive response, and damping on the final deformed shapes of the beams. It was found that contact and momentum transfer mechanisms were responsible for the large permanent beam deflections which were measured postmortem. Additionally, the effects of both damping and material property rate dependence were found to be dominant in determining the final deformed shape of the beams. In fact, our observations suggest that the contributions of material rate dependence and damping are not simply additive, but rather involve a coupling between them that affects the final structure response.

Vibration based damage detection in a uniform strength beam using genetic algorithm

Cantilever steel beams of uniform strength are having various industrial applications. In particular when it is used as leaf spring it undergoes very large deflection in comparison to beam of uniform cross section. The damage occurs in these beams mainly due to fatigue loading. Early detection of damage in such type of beams is very essential to avoid a major failure or accident. In this paper, firstly formulation of an objective function for the genetic search optimization procedure along with the residual force method are presented for the identification of macroscopic structural damage in an uniform strength beam. Two cases have been investigated here. In the first case the width is varied keeping the strength of beam uniform throughout and in the second case both width and depth are varied to represent a special case of uniform strength beam. The developed model require experimentally determined data as input and detect the location and extent of the damage in the beam. Here, experimental data are simulated numerically by using finite element models of structures with inclusion of random noise on the vibration characteristics. It has been shown that the damage may be identified for the said problems with a good accuracy.

Transverse and torsional modes of vibration of American organ reeds

A reed from an American reed organ consists of a brass tongue riveted to a frame with an opening just large enough for the reed to pass through during the course of its oscillation. To a first approximation, the reed tongue can be modeled as a cantilever beam of uniform cross section, and the mode frequencies of transverse and torsional modes can be calculated. The reeds used in this study differ from this simple model in several ways, including a nonuniform thickness along the reed length. In addition, the reed includes a curved spoon-like section at the tip of the reed tongue as well as an asymmetric twist at the tip. Transverse and torsional modes of vibration of a mechanically excited reed tongue have been studied using a proximity sensor (variable impedance transducer), a laser vibrometer, and a TV holography system. The motion of the air-driven reed has also been investigated. Although the motion of air-driven reed is dominated by the first transverse mode, some evidence of higher-order transverse modes and torsional modes has been observed. [Work supported by the NSF from REU Grant No. 0139096.]

A high-frequency theory for thermo-viscoelastic beams

To predict the macromechanical response of beams of uniform cross section, a system of 1-D equations is consistently formulated in both differential and variational forms within the frame of the 3-D theory of thermo-viscoelasticity. First, the basic equations of linear, nonisothermal 3-D theory of thermo-viscoelasticity are expressed in terms of the Laplace transformed field variables by a recent variational principle [Altay and Dokmeci, Acta Mech. 143, 91–111 (2000)]. Next, the temperature increment and displacement fields are expressed by the series expansions of the aerial coordinates of beam cross section. Then, a system of higher-order beam equations is derived by means of the variational principle together with the series expansions of field variables. The system of equations governs the extensional, flexural, torsional, as well as coupled vibrations of thermo-viscoelastic beams at both high and low frequencies. Certain cases involving special types of vibrations, geometry, and material properties are recorded. The uniqueness is investigated in solutions of the system of beam equations. [Work supported in part by TUBA.]

Vibration analysis of ship beams

This paper is essentially addressed to the vibrations of ships which were extensively studied both experimentally and analytically by modeling a ship as a beam of uniform cross section and predicting its vibration characteristics by Euler’s theory of elastic beams. To begin with, the free vibration characteristics of a ship beam are predicted from Timoshenko’s theory of elastic beam of variable cross section by use of Galerkin’s method. The results are evaluated by use of those obtained recently from Euler’s theory [e.g., S. Abrate, J. Sound. Vib. 185 (4), 703–716 (1995)]. Next, a numerical algorithm that is based on the method of moments [e.g., G. A. Altary and M. C. Dökmeci, Z. angew. Math. Mech. 77, 1–17 (1997) and references therein] is developed so as to predict the vibration characteristics of a ship beam at both low and high frequencies. The equations derived include the effects of warping and shear deformations of cross section and, in fact, they incorporate as many higher order effects as deemed necessary in any case of interest. They govern all the extensional, flexural, and torsional as well as coupled motions of ship beam. Also, the results are compared with those of earlier ones. [Work supported in part by TÜBA.]

Crack Localization in Stepped Beams

A method for crack localization, based on so-called Local Modal Crack Sensitivity (LMCS), has recently been proposed by one of the Authors. As a major feature, the method only involves measuring eigenfrequencies. The method has already been used for beams of uniform cross section and homogeneous boundary conditions. In this paper, the method is extended to stepped beams, with both homogeneous and non-homogeneous boundary conditions. Experimental results confirm its usefulness for crack localization. Satisfactory localization of a second crack was also possible, so confirming the effectiveness of the method.

Aeroelastic stability of a beam traveling in a tunnel lined with resonators

The theory of aeroelastic wave propagation along a long beam traveling in a tunnel lined with Helmholtz resonators is developed, and the effect of the lining on the stability of the beam is examined. Assuming an elastic beam of uniform cross section and of infinite length, and taking account of the compressibility of the air, the linear dispersion relation is derived for an air-beam-cavity system on disregarding all dissipative effects. The lining newly introduces the cavity mode of propagation in addition to the acoustic mode and the beam mode. When a restoring force in the transverse direction to the beam's axis is present, it is found that all modes are neutrally stable. Without this force, Kelvin-Helmholtz instability occurs in the beam mode for a long wavelength and a low frequency. But the lining does not take part in this instability. The stability of wave propagation is also examined from the standpoint of the wave energy. It is revealed that the cavity mode turns into the mode of so-called negative energy waves as a wave number increases beyond a critical value. Emergence of negative energy waves implies destabilization of the stable modes by weak dissipative effects. The negative energy waves appear also in the beam mode, if the restoring force is absent, in a very narrow band of the wave number.

Dynamic stability of a tapered cantilever beam on an elastic foundation subjected to a follower force

The equation of motion in matrix form of a tapered cantilever Euler beam subjected to a follower force at the free end is formulated based on the Lagrangian approach and the assumed mode method. The beam is resting on a Winkler-type elastic foundation. The effects of the presence of viscous damping in the foundation, which makes the foundation viscoelastic, and the presence of internal damping in the beam on the critical flutter loads are examined separately to evaluate their relative importance. As expected, internal damping of the beam tends to drastically reduce the critical flutter loads for a beam of uniform cross-section. For tapered beams, the effects are however dependent on the taper ratio of the beam as well as the modulus of the elastic foundation. The critical flutter loads of both tapered beams and beams of uniform cross-section are found to be unaffected by the presence of viscous damping in the elastic foundation. The effect of varying the modulus of the elastic foundation on the critical flutter loads is also discussed in the paper.

Damping effects on the dynamic stability of a rod subjected to intermediate follower loads

The equation of motion in matrix form of an Euler beam of uniform cross-section including the effect of internal damping subjected to an intermediate subtangential follower load is presented. The effect of slight damping on the critical follower load with respect to variation in the location of application of the follower load and the parameter for subtangentiality is examined for a specific example of a clamped-free rod. The modes of instability in the form of flutter or divergence without damping are found to be unaffected by the presence of slight damping although there may be a sharp decrease in the first critical load for instability by flutter.

A Simplified Technique for the Analysis of Multi-Stepped Beams

The availability of software packages that enable algebraic manipulation of equations enables the reconsideration of analytical methods for design optimization which have fallen into disuse because of the difficulty of automating the analysis. Here, the analysis of multistepped beams is simplified by replacing the multi-stepped geometry by a beam of uniform cross-section. A general formula is established to define the induced forces and moments that replace the steps. This enables the singularity function method, which is usually restricted to the analysis of uniform beams, to be easily applied to determine shearing forces and bending moments, and thus simplifies the analysis.

Random vibrations of cantilevered composite beams with torsion-bending coupling

The forced-vibration problem of a layered composite cantilevered beam which includes the effects of torsion and warping is solved using a third-order shear deformation theory. The solution of the free-vibration problem in conjunction with the solution of its adjoint is used to solve for the forced response of the beam. This forced-vibration solution is used to calculate the displacements, zero-crossing rates and normal and transverse shear stress responses for beams subjected to temporally and spatially varying random transverse loads. The effect of including or excluding the effects of shear deformation on the calculated response quantities is examined for beams with different configurations. The results clearly show that the shear deformation effects cannot be ignored in the calculation of response and reliability of composite beams. Although the results are presented only for beams of uniform cross-section, the eigenfunctions developed here can be effectively used as comparison functions in a Rayleigh-Ritz type analysis of nonuniform beams.

Vibration Absorbers for Hysterically Damped Mass-Loaded Beams

A procedure for designing dynamic vibration absorbers for a general mass-loaded beam system of variable cross-sectional area, when it is subjected to an arbitrarily distributed simple harmonic force excitation, is presented. The procedure gives flexibility for choosing the number of absorbers depending upon the number of significant modes to be suppressed. The beam is assumed to be hysteretically damped and each absorber is a spring-mass-damper system. For each absorber, for a selected mass, the stiffness and damping coefficients are optimized so as to minimize the dynamic response corresponding to the resonance frequency at which they are tuned to operate. The interaction between the absorbers is also accounted for in the analysis. This general procedure for designing vibration absorbers is then applied to a space structure modeled as a mass-loaded free-free beam, to suppress the first two resonances when it is subjected to a concentrated simple harmonic force excitation. The frequency response is presented in a graphical form. Furthermore, it is also shown that optimizing the beam system first and then designing vibration absorbers for this beam system will result in a small dynamic response. Although the analysis is general enough to cover beams of nonuniform cross-section, the examples presented in this paper of the above simplified model of the space structure have been restricted to beams of uniform cross-section.

Stress Distribution and Deformation of Adhesive‐Bonded Laminated Composite Beams

A study is reported of the 3-dimensional elasticity solution of flexure and torsion in adhesive-bonded multilayered beams of uniform cross section having one end fixed and the other end loaded by transverse forces. The study found that the elementary beam theory (Beer and Johnston 1981) based on the assumption that plane sections of the beam remain plane after deformation yields an exact elasticity solution for the normal stress for multilayered beams if certain conditions are satisfied. These and other findings are discussed.

Use of Positional Parameter in Rayleigh's Method

The determination of the response of a beam of variable moment of inertia when subjected to an eccentric compressive load is addressed. The beam displacement is approximated in terms of a simple polynomial coordinate function which contains two parameters which allow for two separate minimizations of the governing functional. One of the parameters appears as an exponent of the coordinate function while the other depends upon the origin of the coordinate system. It is shown that the proposed approach yields excellent accuracy in the case of a beam of uniform cross section.