Variable Rectangular Cross-section Research Articles

CLOSED-FORM TORSIONAL COMPLIANCE OF TWO-AXIS FLEXURE HINGES

This paper focuses on closed-form equations of torsional compliance for two-axis flexure hinges with variable rectangular cross-section (VRC) which can serve as the alternatives of flexure spherical hinges in flexure-based spatial compliant mechanisms.

Flat bending shape stability of the beams with variable section width

The paper proposes a methodology for calculating lateral buckling of beams of variable rectangular cross section based on the energy approach. The technique is considered on the example of a cantilever beam of variable width with two sections under the action of a concentrated force. The twist angle function was set in the form of a trigonometric series. As a result, the problem is reduced to a generalized secular equation.

Effect of Bimodularity on the Stress State of a Variable Cross-Section Reinforced Beam

The article is dedicated to the effect of different modulus of the material on the stress state of a beam of the variable rectangular cross section. The height of the beam varies linearly along its length. Formulas for calculating the maximum compressive and tensile stresses and determining the neutral line are obtained. The maximum tensile and compressive stresses are determined for the clamped and simply supported beams. The dependence of the maximum normal stress on the number of reinforcing bars located in the stretched zones is numerically investigated. The stress state of the beam is compared with and without consideration of the bimodularity of the material for simply supported and cantilever beams. It is shown that taking into account the bimodularity of the material significantly affects the maximum tensile and compressive stresses. The magnitude of the tensile stresses is increased by 30%; the magnitude of the compressive stresses is reduced by 21%. As a bimodular material, fibro foam concrete is considered in work.

Modeling and analysis of engagement behavior of one-way clutch with a variable rectangular cross-section torsional spring

This article presents a study on the transient dynamic responses of engagement process of one-way clutch using variable rectangular cross-section torsional spring, which is usually installed on the end of engine for preventing torque being transmitted back. A 2-degree-of-freedom torsional dynamic model is proposed in this article to study the engagement process of clutch under two conditions. Due to the variation of parameters, the torque transfer of the wrap–spring clutch during engagement is described by a multi-parameter equation. The torque transmitted by clutch during engagement process has been described by two phases, spring expanding and spring locking. Physical parameters of spring determining the torque during expanding are discussed. To verify the model, a dynamic experiment on clutch engagement is conducted.

Применение метода конечных элементов для расчета активного сопротивления обмоток электрических машин

Good prospects expected from using innovative AC electrical machines equipped with a compact stator winding are substantiated. A procedure for calculating the resistances of machine winding phases with a variable rectangular cross-section of the stator winding conductors is proposed. A piecewise plane-parallel approximation of the field of turns with a decreased cross-section area of partitions between their active and end-face conductors is applied. The calculation procedure is based on the Laplace equations for electrostatic fields and the finite element method, both implemented in the ELCUT environment. The results from modeling the current density distribution in the turn parts with a variable cross section are given. Application of the procedure for stage-wise determination of the innovative electrical machine’s phase winding resistance is illustrated. The winding is made of a rectangular conductor with the cross section cyclically varying over the conductor length. The necessity of taking into account the influence of nonuniform current spreading in the turns with a variable cross-section in carrying out design calculations of electrical machines with the cores having different length to diameter ratios is considered. An example of estimating the errors in calculating the proposed winding’s phase resistances is given.

CRITERIA OF MINIMUM MATERIALS CONSUMPTION FOR BARS WITH RECTANGULAR CROSS-SECTION AND RESTRICTIONS ON STABILITY OR LIMITATIONS ON THE VALUE OF THE FIRST NATURAL FREQUENCY

Apparatus of mathematical programming is normally used in the most part of research works, dealing with structural optimization. However, the special properties of optimal systems have been identified in several studies. Besides, corresponding criteria, which have been formulated as well, can be used for assessments of proximity of optimal solutions to minimal material consumption. Particularly relevant criteria for bars with rectangular cross-section and restrictions on the stability or limitations on the value of the first natural frequency have been formulated. However, not all the features of some of the criteria have been observed. In addition it seems appropriate to identify relevant criteria for special cases set variable parameters. The distinctive paper contains additional property proximity criterion of optimal solutions to minimal consumption of materials for the bars with a rectangular cross-section and limitations on the value of the first natural frequency, modification of one of the previously proposed criteria and formulation of appropriate criterion for the case where one of the parameters of variable rectangular cross-section is constant along the length of the bar.

Dynamic Analysis and Seismic Response of Planar Circular Arches with Variable Cross-Section

The aim of this paper is to investigate the dynamic response of planar circular arches with variable cross-section subjected to seismic ground motions. Arches have a wide range of application (e.g. bridges, roofs) thanks to their capacity to span large areas by resolving vertical actions into compressive stresses and confining tensile stresses. The full understanding of their dynamic response is a challenging technical and computational problem, especially when seismic loading is considered. For example, the assumption of axial inextensibility simplifies the differential equations but overestimates the vibration frequencies, especially those of shallow arches since axial forces are of paramount importance (as opposed to beams). In lieu of the above, our formulation incorporates the effect of axial extension, and the arches are modeled using a new generic curved beam model that includes both axial (tangential) and transverse (normal) to the arch centerline deformations, and is able to account for variable mass and stiffness properties, as well as elastic support or restraint. The resulting dynamic governing equations of the circular arch are formulated in terms of the displacements, and solved using an efficient integral equation method. Three circular arches with variable rectangular cross-section are analyzed in order to investigate their dynamic properties and seismic performance. Using both time history and modal analysis useful conclusions are drawn with regard to the contribution of each mode on the calculation of different response quantities.

Design and Optimization of Variable Rectangular Cross Section Chassis for On-Road Heavy Vehicles

All the loads generated by other components of heavy vehicle are transferred to its chassis. Chassis related failures are few but the damages to the safety of occupant are huge; sometimes it leads to fatal accidents. In order to overcome this, the chassis has to be optimized based on static and dynamic loads by ensuring a uniform distribution of stress and strain. The shape and cross section of the chassis gives a resistance to the above mentioned loads. The cross section of the chassis structure of all on-road vehicles is uniform despite the variable loads. In this work, variable cross section chassis of an on-road heavy vehicle is designed by keeping optimum sections. Bending moment of the chassis has been mathematically related with section modulus of the chassis. Genetic algorithm based procedures have been used to optimize the height, width and thickness of the chassis cross section. Coding in C language is used to automate the genetic algorithm procedures. For benchmark study, 3D models of optimized and existing chassis of an on-road heavy vehicle were developed. Finite element analysis reveals that the optimized chassis has less failure possibilities due to lower stress values and uniform distribution when compared to those from the model of existing chassis.

Evaluation of the longitudinal parameters of an overhead transmission line with non-homogeneous cross section

This paper presents an accurate method to evaluate the longitudinal parameters of multi-conductor overhead power lines, taking in account skin and proximity effects. This method has the foremost advantage of considering a non-homogeneous ground, composed by separate layers with different conductivities and having surface irregularities, such as hills or valleys. The method splits the system conductors (line and ground) into smaller sub-conductors, with variable rectangular cross-section. The method is validated by comparing the longitudinal impedance results with those obtained using both analytical equations and other numerical methods proposed in the literature. Analytical results presented are concerned to a cylindrical two-wire line and one-phase overhead line. Using the proposed method results are obtained for one-phase overhead line, considering the ground with different conductivity layers and with different topographies.

A mathematical model for fixed-end moments for two types of loads for a parabolic shaped variable rectangular cross section

This paper develops a mathematical model for fixed-end moments for two different types of loads on beams with a parabolic shaped variable rectangular cross section. The loads applied on beam are: 1) a uniformly distributed load and 2) a concentrated load located anywhere along the beam length. The properties of the rectangular cross section of the beam varies along its axis, i.e., the width “b” is constant and the height “h” varies along the beam, this variation follows a parabolic form. The consistent deformation method based on the superposition of the effects is used to solve these problems. The deformation anywhere along the beam is obtained by using the Bernoulli-Euler theory. Traditional methods used to obtain deflections of variable cross section members are any techniques that perform numerical integration, such as Simpson's rule. Tables presented by other authors are restricted to certain relationships. Beyond the effectiveness and accuracy of the developed model, a significant advantage of it is the moments are calculated at any cross section of the beam using the respective integral representations as mathematical formulas.

Two general solutions of torsional compliance for variable rectangular cross-section hinges in compliant mechanisms

Several approaches exist for calculating the torsional compliance of rectangular cross-section beams, but most depend on the relative magnitude of the cross-section thickness and width, which might be changing during the design phase (especially for design optimization) or is variant for variable cross-section beams such as circular flexure hinges and tapered bars. After summarizing current equations and analyzing their computational accuracy, two new equations are proposed, which are thickness-to-width ratio independent, and suitable for variable cross-section beams and optimization design of torsional elements in compliant mechanisms. The closed-form equations for the torsional compliance of elliptical and circular flexure hinges are derived by using the new equations.

3D-curved beam element with varying cross-sectional area under generalized loads

This article deals with the stiffness matrix and the equivalent load vector of a 3D-curved beam element with a variable cross-section under generalized loads. The linear ordinary differential system equation governing its structural behaviour is presented, and the boundary value problem is solved by the application of a recurrence scheme obtained by relating the internal forces and deformation at the two end points of an increase in the centroid line. This solution has a transfer matrix structure. Both the stiffness matrix and the equivalent load vector are obtained arranging the transfer matrix. The calculation method used was compared with other methods in the literature and the exact values demonstrated by means of four examples: a straight cantilever beam with a variable rectangular cross-section, a bi-fixed circular arch, a variable cross-section circular helix and a parabolic arch.

Circular-Hinge Line Element for Finite Element Analysis of Compliant Mechanisms

A three-node six degree-of-freedom per-node line element that is sensitive to axial, bending, and torsional loading is introduced to model single-axis right circular hinges of constant width that are utilized in compliant mechanisms. The Timoshenko model is applied for bending because this particular configuration is virtually short, and provisions are taken that the element is shear-locking free. The Saint Venant theory, which includes warping, is utilized to model torsion of the variable rectangular cross-section circular hinge. The principle of minimum total potential energy is employed to formulate the elemental stiffness and mass matrices, as well as the elemental nodal vector. Static force deflection and modal simulation that are performed based on this finite element model produce results that are in agreement with simulation by commercially available finite element software. The three-node line element is also compared to an analytical model in terms of stiffness and the results are again concurring.

Torsional stiffness of several variable rectangular cross-section flexure hinges formacro-scale and MEMS applications

The paper develops approximate closed-form equations for the torsional stiffness ofseveral variable rectangular cross-section flexure hinges for macro-scale and MEMSapplications. Specifically, corner-filleted, elliptic, parabolic and hyperbolic flexureconfigurations, either longitudinally symmetric or non-symmetric, are studied. Themodel gives the tools for a preliminary assessment of the static/modal responseof flexure-based devices that deform torsionally. Several numerical simulationsare conducted based on the model, which indicate that, for similar values of thegeometric parameters, the hyperbolic flexure is the stiffest, followed by the parabolic,corner-filleted and elliptic configurations. The modal response is studied for atwo-flexure torsional micro-mirror by sequentially considering four different pairs oflongitudinally symmetric flexure designs. The results of the simulation confirmthe stiffness predictions and are also in agreement with finite element analysisresults.

On an one-dimensional bi-layer shallow-water problem

This paper is concerned with the mathematical analysis and the numerical approximation of the system of partial differential equations governing the one-dimensional flow of two superposed shallow layers of immiscible viscous fluid in a channel with variable rectangular cross-section. First, we prove the existence and uniqueness of solution for small data and some smoothness results. Next, a first-order upwind scheme for numerically solving the system is proposed. We apply this scheme to the simulation of some two-layer exchange flows through straits with a sill and a contraction.

Behavior and Modeling of Nonprismatic Members Having T-Sections

This paper emphasizes the discrepancies in conventional methods of analyzing nonprismatic members having T-sections. In practice, the change in the location of the centroidal axis over the length of nonprismatic members is ignored in structural analysis. Commonly used Portland Cement Association tables for fixed-end moment and stiffness factors are based on the assumptions of a straight-line centroidal axis and variable rectangular cross section. However, discontinuities in the centroidal axis produce strong coupling between the end moments and horizontal thrust. Nonprismatic members behave similar to an arch, and this creates an axial force that is neglected in conventional methods. Moreover, beams are considered as T-beams in the structural models because of the effect of RC slabs. Consequently, nonprismatic T-sections require special consideration in structural analysis. The purpose of this paper is to present the behavior of nonprismatic T-section beams based on a 3D finite-element investigation. Based on the results of parametric studies, a simple, practical 2D beam model with an effective length concept is proposed for the analysis of nonprismatic members having T-sections. In the proposed model, the effective length of the 2D beam element representing the haunch evolves as 75% of its actual length. It is concluded that the deviation in results from the results of the finite-element analysis is within 15%. Additionally, different modeling schemes in the analysis of frames with nonprismatic members using the proposed 2D beam representation are investigated.

Determination of the fundamental frequency of vibration of beams of variable rectangular cross-section by means of the integral equation approach

The title problem is solved using the integral equation approach and polynomial approximations which contain an exponential parameter γ which allows for the determination of an optimized value of the fundamental frequency coefficient when using the Ritz method. A lower bound is determined using the method of traces. It is finally shown that the optimized upper bound and the value obtained by the method of traces are extremely close, with less than 0–5% difference in all cases.

Free and mixed convective flow of air in a heated cavity of variable rectangular cross section and orientation

Free and mixed convection in a strongly heated cavity of rectangular cross section have been investigated experimentally, to observe the effects of cavity shape and inclination and of ambient wind on the velocity and temperature distributions. The long edges ( c = 0.533 m) of the cavity were horizontal and parallel to an axis around which the cavity could be rotated. The aperture plane was either vertical (a = 0°), or inclined facing downward at a = 20° or a = 45°. The height of the aperture, b , was always 0.0947 m, while the depth of the cavity, a , was set so that = 0.5, 1.0, or 1.46. The bottom and back walls were electrically heated but the top wall was indirectly heated by conduction and radiation. The absolute average temperature of the wall, T c , was 2.21 times the absolute temperature at infinity, T^, and the Grashof number based on the difference of these two temperatures, with physical properties evaluated at T"^, was 4.2 x 10 7 . The Prandtl number was that of air, Pr « 0.7. In the studies of mixed convection, the axis of rotation was horizontal and normal to the ambient wind, U ^. When the aperture faced directly upstream the inclination angle was a = 0°. The Reynolds number, Re = was bU / v , varied from Re = 120-1100, where free convection was dominant, to Re = 2000-8740, where forced convection was dominant. For both free and mixed convection, wall and gas temperatures were measured with thermocouples and shadowgraph pictures were taken. For pure free convection, three time-averaged velocity components, the corresponding normal Reynolds stress components and one off-diagonal Reynolds stress component were measured by using a two-colour laser-Doppler velocimeter with the assistance of a minicomputer for automated data collection and reduction. The experiments were largely exploratory and revealed a wealth of interesting phenomena. These include instabilities that have previously been seen in more simply bounded flows and periodic oscillations that seem unique to the open cavity. This application of laser-Doppler velocimetry is interesting in its own right, confronting large variations in index of refraction, extraordinarily high percentage fluctuations in velocity and density and great difficulty in assuring the uniform distribution of light-scattering particles.

Stability of a non-uniform beam on a viscoelastic foundation with a shear layer subjected to a non-conservative force

The stability of a cantilever beam of linearly variable rectangular cross-section lying on a viscoelastic foundation with a shear layer subjected to a circulatory force at its free end is investigated. The materials of the beam and the foundation are assumed to be Kelvin-type solids. Approximate values of the critical loads are calculated on the basis of an adjoint variational principle. The influence of geometric and material properties of the system on the critical load is depicted through several graphs. It is found that a shear layer may destabilize the system in certain cases.

A matching superparametric beam element for shell beam systems

A three noded, 18 d.o.f. beam element is developed for beams of variable rectangular cross section, arbitrarily curved in space. The superparametric beam element is combined with the existing eight noded, 40 d.o.f. superparametric shell element for shell beam systems. To show the applicability of the formulations, some simple problems of beams and a four-beam slab are analysed.