Note on axial-shear and contour vibrations of prisms

Exact solutions of the equations of the linear theory of elasticity are given for axial-shear modes of vibration of an isotropic, prismatic bar whose normal section is an equilateral triangle or has the equilateral triangle as a module. A family of contour modes is also described for bars with a rhombic section formed of two equilateral triangles and with sections having the rhombus as a module.     

Circuit parameters from Maxwell's equations

A power expansion in the wave number k is utilized to express the low-frequency behaviour of simply- and doubly-connected conductors as a succession of interior and exterior problems. A general solution is given for the determination of the R, L, C parameters of a conductor of arbitrary shape.     

Asymptotic models for optimizing the contour of multiply-connected cross-section of an elastic bar in torsion

An asymptotic solution is obtained for the problem of maximizing the torsional rigidity of elastic, multiply-connected cylindrical bars for a given area of cross-section. The shapes of the inner contours of the multiply-connected cross-section are specified while the outer contour is determined as a result of the shape optimization. We apply the method of matched asymptotic expansions to construct a first-order asymptotic model. The conditions for unique solvability of the asymptotic model have been established under some restrictions imposed on the location of the inner contours and their polarization matrices. The economy achieved by optimization is estimated.     

Optimal Plastic Design of a Cross Section under Torsion with Small Bending∗

A doubly connected cross section of a bar under torsion with bending is optimized in two variants; either the internal contour is given and the external one is subject to optimization, or only the inside area is given and both contours are optimized. Minimal cross-sectional area is the design objective. The material is assumed to be perfectly plastic, incompressible, and subject to the Huber-Mises-Hencky yield condition. A specially adopted perturbation method is used and only the range of small bending is analyzed. The resulting optimal bars are regarded as thick-walled, hence no wall stability constraints are applied.     

Repeatability of the Contour Method for Residual Stress Measurement

This paper describes the results of a residual stress measurement repeatability study using the contour method. The test specimen is an aluminum bar (cut from plate), with cross sectional dimensions of 50.8 mm?×?76.2 mm (2 in?×?3 in) with a length of 609.6 mm (24 in). There are two bars, one bar with high residual stresses and one bar with low residual stresses. The high residual stress configuration (±150 MPa) is in a quenched and over-aged condition (Al 7050-T74) and the low residual stress configuration (±20 MPa) is stress relieved by stretching (Al 7050-T7451). Five contour measurements were performed on each aluminum bar at the mid-length of successively smaller pieces. Typical contour method procedures are employed with careful clamping of the specimen, wire electric discharge machining (EDM) for the cut, laser surface profiling of the cut faces, surface profile fitting, and linear elastic stress analysis. The measurement results provide repeatability data for the contour method, and the difference in repeatability when measuring high or low magnitude stresses. The results show similar repeatability standard deviation for both samples, being less than 10 MPa over most of the cross section and somewhat larger, around 20 MPa, near the cross section edges. A comparison with published repeatability data for other residual stress measurement techniques (x-ray diffraction, incremental hole drilling, and slitting) shows that the contour method has a level of repeatability that is similar to, or better than, other techniques.     

The lifting force action on a contour in a plane homogeneous vortex flow of an incompressible ideal fluid

The plane problem of homogeneous vortex flows of an incompressible inviscid fluid around a contour is considered. A method is developed for contours having a center or an axis of symmetry for calculating the lifting force acting on the contour which reduces the problem of determining the lifting force to an ordinary problem of a potential flow of a fluid around the given contour.     

Double-moment of spacial curved bars with closed thin-wall cross-section

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具有弹性支座杆件的动力稳定性研究

研究了杆件具有弹性支座时直杆的动力稳定性问题，推导出了杆件的临界频率和动力不稳定边界，分析了弹簧刚度和阻尼对杆件动力稳定性的影响，研究表明杆件的侧向刚度愈大其结构的动力稳定性愈好，阻尼对杆的振动起有利作用，并提出了在实际工程中减小杆件振动的一些措施．     

Determining the Optimal Contour of a Three-Dimensional Supersonic Jet Engine Outlet System

The problem of determining the optimal contour of a two-dimensional jet engine outlet system was solved in [1] using the direct method of the calculus of variations. In this study the method is developed for designing the optimal contour of a three-dimensional outlet system providing maximum thrust in a given direction.     

Influence of breadth in bars with reinforced circular holes under tension

This paper reports stress distributions in epoxy-resin bars with reinforced circular holes under tension. The holes were reinforced by bonding aluminum rings to the bars inside the holes. The stress distributions were determined photoelastically. Various proportions of circular reinforcements and different widths of bars were investigated under pure tension. Stress distributions were determined in the epoxy-resin part of the bar, and the maximum shear stresses were given special attention. The relation between maximum shear stress and the width of the bar was determined, and the ranges for which the theoretical solution for an infinite bar approximates that for a finite bar were defined.     

Construction of compact nonsymmetric nozzles of maximal thrust for given lift

The variational problem of the construction of supersonic nonsymmetric plane nozzles of maximal thrust for given lift and given restrictions on the size is solved. According to the restrictions on the size, the contours are fairly short, so that they do not interact. By virtue of this, the method of an undetermined control contour [1] can be used for the solution. It is shown that the problem is equivalent to a problem of maximal thrust (in the absence of a condition on the lift) in a coordinate system rotated through a certain angle relative to the original coordinate system and with unchanged restrictions on the size. The influence of these restrictions on the solution to the problem is investigated. It is shown that in some cases the control contour method is inapplicable, in particular, due to the possible presence of shock waves in the domain of influence of the contour. Numerical calculations illustrate the influence of the lift on the geometry and thrust. Special cases of this problem have been considered earlier. In [2], the problem of a plane contour of maximal thrust for given lift was solved and in [3] the problem of a nonsymmetric nozzle of maximal thrust without condition on the lift.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 132–136, October–December, 1981.I thank A. N. Kraiko for a helpful discussion of the work.     

Potential model of the coanda effect

Three schemes of potential flow around a convex contour are examined: free jet; in the presence of a jet-directing wall extending to the contour (without a gap); and in the presence of a jet-directing wall not extending to the contour (with a gap). It is assumed that the jet, not branching, flows around the contour and on one side. The existence and (with certain limitations) uniqueness of the solution of integral equations corresponding to the first two schemes are proved. Their analytic solutions are given for the case of a flow of a sufficiently thin jet around a circle. The first problem is calculated numerically in the entire region of the parameters. The scheme with a gap is examined for the case of flow around a half-plane. The problem of closure of the given potential models is dicsussed. In the case of the scheme without a gap the situation is typical for potential flow problems: The position of the separation point is a free parameter of the model. At the same time, in the presence of a gap (in particular, for the case of a free jet) the flow is determined completely by a system ofa priori assigned geometric parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 10–22, July–August, 1978.The author is grateful to M. A. Gol'dshtik for attention to the work and discussion of the results.     

Biomechanical research of joint III. An experimental biomechanical research on the Femur's articular surface of knee of pongidae

In this paper, moiré contour fringes is applied to study the femur's articular surface of the knee of Pongidae. The preliminary division of the femur's articular surface of knee into three types is proposed. The moiré contour fringes ofthe medial condyle is taken as a mark according to the references. Owing to the fact that the moiré contour fringes obtained from experiments after the 2nd order of fringe basically follow a certain rule, an investigation is made on the distribution of the angle α which is defined as the angle of the major axis of the 2nd order's near-oval shaped moiré contour fringe on the medial condyle with the horizontal axis. Preliminary distribution graphs are given in the paper.     

Thermal multipoles for a jet in ambient spathermal multipoles for a jet in ambient space

A general solution of the thermal problem for a non-self-similar axisymmetric jet is determined on the basis of the multiple approach developed for problems of non-self-similar jets in ambient space [1, 2], and several problems of convective heat-transfer in simply- or doubly-connected domains are solved. The advantage of expanding the solution of the convective heat conduction equation in eigenfunctions of the problem is demonstrated. As an illustration, the solutions for a thermal dipole and quadrupole and for a jet flow in a heated tube are presented. The corresponding solutions are also obtained for a turbulent jet in ambient space. The most favorable heat-exchange regime for a jet in a heated tube is predicted on the basis of the particular behavior of the eigenfunctions of the thermal problem.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 40–46, January–February, 1996.     

Unified proof to oscillation property of discrete beam

The oscillation property （OP） is a fundamental and important qualitative property for the vibrations of single span one-dimensional continuums such as strings, bars, torsion bars, and Euler beams. Any properly discretized continuum model should keep the OP. In literatures, the OP of discrete beam models is discussed essentially by means of matrix factorization. The discussion is model-specific and boundary-condition- specific. Besides, matrix factorization is difficult in handling finite element （FE） models of beams. In this paper, according to a sufficient condition for the OP, a new approach to discuss the property is proposed. The local criteria on discrete displacements rather than global matrix factorizations are given to verify the OP. Based on the proposed approach, known results such as the OP for the 2-node FE beams via the Heilinger- Reissener principle （HR-FE beams） as well as the 5-point finite difference （FD） beams are verified. New results on the OP for the 2-node PE-FE beams and the FE Timoshenko beams with small slenderness are given. Through a simple manipulation, the qualitative property of discrete multibearing beams can also be discussed by the proposed approach.     

Force acting on a deformable contour moving in an arbitary fluid stream

A solution is given for the plane nonstationary motion of an arbitary deformable contour in the potential flow of an ideal incompressible fluid. The problem was solved by conformal mapping. A simple formula is obtained for the force acting on a small size contour.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 4–8, January–February, 1973.The author is grateful to L. I. Sedov and Yu. L. Yakimov for formulating the problem and supervising the research.     

Extremal nozzle contours for gas flows with particle lag

The determination of the extremal nozzle contour for gas flow without foreign particles has been carried out in several studies [1–6], based on the calculation of the flow field using the method of characteristics.In [7, 8] the equations are derived for the characteristics and the relations along the streamlines which are required for calculating two-dimensional gas flow with foreign particles. The variational problem for two-phase flow in the two-dimensional formulation may be solved by the method of Guderley and Armitage [9] with the use of equations given in [7] or [8]; however this method is very tedious, even with the use of high-speed computers.In [10, 11] studies are made of two-phase one-dimensional flows by expanding the unknown functions in series in a small parameter, defined by the particle dimensions. In [12] a solution is given for the variational problem (in the one-dimensional formulation) of designing the contour of a nozzle with maximal impulse. However that study does not take account of the static term appearing in the impulse and the solution is obtained in relative cumbersome form. Moreover, the question of account for the losses due to nonparallelism and nonuniformity of the discharge was not considered.The present paper considers in the one-dimensional formulation the flow of a two-phase medium in a Laval nozzle with small particle lags (in velocity and temperature). The variational problem of determining the maximal nozzle impulse is formulated along the nozzle contour for fixed geometric expansion ratio. The impulse losses due to nonparallelism of the discharge are simulated by a function which depends on the ordinates which are variable along the contour and on the slope of the tangent to the contour.The author wishes to thank Yu. D. Shmyglevskii and A. N. Kraiko for helpful discussions and V. K. Starkov for carrying out the calculations on the computer.     

Energy-efficient actuation in infinite lattice structures

Buildup of internal self-stresses in hyperstatic adaptive structures resists actuation. A recent paper by Guest and Hutchinson (2003) shows that periodic infinite truss structures cannot be both statically and kinematically determinate structures; therefore, a rigid infinite lattice bar framework must be hyperstatic. This paper shows that it is possible to design adaptive periodic infinite truss structures that can achieve any state of uniform strain without energy cost by actuating only a subset of the bars in a coordinated fashion. We show that actuation of only 3 bars in two dimensions or 6 bars in three dimensions per unit cell is required. A mathematical apparatus is developed and an example of such a bitriangular lattice structure is given, along with accompanying illustrations. Supporting animations can be found at the authors’ website.     

Minimum-weight design of multi-purpose cylindrical bars

We solve the problem of minimizing the weight of elastic cylindrical bars under given constraints on their torsional and bending rigidities. The possible types of solutions are classed in the space of the design parameters. The necessary optimality conditions are derived and then used in the formulation of a closed-form boundary-value problem for a region with an unspecified boundary—the unknown cross-sectional shape of the optimal bar. An analytical solution of this and other related problems in terms of the given design parameters is obtained.     

Solution of the variational problem of shaping a nozzle to achieve a uniform supersonic flow

A solution is given to the problem of constructing a supersonic flat unsymmetrical nozzle achieving a given uniform flow at the outlet with a minimal length of one of the contours and a given length (or ordinate of the end point) of the other contour.