Deformation of Euler-Bernoulli Beams by Means of Modified Green's Function: Application of Fredholm Alternative Theorem

This article deals with the determination of the static displacement function of an Euler-Bernoulli beam with two guided supports. To this end, the Green's function method is employed and exact solution is obtained. The Green's function of the problem is constructed, using pertinent boundary conditions of the problem. Nevertheless, the problem does not admit a Green's function due to a mathematical contradiction. In order to eliminate the trouble, the Fredholm Alternative Theorem is utilized and the Green's function is modified. In this case, application of this theorem adds a particular term to the Green's function which gives rise to an arbitrary constant in the Green's function. Moreover, it is shown that the problem may have no solution or an infinite number of solutions. Besides, the necessary condition for having any solution is investigated. This requirement, which states a significant rule in the mechanics of solids, is the static equilibrium of vertical forces acting on the beam. Some examples are presented and results are thoroughly discussed.     

Forces and moments acting on bodies rotating about a symmetry axis in a free molecular flow

Analytic expressions are obtained for the forces and moments acting on symmetrically rotating convex figures of revolution moving in a free molecular flow of rarefied gas under the following assumptions: the velocity distribution function of the molecules of the oncoming flow is Maxwellian and the incident molecules have a diffuse—specular interaction with the surface of the body. For bodies with arbitrary piecewise smooth generator, general expressions are found in terms of quadrature for the components of the aerodynamic forces and moments. For a disk, sphere, and cylindrical and conical surfaces, the integration of the forces and the moments, which depend on the rotation of the body, is carried out to the end. For the moments of the forces, graphs are plotted of the errors of the hypothermal approximation as a function of the velocity ratio.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp, 151–155, May–June, 1980.     

Modeling and analysis of rigid multibody systems with driving constraints and frictional translation joints

An approach is proposed for modeling and anal- yses of rigid multibody systems with frictional translation joints and driving constraints. The geometric constraints of translational joints with small clearance are treated as bilat- eral constraints by neglecting the impact between sliders and guides. Firstly, the normal forces acting on sliders, the driv- ing constraint forces （or moments） and the constraint forces of smooth revolute joints are all described by complementary conditions. The frictional contacts are characterized by a set- valued force law of Coulomb＇s dry friction. Combined with the theory of the horizontal linear complementarity problem （HLCP）, an event-driven scheme is used to detect the transi- tions of the contact situation between sliders and guides, and the stick-slip transitions of sliders, respectively. And then, all constraint forces in the system can be computed easily. Secondly, the dynamic equations of multibody systems are written at the acceleration-force level by the Lagrange multiplier technique, and the Baumgarte stabilization method is used to reduce the constraint drift. Finally, a numerical example is given to show some non-smooth dynamical behaviors of the studied system. The obtained results validate the feasibility of algorithm and the effect of constraint stabilization.     

Ambulatory-force measurement using an instrumented-shoe system

A knowledge of the external forces acting on the human lower limbs is necessary for the effective formulation of safety precautions. A new approach of determining lower-leg forces that is more portable than existing measurement systems has been developed. Shoe-mounted load cells measure the orthogonal components of foot-to-ground forces and moments while goniometers continuously monitor the position of the lower leg. The output signals are transposed to a system of loads acting at the lower leg. The axial torque calculated to occur during common turning maneuvers under various conditions is presented and discussed.     

残损航空器搬移拖车悬臂的拓扑优化

首先,采用导重准则法对在固定载荷下以位移为约束的拓扑优化问题进行计算,运用一种新的插值模型推导了在单工况作用下的最小质量拓扑优化迭代算法,并通过一个算例验证了该算法的可行性。然后,将该算法应用于残损航空器搬移拖车悬臂的拓扑优化设计中,将由此获得的悬臂的拓扑形貌与结构优化软件Optistruct得到的拓扑结果进行对比。结果表明,二者的迭代速度差别不大,且导重准则法的优化效果更好。作为概念设计,所得的拓扑形貌为以后悬臂结构的优化设计提供了有效参考。     

Evaluating the Dynamic Load on a High-Speed Railroad Car

The internal dynamic load on a high-speed car following a real rail track with curvature and torsion is evaluated. The centrifugal, gyroscopic, tangential, and Coriolis forces and moments are taken into account. The kinematical parameters were obtained experimentally using an inertial system similar to that used to control space-rocket systems. The Euler-Lagrange equations represented by quaternionic matrices are used to determine the unknown dynamic loads in an explicit analytical form, which makes it possible to evaluate the components of dynamic loads due to the forces and moments acting on the car__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 119–126, March 2005.     

A numerical method for the evaluation of hydrodynamic forces of translating bodies under a free surface

This paper presents a numerical method to evaluate the hydrodynamic forces of translating bodies under a free surface. Both steady and unsteady problems are considered. Analytical and numerical studies are carried out based on the Havelock wave‐source function and the integral equation method. Two main problems arising inherently in the proposed solution method are overcome in order to facilitate the numerical implementation. The first lies in evaluating the Havelock function, which involves integrals with highly oscillatory kernels. Particular integration contours leading to non‐oscillatory integrands are derived a priori so that the integrals can be evaluated efficiently. The second problem lies in evaluating singular kernels in the boundary integral equation. The corresponding non‐singular formulation is derived using some theorems of potential theory, including the Gauss flux theorem and the property related to the equipotential body. The subsequent formulation is amenable to the solution by directly using the standard quadrature formulas without taking another special treatment. This paper also attempts to enhance the computational efficiency by presenting an interpolation method used to evaluate matrix elements, which are ascribed to a discretization procedure. In addition to the steady case, numerical examples consist of cases involving a submerged prolate spheroid, which is originally idle and then suddenly moves with a constant speed and a constant acceleration. Also systematically studied is the variation of hydrodynamic forces acting on the spheroid for various Froude numbers and submergence depths. Copyright © 2000 John Wiley & Sons, Ltd.     

Indentation of a Rigid Body into an Elastic Plate

The problem of a punch shaped like an elliptic paraboloid pressed into an elastic plate is studied under the assumption that the contact region is small. The action of the punch on the plate is modeled by point forces and moments. The method of joined asymptotic expansions is used to formulate the problem of one–sided contact for the internal asymptotic representation; the problem is solved with the use of the results obtained by L. A. Galin. The coordinates of the center of the elliptic contact region, its dimensions, and the angle of rotation are determined. The moments which ensure translational indentation of the punch are calculated and an equation that relates displacements of the punch to the force acting on it is given.     

A robotic model for Codman's paradox simulation and interpretation

A robotic approach based on Denavit–Hartenberg parametrization is proposed for simulating and interpreting Codman's paradox. A 3-degree-of-freedom robot model of the glenohumeral joint, driving the arm reduced to its long humerus, is considered for simulating the two-step rotational sequence of Codman's paradox. We propose to use the classical distinction made in robotics between the joint space, i.e. the inner space of joint angles, and the operational space, i.e. the outer physical space, for interpreting this historical version of the paradox, as there is some kind of confusion between these two spaces to be considered for arm movement definition. In its extended form, developed by MacConnail, the three-step rotational sequence of Codman's paradox would highlight the motor redundancy of the shoulder joint, necessitating for its simulation, according to our robotic approach, a 4-axis model of the shoulder spheroid joint. Our model provides a general prediction of the conjunct rotation angle in full accordance with clinical observation for a two-step or three-step version of Codman's paradox. The relation of the paradox with a possible general law of motion is finally discussed.     

Dependence of the aerodynamic forces on the Reynolds number with the three-dimensional flow of a vortical stream around bodies

The article discusses the dependence of the viscous stresses on the Reynolds number Re in three-dimensional flows around bodies of arbitrary form. It is shown that, with an infinite growth of a vortex with the approach to a body, singular terms appear in an asymptotic expansion in terms of ε=Re^?1/2. The infinite values of the derivatives of the velocity in flows of an incompressible liquid are due to the initial vorticity; in a supersonic flow, they can be connected with the absence of a maximum of the entropy at the critical flow line behind a leading shock wave. The singularity in the tangential stresses brings about the appearance of additional terms in the total aerodynamic forces and moments acting on the body.     

A modified SPH method for simulating motion of rigid bodies in Newtonian fluid flows

A weakly compressible smoothed particle hydrodynamics (WCSPH) method is used along with a new no-slip boundary condition to simulate movement of rigid bodies in incompressible Newtonian fluid flows. It is shown that the new boundary treatment method helps to efficiently calculate the hydrodynamic interaction forces acting on moving bodies. To compensate the effect of truncated compact support near solid boundaries, the method needs specific consistent renormalized schemes for the first and second-order spatial derivatives. In order to resolve the problem of spurious pressure oscillations in the WCSPH method, a modification to the continuity equation is used which improves the stability of the numerical method. The performance of the proposed method is assessed by solving a number of two-dimensional low-Reynolds fluid flow problems containing circular solid bodies. Wherever possible, the results are compared with the available numerical data.     

Interior formulation of axisymmetric Levinson plate theory

In this study, we show that the axisymmetric Levinson plate theory is exclusively an interior theory and we provide a consistent variational formulation for it. First, we discuss an annular Levinson plate according to a vectorial formulation. The boundary layer of the plate is not modeled and, thus, the interior stresses acting as surface tractions do work on the lateral edges of the plate. This feature is confirmed energetically by the Clapeyron's theorem. The variational formulation is carried out for the annular Levinson plate by employing the principle of virtual displacements. As a novel contribution, the formulation includes the external virtual work done by the tractions based on the interior stresses on the inner and outer lateral edges of the Levinson plate. The obtained plate equations are consistent with the vectorially derived Levinson equations. Finally, we develop an exact plate finite element both by a force-based method and from the total potential energy of the Levinson plate.     

Three-dimensional formulation of dislocation climb

We derive a Green's function formulation for the climb of curved dislocations and multiple dislocations in three-dimensions. In this new dislocation climb formulation, the dislocation climb velocity is determined from the Peach–Koehler force on dislocations through vacancy diffusion in a non-local manner. The long-range contribution to the dislocation climb velocity is associated with vacancy diffusion rather than from the climb component of the well-known, long-range elastic effects captured in the Peach–Koehler force. Both long-range effects are important in determining the climb velocity of dislocations. Analytical and numerical examples show that the widely used local climb formula, based on straight infinite dislocations, is not generally applicable, except for a small set of special cases. We also present a numerical discretization method of this Green's function formulation appropriate for implementation in discrete dislocation dynamics (DDD) simulations. In DDD implementations, the long-range Peach–Koehler force is calculated as is commonly done, then a linear system is solved for the climb velocity using these forces. This is also done within the same order of computational cost as existing discrete dislocation dynamics methods.     

Levitation of Magnets and Paramagnetic Bodies in Vessels Filled with Magnetic Fluid

Formulas are obtained for the forces and moments acting on a spherical body made of a paramagnetic material in an uniform applied magnetic field and a magnet in a spherical vessel filled with magnetic fluid. An approximate formula is found for the force acting on bodies in ellipsoidal and cylindrical vessels or in a plane channel with a magnetic fluid in an uniform magnetic field. An analogy between the forces acting on a magnet and a paramagnetic body is demonstrated. The possibility of levitation of magnets and paramagnetic bodies in a vessel with a magnetic fluid is investigated.     

Diffraction of regular waves by an arbitrarily oriented system of vertical circular cylinders

The linear theory of small-amplitude waves is used to construct a solution to the problem of the diffraction of surface gravity waves by a system of arbitrarily oriented vertical circular cylinders. Analytic expressions are obtained for the wave forces and overturning moments acting on each cylinder of the system. A system of two rows of cylinders with three cylinders in each row is considered as an example. It is shown that for certain relationships between the diameter of the cylinders, the distance between them, the angle of approach of the wave, and its wavelength, the maximal values of the hydrodynamic forces acting on the cylinders of the system with allowance for the interaction of the diffracted fields may be appreciably greater than when no allowance is made for it.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 177–181, January–February, 1984.     

Static wetting on flexible substrates: a finite element formulation

In static wetting on an elastic substrate, force exerted by the liquid–vapour surface tension on a solid surface deforms the substrate, producing a capillary ridge along the contact line. This paper presents a finite element formulation for predicting elastic deformation, close to the static wetting line (with angle of contact=90o and σSV=σSL).The substrate deformation is modelled with the Mooney–Rivlin constitutive law for incompressible rubber‐like solids. At the contact line, a stress singularity is known to arise, due to the surface tension acting on a line of infinitesimal thickness. To relive the stress singularity, either (i) the surface tension is applied over a finite contact region (of macroscopic thickness), or (ii) the solid crease angle is fixed. These two options suggest that normal component of Neumann's triangle law of forces, for the three surface tensions, is not applicable for elastic substrates (as for rigid ones). The vertical displacement of the contact line is a strong function of liquid/vapour surface tension and shear modulus of the solid. Copyright 2004 John Wiley & Sons, Ltd.     

船行波对水中直立圆柱作用的数值模拟

针对船行波对水中直立圆柱的作用进行了数值计算。主要包括船行波的计算以及波浪对水中直立圆柱的作用。对于船行波的计算应用薄船理论和Noblesse给出的格林函数的简化形式,计算出Wigley船型在静水中匀速直线运动产生的船行波的远场解。以该船行波为入射波,应用Rankine源方法对船行波对水中直立圆柱的作用力及力矩进行了数值计算,得出了船只经过圆柱周围时圆柱的受力情况,对计算结果进行了定性的分析。     

Two-dimensional sliding frictional contact of functionally graded materials

A multi-layered model for sliding frictional contact analysis of functionally graded materials (FGMs) with arbitrarily varying shear modulus under plane strain-state deformation has been developed. Based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves, the FGM is divided into several sub-layers and in each sub-layers the shear modulus is assumed to be a linear function while the Poisson's ratio is assumed to be a constant. In the contact area, it is assumed that the friction is one of Coulomb type. With this model the fundamental solutions for concentrated forces acting perpendicular and parallel to the FGMs layer surface are obtained. Then the sliding frictional contact problem of a functionally graded coated half-space is investigated. The transfer matrix method and Fourier integral transform technique are employed to cast the problem to a Cauchy singular integral equation. The contact stresses and contact area are calculated for various moving stamps by solving the equations numerically. The results show that appropriate gradual variation of the shear modulus can significantly alter the stresses in the contact zone.     

Finite element methods for a class of continuum models for immiscible flows with moving contact lines

In this paper, we present a finite element method for two‐phase incompressible flows with moving contact lines. We use a sharp interface Navier–Stokes model for the bulk phase fluid dynamics. Surface tension forces, including Marangoni forces and viscous interfacial effects, are modeled. For describing the moving contact lines, we consider a class of continuum models that contains several special cases known from the literature. For the whole model, describing bulk fluid dynamics, surface tension forces, and contact line forces, we derive a variational formulation and a corresponding energy estimate. For handling the evolving interface numerically, the level‐set technique is applied. The discontinuous pressure is accurately approximated by using a stabilized extended finite element space. We apply a Nitsche technique to weakly impose the Navier slip conditions on the solid wall. A unified approach for discretization of the (different types of) surface tension forces and contact line forces is introduced. Results of numerical experiments are presented, which illustrate the performance of the solver. Copyright © 2016 John Wiley & Sons, Ltd.     

Motion of a cylinder in a viscous electroconductive medium with a magnetic field

The method of force sources is used to consider the planar problem of the motion of a circular cylinder in a viscous electroconductive medium with a magnetic field. The conventional and magnetic Reynolds numbers are assumed to be small. Expressions are obtained for the hydrodynamic reaction forces of the medium, acting on the moving cylinder. It is shown that as a result of the flow anisotropy in the medium, caused by the magnetic field, in addition to the resistance forces on bodies moving at an angle to the field, there are deflecting forces perpendicular to the velocity vector. The velocity field disturbances at great distances from the moving cylinder are determined.The problems of viscous electroconductive flow about solid bodies in the presence of a magnetic field constitute one of the divisions of magnetohydrodynamics. Motion of an electroconductive medium in a magnetic field gives rise to inductive electromagnetic fields and currents which interact with the velocity and pressure hydrodynamic fields in the medium [1, 2]. Under conditions of sufficiently strong interaction, the number of independent flow similarity parameters in MHD is considerably greater than in conventional hydrodynamics. This circumstance complicates the theoretical analysis of MHD flow about bodies, and therefore we must limit ourselves to consideration of individual particular flow cases.Here we consider the linear problem of the motion of an infinite circular cylinder in a viscous incompressible medium with finite electroconductivity located in a uniform magnetic field.There are many studies devoted to the flow of a viscous electroconductive medium with a magnetic field about solid bodies (see, for example, [3–5]). Because of this, some of the results obtained here include previously known results, which will be indicated below. In contrast to the cited studies, the examination is made by the method of force sources, suggested in [6]. This method permits obtaining integral equations for the distribution of the forces acting on the surface of the moving body. Their solution is obtained for small Reynolds and Hartmann numbers. Then the nature of the velocity disturbances at great distances from the body are determined. These results are compared with conventional viscous flow about a cylinder in the Oseen approximation.