Angular velocity RCCC-linkage of a spherical four-bar linkageWinkelgeschwindigkeits-RCCC-getriebe eines sphärischen Viergelenkgetriebes

The angular velocity diagram of a spherical four-bar linkage is a tetrahedron in space. Based on this angular velocity tetrahedron an RCCC-linkage may be devised. The driving link of this RCCC-linkage rotates with the same angular velocity as that of the driving link of the original spherical four-bar. The linear displacement and the sliding velocity of the output link of the RCCC-linkage represent respectively the angular velocity and angular acceleration of the output link of the original spherical four-bar. The shape of the equivalent dual spherical four-bar linkage of the RCCC-linkage is identical with that of the original spherical four-bar linkage.     

Stossfreie tourenkupplung für hohe arbeitsgeschwindigkeitenA shock-free,high-speed,rotating coupling

The present paper reports on the development of principles allowing a considerable improvement in the efficiency of rotating couplings. A block diagram is shown in Fig. 2. The constant angular input velocity ω₀ is transformed by a step gear into a progressive stepping motion with the intermittent angular velocity ω_A (0 ? ω_Amax ω_A), and fed to a clutch with torque transmission through a positive connection. Establishing and releasing this connection is performed through response to control signals whenever the angular velocity ω_A = 0. The control signals are generated by a synchronizing unit, a logical operation being carried out between the external coupling instructions and clock pulses identifying the angular velocity ω_A = 0. Since the angular velocity difference between the input and output of the clutch is zero during the coupling action, the driven side is moved harmonically.The three gear drive and the cycloidal step gear for generating the stepping motion, as well as an electromagnetic drive for the clutch, are represented as self-contained function units for the technical implementation of the above principles. The step gears feature a shock- and jerk-free motion transmission with near-dwell. The near-dwell and the coupling time of the clutch are coordinated such that high working speeds are obtained from the rotating coupling.Theoretical considerations about the particular function units are complemented by essential suggestions concerning design and production engineering.The test results listed refer to both the individual function units and the entire coupling. The step gears can be used successfully when step angles of 360° or 180° are desired, with stepping frequencies of up to 50 Hz and higher. The electromagnetic drive enabled coupling times of less than 2 miliseconds to be obtained. In the long-duration test, 2.6 (10)⁶ stepping cycles were attained (for Vf_S = 35 Hz with J_A = 250 gcm²) by means of the full-revolution coupling, and 4(10)⁶ stepping cycles (for f_S = 50 Hz with J_A = 250 gcm²) by means of the half-revolution coupling without disturbances having occured. The results prove the fact that an ingenious combination of mechanism engineering on the one hand and of electromechanics and electronics on the other hand can produce assemblies of high efficiency.     

An unsymmetrical watt-1 linkage generating a family of symmetrical curvesEin unsymmetrisches watt-1 gelenkgetriebe,dass trotzdem eine familie von symmetrische kurven erzeugt

The two minimum transmission angles, 2μ₁ and 2μ₂, and the coupler angle β of a four-bar, determine a family of symmetrical 6-bar curves, traced by non-symmetrical Watt-1 mechanisms, obtained through cognation and generalization of Kempe's focal linkage.The 6-bar linkages obtained, contain a bar that moves in a completely symmetric wayA geometric derivation of these highly useful linkages is presented in the paper in conjunction with a set of expressions for their dimensions. A set of examples, obtained through variation of the three determining parameters and two “double numbers” (δ₁ and δ₂ = ± 1) demonstrates the topological change in the shape of the curves that are traced by a set of points chosen on the symmetrical movinb bar.     

On the dual spherical motions—ISur les mouvements spheriques duaux (lére partie)

In this paper it is shown that if the axes of curvature of the ruled surfaces $\text{U}{}_1\text{(t)}$ in the moving and fixed spaces intersect each other perpendicularly then the dralls (or distribution parameter) of the ruled surfaces, which are generated by the geodesic trihedron U₁, U₂, U₃s of a dual curve $\text{U}{}_1\text{(t)}$, satify the A. Mannheim and R. Hamilton's formulae. Furthermore, the “explicit characterization” of the spherical curve of striction is given in terms of U₃-surface and the skewness of distribution of the generator of U₁-surface.     

A quantitative model to estimate rail surface failure

This paper is concerned with the formulation of an approximate analytical model describing the total wear on both high and low rails as a function of the angle of attack ψ. The rail life γ(ψ) is obtained in this model as follows as: γ(ψ) = l(ψ?m)²+ nψ? 0.0058 radγ(ψ)=Pψ+qψ ? 0.0058 rad where l,m, n, p and q are constants which can be determined for a specific set of boundary conditions. The results of applying this equation to various values of ψ are compared with field data and good agreement is observed.     

Zur geometrie der Joukowsky-transformation und über die kinematische erzeugung von tragflügelprofilenOn the geometry of Joukowski transformation and the kinematical generation of wing profiles

In its first part the paper considers, from an advanced point of view, the essential properties of Joukowski's conformal mapping $\text{J: z}{}^1\text{= z+z?}{}^1$. In general, the point $\text{Z}{}^1\text{(z}{}^1\text{=x}{}^1\text{+iy}{}^1\text{)}$ corresponding to a real point Z (z=x+iy) is uniquely determined, with the exception of the fundamental points situated at the origin O and the circular points $\text{I,}\text{I}$ (Fig. 1). Conversely, a real point $\text{Z}{}^1$ arises from two real points Z, Z′ and two conjugate imaginary points Z″, Z3? are related by the involutary Moebius transformation V: z.z′ = 1.The circle k (1.8), determined by its complex center coordinate M≠0 and radius r≠|m|, is carried by the cubic transformation J into a rational bicircular quartic $\text{K}{}^1$ (1.10), but the same curve is also the mapping of a second circle k′ (1.11), connected with k by V (Fig. 4). The singular foci $\text{M}{}^1$ and $\text{N}{}^1$ correspond to the centers M and N of k and k′. The proper point-pair D, D′ common to k and k′ is mapped onto the proper double point $\text{D}{}^1$ of $\text{k}{}^1$.The second part deals with the kinematic generation, at first by means of an antiparallelogram, of such a quartic $\text{k}{}^1$, determined by $\text{M}{}^1\text{, N}{}^1\text{, D}{}^1$ and an arbitrary simple point $\text{Z}{}^1$. The foci $\text{M}{}^1\text{, N}{}^1$ are the fixed pivots of the four-bar linkage $\text{M}{}^1\text{A}{}_1\text{A}{}_2\text{N}{}^1$, and the position $\text{A}{}_1\text{A}{}_2\text{Z}{}^1$ of the coupler triangle is to be found as the mirror image of the triangle $\text{N}{}^1\text{M}{}^1\text{D}{}^1$ with respect to the perpendicular bisector of $\text{D}{}^1\text{Z}{}^1$ (Figs. 5 and 6). According to the theorem of Roberts, there exist two additional kite linkages generating the same quartic $\text{k}{}^1$. The third fixed pivot $\text{L}{}^1$ corresponds to $\text{D}{}^1$ in the reflection $\text{M}{}^1\text{? N}{}^1$ (Fig. 6).The essential dimensions of all generating linkages are given by formulae (2.2–2.7).     

Solid particle erosion of mullite

The solid particle erosion of mullite (3Al₂O₃·2SiO₂ plus approximately 12% glass phase) was investigated using angular A1₂O₃ particles whose mean diameters D were varied between 23 and 270 μm. A range of impact angles α between 15° and 90° was used and the impact velocity V was varied from 60 to 100 m s^?1. The results of these experiments are in agreement with the general predictions of the two models developed to describe erosion in brittle materials on the basis of the formation of lateral cracks. The velocity exponent of the steady state erosion rate ΔW is between 2.2 and 2.8, being larger for smaller particles. For normal incidence, $\text{ΔW ∝ D}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$ in accord with both theories. For α > 15° only the normal component of velocity need be considered, but for smaller α there is evidence of plasticity both in the α dependence of ΔW and as observed using scanning electron microscopy.     

Die sphärischen analoga zu den ϱ-Kurven von H. AltThe spherical analogy to the -curves of H. Alt

Developing planar coupler mechanisms with a dwell, H. Alt has studied for the first time the “?-curves”. A ?-curve is the sequence of those points of the moving plane, which pass through the position on the path with equal radius of curvature.In this paper the analogous locus curves called the locus cones (λ_?-curves and λ_?-cones) are investigated, which result from the spherical motions; i.e., from motions about a fixed point. These cones are found to be an important special case of the general three-dimensional motion.An introductory part of the paper gives a short survey on the kinematical theory, followed by a brief investigation of the case λ_? = π/2 ( the analogy to the inflection circles of the planar motion) and the planar problem. The general λ_?-curves and corresponding λ_?-cones are treated in detail: their general shape, as well as their curvatures, the spherical subnormals and the positions of the osculating plane of the three-dimensional locus curves for special points are described.     

A theory of plastic anisotropy based on a yield function of fourth order (plane stress state)—I

It is shown from various view points that many of the disadvantages of the conventional theory based on a quadratic yield function can be satisfactorily removed by the use of a yield function of fourth order. Incremental equivalent strain $\text{d}\text{?}{}_{\mathrm{eq}}$is defined by $\text{d}\text{?}{}_{\mathrm{eq}}\text{=σ}{}_{\mathrm{ij}}\text{d}\text{?}{}_{\mathrm{ij}}\text{σ}{}_{\mathrm{eq}}$, and cannot generally be expressed simply by the strain increment components d?_ij. In contrast with the conventional theory, coefficients in the yield function f cannot be determined from the r-values only in uniaxial tensile tests, but yield stresses in these tests and for example in an equi-biaxial tension for the same $\text{}\text{?}\text{ge}{}_{\mathrm{eq}}$ are also required. This fact ensures that the $\text{σ}{}_{\mathrm{eq}}\text{?}\text{?}{}_{\mathrm{eq}}$ curve for arbitrary loading is uniquely determined by the uniaxial tension curve in the rolling direction (R.D.), and thus such an intrinsic difficulty of the conventional theory as dependence of the $\text{σ}{}_{\mathrm{eq}}\text{?}\text{?}{}_{\mathrm{eq}}$ curve on types of loading does not arise. Some formulae for the determination of the coefficients in f are given. Relationships between types of earing in axi-symmetrical deep-drawing and the coefficients of f are examined in detail and it is emphasized that only very special cases are included in the conventional yield function and thus use of it is very limited.     

Determination of an improved gear surface of helical gears for pumping actionZur synthese optimaler maschinenelemente — optimale zahnflächen für schraubenpumpen

An improved gear surface of helical gears for pumping action is obtained mainly by theoretical investigation, and partly by test results. This is a further study following the research on gears[1]. A new theory is presented here by selecting characteristic function sequences

$\text{s = Φ}{}_{\mathrm{n}}\text{(t) = 1 ? (1?t)}{}^{\mathrm{n}}\text{, n = 1, 2, 3, … n}{}_0$

which is monotone increasing. The larger n becomes, the larger rate of mass flow $\text{V}\text{and effective work ratio}\text{=}\text{E.W.R.}\text{=}\text{E}$ increases at the same time, thus the optimum tooth profile of gears (consequently the optimal gear surface) can be determined under the assumption that the minimum available cutting pressure angle (θ_ON_O be given in advance.     

Massenersatz durch punktmassen in räumlichen Getrieben

Dynamic and kinetostatic analysis of three-dimensional mechanisms (i.e. mechanisms with intersecting or skew axes) can be brought about by using mass points. The objective of this paper is to give simple ways of determining such mass points with respect to their quantity and position by means of matrix calculus.Two rigid mass systems will be in quadratic equivalence, if their total masses are equal and their centers of mass coincide and their second moments of mass are equal with respect to a common system of coordinates. Under equal initial conditions, the two bodies will then perform equal motions if equal forces are acting upon them. Thus the total mass of a given three-dimensional rigid body may be substituted for by a finite number of mass points in rigid interconnection (Fig. 1). A mass point of quantity m_i at a point P_i(x_i, y_i, z_i) is determined by four defining quantities (m_i, x_i, y_i, z_i). The condition of equality of the total masses leads to one equation, the coincidence of the centers of mass results in three equations, and six equations follow from the equality of the second moments of mass. Thus the condition of quadratic equivalence will result in a system of non-linear equations (equation (4)) consisting of 1 + 3 + 6 = 10 scalar equations. These must be satisfied by four defining quantities per mass point. Nevertheless a three-dimensional body cannot be substituted for by three mass points with 3 × 4 = 12 defining quantities, since these three mass points are always on the same plane and so can only substitute for a mass system on one plane. The minimum number of mass points necessary is four. Of the total 4 × 4 = 16 defining quantities, 16 ? 10 = 6 can be chosen freely. After introducing suitable values for these six quantities, the ten unknown quantities can be calculated in a definite and explicit way (equations (6)–(15)) in spite of the above-mentioned system of equations being non-linear. It is not necessary to calculate the principle axes of inertia.As two further instances the substitution of a mass system on one plane by three mass points and the substitution of a rectilinear mass system by two mass points is discussed.     

表征漫射体反射特性的一个重要物理参数——双向反射分布函数

本文着重介绍了双向反射分布函数f_b(θ_i,φ_i,θ_r,φ_r)及其在漫射测量中的主要作用,给出了f_b(θ_i,φ_i,θ_r,φ_r)与九种反射比和九种反射比系数之间的关系,同时介绍了f_b(θ_i,φ_i,θ_r,φ_r)的几种测试方法,有关测试仪器的设计准则以及我所研制和使用这种仪器的简单情况.     

The expected volume of a random polytope in a ball*

For any convex body K in d-dimensional Euclidean space E^d(d≥2) and for integers n and i, n ≥ d + 1,1 ≤ i ≤ n, let V^(d)_n-ii(K) be the expected volume of the convex hull H_{n-i, i} of n independent random points, of which n-i are uniformly distributed in the interior, the other i on the boundary of K. We develop an integral formula for V^(d)_{n-i, i}(K) for the case that K is a d-dimensional unit ball by considering an adequate decomposition of V^(d)_n-i, i into d-dimensional simplices. To solve the important case i = 0, that is the case in which all points are chosen at random from the interior of B_d, we require in addition Crofton's theorem on mean values. We illustrate the usefulness of our results by treating some special cases and by giving numerical values for the planar and the three-dimensional cases.     

The high speed size effect in grinding: The role of heat generation

Experimental data on the depth dependence of the specific energy of grinding and of single-point scratching does not agree with the predictions of the ploughing model. In this paper the model is extended to take into account the heat generated beneath the scratching point, the subsequent softening of the material and the influence on the specific energy e and the scratch hardness H_s. For pyramidal points $\text{e = H}{}_{\mathrm{s}}\text{∝ d}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{v}{}_{\mathrm{s}}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$; for spheres $\text{e ∝ d}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>4</mtext>}}\text{v}{}_{\mathrm{s}}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$, whereas H_s shows a different behaviour which depends on the choice of parameters. The dependence on depth d and speed v_s agrees with experimental data on the grinding and scratching of steel.     

Study of ductile mode cutting in grooving of tungsten carbide with and without ultrasonic vibration assistance

In this study, ductile mode chip formation in conventional cutting and ultrasonic vibration assisted cutting of tungsten carbide workpiece material has been investigated through experimental grooving tests using CBN tools on a CNC lathe. The experimental results show that as the depth of cut was increased there was a transition from ductile mode to brittle mode chip formation in grooving both with and without ultrasonic vibration assistance. However, the critical value of the depth of cut for ductile mode cutting with ultrasonic vibration assistance was much larger than that without ultrasonic vibration assistance. The ratio of the volume of removed material to the volume of the machined groove, f _ab , was used to identify the ductile mode and brittle mode of chip formation in the grooving tests, in which f _ab <1 indicates ductile mode chip formation and f _ab >1 indicates brittle mode chip formation. For the same radius of tool cutting edge, the value of f _ab at the ductile-brittle transition region either with or without ultrasonic vibration was less than 1. However, the f _ab value with ultrasonic vibration assistance was close to 1. The experimental results demonstrate that ultrasonic vibration assisted cutting can be used to improve the ductile mode cutting performance of tungsten carbide work material.Nomenclature A amplitude - A ₁ , A ₂ cross-section areas of the ridge - A _V cross-section area of the groove - A _W the value of A _V subtracted by A ₁+A ₂ - f vibration frequency - f _ab ratio of work material removal - t time - v nominal cutting speed - v _u vibration velocity - v _t true cutting speed in ultrasonic cutting - angular frequency     

Elastohydrodynamic analysis of a cylindrical journal bearing with a flexible bearing shell

The effect of the elastic deformation of a bearing shell was considered in the determination of the performance characteristics of a hydrodynamic journal bearing. The finite element method with an iteration scheme was employed to solve the Reynolds equation governing flow in the clearance space and the three-dimensional linear elasticity equations representing the displacement vector field in the bearing shell. For design convenience a nondimensional deformation coefficient ψ relating μ, E_m, U₀ , C, R_j and tis defined. The performance characteristics were obtained in terms of load-carrying capacity, fluid flow, power loss and attitude angle for an aspect ratio $\text{L}\text{D = 1}$, eccentricity ? = 0.6 and for a wide range of deformation coefficients. The results are compared for bearing materials having Poisson's ratio v equal to 0.3 and 0.4.     

Geometric determination of coordinated centers of curvature in network mechanisms through linkage reductionDie geometrische bestimmung der bahnkrümmungen in labyrinthischen gelenkgetrieben mittels zusammenfügungen von jedesmahl zwei einzelgelenke

Coordinated centers of curvature in a network mechanism may be found by way of linkage reduction. This has to be carried out twice, each time in a different way, namely, a first order reduction through joint-joining in order to determine the velocity poles, [1] and a second order reduction, that equally replaces binary bars, but this time preserves instantaneous motion up to the 2nd order.For the reduced linkage, the problem of finding coordinated centers of curvature may be solved by successive application of Bobillier's Theorem in different four-bar loops.In order to show applicability also for linkages not containing four-bars, the method will be demonstrated for an eight-bar linkage that contains only pentagonal loops. The method introduced is a purely geometric one and does not involve velocity or acceleration constructions. Notwithstanding that, the final result may be used also to determine accelerations in linkage mechanisms.     

Computersynthese des viergliedrigen räumlichen koppelgetriebes vom Typ RSSRComputer-aided synthesis of the RSSR spatial four-bar mechanism

During the last several years many scientists have worked in the field of kinematic synthesis of spatial mechanisms; see for example: Levitskii and Shakvazian [1], Hartenberg and Denavit [2], Mohan Rao, Sandor, Kohli, and Soni [7]. In this paper the exact synthesis of the RSSR spatial four-bar mechanism is presented. Both finitely and infinitesimally separated positions of the driver and follower are considered. The analytical development leads to a system of linear equations (6) using the free parameter β. This system of equations has solutions if the system determinant FDET vanishes, eq. (9). Solving this determinant, we obtain all ( α, β) combinations which form a basis for the determination of all geometric parameters of the mechanism. The free parameter β is variable between β_u and β_o and yields an infinite number of theoretical solutions when seven arbitrarily separated input-output positions are prescribed. Therefore, the designer has the possibility of including other constraints such as minimum transmission angle, special geometric demands, realization of an additional input-output position, or velocity or acceleration at a special position of the mechanism. Altogether; eleven separate problems are formulated and solved by use of a digital computer. Some examples are presented which demonstrate this method.     

Network numerical analysis of hydromagnetic squeeze film flow dynamics between two parallel rotating disks with induced magnetic field effects

We study numerically the hydromagnetic squeeze film between two rotating disks using the numerical network simulation method. The external magnetic field, H, generates an induced magnetic field, B, with radial (B_r), tangential (B_θ) and axial (B_z) components between the two disks, which rotate with different angular velocities, Ω₁ and Ω₂, and at time t are separated by a distance D(1−αt)^1/2. The applied magnetic field at the lower disk is assumed to be zero. The conservation equations for mass, momentum and induced magnetic field are reduced to a set of ordinary differential equations using a series of transformations, in terms of four dependent variables, f (axial velocity), g (azimuthal velocity), m (axial magnetic field component) and n (azimuthal magnetic field component) and a single independent variable, η (dimensionless disk separation), with appropriate boundary conditions. The transformed ordinary differential equations have collective order of 10 and are shown to be controlled by rotational Reynolds number (R₁), squeeze Reynolds number (R₂=Re_m/Bt), dimensionless parameter based on the magnetic force in the axial direction (R₃), dimensionless parameter based on magnetic force strength in the azimuthal (tangential) direction (R₄), magnetic Reynolds number (Re_m), disk rotational velocity ratio (S) and Batchelor number (Bt). In the present study we examine the flow regime at various Batchelor numbers (for the case of unity value of the squeeze Reynolds number, Re_m=Bt). Excellent comparison of NSM solutions is achieved with earlier analytical and shooting solutions. The present study finds applications in hydromagnetic lubrication of braking devices, slider bearings, rotating machinery, etc. Applications also arise in hydraulic shock absorbers employing electrically conducting liquids such as sodium where electro-magnetical braking of streams can be achieved in liquid metal cooled nuclear reactors for arresting control rods. Finally in the context of astronautical vehicles, the present study has applications in electromagnetic braking for potential spacecraft in planetary orbits.