Internal Singularities Research Articles

Architecture Selection and Singularity Analysis of a Three-Degree-of-Freedom Planar Parallel Manipulator

A three degree-of-freedom planar parallel manipulator, intended for high-speed, high-precision wire-bonding and electronic-component placement tasks, has been developed in our laboratory. In this paper, the work related to the kinematic manipulator-architecture selection is presented. The reachable workspace and “effective base area” metrics of the parallel manipulator were utilized for selecting the best possible architecture amongst six potential configurations. Constant platform-orientation regions, within the reachable workspace of the selected manipulator, were identified based on the manipulator task requirements. Simulation results for the workspace analyses (reachable workspace, effective base areas, and constant-orientation regions) are presented in this paper. Once the optimal-workspace architecture was selected, both workspace-boundary and internal singularities were further investigated in order to have a clear view of the set of uncontrollable poses of the manipulator. Singularity analyses examples are also included herein.

Singularity Robust Steering Logic for Redundant Single-Gimbal Control Moment Gyros

In this paper we investigate a problem inherent in a system of redundant single-gimbal control moment gyros (CMGs). In particular, we present a simple yet effective way of passing through, and also escaping from, any internal singularities. The proposed approach is based on the so-called singularity robust inverse, but it utilizes a nondiagonal weighting matrix in the mixed, two-norm and weighted least-squares minimization problem. The proposed steering logic effectively generates deterministic dither signals when the CMG system becomes near singular. Because the proposed CMG steering logic is based on the simple minimum two-norm, pseudoinverse solution, it does not explicitly avoid encounters but it rather approaches and rapidly transits unavoidable singularities whenever needed. However, the proposed steering logic is not intended for special missions in which prescribed attitude trajectories are to be exactly tracked in the presence of internal singularities.

Propagation of singularities along a characteristic boundary for a model problem of shell theory and relation with the boundary layer

We consider the propagation of singularities for a differential system which constitutes a simplified model of thin shells with developable middle surface (parabolic case). Extensions of the solutions out of the domain allow us to consider either boundary or internal singularities. The properties of propagation of singularities and their relation with the structure of the boundary layers are given. We remove a mistake in [2], Section 6.1, concerning the analyticity of solutions (in fact they are in the Gevrey class of order 3).

Stokes flow inside a porous spherical shell

Stokes flow of a viscous, incompressible fluid inside a sphere with internal singularities, enclosed by a porous spherical shell, using Darcy's law for the flow in the porous region is discussed. The formulae for drag and torque are found by deriving the corresponding Faxen's laws. It is found that torque does not depend on the thickness of the spherical shell.

Some aspects of four-dimensional black hole solutions in gauss-bonnet extended string gravity

An internal singularity of a string four-dimensional black hole with second order curvature corrections is investigated. A restriction to a minimal size of a neutral black hole is obtained in the frame of the model considered. Vacuum polarization of the surrounding space-time caused by this minimal-size black hole is also discussed.

On tensile cracks close to and across the interface between two welded elastic half-spaces

Tensile cracks are often employed to model magma migration in rift zones or within volcanic edifices through lateral or feeding dykes. In a crack model, the overpressure of magma with respect to the horizontal stress in the host rock is assumed to be responsible for dyke opening and propagation. In this paper the most simple heterogeneous medium is considered, made up of two welded half-spaces, characterized by different elastic parameters. The analytical solutions available for the elementary dislocation problem in such a medium are employed to set up an integral equation with generalized Cauchy kernel, representing the condition for static equilibrium. The unknown in such a problem is the dislocation density distribution, whose singular behaviour is studied near the crack tips and near the intersection with the interface between the two media. The problem for a crack entirely embedded in one of the two media is considered first (case A). The resulting Cauchy integral equation is solved by employing a truncated expansion in Chebyshev polynomials, and ;-analytical solutions are given for the displacement and stress fields provided by a constant overpressure within the crack. The problem of a crack terminating at the interface between the two half-spaces is then considered (case B) and a generalized Cauchy equation is obtained in which additional singularities, localized at the crack tip next to the interface, appear in the integral kernel. The singular stress behaviour in the proximity of the crack tip is affected considerably by the presence of the interface. Within case B, after assuming a vanishing rigidity for the upper medium, a crack open at the free surface of a half-space is obtained (case B1). The singular behaviour of the solution is found to change abruptly in this case (with respect to the case of small, non-vanishing rigidity in the upper medium). Finally, a crack crossing the interface between the two half-spaces is considered (case C). The crack is split into two interacting sections, each embedded in a homogeneous half-space, both open at the interface. A system of generalized Cauchy equations is obtained, which is responsible for the appearance of internal singularities in the dislocation density distribution, localized at the intersection between the crack plane and the interface. The order of this singularity depends only upon the elastic parameters of the media in welded contact. Crack opening, interface displacement and stress components are computed employing the method described for case A, assuming a known overpressure within the crack. It appears that layering can be responsible for stress changes, localized along the interface, which may be considerably higher than the overpressure within the dyke. These results may provide useful hints for the interpretation of induced seismicity in rift zones and in volcanic areas.

The effect of the interface on the interaction of an interior edge dislocation with an elliptical inhomogeneity

The image glide force on an edge dislocation inside an elliptic inhomogeneity is investigated using previously-established general solutions to the plane-elasticity problems of the elastic field produced by an internal line singularity. The matrix-inhomogeneity interface is assumed either perfectly adhering or slipping. The results for the two kinds of boundary are compared by means of equiforce contour plots. It is found that for an adhering interface the force field is qualitatively insensitive to the inhomogeneity shape. On the contrary, for a slipping interface shape effects are essential, ranging from a "matrix-shielding" effect, to the appearance of new stable-equilibrium loci.

Four-Dimensional Dilatonic Black Holes in Gauss-Bonnet Extended String Gravity

An internal singularity of a string four-dimensional black hole with second order curvature corrections is investigated. A restriction to a minimal size of a neutral black hole is obtained in the frame of the model considered. Vacuum polarization of the surrounding space-time caused by this minimal-size black hole is also discussed.

Oscillating flow of a heat-conducting fluid in a narrow tube

Thermoacoustic refrigeration occurs in periodic flow in a duct with heat transfer within the fluid and to the tube. This study considers the periodic limit cycle with large pressure oscillations that is obtained in a tube when prescribed, phase-shifted, periodic velocities at the tube ends, at frequencies lower than acoustic eigenmodes, sweep a length comparable to the tube length. The temperature differences between the two ends are of arbitrary magnitude, heat transfer in the transverse direction within the fluid is assumed to be very effective and the thermal mass of the wall is large. The geometry is two-dimensional, axisymmetric, and conduction is accounted for, not only in the fluid, but also with and within the tube wall. A perturbation solution valid in a local near-isothermal limit determines the equilibrium longitudinal temperature profile that is reached at the periodic regime, the pressure field including longitudinal gradients, and the longitudinal enthalpy flux. Results are presented for tubes open at both ends and also with one end closed. In the latter case, a singularity occurs in the temperature at the closed end, with behaviour identical to Rott's result for acoustic flow with small pressure amplitude. Other new results obtained for tubes open at both ends show that when velocities at both ends are in opposite phase, internal singularities in the temperature profiles may occur.

On integrating vertex singularities using extrapolation

A new approach to the integration of vertex singularities is described. This approach is based on a non-uniform subdivision of the region of integration and the technique fits well to the subdivision strategy used in many adaptive algorithms. A nice feature with this approach is that it can be used in any dimension and on any region of integration which can be subdivided into subregions of the same form. The strategy can be applied both to vertex singularities and internal point singularities. In the latter case this can be done without an initial subdivision of the region in order to put the singular point in a vertex. It turns out that the technique has excellent numerical stability properties.

Internal Lifschitz singularities for one dimensional Schrödinger operators

The integrated density of states of the periodic plus random one-dimensional Schrödinger operator\(H_\omega = - \Delta + V_{per} + \sum\limits_i {qi\left( \omega \right)f\left( {o - i} \right)} \);f≥0,q i (ω)≥0, has Lifschitz singularities at the edges of the gaps inSp(H ω). We use Dirichlet-Neumann bracketing based on a specifically one-dimensional construction of bracketing operators without eigenvalues in a given gap of the periodic ones.

Topological and differential-equation methods for surface intersections

An implicit representation of the intersection of two parametric surface patches is formulated in terms of the zero level set of the oriented distance function of one surface from the other. The topological properties of the vector field defined as the gradient of the oriented distance function (and specifically the concepts of the rotation number of a vector field and the Poincaré index theory) are next used to formulate a condition for the detection of critical points of the field, which may be used to identify internal loops and singularities of the solution. An adaptive search guided by this condition and direct numerical techniques are used to detect and compute critical points. Tensorial differential equations that rely on the implicit representation of the intersection are then developed, and are used to trace intersection segments. The tracing scheme relies on a characterization procedure for singular points that allows the computation of the tangent directions to the intersection curve at such points. The method also allows the tracing of degenerate intersections between surfaces that have a common tangent plane along the intersection curve. Examples drawn from rational spline surface intersections illustrate the above techniques.

Redundancy resolution for the human-arm-like manipulator

The human-arm-like manipulator has a seven-degree-of-freedom kinematic structure obtained by adding a roll joint to the shoulder of the PUMA geometry. This mechanism has higher dexterity than conventional six-degree-of-freedom arms, and allows the elimination of the internal singularities at the shoulder and the wrist. The contribution of the present work is to exploit redundancy to accomplish singularity avoidance and possibly recover the original PUMA design. The inverse for kinematics problem is solved at the velocity level by means of a closed-loop algorithm in the framework of task space augmentation with task priority. Three case studies demonstrate the effectiveness of the proposed technique.

Preferred gimbal angles for single gimbal control moment gyros

This paper deals with torque command generation using single gimbal control moment gyros. The angular momentum and torque envelopes are assumed to be known a priori. A method based on back integration of the gyro torque equation from desired final conditions is used to determine a family of initial gimbal angles that avoid singularities. Each member of this family is defined as a preferred initial gimbal angle set. The pseudoin- verse steering law is used during the numerical integrations. This procedure is demonstrated by means of numerical examples that include attitude control and momentum management of the Space Station Freedom. A feedback control scheme based on null motion is also developed to position the gimbals at preferred angles. ONTROL moment gyros (CMGs) are attractive space- craft attitude-control devices. They require no expendable propellant, which is a limited resource and can contaminate the spacecraft environment. Their fixed rotor speeds minimize structural dynamic excitations. They can be used for rapid slewing maneuvers and precision pointing. For low Earth or- biting spacecraft, momentum dumping can be easily achieved by gravity-gradient torques. From the steering-law viewpoint, it is widely accepted that double-gimbal CMGs (DCMGs) are preferable to single-gimbal CMGs (SCMGs). For DCMGs, steering laws proposed by Kennel 1'2 have been well accepted. The SCMGs have the advantages of possessing relative me- chanical simplicity and producing amplified torques (for low spacecraft angular velocities) on the spacecraft. However, de- velopment of gimbal steering laws for their use is made diffi- cult by the existence of internal singular states. For any system of n CMGs and any direction in space, there exist 2 sets of gimbal angles3 for which no torque can be produced in that direction, and these sets are called internal singularities. Exter- nal singular states correspond to directional angular momen- tum saturation. DCMGs have internal singularities also, but they are easier to avoid. Margulies and Aubrun3 present a geometric theory of SCMG systems. They characterize the momentum envelope of a cluster of SCMGs and identify the internal singular states. Yoshikawa4 presents a steering law for a roof-type configura- tion with four SCMGs. His steering law is based on making all of the internal singular states unstable by providing two jumps with hystereses around the singularities. Cornick5 develops singularity avoidance control laws for the pyramid configura- tion. His technique is based on the ability to calculate the instantaneous locations of all singularities. Hefner and McKenzie6 develop a technique for maximizing the minimum torque capability of a cluster of SCMGs in the pyramid con- figuration. Bauer7 concludes that it is impossible to avoid some singularities and, in general, no global singularity-avoid- ance steering law can exist. Consequently, there will be in- stances when torque demand cannot be met exactly.

Electric monopole with internal magnetic-monopole-like structure

The minimal coupling of a membrane to external electromagnetic vector potentials is governed by its charge-current three- vector. The latter is derivable from a “statistical” gauge field living on the world-manifold. The exclusive spherically symmetric internal configuration is magnetic-monopole-like, intriguingly interpreted as a genuine electric monopole by an outside observer. Electric-charge quantization then follows provided no internal Dirac point singularity exists. The emerging Dirac-type model cannot tolerate a “statistical” Chern-Simons term.

Steering law design for redundant single-gimbal control moment gyroscopes

Two steering laws are presented for single-gimbal control moment gyroscopes. An approach using the Moore-Penrose pseudoinverse with a nondirectional null-motion algorithm is shown by example to avoid internal singularities for unidirectional torque commands, for which existing algorithms fail. Because this is still a tangent-based approach, however, singularity avoidance cannot be guaranteed. The singularity robust inverse is introduced as an alternative to the pseudoinverse for computing torque-producing gimbal rates near singular states. This approach, coupled with the nondirectional null algorithm, is shown by example to provide better steering law performance by allowing torque errors to be produced in the vicinity of singular states.

Investigation of numerical solutions of integral equation methods for multi-element aerofoils

This paper presents a number of integral equation methods, including internal and surface singularity methods. The effect of the curved-element and flat-element models with the computational accuracy and numerical instability is investigated. The analytical solution for the inviscid incompressible flow past the multiple circles, which is conformally mapped onto the aerofoil sections, can be calculated by the application of the Milne-Thomson circle theorem. These results are used to evaluate the accuracy of the numerical solutions.

Nearly one dimensional singularities of solutions to the Navier-Stokes inequality

There exists a functionu satisfying (1)u is a weak solution to the Navier-Stokes equations of incompressible fluid flow in three-space with an external force that reduces the speed at every point, (2) the internal singularities ofu have Hausdorff dimension close to 1.

Universal amplitude ratios of critical sound attenuation

The universal amplitude ratios for ultrasonic attenuation above and below the critical point are determined for uniaxial magnets, the gas liquid critical point, binary mixtures and related universality classes. For uniaxial ferromagnets also the scaling functions are computed. The theory is based on time dependent Ginzburg-Landau models and the ɛ=4−d-expansion. Depending on the universality class the universal amplitude ratio assumes values in the range 1/15 to 15. In systems with continuous internal symmetry Goldstone mode singularities are found.

Internal Lifschitz singularities of disordered finite-difference Schr�dinger operators

The integrated density of states has C∞-like singularities, ln|k(E)−k(Ec)|=−|E−Ec|−v/2 ϕc(E), with ϕc>0, a milder function at the edges of the spectral gaps which appear when the distribution function of the potentialdμ has a sufficiently large gap. The behaviour of ϕc nearEc is determined by the local continuity properties ofdμ near the relevant edge: ϕc(E)=O(1) ifdμ has an atom and ϕ=O(ln|E−Ec|) if μ is (absolutely) continuous and power bounded.