An Improved Upper Bound for the Ring Loading Problem

The Ring Loading Problem emerged in the 1990s to model an important special case of telecommunication networks (SONET rings) which gained attention from practitioners and theorists alike. Given an undirected cycle on $n$ nodes together with non-negative demands between any pair of nodes, the Ring Loading Problem asks for an unsplittable routing of the demands such that the maximum cumulated demand on any edge is minimized. Let $L$ be the value of such a solution. In the relaxed version of the problem, each demand can be split into two parts where the first part is routed clockwise while the second part is routed counter-clockwise. Denote with $L^*$ the maximum load of a minimum split routing solution. In a landmark paper, Schrijver, Seymour and Winkler [SSW98] showed that $L \leq L^* + 1.5D$, where $D$ is the maximum demand value. They also found (implicitly) an instance of the Ring Loading Problem with $L = L^* + 1.01D$. Recently, Skutella [Sku16] improved these bounds by showing that $L \leq L^* + \frac{19}{14}D$, and there exists an instance with $L = L^* + 1.1D$. We contribute to this line of research by showing that $L \leq L^* + 1.3D$. We also take a first step towards lower and upper bounds for small instances

Dynamic buckling of an inextensible elastic ring: Linear and nonlinear analyses

Slender elastic objects such as a column tend to buckle under loads. While static buckling is well understood as a bifurcation problem, the evolution of shapes during dynamic buckling is much harder to study. Elastic rings under normal pressure have emerged as a theoretical and experimental paradigm for the study of dynamic buckling with controlled loads. Experimentally, an elastic ring is placed within a soap film. When the film outside the ring is removed, surface tension pulls the ring inward, mimicking an external pressurization. Here we present a theoretical analysis of this process by performing a post-bifurcation analysis of an elastic ring under pressure. This analysis allows us to understand how inertia, material properties, and loading affect the observed shape. In particular, we combine direct numerical solutions with a post-bifurcation asymptotic analysis to show that inertia drives the system towards higher modes that cannot be selected in static buckling. Our theoretical results explain experimental observations that cannot be captured by a standard linear stability analysis.Comment: 18 pages, 10 figure

Influence of in-plane dynamics of thin compression rings on friction in internal combustion engines

The compression ring-bore conjunction accounts for significant frictional parasitic losses relative to its size. The prerequisite to improving the tribological performance of this contact is a fundamental understanding of ring dynamics within the prevailing transient nature of regime of lubrication. Studies reported thus far take into account ring-bore conformance, based on static fitment of the ring within an out-of-round bore, whose out-of-circularity is affected by manufacturing processes, surface treatment and assembly. The static fitment analyses presume quasi-static equilibrium between ring tension and gas pressure loading with generated conjunctional pressures. This is an implicit assumption of ring rigidity whilst in situ. The current analysis considers the global modal behaviour of the ring as an eigenvalue problem, thus including its dynamic in-plane behaviour in the tribological study of a mixed-hydrodynamic regime of lubrication. The results show that the contact transit time is shorter than that required for the ring to reach steady state condition. Hence, the conjunction is not only subject to transience on account of changing contact kinematics and varied combustion loading, but also subject to perpetual ring transient dynamics. This renders the ring-bore friction a more complex problem than usually assumed in idealised ring fitment analyses. An interesting finding of the analysis is increased ring-bore clearance at and in the vicinity of top dead centre, which reduces the ring sealing effect and suggests a possible increase in blow-by. The current analysis, integrating ring in-plane modal dynamics and mixed regime of lubrication includes salient features which are closer representation of practice, an approach which has not hitherto been reported in literature

Application of the line-spring model to a cylindrical shell containing a circumferential or axial part-through crack

An approximate solution was obtained for a cylindrical shell containing a part-through surface crack. It was assumed that the shell contains a circumferential or axial semi-elliptic internal or external surface crack and was subjected to a uniform membrane loading or a uniform bending moment away from the crack region. A Reissner type theory was used to account for the effects of the transverse shear deformations. The stress intensity factor at the deepest penetration point of the crack was tabulated for bending and membrane loading by varying three dimensionless length parameters of the problem formed from the shell radius, the shell thickness, the crack length, and the crack depth. The upper bounds of the stress intensity factors are provided by the results of the elasticity solution obtained from the axisymmetric crack problem for the circumferential crack, and that found from the plane strain problem for a circular ring having a radial crack for the axial crack. The line-spring model gives the expected results in comparison with the elasticity solutions. Results also compare well with the existing finite element solution of the pressurized cylinder containing an internal semi-elliptic surface crack

Numerical nonlinear inelastic analysis of stiffened shells of revolution. Volume 1: Theory manual for STARS-2P digital computer program

The theoretical analysis background for the STARS-2P nonlinear inelastic program is discussed. The theory involved is amenable for the analysis of large deflection inelastic behavior in axisymmetric shells of revolution subjected to axisymmetric loadings. The analysis is capable of considering such effects as those involved in nonproportional and cyclic loading conditions. The following are also discussed: orthotropic nonlinear kinematic hardening theory; shell wall cross sections and discrete ring stiffeners; the coupled axisymmetric large deflection elasto-plastic torsion problem; and the provision for the inelastic treatment of smeared stiffeners, isogrid, and waffle wall constructions

On the interplay between strain rate and strain rate sensitivity on flow localization in the dynamic expansion of ductile rings

In this work a stability analysis on flow localization in the dynamic expansion of ductile rings is con ducted. Within a 1 D theoretical framework, the boundary value problem of a radially expanding thin ring is posed. Based on a previous work, the equations governing the stretching process of the expanding ring are derived and solved using a linear perturbation method. Then, three different perfectly plastic material constitutive behaviours are analysed: the rate independent material, the rate dependent mate rial showing constant logarithmic rate sensitivity and the rate dependent material showing non constant and non monotonic logarithmic rate sensitivity. The latter allows to investigate the interaction between inertia and strain rate sensitivity on necking formation. The main feature of this work is rationally dem onstrate that under certain loading conditions and material behaviours: (1) decreasing rate sensitivity may not lead to more unstable material, (2) increasing loading rate may not lead to more stable material. This finding reveals that the relation between rate sensitivity and loading rate controls the unstable flow growth. Additionally a finite element model of the ring expansion problem is built in ABAQUS/Explicit. The stability analysis properly reflects the results obtained from the numerical simulations. Both proce dures, perturbation analysis and numerical simulations, allow for emphasizing the interplay between rate sensitivity and inertia on strain localization.The financial support of the Comunidad Autónoma de Madrid (Project CCG10 UC3M/DPI 5596) and of the Ministerio de Ciencia e Innovación de España (Project DPI/2008 06408) is kindly acknowledged.Publicad

Conceptual study of an ICRH traveling-wave antenna system for low-coupling conditions as expected in DEMO

For the central heating of a fusion reactor ion cyclotron radio frequency heating (ICRH) is the first choice method as it is able to couple RF power to the ions without density limit. The drawback of this heating method is the problem of excitation of the magneto-sonic wave through the plasma boundary layer from the antenna located along the wall, without exceeding its voltage standoff. The amount of coupling depends on the antenna excitation and the surface admittance at the antenna output due to the plasma profile. The paper deals with the optimization of the antenna excitation by the use of sections of traveling-wave antennas (TWAs) distributed all along the reactor wall between the blanket modules. They are mounted and fed in resonant ring system(s). First, the physics of the coupling of a strap array is studied by simple models and the coupling code ANTITER II. Then, after the study of the basic properties of a TWA section, its feeding problem is solved by hybrids driving them in resonant ring circuit(s). The complete modeling is obtained from the matrices of the TWA sections connected to one of the feeding hybrid(s). The solution is iterated with the coupling code to determine the loading for a reference low-coupling ITER plasma profile. The resulting wave pattern up to the plasma bulk is derived. The proposed system is totally load resilient and allows us to obtain a very selective exciting wave spectrum. A discussion of some practical implementation problems is added

Structural analysis of a bolted joint concept for the space shuttle's solid rocket motor casing

The Space Shuttle Challenger accident is thought to have been caused by the failure of one of the tang-clevis joints joining together the casing segments of the Solid Rocket Motors (SRM). Excessive displacement between the tang and clevis, possibly unseating the O-ring seals, may have initiated the resulting accident. An effort was made at NASA Langley Research Center to design an alternative concept for mating the casing segments. A bolted flange joint concept was designed and analyzed to determine if the concept would effectively maintain a seal while minimizing joint weight and controlling stress levels. It is shown that under the loading conditions analyzed the seal area of the joint remains seated. The only potential stress problem is a stress concentration in the flange at the edge of the bolt hole, which is highly localized. While heavier than the existing joint, this concept does have some advantages making the bolted joint an attractive alternative