Manual Uniform estimates and blow-up analysis for the Emden exponential equation in any dimension

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MR Zbl pre Pure Appl. MR Zbl Numdam MR Zbl Partial Differential Equations 6 1 Zbl Differential Equations 1 Zbl pre Partial Differential Equations 16 Chang, On a fourth order operator - the Paneitz operator - in conformal geometry, in: Proceedings of the Conference for the 70th of A.

Calderon, in press. Partial Differential Equations 16 2 MR Differential Equations 2 Partial Differential Equations 24 Differential Equations Djadli, A. Malchiodi, Existence of conformal metrics with constant Q -curvature, Ann.

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Esposito, A class of Liouville-type equations arising in Chern-Simons vortex theory: asymptotics and construction of blowing-up solutions, Ph. Paris 36 A local version of the method of moving planes , Comm. The thermal shields that protect the EDM need to be released with a high degree of reliability before reaching the Martian Surface. Therefore, the performance of the EDM separation devices is crucial for the success of the mission and a key point for ESA's contribution to subsequent missions to Mars. Figure 1.

Each unit is basically composed of two brackets that maintain the separating bodies in position. After the command signal, the links are released by a pyrotechnical nut and the bodies immediately jettisoned by a set of springs.

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Developing a tool to validate the design of a separation device is often a challenging task. Not only is it important to focus on the structural behaviour and the performance of the unit itself but also on how it affects to the separating bodies.

In fact, the disturbances introduced on the bodies due to the jettison are to be carefully investigated to avoid potential unbalance, undesired kinematic effects, etc. With this purpose, a set of mathematical models are produced to simulate a large amount of separation scenarios and their consequences. This campaign includes different tests at unit level and at assembly level, among which a set of functional tests are included to demonstrate the capability of the devices to perform safe releases.

With the Qualification of the separation mechanisms, the design, analysis and test activities will be finished. Unlike the usual deterministic analyses considered in the frame of release systems, the paper introduces the novelty of using a statistical approach based on randomly generated cases.

A synopsis

Nastran FEM model. The results from a correlation campaign are also presented in the paper to give evidence of how the analytical model represents Reality with a high degree of similarity. Dytran mathematical model to support the predictions of the test results. Paragraphs 2. To achieve this level, ultralow-shock pyronuts are implemented in the design.

In particular, the axial velocity, the lateral velocity, the ratio of them and the angular rate must not exceed certain values that are verified by analysis and test Performance Requirements The main performance requirements applicable to the mechanisms are: - To provide an adequate structural behaviour in terms of stiffness and strength.

EDM Front-shield and Back-shell separation mechanisms FSSM and SPSM As previously stated, the separation mechanisms allow disconnecting electrically and jettisoning the thermal shields by means of a pyronut and a set of preloaded compression springs. Although the interfaces and the overall sizing of the units obviously are not the same, they share the main design drivers. That is to say, there are two types of mechanisms depending on whether or not they accommodate the connectors: - Type 1: Mechanisms without umbilical connectors.

The only active part of these mechanisms is the pyronut that releases the corresponding pyro-bolt during the separation.

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These mechanisms include the corresponding pyronut and the separation springs that make possible the demating of the electrical umbilical connectors. The preload between both brackets is applied through an intermediate plastic pad by a titanium fastener pyro bolt bolted to the separation nut.

The separation springs of each type 2 mechanism are mounted and compressed in the upper bracket before applying the preload. The nodes include two springs for redundancy purposes.

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The springs are accommodated one inside the other to reduce the mass and volume of the brackets. Each separation spring is sized to provide the load needed for the separation at mechanism level. This implies that only three springs, one per mechanism, could produce the separation. The main reason for the differences between them is that the interfaces in both cases are different and that the SPSM type 2 mechanisms accommodate two umbilical connectors and concentric springs while the FSSM type 2 mechanisms include only one umbilical connector and one pair of springs.

One of the brackets is rigidly attached to the thermal shield and the other one to the Backshell. The separation start is controlled by the ignition of the pyronut, which is required to be as quick as possible to minimise unbalance. Figure 5. The flexibility of these bars could introduce potential disturbances during jettisoning and could produce relative displacements linear and angular over the allowable tolerances of the umbilical connectors.

Thus, a guiding system is implemented in the design to prevent jamming or locking during the disconnection of the umbilical connectors. Umbilical connectors Back Shell Struts Figure 4. SPSM type 2, detail of the guiding system Taking into account the shock limits of sensitive electronic components mounted on the Surface Platform, the pyronut is selected from the ultra-low shock nut series. The purpose of these simulations is to predict the behaviour the test jigs and test article will have during the tests.

This model permits being implemented in higher level functional models to generate simplified separation cases. Some rationale is summarised as follows: - Inputs: The inputs considered in the analysis are the geometric and mass parameters of the bodies, the initial state vectors, the internal and external forces, etc.

Each significant variable is approximated to a statistical function whose mean value is the nominal value of the parameter and the shape of the function depends on the nature of the parameter itself.

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Uniform and Gaussian random distributions are used depending on the type of parameter. Gaussian variables are assumed to be distributed with a probability of success higher than Taking into account that the parameters are usually specified in a range between two values, the statistical distributions are defined based on them.

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With that purpose, all the density functions and hence their statistical parameters mean and standard deviation are obtained from the boundary limits of each variable. Sometimes, a group of inputs maintain a geometrical relationship as in the case of the initial attitude and initial angular rate, which are specified to be linked through an elliptical association see Fig. Figure 7.