Shock transmission and eigenvalue veering within a coupled system.

The operation of dynamical systems in harsh environments requires continuous monitoring. Internal sensors may be used to monitor the conditions in real time. A typical example is the sensor and electronic components used in space structures which, especially during launch, are subject to huge g force. The paper will present an experimental and theoretical study on a simplified model used to analyze the possible cause of high acceleration on the enclosed sensors and equipments due to impulsive loading. The model system consists of two beams coupled using compliant connections. An impulse hammer excites one beam, and vibrations are transmitted to the indirectly driven beam. A theoretical model is developed using a Rayleigh-Ritz approach and validated using experimental results in both the frequency and time domains. Monto Carlo simulation was done with random masses positioned on the indirectly driven beam to determine the worst-case conditions for maximum peak acceleration. Highest acceleration levels were found when mode matching in the two beams led to veering behavior in the coupled modes. The results suggest guidelines for the detailed design of internal components of a structure exposed to shock loading from its environment. [The authors thank Schlumberger Cambridge Research for financial support.].

Continuous spectrum growth and modal instability in swirling duct flow

We present results on the stability of compressible inviscid swirling flows in an annular duct. Such flows are present in aeroengines, for example in the by-pass duct, and there are also similar flows in many aeroacoustic or aeronautical applications. The linearised Euler equations have a ('critical layer') singularity associated with pure convection of the unsteady disturbance by the mean flow, and we focus our attention on this region of the spectrum. By considering the critical layer singularity, we identify the continuous spectrum of the problem and describe how it contributes to the unsteady field. We find a very generic family of instability modes near to the continuous spectrum, whose eigenvalue wavenumbers form an infinite set and accumulate to a point in the complex plane. We study this accumulation process asymptotically, and find conditions on the flow to support such instabilities. It is also found that the continuous spectrum can cause a new type of instability, leading to algebraic growth with an exponent determined by the mean flow, given in the analysis. The exponent of algebraic growth can be arbitrarily large. Numerical demonstrations of the continuous spectrum instability, and also the modal instabilities are presented.

Effects of mechanical and electrical coupling on the parametric sensitivity of mode localized sensors

We compare and contrast the effects of two distinctly different mechanisms of coupling (mechanical and electrical) on the parametric sensitivity of micromechanical sensors utilizing mode localization for sensor applications. For the first time, the strong correlation between mode localization and the phenomenon of 'eigenvalue loci-veering' is exploited for accurate quantification of the strength of internal coupling in mode localized sensors. The effects of capacitive coupling-spring tuning on the parametric sensitivity of electrically coupled resonators utilizing this sensing paradigm is also investigated and a mass sensor with sensitivity tunable by over 400% is realized. ©2009 IEEE.

Optimal H2 order-one reduction by solving eigenproblems for polynomial equations

A method is given for solving an optimal H2 approximation problem for SISO linear time-invariant stable systems. The method, based on constructive algebra, guarantees that the global optimum is found; it does not involve any gradient-based search, and hence avoids the usual problems of local minima. We examine mostly the case when the model order is reduced by one, and when the original system has distinct poles. This case exhibits special structure which allows us to provide a complete solution. The problem is converted into linear algebra by exhibiting a finite-dimensional basis for a certain space, and can then be solved by eigenvalue calculations, following the methods developed by Stetter and Moeller. The use of Buchberger's algorithm is avoided by writing the first-order optimality conditions in a special form, from which a Groebner basis is immediately available. Compared with our previous work the method presented here has much smaller time and memory requirements, and can therefore be applied to systems of significantly higher McMillan degree. In addition, some hypotheses which were required in the previous work have been removed. Some examples are included.

Applications of the dynamic mode decomposition

The decomposition of experimental data into dynamic modes using a data-based algorithm is applied to Schlieren snapshots of a helium jet and to time-resolved PIV-measurements of an unforced and harmonically forced jet. The algorithm relies on the reconstruction of a low-dimensional inter-snapshot map from the available flow field data. The spectral decomposition of this map results in an eigenvalue and eigenvector representation (referred to as dynamic modes) of the underlying fluid behavior contained in the processed flow fields. This dynamic mode decomposition allows the breakdown of a fluid process into dynamically revelant and coherent structures and thus aids in the characterization and quantification of physical mechanisms in fluid flow. © 2010 Springer-Verlag.

Adjoint algorithms for the Navier-Stokes equations in the low Mach number limit

This paper describes a derivation of the adjoint low Mach number equations and their implementation and validation within a global mode solver. The advantage of using the low Mach number equations and their adjoints is that they are appropriate for flows with variable density, such as flames, but do not require resolution of acoustic waves. Two versions of the adjoint are implemented and assessed: a discrete-adjoint and a continuous-adjoint. The most unstable global mode calculated with the discrete-adjoint has exactly the same eigenvalue as the corresponding direct global mode but contains numerical artifacts near the inlet. The most unstable global mode calculated with the continuous-adjoint has no numerical artifacts but a slightly different eigenvalue. The eigenvalues converge, however, as the timestep reduces. Apart from the numerical artifacts, the mode shapes are very similar, which supports the expectation that they are otherwise equivalent. The continuous-adjoint requires less resolution and usually converges more quickly than the discrete-adjoint but is more challenging to implement. Finally, the direct and adjoint global modes are combined in order to calculate the wavemaker region of a low density jet. © 2011 Elsevier Inc.

Impact of heterogeneous link qualities and network connectivity on binary consensus

In this paper we consider a network that is trying to reach consensus over the occurrence of an event while communicating over Additive White Gaussian Noise (AWGN) channels. We characterize the impact of different link qualities and network connectivity on consensus performance by analyzing both the asymptotic and transient behaviors. More specifically, we derive a tight approximation for the second largest eigenvalue of the probability transition matrix. We furthermore characterize the dynamics of each individual node. © 2009 AACC.

Delayed decision-making in bistable models

Switching between two modes of operation is a common property of biological systems. In continuous-time differential equation models, this is often realised by bistability, i.e. the existence of two asymptotically stable steadystates. Several biological models are shown to exhibit delayed switching, with a pronounced transient phase, in particular for near-threshold perturbations. This study shows that this delay in switching from one mode to the other in response to a transient input is reflected in local properties of an unstable saddle point, which has a one dimensional unstable manifold with a significantly slower eigenvalue than the stable ones. Thus, the trajectories first approximatively converge to the saddle point, then linger along the saddle's unstable manifold before quickly approaching one of the stable equilibria. ©2010 IEEE.

A Newton algorithm for invariant subspace computation with large basins of attraction

We study the global behaviour of a Newton algorithm on the Grassmann manifold for invariant subspace computation. It is shown that the basins of attraction of the invariant subspaces may collapse in case of small eigenvalue gaps. A Levenberg-Marquardt-like modification of the algorithm with low numerical cost is proposed. A simple strategy for choosing the parameter is shown to dramatically enlarge the basins of attraction of the invariant subspaces while preserving the fast local convergence.

Coupled neutronic thermo-hydraulic analysis of full PWR core with Monte-Carlo based BGCore system

BGCore reactor analysis system was recently developed at Ben-Gurion University for calculating in-core fuel composition and spent fuel emissions following discharge. It couples the Monte Carlo transport code MCNP with an independently developed burnup and decay module SARAF. Most of the existing MCNP based depletion codes (e.g. MOCUP, Monteburns, MCODE) tally directly the one-group fluxes and reaction rates in order to prepare one-group cross sections necessary for the fuel depletion analysis. BGCore, on the other hand, uses a multi-group (MG) approach for generation of one group cross-sections. This coupling approach significantly reduces the code execution time without compromising the accuracy of the results. Substantial reduction in the BGCore code execution time allows consideration of problems with much higher degree of complexity, such as introduction of thermal hydraulic (TH) feedback into the calculation scheme. Recently, a simplified TH feedback module, THERMO, was developed and integrated into the BGCore system. To demonstrate the capabilities of the upgraded BGCore system, a coupled neutronic TH analysis of a full PWR core was performed. The BGCore results were compared with those of the state of the art 3D deterministic nodal diffusion code DYN3D (Grundmann et al.; 2000). Very good agreement in major core operational parameters including k-eff eigenvalue, axial and radial power profiles, and temperature distributions between the BGCore and DYN3D results was observed. This agreement confirms the consistency of the implementation of the TH feedback module. Although the upgraded BGCore system is capable of performing both, depletion and TH analyses, the calculations in this study were performed for the beginning of cycle state with pre-generated fuel compositions. © 2011 Published by Elsevier B.V.

Implementation of multi-group cross-section methodology in BGCore MC-depletion code

BGCore is a software package for comprehensive computer simulation of nuclear reactor systems and their fuel cycles. The BGCore interfaces Monte Carlo particles transport code MCNP4C with a SARAF module - an independently developed code for calculating in-core fuel composition and spent fuel emissions following discharge. In BGCore system, depletion coupling methodology is based on the multi-group approach that significantly reduces computation time and allows tracking of large number of nuclides during calculations. In this study, burnup calculation capabilities of BGCore system were validated against well established and verified, computer codes for thermal and fast spectrum lattices. Very good agreement in k eigenvalue and nuclide densities prediction was observed for all cases under consideration. In addition, decay heat prediction capabilities of the BGCore system were benchmarked against the most recent edition of ANS Standard methodology for UO2 fuel decay power prediction in LWRs. It was found that the difference between ANS standard data and that predicted by the BGCore does not exceed 5%.

Investigating vibration dyanmics of cross-coupled MEMS resonators for reduced motional resistance

This paper investigates the vibration dynamics of a closed-chain, cross-coupled architecture of MEMS resonators. The system presented here is electrostatically transduced and operates at 1.04 MHz. Curve veering of the eigenvalue loci is used to experimentally quantify the coupling spring constants. Numerical simulations of the motional resistance variation against induced perturbation are used to assess the robustness of the cross-coupled system as opposed to equivalent traditional open-ended linear one-dimensional coupling scheme. Results show improvements of as much as 32% in the motional resistance between the cross-coupled system and its one-dimensional counterpart. © 2013 IEEE.

Wave propagation in a slowly-varying duct with mean vortical swirling flow

The propagation of unsteady disturbances in a slowlyvarying cylindrical duct carrying mean swirling flow is investigated using a multiple-scales technique. This is applicable to turbomachinery flow behind a rotor stage when the swirl and axial velocities are of the same order. The presence of mean vorticity couples acoustic and vorticity equations which produces an eigenvalue problem that is not self-adjoint unlike that for irrotational mean flow. In order to determine the amplitude variation along the duct, an adjoint solution for the coupled system of equations is derived. The solution breaks down where a mode changes from cut on to cut off. In this region the amplitude is governed by a form of Airy's equation, and the effect of swirl is to introduce a small shift in the origin of the Airy function away from the turning-point location. The variation of axial wavenumber and amplitude along the duct is calculated. In hard-walled ducts mean swirl is shown to produce much larger amplitude variation along the duct compared with a nonswirling flow. Mean swirl also has a large effect in ducts with finite-impedance walls which differs depending on whether modes are co-rotating with the swirl or counter rotating. © 2001 by A.J. Cooper, Published by the American Institute of Aeronautics and Astronautics, Inc.