Vibrational modes of circular free plates under tension

The vibrational frequencies of a plate under tension are given by the eigenvalues $\omega$ of the equation $\Delta^2u-\tau\Delta u=\omega u$. This paper determines the eigenfunctions and eigenvalues of this bi-Laplace problem on the ball under natural (free) boundary conditions. In particular, the fundamental modes --- the eigenfunctions of the lowest nonzero eigenvalue --- are identified and found to have simple angular dependence.Comment: 17 pages. To be submitted for publication shortly

Analyticity and criticality results for the eigenvalues of the biharmonic operator

We consider the eigenvalues of the biharmonic operator subject to several homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show that simple eigenvalues and elementary symmetric functions of multiple eigenvalues are real analytic, and provide Hadamard-type formulas for the corresponding shape derivatives. After recalling the known results in shape optimization, we prove that balls are always critical domains under volume constraint.Comment: To appear on the proceedings of the conference "Geometric Properties for Parabolic and Elliptic PDE's - 4th Italian-Japanese Workshop" held in Palinuro (Italy), May 25-29, 201

Experimental Investigation of Injection-Coupled High-Frequency Combustion Instabilities

Self-excited high-frequency combustion instabilities were investigated in a 42-injector cryogenic rocket combustor under representative conditions. In previous research it was found that the instabilities are connected to acoustic resonance of the shear-coaxial injectors. In order to gain a better understanding of the flame dynamics during instabilities, an optical access window was realised in the research combustor. This allowed 2D visualisation of supercritical flame response to acoustics under conditions similar to those found in European launcher engines. Through the window, high-speed imaging of the flame was conducted. Dynamic Mode Decomposition was applied to analyse the flame dynamics at specific frequencies, and was able to isolate the flame response to injector or combustion chamber acoustic modes. The flame response at the eigenfrequencies of the oxygen injectors showed symmetric and longitudinal wave-like structures on the dense oxygen core. With the gained understanding of the BKD coupling mechanism it was possible to derive LOX injector geometry changes in order to reduce the risks of injection-coupled instabilities for future cryogenic rocket engines

Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators

Reflection is a natural phenomenon that occurs when light passes the interface between materials with different refractive index. In many applications, such as solar cells or photodetectors, reflection is an unwanted loss process. Many ways to reduce reflection from a substrate have been investigated so far, including dielectric interference coatings, surface texturing, adiabatic index matching and scattering from plasmonic nanoparticles. Here we present an entirely new concept that suppresses the reflection of light from a silicon surface over a broad spectral range. A two-dimensional periodic array of subwavelength silicon nanocylinders designed to possess strongly substrate-coupled Mie resonances yields almost zero total reflectance over the entire spectral range from the ultraviolet to the near-infrared. This new antireflection concept relies on the strong forward scattering that occurs when a scattering structure is placed in close proximity to a high-index substrate with a high optical density of states

Hidden attractors in fundamental problems and engineering models

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered

Wind, Waves, and Wing Loading: Morphological Specialization May Limit Range Expansion of Endangered Albatrosses

Among the varied adaptations for avian flight, the morphological traits allowing large-bodied albatrosses to capitalize on wind and wave energy for efficient long-distance flight are unparalleled. Consequently, the biogeographic distribution of most albatrosses is limited to the windiest oceanic regions on earth; however, exceptions exist. Species breeding in the North and Central Pacific Ocean (Phoebastria spp.) inhabit regions of lower wind speed and wave height than southern hemisphere genera, and have large intrageneric variation in body size and aerodynamic performance. Here, we test the hypothesis that regional wind and wave regimes explain observed differences in Phoebastria albatross morphology and we compare their aerodynamic performance to representatives from the other three genera of this globally distributed avian family. In the North and Central Pacific, two species (short-tailed P. albatrus and waved P. irrorata) are markedly larger, yet have the smallest breeding ranges near highly productive coastal upwelling systems. Short-tailed albatrosses, however, have 60% higher wing loading (weight per area of lift) compared to waved albatrosses. Indeed, calculated aerodynamic performance of waved albatrosses, the only tropical albatross species, is more similar to those of their smaller congeners (black-footed P. nigripes and Laysan P. immutabilis), which have relatively low wing loading and much larger foraging ranges that include central oceanic gyres of relatively low productivity. Globally, the aerodynamic performance of short-tailed and waved albatrosses are most anomalous for their body sizes, yet consistent with wind regimes within their breeding season foraging ranges. Our results are the first to integrate global wind and wave patterns with albatross aerodynamics, thereby identifying morphological specialization that may explain limited breeding ranges of two endangered albatross species. These results are further relevant to understanding past and potentially predicting future distributional limits of albatrosses globally, particularly with respect to climate change effects on basin-scale and regional wind fields