Zufallsschwingungen von querangeströmten Saiten

Übersicht Am Beispiel einer vorgespannten Saite unter sowohl örtlich als auch zeitlich unkorrelierter Zufallserregung wird eine neue Methode zur Berechnung der Kovarianzfunktionen erklärt. Mit Hilfe einer Greenschen Funktion führt diese Kovarianzanalyse auf Integralgleichungen, deren Lösungen für die vorgegebene Problemstellung geschlossen angebbar sind. Es wird gezeigt, daß sie mit den Reihenentwicklungen der bekannten Modalanalysis übereinstimmen.

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Summary The example of a preloaded string under random excitation uncorrelated in time and in space is used in order to present a new method to calculate the associated covariance functions. By use of a Green's function this covariance analysis leads to integral equations, the solution of which can be given in a closed form in case of the stated problems. It is shown, that the solutions coincide with the expansions of the well-known modal analysis.

     

Stabilitätsbedingungen für Schwinger mit zufälliger oder harmonischer Parametererregung

Übersicht Für ein gedämpftes Schwingungssystem mit einem Freiheitsgrad, dessen Parametererregung schmalbandige, stationäre Zufallsprozesse sind, werden die Bedingungen für die Stabilität der Ruhelage im quadratischen Mittel näherungsweise mit Hilfe der Störungsrechnung ermittelt. Nach dieser Momentenstabilitätsbedingung wird die destabilisierende Wirkung der Zufallserregung um so geringer, je weiter die mittlere Erregerfrequenz ihres Spektrums von der doppelten Eigenfrequenz des Schwingungssystems entfernt ist. Bei einem Grenzübergang zu einer harmonischen Parametererregung liefern die Stabilitätsgrenzwerte den Instabilitätsbereich erster Ordnung, den man bereits von der Mathieuschen Einzeldifferentialgleichung her kennt.

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Summary The conditions of mean-square-stability are determined approximately by means of the perturbation method for a damped system with one degree of freedom, the parametric excitations of which are stationary narrow band processes. According to these conditions the destabilizing effect of a random excitation will be the less, the more distant the average excitation frequency will be from twice the natural frequency of the system. In the special case of a harmonic parametric excitation the limits of mean-square-stability will provide the instability region of the first order, being well-known from the Mathieu equation.

     

Parametric amplification effects in acoustical test facilities

In acoustical test facilities, elastic structures like satellites are exposed to noise excitations in order to detect those parts with a significant risk of failure. Statistically representative test signals are generated from white noise by means of a spectrum shaper. Subsequently, these electrical signals are amplified and transformed into an acoustical reverberant pressure field in the sound chamber. The transformation is due to a parametric amplification effect in the control space of the horns which modulates the signals in the form of a parametrically excited system. In a sense, the Monte Carlo method of non-linear stochastic mechanics is implemented in the laboratory. The paper presents a finite element model of the compressible gas medium in the horn and in the sound chamber. It derives mean values, variances and power spectra of the stationary velocity fluctuations of the flowing gas medium.     

Dependence of the Physical Properties on the Cation Ordering in the New Ternary Phosphide ZnCu2P8

: The air-stable phosphide, ZnCu₂P₈, was synthesized from its elements in gram amounts. It crystallizes in the Cu_2.5In_0.5P₈ structure type [C2/c, Z = 4, a = 1110.8(2) pm, b = 964.7(2) pm, c = 752.7(1) pm, β = 110.027(3)°], and is the first ternary phosphide in the system Zn/Cu/P. Its crystal structure consists of two-dimensional, layered polyphosphide-anions, separated by tetrahedrally-coordinated cations. 135 symmetry independent configurations, each with a different and specific Zn²⁺ and Cu⁺ arrangement, where explored with density functional methods. Whereby, each of the 135 cation ordering pattern has a diverse bandgap, and the total energies vary in the range of 1 eV. As several configurations with low energy are close in energy, an intrinsic disorder is discussed as origin of the promising low lattice thermal conductivity, observed.     

Jump Phenomena and Bifurcations in Stochastic Vehicle-Road Dynamics

When vehicles ride on uneven roads, they are excited to vertical random vibrations whose stationary rms-values (root-mean-square) strongly depend on the velocity of the vehicle. To investigate this vibration behavior, it is appropriate to introduce road models in way domain which are based on the theory of stochastic differential equations and transformed from way to time by means of velocity-dependent way and noise increments. The random base excitations by roads are applied to nonlinear quarter car models. They lead to stationary rms-values of the vertical vehicle vibrations which become resonant for critical velocities and show jump phenomena similar to those of the Duffing oscillator under harmonic excitations. In the stochastic case, jump phenomena are only observable for narrow-banded road excitations. They vanish for increasing car damping and excitation bandwidth. For efficient simulations of the road-vehicle model, the n state equations are utilized to derive n(n + 1)/2 stochastic covariance equations. For small step sizes, their numerical mean square solutions coincide with the nonlinear results of fix-point iterations obtained when the noise terms of the covariance equations are omitted. It can easily be shown, that this deterministic approach leads to the correct stationary covariances in the linear case. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uncommon structural and bonding properties in Ag16B4O10

Ag₁₆B₄O₁₀ has been obtained as a coarse crystalline material via hydrothermal synthesis, and was characterized by X-ray single crystal and powder diffraction, conductivity and magnetic susceptibility measurements, as well as by DFT based theoretical analyses. Neither composition nor crystal structure nor valence electron counts can be fully rationalized by applying known bonding schemes. While the rare cage anion (B₄O₁₀)⁸⁻ is electron precise, and reflects standard bonding properties, the silver ion substructure necessarily has to accommodate eight excess electrons per formula unit, (Ag⁺)₁₆(B³⁺)₄(O²⁻)₁₀ × 8e⁻, rendering the compound sub-valent with respect to silver. However, the phenomena commonly associated with sub-valence metal (partial) structures are not perceptible in this case. Experimentally, the compound has been found to be semiconducting and diamagnetic, ruling out the presence of itinerant electrons; hence the excess electrons have to localize pairwise. However, no pairwise contractions of silver atoms are realized in the structure, thus excluding formation of 2e–2c bonds. Rather, cluster-like aggregates of an approximately tetrahedral shape exist where the Ag–Ag separations are significantly smaller than in elemental silver. The number of these subunits per formula is four, thus matching the required number of sites for pairwise nesting of eight excess electrons. This scenario has been corroborated by computational analyses of the densities of states and electron localization function (ELF), which clearly indicate the presence of an attractor within the shrunken tetrahedral voids in the silver substructure. However, one bonding electron pair of s and p type skeleton electrons per cluster unit is extremely low, and the significant propensity to form and the thermal stability of the title compound suggest d¹⁰–d¹⁰ bonding interactions to strengthen the inter-cluster bonding in a synergistic fashion. With the present state of knowledge, such a particular bonding pattern appears to be a singular feature of the oxide chemistry of silver; however, as indicated by analogous findings in related silver oxides, it is evolving as a general one.

Ag₁₆B₄O₁₀, obtained via hydrothermal synthesis, displays an unprecedented bonding scheme, hosting excess electrons localized pairwise in cluster-like silver subunits.     

Die Elektronen-Lokalisierungs-Funktion in closo-Bor-Clustern

The Electron Localization Function in closo Boron Clusters The structure and the electron density in the closo boron clusters B₄X₄ (X = H, Cl, Br, I), B₆X₆^2? (X = H, Cl, Br, I) and B₁₂H₁₂^2? were determined by pseudopotential Hartree-Fock calculations. The Electron Localization Function (ELF) was used to interpret the bonding characteristics. The regions of high ELF values in all cases have the form of the dual polyhedron of the boron cage. They show perfectly the 3 center 2 electron bonds. The comparison between Hartree-Fock and Extended Hückel calculations point out that semiempirical calculations can also be a good basis for ELF interpretations.     

HB9X9˙ and H2B9X9 (X = Cl,Br, I): Neutral closo-Nonaboranes in the Novel Series BnHn+1, and BnHn+2 – Syntheses,Ab Initio Calculations,and Electronic Structures

Chemical reduction of B₉X₉ (X = Cl, Br, I) with gaseous HI proceeds stepwise to give the neutral paramagnetic clusters HB₉X₉ ^· , and the corresponding diamagnetic clusters H₂B₉X₉. Together they comprise the first neutral derivatives in the series B_nH_n+1 and B_nH_n+2 with n = 9. The EPR spectra of the paramagnetic HB₉X₉ ^· (X = Cl, Br, I) in glassy frozen CH₂Cl₂ solutions showed increasing g anisotropy for the heavier halogen derivatives, illustrating significant halogen participation at the singly occupied MO due to the larger spin-orbit coupling contributions. Temperature dependent ¹H NMR spectra of H₂B₉X₉ (in CD₃CN, X = Cl, Br) indicate the presence of H₂B₉X₉, [HB₉X₉]^–, and [CD₃CNH]⁺ with H₂B₉X₉ acting as a Brønsted acid. The corresponding ¹¹B NMR spectra (in CD₃CN) show the presence of the dianions [B₉X₉]^2– as a result of the protonation of CD₃CN. The ¹¹B resonances of the species H₂B₉X₉ and [HB₉X₉]^– are obscured by superimposition of the two resonance lines of the dianions [B₉X₉]^2–. Temperature dependent ¹¹B{¹H} MAS-NMR spectra of H₂B₉Br₉ show coalescence at 410 K and hence dynamic behaviour of the neutral B₉-cluster in the solid. Cyclic voltammetry experiments of H₂B₉Br₉ in CH₃CN solvent) are compatible with the redox sequence [B₉Br₉]^2––[B₉Br₉] ^{· –}–B₉Br₉. Quantum chemical calculations with the electron localization function (ELF) are described.