Multilayered model in optics and quantum mechanics

Three types of dispersion equations are analyzed that describe the eigenvalues of the effective refractive index of a multilayer plane optical waveguide and the energy eigenvalues of a quantum particle placed in a piecewise constant potential field. The first equation (D1) is derived by setting to zero the determinant of the system of linear equations produced by matching the solutions in the layers. The second equation (D2) is obtained using the well-known method of characteristic matrices. The third equation has been obtained in the general case by the author and is known as a multilayer equation. Simple relations between the three equations are established. It is shown that the set of roots of D2 exactly coincides with the set of eigenvalues of the multilayer problem, while the roots of D1 and the multilayer equation contain those equal to the refractive index in the optical case (or to the potential in the quantum case) in internal layers of the system, which may be superfluous. Examples are presented.     

Load’s temporal characteristics for annulling forced vibrations of linear elastic plates

We consider trapezoidal load-time pulses with linearly increasing and affinely decreasing durations equal to integer multiples of the time period of the first bending mode of vibration of a linearly elastic structure. For arbitrary spatial distributions of loads applied to monolithic and laminated orthotropic plates, it is shown through numerical solutions that plates’ vibrations become miniscule after the load is removed. This phenomenon is independent of the dwell time (i.e., the time duration between the rising and the falling portions) during which the load is kept constant. The primary reason for this response is that for such time-dependent loads, nearly all of plate’s strain energy is concentrated in deformations corresponding to the fundamental bending mode of vibration. Thus plate’s deformations can be studied by taking the mode shape of the 1st bending mode as the basis function and reducing the problem to that of solving a single second-order ordinary differential equation. We have verified this postulate by comparing strain energies computed from the 3-dimensional deformations of different plate geometries and boundary conditions with those determined by using the single degree of freedom (DoF) model. Thus for trapezoidal time-dependent loads applied on plates, the 1 DoF model provides reasonably accurate results and saves considerable computational effort.     

The identification of the vibration control system parameters designed for continuous miner machines

The Continuous Miner machines are exposed to time dependent loads during normal operation of the rock cutting process. These loads cause vibrations, which have a negative influence on the whole structure of the machine. This phenomenon can be eliminated by applying passive or active vibration control systems (VCS). Generally these systems are coupled with additional elements, which provide dispersion or transfer energy. The energy thus acquired can also reinforce the intended function such as rock cutting operation in the case of mining machines. The objective of this paper is to present the method of numerical identification of VCS parameters for Continuous Miner machines. The main function of the presented system is to reduce displacement of cutting drum by using elastic element joined to machine chassis and applying appropriate algorithm of control of the angular velocity of cutting drum. The method described improves efficiency of mining and increases durability of machine. In order to determine mechanical and control parameters of VCS the genetic algorithm optimisation method conjugated with numerical modal analysis was used. Finally the transient dynamic analysis was performed for the full-scale model of Continuous Miner in order to verify VCS in normal working condition.     

Mural inspection by vibration measurements with TV-holography

A commonly encountered problem in the conservation of historical murals is the identification of sections in the plaster that have detached from the wall and thus threaten to fall off. Commonly, walls are inspected by the acoustic response to a gentle finger-tapping (percussion method). Since this is a costly and cumbersome technique, means for a more automatic inspection are searched for. A TV-holography system of increased sensitivity in combination with acoustic excitation of the object is shown to be a new and powerful tool for monitoring of loose areas. It has the advantage of non-contact and remote operation which, for example, is extremely useful in large buildings. Principles of the method, experimental results obtained at an artificial wall in the laboratory, and a thorough comparison of results from historical sites gained by the traditional percussion method and the new technique are presented. The latter shows very good agreement in the assessment of wall quality and thus is evidence of the suitability of the optical equipment for tasks in conservation.     

Global Continuity of the Integrated Density of States for Random Landau Hamiltonians

Abstract

We prove that the integrated density of states (IDS) for the randomly perturbed Landau Hamiltonian is Hölder continuous at all energies with any Hölder exponent 0 < q < 1. The random Anderson-type potential is constructed with a nonnegative, compactly supported single-site potential u. The distribution of the iid random variables is required to be absolutely continuous with a bounded, compactly supported density. This extends a previous result Combes et al. [Combes, J. M., Hislop, P. D., Klopp, F. (2003a). Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Notices 2003: 179--209] that was restricted to constant magnetic fields having rational flux through the unit square. We also prove that the IDS is Hölder continuous as a function of the nonzero magnetic field strength.     

Stereoelectronic structure of MCl4-benzoyl chloride (M=Si,Ge, Sn) systems resulting from ab initio calculations and NQR 35Cl

Results of quantum-chemical calculations of MCl₄–C₆H₅COCl (M=Si, Ge, Sn) systems of 1?:?1 composition using RHF/3-21?G* and MP2/3-21?G* levels as well as those of 1?:?2 composition using the RHF/3-21?G* level have been represented. MCl₄?←?C₆H₅COCl complexes of 1?:?1 composition are energetically more advantageous. They are formed in solid state provided that the M···O distance in individual systems is considerably less than the sum of van der Waals radii of M and O and their total energies are appreciably less than the sum of total energies of components. These conditions are realized only for M=Sn. In systems of 1?:?2 compositions, calculated M···O distances are practically equal to the sum of covalent radii of M and O. Nonetheless, complexes with such composition are not formed in solid state. Total energy of the system which is lower than the sum of its components’ energies is not an indispensable condition for complex formation. The ³⁵Cl nuclear quadrupole resonance (NQR) frequencies and asymmetry parameters of the electric field gradient at the ³⁵Cl nuclei have been evaluated using the results of ab initio calculations.     

Determination of dynamic properties of flax fibres reinforced laminate using vibration measurements

Experimental and numerical methods to identify the linear viscoelastic properties of flax fibre reinforced polymer (FFRP) composite are presented in this study. The method relies on the evolution of storage modulus and loss factor as observed through the frequency response. Free-free symmetrically guided beams were excited in the dynamic range of 10 Hz to 4 kHz with a swept sine excitation focused around their first modes. A fractional derivative Zener model has been identified to predict the complex moduli. A modified ply constitutive law has been then implemented in a classical laminates theory calculation (CLT) routine. Overall, the Zener model fitted the experimental results well. The storage modulus was not frequency dependant, while the loss factor increased with frequency and reached a maximum value for a fibre orientation of 70°. The damping of FFRP was, respectively, 5 and 2 times higher than for equivalent carbon and glass fibres reinforced epoxy composites.