Energy characteristics of a nonlinear layer at resonant frequencies of wave scattering and generation

This work presents a mathematical model, a computational scheme and experimental results describing the electrodynamic characteristics of a nonmagnetic, isotropic, E-polarized, nonlinear layered dielectric object with a cubically polarizable medium. The nonlinear object is irradiated by a quasi-homogeneous field, where the incident field constitutes of a packet of phase-synchronized plane oscillations. In the case under consideration the excitation may consist both of a highly intense electromagnetic field at a basic (fundamental) frequency, which results in the generation of the third harmonic, as well as of weakly intense fields at multiples of the basic frequency which produce no harmonics, but only have an influencing effect on the processes of wave radiation. The investigations were carried out within the setting of a coupled system approach at resonant excitation frequencies determined by the eigenvalues of the induced eigenvalue problems. A verification of the energy balance law is carried out. By means of estimations for the conditionalities of the occuring matrices, the level of degeneration of the induced non-self-adjoint spectral problems as well as the sensitivity of the coupled system of nonlinear boundary value problems with respect to computational errors are verified.

Periodically Forced Nonlinear Oscillatory Acoustic Vacuum

In this work, we study the in-plane oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system.

Diagnostics of a gradient catastrophe for a class of quasilinear hyperbolic systems

A two-stage analysis do detect the appearance of a gradient catastrophe of the solution is proposed for quasilinear systems of hyperbolic equations of special form. Applications of the first stage are considered for the following simple cases: scalar Burgers’ equation and a quasi-orthogonal system generalizing it. The entire two-stage analysis is applied to systems of equations describing one-dimensional electron oscillations in plasma, namely, plane oscillations in the relativistic and non-relativistic cases and also axially symmetric non-relativistic cylindrical oscillations.

On the equivalent systems for concurrent springs and dampers – Part 2: Small out-of-plane oscillations

Concurrent linear springs belonging to systems that perform small out-of-plane oscillations around a stable equilibrium position are considered with a view to obtaining equivalent systems of three mutually orthogonal linear springs. Theorems defining their stiffness coefficients as well as their position, i.e. the position of the principal stiffness axes for which the potential energy does not contain mixed terms, are stated and proven. So far unknown invariants related to the sum of original and new stiffness coefficients are provided. In addition, the equivalent system of three mutually orthogonal dampers is obtained for any system of out-of-plane concurrent linear viscous. The theorem defining their damping coefficients and their directions, collinear with the principal damping axes for which the dissipative function does not contain mixed terms, is provided. The corresponding invariant for damping coefficients is presented, too. An ellipsoid of displacement and an ellipsoid of stiffness are discussed. Three illustrated examples are given.

On the equivalent systems for concurrent springs and dampers – Part 1: Small in-plane oscillations

In this work, concurrent linear springs placed in the system that performs small in-plane oscillations around the stable equilibrium position are considered. New theorems defining how they can be replaced by two mutually orthogonal springs are provided. The same concept is applied to find two mutually orthogonal linear viscous dampers that can replace a system of concurrent linear viscous dampers. The directions of such springs and dampers correspond to the principal stiffness and damping axes, respectively. So far unknown invariants related to the sum of stiffness coefficients and damping coefficient of the original and equivalent systems are presented. A few examples are given to illustrate the use and benefits of this approach. In addition, it is shown how the concept of two mutually orthogonal springs can be beneficially used for analysing problems concerned with oscillations of a particle on elastic frames.