Mode spectra of magnetic and dielectric plasmon spheres obtained from Mie scattering and free oscillation theories

This paper presents Mie scattering theory as compared to rigorous electromagnetic theory of free oscillations in magnetic and electric plasmon spheres. It is shown that the maxima of Mie scattering and absorption spectra well correspond to resonance frequencies of plasmon modes occurring in dielectric and magnetic spheres, similarly as it takes place for ordinary dielectric resonator modes. Mie theory is well applicable to determine resonance frequencies and scattering parameters of spherical plasmons. However, this theory cannot be applied to determine intrinsic properties of modes induced in the object by the incident plane wave, like quality factors. On the contrary, rigorous electromagnetic theory of free oscillations allows one to determine the complex resonance frequency of each mode that can occur in a given object, and the corresponding quality factor accounting for various kind of losses, including medium and radiation losses. The advantage of the free oscillations theory, as shown in this paper, is in the determination of the quality factors of modes occurring in magnetic plasmon spheres made of a strongly dispersive magnetic medium. Graphical Abstract

Oscillations of a soft viscoelastic drop

A soft viscoelastic drop has dynamics governed by the balance between surface tension, viscosity, and elasticity, with the material rheology often being frequency dependent, which are utilized in bioprinting technologies for tissue engineering and drop-deposition processes for splash suppression. We study the free and forced oscillations of a soft viscoelastic drop deriving (1) the dispersion relationship for free oscillations, and (2) the frequency response for forced oscillations, of a soft material with arbitrary rheology. We then restrict our analysis to the classical cases of a Kelvin–Voigt and Maxwell model, which are relevant to soft gels and polymer fluids, respectively. We compute the complex frequencies, which are characterized by an oscillation frequency and decay rate, as they depend upon the dimensionless elastocapillary and Deborah numbers and map the boundary between regions of underdamped and overdamped motions. We conclude by illustrating how our theoretical predictions for the frequency-response diagram could be used in conjunction with drop-oscillation experiments as a “drop vibration rheometer”, suggesting future experiments using either ultrasonic levitation or a microgravity environment.

Toroidal free oscillations of a viscous liquid spherical shell with mixed boundary conditions

The problem of toroidal eigenmodes of free oscillations of a viscous fluid spherical shell is solved under mixed boundary conditions. These conditions are a combination of free surface conditions and solid boundary conditions. Such an artificial combination is used sometimes in the numerical simulation of geodynamo problem. The scheme for calculating the eigenmodes is described. It is shown that the use of mixed conditions leads to the disappearance of the first eigenmode with the smallest eigenvalue. A possible reason for this is the physical incorrectness of the mixed boundary conditions.

Study on free oscillations of a micromechanical gyroscope taking into account the nonorthogonality of the torsion axes

Introduction. The paper is devoted to the study on free oscillations of the sensing element of a micromechanical R-Rtype gyroscope of frame construction developed by the Kuznetsov Research Institute of Applied Mechanics, taking into account the nonorthogonality of the torsion axes. The influence of the instrumental manufacturing error on the accuracy of a gyroscope on a movable base in the case of free oscillations is studied. The work objective was to improve the device accuracy through developing a mathematical model of an R-R type micromechanical gyroscope, taking into account the nonorthogonality of the torsion axes, and to study the influence of this error on the device accuracy. The urgency of the problem of increasing the accuracy of micromechanical gyroscopes is associated with improving the accuracy of inertial navigation systems based on micromechanical sensors.Materials and Methods. A new mathematical model that describes the gyroscope dynamics, taking into account the instrumental error of manufacturing the device, and a formula for estimating the error of a gyroscope, are proposed. The dependences of the state variables obtained from the results of modeling and on the basis of the experiment are presented. Methods of theoretical mechanics and asymptotic methods, including the Lagrange formalism and the Krylov-Bogolyubov averaging method, were used in the research.Results. A new mathematical model of the gyroscope dynamics, taking into account the nonorthogonality of the torsion axes, is developed. The solution to the equations of small oscillations of the gyroscope sensing element and the estimate of the precession angle for the case of a movable base are obtained. A comparative analysis of the developed model and the experimental data obtained in the case of free oscillations of the gyroscope sensing element with a fixed base is carried out. The analysis has confirmed the adequacy of the constructed mathematical model. Analytical expressions are formed. They demonstrate the fact that the nonorthogonality of the torsion axes causes a cross-influence of the amplitudes of the primary vibrations on the amplitudes of the secondary vibrations of the sensing element, and the appearance of an additional error in the angular velocity readings when the gyroscope is operating in free mode.Discussion and Conclusions. The results obtained can be used to improve the device accuracy using the algorithm for analytical compensation of the gyroscope error and the method for identifying the mathematical model parameters.

The Art of Listening

Music and seismology merged in the daily work of the Caltech professor Hugo Benioff, who united the avant-garde technology of the 1920s with a nineteenth-century Helmholtzian aesthetic, cultural, and scientific understanding of music. The transducer facilitated this merger, mediating between science and music and allowing for new ways of listening to waves outside the audible range. Benioff had the capacity to listen—“listening” understood here not as passive perception, but as an active search to distinguish and separate signal from noise, whether from in- or outside of the instrument. For more than forty years, Benioff worked as a sonic expert, perfecting the recording and reproduction of waves and vibrations of all types and frequencies. After tracing elements of Benioff’s biography, I examine how he incorporated the technology of the transducer in his workshop into his seismological and musical instruments, notable not only for the control, austerity, and clarity of lines of their modernist design, but also for a new kind of poetic technology. Benioff’s seismological instruments made it possible to listen to a large variety of previously undetectable phenomena such as the free oscillations of the earth, and his work with the pianist Rosalyn Tureck on electric musical instruments aimed to reproduce the pure sound of traditional instruments. I argue that Benioff’s search for an aesthetic reconciliation of the scientific modern with an enchanted view of the world is very much a product of the social, cultural, technical, and scientific conditions of the interwar period.

EXPERIMENTAL AND COMPUTATIONAL STUDY OF LIQUID OSCILLATION DAMPING IN A FREE SURFACE VESSEL

Studying the issues related to solving the problems of the movement of fluid in partially filled structures of various shapes has always been relevant.This is due, in particular, to the need to ensure the longitudinal and lateral stability of the movement of objects in which the liquid is transported. Contemporarymethods and research tools that allow describing the movement of liquid in a container are overly complex and require in-depth knowledgefrom a scientist. Therefore, development of a mathematical algorithm that would be simple and at the same time ensure sufficient accuracy in determiningthe oscillatory motion in a container is advisable. Each development of a new mathematical algorithm requires experimental research to verifyits adequacy. The purpose of this work is to confirm the feasibility of using the created mathematical apparatus for determining the main parameters offree vibrations depending on the level of liquid in a rectangular prismatic container. The experimental research methodology provides for checking theadequacy of the mathematical algorithms in determining the frequencies of free oscillations of liquid in a container by comparing the theoretical andexperimental values of the oscillation periods, as well as determining the damping decrement of oscillations for liquids of different viscosity. As a result,the adequacy of the formulas for determining the frequencies of free oscillations of liquid in a container is proved; the maximum error does notexceed 4.35%. The decrements of vibration damping are determined experimentally, as well as using theoretical models of the motion of viscous liquidin the so-called partial surface layers, for three liquids of different viscosity, namely, for water, for 20% sugar and water solution, and for vegetable oil.On the basis of experimental studies, the amplitudes and frequencies of forced vibrations of various types of liquids are determined by constructing anamplitude-frequency characteristic. It is shown by calculation how the damping decrement of the liquid affects the value of the horizontal displacementof the surface layer of the liquid in the tank during its transportation by a wheeled tractor.