Frequency Of Longitudinal Oscillations Research Articles

First Frequency of Longitudinal Vibration of a Rod with Mass at one End and Partial Placement on Piecewise-Homogeneous Elastic Foundation

Statement of the problem. Despite the introduction of engineering practice of numerical methods for the calculation, in a case of engineering problems associated with extended structures, the well-known models «rod on an elastic foundation» and «beam on an elastic foundation» are widely used to describe the behavior of structures. Most of the known solutions were obtained in the conditions of an idealized setting and without differences in the properties of the foundation. This work is devoted to the study of the dynamics of a rod of finite length with a point mass at the end and partially placement on a piecewise-homogeneous elastic foundation, formed by two sections of different stiffness. The aim of this study is to obtain analytical dependences for the first frequency of the longitudinal vibrations of a rod with different boundary conditions at its ends: rigid support and free end. The available research results are obtained for special cases, which makes them difficult to apply in practice. To assess the influence of the inertial pro-perties of the foundation, the case with different linear masses of the rod in the pinched and cantilever sections is considered. Results. Using the Ritz method analytical dependences were obtained for the determination of the first frequency of the longitudinal vibrations of a fix and free rod with a point mass at the end and partially placement on a piecewise-homogeneous elastic foundation formed by two sections of different stiffness. Quantitative evaluation shows good agreement with those of other studies. The results for particular cases are obtained. Conclusions. It is shown that the first natural frequency of longitudinal oscillations is in a certain range and in the nonlinear ratios depending on the lengths of the sections various elastic. The inertial properties of the foundation leads to a decrease in the first frequency.

Ribonucleic acid genome mutations induced by the Casimir effect

In this paper, we investigate the Casimir effect within a virus RNA, particularizing the study to the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Then, we discuss the possibility of occurring damage or mutation in its genome due to the presence of quantum vacuum fluctuations inside and around the RNA ribbon. For this, we consider the geometry and the nontrivial topology of the viral RNA as having a simple helical structure. We initially compute the non-thermal Casimir energy associated to that geometry, considering boundary conditions that constrain the zero point oscillations of a massless scalar field to the cylindrical cavity containing a helix pitch of RNA ribbon. Then we extend the obtained result to the electromagnetic field and, following, we calculate the probability of occurring damage or mutation in RNA by using the normalized inverse exponential distribution, which suppresses very low energies, and consider cutoff (threshold) energies corresponding to UV-A and UV-C rays, surely responsible by mutations. Then, by taking into account UV-A, we arrive at a mutation rate per base per infection cycle, which in the case of the SARS-CoV-2 is non-negligible. We find a maximum value of this mutation rate for an RNA ribbon radius, applying it for SARS-CoV-2, in particular. We also calculate a characteristic longitudinal oscillation frequency for the helix pitch value corresponding to the local minimum of the Casimir energy. Finally, we consider thermal fluctuations of classical and quantum nature and show that the corresponding probability of mutation is completely negligible for that virus. Therefore, we conclude that only the nontrivial topology and the geometric attributes of the RNA molecule contribute to the possible mutations caused by quantum vacuum fluctuations in the viral genome.

Исследование по динамике рамных конструкций

Determining the natural and forced oscillations of frame structures simulated by the rods with distributed masses (an infinite number of degrees of freedom) is quite difficult. Therefore, in the article, the frame model is endued with a finite number of degrees of freedom: the mass is placed in a certain number of nodes that elastically interact with rods that have no mass. The rods work only for bending. Longitudinal displacements are not taken into account, since the frequency of longitudinal oscillations is two orders of magnitude higher than the frequency of bending ones. Such a model leads to the construction of expressions of the kinetic and potential energy, which then allows using the Lagrange equations of second kind to obtain a system of differential oscillation equations of the structure. The problem of free oscillations of the G-shaped frame was solved in the article using Green's functions, matrices of stiffness, masses, malleability, etc. The obtained approximate results were compared with little-known exact results and demonstrated good convergence, especially with an increase in the number of degrees of freedom (the number of concentrated masses simulating the distributed mass of the rods of the G-shaped frame).

Advancing asymptotic approaches to studying the longitudinal and torsional oscillations of a moving beam

This paper analyzes the influence of kinetic and physical-mechanical parameters of systems on the characteristics of dynamic processes in moving one-dimensional nonlinear-elastic systems. Improved convenient calculation formulas have been derived that describe the laws of changing the amplitude-frequency characteristics of systems for both a non-resonant case and a resonant one. An important issue of studying the influence of the speed of movement of elements of mechanisms on the oscillations of one-dimensional nonlinear-elastic systems has not been considered in detail until now in the scientific literature. This issue relates to the vibrations of shafts in gears, pipe strings when drilling oil and gas wells, the oscillations of turbine blades and rotating turbine discs, the longitudinal vibrations of the beam as an element of structures. The main reason for this in the analytical study of dynamic processes were the shortcomings of the mathematical apparatus for solving the corresponding nonlinear differential equations that describe the laws of motion of those systems. It was found that in the case of longitudinal oscillations in the moving beam with an increase in the longitudinal speed of the medium to 10 m/s, the amplitude of the oscillation also increases by 13.5 %. However, when the longitudinal velocity of the beam is 5 m/s, the amplitude will increase by only 3 %. It is established that with the growth of the amplitude, the frequency of longitudinal oscillations decreases sharply, and if the system moves at a higher speed, for example, 20 m/s, it reduces the frequency of oscillation by about 13 %. The results reported here make it possible to assess the effect of kinetic and physical-mechanical parameters on the frequency and amplitude of oscillations. The research that involved the asymptotic method makes it possible to predict resonant phenomena and obtain engineering solutions to improve the efficiency of technological equipment.

Using the asymptotic approximation of the Maxwell element model for the analysis of stress in a conveyor belt

The features of the propagation of dynamic stresses in a conveyor belt, the material properties of which correspond to the Maxwell element model, are considered. Analytical expressions are presented for calculating the dynamic elastic modulus, the loss modulus, and the angle of mechanical loss depending on the frequency of longitudinal oscillations in the belt of an extended transport conveyor. To analyze the dynamic stress propagation process, dimensionless parameters are introduced that characterize the specific features of the viscoelastic process in a conveyor belt, the material properties of which correspond to the Maxwell element model. The transition to the dimensionless Maxwell element model is made and the analysis of the relationship between stress and deformation of a conveyor belt element for extremely large and small values of dimensionless parameters is made. The substantiation of the scope of the Maxwell element model is given. It is shown that at sufficiently high frequencies of longitudinal stress oscillations in a conveyor belt, at which the oscillation period is much less than the characteristic oscillation decay time, the relationship between stress and deformation of the conveyor belt element corresponds to Hooke's law. A qualitative analysis of the relaxation time was carried out for a conveyor belt material, the properties of which correspond to the Maxwell element model. The analysis of the propagation of dynamic stresses in the conveyor belt for the characteristic operating modes of the transport conveyor is carried out. The conveyor operating mode with a constant deformation rate of the belt element; the mode in which a constant load is suddenly applied to the belt element; the conveyor operating mode with an instantly applied load to the belt element were investigated. It was determined that in cases where the characteristic process time significantly exceeds the stress relaxation time in the conveyor belt or the longitudinal oscillation period is much less than the stress relaxation time in the conveyor belt, the Maxwell element model can be replaced with a sufficient degree of accuracy by the Hooke element model.

Neutral Shadowing Force Effect on Structural Properties and Oscillations of Dust Particles in Cryogenic Environment

Recently, it was shown that the neutral shadowing interaction can be significantly stronger in the cryogenic complex plasma than in the plasma with neutrals at room temperature. Here, we present the results of the investigation—by molecular dynamics (MD) simulations—of the impact of the neutral shadowing interaction on the radial pair distribution function and the velocity autocorrelation function (VAF) of the charged dust particles in a 2-D layer. The spectrum of VAF was computed to estimate the influence of the neutral shadowing force on the oscillations of dust particles. We found that the neutral shadowing interaction can significantly affect the structural properties and the characteristic longitudinal oscillation frequency of dust particles if the characteristic radius of the neutral shadowing interaction exceeds a mean interdust particle distance. The latter case corresponds to low pressures, $p Pa, and cryogenic temperatures less than 10 K. In addition, the dust acoustic wave taking into account that the neutral shadowing force was derived. It is found that the neutral shadowing force leads to the increase of the sound speed of the dust acoustic wave.

Designing cylindrical springs of minimum weight with restriction on the own frequency of longitudinal oscillations under condition of full and fuzzy information

Formulation of the problem. Elastic elements such as springs are common іn many products of the machine-building, railway, construction industry. They are designed to accumulate or absorb mechanical energy. These elements of various designs and devices have been and are being given enough attention. There is a large number of publications in this field, where questions of calculation and manufacturing are considered. At the same time, a small number of scientific papers are devoted to optimal design, in which the optimization of the characteristics of springs by various criteria is carried out using nonlinear programming methods.This paper is devoted to the problems of optimal design of a cylindrical tension spring by the weight criterion, while its own frequency of longitudinal oscillations is limited. Direct and dual optimization problems are formulated. Realization is provided with the method of Lagrange multipliers and necessary conditions for the existence of an extremum. Purpose of the article. To consider optimization dynamic problems of design of helical cylindrical springs in conditions of complete and incomplete information about the output data. To analyze the influence of the number of active spring turns on the optimal design parameters, as well as the influence of the fuzzy information about weight setting and the value of the natural frequency of the longitudinal oscillations. Conclusion. As a result of solving the direct and dual optimizing dynamic problem for a helical spring it is found that the dependence of the optimal weight from the natural frequency of the intrinsic longitudinal oscillations and the number of active turns is a nonlinear function. The dependence of the optimum frequency of natural oscillations and the diameter of the wire from the given weight is also nonlinear. The estimation of the influence of the fuzzy initial data on the result of the project (the diameter of the wire of the spring) is obtained. The transformation of fuzzy numbers into deterministic ones is performed by the center method (defuzzification operation). Accounting of fuzzy information leads to increasing in the weight parameter and in the wire diameter parameter.

Coaxial carbon nanotubes: from springs to ratchet wheels and nanobearings

Quantum mechanical calculations are performed for double-walled and triple-walled carbon nanotubes (DWCNTs, TWCNTs). We show how the DWCNTs may be modelled by mechanical analogues with interactions mediated by spring forces with weakening elastic constants in addition to a weak van der Waals -like interaction. This enables us to predict the natural frequency of longitudinal oscillations of the DWCNTs, their elastic spring constant and the corresponding interaction between the CNTs. Results of the interaction energy ΔE for the DWCNT system obtained from the mechanical model are in remarkable agreement with those obtained from our quantum mechanical calculations, and we obtain functional dependencies of ΔE on axial elongation and inter-wall separation of the DWCNTs. Our density functional theory calculations reveal that the armchair-armchair systems are significantly more stable than both the corresponding armchair-chiral configurations, as well as the corresponding stacked graphene sheets. The results obtained for translational and rotational motion (slide / roll) of the CNTs indicate a potential use of DWCNTs as springs, ratchet wheels, nanogears and bearings in miniaturized devices. Our work enables us to predict the behavior of DWCNTs of different dimensions and would be of practical importance in the actual construction of nanoscale mechanical parts.

Effect of Dipole‐Dipole Interaction on the Compressional Oscillations in Two‐Dimensional Yukawa liquids

AbstractThe paper presents the study of oscillations in the two‐dimensional Yukawa liquids where the interparticle interaction potential in addition to the Yukawa term has the dipole‐dipole interaction term. The frequency of longitudinal oscillations was studied using the Fourier transform of the velocity autocorrelation function. It was found that in the two‐dimensional Yukawa liquids the longitudinal oscillation frequency of particles becomes sensitive to variations in the coupling parameter if even weak additional dipole‐dipole interaction between particles exists. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Natural frequencies of longitudinal oscillations for a nonuniform variable cross-section rod

The natural frequencies of longitudinal oscillations of a rod such that its Young’s modulus, the density, and the cross-sectional area are functions of the longitudinal coordinate are analyzed. For solving the corresponding problem, an integral formula is used to represent the general solution to the original Helmholtz equation with variable coefficients in terms of the general solution to the accompanying equation with constant coefficients. Frequency equations are derived in the form of rapidly converging Leibniz series for three types of boundary conditions. For these cases the frequency zerothapproximation equations are given to quickly find the lowest natural frequencies with an adequate accuracy.

The effect of magnetomechanical resonance on light polarization in magneto-optical crystals

The peculiarities of changes in the polarization of light passing through a crystal have been investigated for the case of magnetomechanical resonance within the model of longitudinal oscillations of a magneto-optical crystal shaped as a thin rod. The geometry of the magnetic fields is chosen so as to bring about a change in the light polarization by the Faraday and Cotton-Mouton effects. The amplitude and phase-frequency dependences of the variable components of the Faraday effect and the elements of the Stokes vector are plotted in the range of resonant frequencies of longitudinal oscillations of magnetomechanical crystal using the example of yttrium ferrite garnet.

The effect of drift spaces on the longitudinal motion in an ion linac

The effect of drift spaces on the longitudinal motion in a multi-section ion linac at high energy is calculated. It is shown, that if the drift spaces are distributed with a “frequency” that is a multiple of the longitudinal oscillation frequency of the accelerated particles, a resonance build-up of the oscillations can occur that appreciably decreases and distorts the stability region. Practical recommendations are given for the choice and distribution of the drift spaces, in order for undesirable effects to be eliminated.

Solution of spin dynamics equations for 3He superfluid phases in a strong magnetic field

A general method of investigating the solutions of spin dynamics equations of superfluid phases of /sup 3/He in strong magnetic fields is proposed. The effect of longitudinal spin oscillations upon the frequency of precession is taken into account. The motion of a spin near singular points at which the longitudinal oscillation frequency becomes zero is considered. The formulas derived are applied to A and B phases in the open geometry and to the B phase in the plane-parallel geometry.

Lepton spin motion in a weak electromagnetic trap

Averaging methods are used to discuss the spin and orbital motion of charged leptons confined in a trap formed by perturbing a strong uniform magnetic field with a weak nonuniform magnetic or electric field. The principal result is an expression for the gyromagnetic difference frequency correct up to second order in the perturbing field. The motion is also discussed in the resonant case where the difference frequency is an integral multiple of the longitudinal oscillation frequency of the orbital motion. The two most recent precision measurements of the electron $g$-factor anomaly are reanalyzed, resulting in the removal of the apparent discrepancy between them and yielding a corrected value for the electron gyromagnetic anomaly: $a=(1159656.7\ifmmode\pm\else\textpm\fi{}3.5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$.