Energy Mechanisms of Free Vibrations and Resonance in Elastic Bodies

The scientific novelty of this work is determined by the rationale for the participation in transformations, along with the kinetic energy of particles, of four types of elastic energy, identified by the peculiarities of their phase changes in the oscillation process. Two types are converted into kinetic energy, while the other two types change the deformed state of particles in accordance with the equations of motion due to internal sources. The result is obtained based on the use of the superposition principle in the space of Lagrange variables with the imposition of forced and free oscillations, as well as a new model of mechanics based on the concepts of space, time, and energy with a new scale of average stresses that takes into account the energy of particles in the initial state. In such a model of mechanics, a generalized measure of the elastic energy of particles is a quadratic invariant of asymmetric tensor whose components are partial derivatives of Euler variables with respect to Lagrange variables. The concept of kinematic energy parameters is introduced, which differ from the corresponding volumetric energy densities by a multiplier equal to the modulus of elasticity, which is directly proportional to the density and heat capacity of the material, and inversely proportional to the volumetric compression coefficient. Comparison of the values of kinematic parameters shows that most of the energy required for oscillations is associated with the deformation of particles and comes from internal sources. The mechanisms of transformation of forced vibrations into their own for transverse, torsional, and longitudinal vibrations are considered, as well as the occurrence of resonance when free and forced vibrations are superimposed with the same or a similar frequency. The formation of a new free wave after each cycle of external influences with an increase in amplitude, which occurs mainly due to internal, and not external, energy sources is justified.

Superposition of Motions in The Space of Lagrange Variables

The technique of superposition of motions in the space of Lagrange variables is described, which allows us to obtain the equations of combined motion by replacing the Lagrange variables of superimposed (external) motion with Euler variables of nested (internal) motion. The components of velocity and acceleration in the combined motion obtained as a result of differentiating the equations of motion in time coincide with the results of vector addition of the velocities and accelerations of the particles involved in the superimposed motions at each moment of time. Examples of motion and superposition descriptions for absolutely solid and deformable bodies with equations for the main kinematic characteristics of motion, including for robot manipulators with three independent drives, pressing with torsion, bending with tension, and cross– helical rolling, are given. For example, given the fragment of calculation of forces in the kinematic pairs shown the advantages of the description of motion in Lagrangian form for the dynamic analysis of lever mechanisms, allows to determine the required external exposure when performing the energy conservation law at any time in any part of the mechanism.

Linear evolution of the vortex induced by localized temperature disturbance in stratified shear flow

We study the combined effect of the shear flow velocity and its (stable) vertical stratification on the evolution of the three-dimensionally localized vortical disturbance induced by the initial temperature perturbation embedded at the initial time into a local region of the flow. Small geometric scales of perturbations compared to the characteristic scales of velocity and temperature variation of the background flow allow to consider vertical gradients of the horizontal velocity and temperature to be not dependent on the coordinates. Assuming a disturbance sufficiently weak, we use linear theory to calculate fields of vorticity and temperature. The problem is solved analytically using a three-dimensional Fourier transform of the basic set of equations and further transition to a Lagrange variables in the Fourier space. It is shown that the growth of the intensity of the vortex (a measure of which are enstrophy and circulation) is obliged to both stratification and shear. However, the character of this growth (monotonous or oscillating) depends on what of two factors dominates. In the case where the dissipation effects are negligible, enstrophy grows indefinitely (in the framework of the linear theory), but dissipative factors (viscosity and thermal diffusivity) modified this growth and make it only transient, so that eventually the perturbation decays. Perturbation domain stretches along the direction of flow, but its vertical and horizontal movement as a whole in the framework of the linear theory doesn’t occur, since it is the nonlinear effect. Nonlinear evolution of the vortex induced by temperature disturbance is considered in a separate paper.

The momentum of a sound pulse in a slightly dispersive medium

The paper investigates a model of a liquid with some dispersion and some nonlinearity in Lagrange variables. A wavepacket of sound of small amplitude, and of a wavelength long compared with the characteristic wavelength of the dispersion law, will in general split in two, unless the oscillatory amplitude is accompanied by a non-oscillatory motion of an amplitude proportional to the square of the amplitude of the oscillatory part. The total momentum carried is then ambiguous, and it may not be possible to define the total momentum of such a pulse.

A quantum theory for non-viscous fluids in the Lagrange variables

1. Introduction. A number of authors (2,3,7) have recently quantized the motion of an inviscid fluid. The starting point has been the variational principle of Bateman which uses the Clebsch variables. The density turns out to be the canonical conjugate to the velocity potential, and the transition to the quantum theory is then made in the usual way. If the fluid is making small vibrations, ‘phonons’, which are scalar, appear as a result of the quantization.