Superposition of Motions in The Space of Lagrange Variables

Alyushin, Yuriy

doi:10.20944/preprints202101.0225.v1

Cited by 1 publication

(8 citation statements)

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“…The superposition principle, which is successfully used in various problems for absolutely solid and deformable bodies [9,14], and the new model of mechanics with a single elastic modulus (8), confirm the kinematic and energy possibility of joint motion (31) in compliance with the law of conservation of energy. At equal amplitudes q 0 = q 1 , all energy characteristics of the combined oscillation increase by four times in relation to the initial free oscillation.…”

Section: Transverse Vibrationsmentioning

confidence: 84%

“…The energy model of mechanics should use a description of motion of material particles in the form of Lagrange, since only Lagrange variables allow us to consider the change in the energy state of particles at any time interval, and also to consider the transformation of some types of energy into others, including due to deformation and temperature. Let us use the notations [8][9][10] x i = x i (α p , t),…”

Section: Fundamentals Of the Energy Model Of Mechanicsmentioning

confidence: 99%

“…We start counting the time when there is no deformation v(α, t) = 0 and the Lagrangian coordinates coincide with the initial ones. The vibrations, allowed by the law of energy conservation, must correspond to Equation (9), which is converted to the form…”

Section: Transverse Vibrationsmentioning

confidence: 99%

“…In accordance with the superposition principle [9], to obtain the equations of joint motion it is sufficient to replace the Lagrange variables of external (superimposed) motion with expressions for the corresponding Euler variables of internal (nested) motion. In our case, the equations for natural and forced oscillations may differ in the circular frequency ω and the amplitude q, but they are equivalent in their effect on the resulting oscillation.…”

Section: Transverse Vibrationsmentioning

confidence: 99%

“…The purpose of this work is: using a new concept in mechanics based on the concepts of space, time and energy [9,10], to analyze the change in the deformed and energetic state of particles; comparing their phase changes with changes in kinetic energy and energy of external forces, to substantiate the essential role of additional types of internal energy, providing a change only in the deformed state of particles determined by their equations of motion; check the fulfillment of the law of conservation of energy for local volumes and for the body as a whole.…”

Section: Introductionmentioning

confidence: 99%

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Energy Mechanisms of Free Vibrations and Resonance in Elastic Bodies

Alyushin

2021

Physics

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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.

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Section: Transverse Vibrationsmentioning

confidence: 84%

Section: Fundamentals Of the Energy Model Of Mechanicsmentioning

confidence: 99%

Section: Transverse Vibrationsmentioning

confidence: 99%

Section: Transverse Vibrationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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