Energy Mechanisms of Free Vibrations and Resonance in Elastic Bodies

Abstract

The mechanisms of natural oscillations and resonance are described, considering the peculiarities of the transformation of elastic and kinetic energy in the implementation of the law of conservation of energy in local and integral volumes of the body, using the concept of mechanics based on the concepts of space, time and energy. When describing the motion in the Lagrange form, the elastic deformation energy of the particles is determined by the quadratic invariant of the tensor, whose components are the partial derivatives of Euler variables with respect to Lagrange variables. The increment of the invariant due to elastic deformation is represented as the sum of two scalars, one of which depends on the average value of the relative lengths of the edges of the particles in the form of an infinitesimal parallelepiped, the second is equal to the standard deviation of these lengths from the average value. It is shown that each of the scalars can be represented in the form of two dimensionless kinematic parameters of elastic energy, which participate in different ways in the implementation of the law of conservation of energy. One part of the elastic energy passes into kinetic energy and participates in the implementation of the law of conservation of energy for the body as a whole, considering external forces. The second part is not converted into kinetic energy but changes the deformed state of the particles in accordance with the equations of motion while maintaining the same level of the part of the elastic energy of the particles used for this. The kinematic parameters differ from the volume density of the corresponding types of energy by a factor equal to the elastic modulus, which is directly proportional to the density and heat capacity of the material and inversely proportional to the volume compression coefficient. Transverse, torsional, and longitudinal vibrations are considered free and under resonance conditions. The mechanisms of transformation of forced vibrations into their own after the termination of external influences and resonance at the superposition of free and forced vibrations with the same or similar frequency are considered. The formation of a new free wave at each cycle with an increase in the amplitude, which occurs mainly due to internal energy sources, and not external forces, is justified.

Energy Mechanisms of Free Vibrations and Resonance in Elastic Bodies

“…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),…”

“…To determine dδE e we use, by analogy with classical mechanics, the scalar product of the forces δP by the outer surfaces of the particle and their displacements dr. Taking into account the rule of summation by a repeating index, using the concept of the surface force density τ pi , one obtains [8]:…”

“…where I 1,t , I 2,t , and I 3,t are the time derivatives for the linear, quadratic, and cubic invariants (3) of the tensor (2), respectively. From the condition that the energy is invariant with respect to the choice of the velocity reference system, the sum of the last three terms in Equation ( 5) must turn to 0, and the system (1) must satisfy the differential equation [14,15]:…”

“…Then, the parameter e def can be represented via the other dimensionless scalars e e and e s [8,15]:…”

“…The nature of free vibrations and resonance is still not fully understood. There are reasons to believe that the energy basis of free vibrations and resonance, along with the energy of external forces, are internal energy sources [8].…”