TO THE QUESTION OF ANALYSIS OF EQUATIONS OF MOTION OF A RIGID BODY DURING THE MECHANICAL OSCILATIONS

Depending on the current position of the mass in different areas of the spring deformation during the oscillation process the values that determines the natural frequency of free continuous oscillations have opposite signs. It is defined by the change in the direction of acceleration of the mass in these areas, which makes it possible to determine a single inhomogeneous differential equation of the oscillation process in different areas of the movement of the mass. When the oscillation amplitude is much less than the static position of the mass, this inhomogeneous differential equation represents a homogeneous differential equation of free undamped oscillations.

TO THE QUESTION OF ANALYSIS OF EQUATIONS OF MOTION OF A RIGID BODY DURING THE MECHANICAL OSCILATIONS

Depending on the current position of the mass in different areas of the spring deformation during the oscillation process the values that determines the natural frequency of free continuous oscillations have opposite signs. It is defined by the change in the direction of acceleration of the mass in these areas, which makes it possible to determine a single inhomogeneous differential equation of the oscillation process in different areas of the movement of the mass. When the oscillation amplitude is much less than the static position of the mass, this inhomogeneous differential equation represents a homogeneous differential equation of free undamped oscillations.

SOLUTION OF THE MAIN DIFFERENTIAL EQUATION OF DEFORMATIONS OF THE CASING IN A CURVED WELL

In a curvilinear well, the casing functions as a long continuous rod. It is installed on the supports-centralizers and replicates the complex profile of the well, as a result of which it receives large deformations. To describe them, a system of differential equilibrium equations of internal and external forces and moments was composed, which was supplemented to a closed form with a differential equation of curvature. It is non-uniform, because it takes into account the own distributed weight of the rod. Two ways are proposed to solve the problem: by the method of mathematical compression of the system equations into a complex inhomogeneous differential equation or by projecting the equilibrium equations of forces on the global (vertical-horizontal) and on the local (tangent-normal) coordinate systems. It is shown that the first integral of the system can also be found from the equilibrium equations of a portion of a curved rod of finite length. This integral has the form of a second-order inhomogeneous differential equation with variable coefficients and is the main equation that describes the deformation of a long elastic rod under the action of the longitudinal and transverse components of the forces of distributed weight. The main requirement of the technology is the installation of a pipes column on the centering supports, the purpose of which is to ensure the coaxiality of the pipes and the borehole walls and the creation between them a cement ring of the same thickness and strength. Accounting for this requirement allowed us to linearize the main equation. Its solution is the clue to the formulas of deflections, angular slopes, internal bending moments and transverse forces in the rod with the arbitrary arrangement of supports and boundary conditions in their intersections. The solution of the main differential equation of angular deformations of a long bar is found in the form of a linear combination of Airy and Scorer’s functions and in the form of three linearly independent polynomial series in the sum with a partial solution. The obtained formulas of flexure and power parameters allow us to calculate stress and strain in the pipes column during the process of casing the borehole of an arbitrary profile which increases the reliability and durability of the well.

The significance of the Mathieu-Hill differential equation for Newton's apsidal precession theorem

Newton's precession theorem in Proposition 45 of Book I of Principia relates a centripetal force of magnitude μrn-3 as a power of the distance from the center to the apsidal angle θ, where θ is the angle between the point of greatest distance and the point of least distance. The formula θ = π/[Formula: see text] is essentially restricted to orbits of small eccentricity. A study of the apsidal angle for appreciable orbital eccentricity leads to an analysis of the differential equation of the orbit. We show that a detailed perturbative approach leads to a Mathieu-Hill-type of inhomogeneous differential equation. The homogeneous and inhomogeneous differential equations of this type occur in many interesting problems across several disciplines. We find that the approximate solution of this equation is the same as an earlier one obtained by a bootstrap perturbative approach. A more thorough analysis of this inhomogeneous differential equation leads to a modified Hill determinant. We show that the roots of this determinant equation can be solved to obtain an accurate solution for the orbit. This approach may be useful even for cases where n deviates noticeably from 1. The derived analytic results were applied to the Moon, Mercury, the asteroid Icarus, and a hypothetical object. We show that the differential equation that occurs in a perturbative relativistic treatment of the perihelion precession of Mercury also leads to a simplified form of the Mathieu-Hill differential equation.PACS Nos.: 95.100.C, 95.10.E, 95.90, and 02.30.H