The Stability Conditions for a Heavy Solid Motion

A I Ismail,E M Khalil

doi:10.1155/2020/8829439

Abstract

In this paper, the stability conditions for the rotary motion of a heavy solid about its fixed point are considered. The center of mass of the body is assumed to lie on the moving z-axis which is assumed to be the minor axis of the ellipsoid of inertia. The nonlinear equations of motion and their three first integrals are obtained when the principal moments of inertia are distributed as I 1 < I 2 < I 3 . We construct a Lyapunov function L to investigate the stability conditions for this motion. We give a numerical example to illustrate the necessary and sufficient conditions for the stability of the body at certain moments of inertia. This problem has many important applications in different sciences.

Highlights

E author achieved the conditions of stability for this case of studied motion. e authors in [4] studied a branching for the stability of permanent rotatory motions of a rigid body filled with a viscous fluid
E applications such as the Chebyshev–Ritz technique for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures are given by Ebrahimi et al [15]
Our article is searching the conditions of stability of a nonperturbed coupled heavy solid rotating about a fixed point in a uniform field of gravity. e body is assumed to rotate about the minor z-axis of the ellipsoid of inertia with couple λ3

Summary

Definition of the Problem

We define the problem of the motion of a heavy solid rotating about the minor z-axis of the ellipsoid of inertia. Let the body rotates with an angular velocity vector ω (ω1, ω2, ω3) to the system Oxyz. Let the mass center of the body lies on the z-axis (see Figure 1), and the moments of inertia of the body satisfy the conditions: I1 < I2 < I3, z0 ≠ 0,. Where (0, 0, z0) is the position of the mass center for the origin. Us, the nonlinear system of equations of motion for this case and its first integrals are obtained as follows: I1ω_ 1 + I3 − I2􏼁ω2ω3 + λ3ω2 −mgz0c2, I2ω_ 2 + I1 − I3􏼁ω3ω1 − λ3ω1 mgz0c1,. Is is a case of uniform rotation of the solid about the minor axis of the ellipsoid of inertia (z-axis)

Stability Investigation

A Simple Application Example