Stability of stationary rotation of a heavy solid body with two elastic rods: PMM vol. 40, n≗1, 1976, pp. 55–64

Abstract

The example of the stability problem for stationary vertical rotation of a heavy “tuning fork” with a single point is used for showing that the method of stability investigation of a mechanical system stationary motions, based on the solution of the problem of minimum transformed potential energy of a system [1–6], is equally suitable for analyzing “trivial” and “nontrivial” stationary motions for which deformable elements of the system are, respectively, in the undeformed and the deformed state. Two methods for solving this problem are presented. Sufficient stability conditions that impose certain restrictions from below on the stiffness of rods and also from above and below on the angular velocity of the system uniform rotation are obtained and analyzed. The general statement of the problem of motion and stability of an elastic body with a cavity containing a fluid was presented by Rumiantsev [1]. The theorem on stability proved here is an extension of the Routh theorem to systems with distributed parameters, and reduces the problem of stability of stationary motion to that of minimum (transformed) potential energy W of the system. Solutions of a number of problems on the stability of stationary motions of a solid body with elastic rods and fluid in potential force fields appear in [2–5]. Two methods for establishing conditions for positive definiteness of the second variation δ 2 W in investigations of motion stability of mechanical systems consisting of absolutely rigid bodies and material points with attached deformable elastic and fluid bodies are presented and illustrated in [6] on the example of solutions of specific mechanical problems. The problem of stability of stationary motions of mechanical systems with distributed parameters were investigated in [7–10] (see the bibliography in [7–10]). The problem of stability of the “nontrivial” state of relative equilibrium of an absolutely rigid body with elastic rods is considered in [9], where the “trivial” and “nontrivial” equilibrium states of the system are characterized by the undeformed and deformed states, respectively. Only trivial states of relative equilibrium of rigid bodies with elastic rods were investigated in [2–6] and [7, 8]. It is noted in [9] that the previously used method based on the direct Liapunov method cannot be applied for solving problems of stability of the nontrivial state of relative equilibrium of the system, and another method of solution, based on the expansion of elastic displacements of rods in series of some complete system of functions is proposed. A finite number of first terms was retained in such series without any mathematical substantiation, and the system with distributed parameters is essentially reduced to some system with a finite number of degrees of freedom.