On the stability of equilibrium position of three-link rod system under tracking forceload


Аuthors

Baikov A. E.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: alexander@baikov.org

Abstract

The destabilization of stable equilibrium position of a non-conservative system with three degrees of freedom under joint action of potential, non-conservative positional forces and small linear viscous friction with complete dissipation force is considered. It is assumed that the system has an isolated equilibrium position.

The problem of the stability of three-link rod system under non-conservative tracking forceload in a horizontal plane is investigated. Tracking force applied to rigid body forms a constant angle with its axes. The rods are joined by viscoelastic spiral springs.

Stability of equilibrium position is studied in linear approximation using Lyapunov’s theorems. The coefficients of characteristic polynomial are constructed by using Le Verrier’s algorithm. They depend upon the invariants of the matrix of linear approximation equations. Zigler’s effect condition and stabilitycriterion were obtained using perturbation theory. The stabilizationregion and Ziegler’s area werebuilt in the parameter space with the help of computer algebra methods.

Using Lagrange equations of motion formulas for the coefficients of characteristic polynomial are obtained. Stability of three-link rod system’s equilibrium position is investigated when there is no dissipation. Zigler’s area and criterion for the stability of the equilibrium position of a system with three degrees of freedom, in which the friction forces take small values, are constructed.

The results of the study may be useful in stability analysis of a non-conservative system with three degrees of freedom, in particular, of the three-link rod system that can be used as a model of elastic rod.

Keywords:

three-link rod system, tracking force, dissipative forces, asymptotical stability, Zieglers effect, Ziegler’s areas, stabilitycriterion

References

1. Rabinovich B.I. Vvedenie v dinamiku raket-nositelei kosmicheskikh apparatov (Introduction to the Dynamics of a Spacecraft Rockets), Moscow, Mashinostroenie, 1975, 416 p.

2. Ziegler H. Die Stabilität skriterien der Alastomechanik, Ingenieur-Archiv, 1952, vol. 20, no. 1, pp. 4956.

3. Ziegler H. Osnovy teorii ustoichivosti konstruktsii (Principles of Stability Theory of Structures), Moscow, Mir, 1971, 192 p.

4. Seiranyan A.P. Uspekhi mekhaniki, 1990, vol. 13, no. 2, pp. 89–124.

5. Merkin D. R. Vvedenie v teoriyu ustoichivosti dvizheniya (Introduction to the Theory of Motion Stability), Moscow, Nauka, 1976, 319 p.

6. Bruno A.D. Mnozhestvo ustoichivosti mnogoparametricheskikh zadach (The Stability Set of Multiparameter Problems), Moscow, Keldysh Institute preprints, 2010, 14 p.

7. Bruno A.D., Batkhin A.B., Varin V.P. Mnozhestvo ustoichivosti odnoi giroskopicheskoi zadachi (The Stability Set of a Gyroscopic Problem), Moscow, Keldysh Institute preprints, 2010, 30 p.

8. Baikov A.Ye., Krasil’nikov P.S. Prikladnaya matematika i mekhanika, 2010, vol. 74 no. 1, pp. 5160.

9. Markeev A. P. Teoreticheskaya mekhanika (Theoretical Mechanic), Moscow, CheRo, 1999, 572 p.

10. Kwakernaak К., Sven R. Linear Optimal Control Systems, New York, Wiley, 1972, 608 p.


Download

mai.ru — informational site MAI

Copyright © 2000-2021 by MAI

Вход