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IssuesArchive of Issues2004-4pp.110-118

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L. G. Lobas and L. L. Lobas, "Bifurcations, stability, and catastrophes of equilibrium states of a double pendulum subjected to an asymmetric follower force," Mech. Solids. 39 (4), 110-118 (2004)
Year 2004 Volume 39 Number 4 Pages 110-118
Title Bifurcations, stability, and catastrophes of equilibrium states of a double pendulum subjected to an asymmetric follower force
Author(s) L. G. Lobas (Kiev)
L. L. Lobas (Kiev)
Abstract The topological structure of the phase space of an inverted double pendulum with elastoviscous joints is studied. The essential parameters involve the angular eccentricity, the orientation parameter and the magnitude of the follower force, and the stiffness of the elastic clamping of the upper end of the pendulum. Equilibrium manifolds are constructed by means of continuation with respect to a parameter. Bifurcation points are identified. As the eccentricity of the follower force smoothly changes, the pendulum can jump from one stable equilibrium to another. The applicability limits of Euler's static concept for the vertical equilibrium are indicated, provided that an elastic beam compressed by a follower force is modeled by a double pendulum.

Pfluger's attempt [1] to extend Euler's static method to the problem of stability of a cantilever beam compressed by a force that depends on the configuration of the elastic system (remains tangent to the curved axis of the beam) has led to an unexpected conclusion that the critical load does not exist in this problem. This conclusion contradicts the physical considerations.

Pfluger's paradox was solved by Ziegler [2] who studied a discrete model (double mathematical pendulum) instead of a continuous beam. However, another paradox appears - the elastic system loses its stability when an infinitesimal dissipation has been introduced. The destabilizing effect of nonlinear damping in non-conservative systems with follower forces was shown in [3]. The authors of [4] drew attention to the fact that internal and external frictions play different roles. Hermann and Jonh showed that although low damping can produce a destabilizing effect, an appropriate interpretation of the passage to the limit as the damping vanishes yields the value of the critical force coincident with that in the case of the undamped system. According to [6], the destabilization paradox should be understood in a rather restricted sense, since the critical buckling force (the force at which the system looses its stability) does not have a limit as a function of two variables. The influence of the critical value of the follower force by the dissipative forces in the case of partial dissipation was studied in [7]. The stability for almost all initial conditions (in terms of the Lebesgue measure), with the exception, maybe, of two values of the magnitude of the follower force, was proved in [8]. The results of [9] can be regarded as an extension of the familiar results on the dynamic stabilization of the upper equilibrium position of a simple pendulum by means of vertical vibration of the point of suspension to the case of a compound pendulum. The elastic clamping of the upper end of the pendulum [10, 11] requires one more essential parameter (the clamping stiffness) to be introduced. A period-doubling bifurcation cascade and the deterministic chaos were studied in [12]. Normal forms were utilized in [13] to construct limit cycles. Local and global bifurcations of Ziegler's double pendulum in the presence of an asymmetric follower force and connecting springs were investigated in [14-16]. Major attention is given to the neighborhood of a double zero eigenvalue of the linearization matrix.
References
1.  A. Pflüger, Stabilitätsprobleme der Elastostatik, Springer, Berlin, Göttingen, Heidelberg, 1950.
2.  H. Ziegler, "Die Srabilitäskriterien der Elasomechanik," Ing.-Arch, Bd. 20, H. 1, S. 49-56, 1952.
3.  P. Hagedorn, "On the destabilizing effect of the nonlinear damping in non-conservative systems with follower forces," Int. J. Non-linear Mech., Vol. 5, No. 2, pp. 341-358, 1970.
4.  Ya. G. Panovko and S. V. Sorokin, Quasi-stability of elastoviscous systems with follower forces," Izv. AN SSSR. MTT [Mechanics of Solids], No. 5, pp. 135-139, 1970.
5.  G. Herrmann and I.-C. Jonh, "On the destabilizing effect of damping in nonconservative elastic systems," Trans. ASME. J. Appl. Mech., Vol. 32, No. 3, pp. 592-597, 1965.
6.  A. P. Seiranyan, "Stabilization of non-conservative systems by dissipative forces and indeterminacy of the critical load," Doklady AN, Vol. 348, No. 3, pp. 323-326, 1996.
7.  N. I. Zhinzher, "The influence of partial dissipation forces on the stability of elastic systems," Izv. AN. MTT [Mechanics of Solids], No. 1, pp. 149-155, 1994.
8.  S. A. Agafonov, "Stability and self-sustained vibrations of a double pendulum with elastic members under the action of a follower force," Izv. AN. MTT [Mechanics of Solids], No. 5, pp. 185-190, 1992.
9.  S. A. Agafonov, "Stabilization of the equilibrium of Ziegler's pendulum by parametric excitation," Izv. AN. MTT [Mechanics of Solids], No. 6, pp. 36-40, 1997.
10.  R. Scheidl, H. Troger, and K. Zeman, "Coupled flutter and divergence bifurcation of a double pendulum," Int. J. Non-linear Mech., Vol. 19, No. 2, pp. 163-176, 1984.
11.  H. Troger and A. Steindl, Nonlinear Stability and Bifurcation Theory, Springer, Wien, New York, 1991.
12.  Q. Bi and P. Yu, Double Hopf bifurcations and chaos of a nonlinear vibration system," Nonlinear Dynamics, Vol. 19, No. 4, pp. 313-332, 1999.
13.  L. Hsu, L. J. Min, and L. Lavretto, "A recursive approach to compute normal forms," J. Sound and Vibrat., Vol. 243, No. 5, pp. 909-927, 2001.
14.  J.-D. Jin, and Y. Matsuzaki, "Bifurcations in a two-degree-of-freedom elastic system with follower forces," J. Sound and Vibrat., Vol. 126, No. 2, pp. 265-277, 1988.
15.  J.-D. Jin, and Y. Matsuzaki, "Stability and bifurcations of a double pendulum subjected to a follower force," in AIAA/ASME/ASCE/AHS/ASC 30th Struct., Struct. Dyn. and Mater. Conf., Mobile, Ala, 1989: Collect. Techn. Pap. Pt. 1, pp. 432-439, Washington, 1989.
16.  Y. Matsuzaki and S. Futura, "Codimension three bifurcation of a double pendulum subjected to a follower force with imperfection," in AIAA Dyn. Spec. Conf. Long Beach, Calif., 1990: Collect. Techn. Pap., pp. 387-394, Washington (D.C.), 1990.
17.  Y. Shinohara, "A geometric method for the numerical solution of non-linear equations and its application to non-linear oscillations," Publ. Res. Inst. Math. sci., Kyoto Univ., Vol. 8, No. 1, pp. 13-42, 1972.
Received 07 March 2002
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