Abstract
The equilibrium states of an inverted two-link simple pendulum with an asymmetric follower force are classified depending on the characteristics of the springs (hard, soft, or linear) at the upper end and at the hinges. Phase portraits are plotted. The bifurcation points on the equilibrium curves are identified. Emphasis is on fold and cusp catastrophes
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References
V. V. Bolotin, Dynamic Stability of Elastic Systems [in Russian], Gostekhizdat, Moscow (1956).
V. V. Bolotin, Nonconservative Problems in the Theory of Elastic Stability [in Russian], Fizmatgiz, Moscow (1961).
V. A. Dzhupanov and Sv. V. Lilkova-Markova, “Dynamic stability of a fluid-conveying cantilevered pipe on an additional combined support,” Int. Appl. Mech., 39, No. 2, 185–191 (2003).
L. G. Lobas, V. V. Koval’chuk, and O. V. Bambura, “Evolution of the equilibrium states of an inverted pendulum,” Int. Appl. Mech., 43, No. 3, 344–350 (2007).
L. G. Lobas, V. V. Koval’chuk, and O. V. Bambura, “Equilibrium states of a pendulum with an asymmetric follower force,” Int. Appl. Mech., 43, No. 4, 460–466 (2007).
L. G. Lobas, V. V. Koval’chuk, and O. V. Bambura, “Theory of inverted pendulum with follower force revisited,” Int. Appl. Mech., 43, No. 6, 690–700 (2007).
L. G. Lobas, V. V. Koval’chuk, and O. V. Bambura, “Influence of concurrent use of springs with characteristics of different types on the equilibrium of an inverted pendulum,” Int. Appl. Mech., 43, No. 7, 793–798 (2007).
L. G. Lobas and L. L. Lobas, “Bifurcations, stability, and catastrophes of the equilibrium states of a double pendulum with an asymmetric follower force,” Izv. RAN, Mekh. Tverd. Tela, No. 4, 139–149 (2004).
H. Ziegler, “On the concepts of elastic stability,” in: Advances in Applied Mechanics, Vol. 4, Acad. Press, New York (1956), pp. 351–403.
V. A. Dzhupanov and Sv. V. Lilkova-Markova, “Divergent instability domains of a fluid-conveying cantilevered pipe with a combined support,” Int. Appl. Mech., 40, No. 3, 319–321 (2004).
I. Elishakoff, “Controversy associated with the so-called ‘follower forces’: critical overview,” Trans. ASME, Appl. Mech. Rev., 58, No. 3, 117–141 (2005).
P. Hagedorn, “On the destabilizing effect of non-linear damping in non-conservative systems with follower forces,” Int. J. Non-Lin. Mech., 5, No. 2, 341–358 (1970).
J.-D. Jin and Y. Matsuzaki, “Bifurcations in a two-degree-of freedom elastic system with follower forces,” J. Sound Vibr., 126, No. 2, 265–277 (1988).
J.-D. Jin and Y. Matsuzaki, “Stability and bifurcations of a double pendulum subjected to a follower force,” in: Proc. AIAA/ASME/ASCE/AHS/ASC 30th Structures, Structural Dynamics and Materials Conf. (Mobile, Ala, April 3–5, 1989), pt. 1, Washington (1989), pp. 432–439.
W. T. Koiter, “Unrealistic follower forces,” J. Sound Vibr., 194, No. 4, 636–638 (1996).
L. G. Lobas, “Generalized mathematical model of an inverted multilink pendulum with follower force,” Int. Appl. Mech., 41, No. 5, 566–572 (2005).
L. G. Lobas, “Dynamic behavior of multilink pendulums under follower forces,” Int. Appl. Mech., 41, No. 6, 587–613 (2005).
L. G. Lobas and V. V. Koval’chuk, “Influence of the nonlinearity of the elastic elements on the stability of a double pendulum with follower force in the critical case,” Int. Appl. Mech., 41, No. 4, 455–461 (2005).
V. L. Lobas, “Influence of the nonlinear characteristics of elastic elements on the bifurcations of equilibrium states of a double pendulum with follower force,” Int. Appl. Mech., 41, No. 2, 197–202 (2005).
Y. Matsuzaki and S. Futura, “Conimension three bifurcation of a double pendulum subjected to a follower force with imperfection,” in: Proc. AIAA Dyn. Spec. Conf. (Long Beach, California, April 5–6, 1990), Washington (1990), pp. 387–394.
S. Otterbein, “Stabilisierung des n-Pendels und der Indische Seiltrick,” Arch. Rat. Mech. Anal., 78, No. 4, 381–393 (1982).
A. Pflüger, Stabilitäts Probleme der Elastostatik, Springer, Berlin-Göttingen-Heidelberg (1950).
R. Scheidl, H. Troger, and K. Zeman, “Coupled flutter and divergence bifurcation of a double pendulum,” Int. J. Non-Linear Mech., 19, No. 2, 163–176 (1984).
Y. A. Shinohara, “A geometric method for numerical solution of non-linear equations and its application to non-linear oscillations,” Publ. RJMS, Kyoto Univ., 8, No. 1, 13–42 (1972).
Y. Sugiyama, M. A. Langthjem, and B. J. Ryu, “Realistic follower forces,” J. Sound Vibr., 225, No. 4, 779–782 (1999).
H. Troger and A. Steindl, Nonlinear Stability and Bifurcation Theory, Springer-Verlag, Vienna-New York (1991).
H. Ziegler, “Die Stabilitatskriterien der Elastomechanik,” Ing.-Arch., 20, No. 1, 49–56 (1952).
H. Ziegler, “Linear elastic stability” ZAMP, 4, No. 2, 89–121; No. 3, 167–185 (1953).
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Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 115–128, August 2007.
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Lobas, L.G., Koval’chuk, V.V. & Bambura, O.V. Influence of material and geometrical nonlinearities on the bifurcations of equilibrium states of a two-link pendulum. Int Appl Mech 43, 924–934 (2007). https://doi.org/10.1007/s10778-007-0093-8
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DOI: https://doi.org/10.1007/s10778-007-0093-8