Summary
The non-linear static and dynamic stability of a non-linearly elastic, two degree of freedom system under a partial follower load is thoroughly discussed. Conditions for divergence and flutter instability in the large, in the sense of Lagrange, are established. These conditions may be completely different from those corresponding to linearized analyses. It is found that the static method of analysis is applicable for the entire range of the non- conservativeness loading parameter provided that the material non-linearity is accounted for. The stability or instability of critical states depends on the amount of material non-linearity. The unexpected result that the non-linear static critical load may differ substantially from the corresponding non-linear dynamic critical load is deduced with the aid of a numerical integration of the original non-linear equations of motion. The effects of compressibility as well as of other parameters on such a difference are properly examined. New phenomena which contradict well-known results based on linearized analyses reveal the true instability mechanism of non-conservative systems
Übersicht
Diskutiert wird die nichtlineare statische und dynamische Stabilität eines nichtlinear elastischen Systems mit zwei Freiheitsgraden bei teilweise mitgehender Last. Bedingungen für die Stabilität im Großen, im Sinne von Lagrange, bei Knicken und Flattern werden aufgestellt. Sie können völlig verschieden von denjenigen einer lirearisierten Analyse sein. Es stellt sich heraus, daß die statische Betrachtung für den gesamten Parameterbereich der nichtkonservativen Last zulässig ist, vorausgesetzt die Material-Nichtlinearität wird berücksichtigt. Die Stabilität oder Instabilität kritischer Zustände hängt vom Grad der Material-Nichtlinearität ab. Das unerwartete Ergebnis, daß nichtlineare statische und dynamische kritische Last wesentlich voneinander abweichen können, folgt aus einer numerischen Integration der Bewegungsgleichungen. Der Einfluß der Kompressibilität und anderer Parameter auf diese Abweichung wird untersucht. Neue Phänomene, die bekannten Ergebnissen der linearen Analyse widersprechen, enthüllen den tatsächlichen Instabilitätsmechanismus nichtkonservativer Systeme.
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References
Thompson, J. M. T.: A general theory for the equilibrium and stability of discrete conservative systems. Z. Angew. Math. Phys. 20 (1969) 797–846
Kounadis, A. N.: Material-dependent stability conditions in the buckling of nonlinear elastic bars. Acta Mech. 67 (1987) 209–228
Varellis, J.; Kounadis, A. N.: The effects of material nonlinearity and compressibility in the buckling of nonlinear elastic systems. Ing. Arch. 60 (1989) 20–30
Plaut, R. H.: Postbuckling analysis of nonconservative elastic systems. J. Struct. Mech. 4 (1976) 395–416
Plant, R. H.: Branching analysis at coincident buckling loads of nonconservative elastic systems. Trans. ASME. Ser. E: J. Appl. Mech. 44 (1977) 317–321
Herrmann, G.; Bungay, R. W. (1964): On the stability of elastic systems subjected to nonconservative forces. Trans. ASME. Ser. E: J. Appl. Mech. 31 (1964) 435–440
Leipholz, H.: Stability theory. New York: Academic Press 1970
Kounadis, A. N.; Mahrenholtz, O.: Divergence instability conditions in the optimum design of nonlinear elastic systems under follower loads. In: Proc. IUTAM Symp. on shape optimization, Melbourne. Structural Optimization 1 (1989) 163–169.
Nemat-Nasser, S.; Herrmann, G.: Some general considerations concerning the destabilizing effect in nonconservative systems. Z. Angew. Math. Phys. 17 (1966) 303–401
Kounadis, A. N.: The existence of regions of divergence instability for nonconservative systems under follower forces. Int. J. Solids Struct. 19 (1983) 725–733
Burgess, I. W.; Levinson, M.: The post-flutter oscillations of Discrete symmetric structural systems with circulatory loading. Int. J. Mech. Sci. 14 (1972) 471–488
Roorda, J.; Nemat-Nasser, S.: An energy method for stability analysis of nonlinear nonconservative systems. AIAA J. 5 (1967) 1262–1268
Kounadis, A. N. (1989): New instability aspects for nonlinear nonconservative systems with precritical deformation. In: IUTAM Symp. nonlinear dynamics in engineering systems, Stuttgart August 21–26, 1989
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Sotiropoulos, S.N., Kounadis, A.N. The effects of non-linearities and compressibility on the static and dynamic critical load of non-conservative discrete systems. Ing. arch 60, 399–409 (1990). https://doi.org/10.1007/BF00542569
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DOI: https://doi.org/10.1007/BF00542569