Modern Problems of Structural Stability pp 229-283 | Cite as

# Bifurcations of Eigenvalues and Stability Problems in Mechanics

## Abstract

In this Part of the Lecture Notes we study bifurcations of eigenvalues of nonsymmetrical matrix operators depending on parameters with applications to the stability study in different mechanical problems. Section 1 gives general analysis of bifurcations for eigenvalues with geometric interpretation in two- and three-dimensional spaces. Main attention is focused on simple and double eigenvalues, and strong and weak interactions of eigenvalues are distinguished. In Section 2 stability and catastrophes in one-parameter circulatory systems with simple mechanical examples are studied. It is proven that in general they are subjected to catastrophes of three types: flutter, divergence, and transition from flutter to divergence and vice versa. In Section 3 properties of two-parameter circulatory systems are studied, and the explicit formulas describing metamorphoses of frequency curves are derived. These formulas use information on the system only at a merging point of the frequencies, and allow qualitative as well as quantitative analysis of behavior of frequency curves near that point with a change of parameters. In particular, development of “a bubble of instability” is analyzed. In Section 4 the Keldysh problem of aeroelastic stability of a wing with bracing struts is discussed, and the effect of disappearance of flutter instability revealed by Keldysh (1938) is explained. It is shown that this effect is connected with the convexity of the flutter domain in the parameter space. In Section 5 we study a problem of maximizing the critical buckling load of an elastic column of given length and volume assuming elastic supports at both ends of the column. This problem was first formulated by J.-L. Lagrange in 1773 for simply supported columns and has been considered by many authors for various boundary conditions. We obtain the bimodal optimal solutions and investigate their post-buckling behavior. Section 6 concerns instability domains for Hill’s equation with damping under assumption of small excitation amplitude and damping coefficient. It is shown that these domains are halves of cones in the three-dimensional parameter space. One of the important applications of Hill’s equation is connected with the stability study of periodic motions for nonlinear dynamical systems. It is shown how to find stable and unstable regimes for harmonically excited Duffing’s equation.

## Keywords

Parametric Resonance Critical Speed Frequency Curve Monodromy Matrix Jordan Chain## Preview

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## References

- Arnold, V.I. (1983).
*Geometrical Methods of the Theory of Ordinary Differential Equations*. New York: Springer-Verlag.CrossRefMATHGoogle Scholar - Arnold, V.I. (1984).
*Catastrophe Theory*. Berlin: Springer-Verlag.MATHGoogle Scholar - Arnold, V.I. (1989). Bifurcations and singularities in mathematics and mechanics. In
*Theoretical and Applied Mechanics*, Paris: Elsevier. 1–25.Google Scholar - Bolotin, V.V. (1963).
*Nonconservative Problems of the Theory of Elastic Stability*. New York: Pergamon.MATHGoogle Scholar - Claudon, J.L. (1975). Characteristic curves and optimum design of two structures subjected to circulatory loads.
*Journal de Mechanique*14: 531–543.MATHGoogle Scholar - Cox, S.J., and Overton, M.L. (1992). On the optimal design of columns against buckling.
*SIAM Journal of Mathematical Analysis*23: 287–325.CrossRefMATHMathSciNetGoogle Scholar - Fung, Y.C. (1955).
*An Introduction to the Theory ofAeroelasticity*. New York: Wiley.Google Scholar - Grossman, E.P. (1937). Flutter.
*Trudy TsAGI*, no. 284.Google Scholar - Galin, A.M. (1972). On real matrices depending on parameters.
*Uspekhi matematicheskikh nauk*27 (1): 241–242.MATHMathSciNetGoogle Scholar - Hanaoka, M., and Washizu, K. (1980). Optimum design of Beck’s column.
*Computers and Structures*11: 473–480.CrossRefMATHMathSciNetGoogle Scholar - Huseyin, K. (1978).
*Vibrations and Stability of Multiple Parameter Systems*. Alphen aan den Rijn: Sijthoff & Nordhoff.Google Scholar - Kamke, E. (1967).
*Differentialgleichungen*. Leipzig: Verlagsgesellschaft.Google Scholar - Keldysh, M.V. (1938). Vibrations of a wing with bracing struts in airflow.
*Trudy TsAGI*, no. 357.Google Scholar - Keldysh, M.V. (1985).
*Mechanics. Selected Works*. Moscow: Nauka, 304–341.Google Scholar - Kirillov, O.N., and Seyranian, A.P. (1999).
*On the Stability Boundaries of Circulatory Systems*. Moscow State Lomonosov University. Preprint 51–99, Institute of Mechanics.Google Scholar - Kirillov, O.N., and Seyranian, A.P. (2001). Overlapping of frequency curves in nonconservative systems.
*Physics-Doklady*377 (1): 44–49.Google Scholar - Kounadis, A.N., and Katsikadelis, J.T. (1980). On the discontinuity of the flutter load for various types of cantilevers.
*International Journal of Solids and Structures*, 16: 375–383.CrossRefMATHMathSciNetGoogle Scholar - Lagrange, J.-L. (1868). Sur la figure des colonnes. In
*Ouvres de Lagrange*, v. 2. Paris: Gauthier -Villars, 125–170.Google Scholar - Langthjem, M.A., and Sugiyama, Y. (1999). Optimum shape design against flutter of a cantilevered column with an end-mass of finite size subjected to a non-conservative load.
*Journal of Sound and Vibration*226: 1–23.CrossRefGoogle Scholar - Leipholz, H. (1980).
*Stability of Elastic Systems*. Alphen aan den Rijn: Sijthoff & Nordhoff.Google Scholar - Leipholz, H. (1987).
*Stability Theory*. Stuttgart: John Wiley& Sons and B.G.Teubner.CrossRefMATHGoogle Scholar - Mailybaev, A.A., and Seyranian, A.P. (1996). The Keldysh problem on aeroelastic stability of a strut braced wing.
*Physics-Doklady*41 (10): 484–487.MATHMathSciNetGoogle Scholar - Mailybaev, A.A., and Seyranian, A.P. (1998). Aeroelastic stability of a wing with bracing struts (Keldysh problem).
*Fluid Dynamics*33 (1): 124–134.CrossRefMATHMathSciNetGoogle Scholar - Merkin, D. (1997).
*Introduction to the Theory of Stability*. New York: Springer-Verlag.Google Scholar - Nayfeh, A.H., and Mook, D.T. (1979).
*Nonlinear Oscillations*. New York: John Wiley and Sons.MATHGoogle Scholar - Olhoff, N., and Rasmussen, S.H. (1977). On single and bimodal optimum buckling loads of clamped columns.
*International Journal of Solids and Structures*13: 605–614.CrossRefMATHGoogle Scholar - Panovko, Ya.G., and Gubanova, I.I. (1965).
*Stability and Oscillations of Elastic Systems*. New York: Consultants Bureau.Google Scholar - Pedersen, P. (1980). Stability of the solutions to Mathieu-Hill equations with damping.
*Ingenieur-Archiv*49: 15–29.CrossRefMATHGoogle Scholar - Pedersen, P. (1985). On stability diagrams for damped Hill equations.
*Quarterly of Applied Mathematics*42: 477–495.MATHMathSciNetGoogle Scholar - Plaut, R.H. (1972). Determining the nature of instability in nonconservative problems.
*AIAA Journal**10*: 967–968.CrossRefMathSciNetGoogle Scholar - Privalova, O.G., and Seyranian, A.P. (1999). Supercritical behavior of bimodal optimal rods.
*Mechanics of Solids*342 (2): 142–150.Google Scholar - Seyranian, A.P. (1991). Sensitivity analysis of eigenvalues and development of instability.
*Stroinicki Casopis*42: 193–208.Google Scholar - Seyranian, A.P. (1993). Sensitivity analysis of multiple eigenvalues.
*Mechanics of Structures and Machines*21: 261–284.CrossRefMathSciNetGoogle Scholar - Seyranian, A.P. (1994). Bifurcations in single-parameter circulatory systems.
*Izv. RAN. Mekhanika Tverdogo Tela*29: 142–148.Google Scholar - Seyranian, A.P. (1994). Collision of eigenvalues in linear oscillatory systems.
*Journal of Applied Mathematics and Mechanics*58: 805–813.CrossRefMathSciNetGoogle Scholar - Seyranian, A.P. (1995). New solutions to Lagrange’s problem.
*Physics-Doklady*40 (5): 251–253.MathSciNetGoogle Scholar - Seyranian, A.P. (2000).
*The Lagrange Problem on Optimal Column*. Moscow State Lomonosov University, Institute of Mechanics. Preprint No. 60Google Scholar - Seyranian, A.P. (2001). Resonance domains for the Hill equation with allowance for damping.
*Physics-Doklady*46 (1): 41–44.CrossRefGoogle Scholar - Seyranian, A.P., Lund, E., and Olhoff, N. (1994). Multiple eigenvalues in structural optimization problems.
*Structural Optimization*8: 207–227.CrossRefGoogle Scholar - Seyranian, A.P., and Pedersen, P. (1995). On two effects in fluid/structure interaction theory. In Bearman, P.W., ed.,
*Flow-Induced Vibration*. Rotterdam: A.A.Balkema. 565–576.Google Scholar - Seyranian, A.P., and Privalova, O.G. (1999). Supercritical behavior of optimal bars with two modes of buckling.
*Doklady Physics*44 (6): 368–372.MATHMathSciNetGoogle Scholar - Seyranian, A.P., and Sharanyuk, A.V. (1983). Sensitivity and optimization of critical parameters in dynamic stability problems.
*Izv. AN SSSR. Mekhanika Tverdogo Tela*18 (5): 173–182.Google Scholar - Seyranian, A.P., Solem, F., and Pedersen, P. (1999). Stability analysis for multi-parameter linear periodic systems.
*Archive of Applied Mechanics*69: 160–180.CrossRefMATHGoogle Scholar - Seyranian, A.P., Solem, F., and Pedersen, P. (2000). Multi-parameter linear periodic systems: sensitivity analysis and applications.
*Journal of Sound and Vibration*229: 89–111.CrossRefMATHMathSciNetGoogle Scholar - Seyranian, A.P., Solem, F., and Pedersen, P. (2000). Sensitivity analysis for the Floquet multipliers. In Chemousko, F.L., and Fradkov, A.L., eds.,
*Proceedings of 2**nd**InternationalConference on Control of Oscillations and Chaos*. St.Petersburg. 404–407.Google Scholar - Tadjbakhsh, I. and Keller, J.B. (1962). Strongest columns and isoperimetric inequalities for eigenvalue.
*Transactions of ASME*,*Ser.E. Journal of Applied Mechanics*29: 159–164.CrossRefMATHMathSciNetGoogle Scholar - Thompson, J.M.T. (1972). Optimization as a generator of structural instability. Letter to the Editor.
*International Journal of Mechanical Sciences*14: 627–629.CrossRefGoogle Scholar - Thompson, J.M.T. (1982).
*Instabilities and Catastrophes in Science and Engineering*. London: John Wiley & Sons.MATHGoogle Scholar - Thompson, J.M.T., and Hunt, G.W. (1973).
*A General Theory of Elastic Stability*. London: John Wiley & Sons.MATHGoogle Scholar - Thomsen, J.J. (1997).
*Vibrations and Stability. Order and Chaos*. London: McGraw-Hill.Google Scholar - Vishik, M.I., and Lyusternik, L.A. (1960). Solution of some perturbation problems in the case of matrices and selfadjoint and non-selfadjoint equations.
*Russian Mathematical Surveys*15 (3): 1–73CrossRefMATHMathSciNetGoogle Scholar - Yakubovich, V.A., and Starzhinskii, V.M. (1987).
*Parametric Resonance in Linear Systems*. Moscow: Nauka (in Russian).MATHGoogle Scholar - Zevin, A.A. (1988). On the theory of linear nonconservative systems. Journal of Applied Mathematics and Mechanics 52 (3): 386–391.CrossRefMathSciNetGoogle Scholar
- Ziegler, H. (1968).
*Principles of Structural Stability*. Waltham, Massachusetts: Blaisdell.Google Scholar