Advertisement

Applied Mathematics and Mechanics

, Volume 39, Issue 3, pp 305–316 | Cite as

A simple criterion for finite time stability with application to impacted buckling of elastic columns

  • C. Q. Ru
Article

Abstract

The existing theories of finite-time stability depend on a prescribed bound on initial disturbances and a prescribed threshold for allowable responses. It remains a challenge to identify the critical value of loading parameter for finite time instability observed in experiments without the need of specifying any prescribed threshold for allowable responses. Based on an energy balance analysis of a simple dynamic system, this paper proposes a general criterion for finite time stability which indicates that finite time stability of a linear dynamic system with constant coefficients during a given time interval [0, t f ] is guaranteed provided the product of its maximum growth rate (determined by the maximum eigen-root p1 >0) and the duration t f does not exceed 2, i.e., p1t f <2. The proposed criterion (p1t f =2) is applied to several problems of impacted buckling of elastic columns: (i) an elastic column impacted by a striking mass, (ii) longitudinal impact of an elastic column on a rigid wall, and (iii) an elastic column compressed at a constant speed (“Hoff problem”), in which the time-varying axial force is replaced approximately by its average value over the time duration. Comparison of critical parameters predicted by the proposed criterion with available experimental and simulation data shows that the proposed criterion is in robust reasonable agreement with the known data, which suggests that the proposed simple criterion (p1t f =2) can be used to estimate critical parameters for finite time stability of dynamic systems governed by linear equations with constant coefficients.

Keywords

finite time stability buckling impact elastic column Hoff problem dynamic buckling stability criterion 

Chinese Library Classification

O349.9 

2010 Mathematics Subject Classification

34D20 34D30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Dorato, P. An overview of finite time stability. Current Trends in Nonlinear Systems and Control (ed. Menini, L.), Springer, Boston, 185–194 (2006)CrossRefGoogle Scholar
  2. [2]
    Bhat, S. P. and Bernstein, D. S. Finite time stability of continuous autonomous systems. SIAM Journal on Control and Optimization, 38, 751–766 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    Amato, F., de Tommasi, G., and Pironti, A. Necessary and sufficient conditions for finite-time stability of impulsive dynamics linear systems. Automatica, 49, 2546–2550 (2013)CrossRefMATHGoogle Scholar
  4. [4]
    Kussaba, H. T. M., Borges, R. A., and Ishihara, J. Y. A new condition for finite time boundedness analysis. Journal of the Franklin Institute, 352, 5514–5528 (2015)MathSciNetCrossRefGoogle Scholar
  5. [5]
    Lindburg, H. E. Impact buckling of a thin bar. Journal of Applied Mechanics, 32, 315–322 (1965)CrossRefGoogle Scholar
  6. [6]
    Abrahamson, G. R. and Goodier, J. N. Dynamic flextual buckling of rods within an axial plastic compression wave. Journal of Applied Mechanics, 33, 241–248 (1966)CrossRefGoogle Scholar
  7. [7]
    Lindburg, H. E. Little Book of Dynamic Buckling, LCE Science/Software (2003)Google Scholar
  8. [8]
    Hutchinson, J. W. and Budiansky, B. Dynamic buckling estimates. AIAA Journal, 4, 525–530 (1966)CrossRefGoogle Scholar
  9. [9]
    Simitses, G. J. Instability of dynamically-loaded structures. Applied Mechanics Reviews, 40, 1403–1408 (1987)CrossRefGoogle Scholar
  10. [10]
    Ari-Gur, J., Weller, T., and Singer, J. Experimental and theoretical studies of columns under axial impact. International Journal of Solids and Structures, 18, 619–641 (1982)CrossRefGoogle Scholar
  11. [11]
    Weller, T., Abramovich, H., and Yaffe, R. Dynamic buckling of beams and plates subjected to axial impact. Computers and Structures, 32, 835–851 (1989)CrossRefGoogle Scholar
  12. [12]
    Kornev, V. M. Development of dynamic forms of stability loss of elastic systems under intensive loading over a finite time interval. Journal of Applied Mechanics and Technical Physics, 13, 536–541 (1972)CrossRefGoogle Scholar
  13. [13]
    Morozov, N. F., Il’in, D. N., and Belyaev, A. K. Dynamic buckling of rod under axial jump loading. Doklady Physics, 58, 191–195 (2013)CrossRefGoogle Scholar
  14. [14]
    Hoff, N. J. The dynamics of the buckling of elastic columns. Journal of Applied Mechanics, 18, 68–74 (1951)MathSciNetMATHGoogle Scholar
  15. [15]
    Elishakoff, I. Hoff’s problem in probabilistic setting. Journal of Applied Mechanics, 47, 403–408 (1980)CrossRefMATHGoogle Scholar
  16. [16]
    Kounadis, A. N. and Mallis, J. Dynamic stability of initially crooked columns under a time-dependent axial displacement of their support. Quarterly Journal of Mechanics and Applied Mathematics, 41, 579–596 (1988)MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Motamarri, P. and Suryanarayan, S. Unified analytical solution for dynamic elastic buckling of beams for various boundary conditions and loading rates. International Journal of Mechanical Sciences, 56, 60–69 (2012)CrossRefGoogle Scholar
  18. [18]
    Kuzkin, V. A. and Dannert, M. M. Buckling of a column under a constant speed compression: a dynamic correction to the Euler formula. Acta Mechanica, 227, 1645–1652 (2016)MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Davidson, J. F. Buckling of struts under dynamic loading. Journal of the Mechanics and Physics of Solids, 2, 54–66 (1953)CrossRefGoogle Scholar
  20. [20]
    Zhang, Z. and Taheri, F. Dynamic pulse-buckling behavior of quasi-ductile carbon/epoxy and e-glass/epoxy laminated composite beams. Composites and Structures, 64, 269–274 (2004)CrossRefGoogle Scholar
  21. [21]
    Ji, W. and Waas, A. M. Dynamic bifurcation buckling of an impacted column. International Journal of Engineering Science, 46, 958–967 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    Wang, A. and Tian, W. Mechanism of buckling development in elastic bars subjected to axial impact. International Journal of Impact Engineering, 34, 232–252 (2007)CrossRefGoogle Scholar
  23. [23]
    Gladden, J. R., Handzy, N. Z., Belmonte, A., and Villemaux, E. Dynamic buckling and fragmen-tation in brittle rods. Physical Review Letters, 94, 035503 (2005)CrossRefGoogle Scholar
  24. [24]
    Jiao, X. J. and Ma, J. M. Influence of the connection condition on the dynamic buckling of longitudinal impact for an elastic rod. Acta Mechanica Solida Sinica, 30, 291–298 (2017)CrossRefGoogle Scholar
  25. [25]
    Dost, S. and Glockner, P. G. On the dynamic stability of viscoelastic prefect column. International Journal of Solids and Structures, 18, 587–596 (1982)CrossRefMATHGoogle Scholar
  26. [26]
    Giofgi, C., Pata, V., and Vuk, E. On the extensible viscoelastic beam. Nonlinearity, 21, 713–733 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    Wang, S., Wang, Y., Huang, Z. L., and Yu, T. X. Dynamic behavior of elastic bars and beams impinging on ideal springs. Journal of Applied Mechanics, 83, 031002 (2016)CrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of AlbertaEdmontonCanada

Personalised recommendations