Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Auto-parametric resonance of framed structures under periodic excitations

  • Li, Yuchun (Department of Hydraulic Engineering, College of Civil Engineering, Tongji University) ;
  • Gou, Hongliang (Department of Hydraulic Engineering, College of Civil Engineering, Tongji University) ;
  • Zhang, Long (Department of Hydraulic Engineering, College of Civil Engineering, Tongji University) ;
  • Chang, Chenyu (Department of Hydraulic Engineering, College of Civil Engineering, Tongji University)
  • Received : 2016.07.17
  • Accepted : 2016.11.18
  • Published : 2017.02.25
⟨ Previous Next ⟩

Abstract

A framed structure may be composed of two sub-structures, which are linked by a hinged joint. One sub-structure is the primary system and the other is the secondary system. The primary system, which is subjected to the periodic external load, can give rise to an auto-parametric resonance of the second system. Considering the geometric-stiffness effect produced by the axially internal force, the element equation of motion is derived by the extended Hamilton's principle. The element equations are then assembled into the global non-homogeneous Mathieu-Hill equations. The Newmark's method is introduced to solve the time-history responses of the non-homogeneous Mathieu-Hill equations. The energy-growth exponent/coefficient (EGE/EGC) and a finite-time Lyapunov exponent (FLE) are proposed for determining the auto-parametric instability boundaries of the structural system. The auto-parametric instabilities are numerically analyzed for the two frames. The influence of relative stiffness between the primary and secondary systems on the auto-parametric instability boundaries is investigated. A phenomenon of the "auto-parametric internal resonance" (the auto-parametric resonance of the second system induced by a normal resonance of the primary system) is predicted through the two numerical examples. The risk of auto-parametric internal resonance is emphasized. An auto-parametric resonance experiment of a ${\Gamma}$-shaped frame is conducted for verifying the theoretical predictions and present calculation method.

Keywords

Acknowledgement

Supported by : National Science Foundation of China

References

  1. Basar, Y., Eller, C. and Kratzig, W.B. (1987), "Finite element procedures for parametric resonance phenomena of arbitrary elastic shell structures", Comput. Mech., 2, 89-98. https://doi.org/10.1007/BF00282131
  2. Bathe, K.J. (1996), Finite Element Procedures, Prentice Hall, New Jersey.
  3. Bolotin, V.V. (1964), The dynamic stability of elastic systems, Holden Day, San Francisco.
  4. Briseghella, L., Majorana, C.E. and Pellegrino, C. (1998), "Dynamic stability of elastic structures: a finite element approach", Comput. Struct., 69, 11-25. https://doi.org/10.1016/S0045-7949(98)00084-4
  5. Chopra, A.K. (2007), Dynamics of Structures: Theory and Applications to Earthquake Engineering, Prentice Hall, New Jersey.
  6. Clough, R.W. and Penzien J. (2003), Dynamics of Structures, Third Edition, Computers & Structures, Inc., Berkeley.
  7. Dingwell, J.B., Cusumano, J.P., Cavanagh, P.R. and Sternad, D. (2001), "Local dynamic stability versus kinematic variability of continuous overground and treadmill walking", J. Biomech. Eng., 123, 27-32. https://doi.org/10.1115/1.1336798
  8. Elfrgani, A.M., Moussounda, R. and Rojas, R.G. (2015), "Stability assessment of non-Foster circuits based on time-domain method", IET Microw. Anten. Propagat., 9(15), 1769-1777. https://doi.org/10.1049/iet-map.2014.0671
  9. Kr a tzig, W. B., Bsar, Y., and Nawrotzki, P. (1996), "Dynamic structural instabilities", Dynamics of Civil Engineering Structures, Eds. W.B. Kr a tzig, Y. Bsar and P. Nawrotzki, Balkema, Rotterdam.
  10. Kumar, R., Kumar, A. and Panda, S.K. (2015), "Parametric Resonance of composite skew plate under non-uniform in-plane loading", Struct. Eng. Mech., 55(2), 435-459. https://doi.org/10.12989/sem.2015.55.2.435
  11. Li, Y. and Wang, Z. (2016), "Unstable characteristics of twodimensional parametric sloshing in various shape tanks: theoretical and experimental analyses", J. Vib. Control, 22(19), 4025-4046. https://doi.org/10.1177/1077546315570716
  12. Li, Y., Wang, L. and Yu, Y. (2016), "Stability analysis of parametrically excited systems using the energy-growth exponent/coefficient", Int. J. Struct. Stab. Dyn., 16(10), 1750018.
  13. Majorana, C.E. and Pellegrino, C.D. (1997), "Dynamic stability of elastically constrained beams: an exact approach", Eng. Comput., 14(7), 792-805. https://doi.org/10.1108/02644409710188709
  14. Majorana, C.E. and Pomaro, B. (2011), "Dynamic stability of an elastic beam with visco-elastic translational and rotational supports", Eng. Comput., 28(2), 114-129. https://doi.org/10.1108/02644401111109187
  15. Majorana, C.E. and Pomaro, B. (2012), "Dynamic stability of an elastic beam with visco-elasto-damaged translational and rotational supports", J. Eng. Mech., 138(6), 582-590. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000360
  16. Mishra, U.K. and Sahu, S.K. (2015), "Parametric instability of beams with transverse cracks subjected to harmonic in-plane loading", Int. J. Struct. Stab. Dyn., 15(1), 1540006. https://doi.org/10.1142/S0219455415400064
  17. Naprstek, J. and Fischer, C. (2015), "Dynamic stability of a vertically excited non-linear continuous system", Comput. Struct., 155, 106-114. https://doi.org/10.1016/j.compstruc.2015.01.001
  18. Pradyumna, S. and Gupta, A. (2011), "Dynamic stability of laminated composite plates with piezoelectric layers subjected to periodic in-plane load", Int. J. Struct. Stab. Dyn., 11(2), 297-311.
  19. Rosenstein, M.T., Collins, J.J. and Luca, C.J.D. (1993), "A practical method for calculating largest Lyapunov exponents from small data sets", Physica D: Nonlin. Phenomena, 65(1-2), 117-134. https://doi.org/10.1016/0167-2789(93)90009-P
  20. Shastry, B.P. and Rao, G.V. (1984), "Dynamic stability of bars considering shear deformation and rotatory inertia", Comput. Struct., 19, 823-7. https://doi.org/10.1016/0045-7949(84)90182-2
  21. Shastry, B.P. and Rao, G.V. (1986), "Dynamic stability of a short cantilever column subjected to distributed axial loads", Comput. Struct., 22, 1063-1064. https://doi.org/10.1016/0045-7949(86)90166-5
  22. Tondl, A., Ruijgrok, M., Verhulst, F. and Nabergoj, R. (2000), Autoparametric Resonance in Mechanical Systems, Cambridge University Press, Cambridge.
  23. Wang, X. (2003), Finite Element Method, Tsinghua University Press, Beijing. (in Chinese)
  24. Xia, Y. and Fujino, Y. (2006), "Auto-parametric vibration of a cable-stayed-beam structure under random excitation", J. Eng. Mech., 132(3), 279-286. https://doi.org/10.1061/(ASCE)0733-9399(2006)132:3(279)
  25. Xie, W.C. (2006), Dynamic Stability of Structures, Cambridge University Press, New York.
  26. Wolf, A., Swift, J.B., Swinney, H.L. and Vastano, J.A. (1985), "Determining Lyapunov exponents from a time series", Physica D: Nonlin. Phenomena, 16, 285-317. https://doi.org/10.1016/0167-2789(85)90011-9

Cited by

  1. Experimental and Numerical Analyses for Auto-Parametric Internal Resonance of a Framed Structure vol.21, pp.1, 2017, https://doi.org/10.1142/s0219455421500127
  2. Stability analysis for parametric resonances of frame structures using dynamic axis-force transfer coefficient vol.34, pp.None, 2017, https://doi.org/10.1016/j.istruc.2021.09.095