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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Nonlinear Dynamic Response of a Stiffened Imperfect Beam under Primary Resonance Excitation

Journal of Applied Nonlinear Dynamics 11(3) (2022) 667--702 | DOI:10.5890/JAND.2022.09.010

Osama F. Abdel Aal, Mohammad Abdel Aal, Ahmad Al Qaisia

$^1$ Department of Mechatronics Engineering, University of Jordan, Amman, Jordan

$^2$ Department of Basic Sciences, Middle East University, Amman, Jordan

$^3$ Department of Mechanical Engineering, University of Jordan, Amman, Jordan

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Abstract

This work presents an investigation on the effect of an initial geometric imperfection; wavelength, amplitude, and degree of localization on the in-plane nonlinear resonance responses of an imperfect beam. The beam was modeled as an Euler Bernoulli beam resting on an elastic foundation, hinged at one end and supported by a torsional spring at the other end. The governing model accounts for the effect of the axial force induced by mid-plane stretching. The imperfection was introduced as a rise with different shapes, different amplitudes, and wavelengths. A quantitative and qualitative analysis of the steady-state responses and their stability is obtained by computing force-frequency response curves using the Harmonic Balance method $HB$ and the method of Multiple Scales $MMS$. Results have shown that the behavior of the steady-state responses may change from hardening to softening types depending on the geometrical and physical parameters of the beam under consideration. Results are presented in a dimensionless form for the steady-state forced vibration in the aid of time histories, phase planes, frequency spectrums, and Poincare maps for selected values of physical parameters. It is shown that the beam response may exhibit complicated dynamic behaviors including period multiplying and period de multiplying bifurcations, period three and period six motions, jump phenomenon, and chaos.

References

  1. [1]  Chan, H. and Liu, J. (2000), Mode localization and frequency loci veering in disordered engineering structures, Chaos, Solitons $\&$ Fractals, 11, 1493-1504.
  2. [2]  Chiba, M. and Sugimoto, T. (2003), Vibration characteristics of a cantilever plate with attached spring-mass system, Journal of Sound and Vibration, 260, 237-263.
  3. [3]  Breslavsky, I., Avramov, K., Mikhlin, Y., and Kochurov, R. (2008), Nonlinear modes of snap-through motions of a shallow arch, Journal of Sound and Vibration, 311, 297-313.
  4. [4]  Cooley, C.G. and Parker, R.G. (2014), Vibration of spinning cantilever beams with an attached rigid body undergoing bending-bending-torsional-axial motions, Journal of Applied Mechanics, 81.
  5. [5]  Ouakad, H.M. and Younis, M.I. (2011), Natural frequencies and mode shapes of initially curved carbon nanotube resonators under electric excitation, Journal of Sound and Vibration, 330, 3182-3195.
  6. [6]  Giannini, O. and Sestieri, A. (2016), Experimental characterization of veering crossing and lock-in in simple mechanical systems, Mechanical Systems and Signal Processing, 72, 846-864.
  7. [7]  Ouakad, H.M. and Younis, M.I. (2014), On using the dynamic snap-through motion of mems initially curved microbeams for filtering applications, Journal of Sound and Vibration, 333, 555-568.
  8. [8]  Leissa, A.W. (1974) On a curve veering aberration, Zeitschrift f{\"ur angewandte Mathematik und Physik ZAMP}, 25, 99-111.
  9. [9]  Perkins, N. and MoteJr, C. (1986), Comments on curve veering in eigenvalue problems, Journal of Sound and Vibration, 106, 451-463.
  10. [10]  Hodges, C. (1982), Confinement of vibration by structural irregularity, Journal of sound and vibration, 82, 411-424.
  11. [11]  Pierre, C. (1988), Mode localization and eigenvalue loci veering phenomena in disordered structures, Journal of Sound and Vibration, 126, 485-502.
  12. [12]  Triantafyllou, M. and Triantafyllou, G. (1991), Frequency coalescence and mode localization phenomena: a geometric theory, Journal of sound and vibration, 150, 485-500.
  13. [13]  Lee, S.-Y. and MoteJr, C. (1998), Traveling wave dynamics in a translating string coupled to stationary constraints: energy transfer and mode localization, Journal of Sound and Vibration, 212, 1-22.
  14. [14]  King, M.E. and Layne, P.A. (1998), Dynamics of nonlinear cyclic systems with structural irregularity, Nonlinear Dynamics, 15, 225-244.
  15. [15]  King, M. and Vakakis, A. (1995), Mode localization in a system of coupled flexible beams with geometric nonlinearities, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f{\"ur Angewandte Mathematik und Mechanik}, 75, 127-139.
  16. [16]  Chandrashaker, A., Adhikari, S., and Friswell, M. (2016), Quantification of vibration localization in periodic structures, Journal of Vibration and Acoustics, 138.
  17. [17]  Gil-Santos, E., Ramos, D., Pini, V., Calleja, M., and Tamayo, J. (2011), Exponential tuning of the coupling constant of coupled microcantilevers by modifying their separation, Applied physics letters, 98, 123108.
  18. [18]  Manav, M., Reynen, G., Sharma, M., Cretu, E., and Phani, A. (2014), Ultrasensitive resonant mems transducers with tuneable coupling, Journal of Micromechanics and Microengineering, 24, 055005.
  19. [19]  Plaut, R. and Johnson, E. (1981), The effects of initial thrust and elastic foundation on the vibration frequencies of a shallow arch, Journal of Sound and Vibration, 78, 565-571.
  20. [20]  Plaut, R. and Hsieh, J.-C. (1985), Oscillations and instability of a shallow-arch under two-frequency excitation, Journal of Sound and Vibration, 102, 189-201.
  21. [21]  Kenny, S., Pegg, N., and Taheri, F. (2000), Dynamic elastic buckling of a slender beam with geometric imperfections subject to an axial impulse. Finite elements in analysis and design, 35, 227-246.
  22. [22]  Lindberg, H.E. and Florence, A.L. (1987), Impact buckling of bars. Dynamic Pulse Buckling, pp. 11-73, Springer.
  23. [23]  Lacarbonara, W., Arafat, H.N., and Nayfeh, A.H. (2005), Non-linear interactions in imperfect beams at veering, International Journal of Non-Linear Mechanics, 40, 987-1003.
  24. [24]  Wadee, M.A. (2002), Localized buckling in sandwich struts with pre-existing delaminations and geometrical imperfections, Journal of the Mechanics and Physics of Solids, 50, 1767-1787.
  25. [25]  Hamdan, M., Abuzeid, O., and Al-Salaymeh, A. (2007), Assessment of an edge type settlement of above ground liquid storage tanks using a simple beam model. Applied mathematical modelling, 31, 2461-2474.
  26. [26]  Fung, Y. and Kaplan, A. (1952), Buckling of low arches or curved beams of small curvature.
  27. [27]  Al-Qaisia, A. and Hamdan, M. (2009), Nonlinear frequency veering in a beam resting on an elastic foundation, Journal of Vibration and Control, 15, 1627-1647.
  28. [28]  Al-Qaisia, A. and Hamdan, M. (2010), Primary resonance response of a beam with a differential edge settlement attached to an elastic foundation, Journal of Vibration and Control, 16, 853-877.
  29. [29]  Al-Qaisia, A. and Hamdan, M. (2013), On nonlinear frequency veering and mode localization of a beam with geometric imperfection resting on elastic foundation, Journal of Sound and Vibration, 332, 4641-4655.
  30. [30]  DinhDuc, N., Tuan, N.D., Tran, P., and Quan, T.Q. (2017), Nonlinear dynamic response and vibration of imperfect shear deformable functionally graded plates subjected to blast and thermal loads, Mechanics of Advanced Materials and Structures, 24, 318-329.
  31. [31]  Duc, N.D., Quan, T.Q., and Luat, V.D. (2015), Nonlinear dynamic analysis and vibration of shear deformable piezoelectric fgm double curved shallow shells under damping-thermo-electro-mechanical loads, Composite Structures, 125, 29-40.
  32. [32]  Duc, N.D., Tuan, N.D., Tran, P., Quan, T.Q., and VanThanh, N. (2019), Nonlinear dynamic response and vibration of imperfect eccentrically stiffened sandwich third-order shear deformable fgm cylindrical panels in thermal environments, Journal of Sandwich Structures $\&$ Materials, 21, 2816-2845.
  33. [33]  Duc, N.D. (2013) Nonlinear dynamic response of imperfect eccentrically stiffened fgm double curved shallow shells on elastic foundation, Composite Structures, 99, 88-96.
  34. [34]  Cong, P.H., Khanh, N.D., Khoa, N.D., and Duc, N.D. (2018), New approach to investigate nonlinear dynamic response of sandwich auxetic double curves shallow shells using tsdt, Composite Structures, 185, 455-465.
  35. [35]  Nguyen, D.D., Tran, Q.Q., and Nguyen, D.K. (2017), New approach to investigate nonlinear dynamic response and vibration of imperfect functionally graded carbon nanotube reinforced composite double curved shallow shells subjected to blast load and temperature, Aerospace Science and Technology, 71, 360-372.
  36. [36]  Duc, N.D. (2016), Nonlinear thermal dynamic analysis of eccentrically stiffened s-fgm circular cylindrical shells surrounded on elastic foundations using the reddy's third-order shear deformation shell theory, European Journal of Mechanics-A/Solids, 58, 10-30.
  37. [37]  Nayfeh, A.H. (2011), Introduction to perturbation techniques. John Wiley $\&$ Sons.
  38. [38]  Hamdan, M. and Burton, T. (1993), On the steady state response and stability of non-linear oscillators using harmonic balance, Journal of sound and vibration, 166, 255-266.
  39. [39]  Nayfeh, A.H. and Mook, D.T. (2008), Nonlinear oscillations. John Wiley $\&$ Sons.
  40. [40]  Nayfeh, A., Nayfeh, J., and Mook, D. (1992), On methods for continuous systems with quadratic and cubic nonlinearities, Nonlinear Dynamics, 3, 145-162.
  41. [41]  Nayfeh, A.H. and Lacarbonara, W. (1997), On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities. Nonlinear Dynamics, 13, 203-220.
  42. [42]  Benedettini, F. and Rega, G. (1987), Non-linear dynamics of an elastic cable under planar excitation, International Journal of non-linear mechanics, 22, 497-509.
  43. [43]  Al-Qaisia, A. and Hamdan, M. (2001), Bifurcations of approximate harmonic balance solutions and transition to chaos in an oscillator with inertial and elastic symmetric nonlinearities, Journal of sound and vibration, 244, 453-479.
  44. [44]  Nayfeh, A.H. and Balachandran, B. (2008), Applied nonlinear dynamics: analytical, computational, and experimental methods. John Wiley $\&$ Sons.
  45. [45]  Szempli{n}ska-Stupnicka, W. (1988), Bifurcations of harmonic solution leading to chaotic motion in the softening type duffing's oscillator, International Journal of Non-Linear Mechanics, 23, 257-277.
  46. [46]  Al-Noury, S. and Ali, S. (1985), Large-amplitude vibrations of parabolic cables. Journal of Sound Vibration, 101, 451-462.
  47. [47]  Nayfeh, A.H. (1986), Perturbation methods in nonlinear dynamics, Lecture Notes in Physics, pp. 238-314, Springer.

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