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Non-planar vibrations of beams with geometric imperfections and hysteretic boundary conditions

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Abstract

The non-planar vibration response of a beam with initial geometric imperfections is investigated through the development of a geometrically nonlinear beam model. The equations of motion are derived through the application of Lagrange’s equation and accounting for the interaction of the initial deviations from straightness and the transverse bending deformations. The experimental frequency response of a nuclear fuel rod supported by two prototypical spacer grids is described by the response in the driven direction and the response in the non-driven direction. The beam model also includes a hysteretic restoring moment at the ends to represent the constraint of the spacer grid and the associated reduction in resonant frequency as the applied force is increased. A pseudo-arc-length continuation algorithm is used to solve the equations of motions and calculate the periodic solution. Comparison between the nonlinear beam model response and the experimental data shows good agreement around the resonant frequency of the first bending mode for both the driven and companion modes.

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References

  1. Haight, E.C., King, W.W.: Stability of nonlinear oscillations of an elastic rod. J Acoust Soc Am 52(3B), 899–911 (1972)

    Article  MATH  Google Scholar 

  2. Ho, C.H., Scott, R.A., Eisley, J.G.: Non-planar, non-linear oscillations of a beam—I. Forced motions. Int. J. Non-Linear Mech. 10(2), 113–127 (1975)

    Article  MATH  Google Scholar 

  3. Crespo da Silva, M.R.M., Glynn, C.C.: Nonlinear flexural-flexural-torsional dynamics of inextensional beams. I. Equations of Motion. J. Struct. Mech. 6(4), 437–448 (1978)

    Article  Google Scholar 

  4. Crespo da Silva, M.R.M., Glynn, C.C.: Nonlinear flexural-flexural-torsional dynamics of inextensional beams. II. Forced motions. J. Struct. Mech. 6(4), 449–461 (1978)

    Article  Google Scholar 

  5. Crespo Da Silva, M.R.M., Zaretzky, C.L.: Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics. I: primary resonance. Nonlinear Dyn. 5(1), 3–23 (1994)

    Article  Google Scholar 

  6. Zaretzky, C.L., Crespo da Silva, M.R.M.: Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics. II: Combination resonance. Nonlinear Dyn. 5(2), 161–180 (1994)

    Article  Google Scholar 

  7. Carvalho, E.C., Gonçalves, P.B., Rega, G.: Multiple internal resonances and nonplanar dynamics of a cruciform beam with low torsional stiffness. Int. J. Solids Struct. 121, 117–134 (2017)

    Article  Google Scholar 

  8. Aghababaei, O., Nahvi, H., Ziaei-Rad, S.: Non-linear non-planar vibrations of geometrically imperfect inextensional beams, Part I: equations of motion and experimental validation. Int. J. Non-Linear Mech. 44(2), 147–160 (2009)

    Article  MATH  Google Scholar 

  9. Zaretzky, C.L., da Silva, M.R.M.C.: Experimental investigation of non-linear modal coupling in the response of cantilever beams. J. Sound Vib. 174(2), 145–167 (1994)

    Article  Google Scholar 

  10. Ferrari, G., et al.: Nonlinear vibrations of a nuclear fuel rod supported by spacer grids. Nucl. Eng. Des. 361, 110503 (2020)

    Article  Google Scholar 

  11. Balasubramanian, P., et al.: Nonlinear vibrations of beams with bilinear hysteresis at supports: interpretation of experimental results. J. Sound Vib. 499, 115998 (2021)

    Article  Google Scholar 

  12. Painter, B., Ferrari, G., Amabili, M.: Nonlinear vibrations of beams with Bouc-Wen hysteretic boundary conditions. Nonlinear Dyn. 108(4), 2903–2916 (2022)

    Article  Google Scholar 

  13. Ferrari, G., Karazis, K., Amabili, M.: Experiments on the localized interaction at the interface fuel rod/spacer grid in pressurized water reactors. Nucl. Eng. Des. 399, 111998 (2022)

    Article  Google Scholar 

  14. Ferrari, G., et al.: Nonlinear vibrations of a 3 × 3 reduced scale PWR fuel assembly supported by spacer grids. Nucl. Eng. Des. 364, 110674 (2020)

    Article  Google Scholar 

  15. Catterou, T., et al.: Non-linear damping of sodium fast reactor fuel pins: Experimental analysis and numerical modelisation. Nucl. Eng. Des. 364, 110643 (2020)

    Article  Google Scholar 

  16. Wen, Y.Y.-K.: Method for random vibration of hysteretic systems. ASCE J Eng Mech Div 102(2), 249–263 (1976)

    Article  Google Scholar 

  17. Wolfram Research, Inc., Mathematica, Version 13.2, Champaign, Illinois (2022)

  18. Doedel, E.J., Oldeman, B.E.: AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations. Concordia University, Montreal, Canada (2012)

    Google Scholar 

  19. Pai, P.F., Nayfeh, A.H.: Non-linear non-planar oscillations of a cantilever beam under lateral base excitations. Int. J. Non-Linear Mech. 25(5), 455–474 (1990)

    Article  Google Scholar 

  20. Aghababaei, O., Nahvi, H., Ziaei-Rad, S.: Non-linear non-planar vibrations of geometrically imperfect inextensional beams. Part II—Bifurcation analysis under base excitations. Int. J. Non-Linear Mech. 44(2), 161–179 (2009)

    Article  MATH  Google Scholar 

  21. Ferrari, G., et al.: Non-linear vibrations of nuclear fuel rods. Nucl. Eng. Des. 338, 269–283 (2018)

    Article  Google Scholar 

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Acknowledgements

The authors acknowledge the financial assistance of the NSERC CRDPJ 530933-18 Grant and the technical support of Framatome Canada Ltd.

Funding

The work was funded through the Natural Sciences and Engineering Research Council of Canada Grant CRDPJ 530933-18.

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Authors and Affiliations

  1. Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, H3A 0C3, Canada

    Brian Painter & Marco Amabili

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  1. Brian Painter
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  2. Marco Amabili
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Contributions

BP contributed to the conceptualization, data curation, formal analysis, investigation, methodology, software, visualization, writing—original draft, and writing—review and editing. MA contributed to conceptualization, formal analysis, funding acquisition, investigation, project administration, supervision, writing—original draft, and writing—review and editing.

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Correspondence to Marco Amabili.

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Appendix A: Convergence study

Appendix A: Convergence study

A convergence study was made to validate the number of terms in the expansions for the axial and two transverse displacement fields. The nonlinear frequency response was checked to confirm that the model response did not change as more terms were added to the expansions. The nonlinear response is of primary interest due to the sensitivity of the hysteresis model to rotations at the ends of the beam. In the first study, the number of terms in the transverse displacement expansions is M = 7, M = 11, M = 13, and M = 15. In each case, the number of terms in the expansion for the axial displacement is kept equal to the number of terms in the expansions for the transverse displacements. The damping ratio is also kept at a constant value of 1%. The model response for an applied force of 0.1 N and 1.0 N is shown in Fig. 10, and the corresponding resonant frequencies are provided in Table 4. The frequency of the resonance peak of the response changes by less than 0.5% between M = 13 and M = 15.

Fig. 10
figure 10

Convergence of amplitude frequency response curves for harmonic excitations of f = 0.1 N and f = 1.0 N. Solid line, M = 7; dashed line, M = 11; (*), M = 13; ( +), M = 15

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Table 4 Convergence of resonant frequencies, in rad/s, for harmonic excitations of f = 0.1 N and f = 1.0 N
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In the second study, the number of terms in the expansion for the axial displacement is varied while keeping the number of terms in the transverse displacement expansions constant at M = 15. The result was that, when N is greater than 3, the solution is insensitive to additional terms. Therefore, the numerical model with M = 15 and N = 15 is determined to be converged.

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Painter, B., Amabili, M. Non-planar vibrations of beams with geometric imperfections and hysteretic boundary conditions. Nonlinear Dyn 111, 19749–19761 (2023). https://doi.org/10.1007/s11071-023-08916-7

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