Abstract
This study analyzes the vibro-impact behavior of two adjacent cantilever beams subjected to vibration generated by applying harmonic excitation to their rigid base. For the investigation, a dynamic model of two flexible beams that can collide at arbitrary positions along the beam length is proposed. The vibro-impact mechanism of the two flexible beams was considered using the penalty method, and the equations of motion for the proposed model were derived. After discretizing the derived equations of motion using the finite element method, a numerical analysis was implemented to calculate the dynamic responses using the generalized-α time integration method. The contact force due to the impact between the two beams was calculated by applying the well-known penalty method. The calculated contact forces at the impact positions were applied to the discretized equations of motion for response computations. To verify the results, the dynamic responses derived by numerical analysis were compared with the experimentally measured responses. Using the numerical analysis of the proposed dynamics model, the effect of vibro-impact on the dynamic responses was evaluated. Moreover, the relationship between the probability of vibro-impact occurrence and ratio of natural frequency to excitation frequency was examined. Based on the evaluation, a design guide that minimizes the probability of vibro-impact when two adjacent beams are subjected to vibration generated by applying harmonic excitation to their rigid base is presented.
Similar content being viewed by others
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Trapp, M., Chen, F.: Automotive buzz, squeak and rattle: Mechanisms, analysis evaluation and prevention. Elsevier, Amsterdam (2011)
Babitsky, V.I.: Theory of vibro-impact systems and applications. Springer, Berlin (1998)
Jalali, H., Ahmadian, H., Pourahmadian, F.: Identification of micro-vibro-impacts at boundary condition of a nonlinear beam. Mech. Syst. Sig. Process. 25, 1073–1085 (2011)
Vyasarayani, C.P., McPhee, J., Birkett, S.: Modeling impacts between a continuous system and a rigid obstacle using coefficient of restitution. ASME J. Appl. Mech. 77, 021008 (2010)
Kurt, M., Chen, H., Lee, Y.S., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Nonlinear system identification of the dynamics of a vibro-impact beam: numerical results. Arch. Appl. Mech. 82, 1461–1479 (2012)
Andreaus, U., Baragatti, P., Placidi, L.: Experimental and numerical investigations of the responses of a cantilever beam possibly contacting a deformable and dissipative obstacle under harmonic excitation. Int. J. Non-Linear Mech. 80, 96–106 (2016)
Fegelman, K.J.L., Grosh, K.: Dynamics of a flexible beam contacting a linear spring at low frequency excitation: Experiment and analysis. ASME J. Vib. Acoust. 124, 237–249 (2002)
van de Wouw, N., de Kraker, A., van Campen, D.H., Nijmeijer, H.: Non-linear dynamics of a stochastically excited beam system with impact. Int. J. Non-Linear Mech. 38, 767–779 (2003)
Wei, H., Li, G., Guo, P., Zhao, J.: Effect of method type on the response of continuum vibro-impact. Shock Vib. 2019, 2718502 (2019)
Bazrafshan, M., Ahmadian, H., Jalali, H.: Modeling the interaction between contact mechanisms in normal and tangential directions. Int. J. Non-Linear Mech. 58, 111–119 (2014)
Elmegård, M., Krauskopf, B., Osinga, H.M., Starke, J., Thomsen, J.J.: Bifurcation analysis of a smoothed model of a forced impacting beam and comparison with an experiment. Nonlinear Dyn. 77, 951–966 (2014)
Gandhi, P.S., Vyas, V.: On the dynamics of tapered vibro-impacting cantilever with tip mass. J. Mech. Sci. Technol. 31, 63–73 (2017)
Duan, Y., Zhang, D., Hong, J.: Global impact dynamic modeling and verification of a flexible beam with large overall motion. Adv. Mech. Eng. 2013, 362317 (2013)
Abdul Azeez, M.F., Vakakis, A.F.: Numerical and experimental analysis of a continuous overhung rotor undergoing vibro-impacts. Int. J. Non-Linear Mech. 34, 415–435 (1999)
Krishna, I.R.P., Padmanabhan, C.: Experimental and numerical investigations of impacting cantilever beams part 1: first mode response. Nonlinear Dyn. 67, 1985–2000 (2012)
Krishna, I.R.P., Padmanabhan, C.: Experimental and numerical investigation of impacting cantilever beams: second mode response. Int. J. Mech. Sci. 92, 187–193 (2015)
Ervin, E.K.: Vibro-impact behavior of two orthogonal beams. J. Eng. Mech. 135, 529–537 (2009)
Long, X., Liu, J., Meng, G.: Nonlinear dynamics of two harmonically excited elastic structures with impact interaction. J. Sound Vib. 333, 1430–1441 (2014)
Ma, H., Xie, F., Nai, H., Wen, B.: Vibration characteristics analysis of rotating shrouded blades with impacts. J. Sound Vib. 378, 92–108 (2016)
Xie, F., Ma, H., Cui, C., Wen, B.: Vibration response comparison of twisted shrouded blades using different impact models. J. Sound Vib. 397, 171–191 (2017)
Cui, C., Ma, H., Jin, Y., Xie, F., Yang, T. Liu, S.: Numerical and experimental investigation on the vibro-impact responses analysis of shrouded blade. J. Low Freq. Noise Vibr. Act. Control 38, 1188–1201 (2019)
Vijayan, K., Friswell, M.I., Haddad Khodaparast, H., Adhikari, S.: Non-linear energy harvesting from coupled impacting beams. Int. J. Mech. Sci. 96–97, 101–109 (2015)
Fu, Y., Ouyang, H., Davis, R.B.: Triboelectric energy harvesting from the vibro-impact of three cantilevered beams. Mech. Syst. Sig. Process. 121, 509–531 (2019)
Li, W., Wierschem, N.E., Li, X., Yang, T., Brennan, M.J.: Numerical study of a symmetric single-sided vibro-impact nonlinear energy sink for rapid response reduction of a cantilever beam. Nonlinear Dyn. 100, 951–971 (2020)
Li, H., Touzé, C., Pelat, A., Gautier, F., Kong, X.: A vibro-impact acoustic black hole for passive damping of flexural beam vibrations. J. Sound Vib. 450, 28–46 (2019)
Hunt, K.H., Crossley, F.R.E.: Coefficient of restitution interpreted as damping in vibroimpact. ASME J. Appl. Mech. 42, 440–445 (1975)
Machado, M., Moreira, P., Flores, P., Lankarani, H.M.: Compliant contact force models in multibody dynamics: evolution of the Hertz contact theory. Mech. Mach. Theory 53, 99–121 (2012)
Alves, J., Peixinho, N., da Silva, M.T., Flores, P., Lankarani, H.M.: A comparative study of the viscoelastic constitutive models for frictionless contact interfaces in solids. Mech. Mach. Theory 85, 172–188 (2015)
Chung, J., Hulbert, G.M.: A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method. ASME J. Appl. Mech. 60, 371–375 (1993)
Acknowledgements
This work was supported by a grant from the National Research Foundation of Korea (NRF), funded by the Korean government (MEST) (NRF-2021R1A2C2007979).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sim, W., Lee, B., Kim, D.J. et al. Vibro-impact analysis of two adjacent cantilever beams. Nonlinear Dyn 108, 987–1004 (2022). https://doi.org/10.1007/s11071-022-07246-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-022-07246-4