Abstract
Using von Kármán nonlinear beam theory, Hamilton’s principle and D’Alembert principle, the coupled nonlinear dynamic equations of the beam, moving vehicle and tuned mass damper (TMD) are established. The partial differential equation of the beam is transformed into ordinary differential equations of multimode using the Galerkin method. The viscous damping and internal damping of the beam are considered according to the proportional damping and Kelvin–Voigt model, respectively. Based on the Newmark conjunction with Newton–Raphson method, the optimum stiffness, damping and location parameters of the TMD are given for linear and nonlinear beams under the fixed load and moving vehicle. The classical control schemes (Den Hartog, Warburton) and numerical methods are compared. Also, taking the energy dissipation ratio as the control objective, the present results are verified by those of the previous literature.
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References
Basta, E.E., Ghommem, M., Emam, S.A.: Vibration suppression and optimization of conserved-mass metamaterial beam. Int. J. Non-Linear Mech. 120, 103360 (2020)
Kaveh, A., Fahimi Farzam, M., Hojat Jalali, H., Maroofifiazar, R.: Robust optimum design of a tuned mass damper inerter. Acta Mech. 231, 3871–3896 (2020)
Sanches, L., Guimarães, T.A.M., Marques, F.D.: Nonlinear energy sink to enhance the landing gear shimmy performance. Acta Mech. 232, 2605–2622 (2021)
Wang, G.X., Ding, H.: Mass design of nonlinear energy sinks. Eng. Struct. 250, 113438 (2022)
Philip, R., Santhosh, B., Balaram, B.: Analytical and numerical investigations on inerter-based NES absorber system with nonlinear damping. Acta Mech. 233, 4365–4383 (2022)
Yau, J.D., Yang, Y.B.: Vibration reduction for cable-stayed bridges traveled by high-speed trains. Finite Elem. Anal. Des. 40, 341–359 (2004)
Samani, F.S., Pellicano, F.: Vibration reduction on beams subjected to moving loads using linear and nonlinear dynamic absorbers. J. Sound Vib. 325, 742–754 (2009)
Luu, M., Zabel, V., Könke, C.: An optimization method of multi-resonant response of high-speed train bridges using TMDs. Finite Elem. Anal. Des. 53, 13–23 (2012)
Adam, C., Lorenzo, S.D., Failla, G., Pirrotta, A.: On the moving load problem in beam structures equipped with tuned mass dampers. Meccanica 52, 3101–3115 (2017)
Wang, D., Wu, C.Q., Zhang, Y.S.: Study on vertical vibration control of long-span steel footbridge with tuned mass dampers under pedestrian excitation. J. Constr. Steel Res. 154, 84–98 (2019)
Bai, X.Y., Liang, Q.G., Huo, L.S.: Vibration control of beam-model using tuned inerter enhanced TMD. J. Sound Vib. 510, 116304 (2021)
Kani, M., Khadem, S.E., Pashaei, M.H.: Vibration control of a nonlinear beam with a nonlinear energy sink. Nonlinear Dyn. 83, 1–22 (2016)
Wang, Y.R., Feng, C.K., Chen, S.Y.: Damping effects of linear and nonlinear tuned mass dampers on nonlinear hinged-hinged beam. J. Sound Vib. 430, 150–173 (2018)
Xiong, X., Wang, Y., Li, J.Q., Li, F.M.: Internal resonance analysis of bio-inspired X-shaped structure with nonlinear vibration absorber. Mech. Syst. Sig. Process. 185, 109809 (2023)
Sheng, G.G., Wang, X.: Nonlinear vibration of FG beams subjected to parametric and external excitations. Eur. J. Mech-A /Solids 71, 224–234 (2018)
Sheng, G.G., Wang, X.: Nonlinear forced vibration of functionally graded Timoshenko microbeams with thermal effect and parametric excitation. Int. J. Mech. Sci. 155, 405–416 (2019)
Şimşek, M.: Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. Int. J. Eng. Sci. 105, 12–27 (2016)
Ghayesh, M.H.: Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams. Appl. Math. Modell. 59, 583–596 (2018)
Ke, L.L., Yang, J., Kitipornchai, S.: An analytical study on the nonlinear vibration of functionally graded beams. Meccanica 45, 743–752 (2010)
Ghayesh, M.H.: Asymmetric viscoelastic nonlinear vibrations of imperfect AFG beams. Appl. Acoust. 154, 121–128 (2019)
Şimşek, M., Kocatürk, T.: Nonlinear dynamic analysis of an eccentrically prestressed damped beam under a concentrated moving harmonic load. J. Sound Vib. 320, 235–253 (2009)
Chen, H.Y., Ding, H., Li, S.H., Chen, L.Q.: Convergent term of the Galerkin truncation for dynamic response of sandwich beams on nonlinear foundations. J. Sound Vib. 483, 115514 (2020)
Chen, H.Y., Ding, H., Li, S.H., Chen, L.Q.: The scheme to determine the convergence term of the Galerkin method for dynamic analysis of sandwich plates on nonlinear foundations. Acta Mech. Solida Sin. 34(1), 1–11 (2021)
Miguel Leandro, F.F., Lopez Rafael, H., Torii André, J., Miguel, L.F.F., Beck, A.: Robust design optimization of TMDs in vehicle-bridge coupled vibration problems. Eng. Struct. 126, 703–711 (2016)
Den Hartog, J.P.: Mechanical Vibration. McGraw–Hill, New York (1956)
Warburton, G.B.: Optimum absorbers parameters for various combinations of response and excitation. Earthq. Eng. Struct. Dyn. 10, 381–401 (1982)
Rana, R., Soong, T.T.: Parametric study and simplified design of tuned mass dampers. Eng. Struct. 20, 193–204 (1998)
Law, S.S., Bu, J.Q., Zhu, X.Q.: Vehicle axle loads identification using finite element method. Eng. Struct. 26, 1143–1153 (2004)
Nguyen, D.K., Nguyen, Q.H., Tran, T.T., Bui, V.T.: Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load. Acta Mech. 228, 141–155 (2017)
Jafari, P., Kiani, Y.: A four-variable shear and normal deformable quasi-3D beam model to analyze the free and forced vibrations of FG-GPLRC beams under moving load. Acta Mech. 233, 2797–2814 (2022)
Zhu, D.Y., Zhang, Y.H., Ouyang, H.: A linear complementarity method for dynamic analysis of bridges under moving vehicles considering separation and surface roughness. Comput. Struct. 154, 135–144 (2015)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant Nos. 12102066), Natural Science Foundation of Hunan Province China (2022JJ40465). Very thanks to reviewers for their comments on the paper.
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Sheng, G.G., Han, Y., Zhang, Z. et al. Control of nonlinear vibration of beams subjected to moving loads using tuned mass dampers. Acta Mech 234, 3019–3036 (2023). https://doi.org/10.1007/s00707-023-03544-z
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DOI: https://doi.org/10.1007/s00707-023-03544-z