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Dynamic stiffness approach to vibration transmission within a beam structure carrying spring–mass systems

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Abstract

The dynamic stiffness method is developed for the dynamics of a beam structure carrying multiple spring−mass systems. Based on classical Bernoulli–Euler beam theory, three types of vibrations, namely, bending, longitudinal and torsional motions, are formulated in terms of dynamic stiffness matrix. Similar to finite element technique, the local dynamic stiffness matrices for individual beams and spring−mass systems are assembled into global dynamic stiffness matrices so as to address vibration transmission from machines to flexible beamlike foundations. Using our proposed method, the vibration transmission within a beam frame, carrying multiple spring−mass systems is addressed. Through numerical analysis, the calculated vibration responses agree well with those from finite element method, which demonstrates that dynamic stiffness formulation has great potential in modeling the dynamics of built-up beam structures that supports machinery, especially in characterizing vibration isolation due to wave reflections, wave conversions, and other underlying mechanisms.

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References

  • Aksencer, T., Aydogdu, M.: Vibrations of a rotating composite beam with an attached point mass. Compos. Struct. 190, 1–9 (2018)

    Article  Google Scholar 

  • Bao, G., Xu, R.: Dynamic stiffness matrix of partial-interaction composite beams. Adv. Mech. Eng. 7(3), 1687814015575990 (2015)

    Article  Google Scholar 

  • Banerjee, J.R.: Dynamic stiffness formulation for structural elements: a general approach. Comput. Struct. 63(1), 101–103 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Banerjee, J.R.: Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method. Comput. Struct. 69, 197–208 (1998)

    Article  MATH  Google Scholar 

  • Banerjee, J.R.: Dynamic stiffness matrix development and free vibration analysis of a moving beam. J. Sound Vib. 303, 135–143 (2007)

    Article  Google Scholar 

  • Bathe, K.J., Wilson, E.: Numerical Method in Finite Element Analysis. Prentice-Hall, Upper Saddle River (1976)

    MATH  Google Scholar 

  • Bondaryk, J.: Vibration of truss structures. Journal of Acoustical Society of America 102, 2167–2175 (1997)

    Article  Google Scholar 

  • Chang, C.H.: Free vibration of a simply supported beam carrying a rigid mass at the middle. J. Sound Vib. 237(4), 733–744 (2000)

    Article  Google Scholar 

  • Chen, D.W., Wu, J.S.: The exact solutions for the natural frequencies and mode shapes of non-uniform beams with multiple spring−mass systems. J. Sound Vib. 255(2), 299–322 (2002)

    Article  Google Scholar 

  • Hajhosseini, M., et al.: Vibration band gap analysis of a new periodic beam model using GQDR method. Mech. Res. Commun. 79, 43–50 (2017)

    Article  Google Scholar 

  • Karami, G., et al.: A DQEM for vibration of shear deformable nonuniform beams with general boundary conditions. Eng. Struct. 25, 1169–1178 (2003)

    Article  Google Scholar 

  • Kaya, M.O., Ozgumus, O.O.: Flexural–torsional-coupled vibration analysis of axially loaded closed-section composite Timoshenko beam by using DTM. J. Sound Vib. 306(3–5), 95–506 (2007)

    Google Scholar 

  • Lee, J.W., Lee, J.Y.: Free vibration analysis using the transfer-matrix method on a tapered beam. Comput. Struct. 164, 75–82 (2016)

    Article  Google Scholar 

  • Lin, H.-P., Chang, S.C.: Free vibration analysis of multi-span beams with intermediate flexible constraints. J. Sound Vib. 281, 155–169 (2005)

    Article  Google Scholar 

  • Li, H., et al.: Dynamic stiffness formulation for in-plane and bending vibrations of plates with two opposite edges simply supported. J. Vib. Control 24(9), 1652–1669 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Li and Hua: Dynamic stiffness analysis of laminated composite beams using trigonometric shear deformation theory. Compos. Struct. 89, 433–442 (2009)

    Article  Google Scholar 

  • Li and Hua: The effects of shear deformation on the free vibration of elastic beams with general boundary conditions. J. Mechanical Engineering Science 224, 71–84 (2010)

    Article  Google Scholar 

  • Lin, H.Y., Tsai, Y.C.: Free vibration analysis of a uniform multi-span beam carrying multiple spring−mass systems. J. Sound Vib. 302(3), 442–456 (2007)

    Article  Google Scholar 

  • Rossit, C.A., Laura, P.A.A.: Free vibrations of a cantilever beam with a spring−mass system attached to the free end. Ocean Eng. 28(7), 933–939 (2001a)

    Article  Google Scholar 

  • Rossit, C.A., Laura, P.A.A.: Transverse, normal modes of vibration of a cantilever Timoshenko beam with a mass elastically mounted at the free end. Journal of Acoustical Society of America 110(6), 2837–2840 (2001b)

    Article  Google Scholar 

  • Tang, H.B., et al.: Vibration analysis for a coupled beam-sdof system by using the recurrence equation method. J. Sound Vib. 311(3–5), 912–923 (2008)

    Article  Google Scholar 

  • Wang, J.R., et al.: Free vibration analysis of a Timoshenko beam carrying multiple spring−mass system with effects of shear deformation and rotatory inertia. Struct Eng Mech 26(1), 1–14 (2007)

    Article  Google Scholar 

  • Williams, F.W., et al.: Towards deep and simple understanding of the transcendental eigenproblem of structural vibrations. J. Sound Vib. 256(4), 681–693 (2002)

    Article  Google Scholar 

  • Wu, J.S., Chen, C.T.: A lumped-mass TMM for free vibration analysis of a multi-step Timoshenko beam carrying eccentric lumped masses with rotary inertias. J. Sound Vib. 301, 878–897 (2007)

    Article  Google Scholar 

  • Yildirim, V.: Effect of the longitudinal to transverse moduli ratio on the in-plane natural frequencies of symmetric cross-ply laminated beams by the stiffness method. Compos. Struct. 50, 319–326 (2000)

    Article  Google Scholar 

  • Yin, X., Zhang, J.: Modeling the dynamic flow-fiber interaction for microscopic biofluid systems. J. Biomech. 46, 314–318 (2013)

    Article  Google Scholar 

  • Zhi-Jing Wu and Feng-Ming Li: Vibration band-gap properties of three-dimensional Kagome Lattices using the spectral element method. J. Sound Vib. 341, 162–173 (2015)

    Article  Google Scholar 

  • Zhijing, Wu, et al.: Band-gap analysis of a novel lattice with a hierarchical periodicity using the spectral element method. J. Sound Vib. 421, 246–260 (2018)

    Article  Google Scholar 

  • Zhou, D.: Free vibration of multi-span Timoshenko beams using static Timoshenko beam functions. J. Sound Vib. 241(4), 725–734 (2001)

    Article  Google Scholar 

  • Zhou, Ding, Ji, Tianjian: Dynamic characteristics of a beam and distributed spring−mass system. Int. J. Solids Struct. 43, 5555–5569 (2006)

    Article  MATH  Google Scholar 

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Acknowledgements

The authors wish to thank High-Tech Ship Fund from the Ministry of Industry and Information Technology: Deepwater Semi-submersible Support Platform (Grant No.: 2016[546]), High Quality Brand Ship Board Machinery (Grant No.:2016[547]), The Seventh Generation of Ultra-deepwater Drilling Platform Innovation (Grant No.: 2016[24]). This work was also partially supported National Science Foundation of Jiangsu Province-Youth Fund (Grant No.: BK20170217).

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Authors and Affiliations

  1. China Ship Scientific Research Center, 222 Shanshui East Road, Wuxi, 214082, China

    Hui Li, Xue Wen Yin & Wen Wei Wu

  2. National Key Laboratory on Ship Vibration and Noise, Wuxi, 214082, China

    Hui Li, Xue Wen Yin & Wen Wei Wu

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  1. Hui Li
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  2. Xue Wen Yin
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  3. Wen Wei Wu
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Correspondence to Xue Wen Yin.

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Li, H., Yin, X.W. & Wu, W.W. Dynamic stiffness approach to vibration transmission within a beam structure carrying spring–mass systems. Int J Mech Mater Des 16, 279–288 (2020). https://doi.org/10.1007/s10999-019-09474-w

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