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Free Flexural Vibrations of Axially Loaded Timoshenko Beams with Internal Viscous Damping Using Dynamic Stiffness Formulation and Differential Transformation

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Seismic Isolation, Structural Health Monitoring, and Performance Based Seismic Design in Earthquake Engineering

Abstract

The effects of axial compressive load and internal viscous damping on the free vibration characteristics of Timoshenko beams are carried out using the dynamic stiffness formulation and the differential transformation method. The governing equations of motion are derived using the Hamilton’s principle. After the analytical solution of the equation of motion has been obtained, the dynamic stiffness method (DSM) is used and the dynamic stiffness matrix of the axially loaded Timoshenko beam with internal viscous damping is constructed to calculate natural frequencies. Moreover, an efficient mathematical technique called the differential transform method (DTM) is used to solve the governing differential equations of motion. The calculated natural frequencies of Timoshenko beams with various combinations of boundary conditions using the DSM and DTM are presented and compared with the analytical results where a very good agreement is observed.

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References

  • Banerjee JR (1997) Dynamic stiffness for structural elements: a general approach. Comput Struct 63:101–103

    Article  Google Scholar 

  • Banerjee JR (2012) Free vibration of beams carrying spring-mass systems—a dynamic stiffness approach. Comput Struct 104–105:21–26

    Article  Google Scholar 

  • Banerjee JR, Jackson DR (2013) Free vibration of a rotating tapered Rayleigh beam: a dynamic stiffness method of solution. Comput Struct 124:11–20

    Article  Google Scholar 

  • Bao-hui L, Hang-shan G, Hong-bo Z et al (2011) Free vibration analysis of multi-span pipe conveying fluid with dynamic stiffness method. Nucl Eng Des 241:666–671

    Article  Google Scholar 

  • Bozyigit B, Yesilce Y (2016) Dynamic stiffness approach and differential transformation for free vibration analysis of a moving Reddy-Bickford beam. Struct Eng Mech 58(5):847–868

    Article  Google Scholar 

  • Cai C, Zheng H, Hung KC et al (2006) Vibration analysis of a beam with an active constraining layer damping patch. Smart Mater Struct 15:147–156

    Article  Google Scholar 

  • Capsoni A, Viganò GM, Hani KB (2013) On damping effects in Timoshenko beams. Int J Mech Sci 73:27–39

    Article  Google Scholar 

  • Çatal S (2006) Analysis of free vibration of beam on elastic soil using differential transform method. Struct Eng Mech 24(1):51–62

    Article  Google Scholar 

  • Çatal S (2008) Solution of free vibration equations of beam on elastic soil by using differential transform method. Appl Math Model 32:1744–1757

    Article  Google Scholar 

  • Çatal S, Çatal HH (2006) Buckling analysis of partially embedded pile in elastic soil using differential transform method. Struct Eng Mech 24(2):247–268

    Article  MathSciNet  Google Scholar 

  • Chen WR (2014a) Parametric studies on bending vibration of axially-loaded twisted Timoshenko beams with locally distributed Kelvin-Voigt damping. Int J Mech Sci 88:61–70

    Article  Google Scholar 

  • Chen WR (2014b) Effect of local Kelvin-Voigt damping on eigenfrequencies of cantilevered twisted Timoshenko beams. Procedia Eng 79:160–165

    Article  Google Scholar 

  • Chen WR, Hsin SW, Chu TH (2013) Vibration analysis of twisted Timoshenko beams with internal Kelvin-Voigt damping. Procedia Eng 67:525–532

    Article  Google Scholar 

  • Dohnal F, Ecker H, Springer H (2008) Enhanced damping of a cantilever beam by axial parametric excitation. Arch Appl Mech 78:935–947

    Article  Google Scholar 

  • Gürgöze M, Erol H (2004) On the eigencharacteristics of multi-step beams carrying a tip mass subjected to non-homogeneous external viscous damping. J Sound Vib 272:1113–1124

    Article  Google Scholar 

  • Jun L, Hongxing H, Rongying H (2008) Dynamic stiffness analysis for free vibrations of axially loaded laminated composite beams. Comput Struct 84:87–98

    Article  Google Scholar 

  • Kaya MO, Ozgumus OO (2007) Flexural-torsional-coupled vibration analysis of axially loaded closed-section composite Timoshenko beam by using DTM. J Sound Vib 306:495–506

    Article  Google Scholar 

  • Lin SM (2014) Analytical solutions for thermoelastic vibrations of beam resonators with viscous damping in non-Fourier model. Int J Mech Sci 87:26–35

    Article  Google Scholar 

  • Matlab R2014b (2014) The MathWorks, Inc.

    Google Scholar 

  • Nefovska-Danilovic M, Petronijevic M (2015) In-plane free vibration and response analysis of isotropic rectangular plates using the dynamic stiffness method. Comput Struct 152:82–95

    Article  Google Scholar 

  • Ozgumus OO, Kaya MO (2006) Flapwise bending vibration analysis of double tapered rotating Euler-Bernoulli beam by using the differential transform method. Meccanica 41:661–670

    Article  Google Scholar 

  • Ozgumus OO, Kaya MO (2007) Energy expressions and free vibration analysis of a rotating double tapered Timoshenko beam featuring bending-torsion coupling. Int J Eng Sci 45:562–586

    Article  Google Scholar 

  • Sorrentino S, Fasana A, Marchesiello S (2007) Analysis of non-homogeneous Timoshenko beams with generalized damping distributions. J Sound Vib 304:779–792

    Article  Google Scholar 

  • Su H, Banerjee JR (2015) Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams. Comput Struct 147:107–116

    Article  Google Scholar 

  • Xie Z, Shepard WS Jr (2009) An enhanced beam model for constrained layer damping and a parameter study of damping contribution. J Sound Vib 319:1271–1284

    Article  Google Scholar 

  • Yesilce Y (2010) Differential transform method for free vibration analysis of a moving beam. Struct Eng Mech 35(5):645–658

    Article  Google Scholar 

  • Yesilce Y (2013) Determination of natural frequencies and mode shapes of axially moving Timoshenko beams with different boundary conditions using differential transform method. Adv Vib Eng 12(1):90–108

    Google Scholar 

  • Yesilce Y (2015) Differential transform method and numerical assembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and rotary inertias. Struct Eng Mech 53(3):537–573

    Article  Google Scholar 

  • Zhou JK (1986) Differential transformation and its applications for electrical circuits. Huazhong University Press, Wuhan, China

    Google Scholar 

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Authors and Affiliations

  1. Department of Civil Engineering, Dokuz Eylul University, 35160, Buca, Izmir, Turkey

    Baran Bozyigit, Yusuf Yesilce & Hikmet Huseyin Catal

Authors
  1. Baran Bozyigit
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  2. Yusuf Yesilce
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  3. Hikmet Huseyin Catal
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Corresponding author

Correspondence to Baran Bozyigit .

Editor information

Editors and Affiliations

  1. Earthquake Engineering Research Facility, Department of Civil Engineering, Ondokuz Mayıs University, Atakum, Samsun, Turkey

    Azer A. Kasimzade

  2. Kandilli Observatory and Earthquake Research Institute, Bogazici University, Istanbul, Turkey

    Erdal Şafak

  3. Department of Civil Engineering, University of British Columbia, Vancouver, British Columbia, Canada

    Carlos E. Ventura

  4. Farzad Naeim Incorporation , Irvine, California, USA

    Farzad Naeim

  5. Graduate School of Engineering, Kobe University, Kobe, Hyogo, Japan

    Yoichi Mukai

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Bozyigit, B., Yesilce, Y., Catal, H.H. (2019). Free Flexural Vibrations of Axially Loaded Timoshenko Beams with Internal Viscous Damping Using Dynamic Stiffness Formulation and Differential Transformation. In: Kasimzade, A., Şafak, E., Ventura, C., Naeim, F., Mukai, Y. (eds) Seismic Isolation, Structural Health Monitoring, and Performance Based Seismic Design in Earthquake Engineering . Springer, Cham. https://doi.org/10.1007/978-3-319-93157-9_15

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  • DOI: https://doi.org/10.1007/978-3-319-93157-9_15

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-93156-2

  • Online ISBN: 978-3-319-93157-9

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