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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Banerjee JR (1997) Dynamic stiffness for structural elements: a general approach. Comput Struct 63:101–103
Banerjee JR (2012) Free vibration of beams carrying spring-mass systems—a dynamic stiffness approach. Comput Struct 104–105:21–26
Banerjee JR, Jackson DR (2013) Free vibration of a rotating tapered Rayleigh beam: a dynamic stiffness method of solution. Comput Struct 124:11–20
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
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
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
Capsoni A, Viganò GM, Hani KB (2013) On damping effects in Timoshenko beams. Int J Mech Sci 73:27–39
Çatal S (2006) Analysis of free vibration of beam on elastic soil using differential transform method. Struct Eng Mech 24(1):51–62
Çatal S (2008) Solution of free vibration equations of beam on elastic soil by using differential transform method. Appl Math Model 32:1744–1757
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
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
Chen WR (2014b) Effect of local Kelvin-Voigt damping on eigenfrequencies of cantilevered twisted Timoshenko beams. Procedia Eng 79:160–165
Chen WR, Hsin SW, Chu TH (2013) Vibration analysis of twisted Timoshenko beams with internal Kelvin-Voigt damping. Procedia Eng 67:525–532
Dohnal F, Ecker H, Springer H (2008) Enhanced damping of a cantilever beam by axial parametric excitation. Arch Appl Mech 78:935–947
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
Jun L, Hongxing H, Rongying H (2008) Dynamic stiffness analysis for free vibrations of axially loaded laminated composite beams. Comput Struct 84:87–98
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
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
Matlab R2014b (2014) The MathWorks, Inc.
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
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
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
Sorrentino S, Fasana A, Marchesiello S (2007) Analysis of non-homogeneous Timoshenko beams with generalized damping distributions. J Sound Vib 304:779–792
Su H, Banerjee JR (2015) Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams. Comput Struct 147:107–116
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
Yesilce Y (2010) Differential transform method for free vibration analysis of a moving beam. Struct Eng Mech 35(5):645–658
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
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
Zhou JK (1986) Differential transformation and its applications for electrical circuits. Huazhong University Press, Wuhan, China
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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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