Abstract
To carry out a dynamic analysis of vibrating systems, different methods exist and numerical methods take a large place. In this study, a beam with one overhang, having a rotational restraint at one pinned end and a support of variable abscissa is investigated. Many situations are studied using the finite element method based on the Euler-Bernoulli assumptions to analyze beams vibration, without elastic restraints. The purpose of this investigation consists of treating similar beam cases with and without a rotational restraint at one node. The first step of validation is relative to the extreme situations where either the overhang or the intermediate span length is zero (span length approx. zero). In these two studied cases, the rotational restraint is not considered. It is compared to theoretical, energetic Rayleigh method results and those due to the finite element method for the simply supported beam, pinned-clamped and free-fixed beam (cantilever beam). When the rotational restraint value increases the beam is considered first pinned-pinned and then pinned-fixed behavior is obtained. The results as found are in agreement with analytical ones and after introducing the rotational restraint at the right-hand end, an intermediate behavior is initiated by varying the s/L (span length/beam length) ratio, allowing a simplified extraction of the fundamental vibration frequency for various values of Kr (rotational restraint).
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References
Bathe KJ, Wilson EL (1976) Numerical methods in finite element analysis. Prentice-Hall, Englewood Cliffs
Bathe KJ (1996) Finite element procedures. Prentice Hall, Englewood Cliffs
Wilson EL (2008) Three dimensional static and dynamic analysis of structures, 3rd edn. Computers and Structures, Inc., Berkeley
Han SM, Benaroya H, Wei T (1999) Dynamics of transversely vibrating beams: using four engineering theories. J Sound Vib 225(5):935–988
Clough RW, Penzien J (2003) Dynamics of structures, 3rd edn. Computers and Structures, Inc., Berkeley
Paz M (2000) Structural dynamics theory and computation. Chapman & Hall, London
Meenakshi Sundaram M, Ananthasuresh GK (2012) A note on the inverse mode shape problem for bars, beams, and plates. Inverse Prob Sci Eng 21:1–16
Gladwell GML (2004) Inverse problems in vibrations, 2nd edn. Kluwer Academic Publications, Dordrecht
Chu MT (1992) Inverse eigenvalue problems. SIAM Rev 40:1–39
Chu MT, Gene HG (2005) Inverse eigenvalue problems theory, algorithms, and applications. Oxford University Press, Oxford
Murphy JF (1997) Transverse vibration of a simply supported beam with symmetric overhang of arbitrary length. J Test Eval JTEVA 25(5):522–524
Falek K, Rezgui L, Chalah F, Bali A, Nechnech A (2013) Structural element vibration analysis. In: ICA2013, vol 133, no. 5, Part 2 of 2 May 2013, Montreal, Canada
Chalah-Rezgui L, Chalah F, Falek K, Bali A, Nechnech A (2013) Transverse vibration analysis of uniform beams under various ends restraints. 2013 2nd international conference on civil engineering (ICCEN 2013), Stockholm, Sweden
Wu J-S, Hsu T-F (2007) Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered. Int J Mech Sci 49:834–852
Hamdan MN, Jubran BA (1991) Free and forced vibrations of a restrained cantilever beam carrying a concentrated mass. J KAU Eng Sci 3:71–83 (1411 A.H./1991 A.D.)
Lin H-Y, Tsai Y-C (2007) Free vibration analysis of a uniform multi-span beam carrying multiple spring––mass systems. J Sound Vib 302:442–456
Darabi MA, Kazemirad S, Ghayesh MH (2012) Free vibrations of beam–mass–spring systems: analytical analysis with numerical confirmation. Acta Mech Sin 28(2):468–481. doi:10.1007/s10409-012-0010-1
Maurizi MJ, Rossi RE, Reyes JA (1976) Vibration frequencies for a uniform beam with one end spring hinged and subjected to a translational restraint at the other end. J Sound Vib 48(4):565–568
Banerjee JR (2012) Free vibration of beams carrying spring-mass systems—a dynamic stiffness approach. Comput Struct 104–105:21–26
Liu Y, Gurram CS (2009) The use of He’s variational iteration method for obtaining the free vibration of an Euler-Bernoulli beam. Math Comput Model 50:1545–1552
Lai H-Y, Hsu J-C, Chen C-K (2008) An innovative eigenvalue problem solver for free vibration of Euler–Bernoulli beam by using the Adomian decomposition method. Comput Math Appl 56:3204–3220
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Chalah-Rezgui, L., Chalah, F., Djellab, S.E., Nechnech, A., Bali, A. (2015). Free Vibration of a Beam Having a Rotational Restraint at One Pinned End and a Support of Variable Abscissa. In: Öchsner, A., Altenbach, H. (eds) Mechanical and Materials Engineering of Modern Structure and Component Design. Advanced Structured Materials, vol 70. Springer, Cham. https://doi.org/10.1007/978-3-319-19443-1_32
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DOI: https://doi.org/10.1007/978-3-319-19443-1_32
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