Abstract
In this paper, the finite free-form curved beam element is formulated by the isogeometric approach based on the Timoshenko Rcurved beam theory to investigate the free vibration behavior of the curved beams with arbitrary curvature. The non-uniform rational B-splines (NURBS) functions which define the geometry of the curved beam are used as the basis functions for the finite element analysis. In order to enrich the basis functions and to increase the accuracy of the solution fields, the h-, p-, and k-refinement techniques are implemented. The geometry and curvature of free-form curved beams are modelled in a unique way based on NURBS. The gap between the free vibration analysis of the curved beams with constant curvature and those with variable curvature is eliminated. All the effects of the axis extensibility, the shear deformation, and the rotary inertia are taken into consideration by the present isogeometric model. Results of the parabolic and elliptic curved beams for non-dimensional frequencies are compared with other available results in order to show the accuracy and efficiency of the present isogeometric approach. Furthermore, the free vibration analysis of the elliptic thick rings is presented. Particularly, the Tschirnhausen’s cubic curved beam is considered to study the dynamic behavior as an example of free-form curved beams.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Oh SJ, Lee BK, Lee IW (1999) Natural frequencies of non-circular arches with rotatory inertia and shear deformation. J Sound Vib 219:23–33
Oh SJ, Lee BK, Lee IW (2000) Free vibration of non-circular arches with non-uniform cross-section. Int J Solids Struct 37:4871–4891
Rossi RE, Laura PAA, Verniere PL (1989) In-plane vibrations of cantilvered non-circular arcs of non-uniform cross-section with a tip mass. J Sound Vib 129:201–213
Tseng YP, Huang CS, Lin CJ (1997) Dynamic stiffness analysis for in-plane vibrations of arches with variable curvature. J Sound Vib 207:15–31
Tseng YP, Huang CS, Kao MS (2000) In-plane vibration of laminated curved beams with variable curvature by dynamic stiffness analysis. Compos Struct 50:103–114
Huang CS, Tseng YP, Leissa AW, Nieh KY (1998) An exact solution for in-plane vibrations of an arch having variable curvature and cross section. Int J Mech Sci 40:1159–1173
Nieh KY, Huang CS, Tseng YP (2003) An analytical solution for in-plane free vibration and stability of loaded elliptic arches. Comput Struct 81:1311–1327
Wang TM, Moore JA (1973) Lowest natural extensional frequency of clamped elliptic arcs. J Sound Vib 30:1–7
Romanelli E, Laura PA (1972) Fundamental frequencies of non-circular, elastic, hinged arcs. J Sound Vib 24:17–22
Yang F, Sedaghati R, Esmailzadeh E (2008) Free in-plane vibration of general curved beams using finite element method. J Sound Vib 318:850–867
Wu JS, Chiang LK (2003) Free vibration analysis of arches using curved beam elements. Int J Numer Methods Eng 58:1907–1936
Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195
Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, New York
Reali A (2006) An isogeometric analysis approach for the study of structural vibrations. J Earthq Eng 10:1–30
Cottrell JA, Reali A, Bazilevs Y, Hughes TJR (2006) Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng 195:5257–5296
Shojaee S, Valizadeh N, Izadpanah E, Bui T, Vu TV (2012) Free vibration and bucking analysis of laminated composite plates using the NURBS-based isogeometric finite element method. Compos Struct 94:1677–1693
Thai CH, Nguyen-Xuan H, Nguyen-Thanh N, Le T-H, Nguyen-Thoi T, Rabczuk T (2012) Static, free vibration, and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach. Int J Numer Methods Eng 91:571–603
Lee SJ, Park KS (2013) Vibrations of Timoshenko beams with isogeometric approach. Appl Math Model 37:9174–9190
Nagy AP, Abdalla MM, Gürdal Z (2011) Isogeometric design of elastic arches for maximum fundamental frequency. Struct Multidisc Optim 43:135–149
Piegl L, Tiller W (1997) The NURBS book, 2nd edn. Pringer-Verlag, New York
Raveendranath P, Singh G, Rao GV (2001) A three-noded shear-flexible curved beam element based on coupled displacement field interpolations. Int J Numer Methods Eng 51:85–101
Babu CR, Prathap G (1986) A linear thick curved beam element. Int J Numer Methods Eng 23:1313–1328
Day RA, Potts DM (1990) Curved Mindlin beam and axisymmetric shell elements—a new approach. Int J Numer Methods Eng 30:1263–1274
Bouclier R, Elguedj T, Combescure A (2012) Locking free isogeometric formulations of curved thick beams. Comput Methods Appl Mech Eng 245–246:144–162
Benson DJ, Bazilevs Y, Hsu MC, Hughes TJR (2010) Isogeometric shell analysis: the Reissner–Mindlin shell. Comput Methods Appl Mech Eng 199:276–289
Sato K (1974) Free flexural vibrations of an elliptical ring in its plane. J Acoust Soc Am 57:113–115
Lawrence JD (1972) A catalog of special plane curves. Dover, New York
Acknowledgments
This research was supported by National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology through NRF2010-0019373 and 2012R1A2A1A01007405, and by Korea Ministry of Knowledge Economy under the National HRD support program for convergence information technology supervised by National IT Industry Promotion Agency through NIPA-2013-H0401-13-1003. The support is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Luu, AT., Kim, NI. & Lee, J. Isogeometric vibration analysis of free-form Timoshenko curved beams. Meccanica 50, 169–187 (2015). https://doi.org/10.1007/s11012-014-0062-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-014-0062-3