Isogeometric vibration analysis of free-form Timoshenko curved beams

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.

[1]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[2]  David M. Potts,et al.  CURVED MINDLIN BEAM AND AXI-SYMMETRIC SHELL ELEMENTS : A NEW APPROACH , 1990 .

[3]  J. Lawrence,et al.  A Catalog of Special Plane Curves , 2013 .

[4]  Alain Combescure,et al.  Locking free isogeometric formulations of curved thick beams , 2012 .

[5]  Ebrahim Esmailzadeh,et al.  Free in-plane vibration of general curved beams using finite element method , 2008 .

[6]  Kenzo Sato.  Free flexural vibrations of an elliptical ring in its plane , 1975 .

[7]  Kyoung Sub Park,et al.  Vibrations of Timoshenko beams with isogeometric approach , 2013 .

[8]  Arthur W. Leissa,et al.  An exact solution for in-plane vibrations of an arch having variable curvature and cross section , 1998 .

[9]  Les A. Piegl,et al.  The NURBS book (2nd ed.) , 1997 .

[10]  R. E. Rossi,et al.  In-plane vibrations of cantilevered non-circular arcs of non-uniform cross-section with a tip mass , 1989 .

[11]  Y. P. Tseng,et al.  An analytical solution for in-plane free vibration and stability of loaded elliptic arches , 2003 .

[12]  T. Q. Bui,et al.  Free vibration and buckling analysis of laminated composite plates using the NURBS-based isogeometric finite element method , 2012 .

[13]  Thomas J. R. Hughes,et al.  Isogeometric shell analysis: The Reissner-Mindlin shell , 2010 .

[14]  P. Raveendranath,et al.  A three‐noded shear‐flexible curved beam element based on coupled displacement field interpolations , 2001 .

[15]  In-Won Lee,et al.  Free vibrations of non-circular arches with non-uniform cross-section , 2000 .

[16]  Alessandro Reali,et al.  Isogeometric Analysis of Structural Vibrations , 2006 .

[17]  Jong-Shyong Wu,et al.  Free vibration analysis of arches using curved beam elements , 2003 .

[18]  Y. P. Tseng,et al.  In-plane vibration of laminated curved beams with variable curvature by dynamic stiffness analysis , 2000 .

[19]  Hung Nguyen-Xuan,et al.  Static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plates using NURBS‐based isogeometric approach , 2012 .

[20]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[21]  Alessandro Reali,et al.  AN ISO GEOMETRIC ANALYSIS APPROACH FOR THE STUDY OF STRUCTURAL VIBRATIONS , 2006 .

[22]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[23]  Mostafa Abdalla,et al.  Isogeometric design of elastic arches for maximum fundamental frequency , 2011 .

[24]  Gangan Prathap,et al.  A linear thick curved beam element , 1986 .

[25]  In-Won Lee,et al.  NATURAL FREQUENCIES OF NON-CIRCULAR ARCHES WITH ROTATARY INERTIA AND SHEAR DEFORMATION , 1999 .

[26]  J. A. Moore,et al.  Lowest natural extensional frequency of clamped elliptic arcs , 1973 .

[27]  Y. P. Tseng,et al.  DYNAMIC STIFFNESS ANALYSIS FOR IN-PLANE VIBRATIONS OF ARCHES WITH VARIABLE CURVATURE , 1997 .

[28]  P.A.A. Laura,et al.  Fundamental frequencies of non-circular, elastic, hinged arcs , 1972 .