Abstract
In this paper, the generalized differential quadrature (GDQ) method is presented for solving the nonlinear, fully intrinsic equations of geometrically exact rotating and nonrotating beams. The fully intrinsic equations of beams involve only moments, forces, velocity and angular velocity, and in these equations, the displacements and rotations will not appear explicitly. This paper presents the generalized differential quadrature method for solution of these equations. To show the accuracy, validity and applicability of the proposed generalized differential quadrature method for solving the fully intrinsic beam equations, different cases are considered. It is found that the GDQ method gives very accurate results with very few numbers of discrete points and also has very low computational cost as compared to some other conventional numerical methods and therefore this method is very efficient, accurate and fast for solving the fully intrinsic equations.
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Bauchau O.A., Kang N.K.: A multibody formulation for helicopter structural dynamic analysis. J. Am. Helicopter Soc. 38(2), 3–14 (1993)
Simo J.C., Vu-Quoc L.: On the dynamics in space of rods undergoing large motions—a geometrically exact approach. Comput. Methods Appl. Mech. Eng. 66(2), 125–161 (1988)
Hodges D.H.: A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams. Int. J. Solids Struct. 26(11), 1253–1273 (1990)
Patil M.J., Hodges D.H., Cesnik C.E.S.: Nonlinear aeroelastic analysis of complete aircraft in subsonic flow. J. Aircr. 37(5), 753–760 (2000)
Patil M.J., Hodges D.H.: Flight dynamics of highly flexible flying wings. J. Aircr. 43(6), 1790–1799 (2006)
Hodges D.H., Shang X., Cesnik C.E.S.: Finite element solution of nonlinear intrinsic equations for curved composite beams. J. Am. Helicopter Soc. 41(4), 313–321 (1996)
Green, A.E., Laws, N.: A General Theory of Rods, Proceedings of the Royal Society of London. 293, 145–155 (1966)
Reissner E.: On one-dimensional large-displacement finite-strain beam theory. Stud. Appl. Math. 52(2), 87–95 (1973)
Hegemier G.A., Nair S.: A nonlinear dynamical theory for heterogeneous, anisotropic, elasticrods. AIAA J. 15(1), 8–15 (1977)
Hodges D.H.: Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams. AIAA J. 41(6), 1131–1137 (2003)
Palacios, R., Cesnik, C.E.S.: Structural models for flight dynamic analysis of very flexible aircraft. In: Paper Presented at the 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, California, USA, 4–7 May 2009
Traugott J.P., Patil M.J., Holzapfel F.: Nonlinear modeling of integrally actuated beams. Aerosp. Sci. Technol. 10(6), 509–518 (2006)
Patil M.J., Althoff M.: Energy-consistent, Galerkin approach for the nonlinear dynamics of beams using intrinsic equations. J. Vib. Control 17(11), 1748–1758 (2011)
Patil M.J., Hodges D.H.: Variable-order finite elements for nonlinear, fully intrinsic beam equations. J. Mech. Mater. Struct. 6(1-4), 479–493 (2011)
Sotoudeh Z., Hodges D.H.: Incremental method for structural analysis of joined-wing aircraft. J. Aircr. 48(5), 1588–1601 (2011)
Khaneh Masjedi P., Ovesy H.R.: Chebyshev collocation method for static intrinsic equations of geometrically exact beams. Int. J. Solids Struct. 54(0), 183–191 (2015)
Khaneh Masjedi, P., Ovesy, H.: Large deflection analysis of geometrically exact spatial beams under conservative and nonconservative loads using intrinsic equations. Acta Mech. 226(6), 1689–1706 (2014)
Bellman R., Casti J.: Differential quadrature and long-term integration. J. Math. Anal. Appl. 34(2), 235–238 (1971)
Shu C., Richards B.E.: Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 15(7), 791–798 (1992)
Shu C., Richard B.E.: Parallel simulation of incompressible viscous flows by generalized differential quadrature. Comput. Syst. Eng. 3(1), 271–281 (1992)
Shu C., Chew Y.T., Richards B.E.: Generalized differential and integral quadrature and their application to solve boundary layer equations. Int. J. Numer. Methods Fluids 21(9), 723–733 (1995)
Shu C., Chew Y.T., Khoo B.C., Yeo K.S.: Solutions of three-dimensional boundary layer equations by global methods of generalized differential-integral quadrature. Int. J. Numer. Methods Heat Fluid Flow 6(2), 61–75 (1996)
Du H., Lim M.K., Lin R.M.: Application of generalized differential quadrature method to structural problems. Int. J. Numer. Methods Eng. 37(11), 1881–1896 (1994)
Du H., Lim M.K., Lin R.M.: Application of generalized differential quadrature to vibration analysis. J. Sound Vib. 181(2), 279–293 (1995)
Laura, P.A.A., Gutierrez, R.H.: Analysis of vibrating Timoshenko beams using the method of differential quadrature. Shock Vib. 1(1) 89–93 (1993)
Laura P.A.A., Gutierrez R.H.: Analysis of vibrating rectangular plates with non-uniform boundary conditions by using the differential quadrature method. J. Sound Vib. 173(5), 702–706 (1994)
Lin R.M., Lim M.K., Du H.: Large deflection analysis of plates under thermal loading. Comput. Methods Appl. Mech. Eng. 117(3), 381–390 (1994)
Du H., Liew K.M., Lim M.K.: Generalized differential quadrature method for buckling analysis. J. Eng. Mech. 122(2), 95–100 (1996)
Bert C.W., Wang X., Striz A.G.: Static and free vibrational analysis of beams and plates by differential quadrature method. Acta Mech. 102(1–4), 11–24 (1994)
Lin R.M., Lim M.K., Du H.: Deflection of plates with nonlinear boundary supports using generalized differential quadrature. Comput. Struct. 53(4), 993–999 (1994)
Marzani A., Tornabene F., Viola E.: Nonconservative stability problems via generalized differential quadrature method. J. Sound Vib. 315(1–2), 176–196 (2008)
Lal R., Saini R.: On the use of GDQ for vibration characteristic of non-homogeneous orthotropic rectangular plates of bilinearly varying thickness. Acta Mech. 226(5), 1605–1620 (2015)
Bert C.W., Malik M.: Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49(1), 1–28 (1996)
Patil, M., Johnson, E.: Cross-sectional analysis of anisotropic, thin-walled, closed-section beams with embedded strain actuation. In: 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics andMaterials Conference (Reston, Va.) 2005. American Institute of Aeronautics and Astronautics
Volovoi V.V., Hodges D.H.: Theory of anisotropic thin-walled beams. J. Appl. Mech. 67(3), 453–459 (2000)
Cesnik C.E.S., Hodges D.H.: VABS: a new concept for composite rotor blade cross-sectional modeling. J. Am. Helicopter Soc. 42(1), 27–38 (1997)
Yu W., Hodges D.H., Volovoi V., Cesnik C.E.S.: On Timoshenko-like modeling of initially curved and twisted composite beams. Int. J. Solids Struct. 39(19), 5101–5121 (2002)
Sotoudeh Z., Hodges D.H., Chang C.-S.: Validation studies for aeroelastic trim and stability of highly flexible aircraft. J. Aircr. 47(4), 1240–1247 (2010)
Shu C.: Differential Quadrature and Its Application in Engineering. Springer, Berlin (2000)
Simitses G.J., Hodges D.H.: Fundamentals of Structural Stability. Butterworth-Heinemann, Oxford (2006)
Kondoh K., Atluri S.N.: Large-deformation, elasto-plastic analysis of frames under nonconservative loading, using explicitly derived tangent stiffnesses based on assumed stresses. Comput. Mech. 2(1), 1–25 (1987)
Wright A.D., Smith C.E., Thresher R.W., Wang J.L.C.: Vibration modes of centrifugally stiffened beams. J. Appl. Mech. 49(1), 197–202 (1982)
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Amoozgar, M.R., Shahverdi, H. Analysis of nonlinear fully intrinsic equations of geometrically exact beams using generalized differential quadrature method. Acta Mech 227, 1265–1277 (2016). https://doi.org/10.1007/s00707-015-1528-7
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DOI: https://doi.org/10.1007/s00707-015-1528-7