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Year 2009, Volume: 1 Issue: 4, 1 - 25, 01.12.2009

C. H. W. Ng Y. B. Zhao Y. Xiang G. W. Wei

Abstract

References

  • [1] R.E. Bellman, J. Casti, ‘Differential quadrature and long term integration’, J. Math Anal. Appl. 34, 235–238 (1971).
  • [2] R. Bellman, B.G. Kashef and J. Casti, ‘Differential quadrature: a technique for the rapid solution of non–linear partial differential equations’, J. Comp. Phys. 10, 40–52 (1972).
  • [3] R.E. Bellman, B.G. Kashef, ‘Solution of the partial differential equation of the HodgkinsHuley model using differential quadrature’, Math. BioSci. 19, 1–8 (1974).
  • [4] G. Naadimuthu, R.E. Bellman, M. Wang, E.S. Lee, ‘Differntial quadrature and partial differential equations: some numerical results’, J. Math Anal. Appl. 98, 220–235 (1984).
  • [5] J.O. Mingle, ‘The method of differntial quadrature for transient nonlinear diffusion’, J. Math Anal. Appl. 60, 559–569 (1977).
  • [6] F. Civan, C.M. Sliepcevich, ‘Application of differential quadrature to transport processes’, J. Math Anal. Appl. 93, 206–221 (1983).
  • [7] F. Civan, C.M. Sliepcevich, ‘Solution of the poisson equtaion by differential quadrature’, Int. J. Num. Meth. Engng. 19, 711–724 (1983).
  • [8] F. Civan, C.M. Sliepcevich, ‘Differential quadrature multi-dimensional problems’, J. Math Anal. Appl. 101, 423–443 (1984).
  • [9] C.W. Bert, S.K. Jang and A.G. Striz, ‘Two new approximate methods for analyzing free vibration of structural components’, AIAA J. 26, 612–618 (1988).
  • [10] S.K. Jang, C.W. Bert and A.G. Striz, ‘Application of differential quadrature to static analysis of structural problems’, Int. J. Num. Meth. Engng. 28, 561–577 (1989).
  • [11] A.G. Striz, S.K. Jang, C.W. Bert, ‘Nonlinear bending analysis of thin circular plates by differential quadrature’, Thin-Walled Structures. 6, 51–62 (1988).
  • [12] C.W. Bert, S.K. Jang, A.G. Striz, ‘Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature’, Comput. Mech.. 5, 217–226 (1989).
  • [13] P.A.A. Laura, H. Gutierrez, ‘Analysis of vibrating rectangular plates with non-uniform boundary conditions by using the differential quadrature method’, J. Sound Vibr. 173, 703– 706 (1994).
  • [14] C. Shu and H. Du, ‘Implementation of clamped and simple supported boundary conditions in the GDQ free vibration analysis of beams and plates’, Int. J. Solid Struct. 34, 819–835 (1997).
  • [15] C. Shu and H. Du, ‘A generalizes approach for implementing general boundary conditions in GDQ free vibration analysis of plate’, Int. J. Solids Struct. 34, 837–846 (1997).
  • [16] C. Shu and C.M. Wang, ‘Treatment of mixed and non–uniform boundary conditions in GDQ vibration analysis of rectangular plate’, JEST. 21, 125–134 (1999).
  • [17] C.P. Wu and C.Y. Lee, ‘Differential quadrature solution for free vibration analysis of laminated conical shells with variable stiffness’, Int. J. Mech. Sci. 43, 1853–1869 (2001).
  • [18] C. Shu, ‘Analysis of elliptical waveguides by differential quadrature method’, IEEE trans. Microw. Theory and techn. 48, No 2, 319–322 (2000).
  • [19] T.Y. Wu, G.R. Liu, ‘The generalized differential quadrature rule for fourth-order differential equations’, Int. J. Num. Meth. Engng. 50, 1907–1929 (2001).
  • [20] G.R. Liu, T.Y. Wu, ‘In-plane vibration analysis of circular arches by the generalized differential quadrature rule’, Int. J. Mech. Sci. 43, 2597–2611 (2001).
  • [21] T.Y. Wu, G.R. Liu, ‘Application of generalized differential quadrature rule to sixthorder differential equations’, Commun. Numr. Meth. Engng. 16, 777–784 (2000).
  • [22] C. Shu, Differential quadrature and its application in Engineering, Spinger, New York, 2000.
  • [23] C. Shu and B.E. Richard, ‘Application of generalized differential quadrature to solve two– dimensions incompressible Navier–stress equations’, Int. J. Num. Meth. Fluids. 15, 791–798 (1992).
  • [24] C. Shu, Y.T. Chew, ‘On the equivalence of generalized differential quadrature and highest order finite difference scheme’, Comput. Methods Appl. Mech. Engng. 155, 249-260 (1998).
  • [25] G.W. Wei, ‘Discrete singular convolution for solution of the Fokker–Planck equations’, J. Chem. Phys. 110, 8930–8942 (1999).
  • [26] G.W. Wei, ‘A unified approach for solving the Fokker-Planck equation’, J. Phys. A, Math. Gen. 33, 4935-4953 (2000).
  • [27] O. Civalek, ‘Nonlinear analysis of thin rectangular plates on Winkler-Pasternak elastic foundations by DSC-HDQ methods’, Appl. Math. Model.. 31, 606-624 (2007).
  • [28] O. Civalek, ‘Linear vibration analysis of isotropic conical shells by discrete singular convolution (DSC)’, Struct. Engng. Mech.. 25, 127-130 (2007).
  • [29] O. Civalek, ‘Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: discrete singular convolution (DSC) approach’, J. Comput. Appl. Math.. 205, 251271 (2007).
  • [30] O. Civalek, ‘Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method’, Int. J. Mech. Sci.. 49, 752765 (2007).
  • [31] O. Civalek, ‘Free vibration analysis of composite conical shells using the discrete singular convolution algorithm’, Steel and Composite Struct.. 6, 353-366 (2006).
  • [32] O. Civalek, ‘An efficient method for free vibration analysis of rotating truncated conical shells’, Int. J. Press. Vess. Piping. 83, 1-12 (2006).
  • [33] O. Civalek, ‘The determination of frequencies of laminated conical shells via the discrete singular convolution method’, J. Mech. Mater. Struct.. 1, 165-192 (2006).
  • [34] O. Civalek and M. Gurses, ‘Discrete singular convolution for free vibration analysis annular membranes’, Mathematical & Computational Applications. 14, 131-138 (2009).
  • [35] O. Civalek, ‘Discrete singular convolution methodology for free vibration and stability analyses of arbitrary straight-sided quadrilateral plates’, Communications in Numerical Methods in Engineering. 24, 1475-1495 (2008).
  • [36] S. Zhao and G.W. Wei, ‘High order FDTD methods via derivative matching for Maxwell’s equations with material interfaces’, J. Comput. Phys.. 200, 60-103 (2004).
  • [37] Y.C. Zhou, S. Zhao, M. Feig and G. W. Wei, ‘High order matched interface and boundary (MIB) schemes for elliptic equations with discontinuous coefficients and singular sources’, J. Comput. Phys.. 213, 1-30 (2006).
  • [38] Y.C. Zhou and G.W. Wei, ‘ On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method’, J. Comput. Phys.. 219, 228-246 (2006).
  • [39] S.N. Yu and G.W. Wei, ‘ Three-dimansional matched interface and boundary (MIB) method for treating geometric singularities’, J. Comput. Phys.. 227, 602-632 (2007).
  • [40] S. Zhao and G.W. Wei, ‘Matched interface and boundary (MIB) for the implementation of boundary conditions in high-order central finite differences’, Int. J. Numer. Meth. Engng.. 77, 16901730 (2009).
  • [41] G.W. Wei, ‘Wavelet generated by using discrete singular convolution kernels’, J. Phys. A, Math. Gen. 33, 8577–8596 (2000).
  • [42] L. Schwarz, Theore des distributions, Hermann, Paris, 1951.
  • [43] Y.C. Zhou, B. S. V. Patnaik, D.C. Wan and G. W. Wei, ‘DSC solution for flow in a staggered double lid driven cavity’, Int. J. Numer. Methods in Engng. 57, 211-234 (2003).
  • [44] G. Bao, G.W. Wei, and S. Zhao, ‘Numerical solution of the Helmholtz equation with high wave numbers’, Int. J. Numer. Meth. Engng. 59, 389-408 (2004).
  • [45] G.W. Wei, ‘Discrete singular convolution for beam analysis’, Engng Struct. 23, 1045–1053 (2001).
  • [46] G.W. Wei, ‘A new algorithm for solving some mechanical problems’, Comput. Meth. Appl. Mech. and Engng. 190, 2017–2030 (2001).
  • [47] G.W. Wei, ‘Vibration analysis by discrete singular convolution’, J. Sound Vibr. 244, 535-553 (2001).
  • [48] G.W. Wei, Y.B. Zhao and Y. Xiang, ‘The determination of the natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution’, Int. J. Mech. Sci. 43, 1731–1746 (2001).
  • [49] Y.B. Zhao, G.W. Wei, and Y. Xiang, ‘Plate vibration under irregular internal supports’, Int. J. Solids Struct. 39, 1361-1383 (2002).
  • [50] Y.B. Zhao, G.W. Wei, and Y. Xiang, ‘Discrete singular convolution for the prediction of high frequency vibration of plates’, Int. J. Solids Struct. 39, 65-88 (2002).
  • [51] G.W. Wei, Y.B. Zhao and Y. Xiang, ‘Discrete singular convolution and its application to the analysis of plates with internal supports. I Theory and algorithm’, Int. J. Numer. Methods Engng. 55, 913-946 (2002).
  • [52] Y. Xiang, Y.B. Zhao and G.W. Wei, ‘Discrete singular convolution and its application to the analysis of plates with internal supports. II Complex supports’, Int. J. Numer. Methods Engng. 55, 947-971,(2002).
  • [53] Y.S. Hou, G.W. Wei and Y. Xiang, ‘DSC-Ritz method for the vibration analysis of Mindlin plates’, Int. J. Numer. Meth. Engng.. 62, 262-288 (2005).
  • [54] C.W. Lim, Z.R. Li and G.W. Wei, ‘DSC-Ritz method for the high frequency mode analysis of thick shallow shells’. Int. J. Numer. Meth. Engng. 62 205-232 2005
  • [55] G.W. Wei, D.S. Zhang, D.J. Kouri, D.K. Hoffman, ‘Lagrange distributed approximating functionals’, Phys. Rev. Lett. 79, 775-779 (1997).
  • [56] C.H.W. Ng, Y.B. Zhao and G.W. Wei, ‘Comparison of the DSC and GDQ methods for the vibration analysis of plates’, Comput. Methods Appl. Mech. Engng. 193, 2483-2506 (2004).
  • [57] Y. Wang, Y.B. Zhao and G.W. Wei, ‘A note on the numerical solution of high-order differential equations’, J. Comput. Appl. Math. 159, 387-398 (2003).

On the Accuracy and Stability of a Variety of Differential Quadrature Formulations for the Vibration Analysis of Beams

Year 2009, Volume: 1 Issue: 4, 1 - 25, 01.12.2009

C. H. W. Ng Y. B. Zhao Y. Xiang G. W. Wei

Abstract

The occurrence of spurious complex eigenvalues is a serious stability problem in many differential quadrature methods (DQMs). This paper studies the accuracy and stability of a variety of different differential quadrature formulations. Special emphasis is given to two local DQMs. One utilizes both fictitious grids and banded matrices, called local adaptive differential quadrature method (LaDQM). The other has banded matrices without using fictitious grids to facilitate boundary conditions, called finite difference differential quadrature methods (FDDQMs). These local DQMs include the classic DQMs as special cases given by extending their banded matrices to full matrices. LaDQMs and FDDQMs are implemented on a variety of treatments of boundary conditions, distributions of grids (i.e., uniform grids and Chebyshev grids), and lengths of stencils. A comprehensive comparison among these methods over beams of six different combinations of supporting edges sheds light on the stability and accuracy of DQMs

Keywords

Stability analysis Local adaptive differential quadrature method Differential quadrature method Beam Vibration analysis

References

  • [1] R.E. Bellman, J. Casti, ‘Differential quadrature and long term integration’, J. Math Anal. Appl. 34, 235–238 (1971).
  • [2] R. Bellman, B.G. Kashef and J. Casti, ‘Differential quadrature: a technique for the rapid solution of non–linear partial differential equations’, J. Comp. Phys. 10, 40–52 (1972).
  • [3] R.E. Bellman, B.G. Kashef, ‘Solution of the partial differential equation of the HodgkinsHuley model using differential quadrature’, Math. BioSci. 19, 1–8 (1974).
  • [4] G. Naadimuthu, R.E. Bellman, M. Wang, E.S. Lee, ‘Differntial quadrature and partial differential equations: some numerical results’, J. Math Anal. Appl. 98, 220–235 (1984).
  • [5] J.O. Mingle, ‘The method of differntial quadrature for transient nonlinear diffusion’, J. Math Anal. Appl. 60, 559–569 (1977).
  • [6] F. Civan, C.M. Sliepcevich, ‘Application of differential quadrature to transport processes’, J. Math Anal. Appl. 93, 206–221 (1983).
  • [7] F. Civan, C.M. Sliepcevich, ‘Solution of the poisson equtaion by differential quadrature’, Int. J. Num. Meth. Engng. 19, 711–724 (1983).
  • [8] F. Civan, C.M. Sliepcevich, ‘Differential quadrature multi-dimensional problems’, J. Math Anal. Appl. 101, 423–443 (1984).
  • [9] C.W. Bert, S.K. Jang and A.G. Striz, ‘Two new approximate methods for analyzing free vibration of structural components’, AIAA J. 26, 612–618 (1988).
  • [10] S.K. Jang, C.W. Bert and A.G. Striz, ‘Application of differential quadrature to static analysis of structural problems’, Int. J. Num. Meth. Engng. 28, 561–577 (1989).
  • [11] A.G. Striz, S.K. Jang, C.W. Bert, ‘Nonlinear bending analysis of thin circular plates by differential quadrature’, Thin-Walled Structures. 6, 51–62 (1988).
  • [12] C.W. Bert, S.K. Jang, A.G. Striz, ‘Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature’, Comput. Mech.. 5, 217–226 (1989).
  • [13] P.A.A. Laura, H. Gutierrez, ‘Analysis of vibrating rectangular plates with non-uniform boundary conditions by using the differential quadrature method’, J. Sound Vibr. 173, 703– 706 (1994).
  • [14] C. Shu and H. Du, ‘Implementation of clamped and simple supported boundary conditions in the GDQ free vibration analysis of beams and plates’, Int. J. Solid Struct. 34, 819–835 (1997).
  • [15] C. Shu and H. Du, ‘A generalizes approach for implementing general boundary conditions in GDQ free vibration analysis of plate’, Int. J. Solids Struct. 34, 837–846 (1997).
  • [16] C. Shu and C.M. Wang, ‘Treatment of mixed and non–uniform boundary conditions in GDQ vibration analysis of rectangular plate’, JEST. 21, 125–134 (1999).
  • [17] C.P. Wu and C.Y. Lee, ‘Differential quadrature solution for free vibration analysis of laminated conical shells with variable stiffness’, Int. J. Mech. Sci. 43, 1853–1869 (2001).
  • [18] C. Shu, ‘Analysis of elliptical waveguides by differential quadrature method’, IEEE trans. Microw. Theory and techn. 48, No 2, 319–322 (2000).
  • [19] T.Y. Wu, G.R. Liu, ‘The generalized differential quadrature rule for fourth-order differential equations’, Int. J. Num. Meth. Engng. 50, 1907–1929 (2001).
  • [20] G.R. Liu, T.Y. Wu, ‘In-plane vibration analysis of circular arches by the generalized differential quadrature rule’, Int. J. Mech. Sci. 43, 2597–2611 (2001).
  • [21] T.Y. Wu, G.R. Liu, ‘Application of generalized differential quadrature rule to sixthorder differential equations’, Commun. Numr. Meth. Engng. 16, 777–784 (2000).
  • [22] C. Shu, Differential quadrature and its application in Engineering, Spinger, New York, 2000.
  • [23] C. Shu and B.E. Richard, ‘Application of generalized differential quadrature to solve two– dimensions incompressible Navier–stress equations’, Int. J. Num. Meth. Fluids. 15, 791–798 (1992).
  • [24] C. Shu, Y.T. Chew, ‘On the equivalence of generalized differential quadrature and highest order finite difference scheme’, Comput. Methods Appl. Mech. Engng. 155, 249-260 (1998).
  • [25] G.W. Wei, ‘Discrete singular convolution for solution of the Fokker–Planck equations’, J. Chem. Phys. 110, 8930–8942 (1999).
  • [26] G.W. Wei, ‘A unified approach for solving the Fokker-Planck equation’, J. Phys. A, Math. Gen. 33, 4935-4953 (2000).
  • [27] O. Civalek, ‘Nonlinear analysis of thin rectangular plates on Winkler-Pasternak elastic foundations by DSC-HDQ methods’, Appl. Math. Model.. 31, 606-624 (2007).
  • [28] O. Civalek, ‘Linear vibration analysis of isotropic conical shells by discrete singular convolution (DSC)’, Struct. Engng. Mech.. 25, 127-130 (2007).
  • [29] O. Civalek, ‘Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: discrete singular convolution (DSC) approach’, J. Comput. Appl. Math.. 205, 251271 (2007).
  • [30] O. Civalek, ‘Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method’, Int. J. Mech. Sci.. 49, 752765 (2007).
  • [31] O. Civalek, ‘Free vibration analysis of composite conical shells using the discrete singular convolution algorithm’, Steel and Composite Struct.. 6, 353-366 (2006).
  • [32] O. Civalek, ‘An efficient method for free vibration analysis of rotating truncated conical shells’, Int. J. Press. Vess. Piping. 83, 1-12 (2006).
  • [33] O. Civalek, ‘The determination of frequencies of laminated conical shells via the discrete singular convolution method’, J. Mech. Mater. Struct.. 1, 165-192 (2006).
  • [34] O. Civalek and M. Gurses, ‘Discrete singular convolution for free vibration analysis annular membranes’, Mathematical & Computational Applications. 14, 131-138 (2009).
  • [35] O. Civalek, ‘Discrete singular convolution methodology for free vibration and stability analyses of arbitrary straight-sided quadrilateral plates’, Communications in Numerical Methods in Engineering. 24, 1475-1495 (2008).
  • [36] S. Zhao and G.W. Wei, ‘High order FDTD methods via derivative matching for Maxwell’s equations with material interfaces’, J. Comput. Phys.. 200, 60-103 (2004).
  • [37] Y.C. Zhou, S. Zhao, M. Feig and G. W. Wei, ‘High order matched interface and boundary (MIB) schemes for elliptic equations with discontinuous coefficients and singular sources’, J. Comput. Phys.. 213, 1-30 (2006).
  • [38] Y.C. Zhou and G.W. Wei, ‘ On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method’, J. Comput. Phys.. 219, 228-246 (2006).
  • [39] S.N. Yu and G.W. Wei, ‘ Three-dimansional matched interface and boundary (MIB) method for treating geometric singularities’, J. Comput. Phys.. 227, 602-632 (2007).
  • [40] S. Zhao and G.W. Wei, ‘Matched interface and boundary (MIB) for the implementation of boundary conditions in high-order central finite differences’, Int. J. Numer. Meth. Engng.. 77, 16901730 (2009).
  • [41] G.W. Wei, ‘Wavelet generated by using discrete singular convolution kernels’, J. Phys. A, Math. Gen. 33, 8577–8596 (2000).
  • [42] L. Schwarz, Theore des distributions, Hermann, Paris, 1951.
  • [43] Y.C. Zhou, B. S. V. Patnaik, D.C. Wan and G. W. Wei, ‘DSC solution for flow in a staggered double lid driven cavity’, Int. J. Numer. Methods in Engng. 57, 211-234 (2003).
  • [44] G. Bao, G.W. Wei, and S. Zhao, ‘Numerical solution of the Helmholtz equation with high wave numbers’, Int. J. Numer. Meth. Engng. 59, 389-408 (2004).
  • [45] G.W. Wei, ‘Discrete singular convolution for beam analysis’, Engng Struct. 23, 1045–1053 (2001).
  • [46] G.W. Wei, ‘A new algorithm for solving some mechanical problems’, Comput. Meth. Appl. Mech. and Engng. 190, 2017–2030 (2001).
  • [47] G.W. Wei, ‘Vibration analysis by discrete singular convolution’, J. Sound Vibr. 244, 535-553 (2001).
  • [48] G.W. Wei, Y.B. Zhao and Y. Xiang, ‘The determination of the natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution’, Int. J. Mech. Sci. 43, 1731–1746 (2001).
  • [49] Y.B. Zhao, G.W. Wei, and Y. Xiang, ‘Plate vibration under irregular internal supports’, Int. J. Solids Struct. 39, 1361-1383 (2002).
  • [50] Y.B. Zhao, G.W. Wei, and Y. Xiang, ‘Discrete singular convolution for the prediction of high frequency vibration of plates’, Int. J. Solids Struct. 39, 65-88 (2002).
  • [51] G.W. Wei, Y.B. Zhao and Y. Xiang, ‘Discrete singular convolution and its application to the analysis of plates with internal supports. I Theory and algorithm’, Int. J. Numer. Methods Engng. 55, 913-946 (2002).
  • [52] Y. Xiang, Y.B. Zhao and G.W. Wei, ‘Discrete singular convolution and its application to the analysis of plates with internal supports. II Complex supports’, Int. J. Numer. Methods Engng. 55, 947-971,(2002).
  • [53] Y.S. Hou, G.W. Wei and Y. Xiang, ‘DSC-Ritz method for the vibration analysis of Mindlin plates’, Int. J. Numer. Meth. Engng.. 62, 262-288 (2005).
  • [54] C.W. Lim, Z.R. Li and G.W. Wei, ‘DSC-Ritz method for the high frequency mode analysis of thick shallow shells’. Int. J. Numer. Meth. Engng. 62 205-232 2005
  • [55] G.W. Wei, D.S. Zhang, D.J. Kouri, D.K. Hoffman, ‘Lagrange distributed approximating functionals’, Phys. Rev. Lett. 79, 775-779 (1997).
  • [56] C.H.W. Ng, Y.B. Zhao and G.W. Wei, ‘Comparison of the DSC and GDQ methods for the vibration analysis of plates’, Comput. Methods Appl. Mech. Engng. 193, 2483-2506 (2004).
  • [57] Y. Wang, Y.B. Zhao and G.W. Wei, ‘A note on the numerical solution of high-order differential equations’, J. Comput. Appl. Math. 159, 387-398 (2003).
There are 57 citations in total.

Details

Other ID JA65GN47ZY
Journal Section Articles
Authors

C. H. W. Ng This is me

Y. B. Zhao This is me

Y. Xiang This is me

G. W. Wei This is me

Publication Date December 1, 2009
Published in Issue Year 2009 Volume: 1 Issue: 4

Cite

APA Ng, C. H. W., Zhao, Y. B., Xiang, Y., Wei, G. W. (2009). On the Accuracy and Stability of a Variety of Differential Quadrature Formulations for the Vibration Analysis of Beams. International Journal of Engineering and Applied Sciences, 1(4), 1-25.
AMA Ng CHW, Zhao YB, Xiang Y, Wei GW. On the Accuracy and Stability of a Variety of Differential Quadrature Formulations for the Vibration Analysis of Beams. IJEAS. December 2009;1(4):1-25.
Chicago Ng, C. H. W., Y. B. Zhao, Y. Xiang, and G. W. Wei. “On the Accuracy and Stability of a Variety of Differential Quadrature Formulations for the Vibration Analysis of Beams”. International Journal of Engineering and Applied Sciences 1, no. 4 (December 2009): 1-25.
EndNote Ng CHW, Zhao YB, Xiang Y, Wei GW (December 1, 2009) On the Accuracy and Stability of a Variety of Differential Quadrature Formulations for the Vibration Analysis of Beams. International Journal of Engineering and Applied Sciences 1 4 1–25.
IEEE C. H. W. Ng, Y. B. Zhao, Y. Xiang, and G. W. Wei, “On the Accuracy and Stability of a Variety of Differential Quadrature Formulations for the Vibration Analysis of Beams”, IJEAS, vol. 1, no. 4, pp. 1–25, 2009.
ISNAD Ng, C. H. W. et al. “On the Accuracy and Stability of a Variety of Differential Quadrature Formulations for the Vibration Analysis of Beams”. International Journal of Engineering and Applied Sciences 1/4 (December 2009), 1-25.
JAMA Ng CHW, Zhao YB, Xiang Y, Wei GW. On the Accuracy and Stability of a Variety of Differential Quadrature Formulations for the Vibration Analysis of Beams. IJEAS. 2009;1:1–25.
MLA Ng, C. H. W. et al. “On the Accuracy and Stability of a Variety of Differential Quadrature Formulations for the Vibration Analysis of Beams”. International Journal of Engineering and Applied Sciences, vol. 1, no. 4, 2009, pp. 1-25.
Vancouver Ng CHW, Zhao YB, Xiang Y, Wei GW. On the Accuracy and Stability of a Variety of Differential Quadrature Formulations for the Vibration Analysis of Beams. IJEAS. 2009;1(4):1-25.

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