Further Development Of Spline-based Differential Quadrature Method And Nonlinear Vibration Analysis Of Beams

The Spline-based Differential Quadrature (SDQ) is a newly developed numerical method. The main distinction of the SDQ lies in the determination of weighting coefficients on the basis of cardinal B-spline interpolation functions. The application scope of the SDQ has been limited since the method is still in its infancy stage. One of the aims of the present research is to further extend the application of the method to more practical problems in structural analysis. The constructions of the quartic and the sextic SDQ are elaborated and the explicit expressions of weighting coefficients for approximation of derivatives are obtained. A brief summary is given for the development of differential quadrature method using ether odd-order or even-order B-splines. Excellent results are achieved in the provided examples. The SDQ method is shown to exhibit great flexibility and stability. The other main aim of the present study is the nonlinear (large amplitude) vibration analysis of beams. Two beam theories, the Bernoulli-Euler beam theory and the Timoshenko beam theory, are considered. The nonlinear vibrations of prismatic beams and tapered beams are investigated. The conventional differential quadrature method and the SDQ are used to resolve the nonlinear vibration problems. For the Bernoulli-Euler beams, the effects of the different boundary conditions on the formulation and the solution are discussed. For nonlinear vibrations of Timoshenko beams, the corresponding governing equations are established for the first time in present work. The nonlinear terms including the axial stretching, nonlinear bending curvature and shear strain are considered. It is shown that the nonlinear frequency ratio increases with the vibration amplitude and the slenderness ratio. The effects of nonlinear terms on the nonlinear frequency ratio are discussed at length and their contributions are found to be in the following descending sequence: axial stretching, shear strain and bending curvature. The change of nonlinear frequency against the effects of boundary conditions is always in the following descending order: beams with two simply supported ends, beams with two clamped ends and beams with one end simply supported and the other end clamed. For a specific vibration amplitude and a specific slenderness ratio, the effect of the cross-sectional height change on the nonlinear frequency ratio is larger than that of the cross-sectional width change.In addition, the local implementation strategy in the conventional differential quadrature is discussed briefly. It is shown that the philosophy of local implementation can efficiently improve the stability of the conventional differential quadrature and extend its application accordingly.