Application of generalized differential quadrature rule to sixth-order differential equations

A generalized differential quadrature rule (GDQR) has been proposed as a general numerical method to solve high-order differential equations. Applications are given to sixth-order differential equations that govern the free vibration analysis of ring structures. The DQM uses the function values at all grid points as independent variables in the establishment of algebraic equations. This leads to difficulties in implementing the boundary conditions at a point. The GDQR regards the function values at all grid points and their derivatives at grid points wherever necessary as independent variables. The given multiple conditions at any point can therefore be imposed in the GDQR in a direct manner. A procedure is also proposed for calculating GDQR's explicit weighting coefficients for an effective establishment of algebraic equations.