Numerical Methods in Calculating Eigenvalues: Case Studies in Stability of Euler Columns

Abstract

A comparative analysis of well renowned "Shooting Method" with another numerical method "Complementary Functions Method" (CFM) is presented for calculating eigenvalue (λ). Contrary to the shooting method hit and trial approach, CFM exploits the properties of linear ordinary differential equation (LODE). In the case of linear eigenvalue Boundary value Problem (BVP), CFM generates an algebraic equation system with one unknown "λ" and, alone root finding method is sufficient to give required eigenvalue. However, the Shooting Method create a system of algebraic equations containing two unknowns "λ" and "missing initial conditions", that demands an additional numerical technique along with root finding method. These radical differences between two approaches, sets the basis for this comparative investigation. As a case study in Linear Elastic Stability, different cases of Euler columns are investigated by finding eigenvalues for each case numerically, under both methods. Comparison is performed on the basis of results accuracy and cost effectiveness for both numerical techniques while solving linear stability problems.

A powerful numerical approach for the axisymmetric bending response of shear deformable two-directional functionally graded (2D-FG) plates with variable thickness

“…The CFM has been applied to several problems in applied mechanics. [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] Details of the CFM are presented in the Appendix. The matrix form of the governing equations ( 58) to (63) can be written as follows |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}…”

“…The method can be applied to any arbitrary continuous function. This method has been applied in the analysis of several structures [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] and compared with other available methods like FEM, shooting method, etc. The shooting method uses the trial and error method to determine the initial conditions.…”