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.

Keywords

• Shooting method,Linear Elastic Stability,Complementary Functions Method (CFM),eigenvalue,Linear eigenvalue boundary value problem,Euler Columns

References

1. [1] D.J.Jones, “Use of a shooting method to compute eigenvalues of fourth- order two-point boundary value problems” ,Journal of Computational and Applied Mathematics, Volume 47, No. 3, PP. 395-400
2. [2] X.Chen, “The Shooting Method for Solving Eigenvalue Problems”, Journal of Mathemtical Analysis and applications 203, 435]450 1996.
3. [3] S.Li, Y.H.Zhou & X.Zheng (2002), “Thermal Postbuckling of a Heated Elastic Rod with Pinned-Fixed Ends”, Journal of Thermal Stresses, 25:1, 45-56,
4. [4] S.-R. Li, “Vibration of Thermally Post-Buckled Orthotropic Circular Plates”, Journal of Sound and vibration (2001) ,248(2), 379}386
5. [5] Y.ZHU, Y.J.Hu, C.J.Cheng, “Analysis of nonlinear stability and post-buckling for Euler-type beam-column Structure”, Appl. Math. Mech. -Engl. Ed., 32(6), 719–728 (2011)
6. [6] Q.Li, “Shooting method for free vibration of FGM Reissner-Mindlin circular platesresting on elastic foundation in thermal environments”, Journal of Vibroengineering 19(6) • October 2017
7. [7] L.S.Ma, Buckling of Functionally Graded Circular/Annular Plates Based on the First-Order Shear Deformation Plate Theory, Key Engineering Materials 261-263:609-614 • January 2004
8. [8] R.P.Agarwal, “On the method of complementary functions for nonlinear boundary-value problems”, Journal of Optimization Theory and Applications volume 36, pages139–144(1982)

9. [9] A.Miele, “Method of particular solutions for linear, two-point boundary-value problems”, Journal of Optimization Theory and Applications, July 1968, Volume 2, Issue 4, pp 260–273
10. [10] K.N.Murty, K.R.Prasad, Y.S.Rao, “On the Method of Complementary Functions for Linear and Nonlinear Two-Point Boundary Value Problems”, Journal of Mathematical Analysis & Applications, 167, 32-42 (1992)
11. [11] S.Yildirim,N.Tutuncu, “Effect of Magneto-Thermal Loads on the Rotational Instability of Heterogeneous Rotors”, AIAA Journal, March 2019
12. [12] S.Yildirim,N.Tutuncu, “On the Inertio-Elastic Instability of Variable-Thickness Functionally-Graded Disks”, Mechanics Research Communications, 91:1-6, May 2018