Application of Power Method and Dominant Eigenvector/Eigenvalue Concept for Approximate Eigenspace Solutions to Mechanical Engineering Algebraic Systems

Abstract

This paper shows how the concept of dominant eigenvector/eigenvalue and the power method can be used for understanding the solution to practical problems in a broad class of mechanical engineering systems described by linear algebraic equations. An analytical mathematical procedure is developed to obtain a reasonably accurate approximate eigenspace solution to the system Ax=b by transforming the non-symmetric system matrix into a form where the power method is used several times to compute the contribution of only one or two eigenvector/eigenvalue pairs that dominate the solution. The complete set of eigenvalues and eigenvectors is not required. The intent is to provide a novel application of and show the importance of the significance of dominant eigenvector/eigenvalue pairs and to use the power method in the analysis and understanding of mechanical engineering systems for both education and practice. Typically, the concepts of both eigenvector expansion and dominant contribution are not included in mechanical engineering education. The scope of application is a broad area of six practical mechanical engineering problems including translational and rotational dynamics, statics of structures, thermal energy balance, fluid continuity, and feedback control. These general mechanical engineering systems naturally contain numerical A and b matrices that fit the scope suitable for providing feasible accuracy. The systems range from fourth order to two hundredth order. Results indicate that accuracy greatly improves as the A matrix contains elements of the same order of numerical magnitude, either naturally from the physics of the problem, or through a transformation to dimensionless parameters. Motivation for using the power method for dominant eigenvector/eigenvalue calculation comes from methods used in web page rank algorithms and a desire to expand mechanical engineering students’ education in understanding the role of the eigenstate expansion method for the solution of algebraic equations without computing all of its eigenvalues and eigenvectors. An in depth quantitative assessment of the approximate dominant eigenspace method accuracy is obtained by testing the mechanical engineering examples against a near exact solution via a software solver. This error assessment is based on parameters useful in the design and understanding of mechanical systems. The accuracy achieved is feasible for education and for other uses of the method by practicing mechanical engineers such as when a simple analytical based approximate model may be more suited for inclusion into larger system based models.