Application of reciprocal condition number in vibration analysis

Abstract

This paper explores the use of the reciprocal condition number (rcond) in vibration analysis. The method employs the reciprocal of the matrix condition number in 1-norm for efficient calculation of natural frequencies, avoiding the need to solve complex frequency equations that involve transcendental functions. Natural frequencies are shown to correspond to the zero values of the reciprocal condition number, which forms the basis of the procedure for identifying the minima of the discretized rcond function and verifying whether they correspond to natural frequencies, enabling flexible application in the eigenvalue analysis of various structures. To validate this generally applicable approach, the study examines a case involving beams with arbitrary number of cross-section steps, general boundary conditions and various lumped elements, including viscous dampers. By assembling the dynamic stiffness matrix, natural frequencies can be calculated using both the proposed reciprocal condition number approach and the well-established Wittrick-Williams algorithm. A comparative analysis is carried out to evaluate the effectiveness and accuracy of the proposed method. Numerical tests confirm its ability to solve transcendental eigenvalue problems of large matrix systems effectively, showing strong consistency with previous studies and alignment with finite element method results. This establishes the method as a valuable tool for engineering design and structural analysis.