The effect of small changes in mass and stiffness on the natural frequencies and modes of vibrating systems

Eigendesign in brake carriers

“…The first work related directly to this topic was published by Jones,1 in which the effect of small changes in mass and stiffness was developed directly for vibrating systems and applied to a five-mass flexural system. Apart from this first approach, and based on mathematical developments for eigenvalue and eigenvector variations, 2 the perturbation methods took their basis from the derivatives of eigenfrequencies and eigenmodes.…”

“…When only little mass and/or stiffness variations are allowed, and if these variations can be easily reflected within the expression of the terms in mass and/or stiffness matrices, the perturbation methods are especially well suited to estimate the variations of the eigenfrequencies and the eigenmodes of a mechanical system. The first work related directly to this topic was published by Jones, 1 in which the effect of small changes in mass and stiffness was developed directly for vibrating systems and applied to a five-mass flexural system. Apart from this first approach, and based on mathematical developments for eigenvalue and eigenvector variations, 2 the perturbation methods took their basis from the derivatives of eigenfrequencies and eigenmodes.…”

“…The first work related to this topic in the area of mechanical vibrations was published by Fox and Kapoor, 3 in which the derivatives of the natural frequencies were developed with respect to design parameters, and two alternative methodologies were arranged to evaluate the derivatives of the vibration modes; one of them was the same than the one developed in the study by Jones. 1 Since then, and only to cite some of the works in the past 10 years, a lot of papers have been published in the field of the eigensensitivity and the dynamic perturbation methods, coping with particularities of the eigenproblem such as high-order derivatives, repeated eigenvalues, very close eigenvalues, and damped systems involving complex eigenmodes. [4][5][6][7][8][9][10] Any geometrical modification in a mechanical system results automatically in a mass variation and a stiffness variation.…”

A Universal Matrix Perturbation Technique for Structural Dynamic Modification Using Singular Value Decomposition

“…Stetson Meirovitch and Ryland put forward MPTs for holographic vibration analysis [9] and gyroscopic systems [10]. Jones studied the e!ect of small changes in mass and sti!ness on the natural frequencies [11]. Rizai and Bernard analyzed the dynamic e!ects of redesign [12].…”