Self-similar eigenvalue distribution of a binary tree-structured mass–spring–damper system

Abstract

The eigenvalue distribution of a binary tree-structured multi-degree-of-freedom linear oscillator is examined. The block masses and spring stiffnesses in the model are a power-law function of the tree level. A recursive formula is proven for the characteristic polynomial of the undamped system. For the power-law quotient $\sigma =1/2$ the resulting characteristic equation is solved analytically and the eigenvalue distribution is derived as the function of the tree level up to the limit of the infinite tree. The importance of the eigenvalue distribution is demonstrated with examples of the dynamical behavior, when the system is subjected to small mistuning of its block masses. The mistuning perturbs the block masses by drawing a random value from a uniform distribution centered around the base value of each block mass. The width of the uniform distribution for each block mass is given by the product of the base value of the mass and the mistuning parameter $r$. The displacement solutions of the governing differential equation with two sets of initial conditions are compared for different values of the mistuning parameter $r$. Even though the mistuning has only a small effect on the eigenfrequencies of the system, it is concluded that the small perturbation of the block masses can cause a significant shift in the apparent frequency of the vibration.