An approximate formula to calculate the fundamental period of a fixed-free mass-spring system with varying mass and stiffness

A formula to approximate the fundamental period of a fixed-free mass-spring varying mass and varying stiffness is formulated. The formula is derived mainly by taking parts from the general form of the characteristic polynomial, and adjusting the initial approximation coefficient derived from the exact solution of a uniform case. The formula is tested for a large randomly generated structures, and the results show that the approximated fundamental periods the error range of 4% with 90% of confidence. Also, the error is shown to be normally distributed zero mean, and the width of the distribution (as measured by the standard deviation) tends the total number of discretized elements in the system increases. Other possible extensions are discussed, including an extension to a continuous cantilever structure with distributed stiffness. The suggested formula provides an efficient way to estimate the fundamental period structures and other systems that can be modeled as mass-spring systems.