Defect Analysis of 1D Spring-Mass Systems via Laplace Transform and Asymptotics

Spring-mass systems have seen utility for decades in modeling multiple physical phenomena like elastic deformation and wave propagation. Particularly, we focus on linear spring-mass systems and use experimental data to analytically locate and enumerate defects along the chain. We begin by taking the differential equations describing an almost uniform spring-mass system with a predetermined number of blocks into the Laplace domain; there is an arbitrary number of “error” masses somewhere along the chain with higher or lower mass than the remaining blocks. Application of elementary algebra and asymptotic analysis enables us to numerically test each block for defect status using the vibrational data of only the first block in the chain while simultaneously counting the number of defects. However, despite the sound nature of the theory, this process works only for data without measurement/numerical noise and defects beyond the first are undetectable.