Recent extended X-ray absorption fine structure (EXAFS) studies suggest that in skutterudites, the nearly square rings (such as As4 in CeFe4As12) are quite rigid and may vibrate with low energy modes in one direction, similar to "rattler" atom vibrations. That work suggests that the motions of the square rings and the rattler atoms are coupled. In addition, for LnCu3Ru4O12, the second neighbor pairs about Ln have stiffer effective springs than the nearest neighbor pairs. To investigate these systems, a one dimensional, four mass, linear chain spring model is developed to describe the recent experimental results, and provide insight about the low energy vibrations in such systems. Our model solves the resulting coupled network of overlapping weak and strong springs and determines the eigenfrequencies and eigenvectors. The dispersion curves show an acoustic mode, two different low energy optical rattling modes involving both the rattler and square, and a non interacting optical mode. Each rattler mode can couple to the acoustic mode, which generates avoided crossings characterized by flattening of the modes; this has important consequences for thermal transport. From these results we calculate atomic correlation functions and the Debye-Waller-like function used in EXAFS, σ 2 , as a function of temperature. These calculations show that for the rattler-neighbor pairs, σ 2 is a sum over several modes; it is not the result of a single mode. The inverse slope of σ 2 (T) at high T provides a measure of the effective spring constants, and the results show that for small direct spring constants the effective spring constant can be significantly larger than the direct spring constants. The locations of the avoided crossings (between rattler modes and the acoustic mode) in q-space can be tuned by the choice of both the rattler and the square atoms. Consequently, it may be possible to further reduce the thermal conductivity using a mixture of nanoparticles, each with avoided crossings at different positions in q-space.