Theory of Vibration with Applications by William Thompson and Marie Dillon Dahleh. Say you have two cars linked, with some spring constant; | --^^-- [c1] --^^

Theory of Vibration with Applications by William Thompson and Marie Dillon Dahleh. Say you have two cars linked, with some spring constant; | --^^-- [c1] --^^-- [c2] --^^--| where '^^' is a spring and '|' is a wall. The motion of these cars can be written using the spring forces in the system or, alternately, as the harmonic motion of the undamped system with some natural frequency. Setting this up as two simultaneous equations (one for each car) and solving for the roots give you the eigenvalues. The natural frequency is the square root of the eigenvalue. In other words, the eigenvalues help you define the natural frequencies which can be used to characterize the motion of the cars in the more complicated spring-mass system.