Uncertainty Propagation in the Band Gap Structure of a 1D Array of Magnetically Coupled Oscillators

The propagation constant technique has previously been used to predict band gap regions in linear oscillator chains by solving an eigenvalue problem for frequency in terms of a wave number. This paper describes a method by which selected design parameters can be separated from the eigenvalue problem, allowing standard uncertainty propagation techniques to provide closed form solutions for the uncertainty in frequency. Examples are provided for different types of measurement or environmental uncertainty showing the varying robustness of a band gap region to changes in parameters of the same or different order. The system studied in this paper is comprised of repelling magnetic oscillators using a dipole model. Numerical simulation has been performed to confirm the accuracy of analytical solutions up to a certain level of base excitation amplitude after which nonlinear effects change the predicted band gap regions to low energy chaos.