High frequency response to low frequency forcing in a nonlinear mechanical model.

摘要

Recent developments in computational technology have led to independence from oversimplification in more and more mechanical models. Here we focus on a modest yet interesting nonlinear ordinary differential equation modelling vertical motion in a suspension bridge. The examination involves delving into the characteristics of the solutions when the model is subjected to periodic forcing and binges on creating the bifurcation curve of the amplitude of the response versus that of the external force using a continuation algorithm. Surprisingly, steepest descent assists as it has proven to be unreasonably effective in helping to find new responses, including the elusive unstable solutions. Results show that as soon as the motions extend into the nonlinear range, not only do multiple solutions exist including those which have doubled and quadrupled in period, but many exhibit a high frequency component when only low frequency forcing has been introduced. We enlist Floquet Theory and Fourier Analysis to study these behaviors. Lastly, we investigate any variation in these behaviors when we smooth the nonlinearity.