The Fluctuations of Small Elastic Objects in Fluid with Linear and Nonlinear Restoring Forces

As an elastic object, such as a beam, is uniformly reduced in size its resonant frequency increases while its stiffness typically decreases. As technology continues to push towards smaller objects to exploit this favorable combination, we will reach a point where Brownian motion will drive the dynamics into the nonlinear regime. For example, a strongly driven beam will yield a geometric nonlinearity where the stiffness increases cubically with displacement. We explore the stochastic dynamics of elastic objects in fluid for a range of restoring forces from linear to strongly nonlinear. For a linear restoring force the fluctuation-dissipation theorem can be used to provide a deterministic theoretical description. We compare theory with experiment for a beam that is under tension and immersed in a fluid where excellent agreement is found. We numerically probe the role of nonlinearity on the dynamics of elastic objects with a Duffing restoring force using a stochastic differential equation description. We use a finite element approach to study the dynamics of 3D elastic objects whose geometry and properties have been tailored to yield a nonlinear response.