A Timoshenko beam reduced order model for shape tracking with a slender mechanism

We consider a flexible bio-inspired slender mechanism, modeled as a Timoshenko beam. It is coupled to the environment by a continuous distribution of compliant elements. We derive a reduced order model by projecting the governing partial differential equations along the linear modal basis of the Timoshenko beam. The coupling with the substrate allows us to formulate the problem in a control framework, and eventually to treat the system as a sensor to reconstruct the profile of the substrate through the deformation of the body. The coupling is modeled in the framework of two parameters elastic foundations. The convergence of the reduced order model with increasing number of basis functions is addressed in a suitable H1 error norm. A closed loop force control is simulated for shape morphing when the system is coupled with a smooth substrate.