MS04.3: Experiments in Nonlinear Dynamics and Control

Many theoretical studies on post-critical non-planar motions of a cantilevered pipe conveying fluid have been studied for more than forty years. This system belongs to non-self-adjoint system and reveals self-excited oscillation. Therefore, depending on system parameters, a certain mode can reveal dynamical instabilities. It is well known that an end mass is a key parameter to generate non-planar motions and make the dynamics of systems very rich. In this presentation, we experimentally investigate the effects of the end mass on the post-critical non-planar motions. In particular, we focus on the complex non-planar motions which are caused by nonlinear interactions between dynamically unstable modes. The experiments were extensively conducted by changing the end mass and the flow velocity as control parameters. Three-dimensional non-contacting measurements system consists of two high speed video cameras and image processing system which enabled us to capture the non-planar motions caused by nonlinear interactions between unstable modes. Due to mode exchanges and double Hopf bifurcations, in wide range of parameters, we observed complex non-planar motions.

Active vibration control of rotating structures can be a challenging task due to the nonlinearities induced by rotation \textit{i.e} modal frequency shifts or damping variations. In addition, cyclic boundary conditions like a permanent contact of the rotor with a stator element at a particular angle can significantly modify the frequency response between embedded control actuators and sensors within the rotating frame. In this study, a robust observation and control method is proposed and applied experimentally to a rotating spoked structure recreating the cyclic conditions of a vehicle wheel on the road. A special consideration is given to the robust nonlinear observer to ensure its correct tracking of the system states despite high cyclic modeling uncertainties due to the rotation.

Underactuated wheeled vehicles are commonly studied as nonholonomic systems with periodic actuation. Twistcar is a classical example inspired by a riding toy, which has been analyzed using a planar model of a dynamical system with nonholonomic constraints. Previous analyses of the Twistcar model also revealed the possibility of reversing the direction of motion depending on the geometric and mass properties of the vehicle. In this work, we present new experimental results using a two-link robotic prototype of the Twistcar vehicle. An important finding from experimental measurements is that the vehicle’s motion reaches bounded oscillations, in contrast to previous analyses that predicted unrealistic results of unbounded divergence of vehicle’s speed, as well as the amplitude of body angle oscillations. In order to improve the predictive power of our analysis, we consider a modified version of the Twistcar model with added viscous dissipation due to rolling resistance. Numerical analysis of the dissipative Twistcar model leads to convergence to bounded oscillations of the vehicle motion, which agrees with experimental observations. We also present asymptotic analysis, which enables obtaining explicit expressions that highlight the influence of various parameters on the motion, including reversal of the direction.

In experimental nonlinear modal analysis, phase-locked loops with usually heuristically determined controller parameters are used. In this work, approaches to systematically design the control parameters are presented. Three variations of the method with different levels of complexity were developed. While the most simple one relies on severe simplifications, e. g. linear behavior of the structure under test, the most advanced one accounts for the nonlinearity of the system and makes it possible to select amplitudedependent controller parameters. While all methods lead to a stable and robust control loop, the more advanced variants promise significantly shorter settling times of the controller and thus shorter test times.