Initiation and directional control of period-1 rotation for a parametric pendulum

Abstract

We study a time-delayed feedback control for initiating period-1 rotations of a vertically excited parametric pendulum from arbitrary initial conditions. The possibility of controlling the direction of rotation has also been explored. We start with a simple linear time-delayed control for which the control gain corresponding to the most stable period-1 rotation has been obtained using the Floquet theory. This control increases the basins of attraction of rotations, but they do not encompass the full initial condition space. We modify our control law by using a switched control gain that destabilizes all the oscillatory solutions, and the entire initial condition space becomes the basin of attraction of either the clockwise or the anticlockwise rotation. By a suitable modification of the switching condition, we can choose a preferential stable direction of rotation. Hence, we can initiate either clockwise or anticlockwise rotation for a parametric pendulum from arbitrary initial conditions. Performance of our controller in achieving this objective has been demonstrated for different sets of parameters to establish its effectiveness.

Energy extraction from vortex-induced vibrations using period-1 rotation of an autoparametric pendulum

“…It can be seen from figure 12 that the basin of attraction reduces with an increase in μ 2 which has an adverse effect on the harvester performance since rotation will not initiate from the initial conditions for the pendulum motion as suggested by a study of the parametric pendulum which is obtained with μ 2 = 0. However, for using the rotating motion of a pendulum for energy harvesting, we typically employ a controller [18] to guarantee robust initiation of rotation from all initial conditions. In such a situation, the phase shift introduced by incorporating the two-way coupling between the pendulum and the structure becomes irrelevant and we can replace the autoparametric system with a parametric pendulum.…”

“…We finally conclude this study on the horizontal configuration with the two-dimensional slice of the basin of attraction of the period-1 rotation shown in figure 20. It can be seen that the combined basin does not cover the whole initial condition space and, hence, an external controller is required to ensure that rotation can be initiated independent of the initial disturbance in order to ensure good performance of the energy harvester [18].…”

“…Hence, for capturing pendulum rotation, we can neglect the dynamics of the base structure and the vibration source and replace it by a prescribed base motion thereby giving rise to a parametrically forced pendulum. This simplifies further study of the system and development of control strategies to ensure initiation of rotation [4,[16][17][18][19]. Control strategies are required for an effective harvester since a stable rotating solution is always accompanied by other attractors resulting in a limited basin of attraction (for rotation) and special initial conditions are required to ensure a rotating response.…”

“…Hence, it modifies the basins of attraction of the rotating solution in the space of initial conditions for the pendulum. However, with the control strategy developed in [18,19], the entire initial condition space becomes the basin of attraction of the rotating motions of the parametric pendulum and hence, this issue can possibly be circumvented.…”

Performance limit for base-excited energy harvesting, and comparison with experiments

“…Reguera et al [14] proposed a control strategy in which the length of a vertically excited pendulum is varied to achieve continuous rotations. Das and Wahi [15] used time delay control to initiate and sustain full rotations in a parametric pendulum under vertical base excitation. Similarly, Firoozy et al [16] has used time delay control to improve the performance of a maglev energy harvester under base excitation.…”