Start Control of Parametric Pendulum into Periodic Rotation

Abstract

Parametrically excited pendulum inherently demonstrates a conversion from forcing vibration to rotational motion. The onset of periodic rotation strongly depends on the initial condition. We propose a control method to start up a parametric pendulum into periodic rotation based on an external force input with time delay. The feasibility of proposed method is verified numerically and experimentally. This paper advocates that the proposed method is suitable for crossing over a separatrix which governs the dynamics from initial conditions.

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

“…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.…”

Rotating a pendulum with an electromechanical excitation

“…To address this shortcoming, delayed-feedback control, proposed in [24], was introduced in [25] with the aim of initiating the rotational motion regardless of the initial condi-40 tions and avoiding bifurcations that destabilize the rotation motion of pendulum while maintaining stable rotation. It was used for start up control of pendulum in [26], and the tolerance of start up control and delayed feedback control for maintaining periodic rotation was investigated in [27] and [28]. Its variant control strategy, extended time-delayed feedback which was proposed in [29], was 45 used in [30] to control bifurcation for the parametric pendulum under harmonic excitation.…”

Tolerance of start-up control of rotation in parametric pendulum by delayed feedback

“…The stable periodic rotation disappears through the saddle-node bifurcation, denoted by C and D, at both sides of the branch. For appropriate torque N a stable rotational limit cycle exists in the system (4) without the parametric excitation. The Melnikov's method [20] gives the existence condition N > 4γ/π =: N 0 ≈ 0.127.…”

“…Rotation of parametric pendulum [1][2][3] exhibits a conversion from the external vibration into its rotational motion. The converting motion is effective from the viewpoint of applications such as energy scavenging from vibration of nature [4]. In this paper, we propose a control method for establishing the periodic rotation inherent in the parametric pendulum based on a delayed feedback control.…”