This paper introduces node control, whereby discrete direct feedback control forces are placed at the nodes of the N+1th mode (the lowest N modes participate in the response). Node control is motivated by the node control theorem which states, under certain conditions, that node control preserves the natural frequencies and natural modes of vibration of the controlled system while achieving uniform damping. The node control theorem is verified for uniform beams with pinned-pinned, cantilevered, and free-free boundary conditions, and two cases of beams with springs on the boundaries. A general proof of the node control theorem remains elusive.
This paper formulates the equations governing the dynamics and control of electrostatic structures. Using a Lagrangian mechanics approach, a potential energy function composed of a strain component, an electric component, and a gravitational component is defined. The resulting system of nonlinear ordinary differential equations are linearized about the electrostatic equilibrium leading to a linear system of ordinary differential equations characterized by mass, stiffness, damping, gyroscopic, and circulatory effects. In the absence of feedback control, the damping, gyroscopic, and circulatory effects vanish resulting in a symmetric system that admits normal mode vibration. Voltages applied over the charged subsurfaces (control points) of the electrostatic structure can control its shape. In the presence of feedback controls, control gains can be tailored to produce desirable levels of stiffness and damping. Two different control approaches are studied, one using control points that are attached to the electrostatic structure and one where the control points are fixed in space. Example problems illustrate the dynamics and control; specifically, circumstances that lead to instabilities, shape control using attached control surfaces, shape control using fixed control surfaces, and electrostatic damping.
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