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.

1.
Jackson, J. D., 1975, Classical Electrodynamics, John Wiley and Sons, New York.
2.
Meirovitch, L., 1980, Computational Methods in Structural Dynamics, Sijthoff and Noordhoff, Rockville, MD.
3.
Mihora, D. H., and Redmond, P. J., 1979, “Electrostatically Formed Antennas,” General Research Corporation, Santa Barbara, CA, Internal Memo 2222.
4.
Park, K., 1992, “An Investigation of Planar Electrodynamic Structures,” Ph.D. Dissertation, North Carolina State University, Raleigh, NC.
5.
Rhim
W. K.
,
Collender
M.
,
Hyson
M. T.
,
Simms
W. T.
, and
Elleman
D. D.
,
1985
, “
Development of an Electrostatic Positioner for Space Material Processing
,”
Review of Scientific Instruments
, Vol.
56
, No.
2
, pp.
307
317
.
6.
Rhim
N. K.
,
Chung
S. K.
,
Hyson
M. T.
,
Trinh
E. H.
, and
Elleman
D. D.
,
1987
, “
Large Charged Drop Levitation Against Gravity
,”
IEEE Transactions on Industry Applications
, Vol.
IA-23
, No.
6
, pp.
975
979
.
7.
Silverberg, L., 1994, “Electrostatically Shaped Membranes,” U.S. Patent No. 5,307,082.
8.
Streng
J. H.
,
1990
, “
Charge Movements on the Stretched Membrane in a Circular Electrostatic Push-Pull Loudspeaker
,”
Journal of the Audio Engineering Society
, Vol.
38
, No.
5
, pp.
331
338
.
9.
Streng
J. H.
,
1989
, “
Sound Radiation from Circular Stretched Membranes in Free Space
,”
Journal of the Audio Engineering Society
, Vol.
37
, No.
3
, pp.
107
118
.
10.
Yam
Y.
,
Lang
J. H.
,
Staelin
D. H.
, and
Johnson
T. L.
,
1988
, “
The Experimental Computer Control of a Two-Dimensional Hyperbolic System
,”
IEEE Transactions on Automatic Control
, Vol.
33
, No.
1
, pp.
79
87
.
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