LQR Control of Thin Shell Dynamics: Formulation and Numerical Implementation

A PDE-based feedback control method for thin cylindrical shells with surface-mounted piezoceramic actuators is presented. Donnell-Mushtari equations modified to incorporate both passive and active piezoceramic patch contributions are used to model the system dynamics. The well-posedness of this model and the associated LQR problem with an unbounded input operator are established through analytic semigroup theory. The model is discretized using a Galerkin expansion and basis functions constructed from Fourier polynomials tensored with cubic splines, and convergence criteria for the associated approximate LQR problem are established. The effectiveness of the method for attenuating the coupled longitudinal, circumferential and transverse shell displacements is illustrated through a set of numerical examples.

[1]  Venkata R. Sonti,et al.  Dynamic Effects of Piezoactuators on the Cylindrical Shell Response , 1996 .

[2]  H. Kraus.  Thin elastic shells: an introduction to the theoretical foundations and the analysis of their static and dynamic behavior , 1967 .

[3]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[4]  Irena Lasiecka.  Finite Element Approximations of Compensator Design for Analytic Generators with Fully Unbounded Controls/Observations , 1995 .

[5]  Ralph C. Smith,et al.  Modeling and Parameter Estimation for an Imperfectly Clamped Plate , 1995 .

[6]  Scott D. Snyder,et al.  Active control of interior noise in model aircraft fuselages using piezoceramic actuators , 1990 .

[7]  Richard J. Silcox,et al.  Evaluation of piezoceramic actuators for control of aircraft interior noise , 1992 .

[8]  Harvey Thomas Banks,et al.  An H/sup /spl infin///MinMax periodic control in a two-dimensional structural acoustic model with piezoceramic actuators , 1996 .

[9]  田辺 広城,et al.  Equations of evolution , 1979 .

[10]  Ralph C. Smith,et al.  Spline Approximation of Thin Shell Dynamics , 1997 .

[11]  Kazufumi Ito,et al.  Approximation in LQR problems for infinite dimensional systems with unbounded input operators , 1994 .

[12]  Harvey Thomas Banks,et al.  Approximation Methods for Control of Structural Acoustics Models with Piezoceramic Actuators , 1993 .

[13]  Harvey Thomas Banks,et al.  The modeling of piezoceramic patch interactions with shells, plates, and beams , 1995 .

[14]  I. Lasiecka,et al.  Riccati Equations Arising in Acoustic Structure Interactions with Curved Walls , 1998 .

[15]  Zheng-Hua Luo,et al.  On the Existence of Normal Modes of Damped Discrete-continuous Systems , 1997 .

[16]  R. Showalter.  Hilbert Space Methods for Partial Differential Equations , 1979, Electronic Journal of Differential Equations.

[17]  Harvey Thomas Banks,et al.  A Unified Framework for Approximation in Inverse Problems for Distributed Parameter Systems. , 1988 .

[18]  An H 1 / MinMax Periodic Control in a 2-DStructural Acoustic Model with PiezoceramicActuators , 1994 .

[19]  Irena Lasiecka,et al.  Control and stabilization in nonlinear structural acoustic problems , 1997, Smart Structures.

[20]  Harvey Thomas Banks,et al.  Approximation methods for control of acoustic/structure models with piezoceramic actuators , 1991 .

[21]  H. T. Banksy,et al.  Approximation in Lqr Problems for Innnite Dimensional Systems with Unbounded Input Operators , 1990 .

[22]  Harvey Thomas Banks,et al.  The estimation of material and patch parameters in a PDE-based circular plate model , 1997 .

[23]  Kenneth L. Bowers,et al.  Computation and Control III , 1991 .

[24]  Steven F. Griffin,et al.  INVESTIGATION INTO INACCURACIES IN CLOSED-FORM PIEZOCERAMIC ACTUATOR MODELS FOR SMART STRUCTURES , 1997 .

[25]  Michael A. Demetriou,et al.  An H-infinity/MinMax periodic control in a 2-D structural acoustic model with piezoceramic actuators , 1994 .

[26]  G. Da Prato.  Synthesis of optimal control for an infinite dimensional periodic problem , 1987 .

[27]  Harvey Thomas Banks,et al.  Smart material structures: Modeling, estimation, and control , 1996 .

[28]  A. Leissa,et al.  Vibration of shells , 1973 .

[29]  W. Soedel.  Vibrations of shells and plates , 1981 .

[30]  I. Babuska,et al.  Locking effects in the finite element approximation of elasticity problems , 1992 .

[31]  Juhani Pitkäranta,et al.  On the membrane locking of h-p finite elements in a cylindrical shell problem , 1994 .