Spline approximation of thin shell dynamics

Rosario, Ricardo C.H. del; Smith, Ralph C.

doi:10.1002/(sici)1097-0207(19970815)40:15<2807::aid-nme192>3.0.co;2-h

“…Passive patch contributions in the density, moment and force resultants are neglected and the glue bonding layer is assumed to have negligible contribution to the structural dynamics. For systems incorporating passive patch contributions, see 5,7], and discussions regarding incorporation of bonding layers in the model could be found in 5].…”

Section: Shell Modelmentioning

confidence: 99%

Reduced-order model feedback control design: numerical implementation in a thin shell model

Banks

¹

,

Rosario

²

,

Smith

³

2000

IEEE Trans. Automat. Contr.

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Reduced order models employing the Lagrange and POD reduced basis methods in numerical approximation and feedback control of systems are presented and numerically tested. The system under consideration is a thin cylindrical shell with surface-mounted piezoceramic actuators. Donnell-Mushtari equations, modi ed to include Kelvin-Voigt damping, are used to model the system dynamics. Basis functions constructed from Fourier polynomials tensored with cubic splines are employed in the Galerkin expansion of the full order model. Reduced basis elements are then formed from full order approximations of the exogenously excited shell taken at di erent time instances. Numerical examples illustrating the features of the reduced basis methods are presented. As a rst step toward investigating the behavior of the methods when implemented in physical systems, the use of reduced order model feedback control gains in the full order model is considered and numerical examples are presented.

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“…The reader is referred to [7,25] for details concerning the construction of the mass, stiffness and damping matrices M, K and Q.…”

Section: Cylindrical Actuator Modelmentioning

confidence: 99%

Model Development for Atomic Force Microscope Stage Mechanisms

Smith

¹

,

Hatch

²

,

De

³

et al. 2006

SIAM J. Appl. Math.

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In this paper, we develop nonlinear constitutive equations and resulting system models quantifying the nonlinear and hysteretic field-displacement relations inherent to lead zirconate titanate (PZT) devices employed in atomic force microscope stage mechanisms. We focus specifically on PZT rods utilizing d 33 motion and PZT shells driven in d 31 regimes, but the modeling framework is sufficiently general to accommodate a variety of drive geometries. In the first step of the model development, lattice-level energy relations are combined with stochastic homogenization techniques to construct nonlinear constitutive relations which accommodate the hysteresis inherent to ferroelectric compounds. Secondly, these constitutive relations are employed in classical rod and shell relations to construct system models appropriate for presently employed nanopositioner designs. The capability of the models to quantify the frequency-dependent hysteresis inherent to the PZT stages is illustrated through comparison with experimental data. i Report Documentation Page Form Approved OMB No. 0704-0188Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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“…Among the issues which must be addressed when constructing finite element or general Galerkin methods for the shell is the choice of elements which avoid shear and membrane locking and the maintenance of boundary conditions. We summarize here a spline-based Galerkin method developed in [7] for thin shells and direct the reader to that source for details regarding the construction of constituent matrices and convergence properties of the method. Details regarding the use of this approximation method for LQR control of shells utilizing PZT actuators can be found in [8].…”

Section: Cylindrical Actuator Modelmentioning

confidence: 99%

Model Development for Atomic Force Microscope Stage Mechanisms

Smith¹,

Hatch²,

De³

et al. 2005

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In this paper, we develop nonlinear constitutive equations and resulting system models quantifying the nonlinear and hysteretic field-displacement relations inherent to lead zirconate titanate (PZT) devices employed in atomic force microscope stage mechanisms. We focus specifically on PZT rods utilizing d 33 motion and PZT shells driven in d 31 regimes, but the modeling framework is sufficiently general to accommodate a variety of drive geometries. In the first step of the model development, lattice-level energy relations are combined with stochastic homogenization techniques to construct nonlinear constitutive relations which accommodate the hysteresis inherent to ferroelectric compounds. Secondly, these constitutive relations are employed in classical rod and shell relations to construct system models appropriate for presently employed nanopositioner designs. The capability of the models to quantify the frequency-dependent hysteresis inherent to the PZT stages is illustrated through comparison with experimental data. i Report Documentation Page Form Approved OMB No. 0704-0188Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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Spline approximation of thin shell dynamics

Cited by 10 publications

References 19 publications

Reduced-order model feedback control design: numerical implementation in a thin shell model

Reduced-order model feedback control design: numerical implementation in a thin shell model

Model Development for Atomic Force Microscope Stage Mechanisms

Model Development for Atomic Force Microscope Stage Mechanisms

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