By D. Durban, Dan Givoli, J.G. Simmonds

The optimum keep watch over of versatile constructions is an energetic zone of analysis. the most physique of labor during this sector is anxious with the keep an eye on of time-dependent displacements and stresses, and assumes linear elastic stipulations, specifically linear elastic fabric habit and small defor- tion. See, e. g. , [1]–[3], the collections of papers [4, 5], and references therein. nonetheless, within the current paper we reflect on the static optimum keep watch over of a constitution made up of a nonlinear elastic fabric and und- going huge deformation. a major software is the suppression of static or quasi-static elastic deformation in versatile house buildings comparable to elements of satellites by way of regulate quite a bit [6]. sunlight rad- tion and radiation from different resources result in a temperature box within the constitution, which in flip generates an elastic displacement box. The displacements needs to frequently fulfill definite barriers dictated via the allowed operating stipulations of assorted orientation-sensitive tools and antennas within the area motor vehicle. for instance, a parabolic reflector may well stop to be potent whilst present process huge deflection. The elastic deformation will be diminished by way of use of keep watch over so much, that could be imp- mented through mechanically-based actuators or extra smooth piezoelectric units. whilst the constitution into consideration is made up of a rubb- like fabric and is present process huge deformation, nonlinear fabric and geometric results needs to be taken under consideration within the research.

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The balance equations for the bending problem are written as (24) (25) (26) where (27) (28) in which H α is the director moment vector. The elimination of N 3 from the AMB and DMB equations gives the following moment equilibrium equation: (29) where the internal and external moment vectors are defined by (30) When the shearing forces N α 3 are eliminated from Eqs. 2 Kinematics for the bending problem The deformed in-plane base vectors are written as (32) Since the Kirchhoff-Love theory is used, the deformed normal where vector is written as (33) Let θ 2 and θ 3 be the components of rotational variable which transforms A3 such that into (34) The comparison of Eq.

Since the constitutive equations and the compatibility equations are the same as those of Eq. (72), we show the balance equations and the boundary conditions as follows: Constitutive Equations + Compatibility equations (74) Note that, in the case of Hu-Washizu type functional, the moment equilibrium equation is recovered in place of the AMB and DMB equations. With the help of Legendre transformation, we may have the complementary function defined by (75) (76) Substituting Eq. (75) or Eq. (76) into Eq.

M. Zaslavskii, Physics of Chaos in Hamiltonian Systems, World Scientific, 1998. SUETAKE*** * UCLA, 7704 Boelter Hall, Los Angeles, CA 90095-1600, USA. ** Tokyo Denki University, Hatoyama, Hiki, Saitama, Japan. *** Ashikaga Institute of Technology, Ashikaga, Tochigi, Japan. 1 . Introduction The concept of a finite rotation vector has been introduced by Simmonds and Danielson [15, 16] to develop a nonlinear shell theory. Initially, the objective of introducing the finite rotation vector was, to derive a simple form for the governing equations of nonlinear shell theory (see Atluri [3] and Pietraszkiewicz [13]).