The purpose of this monograph is threefold. First, mathematical models of the transient behavior of some or all of the state variables describing the motion of multiple-link flexible structures will be developed. The structures which we have in mind consist of finitely many interconnected flexible ele­ ments such as strings, beams, plates and shells or combinations thereof and are representative of trusses, frames, robot arms, solar panels, antennae, deformable mirrors, etc. , currently in use. For example, a typical subsys­ tem found in almost all aircraft and space vehicles consists of beam, plate and/or shell elements attached to each other in a rigid or flexible manner. Due to limitations on their weights, the elements themselves must be highly flexible, and due to limitations on their initial configuration (i. e. , before de­ ployment), those aggregates often have to contain several links so that the substructure may be unfolded or telescoped once it is deployed. The point of view we wish to adopt is that in order to understand completely the dynamic response of a complex elastic structure it is not sufficient to con­ to take into account the sider only its global motion but also necessary flexibility of individual elements and the interaction and transmission of elastic effects such as bending, torsion and axial deformations at junctions where members are connected to each other. The second object of this book is to provide rigorous mathematical analyses of the resulting models.

Inhalt

I. Introduction.- 1. General Overview.- 2. On the Contents of the Book.- II. Modeling of Networks of Elastic Strings.- 1. Modeling of Nonlinear Elastic Strings.- 2. Networks of Nonlinear Elastic Strings.- 3. Linearization.- 4. Well-posedness of the Network Equations.- 5. Controllability of Networks of Elastic Strings.- 5.1. Exact Controllability of Tree Networks.- 5.2. Lack of Controllability for Networks with Closed Circuits.- 6. Stabilizability of String Networks.- 7. String Networks with Masses at the Nodes.- III. Networks of Thermoelastic Beams.- 1. Modeling of a Thin Thermoelastic Curved Beam.- 2. The Equations of Motion.- 2.1. Some Remarks on Warping and Torsion.- 3. Rotating Beams.- 3.1. Dynamic Stiffening.- 4. Straight, Untwisted, Nonshearable Nonlinear 3-d Beams.- 4.1. Approximation-Generalizations.- 5. Straight, Untwisted Shearable Linear 3-d Beams.- 6. Shearable Nonlinear 2-d Beams with Curvature.- 6.1. Approximation-Generalizations.- 7. A List of Beam Models.- Damping.- 8. Networks of Beams.- 8.1. Geometric Joint Conditions.- 8.1.1. Rigid Joints.- 8.1.2. Pinned Joints.- 8.2. Dynamic Joint Conditions.- 8.2.1. Rigid Joints.- 8.2.2. Pinned Joints.- 9. Rotating Two-link Flexible Nonlinear Shearable Beams.- IV. A General Hyperbolic Model for Networks.- 1. The General Model.- 2. Some Special Cases.- 2.1. String Networks.- 2.2. Networks of Planar Timoshenko Beams.- 2.4. Networks of Initially Curved Bresse Beams.- 2.5. Beams and Strings.- 3. Existence and Regularity of Solutions.- 4. Energy Estimates for Hyperbolic Systems.- 5. Exact Controllability of the Network Model.- 6. Stabilizability of the Network Model.- V. Spectral Analysis and Numerical Simulations.- 1. Preliminaries.- 1.1. Notation.- 1.2. Networks of Strings.- 1.3. Networks of Timoshenko Beams.- 1.4. Networks of Euler-Bernoulli Beams.- 2. Eigenvalue Problems for Networks of 1-d Elements.- 2.1. Introduction.- 2.2. General String Networks.- 2.3. Homogeneous String Networks.- 2.3.1. Examples.- 2.4. Networks of Timoshenko Beams.- 2.4.1. The Case Where ? =0.- 2.4.2. The Case Where ? Belongs to an Individual Beam.- 2.4.3. Eigenvalues for the Entire Graph.- 2.5. Homogeneous Timoshenko Networks.- 3. Numerical Simulations of Controlled 1-d Networks.- 3.1. Introductory Remarks.- 3.2. Networks of Strings.- 3.2.1. Absorbing Controls.- 3.2.2. Directing Controls.- 4. Finite Element Approximations of Timoshenko Networks.- 5. Implicit Runge-Kutta Method: Dry Friction at Joints.- VI. Interconnected Membranes.- 1. Modeling of Dynamic Nonlinear Elastic Membranes.- 1.1. Equations of Motion.- 1.2. Edge Conditions.- 1.3. Hamilton's Principle.- 2. Systems of Interconnected Elastic Membranes.- 2.1. Geometric Junction Conditions.- 2.2. Dynamic Conditions.- 2.3. Linearization.- 2.4. Well-Posedness of the Linear Model.- 3. Controllability of Linked Isotropic Membranes.- 3.1. Observability Estimates for the Homogeneous Problem.- 3.2. A Priori Estimates for Serially Connected Membranes.- 3.3. A Priori Estimates for Single Jointed Membrane Systems.- 3.4. The Reachable States.- 3.4.1. Serially Connected Membranes.- 3.4.2. Membrane Transmission Problems.- VII. Systems of Linked Plates.- 1. Modeling of Dynamic Nonlinear Elastic Plates.- 1.1. Equations of Motion.- 1.2. Edge Conditions.- 1.3. Hamilton's Principle.- 1.4. Additional Kinematic and Material Assumptions.- 1.5. Rotations Associated with Plate Deformation.- 2. Linearization.- 2.1. Linearization of Equations of Motion.- 2.2. Linearization of Edge Conditions.- 2.3. Hamilton's Principle for the Reissner Model.- 2.4. Linearization of the Vector Rotation Angle.- 2.5. The Kirchhoff Plate Model.- 3. Systems of Linked Reissner Plates.- 3.1. Geometric Junction Conditions.- 3.2. Linearization of the Geometric Joint Conditions.- 3.3. Dynamic Joint Conditions.- 3.3.1. Dynamic conditions at a connected joint.- 3.3.2. Dynamic conditions at a hinged joint.- 3.3.3. Dynamic conditions at a semi-rigid joint.- 3.3.4. Dynamic conditions at a rigid joint.- 3.4. Junctions With Masses and Applied Forces.- 3.4.1. Dynamic conditions at a connected joint.- 3.4.2. Dynamic conditions at a hinged joint.- 3.4.3. Dynamic conditions at a semi-rigid joint.- 3.4.4. Dynamic conditions at a rigid joint.- 4. Well-posedness of Systems of Linked Reissner Plates.- 4.1. Function Spaces for Linked Plates.- 4.2. Existence and Uniqueness of Solutions.- 5. Controllability of Linked Reissner Plates.- 5.1. Controllability in Transmission Problems for Thin Plates.- 5.1.1. Observability Estimates and Consequences.- 6. Systems of Linked Kirchhoff Plates.- 6.1. Semi-rigid Joints.- 6.1.1. Interpretation of Theorem 6..- 6.2. Proof of Theorem 6.1.- 6.3. Connected, Hinged or Rigid Joints.- VIII. Plate-Beam Systems.- 1. Introduction.- 2. Modeling of the Plate-Beam Junction: I.- 2.1. Geometric Conditions.- 2.2. Dynamic conditions.- 3. Function Spaces and Well-Posedness.- 4. The Reachable Set.- 4.1. Observability estimates.- 4.2. Proofs of Theorems 4.1 and 4.2.- 5. Limit Model as the Shear Moduli Approach Infinity.- 5.1. Interpretation of the Limit Model.- 6. Modeling of a Plate-Beam Junction: II.- 6.1. Geometric Conditions.- 6.2. Dynamic Conditions.