ABSTRACT The stability of mechanical systems that are subjected to nonconservative forces and small viscous damping is investigated. Necessary conditions for the perturbations to be ideal are derived and asymptotic formulas for critical values are obtained. The structure of matrices that realize the ideal perturbations is studied. Necessary stability conditions and formulas for evaluating the defect of the damping matrix are derived for the case of defective perturbations. Examples are considered.

ABSTRACT In this paper, mathematical properties of variational solutions and boundary integral equation solutions of linear elasticity problems are studied. It is first noted that a variational solution of the three-dimensional linear elasticity problem exists in the Sobolev space [H1(Ω)]3. It is then shown that a unique solution of the boundary integral equation is identical to the variational solution in [H1(Ω)]3. To represent the boundary integral equation, Green's formula in the Sobolev space is utilized on the solution domain, excluding a ball, with small radius ρ that is centered at the point where the point load is applied. By letting ρ tend to zero, it is shown that, for the linear elasticity problem, the boundary integral expression is valid for the variational solution. From this fact, a numerical approximation of the variational solution can be obtained by the boundary element method, even when a classical solution does not exist.

ABSTRACT The symmetry limit theory of the present authors for beam-columns subjected to idealized completely reversed tip deflection cycling programs is extended to a general branching problem of steady-state paths. It is shown that transition to asymmetric steady states may occur from any symmetric steady state whose tip deflection amplitude is greater than or equal to the symmetry limit amplitude but is less than the upper limit of branching with increasing tip deflection amplitude. It is verified that the smallest amplitude among those at which transition to asymmetric steady states may occur coincides exactly with the symmetry limit, yielding a mathematical proof of the validity of the symmetry limit theory.

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This paper presents a variational formulation of constrained dynamics of flexible multibody systems, using a vector-variational calculus approach. Body reference frames are used to define global position and orientation of individual bodies in the system, located and oriented by position of its origin and Euler parameters, respectively. Small strain linear elastic deformation of individual components, relative to their body references frames, is defined by linear combinations of deformation modes that are induced by constraint reaction forces and normal modes of vibration. A library of kinematic couplings between flexible and/or rigid bodies is defined and analyzed. Variational equations of motion for multibody systems are obtained and reduced to mixed differential-algebraic equations of motion. A space structure that must deform during deployment is analyzed, to illustrate use of the methods developed.

ABSTRACT Large axial displacements of an initially straight, simply supported, column that is subjected to an axial compressive force at its movable support are determined, using several variants of axial strain-displacement relationships. Their accuracy and range of applicability are discussed by comparing corresponding results with those of exact elastica analysis, in which the compressibility effect of the bar axis is taken into account. From a variety of results, the following conclusion is drawn! a simplified linear kinematic relation leads to a sufficiently accurate evaluation of the initial part of the postbuckling path, which is of significant importance for structural design purposes.

ABSTRACT ABSTRACT A recursive formulation of the equations of motion of constrained mechanical systems with closed loops is derived, using tools of variational and vector calculus. Kinematic couplings between pairs of contiguous bodies presented in Part 1 of this paper are generalized. Lagrange multipliers are introduced to account for the effects of joints that are cut to define a tree structure. Constraint Jacobian terms are added to the reduced variational equations derived in Part I. Cut-joint constraint acceleration equations are derived, to complete the reduced equations of motion. Lagrange multipliers associated with each cut-joint are eliminated at the first junction body encountered that permits closing the loop that constraints in cut joint. The inductive algorithm developed in Part I is used to calculate accelerations for the system. A multi-loop compressor is analyzed to illustrate use of the method.

ABSTRACT A problem of optimal location, stiffness, and prestress of supports in an elastic frame structure that is subjected to external loads, displacements, and initial distortions is formulated. Sensitivity analysis and optimality conditions are discussed for the assumed objective functional, accounting for conflicting design requirements that correspond to stiffness and stress constraints. A simple example is presented that illustrates the general theory.

ABSTRACT Much study has been made of post-buckling behavior of an elastic structure in which twofold branching occurs, involving construction of a potential energy function in terms of normalized coordinates. In general, cubic terms dominate the behavior, but when symmetry exists in the modes in which the branching occurs, then the potential function is of quartic type. In previous works, imperfection sensitivity is investigated by way of at most two imperfection parameters, while Catastrophe Theory indicates that seven besides the load parameter are needed for a complete investigation. Here, the problem of a beam on an elastic foundation is considered and a complete set of physically meaningful parameters that provide a universal unfolding is constructed.

ABSTRACT A method of simulating structural degradation due to removal of some elements, by introduction of proper, self-equilibrated states of initial stresses is presented. The initial stresses are caused by geometrically incompatible distortions in the structure.