Shape design of pin-jointed multistable compliant mechanisms using snapthrough behavior

A general approach is presented for generating pin-jointed multistable compliant mechanisms using snapthrough behavior. An optimization problem is formulated for minimizing the total structural volume under constraints on the displacements at the specified nodes, stiffnesses at initial and final states, and load factors to lead to snapthrough behavior. The design variables are cross-sectional areas and the nodal coordinates. It is shown in the numerical examples that several mechanisms can be naturally found as a result of optimization starting from randomly selected initial solutions. It is also shown that no local bifurcation point exists along the equilibrium path, and the obtained mechanism is not sensitive to initial imperfections.     

Optimal design of vibrating beams with unspecified support reactions

The optimal design problem concerns vibrating beams having unspecified thicknesses and lengths as well as supports whose positions and stiffnesses are to be determined. The character of optimal solutions is discussed, and it is shown that the optimal support position may correspond to unimodal or multimodal vibrating states.An iterative design procedure is discussed and applied to generate optimal designs in simple cases. Optimal design for more complex cases can be achieved by proper scaling and connecting basic solutions.     

Buckling and initial post-buckling behavior of thin-walled I columns

The behavior of axially compressed thin-walled I columns after torsional buckling is discussed. At first the state of strain in a column subjected to a large angle of twist is outlined. The considerations are based on the following assumptions: (1) cross section is non-deformable in its own plane, (2) shear deformation is negligible, and (3) strains are small and elastic. The initial post-buckling equilibrium paths are determined by utilizing a perturbation approach. The finite-element and analytical procedure is presented. It has been shown that the point of bifurcation for simply supported, or clamped I column with constant cross section is symmetric and stable. Two examples of I columns of variable cross section are also considered. It is worthwhile noticing that in these examples the critical loads are beyond the limits described by the critical force for column with the minimum and the maximum cross section. The points of bifurcation in these cases are also symmetric and stable. This property of the bifurcation point is very important with regard to the sensitivity to the initial geometrical imperfections. In the case of the unstable point of bifurcation a drastic decrease of the value of the critical loads is possible, which does not hold for stable point.     

Vibrations and large postbuckling deflections of optimal pinned columns with elastic foundations

The optimal distribution of material to maximize the critical load of columns has been studied extensively in the past, along with initial postbuckling behavior. Here, large postbuckling deflections are analyzed for optimal columns with pinned ends. Small vibrations of the optimal columns about postbuckled equilibrium shapes are also investigated. A shooting method is utilized to obtain numerical solutions. In some examples, an elastic foundation is attached to the column. The foundation includes the usual transverse resistance and an axial resisting force. The bifurcation is subcritical in some cases, and then the column is imperfection-sensitive. Results are compared to those for the corresponding uniform column with the same total volume.     

FEM-based analysis of the local-plate/distortional mode interaction in cold-formed steel lipped channel columns

This paper reports the results of a numerical investigation concerning the elastic and elastic–plastic post-buckling behaviour of cold-formed steel lipped channel columns affected by local-plate/distortional buckling mode interaction. The results presented and discussed were obtained through analyses performed using the finite element code Abaqus and discretising the columns by means of fine 4-node shell element meshes. The columns analysed (i) are simply supported (end sections locally/globally pinned and free-to-warp), (ii) have cross-section dimensions that ensure equal local-plate and distortional critical buckling stresses, thus maximising the local-plate/distortional mode interaction effects, and (iii) contain critical-mode initial geometrical imperfections that exhibit different shapes but share the same combined amplitude. The numerical post-buckling results reported consist of (i) elastic and elastic–plastic non-linear equilibrium paths, (ii) curves and figures describing how the column deformed configuration (expressed as a linear combination of its local-plate and distortional components) evolves along the elastic post-buckling equilibrium paths and (iii) figures providing a clear visualisation of the (iii₁) evolution of the elastic–plastic column deformed configurations, (iii₂) the growth of the plastic strains and (iii₃) failure mechanisms exhibited by a fairly large portion of the elastic–plastic columns that were analysed in this work.     

Non-linear GBT formulation for open-section thin-walled members with arbitrary support conditions

This paper presents the development and illustrates the application of a beam finite element based on Generalised Beam Theory (GBT) and intended to analyse the elastic post-buckling behaviour of thin-walled steel members exhibiting arbitrary open cross-section and with non-standard support conditions (e.g., in-span bracing systems). After briefly reviewing the main concepts and procedures required to obtain the GBT system of non-linear equilibrium equations, the paper describes the steps involved in the numerical implementation (incremental–iterative strategy) of a non-linear beam finite element that incorporates the influence of the non-standard support conditions. Finally, the application and capabilities of the proposed GBT-based beam finite element are illustrated by means of the presentation and discussion of numerical results concerning the post-buckling behaviour of lipped I-section columns and lipped channel columns and beams with and without localised displacement restraints. For validation purposes, most GBT-based results are compared with values yielded by shell finite element analyses carried out in the commercial code Ansys.     

Stabilization of linear autonomous systems of differential equations with distributed delay

Consideration is given to the problem of optimal stabilization of differential equation systems with distributed delay. The optimal stabilizing control is formed according to the principle of feedback. The formulation of the problem in the functional space of states is used. It was shown that coefficients of the optimal stabilizing control are defined by algebraic and functional-differential Riccati equations. To find solutions to Riccati equations, the method of successive approximations is used. The problem for this control law and performance criterion is to find coefficients of a differential equation system with distributed delay, for which the chosen control is a control of optimal stabilization. A class of control laws for which the posed problem admits an analytic solution is described.     

The large deformation,post-buckling and catastrophic stability of slender cantilever columns

Several solutions, analytical and numerical, exist for the cases of nonlinear bending of beams and post-buckling of columns. However, the solution for the general case of an arbitrarily inclined load does not seem to exist. In this paper, the governing equation of equilibrium for a cantilever of variable cross section under an arbitrarily inclined load is formulated, by using the axial and transverse displacement quantities. The solution is obtained by means of a simple numerical iterative procedure. The results for the post-buckling and non-linear bending of the structure under concentrated tip load, uniformily distributed load and linearly varying load are presented, and compared with the available results, wherever possible. Further, a type of catastrophic behaviour in post-buckling region is observed and studied using the same numerical scheme. The method is general and can be used for any load variation and for any tapered beam-column.     

Buckling and post-buckling analysis of extensible beam-columns by using the differential quadrature method

Buckling and post-buckling analysis of extensible beam-columns is performed numerically in this paper. It was experienced earlier that in some cases the numerical integration would not produce the convergent post-buckling solutions, especially under high loads. Therefore, a new differential quadrature (DQ) based iterative numerical integration method is proposed to solve post-buckling differential equations of extensible beam-columns. Six cases, including five classical Euler buckling cases, are analyzed. Critical loads and convergent post-buckling solutions under different applied loads are obtained. The results are compared with the existing multiple scales solutions. It is found that under high applied loads, the small rotation assumption in obtaining multiple scales solutions is no longer valid. The proposed iterative DQ based numerical integration method can yield reliable and accurate post-buckling solutions even at high applied loads for the cases investigated.     

Optimal shape of a heavy compressed column

By using Pontryagins maximum principle we determine the shape of a heavy compressed rod, stable against buckling. It is assumed that the eigenvalue pair corresponding to the optimal rod is simple. With this assumption (unimodal optimization) it is shown that the cross-sectional area function is determined from the solution of a nonlinear boundary value problem. A variational principle for this boundary value problem is formulated and two first integrals are constructed. The optimal shape of a rod is determined by numerical integration.     

Post-buckling behaviour and strength of cold-formed steel lipped channel columns experiencing distortional/global interaction

This paper reports the results of a numerical investigation concerning the elastic and elastic–plastic post-buckling behaviour of cold-formed steel lipped channel columns affected by distortional/global (flexural–torsional) buckling mode interaction. The results presented and discussed were obtained by means of analyses performed using the finite element code Abaqus and adopting column discretisations into fine 4-node isoparametric shell element meshes. The columns analysed (i) are simply supported (locally/globally pinned end sections that may warp freely), (ii) have cross-section dimensions and lengths that ensure equal distortional and global (flexural–torsional) critical buckling loads, thus maximising the distortional/global mode interaction effects, and (iii) contain critical-mode initial geometrical imperfections exhibiting different configurations, all corresponding to linear combinations of the two “competing” critical buckling modes. After briefly addressing the lipped channel column “pure” distortional and global post-buckling behaviours, one presents and discusses in great detail a fair number of numerical results concerning the post-buckling behaviour and strength of similar columns experiencing strong distortional/global mode interaction effects. These results consist of (i) elastic (mostly) and elastic–plastic non-linear equilibrium paths, (ii) curves or figures providing the evolution of the deformed configurations of several columns (expressed as linear combination of their distortional and global components) and, for the elastic–plastic columns, (iii) figures enabling a clear visualisation of (iii₁) the location and growth of the plastic strains and (iii₂) the characteristics of the failure mechanisms more often detected in the course of this research work.     

On the optimal shape of a compressed column subjected to restrictions on the cross-sectional area

By using Pontryagin’s principle we study the optimal shape of an elastic compressed column with clamped ends and restrictions on cross-sectional area. The restrictions are imposed on either maximum or both minimal and maximum value of the cross-sectional area. We analyzed a column on elastic foundation of Winkler type and a column without foundation. It is shown that the optimization can be both bimodal and unimodal. We determine the transition value between unimodal and bimodal optimization for specified values of parameters.     

Suboptimal control of singularly perturbed systems and periodicoptimization

A traditional approach to singularly perturbed optimal control problems is based on an approximation of these problems by reduced problems which are obtained via the formal replacement of the fast variables by the states of equilibrium of the fast subsystems considered with frozen slow variables and controls. It is shown that such an approximation is valid if and only if certain families of periodic optimization problems admit steady state solutions. It is also shown how the solutions of these problems can be used to construct suboptimal controls for singularly perturbed problems when approximation by reduced problems is not possible     

Homoclinic and heteroclinic orbits underlying the post-buckling of axially-compressed cylindrical shells

A structural system with an unstable post-buckling response that subsequently restabilizes has the potential to exhibit homoclinic connections from the fundamental equilibrium state to itself over a range of loads, and heteroclinic connections between fundamental and periodic equilibrium states over a different (smaller) range of loads. It is argued that such equilibrium configurations are important in the interpretation of observed behaviour, and govern the minimum possible post-buckling loads.

To illustrate this, the classical problem of a long thin axially-compressed cylindrical shell is revisited from three different perspectives: asymptotic conjecture, analogy with nonlinear dynamics, and numerical continuation analysis of a partial spectral decomposition of the underlying equilibrium equations. The nonlinear dynamics analogy demonstrates that the structure of the heteroclinic connections is more complicated than that indicated by the asymptotics: this is confirmed by the numerics. However, when the asymptotic portrayal is compared to the numerics, it turns out to be surprisingly accurate in its Maxwell-load prediction of the practically-significant first minimum to appear in the post-buckling regime.     

Noisy relativistic quantum games in noninertial frames

The influence of noise and of Unruh effect on quantum Prisoners’ dilemma is investigated both for entangled and unentangled initial states. The noise is incorporated through amplitude damping channel. For unentangled initial state, the decoherence compensates for the adverse effect of acceleration of the frame and the effect of acceleration becomes irrelevant provided the game is fully decohered. It is shown that the inertial player always out scores the noninertial player by choosing defection. For maximally entangled initially state, we show that for fully decohered case every strategy profile results in either of the two possible equilibrium outcomes. Two of the four possible strategy profiles become Pareto optimal and Nash equilibrium and no dilemma is leftover. It is shown that other equilibrium points emerge for different region of values of decoherence parameter that are either Pareto optimal or Pareto inefficient in the quantum strategic spaces. It is shown that the Eisert et al. (Phys Rev Lett 83:3077, 1999) miracle move is a special move that leads always to distinguishable results compare to other moves. We show that the dilemma like situation is resolved in favor of one player or the other.     

Post-buckling of cylindrically orthotropic linearly tapered circular plates by finite element method

The post-buckling behavior of tapered circular plates with cylindrically orthotropic material properties is studied in this paper through a finite element formulation. The results in the form of an empirical formula for the radial load ratios are presented for various values of the taper parameter and orthotropy parameter. Both simply supported and clamped boundary conditions are considered.     

Structural optimization for post-buckling behavior using particle swarms

The aim of this paper is to develop a new algorithm based on the particle swarm optimization (PSO) concept and then to apply it in the solution of some new structural optimization problems for post-buckling behavior. Proposed modifications of the algorithm regard both the PSO kernel and the constraints handling. The “controlled reflection” technique is proposed for dealing with inequality constraints. The values of the objective are calculated for some control points chosen along a move vector. The position for which the objective is the smallest one and the constraints are not violated is selected. For the case of equality constraints, the “particle trap” strategy is proposed. First, equalities are transformed into inequalities forming constraint “zone of influence.” If a particle from a swarm drops into this “zone” it remains trapped there and can move further only inside this subspace. Simultaneously, a penalty term is added to the objective function to force particles to be “captured” and constraints to become active at the optimum. The new PSO algorithm has been successfully applied to problems of structural optimization against instability. The standard maximization of the critical load is performed both for single and double buckling loads. The modified optimization for post-buckling behavior is also performed. A new problem of reconstruction of a predicted post-buckling path is formulated. The sum of squared distances between the control points of a given equilibrium path and the reconstructed one is minimized. Another new problem regards the modification of the slope of nonlinear equilibrium curve. This is obtained by adding a set of post-buckling constraints imposed on derivative values calculated for selected control points at the equilibrium curve.This is the full version of the paper presented at the Congress WCSMO6, Rio de Janeiro, 2005.     

Model-free Q-learning designs for linear discrete-time zero-sum games with application to H-infinity control

In this paper, the optimal strategies for discrete-time linear system quadratic zero-sum games related to the H-infinity optimal control problem are solved in forward time without knowing the system dynamical matrices. The idea is to solve for an action dependent value function Q(x,u,w) of the zero-sum game instead of solving for the state dependent value function V(x) which satisfies a corresponding game algebraic Riccati equation (GARE). Since the state and actions spaces are continuous, two action networks and one critic network are used that are adaptively tuned in forward time using adaptive critic methods. The result is a Q-learning approximate dynamic programming (ADP) model-free approach that solves the zero-sum game forward in time. It is shown that the critic converges to the game value function and the action networks converge to the Nash equilibrium of the game. Proofs of convergence of the algorithm are shown. It is proven that the algorithm ends up to be a model-free iterative algorithm to solve the GARE of the linear quadratic discrete-time zero-sum game. The effectiveness of this method is shown by performing an H-infinity control autopilot design for an F-16 aircraft.     

Biaxial buckling and post-buckling of composite laminated plates on two-parameter elastic foundations

A post-buckling analysis is presented for a simply supported, composite laminated rectangular plate under biaxial compressive loading and resting on a two-parameter (Pasternak-type) elastic foundation. The analysis uses a perturbation technique to determine the interactive buckling loads and post-buckling equilibrium paths. The initial geometrical imperfection of the plates is taken into account. Numerical examples are presented that relate to the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated rectangular plates. Typical results are presented in dimensionless graphical form.     

From Pre-Historic to Post-Modern Symbolic Model Checking

Symbolic model checking, which enables the automatic verification of large systems, proceeds by calculating expressions that represent state sets. Traditionally, symbolic model-checking tools are based on backward state traversal; their basic operation is the function pre, which, given a set of states, returns the set of all predecessor states. This is because specifiers usually employ formalisms with future-time modalities, which are naturally evaluated by iterating applications of pre. It has been shown experimentally that symbolic model checking can perform significantly better if it is based, instead, on forward state traversal; in this case, the basic operation is the function post, which, given a set of states, returns the set of all successor states. This is because forward state traversal can ensure that only parts of the state space that are reachable from an initial state and relevant for the satisfaction or violation of the specification are explored; that is, errors can be detected as soon as possible.In this paper, we investigate which specifications can be checked by symbolic forward state traversal. We formulate the problems of symbolic backward and forward model checking by means of two -calculi. The pre- calculus is based on the pre operation, and the post- calculus is based on the post operation. These two -calculi induce query logics, which augment fixpoint expressions with a boolean emptiness query. Using query logics, we are able to relate and compare the symbolic backward and forward approaches. In particular, we prove that all -regular (linear-time) specifications can be expressed as post- queries, and therefore checked using symbolic forward state traversal. On the other hand, we show that there are simple branching-time specifications that cannot be checked in this way.