M. Salama

Optimal placement of active/passive members in truss structures using simulated annealing

Active structural members with built-in sensing, feedback control, and actuation functions are used herein, along with passively damped members, to augment the inherent damping in truss structures. The effective use of such members makes it desirable to distribute them optimally throughout the structure. For simple structural systems, it is possible to place these members with some degree of optimality on the basis of engineering judgment. However, for more complex systems, the number of possible choices is so large that one may have to rely on a more formal optimization technique. This paper deals with the optimal placement of active and passive members in complex truss structures. The problem falls in the class of combinatorial optimization, for which the solution becomes exceedingly intractable as the problem size increases. This difficulty is overcome herein by use of the simulated annealing technique. We adopt the maximization of the cumulative energy dissipated over a finite time interval as the measure of optimality. The selection of nearly optimal locations for both passive and active members is consistently treated through the use of the finite-time energy dissipation criterion within the framework of the simulated annealing algorithm. Numerical examples are used to illustrate the effectiveness of this methodology.

Simulation of Deployment Dynamics of Inflatable Structures

Analytical simulation of the ine ation process of ine atable structures is key to assessing their robust deployment in a space environment. Although the problem seems to be intractable, a simplie ed e nite volume ine ation model is developed, which allows discretized description of the gas e ow, pressure variation, and resulting nonlinear large deformation throughout domains of the ine atable structure. Simulation results are presented along with convergence studies, which provide insight into the deployment of three different packaging/deployment schemes: Z-unfolding, rollout, and extrusion of a cylindrical tube. The proposed ine ation model is not limited to prismatic one-dimensional components and can be applied to ine atable shapes with complex cone gurations. As a validation of the numerical simulation, a laboratory experiment was conducted on the Z-folded cone guration. Generally, good agreement in the overall ine ation dynamics is observed between the analysis and experiment. EVERALspacemissionshaveconsideredtheuseoflightweight ine atable structures for components such as booms, sunshades, solar concentrators, solar sails, and antennas for nearly all aspects of Earth and space explorations. As a prelude to these missions, the ine atable antenna experiment (IAE) was deployed from the Space Shuttle Endeavour in May 1996 to demonstrate the readiness and reliabilityoftheine atabletechnologyforalarge14-mantennastructure in a realistic space environment. 1 One of the urgent technology issues revealed by the brief 80-min IAE e ight experiment was the need to better understand the dynamics of deploying ine atable structuresinspaceandhowthe ine ationprocessisine uenced by the deployment scheme. This includes the initial packaging and subsequent release and ine ation. Depending on the deployment scheme, large ine atable structures can be extremely e exible to the point of instability, especially during the early stages of ine ation. Rigidization can begin only after e nal deployment is achieved. Testing the deploymentof alargeine atableinan Earth environmentin the presence of gravity and air is of limited value for inferring its deploymentbehavior in space. Therefore, analytical models for simulating and predicting the dynamics of the ine ation process are essential tools for understanding ine atables’ deployment behavior in space environment and for guiding and improving future packaging and deployment concepts. To thisend,this paper explores theuse ofa e nite volume ine ation model that allows a tractable simulation of the deployment process of various ine atable cone gurations, from their initial stowed state to full deployment. Of interest here is a description of all states of ine ation as a function of time. Knowledge of all states of ine ation is essential for subsequent assessment of conditions of stability and controllability of the deployment process.

Combined control-structural optimization

ConclusionsAn approach for combined control-structure optimization keyed to enhancing early design trade-offs has been outlined and illustrated by numerical examples. The approach employs a homotopic strategy and appears to be effective for generating families of designs that can be used in these early trade studies.Analytical results were obtained for classes of structure/control objectives with LQG and LQR costs. For these, we have demonstrated that global optima can be computed for small values of the homotopy parameter. Conditions for local optima along the homotopy path were also given. Details of three numerical examples employing the LQR control cost were given showing variations of the optimal design variables along the homotopy path. The results of the second example suggest that introducing a second homotopy parameter relating the two parts of the control index in the LQG/LQR formulation might serve to enlarge the family of Pareto optima, but its effect on modifying the optimal structural shapes may be analogous to the original parameter λ.

Errors in reduction methods

Abstract A mathematical basis is given for comparing the relative merits of various techniques used to reduce the order of large linear and non-linear dynamics problems during their numerical integration. In such techniques as Guyan-Irons, path derivatives, selected eigenvectors, Ritz vectors, etc., the nth order initial value problem of [ /.y = f(y) for t > 0, y(0) given] is typically reduced to the mth order (m ⪡ n) problem of z = g(z) for t > 0, z(0) given] by the transformation y = Pz where P changes from technique to technique. This paper gives an explicit approximate expression for the reduction error ei in terms of P and the Jacobian of f. It is shown that: (a) reduction techniques are more accurate when the time rate of change of the response y is relatively small; (b) the change in response between two successive stations contributes to the errors at future stations after the change in response is transformed by a filtering matrix H, defined in terms of P; (c) the error committed at a station propagates to future stations by a mixing and scaling matrix G, defined in terms of P, Jacobian of f, and time increment h. The paper discusses the conditions under which the reduction errors may be minimized and gives guidelines for selecting the reduction basis vector, i.e. the columns of P.

Shape adjustment of precision truss structures: analytical and experimental validation

The use of a limited number of actuator/sensor pairs to alter the shape or behavior of a truss structure to other desired states is explored analytically and experimentally. Feasibility of the concept is established by numerical simulation. As a demonstration, a sequence of validation tests is performed and the correlation with analysis is discussed. An existing full scale, space-erectable high precision truss structure is used for this purpose. For the most part, the test results agreed well with the analysis. However, micron-level nonlinearities were discovered in the truss behavior. The significance of these nonlinearities in precision structures and their impact on the basic premise of adaptivity is discussed.

A parallel Householder tridiagonalization stratagem using scattered square decomposition

The parallel stratagem in this paper uses scattered square decomposition, introduced by G. Fox, for its data assignment and then exploits parallelism in the solution steps of the sequential Householder tridiagonalization algorithm. One may condense a real symmetric full matrix A of order n into a tridiagonal form by the stratagem in concurrent machines where N(= D2) processors are used. Expressions for efficiency and speedup are given for the evaluation of the stratagem. An alternative stratagem which requires less data transmission but more computations is also discussed. The results shown that the Householder Method of tridiagonalization may be implemented on a concurrent machine efficiently by scattered square decomposition provided that the number of matrix elements contained in each processor is much larger than the number of processors of the concurrent machine, and the ratio of the time to transmit one data item from one processor to any other processor to the time to perform a floating-point arithmetic operation is small enough.

On nonlinear finite element analysis in single-, multi- and parallel-processors☆

Numerical solution of nonlinear equilibrium problems of structures by means of Newton-Raphson type iterations is reviewed. Each step of the iteration is shown to correspond to the solution of a linear problem, therefore the feasibility of the finite element method for nonlinear analysis is established. Organization and flow of data for various types of digital computers, such as single-processor/single-level memory, single-processor/two-level-memory, vector-processor/two-level-memory, and parallel-processors, with and without sub-structuring (i.e. partitioning) are given. The effect of the relative costs of computation, memory and data transfer on substructuring is shown. The idea of assigning comparable size substructures to parallel processors is exploited. Under Cholesky type factorization schemes, the efficiency of parallel processing is shown to decrease due to the occasional shared data, just as that due to the shared facilities.

Shape estimation from incomplete measurements: a neural-net approach

Accurate estimation of the shape of precision space structures is the key to the success of spaceborne optical systems. A combined simulated annealing and neural-network approach is proposed whereby one can infer the current deformed state of the structure from a limited number of on-board measurements. The approach is especially effective when most of the computations must be done on-ground or off-line, and only minimal calculations are allowed for near real-time on-board processing. It is shown that the performance of the network and its ability to estimate the shape accurately is highly dependent upon the off-line training and tuning of the model to a specific family of expected disturbances. Details of the methodology and results of numerical simulations are given for various on-board estimation scenarios.

Nonlinear equations of dynamics for spinning paraboloidal antennas

Abstract The nonlinear strain-displacement and velocity-displacement relations of spinning imperfect rotational paraboloidal thin shell antennas of linearly elastic but anisotropic materials are derived for non-axisymmetrical deformations. Using these with the admissible trial functions (expressed in terms of any number of known yet unprescribed coordinate functions of spatial variables, and undetermined functions of time) in the principle functional of dynamics, the nonlinear equations of stress inducing motion are expressed in the form of a set of quasi-linear ordinary differential equations of undetermined functions by means of the Rayleigh-Ritz procedure. These equations include all nonlinear terms upto and including the third degree. Both translational and rotational off-sets of the axis of revolution with respect to the spin axis are considered. Explicit expression are given for the coefficient matrices appearing in the nonlinear differential equations.

On numerical nonlinear analysis of highly flexible spinning cantilevers

The general nonlinear discretized equations of motion of spinning elastic solids and structures are derived as a set of nonlinear ordinary differential equations for the case when the strain-displacement and velocity-displacement relations are nonlinear up to the second order. It is shown that the cost of generation of such equations is proportional to the fourth power of the number of degrees of freedom. A computer program is written to automatically generate the equations for the case of spinning cantilevers with initial imperfections. The types and the number of the coordinate functions used in the trial solution are parameters of the program.