The Fokker-Planck equation, a convective-diffusion equation governing the evolution of the transition probability density function of a class of dynamical systems subjected to both additive and multiplicative excitations exhibiting Markovian response, has been solved successfully by a finite-element method for certain second and third order systems. The development of post-processing software to compute marginal density functions, n -th order moments of response, and extremal statistics associated with these distributions is now complete. Visualization methods have been used to examine the details of resulting probability flows in the phase space. Current research is directed at higher dimensional systems.

One way to investigate the vibration of a large complex structure is to divide it into a multiplicity of simpler substructures that can be analyzed individually. The vibration characteristics of these subsystems are then used to obtain those of the combined system, thus reducing the analysis of a complex system to the problem of investigating some characteristic matrix whose order depends on the number of degrees of freedom taking part in the interaction between subsystems. The Green's operators of the subsystems are used to obtain the characteristic equation of the system, which can be efficiently solved. This research has encompassed both active and passive systems, with current efforts directed at the synthesis of Green's operators for increasingly complex substructures.

It sometimes may be desirable to monitor the response of one or more modes of a system to warn of the onset of pathological behavior associated with a single mode, such as flutter. This is not straightforward, particularly if the goal is to perform on-line, or ``on-the-fly,'' monitoring in the presence of unmodeled dynamics, unknown external forces, rapidly changing control forces, and various noise sources associated with real measurements. In this context, the efficacy of the concept of modal filtering has been demonstrated, but there remain questions regarding its use for identification of systems having significant uncertainty in their physical, control, and input parameters. These are the areas that we are currently addressing, using the concepts of adaptivity and robustness in the design of the modal filter.

Parameter uncertainty can degrade the performance of an otherwise well-designed control system, sometimes leading to system instability. In the context of structural control, performance degradation and instability imply excessive vibration and even structural failure. The ability of a controller to maintain the stability of a system in spite of parameter uncertainty is measured by its robustness, which can be viewed as a probability measure, wherein the joint distribution is of a dimension equal to the number of uncertain parameters, and the failure hypersurface is defined by the onset of instability in the eigenspace. This observation has led to some recent analyses employing a MATLAB implementation of FORM for systems with multiple failure modes.

An analysis of subsonic and supersonic torsion-bending flutter, including rotary inertia, shear, and hearing effects, of a time-dependent linear viscoelastic lifting surface consisting of either a Bernoulli-Euler or a Timoshenko beam is formulated using aerodynamic strip theory. Complex moduli models for aluminum are characterized as functions of temperature and frequency by fitting Chebyshev polynomials to actual material experimental data. The flutter analysis is carried out in the complex plane and a computerized iterative method for the determination of flutter speeds and frequencies is developed. The influence of viscoelastic material properties (storage and loss moduli), temperature, rotary inertia, and shear effects is evaluated.

Advanced composite laminates are being used in flight vehicles to improve performance by substantial structural weight savings. Present numerical analysis requires computers with large storage and takes long real time to complete the calculations. The method under development uses Laplace transforms and thereby requires computer real time use comparable to elastic anisotropic analyses. Results of various loading conditions compare extremely well with exact analytical solutions. Finite-element analyses for dynamic loadings on anisotropic viscoelastic composites that save extensive computer time and storage have been developed. The numerical results compare extremely well with analytical exact solutions.

The consideration of using orthotropic, composite material structures in aerospace applications is increasing rapidly. The use of these materials in the wing analysis offers some new analytical challenges as well as potential benefits. In this investigation an analytical model is being developed for handling monocoque and semi-monocoque wings composed of laminated, thin-walled construction. The model is applied to the analysis of aeroelastic phenomena, and the effects of the orthotropic properties are examined in detail. The aeroelastic phenomena examined include static problems such as divergence and the dynamic problem of bending-torsion flutter.

The finite-element models being developed applies to general configurations with either open or closed sections and are designed to retain the orthotropic properties in a most general manner for each layer without limitation on a number of layers. The model being developed holds for general geometry sections. Each section is divided into segments. The geometry of the segments can be arranged so as to create any desired open or closed section. Each segment of the section can have a different number of orthotropic material layers, with each layer having its unique orthotropic axes relative to beam coordinates. A set of beam conditions is applied then through the thickness of the segments and the beam section as a whole.

In the application of thin, laminated, orthotropic beams, it is necessary to position the shear center relative to the cross-section. The definition of the shear center, although similar to that for the isotropic beam, does not have the same meaning. In the case of the orthotropic beam, a load can act through the center yet produce twist. The shear center for orthotropic structures is defined relative to shear stress distribution through the thin section.