Digital simulation of random processes and its applications

With the aid of the finite element method, the present paper deals with the problem of structural response variability resulting from the spatial variability of material properties of structures, when they are subjected to static loads of a deterministic nature. The spatial variabilities are modeled as two‐dimensional stochastic fields. The finite element discretization is performed in such a way that the size of each element is sufficiently small. Then, the present paper takes advantage of the Neumann expansion technique in deriving the finite element solution for the response variability within the framework of the Monte Carlo method. The Neumann expansion technique permits more detailed comparison between the perturbation and Monte Carlo solutions for accuracy, convergence, and computational efficiency. The result from such a Monte Carlo method is also compared with that based on the commonly used perturbation method. The comparison shows that the validity of the perturbation method is limited to the c...

Neumann Expansion for Stochastic Finite Element Analysis

Common in the design of wind turbine generators of the multi-megawatt size is a soft shaft in the low speed part of the drive train. The presence of the soft shaft gives rise to a low frequency torsional mode involving an oscillation where the hub and blades swing relative to the generator. It is shown that normal wind variations can excite this mode resulting in large oscillatory fluctuations in the drive train torques as well as in the generated electric power. Methods of improving the dynamic performance are examined.

Dynamic Behavior of a Class of Wind Turbine Generators during Random Wind Fluctuations

We present a generalized polynomial chaos algorithm to model the input uncertainty and its propagation in flow-structure interactions. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as the trial basis in the random space. A standard Galerkin projection is applied in the random dimension to obtain the equations in the weak form. The resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode. This approach is a generalization of the original polynomial chaos expansion, which was first introduced by N. Wiener and employs the Hermite polynomials (a subset of the Askey scheme) as the basis in random space. The algorithm is first applied to second-order oscillators to demonstrate convergence, and subsequently is coupled to incompressible Navier-Stokes equations. Error bars are obtained, similar to laboratory experiments, for the pressure distribution on the surface of a cylinder subject to vortex-induced vibrations

https://www.brown.edu/research/projects/scientific-computing/sites/brown.edu.research.projects.scientific-computing/files/uploads/Stochastic%20Modeling%20of%20Flow-Structure%20Interactions.pdf

Stochastic Modeling of Flow-Structure Interactions Using Generalized Polynomial Chaos

Two methods of system identification of n degrees-of-freedom nonlinear structural dynamic systems are studied and applied to identification of the hydrodynamic coefficient matrices associated with nonlinear drag and linear inertia forces appearing in the equations of motion of offshore structures subjected to wave forces. These two methods, being essentially the methods of state estimation, use nonlinear Kalman filtering algorithms which can be applied to parameter estimation problems by regarding each of the parameters involved in the system as an augmented state variable. One of these methods uses the extended Kalman filter while the other was the iterated linear filter-smoother. Analytical simulation studies are performed for two degrees-of-freedom structural systems on the basis of artificially generated input and output observations under the various output noise conditions. Both methods yield good estimates even under the conditions of fairly large amounts of output noise and are moderately...

Identification of Nonlinear Structural Dynamic Systems

T is becoming increasingly apparent that the development of supersonic/hypersonic vehicles such as the NASP cannot become a reality until the fatigue problem under severe aerodynamic, acoustic, and thermal environment is solved. Furthermore, with an increasing use of light-weight materials such as fiber-reinforced composites, metal matrix composites, and various intermetallic compounds, there is an urgent need for improved analytical methods for nonlinear response and sonic fatigue predictions. The proposed NASP concept involves operations at subsonic, supersonic, and hypersonic speeds. Thus, aerodynamic, acoustic, and thermal loadings should be considered for various phases of flight. In the early stages of the takeoff, the aft surface thermal protection systems will be in the near field of the engine exhaust noise. As the speed of the vehicle increases, the effect of engine exhaust noise (except in the near vicinity of exhaust nozzles) will decrease, and at Mach 1 and higher speeds the acoustic loads are expected to become negligible. However, at supersonic and hypersonic speeds, the fluctuating surface pressures due to convecting turbulent boundary layer will become significant. In addition, local impinging shocks on the structural surface could induce severe dynamic loads. Supersonic/hypersonic vehicles will be subjected to severe surface temperatures exceeding SCOOT.1'4 Structural surface components and control devices can be expected to behave in a highly nonlinear fashion with respect to both geometry and materials.4^8 No reliable analytical procedures have yet been developed which could handle these complexities and give meaningful results that are useful for the design of these vehicles. The thermal surface protection systems of aircraft structures are usually constructed from discretely stiffened panels and stiffened shells. A typical discretely stiffened panel is shown in Fig. 1. Due to anticipated high-surface temperatures of advanced supersonic/hypersonic vehicles, multi-wall and/ or multilayer constructions might need to be utilized for the design of thermal surfaces. High-cycle fatigue failures have occurred in discretely stiffened surface panels with the majority of cracks appearing in the near vicinity of the stiffening element.5'13 Thus, proper dynamic interaction between the panel and various stiffening elements should be taken into account when calculating the nonlinear response of the surface panels. A single bay panel with clamped or simple support boundary conditions might provide reasonable estimates on deformations,14'23 but the predictions of principal stresses of a discretely stiffened panel by a single panel model could be unrealistic. Since fatigue damage is controlled by local stress conditions, accurate stress predictions are essential for fatigue life estimates. A time domain Monte Carlo-type approach has been successfully applied to a variety of problems of a linear and nonlinear nature with complex random inputs.24'33 Utilizing the nonlinear time domain stress response solutions and fatigue information from constant amplitude coupon tests, preliminary fatigue damage models have been constructed. The adverse thermal conditions could result in degradation of strength, stiffness, and fatigue life. In addition, thermal inplane loads could induce buckling and "snap through" type vibrations of surface panels. The time domain procedures could account for these effects in the general formulation of the total response problem. General Features of Time-Domain Method For structural dynamic problems of nonlinear/probabilistic nature where close form or effective approximate solutions are not possible, the time domain Monte Carlo method could provide a feasible approach to construct practical solutions. The recent advent of high-speed digital computers has made this procedure a useful and effective method. To illustrate the basic features of the time domain approach, consider a generalized form of a second-order nonlinear equation

Nonlinear Response and Sonic Fatigue of National Aerospace Space Plane Surface Panels

The vibration of a flexible elastic plate is investigated by a Monte Carlo technique. The response analysis is performed in time domain by numerically simulating the resulting generalized forces. The nonlinear plate deflection and mutual interaction between the plate motion and external and/or internal airflow is included. The fluid perturbations due to the structural motion are described by linear aerodynamic theory. The boundary-layer pressure field is idealized as a homogeneous multidimensional Gaussian random process with mean zero. The plate differential equations of motion and boundary conditions are satisfied in a Galerkin sense by developing a modal solution. Illustrative examples are presented for subsonic and supersonic flow regions using shallow cavity and two plate mode approximations.

Nonlinear Panel Response from a Turbulent Boundary Layer

Vertical and torsional motions of a suspension bridge due to wind loading are studied. Both, self-excited, due to bridge motion, and buffeting, independent of bridge motion, wind loads are included. The self-excited aerodynamic forces are modeled using aerodynamic coefficient or Duhamel integral formulations. For this purpose, experimental information on section models from wind tunnel testing is utilized. The buffeting random loads are determined from specified spectral densities or horizontal and vertical turbulent wind velocity fluctuations. Numerical results are obtained using frequency and time domain formulations. Dynamic bridge stability is also investigated.

Motion of Suspension Bridge Subject to Wind Loads

The vibration of clamped and simply supported elastic panels due to subsonic and supersonic turbulent boundary-layer flows is investigated by a Monte Carlo technique. The resulting generalized random forces are simulated numerically from boundary-layer turbulence spectra, and the response analysis is performed in the time domain. The mutual interaction between panel motion and external and/or internal airflow is included. Response studies are performed with respect to rms response, probability structure, peak distribution, threshold crossing and spectral density. The effect on the response statistics of in-plane loading, static pressure differential, and cavity pressure is investigated.

Nonlinear Panel Response by a Monte Carlo Approach

This paper describes experimental studies of interior noise in a twin-engine, propeller-driven, light aircraft. An analytical model for this type of aircraft is also discussed. Results indicate that interior noise levels in this aircraft due to propeller noise can be reduced by reducing engine rpm at constant airspeed (about 3 dB), and by synchrophasing the twin engines/propellers (perhaps up to 12 dB). Ground tests show that the exterior noise pressure imposed on the fuselage consists of a complex combination of narrow-band harmonics due to propeller and engine exhaust sources. This noise is reduced by about 20-40 dB (depending on the frequency) by transmission through the sidewall to the cabin interior. The analytical model described uses modal methods and incorporates the flat-side geometrical and skin-stringer structural features of this light aircraft.

Rimas Vaicaitis

Papers

Nonlinear Response and Sonic Fatigue of National Aerospace Space Plane Surface Panels

Nonlinear Panel Response from a Turbulent Boundary Layer

Motion of Suspension Bridge Subject to Wind Loads

Nonlinear Panel Response by a Monte Carlo Approach

Investigation of Interior Noise in a Twin-Engine Light Aircraft