Dewey H. Hodges

A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams

Abstract A nonlinear intrinsic formulation for the dynamics of initially curved and twisted beams in a moving frame is presented. The equations are written in a compact matrix form without any approximations to the geometry of the deformed beam reference line or to the orientation of the intrinsic cross-section frame. In accordance with previously published work, when effects of warping on the dynamics and local constraints on the cross-sectional deformation are ignorable, the in- and out-of-plane St Venant warping displacements (which are fully coupled for nonhomogeneous, anisotropic beams) need only be included explicitly in the determination of a suitable elastic law (a two-dimensional problem) and need not be considered explicitly in the one-dimensional equations governing global deformation. In this paper it is presupposed that such an elastic law is given as a one-dimensional strain energy function. Thus, the present equations, which are based on only six generalized strain variables, are valid for beams with closed cross-sections and for which warping is unrestrained. When simplified for various special cases, they agree with similar intrinsic equations in the literature. Although the resulting equations are Newtonian in structure (closely resembling Eulers dynamical equations for a rigid body), the formulation adheres to a variational approach throughout, thus providing a link between Newtonian and energy-based methods. In particular, the present development provides substantial insight into the relationships among variational formulations in which different displacement and rotational variables are used as well as between these formulations and Newtonian ones. For computational purposes, a compact and complete mixed variational formulation is presented that is ideally suited for development of finite element analyses. Finally, a specialized version of the intrinsic equations is developed in which shear deformations is suppressed.

Nonlinear composite beam theory

* Introduction * Elements of System Theory * Mathematical Model of an Aircraft * Outline of Estimation Theory * Regression Methods * Maximum Likelihood Methods * Frequency Domain Methods * Real-Time Parameter Estimation * Experiment Design * Data Compatibility * Data Analysis * MATLAB[registered] Software * Appendices * Index.

On Timoshenko-like modeling of initially curved and twisted composite beams

Abstract A generalized, finite-element-based, cross-sectional analysis for nonhomogenous, initially curved and twisted, anistropic beams is formulated from geometrically nonlinear, three-dimensional (3-D) elasticity. The 3-D strain field is formulated based on the concept of decomposition of the rotation tensor and is given in terms of one-dimensional (1-D) generalized strains and a 3-D warping displacement that is obtained from the formulation, not assumed. The warping is found in terms of the 1-D strains via the variational asymptotic method (VAM). In this paper a Timoshenko-like model is presupposed for a beam with cross-sectional characteristic length h, wavelength of deformation given by l, and the magnitude of the radius of initial curvature and/or twist is taken to be of the order R. First, a solution for the asymptotically correct refinement of classical anisotropic beam theory for initially curved and twisted beams through O(h2/R2) is obtained. Next, the O(h2/l2) correction is computed. It is known that Timoshenko-like theory is not capable of capturing all the O(h2/l2) corrections for generally anisotropic beams. However, if all the O(h2/l2) terms are known, then the corresponding Timoshenko-like theory is uniquely defined. Numerical results are presented to illustrate the trends of the various classical (extension-twist, bending-twist, and extension-bending) and nonclassical couplings (extension-shear, bending-shear, and shear-torsion) as the initial twist and curvatures are varied.

Nonlinear Aeroelasticity and Flight Dynamics of High-Altitude Long-Endurance Aircraft

High-Altitude Long-Endurance (HALE) aircraft have wings with high aspect ratios. During operations of these aircraft, the wings can undergo large de∞ections. These large de∞ections can change the natural frequencies of the wing which, in turn, can produce noticeable changes in its aeroelastic behavior. This behavior can be accounted for only by using a rigorous nonlinear aeroelastic analysis. Results are obtained from such an analysis for aeroelastic behavior as well as overall ∞ight dynamic characteristics of a complete aircraft model representative of HALE aircraft. When the nonlinear ∞exibility efiects are taken into account in the calculation of trim and ∞ight dynamics characteristics, the predicted aeroelastic behavior of the complete aircraft turns out to be very difierent from what it would be without such efiects. The overall ∞ight dynamic characteristics of the aircraft also change due to wing ∞exibility. For example, the results show that the trim solution as well as the short-period and phugoid modes are afiected by wing ∞exibility.

Validation of the Variational Asymptotic Beam sectional Analysis

The computer program VABS (Variational Asymptotic Beam Section Analysis) uses the variational asymptotic method to split a three-dimensional nonlinear elasticity problem into a twodimensional linear cross-sectional analysis and a one-dimensional, nonlinear beam problem. This is accomplished by taking advantage of certain small parameters inherent to beam-like structures. VABS is able to calculate the one-dimensional cross-sectional stiffness constants, with transverse shear and Vlasov refinements, for initially twisted and curved beams with arbitrary geometry and material properties. Several validation cases are presented. First, an elliptic bar is modeled with transverse shear refinement using the variational asymptotic method, and the solution is shown to be identical to that obtained from the theory of elasticity. The shear center locations calculated by VABS for various cross sections agree well with those obtained from common engineering analyses. Comparisons with other composite beam theories prove that it is unnecessary to introduce ad hoc kinematic assumptions to build an accurate beam model. For numerical validation, values of the one-dimensional variables are extracted from an ABAQUS model and compared with results from a one-dimensional beam analysis using cross-sectional constants from VABS. Furthermore, point-wise three-dimensional stress and strain fields are recovered using VABS, and the correlation with the three-dimensional results from ABAQUS is excellent. Finally, classical theory is shown to be insufficient for general-purpose beam modeling. Appropriate refined theories are recommended for some classes of problems.

Nonlinear Beam Kinematics by Decomposition of the Rotation Tensor

A simple matrix expression is obtained for the strain components of a beam in which the displacements and rotations are large. The only restrictions are on the magnitudes of the strain and of the local rotation, a newly-identified kinematical quantity. The local rotation is defined as the change of orientation of material elements relative to the change of orientation of the beam reference triad. The vectors and tensors in the theory are resolved along orthogonal triads of base vectors centered along the undeformed and deformed beam reference axes, so Cartesian tensor notation is used. Although a curvilinear coordinate system is natural to the beam problem, the complications usually associated with its use are circumvented. Local rotations appear explicitly in the resulting strain expressions, facilitating the treatment of beams with both open and closed cross sections in applications of the theory. The theory is used to obtain the kinematical relations for coupled bending, torsion, extension, shear deformation, and warping of an initially curved and twisted beam.

Nonlinear aeroelastic analysis of complete aircraft in subsonic flow

Aeroelastic instabilities are among the factors that may constrain the flight envelope of aircraft and, thus, must be considered during design. As future aircraft designs reduce weight and raise performance levels using directional material, thus leading to an increasingly flexible aircraft, there is a need for reliable analysis that models all of the important characteristics of the fluid-structure interaction problem. Such a model would be used in preliminary design and control synthesis. A theoretical basis has been established for a consistent analysis that takes into account 1) material anisotropy, 2) geometrical nonlinearities of the structure, 3) unsteady flow behavior, and 4) dynamic stall for the complete aircraft. Such a formulation for aeroelastic analysis of a complete aircraft in subsonic flow is described. Linear results are presented and validated for the Goland wing (Goland, M., The Flutter of a Uniform Cantilever Wing, Journal of Applied Mechanics, Vol. 12, No. 4, 1945, pp. A197-A208). Further results have been obtained that highlight the effects of structural and aerodynamic nonlinearities on the trim solution, flutter speed, and amplitude of limit-cycle oscillations. These results give insight into various nonlinear aeroelastic phenomena of interest: 1) the effect of steady-state lift and accompanying deformation on the speed at which instabilities occur, 2) the effect on nonlinearities in limiting the amplitude of oscillations once an instability is encountered, and 3) the destabilizing effects of nonlinearities for finite disturbances at stable conditions.

On a simplified strain energy function for geometrically nonlinear behaviour of anisotropic beams

Abstract An asymptotically exact methodology, based on geometrically nonlinear, three-dimensional elasticity, is presented for analysis of prismatic, nonhomogeneous, anisotropic beams. The analysis is subject only to the restrictions that the strain is small relative to unity and that the maximum dimension of the cross-section is small relative to a length parameter which is characteristic of the rapidity with which the deformation varies along the beam; thus, restrained warping effects are not considered. A two-dimensional function is derived which enables the determination of sectional elastic constants, as well as relations between the beam (i.e. one-dimensional) displacement and generalized strain measures and the three-dimensional displacement and strain fields. Since the three-dimensional foundation of the formulation allows for all possible deformations, the complex coupling phenomena associated with shear deformation are correctly accounted for. The final form of the strain energy contains only extensional, bending and torsional deformation measures—identical to the form of classical theory, but with stiffness constants that are numerically quite different from those of a purely classical theory. Indeed, the stiffnesses obtained from classical theory may, in certain extreme cases, be more than twice as stiff in bending as they should be. Stiffness constants which arise from these various models are used to predict beam deformation for different types of composite beams. Predictions from the present reduced stiffness model are essentially identical to those of more sophisticated models and agree very well with experimental data for large deformation.

Evaluation of computational algorithms suitable for fluid-structure interactions

The objective was to identify and mathematically evaluate suitable methods to transfer information between nonlinear computational fluid dynamics (CFD) and computational structural dynamics (CSD) grids. This data transfer is vital in the field of computational aeroelasticity, where the interpolation method between the two grids can easily be the limiting factor in the accuracy of an aeroelastic simulation. The data to be transferred can include a variety of field variables, such as deflections, loads, pressure, and temperature. For a method to be suitable, it is important that it provide a smooth, yet accurate transfer of data for a wide variety of functional forms that the data may represent. An extensive literature survey was completed that identified current algorithms in use, as well as other candidate algorithms from different implementations, such as mapping and CAD/CAM. The performance of the various methods was assessed on a number of analytical functions, followed by a series of applications that have been or are currently being studied using nonlinear CFD methods coupled with linear representations of the CSD equations (equivalent plate/shell mode shapes and influence coefficient matrices). Two methods, multiquadric-biharmonic and thin-plate spline, are shown to be the most robust, cost-effective, and accurate of all of the methods tested.

Elasticity Solutions Versus Asymptotic Sectional Analysis of Homogeneous, Isotropic, Prismatic Beams

The original three-dimensional elasticity problem of isotropic prismatic beams has been solved analytically by the variational asymptotic method (VAM). The resulting classical model (Eiiler-Bernoulli-like) is the same as the superposition of elasticity solutions of extension, Saint-Venant torsion, and pure bending in two orthogonal directions. The resulting refined model (Timoshenko-like) is the same as the superposition of elasticity solutions of extension, Saint-Venant torsion, and both bending and transverse shear in two orthogonal directions. The fact that the VAM can reproduce results from the theory of elasticity proves that two-dimensioned finite-element-based cross-sectional analyses using the VAM, such as the variational asymptotic beam sectional analysis (VABS), have a solid mathematical foundation. One is thus able to reproduce numerically with VABS the same results for this problem as one obtains from three-dimensional elasticity, but with orders of magnitude less computational cost relative to three-dimensional finite elements.