David A. Peters

Finite state induced flow models. I: Two-dimensional thin airfoil

A new finite state aerodynamic theory is presented for incompressible, two-dimensional flow around thin airfoils. The theory is derived directly from potential flow theory with no assumptions on the time history of airfoil motions. The aerodynamic states are the coefficients of a set of induced-flow expansions. As a result, the finite state equations are hierarchical in nature and have closed-form coefficients. Therefore, the model can be taken to as many states as are dictated by the spatial texture and frequency range of interest with no intermediate numerical analysis. The set of first-order state equations is easily coupled with structure and control equations and can be exercised in the frequency or Laplace domain as well as in the time domain. Comparisons are given with Theodorsen theory, Wagner theory, and other methods. Excellent results are found with only a few states.

Finite state induced flow models. II - Three-dimensional rotor disk

In Part I of this two-part article, we developed a finite state induced flow model for a two-dimensional airfoil. In this second part, we develop a finite state induced flow model for the three-dimensional induced flow for a rotor. The coefficients of this model are found in a compact closed form. Although the model does not presuppose anything about the source of lift on the rotating blades, applications are given in which the Prandtl assumption is invoked. That is, the two-dimensional lift equations are used at each radial station, but with the inflow from the three-dimensional model. The results are shown to reduce (in several special cases) to Prandtl-Golds tein theory, Theodorsen theory, Loewy theory, dynamic inflow, and blade-element momentum theory. Comparisons with vortex-filament models and with experimental data in hover and forward flight also show excellent correlation.

hp-Version finite elements for the space-time domain

A bilinear formulation of elasto-dynamics is offered which includes, as a special case, “Hamiltons law of varying action”. However, the more general bilinear formulation has several advantages over Hamiltons law. First, it admits a larger class of initial-value and boundary-value problems. Second, in its variational form, it offers physical insight into the so-called “trailing terms” of Hamiltons law. Third, numerical applications (i.e., finite elements in time) can be proven to be convergent under correct application of the bilinear formulation, whereas they can be demonstrated to diverge for specific problems under Hamiltons law. Fourth, the bilinear formulation offers automatic convergence of the “natural” velocity end conditions; while these must be constrained in present applications of Hamiltons law. Fifth, the bilinear formulation can be implemented in terms of a Larange multiplier that gives an order of magnitude improvement in the convergence of velocity. This implies that, in this form, the method is a hybrid finite-element approach.

Effects of Radial Immersion and Cutting Direction on Chatter Instability in End-Milling

Low radial immersion end-milling involves intermittent cutting. If the tool is flexible, its motion in both the x- and y-directions affects the chip load and cutting forces, leading to chatter instability under certain conditions. Interrupted cutting complicates stability analysis by imposing sharp periodic variations in the dynamic model. Stability predictions for the 2-DOF model differ significantly from prior 1-DOF models of interrupted cutting. In this paper stability boundaries of the 2-DOF milling process are determined by three techniques and compared: (1) a frequency-domain technique developed by Altintas and Budak (1995); (2) a method based on time finite element analysis; and (3) the statistical variance of periodic 1/tooth samples in a time-marching simulation. Each method has advantages in different situations. The frequency-domain technique is fastest, and is accurate except at very low radial immersions. The temporal FEA method is significantly more efficient than time-marching simulation, and provides accurate stability predictions at small radial immersions. The variance estimate is a robust and versatile measure of stability for experimental tests as well as simulation. Experimental up-milling and down-milling tests, in a simple model with varying cutting directions, agree well with theory.Copyright

Design of helicopter rotor blades for optimum dynamic characteristics

Abstract The mass and stiffness distributions for helicopter rotor blades are to be tailored in such a way to give a predetermined placement of blade natural frequencies. The optimal design is pursued with respect of minimum weight, sufficient inertia and reasonable dynamic characteristics. The finite element technique will be used as a tool. Rotor types include hingeless, articulated and teetering.

On the lateral buckling of uniform slender cantilever beams

Abstract The general lateral buckling equation is developed for a uniform, slender cantilever beam with a load applied at the shear center of the end cross section. This equation is then specialized to include only the first order effect of the principal bending curvature revealing errors in previous first order analyses. These errors resulted from a failure to properly distinguish between the geometric and elastic angles of twist. The correct specialized equation is actually simpler than previously published equations and results in a buckling load formula noticeably different from formulas based on these earlier equations. This present buckling load formula is shown to compare favorably with a numerical solution of the general equation.

Helicopter trim with flap-lag-torsion and stall by an optimized controller

An autopilot is applied to helicopter rotor flap-lag-torsion equations to obtain the control settings for a trimmed flight condition. The rotor aerodynamic description includes a stage-space dynamic stall model for lift and for pitching moments. Thus, the rotor is trimmed for flight conditions in which significant stall and torsional deformations are present. The autopilot is extended to Q-bladed rotors by a series of time-delay terms. As a result, the optimum gains and time constants depend upon the number of blades as well as upon the torsional stiffness.

Lateral-torsional buckling of cantilevered elastically coupled composite strip- and I-beams

The lateral‐torsional buckling of composite strip- and I-beams is considered. The geometrically exact governing equations are simplified by consistently regarding certain configuration parameters as small. The assumption that these parameters are indeed small is equivalent to the assumption that the square of the maximum prebuckling cross-sectional rotation due to bending is small compared to unity. The analysis takes into account various refinements of previously published results, including the Vlasov eAect, elastic coupling, the oAset of the load from the centroid, and, of course, prebuckling deflections. The analysis is thereby reduced to a single fourth-order diAerential equation and boundary conditions, all of which are derivable from a corresponding energy expression. From the form of matched asymptotic expansions of the buckling mode when small parameters are ignored altogether, a single comparison function is found which gives the correct buckling load to within 1% for a wide range of the warping rigidity. Using this comparison function, a formula for the buckling load as a function of the small parameters of the problem is found and validated. With certain exceptions regarding the load oAset parameter, the formula provides results which agree quite well with the numerical solution of the exact equations as long as all the small parameters remain small. However, the load oAset parameter always appears in the governing equations as multiplied by a ratio of stiAnesses, which can become large, especially for composite I-beams. For this case, a special treatment is required. ” 2001 Elsevier Science Ltd. All rights reserved.

Use of Multiblade Coordinates for Helicopter Flap-Lag Stability with Dynamic Inflow

Rotor flap-lag stability in forward flight is studied with and without dynamic inflow feedback via a multiblade coordinate transformation (MCT). The algebra of MCT is found to be so involved that it requires checking the final equations by independent means. Accordingly, an assessment of three derivation methods is given. Numerical results are presented for three- and four-bladed rotors up to an advance ratio of 0.5. While the constant-coefficient approximation under trimmed conditions is satisfactory for low-frequency modes, it is not satisfactory for high-frequency modes or for untrimmed conditions. The advantages of multiblade coordinates are pronounced when the blades are coupled by dynamic inflow.

Asymptotic solutions to a stability problem

This paper is concerned with the lateral stability of a free flying column subjected to an axial thrust with directional control. The stability curve (i.e., eignevaue vs . thrust, in the neighborhood of zero eigenvalues) and the associated eignefunctions of this problem have not been fully understood. Here, asymptotic expansions are used to examine closely, for all values of the thrust directional control parameter, both the intersection of the eigenvalue curves with the zero branch and the associated eigenfunctions of zero and nearly zero eigenvalues. Several analytical proofs are provided substantiating previous numerical findings.