UNSTEADY GUST RESPONSE IN THE FREQUENCY DOMAIN

A. Filippone

UMIST

Department of Mechanical, Aerospace, Manufacturing Engineering Manchester M60 1QD United Kingdom a. filippone@umist. ac. uk

Abstract A theory has been derived to describe the unsteady response of arbitrary two­dimensional bodies in the frequency domain. The theory provides the values of the admittance for side force and yawing moment under sinusoidal gust condi­tions. This approach provides indirectly the power spectra density (PSD), which is often used to characterize systems in unsteady periodic conditions (Filippone and Siquier, 2003, Filippone, 2003). The fbw model is inviscid, with the as­sumption of small perturbations. Results are shown for a squared, a triangular, and a circular plate, as well as some road vehicles. The existence of critical damping is discussed for some cases.

1. Introduction

Lateral gusts have been known to affect the handling of many road vehicles, including large Sport Utility Vehicles (UV). Another example is vehicle pass­ing: at some speeds, this creates a destabilizing wave that can be attributed to a gust-like phenomenon. The destabilizing effect is a function of both the rela­tive speed and the relative size and inertial mass of the vehicles. A high speed train encountering a gust at the exit of a tunnel is another case considered crit­ical (Schetz, 2001). However, some gust events are outright violent. Recently (May 8, 2003), the US National Weather Service reported that straight-line winds were suspected of causing a freight train to derail in Kansas.

Laboratory experiments in this area of aerodynamics are scarce due to the intrinsic difficulty of creating gust conditions in the wind tunnel. In fact, lateral unsteady ft>ws of fixed frequency, shape and speed must be created along with the main wind tunnel ft>w. However, a limited number of studies exist. Bear – man and Mullarkey, 1994, studied the lateral gust response of a family of bluff

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K. C. Hall et al (eds.),

Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, 95-106. © 2006 Springer. Printed in the Netherlands.

bodies resembling road vehicles. These bodies are characterized by different after-body scant angles, from zero to 40 degrees.

Earlier on, Bearman, 1971a, and Bearman, 1971b, performed important experiments on fht plates and circular disks in turbulent and laminar fbw, and calculated the admittance for the drag force (or side force, in the present nomenclature) of these bodies. The systems were placed in a wind tunnel in a position normal to the incoming fbw. These plates were a particular case of bluff bodies, for which fbw separation plays a major role. Bearman’s paper discusses a number of fundamental issues, such as the general behavior of the admittance in turbulent fbw, and the theory used to correlate the wind tun­nel data (Vickery, 1965). Vickery’s experimental and theoretical work led to a semi-empirical relationship for the admittance of fht plates in normal turbu­lent fbw. This correlation is sometimes used to predict the admittance in wind engineering applications.

Howell and Everitt, 1983, considered a high speed train with two degrees – of-freedom (pitch and yaw), in order to identify the risk of a train overturning in high cross winds, and the effects associated to track-side structures and pas­sengers. Larose et al., 1999, have performed wind tunnel experiments to derive the frequency response of very large ships at sea. This study was aimed at pro­viding controls to large vessels at port. Data for aerospace systems could not be found in the technical literature, perhaps because wing controls are considered in terms of effectiveness, rather than admittance.

We will use a development based on an indicial approach, which, in spite of some underlying simplifications, is fast and powerful. This paper will discuss the properties of the admittance functions and their practical meaning.

The theory originates from a mathematical treatment of the results deter­mined by Drischler and Diederich, 1957, who considered sharp-edged travel­ing gusts past two-dimensional wings in a wide range of speeds. The theory allows the calculation of the admittance for the lift force (or side force, if the system is non lifting) and for the pitching moment (or yawing moment, re­spectively). This analysis does not take into account the structural inertia of the system.

Classical analyses in the frequency domain are due to von Karman and Sears, 1938 for the sinusoidal gust on the two-dimensional airfoil. However, it is more common to find analyses in the time domain, for example the classical works of Kussner, 1936, and Wagner, 1925, who derived basic transfer func­tions for the abrupt change in angle of attack (Kussner) and the response to a fixed sharp-edged gust (Wagner). See Leishman, 2000, for a full review.