Mathematical models to approximate the dynamics of aircraft are often generated by considering aircraft structures to be rigid bodies. These models may be unsuitable for evaluating the maneuvering responses and flight characteristics of flexible aircraft. Computational simulations of the dynamics of a flexible aircraft by use of a model that does not account for the elasticity may indicate inaccurate tracking performance, handling qualities, and response bandwidth, especially in evaluations of responses to pilot commands.

It is necessary to account for the elastic as well as the rigid-body dynamics of the SR-71 airplane (see Figure 1) in order to simulate its overall dynamic responses accurately. In particular, a submodel of elastic dynamics must represent the first bending mode of the fuselage. This mode can significantly affect responses because large-amplitude oscillations in this mode are easily excited during standard pilot maneuvers. Thus, it is necessary to model this mode accurately for closed-loop analysis and pilot simulations.

Flight data can be used to generate mathematical models that account for rigid-body and elastic dynamics, according to the method described in "Calibrating Aircraft-Vibration Models from Flight Data" (DRC-95-05), NASA Tech Briefs, Vol. 21, No. 11 (November 1997), page 78. In this method, one considers a general theoretical model and analyzes flight data by use of parameter-estimation algorithms to determine the optimal coefficients for that model. Models of the SR-71 airplane were developed by introducing a simple uniform beam to represent the mode shape of the fuselage in a general model. An optimal model resulting from analysis of flight data was found to be capable of simulating responses much more accurately than did rigid-body models; however, there were some errors in the simulated dynamics of the fuselage.

An additive-beam model has been conceived for use in representing the mode shapes of the elastic dynamics of the fuselage; in other words, the bending shapes of modes of vibration of the fuselage. This additive-beam model is formulated by superimposing the responses of a uniform beam with no fixed ends and a uniform beam with one fixed end. The resulting beam is not restricted to be symmetric about a midpoint and, consequently, can represent complex mode shapes.

A model of the SR-71 is generated by analyzing flight data according to the parameter-estimation method and utilizing a general model that includes the additive beam. Figure 2 presents an example of flight data - acceleration measured near the pilot station during a pitch maneuver - along with the corresponding simulated responses of a rigid-body model and the model that includes the additive-beam submodel. Responses from the additive-beam model are very similar to the flight data and are clearly more accurate than are the responses from the rigid-body model. Thus, the additive beam has been shown to represent accurately the first bending mode of the fuselage.

The additive beam is particularly useful for modeling the fuselage of the SR-71 because of the resulting complicated mode shape. This shape accounts for the variations in structural stiffness that occur along the fuselage as a result of wings and engine mountings. Similarly, additive-beam models can be included in models of other aircraft that have complicated mode shapes. For example, many long-endurance, high-altitude airplanes with long wing spans are affected by low-frequency bending modes with complicated mode shapes that cannot be accurately modeled by simple uniform beams.

This work was done by Rick Lind of Dryden Flight Research Centerand Carla Iorio of West Virginia University. DRC-98-36

##### NASA Tech Briefs Magazine

This article first appeared in the November, 1999 issue of NASA Tech Briefs Magazine.

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