Identification of Linear and Nonlinear Aerodynamic Impulse Responses Using Digital Filter Techniques

This paper discusses the mathematical existence and the numerically-correct identification of linear and nonlinear aerodynamic impulse response functions. Differences between continuous-time and discrete-time system theories, which permit the identification and efficient use of these functions, will be detailed. Important input/output definitions and the concept of linear and nonlinear systems with memory will also be discussed. It will be shown that indicial (step or steady) responses (such as WagnerÕs function), forced harmonic responses (such as TheodorsenÕs function or those from doublet lattice theory), and responses to random inputs (such as gusts) can all be obtained from an aerodynamic impulse response function. This paper establishes the aerodynamic impulse response function as the most fundamental, and, therefore, the most computationally efficient, aerodynamic function that can be extracted from any given discrete-time, aerodynamic system. The results presented in this paper help to unify the understanding of classical two-dimensional continuous-time theories with modern threedimensional, discrete-time theories. First, the method is applied to the nonlinear viscous BurgerÕs equation as an example. Next the method is applied to a three-dimensional aeroelastic model using the CAP-TSD (Computational Aeroelasticity Program Transonic Small Disturbance) code and then to a two-dimensional model using the CFL3D Navier-Stokes code. Comparisons of accuracy and computational cost savings are presented. Because of its mathematical generality, an important attribute of this methodology is that it is applicable to a wide range of nonlinear, discrete-time problems. INTRODUCTION During the early development of mathematical models of unsteady aerodynamic responses, the efficiency and power of superposition of scaled and shifted fundamental responses, or convolution, was quickly recognized. This led to the classical WagnerÕs function1, which is the response of a twodimensional airfoil, in incompressible flow, to a unit step variation in angle of attack. Similar functions such as KussnerÕs function, which is the response of a two-dimensional airfoil to a sharp-edged gust in incompressible flow, were developed as well1. As geometric complexity increased, however, the analytical derivation of these time-domain fundamental functions became quite complicated and, therefore, impractical. Ultimately, frequency-domain aerodynamics for threedimensional configurations became the method of choice for computing linear unsteady aerodynamic responses2. For the case where geometryand/or flow-induced nonlinearities are significant in the aerodynamic response, time integration of the nonlinear equations is necessary, as is done in unsteady CFD codes, particularly for aeroelastic analyses. As CFD codes have grown in complexity and capability, there is a very real need to incorporate these codes into aeroservoelastic (ASE) analyses, loads estimation, and other preliminary design efforts in an efficient and accurate manner. Direct incorporation of a CFD code into the ASE process is currently not practical due to the high computational costs and turnaround time required. As computational speeds improve and as new algorithms are developed to address this problem, the practicality of this approach may improve. At the moment, however, the efficient incorporation of the information provided by a CFD code into disciplines such as ASE remains a problem.