Toward a Mathematical Theory of Aeroelasticity

This paper initiates a mathematical theory of aeroelasticity centered on the canonical problem of the flutter boundary an instability endemic to aircraft that limits attainable speed in the subsonic regime. We develop a continuum mathematical model that exhibits the known flutter phenomena and yet is amenable to analysis non-numeric theory. Thus we model the wing as a cantilever beam and limit the aerodynamics to irrotational, isentropic so that we work with the quasilinear Transonic Small Disturbance Equations with the attached flow and Kutta-Joukowsky boundary conditions. We can obtain a Volterra expansion for the solution showing in particular that the stability is determined by the linearized model consistent with the Hopf Bifurcation Theory. Specializing to linear aerodynamics, the time domain version of the aeroelastic problem is shown to be a convolution-evolution equation in a Hilbert space. The aeroelastic modes are shown to be the eigenvalues of the infinitesimal generator of a semigroup, which models the combined aerostructure state space dynamics. We are also able to define flutter boundary in terms of the "root locus" the modes as a function of the air speed U . We are able to track the dependence of the flutter boundary on the Mach number a crucial problem in aeroelasticity but many problems remain for Mach numbers close to one. The model and theory developed should open the way to Control Design for flutter boundary expansion. Introduction To a mathematician specializing in the problems of stability and control for partial differential equations, Aeroelasticity offers a fertile, if challenging, field of application. Currently, however, to a mathemati'Research supported in part by NASA Grant NCC4-157 2 SYSTEM MODELING AND OPTIMIZATION cian even an applied mathematician Aeroelasticity (to paraphrase Richard E. Meyer, in Introduction to Mathematical Fluid Dynamics [I]) "appears to be built on a quicksand of assorted intuitions" plus numerical approximations. This paper is a first halting step toward a "mathematical theory of Aeroelasticity." The canonical problem of Aeroelasticity is flutter. It is an instability endemic to aircraft wings that occurs at high enough airspeed in subsonic flight and thus limits the attainable speed. The purpose of Control Design is to "expand" this "flutter boundary." Control Design, however, requires a mathematical model that is simple enough for non-numeric analysis and yet displays the phenomena of interest in this case flutter. In contrast almost all the extant work on this problem has been computational (see the review paper by Friedmann [2]). Computational techniques despite their success and universal use, require that numerical parameters be specified and thus cannot contribute to Control Design. The lack of a faithful enough mathematical model is undoubtedly one reason why all attempts at flutter control have failed so far. As we shall show, the kind of models needed require crucially recent advances in boundary control of partial differential equations. Even then many purely mathematical questions relating to the model are unanswered as yet. We begin in Section 2 with the wing model, incorporating in addition a model for self-straining actuators. Section 3 is devoted to the aerodynamic model where we derive the TSD Equation from the Full Potential Equation clarifying the many assumptions made, and allowing for nonzero angle of attack. We linearize the TSD Equation and show it can be solved by the Possio Integral Equation, generalized to include nonzero angle of attack. We also develop a solution to the Linear Nonhomogeneous TSD Equation for zero initial and boundary conditions. Using these results we show how to construct a power series expansion actually a Volterra kernel series expansion for the solution of the nonlinear TSD Equation. We are then able to obtain what is perhaps the most significant result that the stability of the system is determined by the stability of the linear system consistent with the Hopf Bifurcation Theory. In Section 4 we go on to the abstract or time domain formulation of the flutter control problem. It turns out to be convolution-evolution equation in a Hilbert Space for the structure state which is not quite the full state for which we used to go to a Banach Space formulation, enabling us to identify the aeroelastic modes as eigenvalues of the infinitesimal generator of the Banach Space semigroup. Of primary interest on the practical side is the calculation of these modes. This in turn Toward a Mathematical Theory of Aeroelasticity 3 leads to the "root locus" the modes as a function of U and the definition of flutter speed. The dependence of the flutter speed on M is an important unresolved issue here. 1. The Wing Model The wing is modelled as a flexible structure the flexibility is of course the key feature as a "straight" uniform rectangular plate. Identifying the modes of the wing structure is one of the standard activities (vibration testing) in flight centers. The structure model must have the ability to conform to the first few measured modes at least. Following the model initiated by Goland [3] in 1954 we allow two degrees of freedom plunge (displacement) and pitch (angle) about the elastic axis. Let where . ! is the wing span (one sided). Then the Goland model is: where K, is the differential operator and L(s, t ) , M ( s , t ) denote the aerodynamic lift and moment. We are thus modelling the structure as a beam which would imply that the spread 2b ("chord length") is "small" compared to the span e. Following Goland the beam is a cantilever clamped at the root s = 0 and free at the tip s = !, so that we have the end conditions: at the root: 4 SYSTEM MODELING AND OPTIMIZATION where the super primes denote derivative with respect to s and the superdots denote time derivatives, in the usual notation. We will need to change the tip conditions to: if we wish to include a generally accepted model for self-straining actuators, with gh , go 2 0 being the gains. 2. The Aerodynamic Model The aerodynamics is far the more complicated part. To comply with space limitation, the presentation will need to be quite compressed with minimal details of proofs. To begin with, we shall assume the flow to be non-viscous. Next we will assume that it is isentropic and that the Perfect Gas Law applies. In this case, as shown in [4], the flow can be described by a velocity potential q!J(x, y, z , t ) which satisfies the so-called Full Potential Equation given by: where q, is the free stream (far-field) velocity and a, is the free stream (far-field) speed of sound, V denotes gradients in the usual notation, and the far stream Mach number assumed 5 1, and y is the ratio of specific heats. This equation would appear to be complex but fortunately can be simplified since our primary concern is stability. Hence we go one level down to the Transonic Small Disturbance Equation there are various versions [6], [lo] but we shall follow Nixon [5] see also [4]. Thus we assume that 0-003 c p = U is "small" (see below for how it is used) where 4, is the undisturbed or far stream potential: 4, = (xql + 942 + ~ 9 3 ) u, Toward a Mathematical Theory of Aeroelasticity 2 U 14m12, 9: + 9; + 9; = 1. We have then the TSD Equation for cp (see [7, equation 2.221): Note that this is a quasi-linear equation with the right hand side neither elliptic nor hyperbolic, studied by Tricomi [6] , Bers [7], Guderley [8], extensively, specialized to the stationary case. 2.1 The Aeroelastic Problem Our interests are different in that we need to go beyond Transonic Aerodynamics to Transonic Aeroelasticity, as reflected in our preoccupation with the boundary conditions: i) Flow Tangency Condition: where w,(x, y, t) the "downwash" is the normal velocity of the structure. For our structure model of zero thickness, with x(x, y, t ) denoting the instantaneous displacement of the wing along the x-axis, we can calculate that: where x = ab locates the elastic axis of the wing in the xy plane. ii) KuttaJoukowsky Conditions: "Zero pressure jump off the wing and at the trailing edge" (10) 6 SYSTEM MODELING AND OPTIMIZATION Now from [4] we have that pressure p(x, y, x, t ) can be expressed as where p(x, y, z, t ) is the density, and where $(x, y, x, t ) is the acceleration potential Now consistent with our small disturbance assumption, Lu2 2 $(x, Y, x, t ) can be approximated as