THE FUNCTIONAL OPERATORS AND THEIR ALGEBRA

The functional diagrams can be represented algebraically by operators. Let a functional block that converts an “input” Qt into an “output” B0 (Fig. 11.5) be written symbolically as

eB = o e{ (1)

The symbol О is called an operator, and the process of finding B0 from 6( is called an operation on Bt by the operator O. It is to be noted that B„

and may represent quantities of different physical dimensions, for example, Bt an angle, and B0 a force. The physical dimensions of an operator are determined by the dimensions of B0 and B{.

If it is possible to solve the inverse problem of finding the input from a known output, then we say that an inverse of the operator exists, and write

в0 = О 6{, 6( = O-i B0 (2)

If there exists a one-to-one correspondence between Qt and B0 for all allowable B(, the operator is said to be regular* Otherwise it is singular. A regular operator has a unique inverse. In the examples named above, the operator that transforms the angle of attack into lift is regular in ranges of flow speed that excludes the critical-divergence speeds, and is singular at the critical-divergence speeds. Similarly, the operator that transforms the gust into the wing response is regular when the speed of flow is not a flutter speed, and is singular at flutter speeds. In. aero – elasticity, we generally consider regular operators that become singular only at certain special values of a set of parameters (e. g., the dynamic

* Clearly it is necessary to specify the regions in which the quantities 0,- and в0 are considered. However, the explicit statements of the regions of interest in our examples below are generally omitted in favor of conciseness. For example, if the linearized airfoil theory is used, the angle of attack, the downwash, and the lift force, etc., must be small. But the range of applicability of the linearized theory will not be mentioned every time. It is hoped that this lack of rigor will not cause confusion.

pressure, the reduced frequency). The set of parameters at which an operator becomes singular are eigenvalues of that operator. The deter­minations of the response from a regular operator and the eigenvalues of a singular operator are the two main problems in aeroelasticity.

An operator is said to be linear if it has the following property:

0(^i + x2) = Oaq + Ox2 (3)

where x1 and x2 both belong to the region over which the operator О is defined. It is said to be nonlinear if this relation does not hold over the entire region of definition. For mathematical simplicity, only linear operators will be considered hereafter.

Modern mathematical theory of functional transformations can be applied to the functional operators to render the mathematical treatment exact. But for the present purpose a heuristic account will be sufficient.

The operators occurring in aeroelasticity may be algebraic, differential, integral, or integral-differential.* They will be discussed in greater detail later. First let us consider some of their algebraic properties.

Consider again the steady-state lift-distribution problem. Let us designate the operators by subscripts as defined by the following equations:

p = Oi(« + 9), 9 = 02p (4)

where p is the aerodynamic pressure distribution, a is the wing surface without elastic deformation, and 9 is the elastic deflection corresponding to p. Obviously we may consider 9 as being generated by two successive operations Oj and Oa and write the entire process in a single equation:

9 = 0201(a + 9) (5)

The successive operations 020j may be regarded as a single operation generated by Oa and Ox in the specified order. It is called a composition product of the original operators. Note that it is in general noncommuta- tive; i. e., 020г Ф 0X02. Strict attention must be given to the order in which a composition product is formed.

Let us assume that and 02 are both linear. Equation 4 can then be written as

p = Ога + Oj0

and

9 = 02p = 02(0га + OjO) = 020ja -f

* An element of a functional diagram may represent a single number, a continuous function, a matrix of numbers, or a matrix of continuous functions of space and time. In particular, the interpretation of the variables as matrices is very important. By such an interpretation the operational equations are made very concise.

OA(« + 0) – 02(V + OA0 (6)

Therefore the composition product of two linear operators is linear.

The lift distribution on the elastic wing as represented in Fig. 11.2 can then be characterized, under the linearity assumption, by the equation

0 = 020]a + О2ОХ0 or

0 — O2OX0 = 020ja

Let us write I as an identical operator, which transforms any quantity into itself:

10 = 0 (7)

The relation between 6 and a can then be written as

(I – OjOJfl = 020,a (8)

A formal solution of this equation is

0 = (I — 020j)_1 OjjOjoc (9)

where (I — 0,00 і 1S the inverse operator (assumed unique) of I — 020x. In order to interpret the meaning of the inverse operator (I — 020j)_1, let us develop the expression formally into a power series in 020x by means of the binomial theorem:

(і – oa)-1 = і + o2ox + (02ox)2 + (OA)3 + • • • (io)

where

(O A)2 = (0,00(020,)

(O, oo*+1 = (OAXOA)* (k = 2, з, • • •) (11)

Since 020j transforms an angle into an angle, so do all the successive powers of OA – Applying Eq. 10 to Eq. 9, we may write

(12)

If the process can be justified, the solution 0 can be obtained by summing the infinite series.

The summation of the infinite series 12 actually amounts to a process of successive approximations. The first term gives the elastic deformation of the wing corresponding to the lift acting on a rigid wing. The second term gives the increment of the elastic deformation due to the change of

lift corresponding to the elastic deformation first computed. The third term gives the second increment of elastic deformation due to the lift corresponding to the first correction of the elastic deformation, and so on. Hence, in this particular case, the process 10 is justifiable for sufficiently small dynamic pressure of the flow. The series 12 is known to converge (Chapter 3) whenever the dynamic pressure q is less than

Example 1. The Lift Acting on a Two-Dimensional Airfoil {Fig. 11.2). Consider the two-dimensional case of § 3.1. Here

Ma — Lee = qc2ea(v. + 9)

Me = K9 = Ma

Hence,

Oj = qc2ea

o2 = 1 IK

Equations 12 and 9 gives at once

The lift per unit span is given by

L = qca(p. + 0) = in agreement with § 3.1.

Example 2. Divergence (Fig: 11.3). Here the functional relation is represented by

0^9 = 9

In the two-dimensional case (Ex. 1), we obtain at once the critical condition

4c’iea j = 0

К

which yields

_ JL

Чйіч ~ c2ea

Example 3. Gust Loading (Fig. 11.4). Let the gust be represented by G and the other symbols denote quantities as shown in Fig. 11.4. Let the operators be defined as follows:

L — OloG + О Lq9 F= Oj9 9 = О e(L + F)

From the loops shown in Fig. 11.4, we obtain

Oa[(0 loG + 0»9) + О70] = в
О® O^G = (I — ОЙО£0 — ОвО,) 0

в — (I — 0Е0Ев — О jjOjr)-1 OeOlgG

Example 4. Flutter. If G = 0, the last equation of Ex. 3 becomes a homogeneous one:

(I — ОЙО£0 — OjgOjr) 0 = 0 (13)

which gives the critical-flutter condition. (Divergence may be considered as flutter of zero frequency, and hence is included in the above equation.) Since Oe depends on the rigidity of the wing, while Oie depend on the dynamic pressure and the reduced frequency, the problem of flutter is to determine the eigenvalues of the rigidity, dynamic pressure, or reduced frequency at which Eq. 13 has a nontrivial solution.