Ingeniero

Longitudinal Aircraft Dynamics Previously we looked at a “pinned” aircraft motion in pitch and found the following differential equation of motion: (1) and its associated characteristic equation: (2) We now seek a method to extend our capabilities beyond a second order equation. To do this, we will see how we can represent this same motion (pinned aircraft in pitch) in another form that is easilyextended to a higher dimension system. The method of approach is to replace the single second order equation with two first order ordinary differential equations. We do this as follows: (3)

Equation (3) is equivalent to Eq. (1). However in this form it is convenient to write these equations in matrix form:

(4)

We consider the variables whose derivatives appear to be “state” variablessince they tell us the “state” of the system. Here the state variables are . In order to determine a solution to these ordinary differential equations, we can proceed just as we did for the second order equation, we will assume solutions of the form: (5) If we substitute these guesses into Eq.(4) and note that the scalars they are non-zero), we arrive at the following : divide or cancel out (since(6)

We can rearrange this matrix by putting everything on the left hand side of the equation to get:

(7)

Here we note that this matrix is just the negative of the original matrix with &s added to the diagonals. If we seek a solution to Eq. (6) for the , i = 1,2 , for the case where the right hand side is zero, then the only way we can get non-zero solutions for coefficient matrix iszero. That is we require is if the determinant of the

(8)

Performing the determinant operation we get the characteristic equation: (9) which is exactly the same as Eq. (2), the characteristic equation of the second order system! Consequently, although the procedures were different, we arrived at the same result. However in this form, the method can be extended to higher order systems since thematrix can be of any dimension. We can summarize the above results in a concise manner. If we consider the state vector x to be a vector whose components are the state variables. In this case . Then we can write Eq. (4) as: (10) where A is called the system matrix (here it is a 2 x 2 matrix), All the properties of the solution are contained in the matrix A. These are extracted by forming thecharacteristic equation as described in Eqs. (7) and (8), that takes the form: (11)

where

is an n x n identity matrix that is a matrix with 1s on its diagonal, and zeros elsewhere.

So the matrix in Eq. (11) has on the diagonals with the negative of the elements of the system matrix elsewhere and on the diagonals. We can now extend this approach to the general longitudinal dynamics of anaircraft. The procedure is similar to that used to develop all the equations of motion we used previously. First we write the general equations of motion, then we examine a reference flight condition, and then look at small motions away from that reference flight condition. Longitudinal Flight Equations The longitudinal flight equations of motion can be written in the following fashion using the forceequations along and perpendicular to the velocity. Then the general equations of motion become:

(12)

where V = airspeed = flight path angle (angle between velocity and local horizontal) T = thrust D = drag m = mass M = pitch moment q = pitch rate = pitch angle Using the definition of state variables we introduced earlier, the state here is x = [ V, . q, ]T. The other variables in Eq. (12)are not state variables, but are functions of the state variables (and another type called control variables). It is necessary for us to assume what variables these functions contain. Based on past experience, we will assume that the functions that appear in Eq.(12) take the following form:

(13)

Note that in the functions we introduced three new variables,

. The first two are

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