THE LINEAR EQUATIONS OF MOTION

When (4.9,17) are substituted into (4.9,7 to 4.9,10) two of the equations [(4.9,7c) and (4.9,86)], contain w terms on the right-hand side. In order to retain the desired form with first derivatives of the dependent variables on the left, we have to solve these two equations simultaneously for w and q. The result is presented in matrix form in (4.9,18 and 4.9,19). Here the equations are divided into two groups, termed longitu­dinal and lateral, for reasons that are explained as follows.

As a consequence of the simplifying assumptions made in their derivation, the pre­ceding equations have the exceedingly useful property of splitting into two indepen­dent groups. Suppose that ф, v, p, г, ДYc, ALC and ANC are identically zero. Then (4.9,19) are all identically satisfied. The remaining equations (4.9,18) form a com­plete set for the six homogeneous variables Am, w, q, A6, AxE, AzE. Thus we may conclude that modes of motion are possible in which only these variables differ from zero. Such motions are called longitudinal or symmetric, and the corresponding equa­tions and variables are likewise named. Conversely, if the longitudinal variables are set equal to zero, the remaining six equations (4.9,19) form a complete set for the de­termination of the variables ф, ф, v, p, r, yE. These are known as the lateral variables, the corresponding equations and motions being likewise named.

It is worthwhile recording here the specific assumptions upon which this separa­tion depends. A study of the various steps that have led to the final equations reveals these facts—the existence of the pure longitudinal motions depends on only two as­sumptions:

1. The existence of a plane of symmetry.

2. The absence of rotor gyroscopic effects.

The existence of the pure lateral motions, however, depends on more restrictive ap­proximations; namely

1. The linearization of the equations.

2. The absence of rotor gyroscopic effects.

3. The neglect of all aerodynamic cross-coupling (approximation 1 p. 110).

If the equations were not linearized, then there would be inertial cross-coupling between the longitudinal and lateral modes, as evidenced by terms such as mpvE in (4.7,1c) and rp{lx – /,) in (4.1,2b). That is, motion in the lateral modes would induce longitudinal motion.

Equations (4.9,18 and 4.9,19) are both in the desired first-order form, commonly referred to as state vector form, conventionally written in vector/matrix notation as

x = Ax + Be (4.9,20)

Here x is the state vector, c is the control vector, and A and В are system matrices. The state vectors for the longitudinal and lateral systems are, respectively:

x = [Am w q Ав]т

x = [v p г ф]г (4.9,21)

and the matrices A for the two cases can be inferred from the full equations. The de­pendent variables xE, yE, zE and ф are not included in the state vectors because they do not appear on the right-hand side of the equations. The matrix В will be discussed later when we come to the analysis of controlled motions.

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