CHARACTERISTIC COORDINATES

In this section we show how the given system of simultaneous, or coupled, real differential equations can be transformed into a new set of separate or uncoupled equations, one for each of the new variables. This decoupling is produced by in effect selecting the eigenvectors as the coordinate system for the state space instead of the original coordinates, the yt.

Let the n X n matrix formed of the n eigenvectors be

U = [Ulu2 • • • uj (3.3,38)

Now let us define a new set of system variables (state space coordinates) q( by the transformation

y = Uq; q = U^y (3.3,39)

(Recall that for self-adjoint systems, U is an orthogonal matrix and U7′ = U-1; the above transformation is then orthogonal. In general, however, this is not the case.) It follows from (3.3,39) that

y(«) = 2 u M*) (3.3,40)

i=1

i. e. that the state vector is a superposition of n vectors parallel to the eigen­vectors. The qft) are the characteristic coordinates. Comparison with (3.3,10) shows that they must be of the form oqe*** where oq are arbitrary constants. Substitution of (3.3,39) into the differential equation of the system, (3.3,1) then yields

Uq = AUq

or, premultiplying by U_1,

q = U_1AUq (3.3,41)

Подпись: (3.3,42)We must now examine the matrix THAU. Using (3.3,38) we have AU = A[ujU2 • • • u J = [AujAu2 • • • Au„]

But the defining condition on the eigenvectors is

where, as can be verified by direct expansion, exp Лі gives the diagonal matrix of the exponential coefficients. Comparison of (3.3,49) with (3.3,13) shows that

eAl = Ue^U-1 (3.3,50)

The usual situation in vehicle dynamics is that some of the eigenvalues and eigenvectors occur in conjugate complex pairs. Thus some members of (3.3,46) will correspondingly be complex pairs. These may be transformed into a set of second-order equations, one for each complex pair of qt. Thus let qs and qj+l = q* be such a pair. The corresponding equations are

Let

4t

II II •<5

(3.3,51)

b = oq + і,3j

and

= »,- + iajj

(3.3,52)

The oq and /3,- are now real linear combinations of the original variables yt that can be calculated by expanding (3.3,39). The pair of conjugate equations are now expanded by means of (3.3,52) to give

<*i + = K – + + *&)

= К — – tfi)

Taking real and imaginary parts of either of the above leads to the alternative pair of first-order coupled equations

&j = «,■(*,■ — co,/3,-

Д,- = oj/j. j + rijfij (3.3,53)

Finally, by eliminating oq or /3,- we get a pair of uncoupled real second-order equations

oq — 2nd.) + (та2 + co2)cq = 0

/З,- – 2+ (та2 + «и2)/?, = 0 (3.3,54)

These equations for the a, /5 replace the original pair of complex first-order equations (3.3,51). However, the number of arbitrary constants in the solutions of (3.3,54) is still only two, i. e. cq(0) and /3,(0), since (3.3,53) fix the inital values of oq and

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