LATERAL-DIRECTIONAL EQUATIONS FOR THE CHEROKEE 180
The three equations governing the lateral-directional motion and control of the Cherokee 180 are obtained by substituting the calculated stability and control derivatives into Equation 10.17. This reduces to:
30.4/3 + 0.396/8 + 0.039<b – 0.543$ + 30.2r = 0.117 Sr (10.27a)
0.0993/3 + 0.153$ + 0.429$ – 0.198r = 0.0531 Sa + 0.0105 8r (10.27b)
-0.0672/8 + 0.0735$ + 1.18f + 0.0873r = -0.0509 8r (10.27c)
The characteristic equation for this set of simultaneous, linear differential equations is obtained from the determinant.
(0.153a-2 + 0.429a) -.198 + 0.0735a – (1.18a-+0.08.73)
This determinant reduces to the following.
a4 + 2.90a3 + 0.381a2 + 0.215a – 0.000454 = 0
Since the constant is negative, it is obvious that Equation 10.28 will have a positive real root. Thus the Cherokee is predicted to possess at least one mode of the lateral-directional motion that is unstable.
Typical of lateral-directional motion, Equation 10.28 has two real roots and a pair of complex roots. The real roots can be found from trial and error,
by graphical means, or otherwise to equal
<r, = -2.79 0-2 = 0.00210
(cr — cr,) can be divided into Equation 10.28 (simply follow the same procedure as is done in a long division problem) to obtain a cubic. The cubic is then divided by (cr — cr2) to obtain a quadratic, which can be solved for the pair of complex roots
0-3 =-0.055 + 0.277/ o-4= -0.055 -0.277 і