# The Quintic Characteristic Equation

If we account for the perturbation of the speed of the forward motion, the corresponding equations can be written as

=(Ct° -2C*o)w – as – a-h – cej – ckx (11.18)

d**[69]**h • ~ • •

^ = 2 U’Cyo + Ceye + Ceye + Chyh + C$h, (11.19)

(i2n _ ….

= Vt(2CX0 – C?)U’ + meJ + mhzh + mhzh + meJ. (11.20)

In these equations, U’ represents the relative perturbation of the cruise speed, Cf is a derivative of the thrust coefficient with respect to the relative speed of forward motion, Cx is a static drag coefficient,2 and yt is the vertical distance of the thrust line from the vehicle’s center of gravity. As earlier, all quantities are rendered nondimensinal using the cruise speed U0 and the root chord C0. The drag coefficient is related to the instantaneous cruise speed U0(l + Uf). Excluding two of the three unknown parameters, we obtain the following fifth-order (quintic) characteristic equation of the perturbed system:

HD – (Cf – 2CXn ) CtD + СЇ Ф + Свх

-2 Cyo №>-ф-СЬ) ~(C°yD + Cey)

-yt(2CX0 – Cf) ~{mhzD + mz) (^izD2 – mezD – mez)

We write the quintic characteristic equation (11.21) as

D5 + BiD4 + B2D3 + B3D2 + BaD + B5 = 0. (11.22)

The corresponding necessary and sufficient requirements for the stability of the system will be

Bi> 0 (i = l,…,5), B1B2-B3>0; (11.23)

(ВгВ2 — B3)(B3B± — B2Br>) — (BB4 — B3)2 > 0. (11.24)

Coefficients Bi can be found in the Appendix to Chapter 11.

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