Stability Derivative Models

The linearized equations for the rigid body and rotor dynamics, in the state-space form, can be expressed as

Xb = AbXb + Abrxr + Dbu

xr = Arxr + ArbXb + Dru

where xb represents the basic 6DOF rigid body state vector, xr represents the higher – order dynamics that may include flapping and lead-lag dynamics, and u represents the control input vector.

xb = (u, v, w, p, q, r, f, U, C, h)

xr = (u, v, w, p, q, r, f, U, C, h, as, bs, … ) (5.42)

u (dlon, dlat, dcol, dped)

Ab and Db matrices denote the rigid body dynamics in the absence of rotor dynamics while Ar and Dr matrices represent the rotor flapping dynamics. Arb and Abr matrices represent the rotor-body and body-rotor coupling.

Taking Laplace transform of Equation 5.41 and solving for xr, we get

sxr = ArbXb + Arxr + Dru

(si — Ar )xr = ArbXb + Dru (5.43)

xr = (si — Ar )—1Arbxb + Dru

Expanding (si — Ar)—1 in a series form, we get

(si — Ar)—1 = – A—1 [i + A—:‘s + A—2s2 + A—3s3 + •••] (5.44)

Substituting the above series expansion for (si — Ar)-1 in Equation 5.43 and taking inverse Laplace yields the following solution for Xr:

xr A— Arbxb A— Arb-_b A— Arbxb ••• A— Dru

— A—2Dru — A—3DrX—————————————————– (5.45)

Equation 5.45 contains the control input and body states along with their higher – order derivatives. At equilibrium, the higher-order derivatives vanish and the rotor state xr can be expressed as

Substituting xr from Equation 5.46 into Equation 5.41 yields the following quasi­static model for xb:

Xь — Аьхь A Abr [ A – AffoXfo A – DruJ A Dьu Simplifying the above equation

Хь — [A – AbrAr 1Агь]хь + [Db – A^A,. 1D^|u (5.47)

or

-Хь — Aqs хь + Bqsu (5.48)

where AqS and BqS represent the quasi-static state and control matrices that comprise the rotor and rigid body stability and control derivatives.

5.6.2 Rotor-Response Decomposition Models

These models are formulated by decomposing the response of higher-order states into two components: (1) self-induced response and (2) response from rigid body motion. For example, the rotor state can be decomposed as

Xr — XrMR + Хгв (5.49)

where xr indicates the response due to main rotor dynamics and xr is the response due to body dynamics. Equation 5.41 for the rotor state can be expressed as

XrMR A XrB Ar(xrMR A XrB) A ArЬXЬ A Dru (5.50)

In the decomposed form, we have

XrMR — ArXrMR A Dru – XrB ArXrB A ArЬXЬ

The approach used for obtaining the expression for Xr in Equation 5.45 can be applied to find a solution to Equation 5.51. The contribution to the rotor state from body motion is then given by

XrB Ar ^ь-^ Ar ArЬ-XЬ Ar ^ь-Ть ”’ (5.52)

Assuming that the body dynamics are low-frequency dynamics compared to the rotor dynamics, the higher-order derivatives in Equation 5.52 can be neglected to obtain the instantaneous rotor response to body motion. Substituting Xr in Equation 5.49 by retaining only the first term in Equation 5.52, we get

Substituting xr from Equation 5.53 into the expression for xb in Equation 5.41, we obtain

xb Abxb A Abr [xr^R Ar Arbxb] A Dbu

On simplification, we get

xb [Ab Ar Arb]xb A AbrxrMR A DbU (5-54)

Comparing Equation 5.54 with Equations 5.47 and 5.48, we observe that the first term of Equation 5.54 is the quasi-static state matrix AqS. The rotor-response decomposition model can now be expressed as

xb AQSxb A AbrxrMR A Dbu

-^rMR = ArxrMR + DrU

The elements of the matrices in Equation 5.55 can be identified either from wind-tunnel testing or flight testing.