9780071086806, Chapter 8, Problem 3

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Chapter 8, Problem 3 Problem 3. In the case of a variable atmospheric density, becomes a function of altitude h. Modify the linear models given in Section 8.1.2 for the aerodynamic forces and moments to include the effect of variable density. In this case, will altitude be a state variable or just a response variable in the state-variable representation of the equations of motion? Step-by-step solution Step of 1 Solution: Consistent with Eqns. 6.272, the linear model for the perturbation aerodynamic forces is where the additional terms due to variable density have been And from Eqns. 8.18, these additional terms are ah Pm = From Eqns. 6.273, the linear model for the perturbations in the aerodynamic moments is = = Analogous to Eqns. 8.18 given above, the additional terms in the model for the moments due to variable density are m aN This result assumes that there are no propulsive moments present in the reference flight condition, and so the reference values for all three aerodynamic moments are zero. Consequently, the linear model for the aerodynamic moments will remain unchanged. From Eqns. A.6 in Appendix A, the density is given as (h) where PRef and KRef are constants defined in Eqns. A.6 for three regions of the atmosphere. Consequently, the density gradient is др. = and Eqns. 8.18 become = = др. = Consequently, the linear model for the aerodynamic forces becomes = where the additional terms are underlined. Since altitude now appears explicitly in the model for these forces, it will also appear in the equations governing the perturbations in translational velocity (i.e., the and equations). Hence, altitude is now a state variable, and the column of the A matrix corresponding to altitude will no longer be null.