Aircraft Dutch Roll with No Dihedral?

Months ago, an excellent question appeared on the Modeling Sciences sub-forum of rcgroups.com concerning an flight-dynamics aerospace engineering topic: Is dutch roll possible in an aircraft with zero effective dihedral?

The answer is yes.

Simplified Yaw-only Analysis

The dutch roll flight mode shows up in a yaw only behavior driven in frequency by the yaw stiffness Nβ and in damping by yaw damping Nr. A pilot would identify the behavior as a snake dominated dutch roll behavior. With zero effective dihedral, we could also reasonably expect only little to modest yaw-roll coupling through the rate terms, which would be primarily driven by the vertical offsets of surfaces. For the engineers, this simplified 2DOF model of dutch roll has a frequency and damping term approximated as: (derivation)

Interestingly enough, the dutch roll behavior seems to appear even if the aircraft has zero effective dihedral AND zero effective yaw stiffness, provided the product of yaw damping and sideforce derivatives are positive. Both Nr and Yβ are almost always expected to be negative.

Coupled Roll-Yaw Analysis

The dutch roll flight modes show up in higher fidelity dynamics models. The lateral 4DOF model below contains the spiral, roll, and dutch roll modes with sideslip, roll rate, yaw rate, and roll angle perturbation states:

Theoretically, we could find a generic analytical eigen-solution (i.e. eigenvalues and eigenvectors) of this coupled non-symmetrical 4×4 matrix; realistically, such a derivation is not feasible. Either we simplify the model (i.e. yaw only, as above) or introduce numerical solutions.

to be continued

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