# Can having too small of a roll or yaw stiffness destabilize the dutch roll mode?

For a dutch roll mode, can having too small of a roll or yaw stiffness destabilize it?

I would think not because you want a lower value to maintain it?

## 1 Answer

To estimate whether the dutch-roll mode goes unstable, we can assess the real-part of the dutch-roll eigenvalue: negative is stable and positive is unstable. From Etkins, Dynamics of Flight, one illustrative estimate (although not necessarily numerically accurate) is as follows:

$$n_{DR}\approx\frac{1}{2} \left[ \frac{Y_v}{m} + \frac{N_r}{I'_z} + \frac{I'_zL_v}{I'_xN_v}\left(\frac{N_p}{I'_z} - \frac{g}{u_0} \right) \right]$$

where $$n_{DR}$$ is the real-part of the eigenvalue, $$Y_v$$ is the dimensional side force sideslip derivative, $$N_r$$ is the dimensional yaw moment damping, $$L_v$$ is the dimensional negative roll stiffness (dimensionless form: $$C_{l_{\beta}}$$), $$N_v$$ is the dimensional yaw stiffness (dimensionless form: $$C_{n_{\beta}}$$), $$N_p$$ is the yaw-roll cross-coupling moment, $$u_0$$ is the trim airspeed, $$g$$ is gravitational acceleration, $$m$$ is aircraft mass, all the $$I'$$ are inertia terms about the respective axes. Note that all the inertia terms are about the principle axes for simplicity.

The first term ($$Y_v$$) is always negative, and so is the second term ($$N_r$$). Ignoring $$N_p$$ for now, positive roll stiffness requires $$L_v$$ to be negative, and positive yaw stiffness requires $$Y_v$$ to be positive; this means that the third term is net positive. Therefore, we can draw the following conclusions:

1. Dutch-roll damping worsens at lower airspeed
2. Roll stiffness worsens the dutch-roll damping
3. Yaw stiffness improves the dutch-roll damping

To answer your question: yes, it's entirely possible for dutch-roll to be unstable due to sufficiently low yaw stiffness.

• Please explain why $Lv$ is expressed as "negative roll stiffness" and $Nv$ as only "yaw stiffness". Is the negative sign added to the roll stiffness term for the equation, or does it mean a direction of roll (away from level/returning to level)? – Robert DiGiovanni Mar 18 '20 at 13:45
• @RobertDiGiovanni Pleas read third paragraph. – JZYL Mar 18 '20 at 13:47
• I would expect $Lv$ to be numerically positive, because $-g/u0$ creates the negative in the eigenvalue calculation. $Np$ certainly does make it less negative! – Robert DiGiovanni Mar 18 '20 at 13:57
• @RobertDiGiovanni Huh? These are stability derivatives – JZYL Mar 18 '20 at 13:59
• Yes, and safe planes are design detivatives. Notice how roll/yaw coupling (from swept wings) makes the third term less negative. – Robert DiGiovanni Mar 18 '20 at 16:41