# Stability

The aim of stability analysis is to determine if a design is statically and dynamically stable.

### The pitching moment coefficient and the concept of the neutral point

The moment about the y-axis of the body-fixed coordinate system affects the stability of the design. To have a design that is "statically stable" the moment about the y-axis should be restoring. What is meant by a restoring moment?

Consider the image below. As the angle of attack is increased, the moment about the y-axis $M_{y}$becomes more and more negative. This means that the design has a tendency to return to its original angle of attack. This tendency is called "static stability".

Since the moment about the y-axis $M_{y}$depends on the center of gravity location, moving the center of gravity location forward or aft will affect the stability of the design. The image blow shows the pitching moment coefficient $C_{m}$as a function of the angle of attack. For a stable design the derivative of $C_{m}$with respect to $\alpha$must be negative $\frac{\partial C_{m}}{\partial\alpha}<0$.

Movement of the center of gravity front to aft affects the value of$\frac{\partial C_{m}}{\partial\alpha}$. The derivative is referred to as the pitch static stability derivative and must be negative if the design is to be stable. It is interesting to determine as the center of gravity is moved at what point the value of $\frac{\partial C_{m}}{\partial\alpha}$becomes 0. This point (or position of the center of gravity) is referred to as the neutral point.

If we have the values of $C_{m}$ and $C_{Z}$at two different angles of attack we can determine the location of the neutral point using the following equation:

$C_{m}^{cg}=C_{m}-\frac{x_{cg}}{c_{ref}}C_{Z}\text{.}$

The above equation calculates the pitching moment about an arbitrary point $x_{cg}$along the x-axis of the body-fixed coordinate system. Since we are interested in finding at what point $\frac{\partial C_{m}}{\partial\alpha}$becomes zero we take the derivative of the equation with respect to the angle of attack and equate it to zero.

$\frac{\partial C_{m}^{cg}}{\partial \alpha}=\frac{\partial C_{m}}{\partial \alpha}-\frac{x_{cg}}{c_{ref}}\frac{\partial C_{Z}}{\partial \alpha}=0$

Rearranging the equation for $x_{cg}$gives:

$x_{cg}=x_{np}=-c_{ref}\frac{\frac{\partial C_{m}}{\partial \alpha}}{\frac{\partial C_{Z}}{\partial \alpha}}\approx\frac{\Delta C_{m}}{\Delta C_{Z}}\approx\frac{C_{m}\rvert_{\alpha=5}-C_{m}\rvert_{\alpha=0}}{C_{Z}\rvert_{\alpha=5}-C_{Z}\rvert_{\alpha=0}}\text{.}$

Solving the above equation will give the location of the center of gravity for which $\frac{\partial C_{m}}{\partial\alpha}=0$. This location is referred to as the neutral point.

If the the center of gravity is behind the neutral point, the design will be statically unstable. If the center of gravity is in front of the neutral point, the design will be statically stable.

The static margin $SM$is a measure of how statically stable the design is. The static margin can be calculated from the from the following equation:

$SM=\frac{x_{cg}-x_{np}}{c_{ref}}$

### Example

The following example shows how to determine the neutral point location of a fixed-wing UAV with reference area $𝑆_{ref}$of $0.7729\text{ }m^2$ and reference chord $c_{ref}$ of $0.2544\text{ }m^2$. A body-fixed coordinate system is placed at the nose. The reference speed is $|V_{ref} |=25\text{ }m/s$ and the reference density, pressure and viscosity are evaluated at an altitude of $h=0\text{ }m$(ISA).

Both fixed-wake and free-wake simulations of the fixed-wing UAV were performed at $\alpha=0$and $\alpha=5$deg.

The table below summarises the values of the $C_{Z}$and $C_{m}$coefficients obtained from the simulations.

Simulation type | $\alpha^\circ$ | $C_{Z}$ | $C_{m}$ |

Fixed-wake | 0 | −0.5357 | −1.1156 |

Fixed-wake | 5 | −1.0417 | −2.2843 |

Free-wake | 0 | −0.2774 | −0.6383 |

Free-wake | 5 | −1.0280 | −2.2641 |

The neutral point is calculated from the previously derived expression:

$x_{np}=-c_{ref}\frac{\Delta C_{m}}{\Delta C_{Z}}=-c_{ref}\frac{C_{m}\rvert_{\alpha=5}-C_{m}\rvert_{\alpha=0}}{C_{Z}\rvert_{\alpha=5}-C_{Z}\rvert_{\alpha=0}}$

For the fixed-wake simulations the neutral point is located at $x=-0.5866\text{ }m$whereas for the free-wake simulations the neutral point is located at $x=-0.5501\text{ }m$. The absolute difference amounts to approximately 6.5 percent.

The free-wake simulations give more accurate prediction for the neutral point location as they account for the wake deformation. This is confirmed by forced-pitch oscillation simulations. The neutral point locations obtained from forced-pitch oscillation simulations and from free-wake simulations agree well.

Once the neutral point location is determined, the location of the center of gravity can be calculated using the following equation:

$x_{cg}=x_{np}+SMc_{ref},$

where $SM$is the static margin. The greater the static margin the more statically stable the UAV is in pitch. For a 10% static margin ($SM=0.1$) $x_{cg}$is $-0.5247\text{ }m$.

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