Directional stability

Directional stability is stability of a moving body or vehicle about an axis which is perpendicular to its direction of motion. Stability of a vehicle concerns itself with the tendency of a vehicle to return to its original direction in relation to the oncoming medium (water, air, road surface, etc.) when disturbed (rotated) away from that original direction. If a vehicle is directionally stable, a restoring moment is produced which is in a direction opposite to the rotational disturbance. This "pushes" the vehicle (in rotation) so as to return it to the original orientation, thus tending to keep the vehicle oriented in the original direction. Directional stability is frequently called "weather vaning" because a directionally stable vehicle free to rotate about its center of mass is similar to a weather vane rotating about its (vertical) pivot. With the exception of spacecraft, vehicles generally have a recognisable front and rear and are designed so that the front points more or less in the direction of motion. Without this stability, they may tumble end over end, spin or orient themselves at a high angle of attack, even broadside on to the direction of motion. At high angles of attack, drag forces may become excessive, the vehicle may be impossible to control, or may even experience structural failure. In general, land, sea, air and underwater vehicles are designed to have a natural tendency to point in the direction of motion.

Arrows, darts, rockets, and airships have tail surfaces (fins or feathers) to achieve directional stability; an airplane uses its vertical stabilizer for the same purpose. A road vehicle does not have elements specifically designed to maintain stability, but relies primarily on the distribution of mass.

These points are best illustrated with an example. The first stage of studying the stability of a road vehicle is the derivation of a reasonable approximation to the equations of motion. The diagram illustrates a four wheel vehicle, in which the front axle is located a distance ahead of the centre of gravity and the rear axle is a distance aft of the cg. The body of the car is pointing in a direction (theta) whilst it is travelling in a direction (psi). In general, these are not the same. The tyre treads at the region of contact point in the direction of travel, but the hubs are aligned with the vehicle body, with the steering held central. The tyres distort as they rotate to accommodate this mis alignment, and generate side forces as a consequence. The net side force Y on the vehicle is the centripetal force causing the vehicle to change the direction it is traveling:

where M is the vehicle mass and V the speed. The angles are all assumed small, so the lateral force equation is:

The rotation of the body subjected to a yawing moment N is governed by:

where I is the moment of inertia in yaw. The forces and moments of interest arise from the distortion of the tyres. The angle between the direction the tread is rolling and the hub is called the slip angle. This is a bit of a misnomer, because the tyre as a whole does not actually slip, part of the region in contact with the road adheres, and part of the region slips. We assume that the tyre force is directly proportional to the slip angle This is made up of the slip of the vehicle as a whole modified by the angular velocity of the body. For the front axle:

whilst for the rear axle:

Let the constant of proportionality be k. The sideforce is, therefore:

The moment is:

Denoting the angular velocity , the equations of motion are:

Let (beta), the slip angle for the vehicle as a whole:

Eliminating yields the following equation in :

This is called a second order linear homogeneous equation, and its properties form the basis of much of control theory.

The forces arising from slip depend on the loading on the tyre as well as the slip angle, this effect has been ignored, but could be taken into account by assuming different values of k for the front and rear axles. Roll motion due to cornering will redistribute the tyre loads between the nearside and offside of the vehicle, again modifying the tyre forces. Engine torque likewise re distributes the load between front and rear tyres. A full analysis should also take account of the suspension response. The complete analysis is essential for the design of high performance road vehicles, but is beyond the scope of this article.

Directional Stability about the aircraft’s vertical axis is also referred to as yawing. This is primarily achieved by the area of the vertical stabilizer and the sides of the fuselage aft of the center of gravity. When an airplane is flying in a straight line while hit by a side gust of wind, the left/right yawing motion will be stopped by the air striking at the right/left side of the vertical stabilizer.