The dihedral plays a role in the roll (lateral) stability, or to be more precise in the spiral mode stability.
When the aircraft enters in the sideslip, a crosswind component appears. In a dihedral, the lower wing benefits from this oblique airflow more than the higher wing due to an increased angle of attack, creating a recovery moment.
Entering in a sideslip and creating a crosswind
The resultant lift of wings is the sum of individual lift vectors, and is in the plan of symmetry of the wings, i.e. vertical.
If a disturbance causes the airplane to roll on the right wing, the resultant lift vector will be rotated with the wings. The rotated vector can be seen as the sum of two components:
- One vertical which continues to oppose the aircraft weight, albeit it is a bit smaller, so now the aircraft is descending.
- One cross which pulls the aircraft on the right side and creates the sideslip motion.
As the aircraft is now also moving sideways, the relative wind is no more coming from the front, but a bit from the right. This cross wind is key to recovering from the roll.
Creating different angles of attack to create a restoring force
If the wing is dihedral, after the roll the lower right wing exposes a larger angle of attack (α) than the other wing to the now oblique wind. This right wing generates more lift. Similarly the left wing AoA decreased and generate less lift. This creates a rolling moment which restores the level attitude. While this principle is quite simple, the reason which changes the angle of attack in an asymmetric way is less obvious.
Rather than a mathematic demonstration, let's just look at dihedral wings. To make the difference more visible, let's add wings with an increasing dihedral angle to the picture (on the left).
Because of the geometry, when we look at the wings from where the wind is coming (right picture), we see a bit of the bottom area of the right wing, the higher the dihedral angel, the more we see. We would see the same from the left wing if it was still in the same plan than the right wing. But it has been folded, and the more the folding angle, the less we see its bottom side.
The folding has no roll effect when the relative airflow is parallel to the roll axis, each wing presenting in this case the same AoA to the airflow. However when the airflow is oblique the AoA — which is, by definition, the angle between the chord and the airflow — is now different because the chord of each wing is oriented differently relative to the airflow, due to the dihedral.
The larger the dihedral angle, the larger the difference in angles of attack between the two wings and the stronger the restoring moment. The effect exists as soon as the dihedral angle is not null. There is a similar effect when the wings are high and the angle is negative (anhedral wings).
Note that the dihedral restoring force is dependent on the fact that the wind comes in oblique, said otherwise on the existence of the sideslip.
Additional role payed by the chord orientation relative to the airflow
The chord line of the airfoil can be approximated to be perpendicular to the leading edge. The airflow can be arbitrarily seen as having two components, one parallel to the chord, one perpendicular to the chord.
Lift is generated taking into account the airflow parallel to the chord which is accelerated. Air moved in the perpendicular direction is not accelerated and doesn't create any lift, see left image:
By the way, that means a swept wing decreases the amount of lift created (this is compensated by other benefits that make it useful anyway).
Now if the swept wing receives wind from an oblique direction, like during a sideslip, the available air energy will not be lost in the same proportion for each wing (see image on the right, above).
The chord of the right wing is better oriented in the airflow coming from the right, and can a larger ratio of air to generate lift than when the airflow comes frontally. This is the contrary for the left wing.
To sum up: The lower wing generates more lift for two reasons: its increased angle of attack due to the dihedral angle and its better efficiency due to the swept wing angle.
Limiting spiral mode is part of the roll stability
The dihedral angle participates to the roll stability, but other factors contribute too. The area where the dihedral plays a critical role is the stabilization of the spiral mode (or spiral divergence). The spiral mode, like the Dutch roll and the phugoid, is an oscillatory mode that can self-decay with time (stable) or constantly increase (unstable). The unstable spiral mode happens this way:
- The disturbance creates a small roll moment and sideslip to the right.
- The sideslip creates a crosswind component from the right.
- The vertical stabilizer AoA increases and creates lift to the left.
- Lift creates a yaw moment and turns the nose to the right.
- The yaw moment increases the roll moment and the sideslip to the right.
- A new cycle has begun.
If this effect is not detected and corrected, which can easily happen in IMC when the natural horizon is not visible, the aircraft continues to sideslip and yaw, while the vertical component of the lift decreases due to the roll, creating a dangerous spiral downwards which can lead to structural damages or ground collision.
The cycle is the result of all dynamic forces in action on the aircraft, in particular the lift on each wing and the position of the center of pressure. The use of a dihedral wings affects the forces and their relative timing, and transforms an unstable spiral mode into a stable one. This is facilitated by also using a smaller vertical stabilizer and rudder, which in turn can create an unstable Dutch roll, or a shorter cabin.
Thanks to ahmetsalih for the Learjet 3D model available at TF3DM.