# How does the dihedral angle work?

I've been looking for this topic on the internet but I don't have enough concrete answers. Suppose we have a plane with dihedral and it has a suddenly swinging to the right (watching from the nose of the plane), so the right wing goes down. I'm trying to understand why the right wing generates more lift than the left wing when it has a sideslip. I have seen in some sites that the sideslip induces a flow from the tip to the root and this makes the right wing increase locally the angle of attack, hence the lift of this wing increases too.

But, why the right wing increases the angle of attack? I think it couldn't be possible because the sideslip flow is in a different plane respect the mainstream.

• start from here – GHB Apr 8 '16 at 11:30
• To me this is not a duplicate. We have no question on the site explaining the principle behind anhedral/dihedral. I concede, though, that some rewording might be required. – Federico Apr 8 '16 at 11:53
• sorry, my native language is not English, so I know I have a lot of mistakes. Anyway, I know "watching from the nose" is not very clear. What I meant is " seen in the same direction that leads the mainstream " – kuvala Apr 8 '16 at 14:13

Basically, dihedral effect is that during banking, the 'lower' wing will experience a higher angle of attack compared to the 'higher' wing, and a result, a greater lift. The resulting net force and moment reduces the banking angle, reducing stability.

Consider a wing with a dihedral angle $\Gamma$ with a forward airspeed of $u$. If the sideslip angle is $\beta$, the wind due to sideslip is $u \cdot sin\beta$. From geometry, the normal velocity induced due to dihedral, $v_{n}$ becomes $u \cdot sin\beta \cdot sin\Gamma$.

Note: The notations are different in the figure; but the principle is the same.

For our purposes, we can take the sideslip velocity ($u \cdot sin\beta$) as $v_{y}$. Now, consider two sections from the wing- one each from the 'lower' and 'higher' sides. The induced velocity is of the same magnitude in both the sides, while the direction differs, as can be seen from the above figure.

Image from people.rit.edu

For small angles, $v_{y}$ is nearly equal to $u \beta$. The induced angle can be given as,

$\Delta \alpha = \frac{v_{n}}{u}$.

From the earlier relations, we have,

$\Delta \alpha_{1} = \beta \cdot sin\Gamma$, and $\Delta \alpha_{2} = -\beta \cdot sin\Gamma$.

Because of these induced angles, the lift on the downgoing wing increases by $\Delta L$, while of the other one decreases by $\Delta L$. The net result is that the 'lower' wing experiences an increases lift, causing a rolling moment, which causes the banking angle to reduce.

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.

Visual demonstration

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.

• You've got me really confused. The disturbance creates the small sideslip to the right. but The sideslip creates a crosswind component from the right. This seems backwards. If you yaw right the crosswind would be from the left side. – TomMcW Apr 8 '16 at 20:31
• When you say sideslip to the right do you mean the tail goes to the right or the nose? – TomMcW Apr 8 '16 at 20:47
• My brain is hurting! If the nose turns to the right wouldn't the crosswind component be from the left side? – TomMcW Apr 8 '16 at 21:11
• Plus, I thought a roll caused an adverse yaw – TomMcW Apr 8 '16 at 21:13
• @TomMcW: The aircraft sideslips to the right ⟹ the aircraft now moves to the right ⟹ a relative crosswind from the right ⟹ the tail creates a nose yaw to the right. For the adverse yaw, yes but is cancelled by the crosswind on the tail. I added a picture for this effect. Remember this is a mechanism without dihedral, the dihedral being the solution to the unstable spiral. – mins Apr 8 '16 at 22:48

This is a very exaggerated diagram of a fuselage with dihedral wings.

When the plane is flying normally (top), both wings produce the same lift vectors.

When the airplane is disturbed in the roll axis, and one wing goes higher than the other (bottom), the vertical lift vectors are different, with the "down" wing producing more vertical lift than the "up" wing. This increased vertical lift on the down wing and decreased vertical lift on the up wing, pushes the down wing up, and helps right the airplane.

Note: everything in this diagram is schematic and is not meant to indicate any specific mathematical or physics laws or formulas.

• Note that in the second image, while the vertical is smaller, the horizontal, which is now above the center of gravity, would actually contribute to more bank, and possibly even overpower... Is this the cause for overbanking tendencies in steep turns? – falstro Apr 8 '16 at 15:10
• There is no rolling moment if the lift intensity is the same measured perpendicularly to the surfaces. – mins Apr 8 '16 at 16:19
• I'm not sure if your explanation is correct, as it does not include a sideslip component. As I understand it, both vectors remain the same in relation to the aircraft (thus no roll moment) unless a sideslip is involved. – TomMcW Apr 8 '16 at 19:25
• This is an oft repeated fallacy. People think that gravity is something special and lift is only measured in relation to the ground. That is false reasoning. The lift generated perpendicular to the wings are the same therefore there is no net torque to correct the bank. If we add sideslip however the higher wing would have less lift due to lower AOA. Note that if the plane does not initially have any sideslip the higher wing will contribute a net side force that will create a sideslip - so yes, even without sideslip dihedral will self-correct because it will self-create the required sideslip. – slebetman Apr 11 '16 at 7:35
• @slebetman: Thank you for pointing this out. Just one nitpick: Your last sentence could be misunderstood that dihedral produces sideslip. The sideslip does not require dihedral - a roll angle will do it already, regardless of dihedral. The dihedral is needed to produce rolling from sideslip. – Peter Kämpf Apr 11 '16 at 10:41