How does the dihedral angle work?

Question

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.

Answer 1

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 $u \cdot \sin\beta \cdot \sin\Gamma$.

Image from Stability and Control of Aerospace Vehicles

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

Image from Stability and Control of Aerospace Vehicles

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.

Image from Stability and Control of Aerospace Vehicles

Answer 2

The explanation lies in the fact that a rolled wing creates a oblique relative wind, and that a wing with a dihedral angle seen from an oblique direction is having a larger angle of attack on the side in this direction:

^{Because of the dihedral angle, for the wind coming from an oblique direction on the right of the aircraft, the right wing shows a larger AoA. This is even more important for a larger dihedral angle.}

Visual demonstration

On the image below:

• On the left side, there is an aircraft which is horizontal, flying level, and in a relative headwind. Obviously the action of the wind will be the same on both wings, whatever the dihedral angle, and the lift vector is oriented vertically (in blue).

• On the right side, the aircraft has been disturbed and for some reason is now rolled to the right without pilot action. Imagine the heading is still the same.

The key to see what will happen is to understand the right wing now develops more lift than the left wing, the difference being proportional to the dihedral angle. As soon as this is clear, we can anticipate the roll angle will be cancelled automatically, without pilot action.

Let's look at the rolled aircraft:

• The lift vector, which is still normal to the wing, is no more vertical. From a mathematical point of view it can be split into two components along arbitrary directions. If we choose a resolution along the vertical and horizontal axes, we see the vertical component is now smaller (therefore the aircraft starts descending) and a horizontal component appeared in the process.

• The horizontal component pulls the aircraft to the right side. As the heading is unchanged, the aircraft is not in a turn, and no centrifugal force opposes this horizontal component of the lift, therefore the aircraft starts side-slipping and the relative wind is no more a headwind, there is some crosswind from the right side.

Angle of attack seen from the relative wind standpoint:

• When the aircraft was flying in a headwind, the angle of attack was the same for both wings.

• With the crosswind component, the angle of attack of the right wing is higher than the angle of attack of the left wing. The difference is small when the dihedral angle is small, and increases with its value. To make this apparent, I added wings with higher dihedral angles to the aircraft:

Note: The difference appears only when the wind is off axis. This means the dihedral effect on the angle of attack exists only when there is a sideslip.

Of course because the angle of attack is larger on the right, a recovery moment starts developing and counteracts the initial roll. The aircraft returns to the horizontal after some damped oscillations around the longitudinal axis.

The lateral stability is of prime importance for general and commercial aviation aircraft. The dihedral angle is the most simple mean to obtain this stability, there are others.

Stability from swept wing, due to spanwise flow

Lift is generated taking into account the airflow parallel to the chord which is accelerated. Air moved in a perpendicular direction is not accelerated and doesn't create any lift, see left image:

(By principle, a swept wing decreases the amount of lift created, this is compensated by other benefits that make it useful anyway).

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 a larger ratio of air generates lift compared to when the airflow comes frontally. This is the contrary for the left wing. This effect also contributes to lateral stability.

Preventing spiral

The dihedral angle participates to the roll stability, along with other factors. 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 either self-decay with time (stable) or constantly increase (unstable). The unstable spiral mode happens this way:

• (1) The disturbance creates a small roll moment and sideslip to the right.
• (2) The sideslip creates a crosswind component from the right.
• (3) The vertical stabilizer AoA increases and creates lift to the left.
  
  
  
• (4) Lift creates a yaw moment and turns the nose to the right.
• (1) 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.}

Answer 3

Dihedral generates a stabilizing roll torque due to the difference in angle-of-attack experienced by the left and right wings during a sideslip.

Furthermore, it's important to note that sideslip cannot be explained simply by noting that when an aircraft is banked, the tilted lift vector contains a sideways component, or that "from the point of view of the aircraft, lift is still acting in the plane of symmetry, but gravity does not and will cause it to sideslip", as is sometimes stated. (For example, we find something close to this in Martin Simons' well-known book "Model Aircraft Aerodynamics".) Those are essentially Aristotelian concepts rather than Newtonian concepts. A continuous unbalanced sideways force component causes a turn, not a sideslip. Force causes acceleration, not steady sideways motion, and turning is a curvature in the flight path which is a form of acceleration.

Rather, sideslip is a result of not being pointed the same direction you are actually going. The reason that banking tends to cause sideslip has to do with the "curving" nature of the relative wind in a turn. Since the aircraft is rotating as well as translating, different parts of the aircraft are moving through the airmass in different directions at any given instant, which means if we map out the relative wind felt by various parts of the aircraft at any given instant, we get curved lines, not straight lines. Even if the vertical fin were perfectly streamlined to the flow at any given instant, more forward parts of the aircraft-- including the wing-- would be experiencing some sideslip. This effect is especially pronounced in aircraft with low "scale speeds"-- i.e. forward airspeed divided by fuselage length.

Yaw rotational inertia can also play a role in promoting sideslip immediately after an increase in bank angle, but this is probably a minor effect.

There is one other nuance to be pointed out that is probably a only very minor effect in most cases. Imagine an aircraft with zero dihedral flying at 10 degrees of angle-of-attack. Imagine that the aircraft abruptly rolls 90 degrees and the rolling motion is about the aircraft's lateral axis, not about the airspeed vector. The angle-of-attack will be converted to sideslip-- the aircraft will end up with 10 degrees of sideslip and no angle-of-attack. Now if we add dihedral, we'll see we end up with a roll torque. However this dynamic is probably trivial in normal roll stability dynamics which involve low rates of roll, allowing the aircraft's inherent pitch stability dynamics to maintain a constant average angle-of-attack, and allowing the aircraft's inherent "weathervane" stability to oppose sideslip, so that angle-of-attack does not end up being converted into sideslip simply by virtue of the rolling motion.

Answer 4

Coming to the aid of the -5 @rbp answer, having been there, and a few items to improve the answer, and to respond to the question "how does dihedral work", as follows:

What is missing in our exaggerated 45 degree model is an evaluation of total lift, vertical lift, and center of lift relative to center of gravity.

One of the quirks of physics is that a 45 degree angled wing still produces 70.7% vertical lift of a zero degree angled wing (relative to the earths surface). This means that both wings at 45 degrees produce 42% more vertical lift than one wing at zero degrees and one at 90 degrees.

What happens when the plane rolls? Vertical lift is lower and the plane SINKS. The aircraft now has a vertical direction component, there for a change in "relative wind".

Now, where is the roll torquing force around the center of gravity coming from? Many have correctly stated that it can not be from the lift vectors, and many have correctly stated it comes from the "slip".

Notice what effect a VERTICAL component will have on the wings. Obviously the zero degree wing will be rolled up from the relative wind (vertical component) until its angle is equal to the opposite wing, restoring the original attitude and lift condition.

The side force, created by the 90 degree wing, also adds side motion. The net motion of the plane is a slip down and to the side until the wings re-level. That's the aerodynamic part, but that's not all!

When the plane rolls, the center of vertical lift, relative to the center of gravity, moves out of alignment, creating a "yin and yang" roll torque effect which also helps right the aircraft.

Dihedral is present in many aircraft designs where cruising comfort is preferred and flight other than straight and level is uncommon.

Answer 5

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.