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I recently started to study paragliding and I am confused about the rolling movement created when a brake is maintained.
I was reading this answer How do paraglider controls work? and I am not clear which force should balance the rolling created by the side force on the levered arm during the initial phase of the turn and stop the rolling motion.
In essence the question is "In a turn using brakes, when a paraglider pilot applies brake on one side and holds it indefinitely, why does the roll rate eventually come to zero? Why must be substantial brake be applied just to maintain the bank angle in the turn?"
While turning using brakes1, to maintain the desired bank angle once it is established, paragliders do in fact typically require the pilot to maintain a substantial control input (brake pressure) in the same direction as was used to initiate the turn. This is quite different from what we see in most conventional airplanes, and in modern hang gliders, but we do see similar dynamics in airplanes with lots of dihedral that lack ailerons and turn using the rudder or spoilers (examples-- "Gentle Lady" or "Radian" radio-controlled sailplane, Quicksilver MX Ultralight). We also see similar dynamics in older hang gliders, where ample roll stability2 is created by the highly swept shape of the delta or modified delta wing. In many of these older hang gliders, additional factors contribute to roll stability. These factors include a) some amount of dihedral, and b) a point of connection between the pilot's harness and the airframe that is well below the center of mass of the airframe, so that the resulting low CG contributes a strong stabilizing "pendulum effect" even when the pilot exerts zero force on the control bar.
For a basic description of paraglider turn dynamics, see this ASE answer -- How do paraglider controls work? . The answer explains how applying "brake" on one side modifies the lift vectors generated at various points along the arc of the wing in a way that generates a roll torque.
The present question is asking not what causes the roll torque into the turn when the brakes are applied, but rather why does the roll rate eventually drop to zero even with the brakes still applied. Clearly some aerodynamic force or torque is "trying" to roll the paraglider back toward wings-level.
The answer is that the turn involves some sideslip. The aircraft is yawed to point slightly toward the outside or high side of the turn, relative to the direction of the flight path at any given instant. One cause of this sideslip is the simple fact that during turning flight, the outboard wingtip travels along a larger circumference and therefore must move through more air in a given unit time than the inboard wingtip, so it tends to experience more drag than the inboard wingtip. The extra drag from the deployed brake on the inboard wingtip partly compensates for this, but not fully. This makes the wing fly in an attitude where it is yawed to point slightly toward the outside of the turn relative to the actual direction of the flight path at any given instant. In other words, the wing is constantly experiencing a sideways airflow component, toward the outside of the turn.
Notice that the arched shape of a paraglider wing exposes a tremendous amount of surface area to any sideways component in the airflow. Ordinarily when we give an aircraft an anhedral geometry by lowering the wingtips in relation to the wing root, this interacts with any sideways airflow to generate a de-stabilizing roll torque, in the "upwind" direction, which in a slipping turn, would be away from wings-level. This destabilizing roll torque component exists in the paraglider case too, but it is dwarfed by the stabilizing "downwind" roll torque caused by the fact that the center of area of the wing is so high above the CG of the whole system. Since the wing is so high above the CG of the whole system, any aerodynamic sideforce generated by sideways airflow against the wing during the (slipping) turn will contribute a strong dihedral-like roll torque in the "downwind" direction, toward wings-level. So will the wing's drag vector, because the glider is "pointing" a different direction than it is actually flying, and the drag vector acts parallel to the flight path. Since these effects are both related to the fact that the CG of the whole system is located far below the center of area, they are sometimes collectively described as a "pendulum" effect, though we need to be cautious not the invoke the idea that the aircraft is somehow directly "feeling" the direction of the weight vector in a way that is not dependent on sideslip.3
So the answer to your question of why the bank angle does not continue to increase even when the brake application is maintained, is essentially this:
Turning flight in paragliders, as in many other rudderless aircraft, always involves some sideslip. The resulting sideways component in the relative wind acts on the center of area of the wing, high above the CG of the whole system, to generate a roll torque towards wings-level.
Here's one more nuance to consider in closing: we should note that when we initially apply "brake" in wings-level flight, the increased drag on the "braked" wing initially will "steer" (yaw) the glider into a skid in the intended direction of turn, so that the glider is actually pointing slightly toward the inside of the turn relative to actual flight path at any instant. At this point the glider is "feeling" a sideways component in the relative wind toward the inside of the turn. So at this point the interaction between the sideways airflow and the wing surface area, high above the CG, is actually generating a torque to help roll the wing toward a steeper bank angle. Only once we establish a significant roll rate, and/ or a significant bank angle and turn rate, do we see the glider adopt an attitude where it is yawed slightly toward the outside of the turn, so that is "slipping" rather than "skidding". (This answer has focused on how turning tends to cause sideslip; to read about how rolling also tends to cause sideslip see this section on adverse yaw from the excellent "See How It Flies" website.) All the same can be said of an airplane with lots of dihedral that has no ailerons but rather uses the rudder for roll control-- like many radio-controlled sailplanes-- when we first apply rudder to enter a turn, we know the aircraft must be skidding, but as the bank angle increases and we settle into a steady turn at constant bank angle, we know the aircraft must in fact be slipping, even though we are continuing to hold some rudder input into the turn. If the aircraft were not slipping at this point, the net roll torque could not be zero, due to the fact that the wing on the outside of the turn is moving faster through the air, and tending to generating more lift, than the wing on the inside of the turn.
Highly related -- Does "pendulum effect" apply to hang gliders or any aircraft?
Footnotes --
1 -- It is not the intent of this answer to suggest that a turn in paragliders is always, or typically, or ideally, accomplished using brakes alone with no weight-shift. Weight-shift, where the pilot leans in the harness to rock his body to one side and load up one side of the suspension system and unload the other side, is another way to turn. Both methods may be used in combination. However the original question was focused on a turn carried out with the brakes, so this answer is as well.
2 -- This answer uses "roll stability" to mean a tendency to roll toward wings-level. Other alternative terms might be used to avoid suggesting that the aircraft is tending to maintain a given bank angle, but "roll stability" is the phrase we'll stick with for this answer.
3 -- "Keel effect" is another term that is sometimes used to describe the stabilizing effect of a low CG position, though this is also somewhat problematic, for two reasons. A) It would be better to call it the "keel weight effect", because an unweighted keel (think of a lightweight centerboard that may be raised or lowered at will) actually contributes a net hydrodynamic roll torque in the opposite direction during sideslip (sideways drift), tending to roll the boat in the upstream direction. B) The righting effect of a heavy keel weight is actually not dependent on sideslip at all-- buoyancy creates a righting moment even when total velocity, and therefore sideslip velocity, is zero-- so we still have the issue that we were trying to avoid by not using the term "pendulum effect", namely that we'd like to avoid suggesting that a (non-buoyant, e.g. heavier-than-air) aircraft in flight can somehow "feel" the direction of the gravity vector and therefore "sense" when the aircraft is tilted away from level, even if sideslip is somehow completely eliminated by appropriate rudder usage or by any other means.
Addendum--
This answer has referenced a "skid" as the turn is first initiated with brake, and a "slip" once the turn is established. This begs the question, "what about the roll torque created as the pilot's body tends to swing to the left or the right, just as a slip-skid (inclinometer) ball would, due to the apparent forces created by the slip or the skid? Doesn't this play a key role in the balance of roll torques on a paraglider?"
Or in other words, "Isn't there some additional sort of 'pendulum effect' that we have not yet fully considered?"
At least one answer to the related question "How do paraglider controls work?" has in fact focused on apparent inertial forces caused by the pilot's body "not want[ing] to turn (yet)", purportedly causing the pilot's body to "swing" against the intended direction of turn as the turn is being initiated, thus creating a roll torque.
The truth is that we can consider the paraglider and pilot to be a single, essentially rigid sytem. The lines do not go slack in normal flight. The pilot is fixed in position under the wing by the triangular geometry of the multiple suspension lines. If we want to analyze the motion of the whole body, it is not necessary to take into account the fictitious "centrifugal force" created by turning, or any other apparent inertial force. (An exception would arise if the aircraft were rotating very rapidly about one of its axes-- for example a very high rate of yaw rotation, somewhat similar to a flat spin in a conventional aircraft, might tend to flatten out the canopy into more of a horizontal line, due to "centrifugal force". Similarly, very high rates of pitch rotation play a key role in the dynamics of certain aerobatic maneuvers, such as spiral dives, in paragliders.)
The apparent tendency of the pilot to "swing" toward the inside of the turn in a slip, and toward the "outside" of the turn in a skid, thus imparting a roll torque to the wing, is in fact already fully accounted for in this answer. We account for that roll torque when we note that the sideforce created by the impact of the sideways airflow against wing acts high above the CG of the whole system and thus creates a roll torque. The apparent sideways force experienced in the reference frame of the pilot in a slip or skid, is in fact simply the mirror image of the sideways component of the real aerodynamic force generated by the aircraft during the slip or skid. If the aircraft could somehow slip or skid sideways through the air without generating any aerodynamic sideforce, then no roll torque would result. Also, if the aircraft could somehow slip or skid sideways through the air without generating any aerodynamic sideforce, then the pilot would experience no apparent tendency to swing to one side.
After noting that during a slip or skid, the sideways airflow against the wing generates an aerodynamic sideforce that acts high above the CG of the whole aircraft-pilot system and thus generates a roll torque, it would be erroneous to suggest that the apparent tendency of the pilot's body to "swing to one side" in a slip or a skid somehow contributes an additional roll torque about the wing's center of lift or about any other point.
Additional content on ASE relating to the ideas in this addendum--
