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I have a reasonable understanding of what a V-n diagram shows and what the envelope and the boundaries mean. However, the question I have is how does the weight of the aircraft specifically affect the diagram?

I know that reducing the weight will mean a lower load factor and therefore potentially more manoeuvrability. But is there a mathematical way of showing this? Also, how does the weight relate to lift? Again, lower weight means less lift is needed. But how does that specifically affect the V-n diagram?

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Well, answering your first question, the airplane weight does not really affect the V-n diagram because this diagram is developed knowing the structural forces that the airplane can take without being damaged.

Answering your question about the relation between the maneuverability and the airplane weight, the relation is just a "simple" physics problem. You just need to know the airplane characteristics like the center of gravity, the different aerodynamic factors and the effect of deflecting the control surfaces. When you solve the six equations with six variables of the differential system you can find how any characteristic of the airplane affects its maneuverability.

Differential equation system of an airplane flight mechanics

Finally, as you say, when the weight is low, the needed lift is less but this does not affect the V-n diagram because n stays more or less constant because: $n=\frac{L}{W}$. This diagram just shows us which are the relation between the aerodynamic forces and the weight. We all know that the weight is one of the problems for an airplane but it also helps to alleviate the bending moment on the wing root, that is why the last fuel tanks that are emptied on an aircraft are the ones that are further away from the fuselage.

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When an aircraft is heavier (say wing tanks or fuselage tanks full of fuel), a given amount of pounds of force generated by the wings results in a lower G-load and thus less force on components of fixed weight, like the battery or engine(s). Therefore the mounts holding these parts of the aircraft in place are subjected to less stress. Therefore the maximum amount of force the wing can be permitted to exert --measured in pounds or Newtons, not G's-- can be raised-- again assuming it's the stress on things like the motor mounts, etc that we're concerned about-- and that's why the maneuvering speed (Va) in many aircraft increases as aircraft weight is increased. (Below maneuvering speed, the wing will stall before generating some critical amount of force that has been judged by the designer to be too much.)

A simpler way to state this is that if we are setting the maneuvering speed to protect the mountings of items of fixed weight, then all that matters is the max G-load that we allow the aircraft to develop. When the aircraft is heavier, for any given G-load, the stall speed will occur at a higher airspeed. Maneuvering speed is the speed where the aircraft will stall before exerting so much force that it damages the structure. So if we are setting the maneuvering speed to protect the mountings of items of fixed weight by not allowing the aircraft to exceed some given G-load, then we can increase the maneuvering speed as we increase the weight of the aircraft.

On the other hand, if were worried about ripping the wings off the fuselage-- if that was the limiting factor in setting our limiting speed-- then it wouldn't make any sense to raise our limiting speed as we increase aircraft weight, at least if all the increased weight was going into the fuselage. In a simplified case where the weight of the wing is negligible compared to the weight of the fuselage, when the wing is generating X pounds of lift, the same amount of force is being transferred from the wing to the fuselage, regardless of how heavy the fuselage is and therefore what the G-load is. If the weight of the wing is not negligible compared to the weight of the fuselage, then adding weight to the fuselage means a lower percentage of the wing's lift force will be "absorbed" by the wing itself, and for a given X pounds of lift force generated by the wing, the force exerted by the wing on the fuselage, and the stress on the wing-fuselage connection, will go up as we increase the aircraft weight. In such a case, if the wing-fuselage connection is our critical concern, then it would make sense for the maneuvering speed to go down as the aircraft weight is increased. On the other hand if the extra weight is going into the wing (fuel, external stores hung from the wings) then for a given X pounds of lift generated by the wing, some of the wing's lift force will be "absorbed" by this weight and the total G-load in any given situation will be less and there will be less force transferred from the wing to the fuselage and less stress on the wing-to-fuselage mounting, so again it would make sense to raise the maneuvering speed as we increase the aircraft weight, if areas such as the wing-to-fuselage mounting are the critical concern.

Similarly, if more of the weight is distributed along the wingspan, the bending stress on the wing spars will be less, for a given total force in pounds generated by the wing. So if the wing spars are the critical component of concern that governs our choice of maneuvering speed, then if we increase weight by adding it to the wing, the maneuvering speed should go up, but if we increase weight by adding it to the fuselage, the maneuvering speed should go down.

So, it's complicated. The simplest case is when the limiting concern is the stress on the mountings of items of fixed weight, such as motor mounts, battery brackets, etc, as described at the start of this answer. My understanding is that that is in fact this most common case and explains why on the Vn diagram, the maneuvering speed typically goes up as the aircraft weight goes up. Again, in this case we are simply setting a maximum allowable G-loading.

"Maneuvering speed" is not explicitly shown on the figure linked in the question, but generally, it occurs at the point where the line representing the max G-load allowable meets the curved left edge of the envelope representing the stall, with some extra safety margin added. The discussion above of whether the maneuvering speed Va should be raised or lowered as we add weight to the aircraft, depending on where we are adding the weight, is exactly equivalent to a discussion of whether we should raise or lower the limiting G-load, or neither, as we add weight to the aircraft. Note that on the V-n diagram, adding weight to the aircraft will shift the curved left edge of the envelope representing the stall further to the right. You can see how, if our goal is simply to set the maneuvering speed in a way that prevents the aircraft from exceeding some fixed maximum allowable G-loading, then an increase in weight will automatically change the V-n diagram in a way that increases the maneuvering speed in proportion to the square root of the increase in the weight. However, if the goal is to limit the stress on the wing spars, or the wing-to-fuselage connection, then the situation may be completely different, depending on where we are adding the weight.

(The diagram linked in the answer doesn't show this, because it doesn't show the stall speed for different weights. Rather, it shows a change in IAS stall speed with altitude, which is likely related to Mach effects, and not something that we have to concern ourselves with in lower-performance aircraft.)

Related ASE questions and answers--

Q: What is Vg in this VG diagram?

A: What is Vg in this VG diagram?" -- in this case the aircraft (glider) was assigned a higher Vg, the maximum speed allowed in gusty conditions, a well as a higher Vne, when it was carrying less weight. The opposite of what we often see in regard to the maneuvering speed, Va, in light airplanes. So-- as indicated in the present answer-- "it's complicated".

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