[gurps] Dragon Aerodynamics

[midnightwind at comcast.net]

Anthony (and the list):

This is so cool.  I even understood (most of) it.  I could never reproduce it, however.
And I don't dare try and "help you" on it.  Damn, I wish I had become an engineer, sometimes.

Incidentally, I noticed Warehouse 23 is selling the 4e Starship Builder.  Before I buy it, any inside scoops on it?

-vk
 -------------- Original message ----------------------
From: Anthony Jackson <ajackson at iii.com>
> Having just been reading a little about aerodynamics, I decided to amuse 
> myself by trying to apply them to dragons (yes, this is slightly 
> pointless). Since I'm applying rather basic principles to the rather 
> complex subject of winged flight, this may not be a perfect translation, 
> but it's good enough to amuse me, so I thought I'd share it and also see 
> if anyone has helpful comments:
> 
> Applying Aerodynamics to Dragons
> 
> It is generally obvious that a dragon, if made of normal materials and 
> flying by normal principles, cannot fly. The usual solution taken by 
> people who want dragons anyway is to give them some exotic method of 
> flight, such as magical levitation. However, just as it's obvious that a 
> dragon can't fly, we also know that a giant made of normal flesh and 
> bone cannot stand up, so supposing that dragons are not made of normal 
> flesh and bone is hardly strange.
> 
> So, if we accept dragons which aren’t made of normal flesh, what can we 
> compute about a dragon? We'll start with a dragon that's 30' long. For 
> the body, we'll give it about the same build as a komodo dragon, which 
> for an average specimen
> might be 8' and 150 lb. Scaling up to 30' we get about 8,000 lb. Note 
> that this corresponds to about 4' of head/neck, 11' of body, 15' of 
> tail. Body width is assumed to peak at about 4'6" wide and 3' tall, for 
> an area of 10.6 sf or about 1 square meter.
> 
> Now, for the wings, we'll go with a wingspan of 40' (12m) with an aspect 
> ratio of 8, which seems like a fair approximation of typical dragon 
> wings. Including some lost area in the body and variation in shape, the 
> maximum width of the wing is probably 8' or so.
> 
> Now, let's start with just how strong the wings are. For simplicity, let 
> us assume that wing flapping is somewhere between the level of effort of 
> flapping your arms (up and down) and doing that with a 5 lb weight in 
> your hand.  This corresponds to a torque of somewhere up to about 15 lb*ft.
> 
> Now, we'll set the dragon wings as a structure five times larger than a 
> human arm/shoulder combination, making it somewhat larger (as compared 
> to the torso, which is only about 4x larger) than a human shoulder. The 
> maximum torque of a rod is proportional to thickness^3, so the natural 
> maximum sustained torque is about 1,875 ft*lb. Since the dragon has a 
> weight of 8,000 lb (4,000 lb per wing) with the an average separation 
> (due to flapping mechanics) of about 15', the average torque is about 
> 60,000 lb per wing, or 32x greater. Flapping is probably actually more 
> stressful than raising and lowering the arm (unless you do it very 
> fast), so we might want to double it.
> 
> If we assume the same numbers for other limbs, an 7.5' komodo dragon 
> probably has a bite ST of 10, which means our 30' dragon would have a 
> basic bite ST of 40, multiplied by sqrt(32) for stronger materials, or 
> 226. Dragon arms usually seem to be shorter and smaller, compared to the 
> torso size, than human arms, so we'll multiply by 0.75 for a limb ST of 170.
> 
> However, flapping is still pretty slow movement relatively speaking, so 
> it might be better to assume a 32x Power, rather than 32x Force. This 
> changes the scaling from X^1/2 to X^1/3, and gives us a slightly more 
> reasonable figure of 130 bite ST, 95 limb ST.
> 
> The speed of an action with a limb is proportional to 
> sqrt(force/weight*length). We're assigning our dragon arms 3x length, 
> 27x mass, and 90x the force allowed for a human arm, so the dragon can 
> strike about as quickly as a human would punch -- though the velocity of 
> the arm is three times greater. This is broadly compatible with the 
> damage the dragon does, though it's quite a bit faster than any real animal.
> 
> Going onward to actual aerodynamics, the lift equation is:
> L = Cl * S * 1/2rho * V^2, where L is lift, Cl is the coeffient of lift, 
> S is wing area, rho is atmospheric density, V is velocity.  For our 
> sample dragon, S is a value of 18 m^2, rho is 1.225 kg/m^3, and the 
> remaining values can vary.
> 
> Ordinary wings have a maximum Cl that is typically 1.5 to 2, but to 
> handle the additional lift that flapping can grant, we're going to treat 
> the wings as allowing a Cl of up to 3. Setting L to 36,000N and solving 
> for V^2, we discover a stall speed of 33 m/s or Move 36.
> 
> Now, we want to know drag. Looking at some handy formula:
> Di = kL^2/(1/2*rho*V^2*S*pi*Ar), where Di is induced drag, k indicates 
> the degree to which the wing is worse than elliptical (probably 
> significant for a dragon wing, but we'll set it to 1.1), Ar is aspect 
> ratio (8, as above), and other terms are as before. At stall speed, it 
> is equal to
> 1.1*35584^2/(0.5*1.225*33^2*18*3.1416*8) or 4600N.
> Ds = 0.5*rho*V^2*Cd*A, where Cd is the coefficient of drag and A is 
> frontal area. Neither of these numbers is particularly easy to compute 
> for a dragon, but for now we'll set Cd*A at 0.2 square meters. At stall, 
> this gives us a drag of 0.5 * 1.225 * 33^2 * 0.2 or 133N.
> 
> Now, the basic power requirement for movement is equal to Drag*Velocity, 
> with some additional inefficiency caused by the mechanics of propulsion 
> that is fairly hard to calculate, and is probably fairly small for 
> wings, as they are quite large. Thus, the basic power requirement is 
> equal to 4800N*33m/s or 160 kilowatts (210 horsepower). As a dragon is 
> about six times more massive than a horse, this is 35x the power to 
> weight ratio, which is reasonably comparable to our strength ratio 
> (derived above) and thus agrees with the idea that the strength might be 
> relatively slow pull muscle.
> 
> Other than stall speed, there are two other numbers of special interest. 
> These are the speed at drag (and thus work per unit distance) is 
> minimized (ideal cruising speed) and the speed at which power is 
> minimized (ideal loiter speed). Drag is minimized at the speed where 
> induced and static drag are equal; power is minimized at a speed equal 
> to the minimum drag speed / sqrt(sqrt(3)) (this will not be derived 
> here). We'll also give it a sprint speed that is 25% faster than 
> cruising speed. As it happens, I have a program which is solving this, 
> which gives us:
> Stall Speed          :  32.8 m/s, Drag  4804N, Power 1.576e+05W
> Minimum Power/Dist   : 80.04 m/s, Drag  1569N, Power 1.256e+05W
> Minimum Power        : 60.81 m/s, Drag  1812N, Power 1.102e+05W
> Sprint Speed         :   100 m/s, Drag  1728N, Power 1.728e+05W
> 
> There is one more category of interest: hovering. The lift for a column 
> of air is equal to V^3*A*rho, where V is the speed of the air, A is the 
> cross-section of the column, and rho is air density. Assuming some 
> sculling motion to allow a cross-section wider than wing area, but also 
> some loss due to flapping, an effective area of 25 square meters results 
> in an air speed of 10.5 meters per second and would require a power of 
> 190 kW. As the airspeed is actually a bit unbalanced, the actual power 
> requirement is more on the order of 250 kW. It is possible that a dragon 
> can maintain that for a few instants, but it probably cannot hover in a 
> sustained way.
> 
> We are also ignoring one key effect: body lift. Depending on body 
> structure, this could be substantial, especially if the body and tail 
> can be flattened or made concave (model after a flying snake). However, 
> while the body has a potentially fairly large area (6 square meters is 
> not hard to imagine), it has a very poor aspect ratio. If we treat the 
> belly as half as effective as ordinary wings, we reduce stall speed but 
> don't affect other numbers:
> Stall Speed          : 30.37 m/s, Drag  5564N, Power 1.69e+05W
> This is mostly useful because it means the dragon can keep its tail up 
> without resorting to extremes of strength.
> 
> All of this results in a dragon that, while not grossly large (30' long, 
> 4 tons) is terrifyingly powerful, at least by low tech standards (230 
> horsepower in a 4 ton car isn't very impressive; 230 horsepower in a 4 
> ton helicopter would be very low). If we figure a komodo dragon has DR 
> 1, and make the scale materials stronger in the same ratio as we made 
> its muscles, net DR is 1(base)*4(4x bigger)*32^1/3) or 13, and bite 
> damage is on the order of 13d (DR may be low; realistically humans 
> probably have DR 1, and weapon damages have been adjusted to compensate. 
> That would give an actual DR of 25).  This may well be beyond the 
> reasonable limits for fantasy PCs, and discovering that a dragon _runs_ 
> twice as fast as a racehorse will likely strike PCs as a bit odd (it's 
> also unlikely for something with this sort of build; the strength ratios 
> only support a peak running speed of about 30 mph.
> 
> So, what numbers can we tweak to make this a bit less absurd? We don't 
> have much in the way of options to change the shape, but the weight is 
> probably subject to change. In general, multiplying weight by X 
> multiplies the speed of stall, minimum drag, and minimum power by 
> sqrt(X). It multiplies power consumption at each of those speeds by 
> X^1.5. It also multiplies calculated ST by either sqrt(X) or X^1/3, 
> depending on which assumptions were being used on muscle. Thus, if we 
> take our 8,000 lb dragon and cut it down to 2,000 lb, we halve all 
> velocities and reduce ST to 80 body, 60 limb; quickness is unaffected; 
> DR remains at a constant of about ST/10(8) (or, with the 'humans are DR 
> 1' theory, go with ST/5-1, or 15).  The new stats are as follows:
> Stall Speed          :  16.4 m/s, Drag  1201N, Power 1.97e+04W
> Minimum Power/Dist   : 40.02 m/s, Drag 392.4N, Power 1.57e+04W
> Minimum Power        : 30.41 m/s, Drag 453.1N, Power 1.378e+04W
> Sprint Speed         :    50 m/s, Drag 431.9N, Power 2.16e+04W
> Again, with the body lift assumptions:
> Stall Speed          : 15.18 m/s, Drag  1391N, Power 2.112e+04W
> 
> We still have 18x the power to weight and 5x the strength to weight of a 
> human, but this is at least probably closer to something that is playable.
> 
> Of course, a dragon that's 10 hexes long, half of which is tail, and is 
> probably no more than 8' at the shoulder, isn't all that impressive. So, 
> how do these numbers change for a dragon of a different size? It turns 
> out that there is a very simple scaling: if you multiply size by X, 
> multiply weight by X^3, speed (all) by X^1/2, power consumption by 
> X^3.5, and ST/DR by X^3/2.  Thus, an immense 100' dragon would be 37 
> tons, stall 30 m/s, minimum P/D 73, minimum P 55, sprint 90, ST 370/490, 
> DR 49, and uses 1.35 megawatts on takeoff.
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