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 <[EMAIL PROTECTED]> > 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 arent 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. > _______________________________________________ > GurpsNet-L mailing list <[email protected]> > http://mail.sjgames.com/mailman/listinfo/gurpsnet-l _______________________________________________ GurpsNet-L mailing list <[email protected]> http://mail.sjgames.com/mailman/listinfo/gurpsnet-l
