Wing Loading, Bah! Land It Like A Man! Full-Throttle and Nose Up!

Recently, we have heard lots of talk about our normally well loaded wings. The balsa versions of our planes work great, and have normal stall speeds around 15-20mph. That is great for normal flying. They are hybrid flying wings and should fly like a Dutch Roll resistant frisbee.

I have been building the carbon fiber and Aluminum versions. They are a bit heavier than we expected, but that is due to real materials. We certainly will take out some of the fat in future revisions, but I think that the issue here is approach speed. You bunch of sissy girls. These planes are UAS, unmanned means the computer should be doing the work to bring the plane in on glide path. I hope our spars are strong enough for the cut the power approach.

What the heck am I talking about? Nope, never been asked that in polite company either. Wing loading is a relative measure that comes out of the basic low-speed aerodynamics of any fixed-aircraft. In some ways, it is a measure of the relative performance of a device with respect to constant thrust. It is most easily expressed as the mass of the aircraft divided by the wing area.

This essentially describes how strong the pressure difference must be between the surfaces to keep the plane in the air. Considering that the higher the wing loading, the larger the drag due to lift turn will be. This will change the trim characteristics of the plane in cruise and require more thrust to keep the plane above stall. Any F-4 or F-15 driver will tell you that more thrust is the answer to everything.

"The critical limit for bird flight is about 5 lb/ft² (25 kg/m²)^[3]. An analysis of bird flight which looked at 138 species ranging in mass from 1x10^-2 to 10 kg, from small passerines to swans and cranes found wing loadings from about 1 to 20 kg/m^2[4]. The wing loadings of some of the lightest aircraft fall comfortably within this range. One typical hang-glider (see table) has a maximum wing loading of 6.3 kg/m², and an ultralight rigid glider^[5] 8.3 kg/m²." - wikipedia

The wing loading also changes the stall speed. If you have to use forward speed to generate enough lift for a given flight regime, you have to go faster to balance the weight of the aircraft with the lift generated. In a flying wing, you cannot just pick the nose up, roll control has to be gentle. The old adage "Little planes add flap, big planes add power" is our friend. If you were interested the effect of wing loading on stall speed is expressed as the following.

v^2 is the stall speed

g is the acceleration due to gravity

Ws is the wing loading, mass/wing area

rho is the density of air

CL is the coefficient of lift of the net wing

Another interesting equation, is the rate of climb. This is just a force balance between the net acceleration, lift generated and the weight of the device.

ac is the climbing acceleration

Ws is still the mass wing loading

vc is the new airspeed

rho is the density of the free stream air

CL is the coefficient of lift of the net wing

g is the acceleration due to gravity

The wing loading term is in the denominator, so if you want to climb faster you need to lower the wing loading, or increase speed. Increasing the speed is way more fun than having gossamer wings. More Power!

We are not so bad, we are in the 23-35 kg/m^2 range for our wing loading depending on the equipment load out. Less than a some gliders. We however, cannot skimp on pimping the power plant. As if we would do that.

Aircraft                Wing Loading (kg/sq m)
swan           10
Buzz Labs Schoolgirl UAV         23
Nieuport 17         38
Cessna 152         51
B-17                    190
F-104                  514
A380                  who cares, it is an airbus
B747                   740

We already know you can do your approach at a reasonable speed, but why? Sensible approaches are for people who do not think that 10 ft/s  sink rates are for roller coasters. Land It Like a Man, Full Throttle and Both Hands on the Stick. Or, just let the autopilot do it, it can tell how far it is from the ground and cut the power at stall two inches off the ground.

If the women don't find you handsome, they ought to find you handy!

Basic Quantities and Some Trigonometry

So You Want to Build a DIY Autopilot

Accelerations

One of the most important quantities for you to measure are accelerations. If your two or more-axis accelerometer is mounted along the traditional axes of the aircraft it will be the easiest to code for. The three traditional axes are:

• From the nose through the center of gravity on the line of symmetry, the Y-axis, roll
• From the center of gravity out of the fuselage toward the tip of the right wing, the X-axis, pitch
• From the center of gravity away from the earth, the Z-axis, yaw
  
     

Figure of Rigid Aircraft Axes

So, now for some basic Trigonometry, everyone remembers SOH CAH TOA.

In general, the following picture is true for a vehicle moving through space.

Figure of the Direction of the Force of Gravitation on a Body

As you can see from the image, the angle of pitch relative to the surface of the earth is the same angle offset of the weight vector relative to the z-axis in the body frame of reference. If we put our accelerometer so that one of its axes is parallel to the body's z-axis at its center of gravity we are measuring this offset vector. Which is really neat, because it means that we can express the angle of the body in level, non-accelerating motion as ratios of the accelerations

Pitch: 

The angle theta between the actual gravity vector and the measured gravity is related to the pitch of the aircraft (pitch = theta + 90°). If we know theta, we know our pitch! Since we know the magnitude of the earth’s gravity, simple calculus gives us our pitch angle:

accelerometer = cos (theta) * gravity

theta = acos (accelerometer / gravity)

And since pitch = theta + 90°

pitch = asin (accelerometer / gravity)

Woot, we calculated the pitch orientation of our airplane using an accelerometer. Pretty easy, huh?

The real formula that we need to use for the software looks like this:

pitch = atan2(accelerometer / gravity, z / gravity)

Common piezo-electric accelerometers return in units of, g, 32.17 ft/s^2 or 9.81 m/s^2. We also know some more things about the flight that let us calculate the angles relative to the ground. More on this later, it is a bit more than basic trigonometry to describe. These equations assume non-accelerating flight. You can use a magnetometer to get the relative plane in space with less math, but magnetometers generally take more interface programming in my experience.

Roll:

Roll needs a second accelerometer with an axis perpendicular to the first so that we can figure out the resultant vector between them and then the angle. Essentially the vector between the accelerometers becomes the "gravitational" acceleration and the relative readings lets us calculate the angle with an atan2 function. The second accelerometer will have some other things to manage such as the effects of the distance between them on the accelerations measured. Physics fun and none of the boring class.

Yaw:

Yaw is the hardest of the angles to measure. The only answer is to use a magnetometer or a compass. In many ways, yaw can be solved by dead reckoning. Dead reckoning is all that is important for most of the projects in the DIY garage. They will be covered later.

Next we will discuss gyroscopes and the beauty of rates and integral calculus