Understanding Lift by Wings: The Lift-To-Drag Ratio

Some ways to generate lift can be categorized as follows:
-Mass Ejection (e.g. rockets)
-Buoyancy (e.g. dirigibles)
-Deflection (e.g. airplanes)

We will talk about wings, which falls under the category of Deflection.

There is a distinction to be made between mechanism and result. The finer details of fluid dynamics explain the necessary phenomena and mechanisms (i.e. laminarity, separation, etc.) for the wing to successfully act as a deflector. Force production (lift, drag) can be explained by the conservation of momentum and Newton's Third Law. We will be looking at the force production in a wing.

Wings work by deflection of the freestream air. You can treat the wing as tilting the net momentum vector of the portion of the oncoming freestream that it influences. So, the momentum vector of the air before it reaches the wing points in one direction and the momentum vector after it has passed the wing is tilted by some angle Φ. Keep in mind, this angle is not necessarily equivalent to angle of attack!

The conservation of momentum states that the momentum vector does not change in magnitude and is only redirected, as the wing does not produce energy or momentum. You can represent the new components in the horizontal and vertical directions of the tilted momentum vector by simple trigonometry. By Newton's Third Law (for every action there is an opposite and equal reaction), finding these components tell you the drag and lift.

The difference in these components is what makes flight by wings (deflection) so efficient. In effect, it is a mechanical advantage by a trigonometric trick! Instead of generating lift directly, we only have to overcome the drag necessary to sustain the momentum deflection.

As you may know, as you increase an angle from 0 degrees the sine component changes relatively rapidly, whereas the cosine component does not. This is reflected in the so-called small angle approximations. Thus we can expect that small angles of momentum deflection will be more efficient, or in other words, have a higher lift-to-drag ratio.

The equations to represent the relative lift and drag components are simple. The functions for normalized lift and drag are represented as L and D, respectively and are function of the momentum tilt angle, Φ. The resulting lift-to-drag ratio is represented as R.

There is something wrong with the above ratio equation. As Φ approaches 0, the equation becomes indeterminate (0/0). We know that in fact it should be 0. There actually should be a skin friction drag term in the denominator, because we know that even when the wing is not producing lift it still generates drag. We will denote this as f, and though it is a small number that becomes negligible as Φ increases, it is a function of Φ. The following is a corrected ratio equation.

From this we basically get the maximum L/D ratios possible for a wing to achieve. Below I have plotted the equation for several constant skin friction values (f=constant).

The lift-to-drag ratio converges at sufficiently high freestream deflection angle, and the skin friction term is important for the lower momentum deflection angles under the assumption of constant skin friction values. We see that this method of analysis reveals the range of maximum lift-to-drag ratios we are accustomed to seeing for typical airfoils.

Remember that this analysis is for when the wing behaves as it should; a deflector of freestream momentum. A wing cannot act as an effective deflector in the case of flow separation.

Cutting Aluminum Sheet Metal: Plasma Cutters and Jig Saws

I have been working with a lot of thin aluminum sheet metal of 0.025" thickness, using a jigsaw with great efficacy. The cutting speed is not too slow, and the cut edges are very clean. There are no heat-affected zones; the jig saw does not produce excess heat that would effect the material properties in the proximity of the cut. There are some minor inconveniences however. The main one is limited maneuverability. As you may realize, a jig saw cannot do sharp corners or turns, but rather one long continuous cut. Also, sometimes when cutting from a large sheet it is inconvenient and slow to maneuver around, since for every cut you have to clamp down the work piece and shift the work piece around so that you do not cut into the table or whatever it is the work piece is resting on.

One of the most important advantages to the jig saw is that the tool and consumables (blades) are very cheap! The one I bought, pictured below, was only $30, with the set of 3 blades being only around $5.



I have also considered a plasma cutter. This much more high-tech option ionizes compressed air via an electric arc and turns it into plasma. The high-speed plasma jet is then used as the cutter by melting through the metal work piece very fast. Pictured below is one $650 model from Harbor Freight, which seems to be the most economical option for the average DIY-er. The advantages the plasma cutter has to offer is fast cutting and maximum maneuverability.



There remains questions for my aluminum sheet application whether the plasma cutter will produce a heat-affected zone and if the cut will be clean. I will have to try it out and report back later.

All in all, the jig saw is the best option for cutting aluminum sheet metal by far, because it is fast, cleanly cuts, and is extremely economical. I only considered the plasma cutter because I have a lot to cut, and I need it for other purposes and materials.

Drilling Large Holes in Sheet Metal: The Step Drill Bit

Have you ever tried to drill large holes in sheet metal, for example, 0.5 inches in diameter? If you did with a regular drill bit (AKA twist bit), you may have had to quit due to the intense vibration and/or the hole turning out more like a Reuleaux triangle (below) rather than a perfect circle. The difficulty is special to thin sheets; the same drill bit will work on the same material of greater thickness.



I had many troubles with this, on both 7075-T6 aluminum (aircraft grade, 0.025") and stainless steel sheets (~24 gauge). Drilling large holes with twist bits seemed practically impossible. I tried different RPMs on my drill press as well as pilot holes, but nothing seemed to work well. I began to think that my 1/3 hp drill press was too weak. Before considering buying a new drill press, I decided to try the step drill bit from Amazon for around $12 (pictured below). The bits go to diameters of 1/2" and 3/4".



These things worked like a charm. Vibration and noise were reduced to the level of drilling small holes. Progressively larger diameters are drilled out step by step. These are the clear solution to drilling large holes in sheet metal. They also can be used to deburr rough holes and can function as all-in-one drill bits for materials as thick as the steps.

Mechanical Fasteners: (Blind) Rivets

I have recently become aware of a strong, efficient, and economical way to fasten metals together: the blind rivet. I have always heard about how rivets were used in aircraft production, and for some reason I never seriously considered using them in my own mechanical projects. I wrongly assumed the cost and complexity was too high for someone with a hobbyist's budget. It turns out blind rivets are very practical and economical even for me.

As I am working with aircraft-grade aluminum (7075 T6), I cannot weld pieces together. Aluminum is notorious for being difficult to weld well, and this applies even more so for the stronger aircraft-grade aluminum alloys. Welding these alloys are usually deemed ineffective, especially when the resulting material strength is important. With welding out the window, I racked my brain for alternatives to heavy screw and nut arrangements, and came across the blind rivet.

As you noticed, there is a specific type of rivet I am referring to: the 'blind' rivet. A regular rivet is a solid pin-looking thing where you have to hammer down and flatten one end. A blind rivet is a rather complicated-looking thing if you have never seen one before, but it can quickly and easily be implemented with a tool that you can buy for less than $20 at your local hardware store. Also, with blind rivets you do not require access to both sides of the joint, only one side is required, hence the name 'blind' rivet. You can Google up many great explanations and visual demonstrations of how blind rivets work.

Blind rivets are economical. On McMasterCarr, I can buy 1/8" diameter aluminum blind rivets at 250 for less than $10. It was actually more economical than the alternative for me, which were socket cap screws of similar size.

I would argue that the space and weight efficiency, for a given bore size, of the rivet is greater than that of screws. Strength of screws are diminished because of the required space for the threads. The screw nuts can be heavy when many screws are needed.

The downside of a rivet compared to a screw is more complicated dis-assembly. You have to destroy the rivet if you want to unfasten something, but it is not too difficult to remove a rivet.

Blind rivets today are the product of many years of improvements from field use. They are extremely effective if welding is not an option and you need a strong, lightweight bond between metal plates.