Being a good-for-nothing sophomore in aeronautical engineering (you know how I got myself involved in that, given no other known choice), I always feel sleepy while writing most of my exams (as a matter of fact, my end semester has already begun, earlier this month). Because, we already know what questions are gonna show up in the exam (oh, that’s quite easy to figure out, just by seeing the activities and responses given by the teacher during those exam-nearing days) and that scheme doesn’t put anyone into a challenge during the exam. Sometimes, itsy-bitsy challenges do show up this way – like plugging the right formulas (which you’ve rote like hell) into a specific problem in the right way.
While I know it requires talent (I’m out of those), I don’t consider them as a challenge, nor a way of evaluating knowledge and skill. They’re just another crackpot way of solving a jigsaw puzzle, where you bring most of the pieces from your home. In my third semester, we have a subject called “Principles of Flight”. It’s actually an interesting one (probably the interesting of all, keeping aside the professor’s activities, who’s slightly weirdo), where you learn about the aircrafts, their motion, their design, their structure, almost all of their clockworks other than those already occupied by the specific fields. For instance, you don’t learn about the working of engines (propulsion covers it), how a wing bends under its own weight (solid mechanics), materials currently used for aircraft’s design (materials science), how the thing moves through air (fluid dynamics, probably the most scared and the most respected field of all, along with a pinch of numerical approximations) and the queue goes on…
So this time, our “Principles of Flight” exam had a nice question, which was attractive enough to trigger my physics monster. Even with my old irregular “Bradley-hand size-12″ font, it took 3 pages for me to finish it off (the grader’s gonna have a hard time grading my sheet). Once I reached home, I figured out that all of which I’ve been writing in the exam was
more or less this square-cube law (applied to all mechanical and biological objects). It follows Joe Walcott’s quote, “The bigger they are, the harder they fall”. Anyways, the question is this…
What are nature’s creations that can move (or) fly through air? How are our aircrafts different from them?
At that time, my dumb brain could only think of insects and birds. Because, it spent all of the time simulating the flight of those organisms and aircrafts (in my mind) at the same time. So, I can’t think of the things that move through air (like the Dandelion with which we all play in our childhood). Anyways, my comparison was quite good. And yesterday, I was wholly (not really) concentrating on collecting1 the required data (via googling) for plotting my guesswork…
What if you scale yourself up?
Say, I have a box. It’s three-dimensional (we associate a few parameters – length, breadth & height to define the box in our world). Any one of those parameters gives the length covered by the box in that dimension. The product of any two of those, gives the area of the box in a particular plane. The product of all the three, gives you the volume covered by the box. Pretty boring it may seem, but that’s the baseline for our topic – Scaling (of biological objects, to be specific)…
If you scale up an object (i.e) increase its size, its area goes with the square of its length, whereas its volume goes with the cube of it. This is what Galileo found in the 17th century. This is also hopefully, our issue. Now, what’s mass? Things like these are very abstract. But, lets go with the usual definition – the intrinsic property of an object that causes it to have some weight in a gravitational field (of say, Earth). Mass depends on the volume. For example, you go to the market and buy a kilo of carrot. How do you know that it weighs one kilo? (assuming that it’s a honest situation, that the shopkeeper doesn’t cheat you with his sorcery) You weigh it with a spring balance. The spring deforms, pointer moves, physics comes in, etc. but what determines its mass?
The volume, for sure. Say, you’re digging out the insides of a carrot, so perfectly that only the skin with the exact shape (a mechanical impossibility) is left out. If it were otherwise (i.e) mass depends on say, area – the carrot should still weigh the same, even after your fantastic sculpturing artwork. Bad luck, nature doesn’t work that way..!!! So, mass goes cubic with size. If you’re gonna double the size of something, the mass goes by a factor of eight.
By measuring how much his biceps contract, right? Imagine cutting the biceps at the stagnant point (highest point in the muscle). Now, think about the same issue. The more the cross-sectional area, the larger the muscle becomes. In other words, for two muscles having variable length but equal cross-sectional area, the force exerted by a longer muscle is no larger than the shorter one. Now, how is this different from a volumetric increase? The muscle simply doesn’t get longer. It’s just getting wider. So, muscles go squared with size increment. You scale up an organism twice, its muscles go only by factor of four. Simply put, the muscles can’t catch up with the increasing mass.
The plot above shows how the area between the and increase with . For larger x-values, the area increases spontaneously. What does this horror mean? It means that the muscles can’t hold the body on any proportion larger than 1:1. In other words, if the size of the body is doubled, the muscles need to be reinforced by 2 times (other than its 4-times increment) to lift the factor-8 mass. If the size is tripled, the muscles should be 3 times stronger to lift the factor-27 mass. Because, that’s the only way the proportion 1:1 can be reached. Hence, the organism can’t be stable. It can’t even lift its own finger.
Here’s another thing to convince you – Oxygen. A living organism needs oxygen (i.e) the cells of the organism needs to breathe. The amount of oxygen it needs, depends on the volume of the cell whereas “how much it consumes”, depends on the surface area of the cell membrane. Because, oxygen is just absorbed by the membrane. If the size is doubled, it needs 8 times more oxygen. But, it can consume only 4 times (using its given surface capabilities). Sad, it has to die anyway..!!!
Bigger organisms are just lazy…
Now you ask, “Does this square-cube law mean that bigger organisms cease to exist?“. I didn’t mean that. If that’s the case, how do you think elephants and blue whales even exist? Of course, they’re the largest mammals (compared to us). Till now, I’ve been saying that you can’t simply size an object like the ones you see in movies. Movies like the Wrath of the Titans, King Kong, etc. have gigantic organisms that have been scaled up from the real-worldly ones. While you can’t scale up an object, you can still create one with the right proportion that can be like say, Bumble Bee. Then you ask, “How do those things get bigger?” They’re just growing. We don’t grow by scaling ourselves up. When we grow, our cells don’t become larger. Instead, they divide. The size of the cells are still the same.
This muscle-scaling-as-square thingy has marvelous effects. That’s the ultimate reason why a mouse’s heart beats at 8.33 Hz, whereas yours is at 1.2 Hz, and a blue whale’s is at 0.1 Hz. Because, the lub-dub you hear, is determined by the muscle that makes it go up & down, pumping blood through the circulatory system. How something walks, talks, acts, responds – everything depends on the scaling. Choose a day when you’re sitting lonely somewhere and spot a smaller organism. Compare its activity with yourself (i.e) how fast it moves, how fast it reacts to your disturbance, etc. Having done the experiments lot of times, I can safely declare that they have a fast response. Our introduction to the scale factor is now a done-deal. This stuck my mind during the exam.
Guesswork on the wing-beat frequency…
If each response depends on scaling, so should the wing-beat frequency of birds or insects. When I was writing the exam, I guessed it to be a few millihertz. Under those circumstances, my brain was able to guess the frequencies for commonly seen species like butterfly, parrot, crow, fly, mosquito, etc. I don’t wanna theorize with my guess now. I wanna prove it to you. I’ve collected data for several species of insects & birds, and plotted the information based on several core variables. So, I’ve been busy.
Basically, the frequency should depend on mass of species, its wing span and its wing area. I included the last one because, the plot had fluctuations even after plotting the former two. Never mind, here’s the numbered list. The number is necessary because, the species are ranked according to their mass.
- House Fly
- Honey Bee
- Large White Butterfly
- Bumble Bee
- Dragon Fly
- Scorpion Fly
- Hummingbird Moth
- Giant Hummingbird
- Laughing Gull
- Cape Pigeon
- Black Vulture
- Bald Eagle
- Whooper Swan
- Wandering Albatross
- Plot 1: Frequency vs. Mass
There are 18 species. I can’t markup the giant plot-legend into the graph (at least, I don’t have sufficient knowledge in Mathematica). So, almost all points lie along a particular line, showing linear variation. From this “swallow-a-jelly” like plot, we can infer that the frequency decreases as mass increases. The butterfly (4), moth (8), puffin (12) and gull (10) show some fluctuations because they’re special cases, which we’ll clarify in the next two plots.
- Plot 2: Frequency vs. Wing Span
Now, this plot can be quite confusing. Because, I’m still following the legend, which I had defined above. The numbers are sorted based on the mass of the species, which is why they appear as if they were misplaced. Now, moth (8) and puffin (12) lie on the line whereas new fellas like dragonfly (6), scorpion fly (7) and bumble bee (5) rush into the fluctuation problem.
- Plot 3: Frequency vs. Wing Area
Before getting into the plot, let me emphasize that the plot has a few assumptions. For birds, I got the details accurately from scientific papers. For insects, I had to guess the areas by considering their wing as rectangles with sides – half the wingspan and more or less half of “half the wingspan”. I corrected my guessed values by comparing those with a normal-viewed photograph and measuring them using pixels in the screen. Anyways, you don’t have to worry. The values are very very near to the actual values. Now, lets discuss the fluctuations of few species…
- Simply, the muscle of the butterfly (4) isn’t good enough to flap its gigantic wings. Its body is small in size (so does its mass & muscle), whereas its wings are giant, having a large frontal area, may require a titanic force. But, its muscle is capable of flapping those wings only at 4 or 5 beats per second, because large area induces more drag (resistance caused by fluid flow) and the muscle can’t shake it fast. But, this drag is far enough to lift the butterfly smoothly in the air. Because of this design, the butterfly can’t swoosh through air faster, compared to other insects.
- The moth (8) is big and massive (more than 10 times compared to butterfly). Its wingspan is great but, the wing area is not quite (at least compared to the butterfly) which implies, less drag – so, it has to flap more to keep itself in the air. Because of its great size, its muscle can provide such a large force to its wings which is just sufficient to operate the thing.
- The body of our gull (10) is small, whereas its wings are not. The wingspan is large, so does its wing area. Apply the former principles of butterfly, and you’ll find that it flaps its wings slowly (i.e) its muscles aren’t good.
- The puffin (12) is my favorite bird. It’s a beautiful one. Though it’s small in size, it’s massive (what else do you think? Have a look at its large belly). In our language, its muscles are badly designed to carry the sluggish thing, which explains the reason why these birds are considered a delicacy in the Northern Pacific and the Atlantic. They’re easy to catch. The trap isn’t complex. Just point the net in its way. Poor beautiful thing can’t brake. It slides into the trap.
I hope that’s enough. You can guess the other things by your own. Because, the above principles, when shuffled back & forth, will be able to explain the other species’ fluctuations…
What if an aircraft flapped its wings?
First of all, did you notice that those were linear plots? I told you all the way down that mass goes with the cube of scaling (volume), whereas muscles go with the square (area). Now, all you see is a not-so-interesting linear plot. Check it again. It’s linear, but both the variables are plotted in a log scale. It’s a log-log plot. Logarithmic plots are extremely useful (they’re magnificent) when power relationships appear in an expression. Say you have a function . Taking log on either side gives you, . Whatever can be, 2, 3, 10, 20, 100 (??), logarithm doesn’t care. It always comes out as a linear plot.
Careful examination may lead you to the conclusion that the beating frequency is inversely related to the core variables mass, wingspan and wing area (I’ve neglected a few variables, which I guessed unwanted. It includes density of air, velocity of species, moment of inertia of the wing, etc. But, we’re not interested in examining those stuff. We’re waiting for the title, the heart of this post – what if an aircraft had flapping wings? What would be its frequency?
I can’t plot those in the log-log plot. Because, that may make it weird. I’ll just give you the values based on the power relationship with the core variables. I chose an average line along the plot and extended the details to our aircrafts (neglecting density of air which is hopefully “almost-1”, and other negligible values) so that their frequencies lie on the line (which is also based on a crude guess). By the way, if you really wanna know how the frequency depends on the variables, I suggest you to analyze it using the powerful dimensional analysis, like the fellas who’ve done in their paper (which is also the one that provided me with the details for albatross and pigeon).
|Aircraft||Takeoff Weight||Wingspan||Wing Area||FM||FS||FA|
|Lockheed Martin F-22 Raptor||38,000 kg||13.5 m||78 m²||0.05||0.24||0.12|
|Boeing 787-10 Dreamliner||251,000 kg||60 m||325 m²||0.02||0.04||0.05|
|Antonov An-225 Mriya||640,000 kg||88.4 m||905 m²||0.012||0.025||0.025|
This is what I obtain if I go along the linear log-log plot. The FM, FS & FA represent the frequencies that correspond to the aircraft’s takeoff weight, wingspan and wing area. “Why are there variations in the frequency?” There should be variations. Because, we’re just going along with our crudely guessed line. Duh, and by the way, we don’t need all the three frequencies. After all, we’re doing a guesswork. Right? So, lets take the average. Not the usual average. Instead, the powerful RMS which has its roots in Euclidean space (Pythagoras theorem).
An absolute value can be obtained from the Root Mean Square of those frequencies and, they turn out to be 160 mHz, 40 mHz and 20 mHz, which explicitly means that it takes 6.25 seconds for the F-22, 25 seconds for Boeing and 50 seconds for Antonov to flap the wings once. So, I was right with the millihertz after all. But, now I realize that I’ve answered WHY an aircraft is different from the birds or insects. Not HOW..!!! (which is what I was asked to investigate during the exam)…
Anyways, the conclusion is this. I was just examining the stability of objects. This is totally for fun and the situations are entirely hypothetical. In reality, no one can design a flapping Antonov (given current capability of technology and materials), because the first flap itself rips off everything from the aircraft structure.
: Basically, this post was inspired by a discussion with Chris White (a graduate student) in our chat room and also, his answer – at Physics Stack Exchange. (Even the usage of log-log plot was suggested by him)…