Work Text:
Okay, so aerospace engineering wasn’t the best decision for a university major. It was too late for regrets, though—he was already in his second year. Mechanics of materials and structures has no reason to be this difficult, nor should aircraft engineering. Alas, universities apparently enjoyed seeing their students suffer even more than high schools did.
A sharp exhale leaves his nose, dragging a hand through his hair in frustration. He’s been staring at his unfinished assignment—a set of physics and mathematical problems related to aerospace engineering—for around two hours now. It isn’t the numbers that tires him, but rather the length of the questions. There are five problems and a total of eight pages.
As of the moment, he is currently stuck on page seven, which is actually decent considering he’s been tackling this assignment since he returned from university. He doesn’t dare check the time lest he loses his drive to finish these damn worksheets.
Question 3.
Bearing stresses are very common in aerospace engineering. It occurs on the surface of contact between any two structural members. Bolts, pins, and rivets are classic examples of where bearing stresses occur. You will find these in many flight vehicles.
A simplified example of this is a bolt pressed against the flat surface of a bar. This might occur during flexure testing of a material. The stresses in an elastic material under a concentrated line load P (units of [Force/Length]) are given by the following equations:
σx = (− 2P/πx) · cos4θ = (− 2P/π) · x3/(x2 + y2)2
σy = (− 2P/πx) · sin2θ cos2θ = (− 2P/π) · xy2/(x2 + y2)2
τxy = (− 2P/πx) · sinθ cos3θ = (− 2P/π) · x2y/(x2 + y2)2
Using non-dimensional coordinates: x = x/a, y = y/a
- Make a three-dimensional surface plot and a two-dimensional (non-flooded) contour line plot of the absolute maximum non-dimensional shear stress (τ/[P/a]) as a two-dimensional function of x and y. Use the ranges: 0 ≤ x ≤ 3 ; -2 ≤ y ≤ 2
- Comment on your results
With a sigh, he flips the sheet over. For some ridiculous reason, the problem on the final page of the assignment is easier than the one before.
Question 4.
Solve the system of differential equation given using Laplace Transform:
v′ + 2w′ = 3e−1
−2u′ + 2v′ + w = 0
2u′ + 2v′ + w′ = 4t
given the initial conditions, u(0) = v(0) = 2 and w(0) = 2
It’s easy enough to solve. Yawning, he picks up his pen and begins writing on the paper. His strokes, usually neat and legible, are scratchy and messy, with some of the ink smudging. The final problem is of similar difficulty. It doesn’t take him very long to finish solving the last two questions. Now, all he has left is the massive third question.
Genuinely, he needs a break. It’s just one more problem, he tells himself, twirling his pen between his fingers, another yawn slipping from his mouth. It’s what has been repeating inside his brain since he began working through his homework, and it’s kept him going so far.
Yet, now that he’s actually reached the final problem, his brain and body are giving up on him. His eyes are drooping and his neck hurts from being bent over papers. Needless to say that the dining table is a mess from all of his stuff scattered about.
“Hirano-san,” a sleepy voice calls out, slippers scuffling on the wood-panelled flooring. “You should’ve told me earlier if you were gonna stay up,” there’s clear disapproval in the tone the blonde’s lover uses.
“I didn’t wanna disturb your sleep,” Hirano mutters, another yawn following. Dammit, exactly how sleepy am I? “You have to leave early tomorrow.”
“It’s already tomorrow,” Kagiura sounds closer, gentle and concerned. “I really think you should sleep,” he adds, glancing at the mess on the table. “You can finish these in the morning since you’ve got time.”
“I’m nearly done,” the blonde argues, involuntarily slumping in his chair, “it’ll be better if I finish it now; I’ll just be pissed off if I leave it for the morning.”
There’s a sigh, then a small clatter as Kagiura sets an object down on the glass surface. “Then at least have something that’ll keep you awake instead of forcing yourself,” the taller says.
Turns out, Kagiura had brought the older a glass of black tea. The dark color of the liquid meant that it was stronger than the sort Hirano usually drank. “...Thank you very much,” Hirano murmurs, taking a sip of the warm beverage. He didn’t feel anymore awake but the taste, although bitter, was strangely soothing. “When did you make this?”
“While you were finishing your work,” Kagiura answers, resting his hands gently on his lover’s shoulders. “You were so focused, I didn’t wanna disturb.”
A chuckle leaves Hirano’s mouth. “And you decided now would be a good time to interrupt me?”
“Yeah,” Kagiura replies cheekily, his voice slightly muffled from where he was pressing his face to the blonde’s head. “Come to bed soon,” he murmurs, pressing a faint kiss to Hirano’s forehead, “it’s cold without you.”
The older reaches out to ruffle Kagiura’s dark hair. “Dumbass,” he says affectionately, “go sleep. I’ll be there in a bit.”