And now it’s all this

I’ve been thinking about 1984 lately—the year, not the novel. What got me thinking about it was the reduction in gas prices over the past few weeks, and this article in the New York Times about Donald Trump’s declining poll numbers among white working-class voters and how that might affect November’s elections.1 Forgive me for not being especially optimistic, but one of my distinct memories from 1984 makes me think the polls are lagging. A news story on one of the networks back in 1984 included…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] At the end of yesterday’s post about the combinatorics of the World Cup group stage, I said “this is good enough for me.” Turns out that was a lie. Today I rewrote some of the code to come up with a slightly more detailed result. The last bit of work in that post was to generate the 40 unique point totals that can come from a group. These are the possible…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] We’re in the middle of the first set of games in the group stage of the 2026 World Cup, and I’ve been thinking about how many ways the points can be distributed among the teams in a given group. I used Python to help with the enumeration. Here’s a quick summary of how the group stage works: The teams are split into groups of four. Within each group, all…

I took a bike ride this morning on a portion of the I&M Canal Trail from Romeoville to Joliet and back. It’s a fairly short ride, eight miles each way, and that distance got me thinking about using Mathematica to do some calculations after I got back home. The trail has mile markers with little snippets of information about the canal and the surrounding area. Here are the markers I passed this morning: The mileage figures on the markers increase as you go south and west, which is downstream.…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] After my last post—the one about using Fourier series—I started thinking about how to use Mathematica to develop Fourier series.1 I could, of course, use the Integrate function to determine the Fourier coefficients, but Mathematica has other functions that can do the job directly once you understand how they work. Mathematica has several Fourier functions,…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] We’re finally here, at the end of all things. In this post, we’ll use a Fourier series to get the formula for the center deflection of a simply supported beam with a uniformly distributed load. We’ll see some of the same math that we saw in the previous Fourier series solution, but the fundamental approach will be different. Let’s start with the…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] This would have been the last post in the series if I hadn’t realized recently that another method deserved a post. So this is the penultimate. It was described in this 1943 paper by Nathan Newmark and is known—or was known—as Newmark’s method. I say was because this method is dead. How dead? Forty-five years ago, when I was an undergraduate taking…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] We’re in the home stretch of this series, so ANIAT will soon go back to complaining about Apple’s UI choices.1 Next week’s WWDC keynote should provide some inspiration. But today’s post covers our eleventh method for deriving the center deflection of a uniformly loaded simply supported beam: the dummy unit load method. When I was an undergraduate, this was…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] The biggest problem I face when writing posts in this series is deciding how much background explanation to put in. If I included only the stuff I do to get the answer, the posts would be very terse: one sketch and a dozen or fewer lines of math. A full explanation, though, could easily turn into several semesters’ worth of structural analysis theory. I’m…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Continuing our series on the many ways to get the center deflection of a uniformly loaded beam, we come to another energy-based technique: Castigliano’s method. Castigliano’s second theorem provides a relationship between displacements, forces, and the strain energy in a linearly elastic system. The strain energy, U, is the potential energy of the system…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Today we’ll use the Rayleigh-Ritz method again, but this time we’ll avoid dealing with an infinite sum. In case you’ve forgotten, this is our problem: We’ll express the shape as a polynomial. The form of the governing differential equation, EIyiv=w tells us that our solution won’t have any terms of higher power than x4. That gets rid of the infinity…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] I just learned that people are listening to music pitched slightly down because it makes them feel better. Instead of the A above middle C being set at 440 Hz, they have it tuned down to 432 Hz. This strikes me as odd, but how you feel is how you feel. Do whatever you want, as long as it doesn’t hurt anyone. I was interested, though, in the math behind…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Continuing our trip through various methods to derive the equation for the center deflection of a uniformly loaded simply supported beam, today we’re going to do the first of two solutions using the Rayleigh-Ritz method. Of all the possible shapes a beam can deform into, the shape it will deform into is the one that minimizes the potential energy of the…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] The sixth way we’ll derive the formula for the center deflection of a uniformly loaded simply supported beam is the Myosotis method, which I wrote about over a decade ago. This is the method popularized1 by J.P. Den Hartog in his Strength of Materials textbook. Image from Wikipedia. Myosotis is the genus of the forget-me-not flower, and the idea behind the…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] The next technique we’ll use to derive the formula for the center deflection of a simply supported beam with a uniform load is the slope-deflection equation: MA=2EIL(2θA+θB−3ψ)−FEMA Let’s start by explaining where all the terms come from. Here’s a beam of length L with arbitrary end supports (could be simple, fixed, free, or sprung) and an arbitrary…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] The fourth way we’re going to derive the formula for the center deflection of a simply supported beam with a uniform load is the conjugate beam method. This is probably tied with the moment-area method for the simplest and fastest way to get the formula—at least if you’ve memorized the properties of parabolas. I wrote about the conjugate beam method last…

In Paul Krugman’s post today, he includes two charts. One is fine, the other isn’t. Here’s the first one: Generally, I’m not a fan of putting stuff in the margins that could be within the body of the plot itself, but at least it’s clear which label goes with which data series. Too bad that’s not the case with Chart 2: The two series end at almost the same spot, so putting the labels out in the margin means they can’t both be centered on their series. The little leader lines that run from the…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Continuing our odyssey through various ways of calculating the deflection at the center of a simply supported beam with a uniform load, today we use the moment-area method. There are two moment-area theorems used to calculate the slopes and deflections of a beam. Here’s how they’re given in the textbook I used as an undergrad, the 3rd edition of Elementary…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] In the introductory post to this series, I mentioned that the shear is the derivative of the moment. It’s easy to see that for our specific problem, the simply supported beam with a uniform distributed load, but it’s also true in general. A further relationship is that the distributed load function—let’s call it q—is the derivative of the shear (with a…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Here’s the first of the derivations for the center deflection of a simply supported beam with a uniform load. We start with the differential relationship between the bending moment, M, and the deflection, y: M=−EId2ydx2 The second derivative of y is the curvature of the beam (for small deflections, which is one of the fundamental assumptions of beam…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] A few years ago, I wrote a post describing how I asked ChatGPT to solve a couple of elementary beam bending problems and how its answers were persistently wrong, even after I told it the mistakes it had made. For the first problem, determining the deflection at the center of a simply supported beam with a uniform load, ChatGPT gave the correct…

I mentioned last week that some wording in a news article struck me as odd. A similar thing happened when I read this Scientific American piece (that’s an Apple News link—I don’t have a SciAm subscription). In this case, I didn’t even have to read the article; the oddity is right there in the headline: “A SpaceX rocket booster is on track to hit the Moon at several times the speed of sound.” I know what the writer means, of course. He’s telling us that the SpaceX debris will hit the Moon at…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Earlier this week, I saw this article in Apple News. It’s from the San Francisco Chronicle and discusses a recent report on the safety of the Golden Gate Bridge. The report is one of several reports spurred by the Francis Scott Key Bridge collapse a couple of years ago. Image from Wikipedia. Spoiler: the report finds the Golden Gate Bridge safe—quite…

After yesterday’s post, I got an email from reader Ron Sprague outlining two ways to block sender addresses from the list view in iOS Mail. Both of these methods are efficient, and neither starts with choosing a command with a stupid name like View Contact Card. I knew about one of these methods but not the other; this post will describe both. (My first thought was that this would be an update to yesterday’s post, but I soon realized it was going to be too long for that.) The first method…

Complaints about Apple’s design choices usually involve transparency, color, and other legibility concerns. These criticisms are legitimate, but the poor design choices that usually set me off are the ones involving the mechanics of commands—what I have to do to get something done. I think of these as the “design is how it works” mistakes. Yesterday, for example, I was going through my mail on my iPhone and came upon a piece of spam. I put it in the Junk folder with the hope that future emails…

I meant to write this last week, shortly after I finished listening to Episode 612 of Upgrade. But things didn’t work out the way I hoped, so now I’m rushing to finish these few paragraphs before this week’s episode comes out (they’re recording as I type). I have nothing to say about the Tim Cook/John Ternus news that hasn’t already been said. I want to focus on Jason and Myke’s choices—made and unmade—in their Apple at 50 Draft. My favorite picks were the oddballs, the products that weren’t…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] In this morning’s blog/newsletter,1 Paul Krugman included a couple of unusual graphs that I want to talk about. It took me a little while—longer than it should have—to figure out what he was doing and why, and I’m still not sure I agree with his plotting choices. Let’s go through them and you can decide for yourselves. The two graphs of interest are made…

During my seven-week Spotlight trial, I was reminded of how easy it is to make file and folder launchers in Keyboard Maestro. In case you’ve also forgotten, here’s a short post on how to do it. There are three items that I open quite often and that Spotlight was slow to find: My blog-stuff directory, which is where I keep all the text files, scripts, images, and other components that go into making blog posts. Each post, or set of posts, gets its own subdirectory in blog-stuff. My notebook…

I started using launchers shortly after returning to the Mac (from Linux) in 2005. The first one I used was the great Quicksilver. I’m sure I learned about it from Merlin Mann, who was Quicksilver’s biggest advocate, but I can’t point to which of his many posts on QS got me started. When Nicholas Jitkoff (Alcor) stopped developing Quicksilver in 2007 or so, I switched to LaunchBar, and that’s been my main launcher ever since.1 I gave Alfred a workout for a few months—inspired, I think, by this…

Yesterday’s post included some behind-the-scenes calculations that I figured were worth talking about. They were all done in Mathematica, and here’s the notebook I used: