Freddie Mercury is back in the news with the release of the new biopic BOHEMIAN RHAPSOSDY. ( Mathspig gives it ⭐⭐⭐⭐⭐. But I’m a Freddie fan. ) So Freddie and the Opera House? What do they have in common?
NOTE: Uncanny likeness of biopic actors to the real Queen!
According to intmath The Sydney Opera House is a very unusual design based on slices out of a ball. Many differential equations (one type of integration) were solved in the design of this building.
You will never see a better parody of Queen’s ICONIC song BOHEMIAN RHAPSODY than Calculus Rhapsody By Phil Kirk & Mike Gospel (below).
And if you need to be reminded of the maths you will find links to texts here.
Or search topics on Youtube eg. Introduction to Conics Khan
Why Does Mathspig like The Kahn Academy? Because it’s:
1. Free.
2. Quick
3. Clear
4. Low-key
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Salman gives digital chalk-and-talk tutes like a teacher. He hand draws the equations and graphs. He uses a calculator from time to time, but he tends not to use whizz bang spreadsheet graphs or perfectly presented textbook equations. It’s a bit wobbly and it’s all coming from Salman’s head.
And students like this approach. It makes maths look do-able.
The tutes on Maths Online are produced by Aussie Maths teachers and they are very good. Mathspig, of course, imagined the quadratic function tutes might look something like this:
The world is in turmoil mathspigs with earthquakes, tsunamis and nuclear power plant meltdown worries. It is the job of engineers to calculate and incorporate – as far as possible- safety margins into the structures of buildings, dams and power plants. Observers have noted that the skyscrapers in Fukushima wobbled during the recent 8.9 magnitude earthquake in Japan.
This is intentional, as rigid structures can snap in strong winds or during earthquakes.
But the maths used to calculate SKYSCRAPER SWAY is straightforward.
Short, rigid buildings are damaged in earthquakes because they shake very fast. 10 story buildings have a period of oscillation of about 1 second the same as the earthquake pulse. This is VERY dangerous.
Tall, flexible buildings can withstand an earthquake because they can sway. They are like a very large, slow moving tuning fork. If they are TOO RIGID they snap. If they are too flexible the people on the 100th floor would be throw all over the place.
The 59-story steel-construction Citicorp Centre, NY (pictured) has an oscillation time of 6.7 seconds. DetailsGoogle Books.
The 102-story brick clad Empire State Empire Building sways about 8cm ( 3 inches) whereas the 110-story steel -mesh World Trades Centre Towers, NY, before they collapsed swayed over 1 m ( 3 ft 5 inches).
One more thing. You want buildings to have springy foundations so they don’t snap at the base and fall over.
Earthquake Engineering
The idea is not to strengthen the building, but to reduce the earthquake generated seismic forces acting upon it. This can be done in 3 ways.
1. Base Isolation. Rubber pads or Rollers. Are used so the base does not feel the full shake or jump off foundations.
3. Active Tuned Mass Dampers use a computer controlled counter moving weight to actively move against the building sway.
The 508m (1,667-foot) Taipei 101 Tower would sway back and forth up to 60cm (2 feet) each way within five seconds. This according toWiredmagazine is highly barfogenic or spewdacious, perhaps, in Aussieland.
The Taipei 101 engineers included a 662 tonne (730-ton) counter giant pendulum to act as a counter weight.Some buildings use a big block of concrete.
It is pushed in the opposite direction to the building sway to dampen the oscillation.
Earthquake Engineering Maths
Take 1:
Wiredmagazine includes the equation for Skyscraper Sway acceleration (See definition of terms @ Wired link):
Mathspigs, you can just look at this equation and see how to change it to make a building EARTHQUAKE SAFE. Keep in mind that k, the stiffness constant actually decreases for taller buildings.
Imagine you are designing a building to withstand the 8.9 magnitude earthquake. You have already added base isolation. Now you have three options to work with: building mass (m), damping constant (c) and stiffness constant (k). Remember the earthquake force is constant. If you change just the stiffness of the building (k) what happens to the distance of sway(x)?
Engineers have to come up with the optimum design for the strongest structure with least acceleration (but enough building mass for strength), greatest damping and least sway at the lowest cost.
Earthquake Engineering Maths
Take 2:
Structural Engineer Ron Klemencic explained on the Discover Newsthat a simple rule of thumb for calculating skyscraper sway was to simply divide the buildings height in by 500 because the building codes demand the building fit a 1:500 sway ratio.
The tallest building in the world at 2,716 feet (828m), the Burj Khalifa, Dubai, would sway back and forth about 5.5 feet or 1.7 m.
Ahhhhhhhhh! But you would have to drag Mathspig onto the 168th floor screaming.
But mathspigs you can work out the sway on the top ten tall buildings in the world.
Mathspig Blog 