Transcript Some Interesting Curves

Some Interesting Curves
John D Barrow
Swiss Re Building
30 St. Mary Axe
‘The Gherkin’
Norman Foster & Partners
The Swiss Re Building
180 metres high
40 floors
2003
Design Factors
Sky visible
Low winds on ground
Slow and smooth airflow
Wedges bring in air and light
Six on each floor,
Offset creates spiral effect
Helps bring air in
All surfaces flat – cheaper!
Torre Agbar
Barcelona
(2005)
142m
Tatiana
Tatiana’s
House
The San Francisco Zoo Disaster
Projectiles
y
v
u
Launch velocity = (u, v)
x = ut
y = vt - ½ gt2 = vx/u – ½ gx2/u2
y vs x is a parabola
dy/dt = 0 at maximum height:
tmax = v/g
ymax = ½ v2/g
x
Mt Etna
Crouching Tiger
h
V
x
V2 = g (h +(h2 + x2))
h = 3.8 metres
x = 10 metres
Over short distances on the flat a
tiger can reach top speeds of more than
22 metres per sec (ie 50 miles per hr).
From a 5 metre start it can easily reach a
launch speed of 14 metres per sec.
Only V > 12 metres per sec launch speed
needed for the tiger to clear the wall
The hanging chain, ‘catenaria’: Leibniz and Huygens 1691
y = Acosh(x/A) = ½ A{ex/A + e-x/A}
The portion AP is in equilibrium under the horizontal tension H at A,
the tension F directed along the tangent at P,
and the weight W of AP. If the weight of the string is w per unit length
and s is the arc AP, W = ws; and from the force triangle, tan ψ = ws/H = s/c,
where c = H/w is called the parameter of the catenary is determined by
dy/dx = s/c
With solutions
y = c cosh(x/c)
s = c sinh(x/c)
Half A Catenary
The Rotunda was originally a tent put up in London as part of the
festivities to celebrate the defeat of Napoleon. Designed by John
Nash, It was moved to Woolwich in 1816 and converted in 1920 into
a permanent structure. It is now the Royal Artillery Museum.
Inverted Catenary Arches
The Gateway Arch, St Louis, MS
630 ft x 630 ft
y = -127.7 ft cosh(x/127.7ft) + 757.7ft
Robert Hooke 1671 Latin anagram (revealed in 1705):
‘As hangs the flexible chain, so inverted stand the touching pieces of an arch.’
Can You Ride A Bike With Square Wheels?
Stan Wagon Demonstrates
For the rolling square:
the shape of the road is the catenary
truncated at x = + sinh-1(1)
For regular n-sided polygonal wheels the curve of the
road is made from
catenaries with
y = -Acosh(x/A)
A = Rcot(/n)
Clifton Suspension Bridge (1865)
Suspension Bridges are Parabolas
Constant weight per unit length p
Gustave Eiffel’s Tower (1899)
‘moulded in a way by the action of the wind itself”
300m = 894 ft high
f(x) x f(t) dt = f (x) x (x - t) f(t) dt  f(x) = Aebx
Watkin’s Great Wembley Folly
Sir Edward Watkin, Chairman of Metropolitan Railway saw Eiffel’s 894 ft Tower
He wanted a bigger one (1200 ft) on his land in Wembley Park
Eiffel refused. Benjamin Baker completed stage 1 (155 ft) in Sept 1895.
Opened 18th May 1896 but never went higher: marshy ground and shifting foundations.
Tea shop for the new Underground Station, few visitors
Declared unsafe in 1902. Demolished 1904-7. Iron sold for scrap.
Wembley Stadium built on the site in the 1920s
Roller Coasters
Millennium Force, Cedar Point
A Tale of Two Forces
You feel
Force of Gravity
Weight = Mg

v
You feel radially outward
Centrifugal Force
Mv2/r
Mv2/r
Staying in your Seat at the Top
Radius
r
Fall from height h under gravity from rest
½mVb2 = mgh
At bottom: Vb =2gh
Ascend to the top of the circular loop of radius r.
Arriving there with speed Vt needs
Energy = 2mgr + ½ mVt2
So: mgh = ½ mVb2 = 2mgr + ½mVt2
Net Upward Force on rider (mass m) at top = mVt2/r – mg > 0
So: we need Vt2 > gr or you fall out of the car !
h > 2.5r
Staying Conscious At the Bottom!
If h > 2.5r you reach the bottom with speed
Vb = (2gh) > (2g2.5r) = (5gr)
The net downward force on you at the bottom will be
Weight + Centrifugal force
mg + mVb2 /r > mg + 5mg = 6mg
A 6-g force will probably render you unconscious
Oxygen would not get to the brain
Circular roller coasters seem to fail their Risk Analysis
A Recipe for Success
We want Vt2 /r big at the top to hold us in
But Vb2 /r small at the bottom to reduce the g force on the riders
Make ‘r’ small at the top and big at the bottom
Ellipses first used in 1901 at Coney Island
Clothoid loop
Shockwave roller coaster at Six Flags Over Texas, Arlington
Werner Stengel’s first use of the Clothoid in 1975
Clothoid loop
Loopen at Tusenfryd in Norway (Vekoma, Corkscrew, 1988)
Clothoid curvature varies linearly with arc length t
Velocity and Acceleration:
Arc Length :
Curvature:
Motorway Junctions
An arc of a clothoid has variable curvature, proportional to the distance
along the curve from the origin.
It provides the smoothest link between a straight line and a circular curve.
It is used in roads and railroads design: the centrifugal force actually varies
in proportion to the time, at a constant rate, from zero value (along the straight
line) to the maximum value (along the curve) and then back again to zero.
A vehicle following the curve at constant speed
will have a constant rate of angular acceleration.
At constant speed you can simply rotate the steering wheel at a constant rotation rate.
If the bend was a different shape then you would need to keep adjusting the rate of
movement of the steering wheel or the speed of the car
Möbius and His Bands
August Möbius, notebook 1858
Möbius Belts, Tape-drives and Conveyor belts
US Patent 3991631
The Möbius Universal Recycling Symbol
Not a trademark!
Gary Anderson, Student at USC, design competition winner, 1970
Taiwanese Recycling Symbol
Maurits Escher, woodcut
Moebius Strip II (Red Ants), 1963
Robert Wilson, Fermilab, Batavia. Illinois