Some Interesting CurvesJohn D Barrow

Swiss Re Building30 St. Mary Axe‘The Gherkin’

Norman Foster & Partners

The Swiss Re Building180 metres high

40 floors2003

Design FactorsSky visible

Low winds on groundSlow and smooth airflow

Wedges bring in air and lightSix on each floor,

Offset creates spiral effect Helps bring air in

All surfaces flat – cheaper!

Torre Agbar Barcelona

(2005)142m

Tatiana

Tatiana’sHouse

The San Francisco Zoo Disaster

Launch velocity = (u, v)

x = ut

y = vt - ½ gt2 = vx/u – ½ gx2/u2

y vs x is a parabolady/dt = 0 at maximum height:

tmax = v/gymax = ½ v2/g

Projectilesx

y

uv

Mt Etna

Crouching TigerCrouching Tiger

V2 = g (h +(h2 + x2))

Vh

x

h = 3.8 metresx = 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)         

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.

Half A Catenary

The Gateway Arch, St Louis, MS630 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.’

Inverted Catenary Arches

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

f(x) x f(t) dt = f (x) x (x - t) f(t) dt f(x) = Aebx

Gustave Eiffel’s Tower (1899)‘moulded in a way by the action of the wind itself”

300m = 894 ft high

Watkin’s Great Wembley FollySir Edward Watkin, Chairman of Metropolitan Railway saw Eiffel’s 894 ft Tower

He wanted a bigger one (1200 ft) on his land in Wembley ParkEiffel 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

You feel radially outwardCentrifugal Force

Mv2/r

v

Mv2/r

Fall from height h under gravity from rest ½mVb

2 = 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 + ½mVt

2

Radiusr

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 in your Seat at the Top

Staying Conscious At the Bottom!

If h > 2.5r you reach the bottom with speed

Vb = (2gh) > (2g2.5r) = (5gr)

The net downward force on you at the bottom will beWeight + Centrifugal force

mg + mVb2 /r > mg + 5mg = 6mg

A 6-g force will probably render you unconsciousOxygen 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

Shockwave roller coaster at Six Flags Over Texas, ArlingtonWerner Stengel’s first use of the Clothoid in 1975

Clothoid loop

Loopen at Tusenfryd in Norway (Vekoma, Corkscrew, 1988)

Clothoid loop

Clothoid curvature varies linearly with arc length t

Velocity and Acceleration:

                                                                                     

                                                          Arc Length  :

                                                   

                                                               

            Curvature:                

                                            

    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.

Motorway Junctions

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

August Möbius, notebook 1858

Möbius and His Bands

US Patent 3991631

Möbius Belts, Tape-drives and Conveyor belts

The Möbius Universal Recycling Symbol

Gary Anderson, Student at USC, design competition winner, 1970

Not a trademark!

Taiwanese Recycling Symbol

Maurits Escher, woodcutMoebius Strip II (Red Ants), 1963

Robert Wilson, Fermilab, Batavia. Illinois