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EXQ2: 
Numerical Methods 2014/2015 - Example Tripos Question.  Time allowed 30 minutes.

THIS QUESTION IS PERHAPS A LITTE TOO LONG TO DO THE IN THE STANDARD 30
MINUTE ALLOWANCE. WE MIGHT TYPICALLY PROVIDE THE STATE VECTOR TO SPEED
UP THE ANSWERING PROCESS.


A children's roller coaster consists of four buggies each of mass M +
mN where N is the number of children in the range 0 to 5 in the
buggy. They are interconnected by springs with elasticity k.

A safety condition is that at all times a buggy must assert positive
force on the track.

The track has length L and is looped such that it has periodic height
profile H(x mod L) where x is the horizontal distance along the track
from an origin. The track also bends to the left in at a rate B(x mod
L) such that the integral of B over a cycle is 4 Pi (720 degrees).

Power is injected as follows: if at any time a buggy's track speed
starts to drop below v_t metres per second a hook drops down to engage
with the track ratchet chain that propels the buggy at v_t metres per second.

Initially a 2-D FDTD simulation of the roller ride is required that
ignores the lateral movement (ignores B).

a) Give a suitable state vector (the variables that need be held from
one time step to another) for the 2-D simulation and give also the
simulation parameters that must be set for each run. Define any
further details of the real ride required. [6 Marks]

[Hints: Assume H'(x) = dH(x mod L)/dx is available.  The height and
vertical velocity of each buggy are not part of the state vector:
these are always given directly by the track profile functions.]


b) Sketch suitable FDTD forwards-difference update assignments that advance the state
vector by one time step and that detect a safety violation.  

Feel free to ignore the following two minor effects:
   1. the triangle effect that reduces the horizontal distance between buggies for a given spring extension,
   2. the vertical force applied by one buggy on its neighbour through the spring.

[ 8 Marks ]


c) The simulation must now be run on a 1970's games console that uses
an 8-bit microprocessor where, apart from bit shitfs, the most complex
arithmetic primitive is a bytewide (8-bit) add or subtract.  Hence
multi-digit arithmetic and multiplications involve software
overhead. Briefly discuss appropriate storage resolutions, simulation
timesteps and simplifications to the simulation. [4 Marks]

d) Do any additional components need adding to the state vector to create a 3-D simulation and does this
make any difference to the 2-D behaviour ?    [2 Marks]

Note: force = mass * acceleration.  Weight = mass * g. A spring with
elasticity k (a constant in Newtons per Metre) asserts force F = k
(x-x_0) where x is its current length and x_0 is its (constant)
natural length.  The forwards acceleration due to gravity on a buggy
on a slope of gradient H' is g * H'.


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