------------------------------------------------------------------------------------------- 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'. ------------------------------------------------------------------------------------------- END