The assignment requires the submission of a report detailing your solutions to the following tasks and contributes 50% to your overall module mark. It should include a disk with all your simulation files upon it with instructions on how to run it in the appendix. The marks shown on the main assignment indicate the maximum possible marks that each element contributes to the final mark and the mark awarded will depend upon the quality of your findings and their reporting. Further break-down of the mark allocation is provided in the attached marking scheme.

The finished report should be submitted to the resource centre (Prospect Building, St Peter's) on or before the stipulated hand-in date.
Whilst it is expected that you will use a variety information sources to complete your research it must be presented in your own words and not taken directly from any other source. All sources you have used in your search must be clearly and appropriately referenced using the Harvard referencing system. Using sources which have not been appropriately referenced or taking any data directly from a source and included in your work will be regarded as plagiarism and may result in you failing this part of the report or even the whole module. It is acceptable to use photographs and/or diagrams from other sources, where appropriate, but these must be properly referenced. Quotations from external sources must be less than 200 words in total for any section of your report.Assignment : Spring Suspended Mass

The sprung mass system shown opposite is an important physical case study in simple harmonic motion. Assuming that the applied force, F , is purely due to gravity and that the acceleration due to
gravity is 10m/s2
then :
K x S
F
S
M
Zero Position Reference (x = 0)
mass (
F
Section A
You are given that the differential equation that describes the above systems dynamic behaviour
is
x
M
K
F
M
x
S=1&&
1) Implement a Simulink model of the above equation and, assuming that it is possible to step change the applied force by instantaneously adding an extra mass, M*of 0.5kg, evaluate its step response. You are given that the original mass, M, is 0.5kg and that the spring stiffness, KS, is 25 kg/s2.

2) The response obtained in 2 above should be of the form x = Asin(ωt) where A and ω are constants which are functions of the system variables KS and the total applied mass MT . Assuming that the initial mass ( and hence the initial conditions) is zero (and therefore MT = M* ), use your simulation to evaluate the relationships that define

i) A ( peak to peak )
ii) ω ( Note that ω = 2π/Τ where T is the period of the sine wave in seconds)

Section B

In practice damping occurs due to frictional forces and therefore the oscillation amplitude decays with time :
1) Show how the differential equation provided in Section A can be adapted to allow for this constant damping ( with damping coefficient b kg/s ) by developing the new differential equation from first principles.

2) Evaluate what value of damping is present if in an experiment the recorded oscillations decayed exponentially with a time constant, τ, of 5 seconds. You may assume the simulation parameters used in Section A part 1 and that the decay amplitude is expressed via
Ae ( t) t ω τ sin - /

3) Use your simulation to evaluate an expression that relates the damping constant, τ, to the system parameters (again assume M = 0 initially).

4) What other observations can be seen in the simulation when damping is included?