Beam bending
A uniformly thick beam of length L with square cross-section with width a has a distributed force,
N/m,
applied to (Q is a constant). A point force P is applied at .

Choose the beam dimension a to reduce the deflection to below 1 cm whilst minimising the mass of the beam. Sketch distributions of shear forces, moment and deflection along the beam length.
You are expected (but not required) to use MatLab to solve this problem.
Values of L, E, Q and P for each group are calculated as follows:(here N is group ID number) N= 18
(m)
(kN)
(kN/m)
(GPa)
Marking scheme:
Finding correct expressions for shear and moment ANALYTICALLY (not using Matlab, please show your working) (8 marks)
Finding correct expression for deflection (analytically or using Matlab) (8 marks)
Finding the optimum beam dimensions (analytically or using Matlab) (6 marks)
Plots of shear force, momentum and deflection (3 marks)
Car suspension
The car suspension helps to reduce the amount of bounce when car is going over a bump. Cars drive on roads that have bumps or holes on them – large ones (sleeping policemen), medium sized ones (potholes) or even small sinusoidal bumps.
Performance of car suspension is often evaluated using a complex system of springs and dampers, e.g. (Shirahatt et al, J. of the Braz. Soc. of Mech. Sci. & Eng. 2008):

We will use a simplified model of a quarter car: one wheel and its suspension that carries a quarter of weight of a car. The wheel axle is connected to the remaining quarter car by a spring and a shock absorber (damper):

The mass of the car is M, the mass of the wheel is m, the height of the car body is y(t) above the level h = 0 (not above the road surface!), the height of the wheel axel above h = 0 is s(t) and the length of the string in the “unloaded” state is l.
We make the following assumptions:
(1) Purely horizontal forces F1 and F2 are applied in such a way as to maintain the quarter car…