Case Study Analysis

All questions within a case studied must be answered (equivalent wording: 1500 words)

Case Study 1

You are an engineer assigned to the design team of a machinery driveshaft that can connect itself to multiple torque-transmission stages, as shown in the diagram of Figure 1.

Figure 1 – Torques applied to the driveshaft

1. Draw the internal torque diagram for this driveshaft, specifying the most sensitive stretch (segment) of the shaft.
2. Derive an expression for the twist angle of the shaft under torque, in terms of the geometric properties of the shaft and the applied torque
3.  Calculate the diameter of the shaft, if the twist angle should not exceed 3.5°.
4.  Calculate the maximum shear stress due to torsion. Consider G = 90 GPa.

*Assume that the length between each segment is 500mm.

Case Study 2

You were called to analyse the efficiency of some mechanical engineering systems. After measuring the power produced at the steam turbine of a power plant, you notice that, for every 50 MW at the turbine’s shaft, 150 MW of heat are supplied to it, by means of the fuel burning.

1. Calculate the efficiency of the thermal cycle. What happens to the remaining amount of energy supplied to the cycle?
   
   
2. As an engineer, you are requested to design a hydraulic circuit for transferring water from one reservoir to a second one, located 6 m below the first one. Another engineer involved in the project said to you that three commercial pipe diameters are available: 35 inches, 40 inches and 50 inches. The pipes are made of steel, with a friction coefficient of 0.05. Make your selection of which pipe diameter to deploy and justify the choice.
   
   
3. Given that a rectangular duct (50 x 30 mm) with a length of 2m, carrying water through a thermal energy storage store at a velocity of 500cm/s. Determine the Reynolds number and type of flow through this duct.
   
   
4. As an engineer, what is the relevance of the Reynolds number in fluid dynamics? Also prove that the Reynolds number is dimensionless.

Case Study 3

Figure 2 shows a model of a damped vibrating system. The new symbol is an idealized dashpot - a piston in a cylinder providing damping but no mass or stiffness. (Real dampers can be constructed rather like this, with a piston usually moving through oil, although of course damping in real systems has many other sources. Nevertheless in mathematical models of vibrating systems, damping, no matter what its source, is often represented diagrammatically by a dashpot.) The physical parameters are m = 35 g, k = 10 N m-¹, c = 0.003 N s m-¹. Calculate the frequency and damping ratio and sketch the natural vibration.

Figure 2: Damped vibrating system.

1. Based on Figure 2, DERIVE the equation for modelling the freedamped vibrations of the system.
2. Explain the effects of the damping introduced. Using a table, state clearly using equations where necessary the differences between forced damping vibration and free damping vibration.