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Guide to Completing Exam Paper (Parts A, B, and C)

Part A

Please note that the examination paper comprises three parts (A, B and C), all three parts should be completed.

Part A contains two compulsory questions both of them worth 25 marks.

Part B and C contains two questions each.

You need to answer both questions from part A (total of 50 marks).

You need to answer both questions from part B (worth 50 marks) and both questions from part C (worth 50 marks).

Part A

Question A.1 [25 Marks]

Question A.2 [25 Marks]

Question A1

A block of 20 kg is connected as shown in Figure QA1.

Numerical values: k = 600 N/m, c = 40 N.s/m

Figure QA1. Mass connected between two springs

a. Write the equation of motion of the system. [5 marks]

b. Determine the damping ratio of the system and establish what type of motion occurs (underdamped, overdamped, critically damped) [5 marks]

c. The mass is displaced 1 cm upward and given a downward speed of 0.02 m/s. Write the equation of the displacement of the mass as a function of time .Consider the positive direction upward. [10 marks]

d. Determine the value of the displacement of the mass after 1/4 sec. [5 marks]

Question A2 [Note that, the solution for section i is independent from section ii]

The tubular propeller shaft of the tugboat is subjected to the compressive force N=10 kN, and torque T=2kNm as shown in Figure QA2.1. The shaft has an inner diameter of 100mm and an outer diameter of 150 mm.

a. Determine:

1. The normal stress at point A due to the normal force (N) [2 Marks]

2. The polar moment of inertia of the shaft’s cross-section (J) [2 Marks]

3. The torsional shear stress due to the torque applied (T) [2 Marks]

b. If due to the applied loading on the shaft the element at point A (see figure QA.2.1) is subjected to the state of stress shown in Figure QA2.2,

4. Determine the average normal stress and the radius of the Mohr’s circle [4 Marks]

5. Sketch the Mohr’s circle, clearly highlight the coordinates of the centre and the reference point used. [6 Marks]

6. Determine the principal stresses and highlight them on the Mohr’s circle [4 Marks]

7. The shaft is made from a brittle material having an ultimate strength of , determine if the shaft fails according to the maximum-normal-stress theory. [5 Marks]

## Question A.1

Part B

Question B.1 [25 Marks]

Question B.2 [25 Marks]

Question B1 (this part is made of three short questions)

(Topics assessed: EOM of a 1DOF rotational system, damped harmonic motion experimental data evaluation and analysis, modelling)

1. A uniform rod of mass = 1 kg and length = 0.25 m (Figure QB1.1) is supported by a pin joint at A and a spring with stiffness  = 300 N/m at B. The mass moment of inertia of the rod about point A is:

The beam is given a small rotation and vibrates freely. Assume the angular rotation (  about point A, is small and that the positive direction is clockwise.

a. Write the force in the spring at B as a function of angular displacement . [4 marks]

b. Write the equation of motion for small vibrations about point A [6 marks]

2. In Figure QB1.2 are represented the free vibration responses of two mass-spring-damper systems. The two systems have the same mass and the same spring. Which one of the two systems (system 1 – dashed line or system 2 – continuous line) has a higher damping ratio? Justify your answer.  [4 marks]

3. A 75-kg electric motor (Figure QB1.3) turns an eccentric flywheel, which is equivalent to an unbalanced 0.12-kg mass located 260 mm from the axis of rotation.

a. If the static deflection of the beam is 25 mm because of the mass of the motor, determine the angular velocity of the flywheel at which resonance will occur. Neglect the mass of the beam. [4 marks]

b. What will be the amplitude of steady-state vibration of the motor if the angular velocity of the flywheel is 22 rad/s [7 marks]

Question B2

(Topics assessed: experimental data evaluation and analysis – frequency response, logarithmic decrement and vibration due to a rotating unbalanced mass, equivalent rotational system)

1. In Figure QB2.1 are represented the frequency response plots of two mass-spring-damper systems. The two systems have the same mass and the same spring. Which one of the two systems (system1 or system2) has a higher damping ratio? Justify your answer. [4 marks]

2. The free vibration response of a system modelled as a spring-mass-damper system is shown in Figure QB2.2

The first two points of maximum displacement occur (as shown in the figure) at:

 1st maximum point 2nd maximum point Time 0.15 s 0.78 s Displacement 0.0805 m 0.031 m

## Question A.2

a. Determine the damping ratio of the system [6 marks]

b. Determine the natural frequency of the system [5 marks]

c. Write the equation of motion of the system in standard form using numerical values [5 marks]

3. In Figure QB2.3 is represented the model of a rotational system as a beam pinned at point A and supported by a spring at point B. A mass is added to the beam at point O.

In order to obtain an equivalent translational system you need to move the spring from point B to point O as shown in Figure QB2.4.

What is the value of the new elastic constant ?  [5 marks]

Part C

Question C.1 [25 Marks]

Question C.2 [25 Marks]

Question C1

The thin-walled pipe (Figure QC1) has an inner diameter of 10 mm and a thickness of 0.5 mm. If the pipe is subjected to an internal pressure of 3.5 MPa and the axial tension and torsional loadings shown:

a. What is the area of the pipe cross-section [2 marks]

b. What is the polar second moment of area “J” [2 marks]

c. Determine the longitudinal stress due to the normal load [2 marks]

d. Determine the longitudinal stress due to the applied pressure, [2 marks]

e. What is the combined longitudinal stress due to both normal load and applied pressure [2 marks]

f. Determine the hoop stress [4 marks]

g. What is the shear stress due to the applied torsional load [4 marks]

h. Sketch an element of the thin walled pipe and show the normal and shear stress applied [3 marks]

i. Determine the principal stresses [4 marks]

Question C2

The member BC is subject to a uniformly distributed load of intensity w. Assume that the member AB is made of steel and is pinned at its ends for x-x axis buckling and fixed at its ends for y-y axis buckling (Figure QC2). Use a factor of safety with respect to buckling of 3. Yield stress

Determine:

a. The moment of inertia about the x-x axis [2 marks]

b. The moment of inertia about the y-y axis [2 marks]

c. The allowable linear distributed force ‘w’ about the x-x axis [6 marks]

d. The allowable linear distributed force w about the y-y axis [6 marks]

e. The maximum critical normal stress [6 marks]

f. Is the Euler's formula valid? Why? [3 marks]