Calculate the magnitude and direction of the torque

Question

Part -1:

Question 1:

Fig. 1(All dimensions in mm)

The component shown in Fig 1 is made from a material with the following properties and is subjected to a compressive force of 5kN.

Material Properties :

Young's Modulus of Elasticity - 200 GNm^-2

Modulus of Rigidity - 90 GNm^-2

Poisons ratio - 0.32

Calculate:
(a) The stress in:
(i) the circular section
(ii) the square section

(b) The strain in:
(I) The circular section
(ii) The square section

(c) The change in length of the component

(d) The change in diameter of the circular section

(e) The change in the 40mm dimension on the square section

(f) If the same component were subjected to a shear force of 7 kN as shown in FIG 2, calculate the shear strain in:
(i) The circular section
(ii) The square section

Fig. 2

Question 2:

When the 5mm diameter bar shown in FIG 3 is subjected to a tensile force F, yield occurs when the bar has extended by 4μm.

Calculate :
(a) The yield stress of the material
(b) The force required to produce yield.

Young's Modulus for the bar material is 150 GNm^-2

Fig. 3

Question 3:

A material is formed into a solid sphere and has a diameter of 100mm when at a pressure of 2MPa. If the diameter of the sphere reduces by 0.1mm when the pressure is increased to 6MPa, determine the bulk modulus of the material.

Question 4.

A material has a modulus of rigidity of 100 GNm^-2 and a Young's Modulus of 250 GNm^-2. Calculate the expected value of poisons ratio for the material.

Question 5.

The simply supported beam shown in FIG 4 is 5m long with a Young's Modulus of 210 GNm^-2. The cross section of the beam is as shown in FIG 5.

FIG 4

FIG 5

(a) Draw the shear force diagram for the beam
(b) Determine the position and magnitude of the maximum bending moment.
(c) Plot a graph of deflection along the length of the beam (calculate the deflection at 1m intervals).

Question 6. A cylindrical vessel 2m internal diameter and 4m long has a wall thickness of 6mm. Strain gauges are installed on the vessel to measure hoop strain (see FIG 6).

FIG 6

E = 290 GNm^-2

Yield Stress = 500 MPa

(i) What is the maximum allowable pressure if a factor of safety of 4 is to be used?
(ii) What pressure would a strain of 40 με indicate?

Part -2:

Question 1. A pulley 150 mm diameter is driven directly by an electric motor at 250 revs min^-1. A V-belt is used to transmit power from this pulley to a second pulley 400 mm diameter against a load of 200 Nm.

The distance between the centre of the pulleys is 600 mm, the included angle of the pulley groove = 40°, the coefficient of friction between the belt and pulley is 0.4 and the ultimate strength of the belt is 8 kN.

(a) Calculate the actual power transmitted to the second pulley.

(b) Calculate the power which can be transmitted if the maximum tension in the belt is limited to half of the ultimate strength of the belt.

(c) What would be the effect of the following factors on the maximum power which can be transmitted (give reasons for your answer):
(i) increasing the coefficient of friction
(ii) increasing the included angle of the pulley groove.

(d) What would be the effect on the following if the load torque is increased and the speed maintains constant (give reasons for your answer):
(i) the tension in the tight side of the belt
(ii) the tension in the slack side of the belt
(iii) the power transmitted.

Question 2. A gear train is to be designed to transmit power from shaft 1 to shaft 2 with the following condition:

(i) Shaft 1 (driven by an electric motor) rotates at 100 revs min^-1.

(ii) Shaft 2 is to rotate at 400 revs min^-1 in the opposite direction to shaft 1 against a load of 200 Nm.

(iii) The centre of shaft 1 is 300 mm from shaft 2.

(iv) The minimum number of teeth on any gear is 15, all gears must have a multiple of 5 teeth and have a module of 2 mm.

(v) The maximum number of gears permissible is 4 gears and the diameter of the maximum gear must be minimised.

(vi) The centres of the gears should lie on a line which is as close to straight as possible.

(vii) All shafts have a frictional resistance of 5 Nm. Carry out the following:

(a) Design a gear train which satisfies the above criteria. Use a sketch to illustrate your design, label the gears A, B, C, etc. from the driver to the driven gear, and state the number of teeth on each gear.

(b) Determine the input power required at shaft 1.

(c) Specify the efficiency of the gear train as a percentage.

(d) Determine an equation for the efficiency of the gear train in terms of the load (torque) on shaft 2 (all other factors remaining constant).

Part -3:

Question 1. (a) For the mechanism shown in FIGURE 1 determine for the angle θ = 45°:

(i) the velocity of the piston relative to the fixed point O (V_BO)
(ii) the angular velocity of AB about point A (i.e. ω_AB)
(iii) the acceleration of point B relative to A (α_BA).

FIG. 1

Note: Link AB is horizontal when θ = 45°
(b) Determine the value of the angle 0 (measured from vertical) when:
(i) the velocity of point B = 0
(ii) the angular velocity of link AB a maximum.
(c) What is the maximum angular velocity of link AB?

Question 2. (a) A shaft 2 in long rotates at 1500 revs min^-1 between bearings as shown in FIGURE 2. The bearings experience forces of 5 kN and 3 kN acting in the same plane as shown. A single mass is to be used to balance the shaft, so that the reactions are zero. The mass is to be placed at a radius of 200 mm from the shaft centre, 180° from the direction of the bearing reactions. Determine the size and position (a and b) of the mass to be used.

Fig. 2

(b) The shaft in part (a) is to be balanced using two masses (m₁ and m₂) placed 0.5 m and 1.5 m from end A and 180° from the direction of the bearing reactions, each on radius arms 100 mm long. Calculate the sizes of m₁ and m₂.

Fig. 3

Question 3. A gearbox and flywheel are as shown in FIGURE 4. The output shaft rotates in the opposite direction to the input shaft at 5 times its speed. The gearbox has an efficiency of 92%.

If the flywheel is solid, has a mass of 50 kg, a diameter of 1.5 m and is to accelerate from rest to 300 revs min-1 in 1 minute:

(a) Calculate the torque required at input T₁.

(b) Calculate the magnitude and direction of the torque required to hold the gearbox stationary (holding torque Th). Show the direction of the holding torque applied to the shaft with the aid of a sketch.

(c) Plot a graph of the input power against time when taking the flywheel from rest to 300 revs min^-1.

Fig. 4