/THE INSTITUTION OF ENGINEERS, SRI LANKA

PART II EXAMINATION -JUNE 2013

213 MECHANICS OF MACHINES

Answer FIVE Questions only. All Questions carry equal marks.

Time Allowed: Three (03) Hours

QUESTION 01

(a) Show that the gyroscopic torque (1) on a wheel of moment of inertia /, rotating at anangular velocity OJ and precessing with an angular velocity OJp is given by

T = I((J((J p

(b) A disc supported between two bearings on a shaft has a mass of 100 kg and a radius ofgyration of 250 mm. The distances of the disc from the bearings are, 300 mm to the rightfrom the left hand bearing and 500 mm to the left from the right hand bearing. Eachbearing is supported by a thin vertical cord. When the disc rotates at 1000 rev/min in theclockwise direction as viewed from the left hand bearing, the cord supporting the lefthand bearing breaks. Find the magnitude and the direction of the angular velocity ofprecession of the disc at the instant of breaking the cord.

QUESTION 02

A rope lifting a load of 250 kg is wound around a drum of diameter 1.2 m. The mass of thedrum is 50 kg and its radius of gyration is 450 mm. The bearing friction torque on the drumshaft is 80 Nm. The drum is driven by an electric motor through a two-stage reduction gear.The rotating parts of the intermediate shaft have a moment of inertia of 4 kgrrr' and theintermediate shaft experiences a frictional torque of 20 Nm. The intermediate shaft runs atfour times the drum speed. The rotating parts of the motor have a moment of inertia of 0.5kgnr' and the motor shaft experiences a frictional torque of 5 Nm. The motor exerts a torqueof75 Nm.

Determine the following.

(a) Gear ratio between the motor and the intermediate shaft for maximum acceleration of theload, and

(b) Magnitude of this acceleration.

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QUESTION 03

The spacing of the four cylinders 1,2,3 and 4 of a vertical in-line engine is 650 mm, 500 mmand 650 mm respectively. The reciprocating masses of the four cylinders, taken in order, arem kg, 80 kg, 80 kg and m kg. The cranks of the cylinders 2 and 3 are at 60° to one another.The stroke of each piston is 325 mm and the length of each connecting rod is 600 mm.

(a) Find the value of m and the relative angular positions of all the cranks if primary forcesand primary couples are balanced.

(b) Also find the maximum unbalanced secondary force when the engine speed is400 rev/min.

QUESTION 04

The following data refer to an open belt drive.

Diameter of the larger pulleyDiameter of the smaller pulleyDistance between the parallel shafts to whichthe pulleys are attachedCoefficient of friction between the belt andthe pulleyArea of cross section of the beltLinear density of the beltMaximum permissible stress in the beltInitial tension of the belt

- 675 mm- 450 mm

- 1800mm

- 0.28- 440 mnr'- 2.5 kg/m- 5N/mm2- 1600N

(a) Determine the maximum power that can be transmitted through the belt at the aboveinitial tension.

(b) Which of the following alternatives. would be more effective in increasing the powerwhich could be transmitted through the belt drive?

(i) Increasing the coefficient of friction by 10%(ii) Increasing the initial tension by 10%

QUESTION 05

A Porter governor has equal arms, each arm being 240 mm long. The arms are pivoted on theaxis of rotation. Each ball has a mass of 5 kg and the load on the sleeve is 18 kg. The radiusof rotation of the balls is 150 mm when the sleeve begins to rise and it is 200 mm at themaximum speed.

(a) Determine the range of speeds.

(b) If the friction at the sleeve is equivalent to a force of 10 N, determine the new range ofspeeds and the sensitiveness of the governor.

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QUESTION 06

The gear wheel A and the sun wheel C of the epicyclic gear are fixed to the input shaft P ofthe gear train shown in Fig. Q6. Wheel C meshes with planet wheelD, carried by the outputshaft Q. Gear wheel E has both internal and external teeth and wheel D is in mesh with theinternal teeth of wheel E. Wheel A is in mesh with wheel B which is fixed to the same shaftas the wheel F. Wheel F meshes with external teeth of wheel E.

All the wheels have the same module and the number of teeth on the wheels are:A -75, B -75, C - 40, F - 30, E(external) - 120, E(internal) - 100.

If the speed of the input shaft is 112 rev/min, find the speed of the output shaft.

1-",:E ~~

-A D

~ ~Q-p

~

C ~

~

c--

---~

~~E0.~

F

~

c--

B,

Fig. Q6

QUESTION 07

Fig. Q7 shows a spring-mass system in which the mass ml is supported by two springs eachof stiffness kl' and the mass '!l2 is supported by a spring of stiffness k2•

(a) Derive the equations of motion of the system.

(b) Show that the frequency equation for small oscillations of the system can be expressed as

Q)4 _[2kl + (kl +k2)]Q)2 + kl(kl +2k2) =0ml m2 mlm2

where (J) is the natural frequency of oscillations of the system.

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(c) If, mi = Zm, = 2m and kl =2k2= 2k, determine the frequencies of oscillations in terms ofk and m and the corresponding ratios of amplitudes of motion of the two masses.

Fig. Q7

QUESTION 08

A two rotor system with fixed shaft ends is shown in Fig. Q8. The rotors have moments ofinertia II and h, and each of the three shaft sections has a stiffuess of K as indicated in thefigure.

(a) Derive the equations of motion of the system and obtain the frequency equation fortorsional oscillations of the system.

(b) Show that if, h = 211 = 21, the frequency equation reduces to

m4 -3am2 +1.5a2 = 0where ())is the natural frequency of torsional oscillations of the system and a = K .

I(c) Hence determine the natural frequencies of torsional vibrations of the system and the

corresponding mode shapes.

,.-

K K ~z? K

~~~~

L-

IIL-

h

Fig. Q8

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