## Mechanical Vibrations - Dec 2013

### Mechanical Engineering (Semester 6)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1 (a)** Fig. 1 shows a rectangular block of mass 'M' resting on top of a semi-cylindrical surface. If block is slightly tipped at same end, find frequency of oscillations.

(8 marks)
**1 (b)** A door along with 250cm high, 80cm wide and 4.5cm thick and weighing 40kg is fitted with an automatic door closer. The door opens against a spring with a modulus of 1kg-cm/radian. If the door is opened 90° and released, how long with the door to be within 1° of clearing (Return of door is critically damped).(8 marks)
**1 (c)** Write a note on critical speed of shaft.(4 marks)
**2 (a)** The clender bar in the system is rotated 5° from equilibrium and released. Determine time independent response of the system.

m=2kg, L=80cm, r=10cm, K=20000N/m, C=300Ns/m.

(12 marks)
**2 (b)** An accelerometer has natural frequency 15kHz. Determine the highest frequency it can measure with 1% accuracy. Assume damping ratio 0.7.(8 marks)
**3 (a)** An aircraft instrument of mass 20kg is to be isolated from engine vibration. The engine runs at speed ranging from 1800rpm to 2500rpm. Natural rubber isolator with negligible damping is used. Determine the rubber stiffness for 90% isolation.(10 marks)
**3 (b)** Using Lagrange's equation, find the frequency of the system shown in fig. 3

(10 marks)
**4 (a)** Find the Eigen values and Eigen vectors of the system shown in fig. 4

(12 marks)
**4 (b)** Derive the equation of motion and find the steady state amplitude of mass 'm'.

(8 marks)
**5 (a)** The natural frequency of transverse vibration of beam in fig. 6 is 20rad/s. Find the natural frequency of vibration if another 30N is added at 40cm from the left support.

Hint:

where U_{ij} is the deflection of section i due to unit load at section j and S_{i} is the distance of section i from left support and Z_{j} is the distance of section j from right support.

(15 marks)
**5 (b)** Explain the 'motion transmissibility' with frequency response curve and formulae.(5 marks)
**6 (a)** A shaft carries five masses A, B, C, D and E which revolve at the same radius in planes which are equidistant from one another. The magnitude of masses in plane A, C and D are 50kg, 40kg and 80kg respectively. The angle between A and C is 90° and that between C and D is 135°. Determine the magnitude of masses in planes B and E and their positions to put the shaft in complete rotating balance.(10 marks)
**6 (b)** Explain follow jump phenomenon in cam.(6 marks)
**6 (c)** What do you mean by static and dynamic balancing?(4 marks)
**7 (a)** A five cylinder in-line engine running at 750rpm has successive cranks 144° apart, the distance between cylinder centre-lines being 375mm. The piston stroke is 225mm and the ratio of connecting rod to the crank is 4. Estimate the engine for balance of primary and secondary forces and couples. Find the maximum values of these and positions of central crank at which these maximum values occur. The reciprocating mass for each cylinder is 15kg.(10 marks)
**7 (b)** A helical spring has both ends securely fixed one vertically above the other and a mass is attached to the spring at some intermediate point. Show that the frequency at vibrations is minimum when the load point is at the centre of fixed ends(y=0.5l) where y=distance between fixed end and load point and l=distance between fixed ends.(6 marks)
**7 (c)** What is the basic principle used in Holzer's method.(4 marks)