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## Mechanical Vibrations - May 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)** Find the time period of small oscillations of an inverted pendulum and spring system shown in 'fig. 1(a)' given that the pendulum is vertical in the equilibrium position. Is there any limitation on the value of k?

Discuss -

Compare the time period at vibration at the above system with that of one in 'fig. 1(b)'.

(8 marks)
**1 (b)** A heavy door with door closing system as shown in fig. 2, has moment of inertia of door panel about hinge is 20kg-m^{2}. If the torsional stiffness is 25Nm/rad, find the value of c.

(8 marks)
**1 (c)** Determine torsional stiffness of 80cm long annular aluminium shaft [G=40N/mm^{2}] of inner radius 30mm and outer radius 40mm.(4 marks)
**2 (a)** A mass of 500kg is suspended with a spring. The system vibrates with a natural frequency of 3rad/s. If the initial amplitude is 24mm and subsequent half amplitudes are 20 and 16mm. Determine the stiffness of spring and coulomb damping force. Also find the number of cycles corresponding to 50% reduction of its initial amplitude.(10 marks)
**2 (b)** Find the natural frequency of the system shown in fig. 3.

(10 marks)
**3 (a)** Analyze the problem in fig. 4 for steady state response of the mass.

(12 marks)
**3 (b)** A vibrating system is supported on 4 springs. It has a mass of 10kg. The mass of reciprocating parts 2kg, which move through a vertical stroke of 100mm with SHM. Neglecting damping, find out the combined stiffness of springs, so that the force transmitted to the foundation is 1/20^{th} of the impressed force. The system crank shaft rotates at 1000rpm. If under actual working conditions the damping reduces the amplitude of successive vibration by 30%. Find:

i) The force transmitted to the foundation at 1000rpm

ii) The force transmitted to the foundation at resonance

iii) The amplitude of vibration at resonance(8 marks)
**4 (a)** Use Lagrange's equation to derive the differential equations governing the motion of the system of fig. 5 using x and θ as generalized co-ordinates. Write the differential equation in matrix form.

(12 marks)
**4 (b)** A commercial type vibration pick up has a natural frequency of 5.75Hz and a damping factor of 0.65, what is the lowest frequency beyond which the amplitude can be measured within 2% error.(8 marks)
**5 (a)** 40N at 20cm, 30N of 60cm, 10N of 110cm from fixed end are the loading on the cantilever. The deflection under 30N load is 3mm, what would be the natural frequency of transverse vibration if 20N is added at 90cm from the fixed end. The deflection at section is due to unit load at j is given by

Where s=distance of section from the fixed end.(14 marks)
**5 (b)** Explain the construction and working of seismic vibration measuring instrument.(6 marks)
**6 (a)** Using Holzer's method, determine the natural frequency for torsional vibration system in range of 1 to 5rad/s. Draw mode shapes for frequency you have found.

(12 marks)
**6 (b)** Discuss the balancing of V engine.(8 marks)
**7 (a)** Explain follower jump phenomenon for cam.(6 marks)
**7 (b)** The firing order of a 6-cylinder vertical four stroke in line engine is 1, 4, 2, 6, 3, 5. The piston stroke is 80mm and the length of each connecting rod is 180mm. The pitch distances between the cylinder centre lines are 80mm, 80mm, 120mm, 80mm and 80mm respectively. The reciprocating mass per cylinder is 1.2kg and the engine speed is 2400rpm. Determine the out of balance primary and secondary forces and couples on the engine taking a plane midway between the cylinder 3 and 4 as a reference plane.(10 marks)
**7 (c)** What do you mean by static and dynamic balancing? (4 marks)