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## Mechanical Vibrations - May 2012

### 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)** A circular cylinder of mass 'm' and radius 'r' is connected by a spring of stiffness 'k' as shown in fig. If it is free to roll on the horizontal surface without slipping, find the natural frequency of vibration:

i) By Energy method

ii) By Newton's law of motion

(10 marks)
**1 (b)** Prove that the resultant unbalanced force is minimum when half of the reciprocating masses are balanced by rotating masses. (i.e. when c=1/2)(10 marks)
**2 (a)** Explain with the help of transmissibility plot, why normally isolators are designed for frequency ratio higher than √2.(5 marks)
**2 (b)** 20N at 30cm, 30N at 60cm, 10N at 100cm from the fixed end are the loading on the cantilever. The deflection under 30N load due to all loads is 2mm. What would be the natural frequency of transverse vibration if 20N is added at 80cm from fixed end. Also find the frequency by Dunkerley's method. The deflection at section 'i' due to unit load at section 'j' is given by -

where 's' is the distance of section from fixed end.(15 marks)
**3 (a)** Find effective spring constant with proper justification for a spring of stiffness 'k' inclined at angle 'θ' to the direction of motion.(5 marks)
**3 (b)** A heavy machine weighing 9810N is being lowered vertically down by a winch at a uniform velocity of 2m/s. The steel cable supporting the machine has a diameter of 0.01m. The winch is suddenly stopped when steel cable's length is 20m. Find the period and amplitude of the ensuring vibration of the machine (Assume E=200GPa).(10 marks)
**3 (c)** State the difference between Coulomb damping and Viscous damping.(5 marks)
**4 (a)** A door 2m high, 1m wide, 40mm thick and weighing 350N is fitted with an automatic door closer. The door opens against a spring with a modulus of 0.1Nm/rad. If the door is opened 90° and released, how long will it take the door to be within 2° of closing? Assume the return spring of the door to be critically damped.(10 marks)
**4 (b)** A mass of 50kg is suspended by a spring of stiffness 12kN/m and acted on by a harmonic force of amplitude 40N. The viscous damping coefficient is 100Ns/m. Find -

i) Resonant amplitude

ii) Peak amplitude

iii) Peak frequency

iv) Resonant phase angle

v) Peak phase angle(10 marks)
**5 (a)** Determine the natural frequency and mode shapes of the system shown in fig. by Holzer's method (If in the range of 100 to 150rad/s)

J_{1}=5kgm^{2}, J_{2}=10kgm^{2}, J_{3}=20kgm^{2},

K_{t1}=1*10 ^{5}Nm/rad, K_{t2}=2*105Nm/rad(10 marks)

**5 (b)**An accelerometer has a suspended mass of 0.01kg with a damped natural frequency of vibration of 150Hz. When mounted on an engine undergoing as acceleration of 9.81m/s

^{2}at an operating speed of 6000rpm, the acceleration is recorded as 9.5m/s

^{2}by the instrument. Find the damping constant 'c' and spring stiffness 'k' of the accelerometer.(10 marks)

**6 (a)**Find the two natural frequencies and mode shapes for the system shown in figure.

m

_{1}=1.5kg, m

_{2}=0.8kg, k

_{1}=k

_{2}=4000N/m.(10 marks)

**6 (b)**Explain the response of undamped cam mechanism with a mathematical model. Also explain in brief 'Jump Phenomenon'.(10 marks)

**7 (a)**A shaft of span 3m between two bearings carries two masses of 15kg and 30kg acting at the end of arms 0.5m and 0.6m respectively. The planes in which these masses rotate are 1m and 2m from the left end bearing. The angle between arms is 60°. The speed of rotation of the shaft is 240rpm. If the masses are balanced by two counter masses rotating with the shaft acting at the radii of 0.25m and placed at 0.3m from each bearing centre. Determine the magnitude of two balance masses and their orientation with respect to 15kg mass.(12 marks)

**7 (b)**i) Explain the critical speed of the shaft.

ii) Explain logarithmic decrement for spring-mass underdamped system.(8 marks)