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VTU Mechanical Engineering (Semester 7)
Mechanical Vibrations
December 2016
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) Add the following harmonic motions analytically and check the solution graphically:$$x_1=4\cos \left ( wt + 10^0 \right ),x_2=6 \sin \left ( wt +60^0 \right ).$$/
10 M
1(b) Represent the periodic motion given in the Fig.Q.1(b) by harmonic series.
!mage
10 M

2(a) Determine the natural frequency of spring-mass system taking the mass of the spring in to account.
10 M
2(b) Using energy method find the natural frequency of the system shown in the Fig.Q.2(b).
!mage
10 M

3(a) Obtain the response of viscous damped system for critically damped case.
10 M
3(b) The disc of a torsional pendulum has a moment of inertia of 0.06kg m2 and is immersed in viscous fluid. The brass shaft attached to it is of 100mm diameter and 400mm long when the pendulum is vibrating. The observed amplitude on the same side of neutral position for the successive cycles are 9°,6° and 4°. Determine: i)Logarithmic decrement
ii) Damping torque at unit velocity.
iii) Periodic time of vibration.
iv) The frequency if the pendulum is removed from the viscous fluid. Assume for brass shaft G = 4.4×1010N/m2.
10 M

4(a) Show that for a spring mass damped system the peak amplitude occurs at a frequency ration given by the expression $$\frac{w}{w_n}=\sqrt{1-2\xi ^2}$$/, when the system is excited by a harmonic force. Also determine the expressions for peak amplitude and the corresponding phase angle.
10 M
4(b) A machine of mass one tonne is acted upon by an external force of 2450N at a frequency of 1500 rpm. To reduce the effects of vibration isolator of rubber having a static deflection of 2mm under the machine load and an estimated damping factor of 0.2 are used. Determine: i) Force transmitted to the foundation ii) Amplitude of vibration of the machine iii) Phase lag of the transmitted force with respect to external force.
10 M

5(a) Explain Frahm's reed tachometer.
5 M
5(b) Explain Vibrometer.
5 M
5(c) A horiziontal shaft of 25mm diameter carries a mass of 12kg mounted midway. The shaft is supported at the ends by two bearings. The span between the bearings is 900mm. The mass centre is 0.02mm from the axis of the shaft. Determine the amplitude of steady state vibrations and the dynamic force on the bearings when the shaft rotates at 3000 rpm. Take E = 2000GPa. Neglect damping and mass of shaft.
10 M

6(a) What is a "Semi definite system"? Explain.
8 M
6(b) Determine the natural frequencies of the system as shown in the Fig.Q.6(b) if k1=4c×103N/m, k2=50×103N/m, k3=60×103N/m, m1=10kg, m22=12kg, r1=0.10m and r2=0.11m.
!mage
12 M

7(a) Using Stodala's method find the natural frequenciesof the four mass system as shown in Fig.Q.7(a) if k = 1 N/m and m = 1kg.
!mage
15 M
7(b) Use Dunkerley method to find the fundamental natural frequency of transverse vibration for the system shown in Fig.Q.7(b).
!mage
5 M

8(a) Briefly explain the hardware of an equipment necessary for experimental modal analysis.
10 M
8(b) Explain machine condition monitoring techniques.
10 M

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