VTU Mechanical Engineering (Semester 7)
Mechanical Vibrations
May 2016
Total marks: --
Total time: --
(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) The motion of the a particle is described as \[\begin{align*}x=2\sin \left ( wt+\frac{\Pi }{6} \right )\end{align*}\]. If the motion has two components, one of which is \[\begin{align*} {x_1}=\sin \left ( wt-\frac{\Pi }{3} \right )\end{align*}\]
6 M
1(b) Explain the "BEATS" phenomenon.
4 M
1(c) Find the Fourier series for the saw tooth curve as in fig.Q1(c) !mage
10 M

2(a) Determine the natural frequency of the simple pendulum neglecting the mass of the rod.
5 M
2(b) Find the natural frequency of oscillation for the system shown in fig. Q2(b) assuming bell crank lever to be light and stiff and mass 'm' to be concentrated.
5 M
2(c) The connecting rod shown in fig.Q2(c) is supported at the wrist pin end. It is displaced and allowed to oscillate. The mass of the rod is 5kg and CG is 0.2m fromt the pivot point 'O'. If the frequency of oscillation is 40 cycles/minute, calculate the moment of intertia of the system about its center of gravity(CG). !mage
10 M

3(a) Explain over damped, critically damped and under damped systems.
6 M
3(b) Damped vibration of a spring mass dashpot system gives the following information : Amplitude of second cycle =12mm;
Amplitude of third cycle = 10.5mm ;
Spring constant K = 7840 N/m;
Mass of body, m = 2kg. Determine the damping coefficient.
4 M
3(c) A spring mass dashpot system consists of spring of stiffness 343 N/m. The mass is 3.43kg. The mass is displaced 20mm beyond the equilibrium position and released. Find the equation of motion for the system if the damping coeffficient for the dashpot is
i) 137.2NS/m
ii) 68.6NS/m.
10 M

4(a) Define the following terms :
i) Forced vibration
ii) Magnification factor
iii) Vibration isolation
iv) Transmissibility.
8 M
4(b) A machine of mass 1000kg is acted upon by an external force of 2450N at a frequency of 1500rpm. To reduce the effect, vibration isolators of rubber having a static deflection of 2mm under the machine load and an estimated damping factor 0.2 are used. Determine
i) Force transmitted to the foundation
ii) The amplitude of viabration of machine
iii) The phase lag.
12 M

5(a) Explain the principle of " seismic" instrument. Explain how it can be used as vibrometer.
6 M
5(b) A commercial vibration pick up has a damped natural frequency of 4.5Hz and a damping ratio of 0.5. What is the range of impressed frequency at which the amplitude can be read directly from the pickup with an error not exceeding 2 percentage of the actual amplitude.
6 M
5(c) A power transmitting shaft has diameter of 30mm and 900mm long and simply supported. The shaft carries of rotor of 4kg at its mid span. The rotor has an eccentricity of 0.5mm. Shaft rotates at 1000 rpm. Neglecting mass of the shaft, calculate the following :
i) Deflection of the shaft at mid span
ii) Critical speed of the shaft
iii) Amplitude of steady state vibration
iv) Dynamic force on bearing.
8 M

6(a) Fig.Q6(a), shows a vibrating system having two degrees of freedoms. Determine the two natural frequencies of vibrations and the ratio of amplitudes of the motion of m1 m2for the two modes of vibration. Given !mage
10 M
6(b) Explain the principle of dynamic vibration absorber. Also explain the demerits of it.
10 M

7(a) Find the frequency equation of a uniform beam fixed at one end and free at the other for transverse vibrations.
10 M
7(b) Derive suitable mathematical expression for longitudinal vibrations of a bar of uniform cross section as shown in fig.Q7(b). !mage
10 M

8(a) Use the Stodola method to find the fundamental mode of vibration and its natural frequency of the spring mass system as shown in fig.Q8(a).
Take K1=K2 =K3=1,
8 M
8(b) Determine the natural frequency and the mode shape of the system shown in fig.Q8(b) by Holzer's method.
K2=10N/m. !mage
12 M

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