## Mechanical Vibrations - Dec 2015

### Mechanical Engg. (Semester 7)

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)** Add the following harmonic motion analytically and check the solution graphically:

x_{1}=4cos(wt+10°)

x_{2}=6sin(wt+60°)(10 marks)
**1 (b)** Develop the Fourier series for the curve shown in FigQ1(b)
(10 marks)
**2 (a)** Explain the energy method of finding natural frequency of spring mass system.(8 marks)
**2 (b)** Find the natural frequency of the spring controlled simple pendulum shown in FigQ2(b) Neglect the mass of the rod
(6 marks)
**2 (c)** For the system shown in figQ2(c) find mass m if the system has a natural frequency of 10Hz

take k_{1}=2N/mm

k_{2}=1.5N/mm

k_{3}=3N/mm

k_{4}=k_{5}=0.5 N/mm
(6 marks)
**3 (a)** Show that the ratio of successive amplitudes of mass in a underdramped, viscously damped spring -mass system is given by

\frac{x_{0}}{x_{1}}=e \ where\ \delta=\frac{2\pi \xi }{\sqrt{1-\xi ^{2}}}(10 marks)
**3 (b)** A machine of mass 20kg is mounted on a spring and dashpot. The spring stiffness is 10N/mm and damping is 0.15 N/mm/s. If the mass is initially at reset and a velocity of 100mm/s imported to it, determine; i) displacement and velocity of mass as a function of time ii) displacement and velocity when time is equal to one second.(10 marks)
**4 (a)** Derive an expression for steady state amplitude of vibration of mass in a spring-mass- damper system when the mass is subjected to harmonic excitation. Also find the phase angle between the mass and excitation(10 marks)
**4 (b)** A vibratory body of mass 150kg supported on springs of total stiffness 1050kN/m has a rotating unbalance force of 525 N at a speed of 6000RPM. If the damping factor is 0.3, determine:

i) Amplitude of vibration and phase angle.

ii) Transmissibility ratio and

iii) Force transmitted to the foundation.(10 marks)
**5 (a)** Explain the principle of working of

i)Vibrometer(Seismometer); ii) Accelerometer(10 marks)
**5 (b)** A rotor od mass 9.5 kg is mounted on a12mm horizontal steel shaft midway between bearing that are 06m apart. The mass centre of the disc is 6mm from its geometric centre. If the damping factor is 0.1 and the shaft rotates at 690RPMm determine the maximum stress in the shaft and compare it with the dead load stress in the shaft. For steel shaft take E=1.96×10^{11}N/m^{2}(10 marks)
**6 (a)** Explain the principles of dynamic vibration absorber. Derive the necessary equation(10 marks)
**6 (b)** For the system shown in FigQ6(b) find the natural frequencies and amplitude ratios. Given m_{1}=10khg, m2=15kg and k=320N/m
(10 marks)
**7 (a)** Find the first natural frequency and draw the mode shape for the system in figQ7(a) by matrix iteration method. Take k_{1}=k_{2}=k_{3}=k and m_{1}=m_{2}=m_{3}=m
(10 marks)
**7 (b)** Using Stodola's method determine the lowest natural frequency of the torsional system shown in FigQ7(b)
(10 marks)
**8 (a)** Explain the role of i)Exciter; ii)Transducer iii)Signal conditioner and iv) Analyser used in experimental modal analysis(10 marks)
**8 (b)** Describe the three types of maintenances scheme given below

i)Breakdown maintenance

ii)Preventive maintenance

iii)Condition-based maintenance(6 marks)
**8 (c)** Explain briefly the following methods of condition monitoring

i) Wear debris monitoring

ii) Vibration analysis(4 marks)