53202-mt----mechanical Vibrations

NR

Code No: 53202/MT

M.Tech. – II Semester Regular Examinations, September, 2008 MECHANICAL VIBRATIONS (Machine Design) Time: 3hours

Max. Marks:60 Answer any FIVE questions All questions carry equal marks ---

1. a) A uniform disk of radius ‘r’ rolls without slipping inside a circular track of radius ‘R’ as shown in figure.1 derive the equation of motion for arbitrarily large angle θ. Then show that in the neighborerhood of the trivial equilibrium θ = 0 the system behaver like a harmonic oscillator, and determine the natural frequency.

b) In a spring mass damper system the mass completes 5 oscillations in 0.5 seconds and the amplitude decays to 5% of the initial value during these 5 oscillations. If the stiffness of the spring is 5 N/mm determine the mass and damping coefficient. If the amplitude is to be brought down to 2% in 2 cycles what must be the damping coefficient? 2. a) Derive the expression for the response of a single degree of freedom system subjected to rotating unbalance. Plot it as a function of frequency ratio and explain the salient features. b) The spring of an automobile trailer are compressed 10.16 cm under its weight. Find the critical speed when the trailer is traveling over a road with a profile approximated by a sine wave of amplitude 7.5 cm and wave length 15 m. What will be the amplitude of vibration at 65 Kmph (neglect damping) Contd…2

Code No: 53202/MT

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3. a) Starting from first principles derive the expression for the response of a spring mass damper systems for an unit impulse. b) An undamped spring mass system M,K is given a force excitation F(t) as shown in figure(3b) determine the response for (i) t < t0 and (ii) t > t0

4. For the two degrees of freedom systems shown in figure4 a) Derive the equations of motion. b) Obtain the natural modes and sketch them. c) Identify the initial conditions to set the system in to first and second modes respectively.

5. Explain the procedure clearly for obtaing the free vibration response for a multi degree freedom system by using modal analysis. Clearly indicate all the variables and parameters involved. 6. By considering a suitable example with at least three degrees of freedom explain a numerical method of your choice to obtain the natural frequencies and mode shapes. 7. Derive the equation of motion for the bending vibration of beams and derive the frequency equation for a beam hinged at both the ends obtain the first three or four modes and sketch the same. Contd…3

Code No: 53202/MT

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8. Write short notes on the following a) Stability. b) Influence of Nonlinear spring forces. c) Critical speeds of shafty. d) Secondary critical speed.