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I Semester M.E. (Civil) Degree Examination, March 2013 Earthquake Engineering 2K8EQ101 : THEORY OF VIBRATIONS Time : 3 Hours

Max. Marks : 100

Note : 1) Answer five full questions. 2) Additional data if necessary may be assumed. I. a) 200 kg machine is attached to the end of a cantilever beam of length L = 2.5 m, elastic modulus E = 200×109 N/m2, and cross sectional moment of inertia 1.8 × 10 –6m4. Determine the circular frequency, time period of vibration.

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b) What is the equivalent stiffness of a system shown in fig (1) ? Given Kb = 4 L3/243 EI, E = 210 ×109 N/m2, I = 5 ×10 –4m4, and K = 1×108 N/m.

I.(b)

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Fig. (1)

II. A harmonic force of amplitude 200 N and frequency 5 Hz acts on a mass of single degree of freedom system having M = 10 kg, K = 2000 N/m, and sec C = 50 N m . Determine the complete solution representing the motion of the mass if the initial displacement and velocity of the mass are 10 mm and 5 m/s sec respectively.

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P.T.O.

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III. a) A free vibration test is run on a cantilever beam shown in fig (2), the ratio of amplitude on successive cycles is 2.5 to 1.0. Determine the response of the machine due to rotating unbalance of magnitude 0.25 kg-m. when the machine operates at 2000 rpm, and the damping is assumed to be viscous.

III.(a)

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Fig. (2)

b) What do you understand by transient vibrations and forced vibrations ? IV. a) Distinguish between generalised and discrete co-ordinates.

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b) Show that the logarithmic decrement is also given by the equation δ=

1 ⎛ x0 ⎞ ln ⎜ ⎟ n ⎜⎝ x n ⎟⎠ .

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c) The damping ratio ξ = 0.2 , for a system having stiffness K = 2000 N/m and mass M = 10 kg. What is i) the damped frequency ii) logarithmic decrement iii) ratio of successive amplitudes ?

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V. a) Determine the response of a single DOF system to the step excitation Fo for an undamped system using convolution integral as x ( t ) = Fo (1 − cos ω n t ) . K

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b) What is the minimum static deflection of an undamped isolator to provide 75% isolation to a pump that operates at speeds between 1500 rpm and 2000 rpm.

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VI. a) An undamped two degree of freedom spring mass system shown in fig (3), Determine : i) The governing equation of motion for the system ii) Natural frequencies of the system iii) The normal modes of the system.

VI.(a)

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Fig. (3)

b) Distinguish between continuous and discrete system.

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VII. a) Determine the lowest natural frequency of longitudinal motion for the system shown in fig. (4). 15

VII.(a)

Fig. (4)

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b) Show that the given modes are orthogonal ⎧1. 000 ⎫ ⎧ 1. 00 ⎪ ⎪ ⎪ φ1 = ⎨1. 379 ⎬ ; φ2 = ⎨ 0 .148 ⎪1. 275 ⎪ ⎪− 1. 29 ⎩ ⎭ ⎩

⎫ ⎧1. 000 ⎫ ⎪ ⎪ ⎪ ⎬ φ 3 = ⎨1. 656 ⎬ ⎪ ⎪ 1 .97 ⎪ ⎭ ⎩ ⎭

0 0 ⎤ ⎡19. 29 ⎢ Μ= 0 18. 78 0 ⎥. ⎢ ⎥ ⎢⎣ 0 0 9.39 ⎥⎦

VIII. Write short notes on four of the following : i) Structural damping ii) Damping ratio iii) Characteristics of normal modes iv) Static condensation v) Duhamel’s Integral. ———————

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(5×4=20)