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Indian Institute of Technology, Madras Machine Design Section ME 7360 Theory of Vibrations Assignment on Continuous systems Date of Submission:15/11/2011 1. Show that the equations of motion for the torsional vibration of a shaft and the axial vibration of a rod are similar to that for the transverse vibration of a string. Find the expression for the natural frequencies and the natural mode shapes for the torsional vibration of the shaft and the axial vibration of the rod if the ends are xed. Find the fundamental frequency for torsional vibration of a shaft of length 2m and diameter 50mm when both the ends are xed. The density of the material is 7800kg/m3 and the shear modulus of rigidity is 0.8 1011 N/m2 . 2. Determine the characteristic equation for obtaining the natural frequencies of a uniform cantilever beam and the corresponding natural modes. Plot the rst three modes. Explain a method by which the transcendental equation can be solved. 3. For the bending vibration of a beam show that the normalized natural modes (Wr (x)) satisfy the following orthogonal relations l mWr (x)Ws (x) dx = rs 0 l 2 EIWr (x)Ws (x) dx = r rs 0 r =1 Wr (x)qr (t), explain how the response of the beam EI expanding w(x, t) = 4 w(x, t) 2w w + m 2 = f (x, t) +c x4 t t can be obtained using the normal mode method. 1

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