Determine the steady state amplitude of vibration, Mechanical Engineering

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1. (a) Sketch frequency-response curves for a damped single degree of freedom system subjected to a harmonic excitation under three different damping ratios. The system has a natural frequency of ωn while the forcing frequency of the excitation is ω. Explain the relation of static and dynamic deformation responses for   

(i)  ω/ωn << 1

(ii)  ω/ωn >> 1

(iii)  ω/ωn = 1

Also state whether the dynamic response is controlled by the mass, stiffness or damping of the system for each case.  

(b) A single degree of freedom building system is excited by a sinusoidal force as shown in Figure 1. Assume that the girder is rigid whereas the columns are flexible to the lateral deformation but rigid in the vertical direction. Use E = 30 GPa, I = 60 x 106 mm4 and 5 % damping and neglect the mass of columns. Determine

(i)  The steady state amplitude of vibration,

(ii)  The maximum shear force in the column,

(iii)  The maximum bending moment.

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