The shear frame shown in Figure 1 is subjected to a harmonic loading with a forcing frequency of due to the action of reciprocating machinery.

(a) If the column bases are assumed to be pinned, calculate the natural frequencies 1 and 2 in terms of the mass m and the stiffness k = 12EI/L 3 , together with the corresponding natural modes for free, undamped vibrations. [5 points]

Problem 1: The shear frame shown in Figure 1 is subjected to a harmonic loading with a forcing frequency of o due to the action of reciprocating machinery. (a) If the column bases are assumed to be pinned, calculate the natural frequencies of and m2 in terms of the mass m and the stiffness k=12E1/L, together with the corresponding natural modes for free, undamped vibrations. [5 points) m Rigid beam EI El p.sincet 2m Rigid beam 2E1 2011 Figure 1 (b) Let the machinery be operating under steady state conditions in which o=0.801, and assume that the structure suffers Rayleigh damping at 5% of the critical level in each of the natural modes. If L = 4 m, EI = 8x10N m, m = 12x 109 kg, and po = 2 kN, find the square root sum of square estimate of the maximum floor displacements and column bending moments. [15 points) (c) Based on the information given in part (b), determine the displacement response of the system. Plot, as a function of the time (up to 10 sec), the displacements u and uz due to each vibration mode separately and combined, using Matlab. [20 points) Problem 1: The shear frame shown in Figure 1 is subjected to a harmonic loading with a forcing frequency of o due to the action of reciprocating machinery. (a) If the column bases are assumed to be pinned, calculate the natural frequencies of and m2 in terms of the mass m and the stiffness k=12E1/L, together with the corresponding natural modes for free, undamped vibrations. [5 points) m Rigid beam EI El p.sincet 2m Rigid beam 2E1 2011 Figure 1 (b) Let the machinery be operating under steady state conditions in which o=0.801, and assume that the structure suffers Rayleigh damping at 5% of the critical level in each of the natural modes. If L = 4 m, EI = 8x10N m, m = 12x 109 kg, and po = 2 kN, find the square root sum of square estimate of the maximum floor displacements and column bending moments. [15 points) (c) Based on the information given in part (b), determine the displacement response of the system. Plot, as a function of the time (up to 10 sec), the displacements u and uz due to each vibration mode separately and combined, using Matlab. [20 points)