Question

A single story frame consists of two rigid columns (with negligible masses), a rigid beam of mass m and a spring-dashpot system as shown in Fig. The ground is forced to vibration by an earthquake; the acceleration ¨uE = b0cos Ωt is known from measurements. Determine the maximum amplitude of the steady state vibrations. Assume that the system is underdamped and that the vibrations have

Answer 1

We assume that the amplitudes of the vibrations are small. Then the elongation of the diagonal is obtained as

The elongation produces the force

in the spring-dashpot system (see the figure). The equation of motion of the beam is given by

Thus, the relative displacement y=x-υEis described by

The maximum amplitude A is obtained for η = Ω/ω ≈ 1 (resonance!). In the case of small damping (ζ ≪ 1) we obtain