This tutorial provides a solution to the problem of the free vibration of a damped single-degree-of-freedom system. **Question**
A one-story building is idealized as a rigid girder supported by weightless columns, (See Figure in the video youtu.be/s3T331iq3ig). In order to evaluate the dynamic properties of this structure, a free-vibration test is made, in which the rigid girder is displaced laterally by a hydraulic jack and then suddenly released. During the jacking operation, it is observed that a force of 88.90 kN is required to displace the girder 5.08 mm. After the instantaneous release of this initial displacement, the maximum displacement on the first return swing is only 4:064 mm, and the period of this displacement cycle is TD=51.4 seconds. From the experimental data, the following dynamic properties are determined.
(i) Effective mass M of the girder
(ii) Damped frequency
(iii) Damping properties
(iv) Amplitude after 6 cycles