Let us follow one cycle of vibration of the structure in Fig.
8. The mass in its equilibrium position at 1 moves to the right reaching the maximum (positive) displacement at 2, at which time the displacement begins to decrease and it returns back to its equilibrium position 3, continues moving to the left reaching the maximum (negative) displacement at 4, and then the displacement decreases again with the mass returning to its equilibrium position 5. One cycle of motion is described by the portion
1-2-3-4-5 of the displacement-time curve. At point 5, the
state (displacement and velocity) of the mass is the same as it was at point 1, and the mass is ready to begin another cycle of vibration.
The amplitude of the simple harmonic motion, as shown in Fig. 8, depends on the initial displacement and velocity. Because the structure is undamped, the motion does not decay, i.e., the displacement amplitude is the same in all vibration cycles.
The natural period of vibration T (sec) of the structure is the
time required for one cycle of free vibration. It is related to the natural circular frequency of vibration w (rads/sec) and the natural cyclic frequency of vibration f (cycles/sec or Hz) as follows: