**Free Vibration**

**Free vibration **: A structure is said to undergo free vibration when it is disturbed from its static stable equilibrium position by an initial displacement and/or initial velocity and then allowed to vibrate without any external dynamic excitation.

**Period of vibration: ** The time required for an undamped system to complete one cycle of free vibration is the natural period of vibration of the system.

**Natural frequency **: The number of cycles of free vibration of an undamped system in one second is termed the natural frequency of the system, and is the inverse of the period of vibration.

**A SDOF system **: If the displacement of a system can be uniquely determined by a single variable, this system is called a single-degree-of-freedom (SDOF) system. Normally it consists of a mass, a spring and a damper. The square of the natural frequency of a SDOF system is proportional to the stiffness of the system and the inverse of its mass.

**A generalised SDOF system **: Consider a discrete system that consists of several masses, springs and dampers, or a continuous system that has distributed mass and flexibility. If the shape or pattern of its displacements is known or assumed, the displacements of the system can then uniquely be determined by its magnitude (a single variable). This system is termed as a generalised SDOF system. The analysis developed for a SDOF system is applicable to a generalised SDOF system.

- For a structure with a given mass, the stiffer the structure, the higher the natural frequency.
- The larger the damping ratio of a structure, the quicker the decay of its free vibration.
- The higher the natural frequency of a structure, the quicker the decay of its free vibration.
- The fundamental natural frequency reflects the stiffness of a structure. Thus it can be used to predict the displacement of a simple structure. Also the displacement of the structure can be used to estimate its fundamental natural frequency.