If the damping factor is less than one or the damping coefficient is less than critical damping coefficient , then the system is said to be an under-damped system.
- We know that roots of differential equations are:
- But for ; the roots for under-damped system are given by and as below:
Where is the imaginary unit of complex root
- The roots are complex and negative, so the solution of differential equation is given by
- According to Euler’s theorem, above equation can be written as:
Above equation shows the equation of motion for an underdamped system, and the amplitude reduces gradually and finally becomes zero after some time.
Amplitude decreases by
The natural angular frequency of damped free vibrations is given by:
- Time period for under-damped vibration is given by: