If the damping factor ξ is less than one or the damping coefficient c is less than critical damping coefficient cc, then the system is said to be an under-damped system.
ξ<1ORccc<1⟹c<cc
We know that roots of differential equations are:
S1=[−ξ+ξ2−1]ωnS2=[−ξ−ξ2−1]ωn
But for ξ<1; the roots for under-damped system are given by S1 and S2 as below:
S1=S2=[−ξ+i(1−ξ2)]ωn[−ξ−i(1−ξ2)]ωn
Where i=−1 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:
x=Xe−ξωnt[sin(ωdt+∅)]WhereXand∅ are constants
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 X.e−ξωnt
The natural angular frequency of damped free vibrations is given by:
ωd=(1−ξ2)ωn
Time period for under-damped vibration is given by:
tp=ωd2π=(1−ξ2)ωn2πsec.
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