Question

The equation of motion of system is given by $m\ddot{x}+c\dot{x}+kx=0$. The damped vibration, ${\omega }_d$ can be written as:

• (A) ${\omega }_d=\left(\sqrt{1-{\xi }^2}\right){\omega }_n$
• (B) ${\omega }_d=\sqrt{{\left({\omega }_n\right)}^2-{\left(\frac{c}{2m}\right)}^2}$
• (C) Both A and B
• (D) Neither A nor B

Answer 1

The Correct Option is C

Explanation:
The general case of damped harmonic motion:$F_{net}=m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=0$Where $k=$ the stiffness of spring, $c=$ damping coefficient and $m=$ massDamped vibration of a system is defined as:${\omega }_d=\sqrt{{\left({\omega }_n\right)}^2-{\left(\frac{c}{2m}\right)}^2}\cdots \cdots$(1)Damping ratio $(\xi )$ : It is defined as the ratio of actual damping to the critical damping.$\xi =\frac{\text{actual damping}}{\text{critical damping}}=\frac{c}{c_c}$$\Rightarrow \xi =\frac{c}{c_c}=\frac{c}{2m{\omega }_n}=\frac{c}{2\sqrt{km}}\frac{c}{2m}=\xi {\omega }_n$Substitute the value of $\xi$ in equation 1, we get,${\omega }_d=\sqrt{{\left({\omega }_n\right)}^2-{\left(\frac{c}{2m}\right)}^2}=\sqrt{{\left({\omega }_n\right)}^2-{\left(\xi {\omega }_n\right)}^2}=\left(\sqrt{1-{\xi }^2}\right){\omega }_n$Hence the correct option is (C).

Answer 2

The oscillation in which the amplitude decreases gradually with time is called damped oscillation. 

• The decrease in amplitude is due to air drag and friction.
• A damped oscillator is approximately periodic with decreasing amplitude.

Equation of Damped Oscillation

The equation of damped oscillation is given by

\[m\frac{d^2\,x}{dt^2\,x}+b\frac{dx\,}{dt\,}+kx=0\]

Where

• m is the mass
• k is restoring force constant or spring constant
• b is a positive constant depends on the characteristic of the medium and size and shape of the block etc.

The solution of the above equation is given by

\(x(t)=Ae\frac{bt\,}{2m\,}cos\)

Where

• A’ = Ae^(-bt/2m) represents the amplitude of the damped oscillation.
• ω’ = angular frequency of the damped oscillation.

The angular frequency of the damped oscillation can be represented by

w'=√(k/m-b^2/4m^2)